FNCE_235_725_Fall2010_Ch13_17

FNCE_235_725_Fall2010_Ch13_17 - Adventures in Debentures...

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Unformatted text preview: Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Adventures in Debentures Part Three More Modern Methods for "These are sweeping changes, and those who are committed Interest Rate Risk 13: Vasicek 1: Properties of the Short-Term Rate . . . 647 14: Vasicek 2: The Term Structure . . . . . . . . . . 697 15: Vasicek 3: More Term Structure . . . . . . . . . 737 to traditional choices and practices naturally resist them." -- B. F. Skinner 16: Vasicek 4: The Greeks . . . . . . . . . . . . . . . 791 17: Valuation by Monte Carlo Methods . . . . . . . . 837 Part Three: More Modern Methods for Interest Rate Risk Page 645 Part Three: More Modern Methods for Interest Rate Risk Page 646 Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Adventures in Debentures Chapter 13 Vasicek 1: Properties of the Short-Term Rate Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649 Announcements and Assignments . . . . . . . . . . . . . . 654 Lecture Notes . . . . . . . . . . . . . . . . . . . . . . . . . 655 A: Estimating Vasicek Parameters from Historical Data . 682 B: More Details about the Vasicek Tree . . . . . . . . . . 685 Practice Questions with Solutions . . . . . . . . . . . . . . 689 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . 695 Chapter 13: Vasicek 1: Properties of the Short-Term Rate Page 647 Chapter 13: Vasicek 1: Properties of the Short-Term Rate Page 648 Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Adventures in Debentures short-term rates, and the Vasciek model captures this very basic idea. This chapter is organized around the following conceptual path. First, we will try to characterize reasonable properties for forecasts of short-term interest rates. Second, we will propose a model that specifies how short-term interest rates could be generated. Finally, we will confirm that given this particular way of generating shortterm rates there is an obvious way to forecast interest rates. This obvious forecasting method has the reasonable properties that we claimed in the first step. We will spend approximately four chapters (including today's lecture) discussing the basic ideas of the term structure based on the Vasicek model. After the next four chapters, we will turn out attention to options or securities which contain implicit options. To understand options on fixed income securities, models like Vasicek are critical. We will discuss basic options and impicit options for five chapters after we complete the Vasicek model. Preface to Vasicek 1: Properties of the Short Term Rate 1.) Suggested Preparation Before Attending This Session. Before coming to class, think what you might do if you were asked to forecast the short-term interest rate. Consider what you expect the short-term rate to be tomorrow, in one year, and in 20 years. Also think about how precise you think each of these forecasts are. 3.) Limitations of the Vasicek Model. 2.) Summary of Chapter. Our goal for this chapter is to develop a method which generates tractable binomial trees for purposes of modeling the uncertain evolution of the short-term rate.1 We will refer to the methodology that we use to construct a particular type of binomial tree as the Vasicek model, which is one of the well known models of the term structure. While there are many other models of the term structure that are available, we are going to focus on the Vasicek model for pedagogical purposes.2 The Vasicek model will provide one explanation as to what determines the shape of the term structure. Clearly, today's long-term rates reflect possible values for future 1 As is true with any model, the Vasicek approach does not explain all empirical facts about the term structure. There is a tradeoff between tractability and factual accuracy. While we could generate models that are more realistic, we would hide some important intuition in the complexity. The Vasicek model suffers from two important defects. First, the Vasicek approach assumes that interest rate volatility over a given horizon is constant. That is, if you are trying to forecast interest rates to be observed in one year, your forecast accuracy is always the same no matter what the current economic environment is. For example, if you are trying to forecast interest rates to be observed in one year, your forecast accuracy is the same whether or not today's interest rates are high or low. Of course, the Vasicek model does not claim that your forecast accuracy is the same when you are forecasting interest rates to be observed next month versus next year. Second, the Vasicek approach assumes that interest rate forecasts are only determined by one source of information -- in particular, current interest rates are used to forecast future interest rates. You can easily imagine that forecasts of future interest rates should be affected by a variety of macroeconomic considerations beyond the level of current interest rates. Chapter 13: Vasicek 1: Properties of the Short-Term Rate, Preface Page 650 2 In the Vasicek model of the term structure, we can use any maturity as the focal point of our discussion. We will adopt the admittedly arbitrary convention of focusing on the shortest maturity. In the case of the continuously compounded yield curve, we will examine the intercept on the term structure (or the "overnight rate"). There are a number of alternative models that could be used. All of these models (including Vasicek) attempt to capture some of the important stylized empirical facts about the behavior of the term structure. However, all of these models will fail to explain all the stylized facts in an effort to maintain some computational tractability. Fortunately, the basic concepts for most of these models can be explained in the context of the Vasicek model. All of these models (including Vasicek) attempt to formulate a framework where arbitrage profits are not present. Chapter 13: Vasicek 1: Properties of the Short-Term Rate, Preface Page 649 Adventures in Debentures Adventures in Debentures Both of the above problems have been analyzed by term structure models that extend the Vasicek framework. Nevertheless, the basic concepts from these more complicated models are quite similar to those in the Vasicek framework. At the very least, we will view the Vasicek model as a convenient and efficient way to list out a complete set of plausible interest rate scenarios for the future. 7.) References (not suggested reading). The following articles provide alot of the mathematical justification for what we will do in this course. However, I provide these references only for completeness. You are not expected to read these articles. Vasicek. 1977. "An Equilibrium Characterization of the Term Structure." Journal of Financial Economics 5. Pages 177 188. (This is the original paper deriving the model which we will use for the rest of the semester. If you decide to examine this paper, you should note that my parameterization of the model for this course is different than what appears in this paper by Vasicek. While the parameters appear different, the basic model is identical to the one we are using.) Hull and White. 1990. "Pricing Interest-Rate-Derivative Securities." Review of Financial Studies 3. Pages 573 592. (This paper provides a useful extension of the Vasicek model to the case where the parameters are time dependent.) Nelson and Ramaswamy. 1990. "Simple Binomial Processes as Diffusion Approximations in Financial Models." The Review of Financial Studies 3. Pages 393 430. (This paper discusses a general approach for approximating models like the Vasicek term structure using binomial trees that are computationally tractable. Most of what we will do in the rest of this course would not be feasible without this paper.) Tian. 1992. "A Simplified Binomial Approach to the Pricing of Interest Rate Contingent Claims." Journal of Financial Engineering 1. Pages 14 37. (This paper provides a useful correction to the Nelson and Ramaswamy paper.) 4.) Road Map for Chapter. The topics in this chapter will be organized as follows: 1.) Give some background and motivation for the methodology. 2.) Discuss reasonable properties for forecasts of the short-term rate. 3.) Describe a method to generate binomial trees for the short-term interest rate. (These trees are desigined to be consistent with reasonable properties for forecasts.) 4.) If time remains, link the parameters for the binomial trees to the implied forecasts and vice versa. 5.) Required Reading. After attending this session, you should read: Appendix A, "Estimating Vasicek Parameters from Historical Data." 8.) New Vocabulary Used in this Chapter. 6.) Supplemental Reading. After attending this session, you may find the following reading useful: Appendix B, "More Details about the Vasicek Tree." Chapter 13: Vasicek 1: Properties of the Short-Term Rate, Preface Page 651 The following buzz words will be used in the lecture notes, the readings, and/or the problem sets: Binomial tree, mean reversion, mu (), phi (), sigma (), standard deviation, and Vasicek model. Chapter 13: Vasicek 1: Properties of the Short-Term Rate, Preface Page 652 Adventures in Debentures Adventures in Debentures Copyright c 2004 by Michael R. Gibbons 9.) Summary of Important Equations. -2 ln h , STEP = Announcements and Assignments . qvt = 1 ( - t rt+h ) h + 2 8 - ln For small h, Expected Value of t rt+h as of time s is [(1 - (t-s) )] + [s rs+h ((t-s) )] , where is between zero and one. Note that in the above equation as t becomes large the expected value converges to . For small h, Variance of t rt+h as of time s is 2 (1 - 2(t-s) ) . Keep in mind that this variance is really a measure of the precision of the forecast of the short-term rate. Chapter 13: Vasicek 1: Properties of the Short-Term Rate, Preface Page 653 Chapter 13, Announcements and Assignments Page 654 Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Adventures in Debentures ecture notes L for Vasicek 1: Properties of the Short Term Rate A Road Map 1.) Give some background and motivation for the methodology. 2.) Discuss reasonable properties for forecasts of the short-term rate. 3.) Describe a method to generate binomial trees for the short-term interest rate. (These trees are desigined to be consistent with reasonable properties for forecasts.) 4.) If time remains, link the parameters for the binomial trees to the implied forecasts and vice versa. Why Focus on the Short-Term Rate? Today's long-term bonds will become short-term bonds sometime in the future. Thus, the market's expectation about short-term rates to be observed in the future should affect the desirability of holding long-term bonds (and hence the current price or current yield for long-term bonds). Thus, we are taking one step back (and hopefully we will make two steps forward) and modeling the possible values for the short-term rate to be observed in the future. In the upcoming sessions, we will take our model for possible shortterm rates to infer what these possible short-term rates imply for current long-term bonds. Chapter 13: Vasicek 1: Properties of the Short-Term Rate, Lecture Notes Page 655 Chapter 13: Vasicek 1: Properties of the Short-Term Rate, Lecture Notes Page 656 Adventures in Debentures Adventures in Debentures Which Short-Term Bond To Be Observed When? For most of our analysis we will examine trt+h, which is a continuously compounded yield for a h period bond observed on date t. Recall our notation described in earlier chapters. s t (t + h) t r t+h (t + 1) (t + 1 + h) (t + 2) .. .............................................. .............................................. .. . .. Tracking the Milestones 1.) Give some background and motivation for the methodology. 2.) Discuss reasonable properties for forecasts of the short-term rate. 3.) Describe a method to generate binomial trees for the short-term interest rate. (These trees are desigined to be consistent with reasonable properties for forecasts.) 4.) If time remains, link the parameters for the binomial trees to the implied forecasts and vice versa. .. .............................................. .............................................. .. . .. t+1 r t+1+h .. ........................................................................................... .......................................................................................... . .. .. ... t r t+1 .. ...................................................................................................................................................................................... ..................................................................................................................................................................................... .. . .. t r t+2 (As h 0, trt+h = tr.) Standing at time period s (where s < t), we are trying to forecast trt+h, t+1rt+1+h, etc. Chapter 13: Vasicek 1: Properties of the Short-Term Rate, Lecture Notes Page 657 Chapter 13: Vasicek 1: Properties of the Short-Term Rate, Lecture Notes Page 658 Forecasts of Short-Term Rate: Numerical Exmple Adventures in Debentures Very Near Future Current Rates Are Low By Historical Standards Chapter 13: Vasicek 1: Properties of the Short-Term Rate, Lecture Notes 1 2 Year From Now 1 Year From Now 2 Years From Now Very Distant Future Current Rates Are Typical By Historical Standards Page 659 Current Rates Are High By Historical Standards Forecasts of Short-Term Rate: Some Notation Very Near Future [(t - s) 0] Year From Now [(t - s) = 0.5] 1 2 Adventures in Debentures 1 Year From Now [(t - s) = 1] 2 Years From Now [(t - s) = 2] Very Distant Future [(t - s) )] Chapter 13: Vasicek 1: Properties of the Short-Term Rate, Lecture Notes Page 660 Current Rates Are Low By Historical Standards sr [t-s s r + (1 - t-s )] [s r + (1 - )] [t-s s r + (1 - t-s )] Current Rates Are Typical By Historical Standards sr [t-s s r + (1 - t-s )] [s r + (1 - )] [t-s s r + (1 - t-s )] Current Rates Are High By Historical Standards sr [t-s s r + (1 - t-s )] [s r + (1 - )] [t-s s r + (1 - t-s )] (NOTE: Today is date "s.") Adventures in Debentures Adventures in Debentures Measuring Forecast Accuracy As our forecast horizon becomes shorter and shorter, our forecast accuracy should become greater and greater. That is, the variance of our forecast error should become zero. When the forecast horizon lengthens, the variance of our forecast error should increase. Let 2 be the variance of our forecast error in the very distant future. It seems reasonable that the variance of our forecast error for intermediate horizons should be some weighted average of the variance of our very distant forecast (that is, 2) and the variance of our very near term forecast (that is, zero). Consider a simple weighting scheme similar to what we did in creating our forecast. If today is date s and if we are forecasting out to date t, then we could say the variance of our forecast error is: VARs(trt+h) = [ 2 (1 - 2(t-s))] + [0 (2(t-s))] . As expected, the prior formula implies the variance of the forecast error is 2 as t becomes large, and the variance of the forecast error is zero as t - s becomes small. When the forecast horizon is one year (that is, t - s = 1), then the weight on 2 is 1 - 2 and the weight on 0 is 2. That is, VARs(s+1rs+1+h) = [ 2 (1-2)]+[0(2)] = 2(1-2) . When the forecast horizon is 2 years (that is, t - s = 2), then the weight on 2 is 1 - 22 and the weight on 0 is 22. That is, VARs(s+2rs+2+h) = [ 2 (1-4)]+[0(4)] = 2(1-4) . When the forecast horizon is 6 months (that is, t - s = .5), then the weight on 2 is 1 - 2.5 and the weight on 0 is 2.5. That is, VARs(s+2rs+2+h) = [ 2 (1 - 1)] + [0 (1)] = 2(1 - ) . Chapter 13: Vasicek 1: Properties of the Short-Term Rate, Lecture Notes Page 661 Chapter 13: Vasicek 1: Properties of the Short-Term Rate, Lecture Notes Page 662 Adventures in Debentures Adventures in Debentures Convenience Versus Flexibility To understand the entire term structure of interest rates, we need to have the ability to study a wide variety of maturities. (In fact, an infinite number of maturities are available in principle.) Each maturity on the term structure will be related to forecasts about the short-term rate at different points in the future. Thus, we need a convenient way to describe a large number of possible forecasts about the short-term rate. The obvious solution is to describe a large number of possible forecasts with a few parameters. Of course, if we have only a few parameters to describe many forecasts, we must have some limitations as to how flexible we can be in describing these forecasts. Our approach will be very convenient by relying on only three parameters (, , and ) to describe our forecasts of the short-term rate for all possible points in the future. Of course, our approach may lack the flexibility to properly characterize forecasts that may be more appropriate in certain settings. Interpreting Vasicek Parameters: A Summary is the long term forecast of the short term rate. measures the uncertainty (i.e., the standard deviation) in the forecast error when predicting the short term rate to be observed in the distant future. determines how quickly the short term forecast of the short term rate converges to the long term forecast of the short term rate. must be greater than or equal to zero, and must be less than one. characterizes the extent of "mean reversion." Chapter 13: Vasicek 1: Properties of the Short-Term Rate, Lecture Notes Page 663 Chapter 13: Vasicek 1: Properties of the Short-Term Rate, Lecture Notes Page 664 Adventures in Debentures Adventures in Debentures Forecasting and Tree Diagrams Given our forecasts about the short-term rate, we need to develop a set of possible future scenarios (and the relevant probabilities) that are consistent with our forecasts. We also need to structure our possible future scenarios to make it convenient to value various financial securities. We will achieve this with binomial tree diagrams . . . Tracking the Milestones 1.) Give some background and motivation for the methodology. 2.) Discuss reasonable properties for forecasts of the short-term rate. 3.) Describe a method to generate binomial trees for the short-term interest rate. (These trees are desigined to be consistent with reasonable properties for forecasts.) 4.) If time remains, link the parameters for the binomial trees to the implied forecasts and vice versa. Chapter 13: Vasicek 1: Properties of the Short-Term Rate, Lecture Notes Page 665 Chapter 13: Vasicek 1: Properties of the Short-Term Rate, Lecture Notes Page 666 Adventures in Debentures Adventures in Debentures Tree Structure for Vasicek Model To value various financial securities the following tree structure is very useful. Further, to maintain computational tractability, it also turns out to be important that the inner nodes of the tree always touch. (That is, going to high rates in period 1 and low rates in period 2 puts you at the same point in period 2 as when you have low rates in period 1 followed by higher rates in period 2.) The following tree diagram illustrates the basic structure we need to work with: . . ... ... ... ... Given the inner nodes of the tree touch, one way to restate the above tree is: . .. ... ... ... ... 0rh ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . . qv0....................... . . ... ... ... ... 0rh + STEP 0rh - STEP . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. ... ... ... ... ... ... ... ... ... vh ....... .. ... . .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . qvh...................... 0rh + 2 STEP q 0rh 0rh - 2 STEP 0rh . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . qv0....................... . . ... ... ... ... h r 2h h r 2h ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... vh............. .. ... ... ... . . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . qvh....................... 2h r 3h q 2h r 3h 2h r 3h Chapter 13: Vasicek 1: Properties of the Short-Term Rate, Lecture Notes Page 667 Chapter 13: Vasicek 1: Properties of the Short-Term Rate, Lecture Notes Page 668 Adventures in Debentures Adventures in Debentures Parameters of a Vasicek Tree We need to set the parameters of the Vasicek tree to correspond to a desired set of forecasts about future interest rates. Let T be the total amount of time to be modeled with the tree, and h is the length of time between each time period on the tree diagram. We will let the size of the "up" and "down" "STEPS" in r be: STEP = -2 ln h . We will assign the probability of an "up" movement to be qvt, while the "down" movement will be 1 - qvt, where: qvt = and qvt is given by: qvt if 0 qvt 1 1 if qvt > 1 0 if qvt < 0 Numerical Illustration of Vasicek Model Assume 0rh = 8%, = 5%, = 6%, and = .7. Also assume T = 1 year and h = 1 year. Then: 2 4 STEP = .06 -2 ln(.7) .25 .0253 . = Probability of up move when current 3 month rate is 8%: 1 (.05 - .08) .25 - ln(.7) qv,0 = + = .45 . 2 .06 8 Probability of up move when current 3 month rate is 10.53%: 1 (.05 - .1053) .25 - ln(.7) + qv,1 = + = .40 . 2 .06 8 Probability of up move when current 3 month rate is 5.47%: 1 (.05 - .0547) .25 - ln(.7) - qv,1 = + = .49 . 2 .06 8 1 ( - trt+h) h qvt = + 2 8 - ln . Note: the probability (qvt) is not the same for the whole tree. qvt changes as a function of the interest rate (trt+h). Chapter 13: Vasicek 1: Properties of the Short-Term Rate, Lecture Notes Page 669 Chapter 13: Vasicek 1: Properties of the Short-Term Rate, Lecture Notes Page 670 Adventures in Debentures Adventures in Debentures 8% . ... ... .. .. ... ... ... ... ... ... .. ... ... ... .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. . 8 + 2.53 = 10.53 .45 .55 8 - 2.53 = 5.47 . . .. .. ... ... ... ... ... ... .. ... ... ... .. .. ... ... ... ... ... ... .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. . . .. ... ... ... ... ... ... .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. . .40 8 + (2 2.53) = 13.07% Comments about Previous Illustration 1.) When the short term rate exceeds , the probability of an up move is less than 50%. .60 .49 8% .51 8 - (2 2.53) = 2.94% 2.) Note how the nodes keep connecting up. This feature is important for trees with a large number of nodes (i.e., where h is small or T is large or both) to maintain computational tractability. 3.) Unfortunately, this model permits interest rates to become negative! 4.) Probabilities can go to 1 or 0. When probability hits 0, you can carry along the branch, but the branch becomes irrelevant. Chapter 13: Vasicek 1: Properties of the Short-Term Rate, Lecture Notes Page 671 Chapter 13: Vasicek 1: Properties of the Short-Term Rate, Lecture Notes Page 672 Adventures in Debentures Adventures in Debentures Tracking the Milestones 1.) Give some background and motivation for the methodology. 2.) Discuss reasonable properties for forecasts of the short-term rate. 3.) Describe a method to generate binomial trees for the short-term interest rate. (These trees are desigined to be consistent with reasonable properties for forecasts.) 4.) If time remains, link the parameters for the binomial trees to the implied forecasts and vice versa. Linking Forecasts and Tree Diagrams I will claim, but not prove rigorously, that the methodology which generates the trees provides a framework consistent with a reasonable (usually) set of forecasts. Some of the various appendices for this and other sessions will try to provide further support for this claim, but again the appendices will not provide formal and rigorous proofs.1 Essentially, we can make the mathematical connections between the forecasts and the trees work as h becomes smaller and smaller. Because of these mathematical details, which we have ignored, it is useful to define the continuously compounded yield on a bond that matures in the next instant. (Perhaps you want to think of this as the "overnight" rate.) Such a rate would appear as the intercept in the yield curve for zero coupon bonds. We will use tr to indicate this continuously compounded yield on a bond with an instant until maturity, where this yield is observed on date t. 1 See the Preface to this session for a reference to a 1990 paper by Nelson and Ramaswamy if you want more mathematical details than I provide. Page 674 Chapter 13: Vasicek 1: Properties of the Short-Term Rate, Lecture Notes Page 673 Chapter 13: Vasicek 1: Properties of the Short-Term Rate, Lecture Notes Adventures in Debentures Adventures in Debentures Forecasts Versus Vasicek Parameters Our framework can be thought of as a parsimonious way to describe a set of forecasts about the short-term rate to be observed at various dates in the future. You can approach the parameter settings in one of two ways. First, you have a set of forecasts, and you need to pick values for , , and that are reasonably consistent with those forecasts. This approach is the illustrated in one of the problem set questions. Second, you have picked values for , , and , and now you want to confirm the implied forecasts are reasonable. This is the task of the next example. The following equations are reminders of what we established earlier in this session about Vasicek forecasts made at time period s: Expected Value of trt+h is [(1 - (t-s))] + [srs+h((t-s))] , Variance of trt+h is 2(1 - 2(t-s)) . (Keep in mind that this variance is really a measure of the imprecision of the forecast of the short-term rate.) Numerical Illustration of Implied Forecasts and Forecast Precision Consider the case where = 8%, = 4%, and = .5. At time s, you are constructing a forecast of tr. If sr = 6%, then: (t - s) 1/12 2/12 1 5 Expected Value of tr 6.11% 6.22% 7.00% 7.94% 8.00% Std Dev of tr 1.32% 1.82% 3.46% 4.00% 4.00% s s + 2/12 s+1 s+5 Chapter 13: Vasicek 1: Properties of the Short-Term Rate, Lecture Notes Page 675 Chapter 13: Vasicek 1: Properties of the Short-Term Rate, Lecture Notes Page 676 Adventures in Debentures Adventures in Debentures Fig 1: Low Current Rates. In all cases, the density is centered around the best forecast of the future rate given the current rate. Further, by substracting the best forecast from the values on the horizontal axis, we would also have the distribution of the forecast errors. prob density 0 0.05 0.1 0.15 0.2 0.25 0.3 Fig 1c: rates in 20 years Chapter 13: Vasicek 1: Properties of the Short-Term Rate, Lecture Notes Figure 1 looks at the distribution of future rates when the current rate is "low" (in particular, the current short-term rate is 5%). Figure 2 looks at the distribution of future rates when the current rate is "typical" in the sense that it equals . Figure 3 looks at the distribution of future rates when the current rate is "high" (in particular, the current short-term rate is 25%). prob density 0 0.05 0.1 0.15 0.2 0.25 0.3 Fig 1b: rates in 1 year In the figures that follow, we will see possible distributions of the short-term rate to be observed in the future. For all the figures, the following parameters are used: = 15%, = 3%, and = 0.8. Figures 1a, 2a, and 3a are rates to be observed in 2 months from the present. Figures 1b, 2b, and 3b are rates to be observed in 1 year from the present. Figures 1c, 2c, and 3c are rates to be observed in 20 years from the present. In all cases the figures reflect a normal distribution. prob density 0 0.05 0.1 0.15 0.2 0.25 0.3 Fig 1a: rates in 2 months The Distribution of Future Short-Term Rates Chapter 13: Vasicek 1: Properties of the Short-Term Rate, Lecture Notes Page 677 Page 678 Adventures in Debentures Adventures in Debentures Fig 2: Typical Current Rates. Fig 3: High Current Rates. prob density 0 0.05 0.1 0.15 0.2 0.25 0.3 Fig 2a: rates in 2 months prob density 0 0.05 0.1 0.15 0.2 0.25 0.3 Fig 3a: rates in 2 months 0 0.05 0.1 0.15 0.2 0.25 0.3 Fig 2b: rates in 1 year prob density 0 0.05 0.1 0.15 0.2 0.25 0.3 Fig 3b: rates in 1 year 0 0.05 0.1 0.15 0.2 0.25 0.3 Fig 2c: rates in 20 years prob density 0 0.05 0.1 0.15 0.2 0.25 0.3 Fig 3c: rates in 20 years Page 679 Chapter 13: Vasicek 1: Properties of the Short-Term Rate, Lecture Notes Chapter 13: Vasicek 1: Properties of the Short-Term Rate, Lecture Notes prob density prob density Page 680 Adventures in Debentures Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Worksheet Appendix A: Estimating Vasicek Parameters from Historical Data 1.) Introduction. In this appendix, we will study how to estimate the Vasicek parameters (i.e., , , and ) using historical data on the short-term rate. If we have enough observations and if we believe the Vasicek model with constant parameters is a good description of the historical data, then it is reasonable to proceed with historical estimation.1 Furthermore, if we are using the estimates of the Vasicek parameters to value a security as of today, then we must also believe that history is a reliable guide to the future. It is important to remember that the Vasicek model could be a very useful approach even if it is not a good description of historical data. There is nothing in the theory which requires (or even assumes) that historical parameter estimates are useful. The valuation approach only requires that the Vasicek model seems to be a reasonable description of your current forecasts about future short-term interest rates. The parameter values that you use for a valuation exercise may be quite different from parameter estimates based on historical data. However, even if the Vasicek model with constant parameters is not a good description of historical data, it may be a useful pedagogical exercise to understand this appendix. By working through some of the ideas here, your intuition about the Vasicek model should improve -- even though you may never use historical estimates for valuing a security. 1 If we are willing to specify how the parameters change, then we could also estimate the Vasicek model under weaker assumptions. Chapter 13: Vasicek 1: Properties of the Short-Term Rate, Lecture Notes Page 681 Chapter 13: Vasicek 1: Properties of the Short-Term Rate, Appendix A Page 682 Adventures in Debentures Adventures in Debentures 2.) Calculating Historical Summary Statistics. An obvious way to estimate and is to take the sample mean and sample standard deviation2 of a time series on the short-term rate. If the historical data are properly described by a constant mean and standard deviation, then our sample estimates should converge to and , respectively, as our sample size increases. A measure of autocorrelation should also provide some information about . If the historical data are described by a Vasicek model, then we expect to see yesterday's short-term rate correlated with today's short-term rate. In fact, the sample correla tion3 between s rs+h and s+t rs+t+h should be equal to t with a large enough sample. When t equals one, this means you have annual observations on the short-term rate (with maturity of h), and the sample correlation is an estimate of . When t is not equal to one, this means you have data that are not observed yearly. In this case, the sample correlation has to be raised to the power of 1 to find an estimate of . t regression: s+t r s+t+h = 0 + 1 s rs+h + us+t , where s2 is the sample variance of the residuals (i.e., us+t ) in the above regression. ^ It turns out that 1 is an estimate of . 1/t provides an estimate of ; 0 1-1 is an estimate of ; and s2 ^ 2 1-1 4.) Which Short-Term Rate? When using these historical estimates as a guide to reasonable values for a Vasicek tree, we need to decide on the maturity to specify for the short-term rate. In the context of a Vasicek tree, there is an obvious answer. Use that short-term rate with a maturity equal to h. 3.) A Regression Interpretation. Sometimes it might be more convenient to run a regression to compute the needed estimates of the Vasicek parameters. If you run a regression of s+t rs+t+h on s rs+h , then you may compute all the relevant Vasicek parameters. Consider the following 2 5.) Overlapping Data.4 If you gather a time series on s rs+h , this corresponds to a continuously compounded yield on a bond that matures in h periods. If you have (say) annual observations and h equals one or less, then there is no overlap in the data. In contrast, if you have (say) quarterly observations and h is greater than 0.25, then you have overlapping data -- that is, the yield for one data point is observed before the yield of the prior data point is actually realized. Overlapping data do not affect our interpretation of the parameter estimates given in prior sections. However, overlapping data can be misleading. If you have 100 nonoverlapping observations, this should provide a more precise estimate of a parameter than if you have 100 overlapping observations holding everything else constant. Large numbers of overlapping observations may actually be less valuable than a smaller number of non-overlapping observations in terms of the precision of your parameter estimates. Essentially, overlapping observations may be less informative than you think -- especially relative to non-overlapping data. The usual formula for the sample mean is: z= 1 M M zi , i=1 where zi is the ith observation on some variable (in this case the short-term rate). Similarly, the usual formula for the sample standard deviation is: 1 M M s= (zi - z )2 , i=1 3 where sometimes the divisor outside the sum is taken to be M - 1, not M . A typical formula for the sample autocorrelation is: 1 M -1 M i=2 (zi - z )(zi-1 - z ) 4 s2 This section is for the emotionally insecure who need to know more than necessary for purposes of this course. Chapter 13: Vasicek 1: Properties of the Short-Term Rate, Appendix A Page 683 Chapter 13: Vasicek 1: Properties of the Short-Term Rate, Appendix A Page 684 Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Adventures in Debentures The probability of an "up" movement is qvt , while the probability of a "down" movement will be 1 - qvt , where: Appendix B: More Details about the Vasicek Tree and qvt is given by: qvt = qvt 1 0 if 0 qvt 1 if qvt > 1 , if qvt < 0 1.) Introduction. The purpose of this appendix is to provide more intuition as to why the Vasicek tree is a useful way to generate future interest rates that are consistent with certain properties of reasonable forecasts for short-term rates. qvt = 1 ( - t rt+h ) h + 2 8 - ln . 3.) Near-Term Forecasting of the Short-Term Rate. 2.) A Short Review of a Vasicek Tree Methodology. Consider the usual tree structure for the Vasicek model: Standing at period s (and knowing s rs+h ), we want to calculate the expected value1 of s+h rs+2h : (qvs )[s rs+h + STEP] + (1 - qvs )[s rs+h - STEP] . + 2 STEP This last expression can be simplified to: s r s+h + STEP(2qvt - 1) . 0 rh . .. ... ... ... ... ... ... v0 ............... . .. ... ... .. ... ... ... ... ... .. ... .. . ........ ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. . q 0 r h + STEP .. ... . . ........ . . ... ... ... ... ... ... ... ... qvh...................... . . ... ... ... ... 0 rh 0 r h - STEP ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. .. . ... ... ... ... ... ... ... ... vh........... . ... ... ... ... ... ... ... ... ... ... . . ........ ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. . q 0 rh Substituting in the definitions for qvt and STEP and then simplifying, we can restate the last expression as: [h ln()] + s rs+h [1 + h ln()] . 0 rh - 2 STEP Now note that [1+h ln()] is approximately equal to h . This approximation becomes more and more accurate as h becomes smaller and smaller.2 For example, assume equals 0.50. The following table demonstrates this approximation works: 1 2 Let T be the total amount of time to be modeled with the tree, and h is the length of time between each time period on the tree diagram. The size of the "up" and "down" "STEPS" in r is: STEP = -2 ln h . Chapter 13: Vasicek 1: Properties of the Short-Term Rate, Appendix B Page 685 The expected value is the forecast of the short-term rate to be observed at (s + h). One way to demonstrate this is to use a first-order Taylor Series approximation around h equals zero. Chapter 13: Vasicek 1: Properties of the Short-Term Rate, Appendix B Page 686 Adventures in Debentures Adventures in Debentures h 1.00000 0.50000 0.10000 0.01000 0.00100 0.00010 0.00001 [1 + h ln()] 0.3068528 0.6534264 0.9306853 0.9930685 0.9993069 0.9999307 0.9999931 h 0.5000000 0.7071068 0.9330330 0.9930925 0.9993071 0.9999307 0.9999931 The prior paragraphs interpret STEP in the context of the tree. What is the relationship between STEP and the (1 - 2h ), which is the formula we discussed within the lecture notes as a sensible way to construct our measure of the accuracy of our forecasts for short-term interest rates? It turns out that STEP is a good approximation3 to (1 - 2h ) for small values of h. The following table demonstrates that there is little difference between (1 - 2h ) and -2h ln as h becomes smaller and smaller. As an illustration, consider the following table where equals 0.03 and equals 0.60: h 1.00000 0.50000 (1 - 2h ) 2.4000000% 1.8973666% 0.9349203% 0.3024575% 0.0958654% 0.0303223% 0.0095890% -2h ln 3.0323030% 2.1441620% 0.9588984% 0.3032303% 0.0958898% 0.0303230% 0.0095890% With this approximation, we can rewrite the last expression as: [1 - ] + s rs+h [ ] . h h 0.10000 0.01000 0.00100 0.00010 0.00001 This is one of the characteristics we discussed for forecasting the short-term rate. We have confirmed that the Vasicek tree produces this charateristic with greater and greater accuracy as h becomes smaller and smaller. 4.) Standard Deviation of Near-Term Forecast Error. Standing at period s (and knowing s rs+h ), we want to calculate the expected value of s+h r s+2h . However, assume h is very small. In this case, the best forecast of s+h r s+2h is approximately equal to s rs+h since h 1 when h is small. Now calculate the variance of the forecast error from such a forecast: Es [(s+h rs+2h - s rs+h )2 ] = qvt (STEP)2 + (1 - qvt )(STEP)2 , where "Es " is the expected value given s rs+h . Clearly, this last expression is equal to STEP2 , or just STEP after taking the square root. Thus, STEP provides a measure of the standard deviation of the forecast error when forecasting h periods ahead. Chapter 13: Vasicek 1: Properties of the Short-Term Rate, Appendix B Page 687 Thus, as h becomes smaller and smaller, STEP provides the link between the analytical formula in the lecture notes for forecast accuracy and the actual uncertainty created in the tree diagrams. 3 The formal derivation of this approximation is just a first order Taylor series expansion around h equals zero. Chapter 13: Vasicek 1: Properties of the Short-Term Rate, Appendix B Page 688 Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Adventures in Debentures SOLUTION: Practice Questions with Solutions The degree of difficulty of each question is indicated. The easiest questions are marked by "($)," and the hardest questions are indicated by "($$$$$)." STEP = qvt = -2 ln h = .02 -2 ln(.6065307) = .02 1 + 2 1 + 2 - ln h ( - t rt+h ) 8 - ln(.6065307) (.12 - t rt+h ) .02 8 1. ($) Today is year 0, and 0 r1 = 12%. Assume the Vasicek model is correct. The appropriate values for the parameters are: = 0.12, = .02, = 0.6065307, h = 1 year, and T = 2 years. Draw a tree diagram (including the probabilities) which provides the possible values for t rt+1 for t = 1 and t = 2. = = 1 + 12.5(.12 - t rt+h ) 2 12% .. .. ... ... ... ... ... ... ... ... ... ... . .......... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. . .5.................... . . . ... .. ... ... ... ... ... ... 14% .5 10% .. .. ... ... ... ... ... ... ... ... ... ... . .......... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . . ... ... ... ... ... ... ... ... ... ... ... ... . . ... ... ... ... ... ... ... ... ... ... ... ... ... ... . .......... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. . .25.................... . . . ... ... ... ... ... ... ... ... 16% .75 .75 12% .25 8% 2. ($$$) In each of the following parts, you will be asked to interpret a series of graphs. Chapter 13: Vasicek 1: Properties of the Short-Term Rate, Practice Questions with Solutions Page 689 Chapter 13: Vasicek 1: Properties of the Short-Term Rate, Practice Questions with Solutions Page 690 Adventures in Debentures Adventures in Debentures Figure A 0.14 short term rate 0.12 0.1 0.08 0.06 0 time short term rate 0.25 0.2 0.15 0.1 0.05 Figure C time Figure B 0.1 0.25 short term rate short term rate 0.08 0.06 0.04 0.02 0 time 0.2 0.15 0.1 0.05 0 Figure D time Part a. Figures A and B provide time series plots of the short term rate for two different economies. In both economies, the entire history of bond prices are consistent with the Vasicek model. However, the value of was not the same for each economy. In which figure (A or B) was larger? (Only a brief explanation is required.) Part b. Figures C and D provide time series plots of the short term rate for two different economies. In both economies, the entire history of bond prices are consistent with the Vasicek model. However, the value of was not the same for each economy. In which figure (C or D) was larger? (Only a brief explanation is required.) Chapter 13: Vasicek 1: Properties of the Short-Term Rate, Practice Questions with Solutions Page 691 Chapter 13: Vasicek 1: Properties of the Short-Term Rate, Practice Questions with Solutions Page 692 Adventures in Debentures Adventures in Debentures Figure E SOLUTION: 0.2 short term rate 0.175 0.15 0.125 0.1 0.075 time All six figures included a horizontal line which was positioned at the appropriate value of . This horizontal line should have helped you in reaching your conclusion for each part below. Part a. Figure A represents the larger . (By drawing a horizontal line through the midpoint of the time series plot, you can see the is about 10% in Figure A and about 5% in Figure B.) For the record, in both Figures A and B, is 2% and is 0. Part b. Figure C represents the larger . (Clearly, the volatility in Figure C is greater than the volatility in Figure D. Just study the deviations about the horizontal line. In fact, in Figure C is 5% while in Figure D is 3%.) For the record, in both Figures C and D is 15% and is 0. Part c. Figure E represents the larger . (The slow drift downward in Figure E indicates the speed of convergence to the long-term forecast is slow. Slow convergence is consistent with a high . In fact, in Figure E is 0.95 while in Figure F is 0.05.) For the record, in both Figures E and F is 10% and is 3%. Figure F short term rate 0.15 0.1 0.05 0 time Part c. Figures E and F provide time series plots of the short term rate for two different economies. In both economies, the entire history of bond prices are consistent with the Vasicek model. However, the value of was not the same for each economy. In which figure (E or F) was larger? (Only a brief explanation is required.) Chapter 13: Vasicek 1: Properties of the Short-Term Rate, Practice Questions with Solutions Page 693 Chapter 13: Vasicek 1: Properties of the Short-Term Rate, Practice Questions with Solutions Page 694 Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Adventures in Debentures Questions The degree of difficulty of each question is indicated. The easiest questions are marked by "($)," and the hardest questions are indicated by "($$$$$)." Even if you decide not to submit your answer to this problem set, you should review my solution only after you attempt the questions to know if you could do them or not. Many times my solution is obvious AFTER you see it, but you need to know if the solution was obvious to you BEFORE you are told the solution. If you submit your answers to this problem set, please keep the following in mind: I sometimes fall behind where I expect to be in the lectures and an assignment is due before I get to some of the relevant material. If you believe the lectures have not yet covered the material necessary to answer a particular question and the required reading provides no guide, then you should indicate this on your solution and skip the question. Names of students along with their course and section numbers should be clear. The number of students in a study group should be less than or equal to five. Homework should be stapled. The final numerical answer should be "flagged" in some manner. Boxing, highlighting, and/or underlining the number are appropriate. Pages of spreadsheet printouts should be kept to a minimum. Only the essential information should be incorporated. You should not spend an excessive amount of time trying to solve any particular question. If you cannot complete a question, just describe what you tried to do. 1. ($$$) Today is year 0. Assume that the Vasicek model is true and h is extremely small. Today is year zero. The intercept of the term structure is 15%. (If you prefer, you can think of the intercept as the "overnight" rate.) You are given the following forecasts (rounded to 2 decimal places) about the short-term rate for years 1 through 9. All forecasts were made on year 0. Year 1 2 3 4 5 6 7 8 9 Forecast 12.00% 10.80% 10.32% 10.13% 10.05% 10.02% 10.01% 10.00% 10.00% Std Dev of Forecast Error 5.50% 5.92% 5.99% 6.00% 6.00% 6.00% 6.00% 6.00% 6.00% Based on the above numbers, can you determine the appropriate values for , , and ? If so, what are the values? 2. ($$) Assume = 0.06, = 0.40, and = .025. Further T is 1 week, and h is 1 week. The current annualized interest rate (with continuous compounding) for a 1 week zero coupon bond is 5%. Part a. Draw a Vasicek binomial tree for the annualized interest rate (with continuous compounding) for a 1 week zero coupon bond. Part b. As of today, what do you expect the annualized interest rate (with annual compounding) for a 1 week zero to be in one week? Chapter 13: Vasicek 1: Properties of the Short-Term Rate, Questions Page 695 Chapter 13: Vasicek 1: Properties of the Short-Term Rate, Questions Page 696 Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Adventures in Debentures Top Ten Signs that you're not going to win a nobel prize: 10. You think the capital of Sweden is Sweden City. Chapter 14 9. You built an artificial heart, but it's the size of a bread truck. Vasicek 2: The Term Structure 8. Closest you've ever come to doing a scientific experiment -- putting a sleeping friend's hand in warm water. 7. Despite all your brilliant ideas, the nurses won't let you have anything sharp to write them down. 6. You're the CBS executive who picked the new fall lineup. Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . 699 Announcements and Assignments . . . . . . . . . . . . . . 702 Lecture Notes . . . . . . . . . . . . . . . . . . . . . . . . . 703 A: Information about the Finance Museum . . . . . . . . 722 Practice Questions with Solutions . . . . . . . . . . . . . . 725 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . 732 5. For the past 10 years, your left thumb has been stuck in a test tube. 4. Title of your doctoral dissertation: "Yee-ouch! Them Pins is Pointy!" 3. Your theory of relativity is E = MC Hammer. 2. You're known around the University as "Professor Gump." 1. Your first name is Boutros Boutros -- but your last name ain't Ghali. -- David Letterman, Late Night With David Letterman Chapter 14: Vasicek 2: The Term Structure Page 697 Chapter 14: Vasicek 2: The Term Structure Page 698 Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Adventures in Debentures Preface to Vasicek 2: The Term Structure 1.) Suggested Preparation Before Attending This Session. At the beginning of this chapter (starting on page 704), we will develop a binomial tree diagram to describe the evolution of the short-term rate for a specific set of parameters. Before coming to class, you may want to verify that you can generate this tree as a way to confirm your understanding of the material from the chapter titled "Vasicek 1: Properties of the Short Term Rate." (This same example will be used in the next two chapters as well.) the real objective is to encourage you to think about the implications of future interest rate scenarios. We use the valuation issue merely as a useful pedagogical framework, for it requires you to be very systematic and complete in your analysis. For our purposes the empirical accuracy of the valuation based on the Vasicek model is not of critical importance, for there are a wide variety of alternative models which may be more accurate. Fortunately, all of the available models are similar in concept to the Vasicek model, which we use here because of its tractability. At the very least, we will view the Vasicek model as a convenient and efficient way to list out a complete set of plausible interest rate scenarios for the future. 3.) Road Map for Chapter. The topics in this chapter will be organized as follows: 1.) Provide another numerical illustration of how to generate the evolution of the short-term rate. 2.) Discuss the Vasicek methodology for risk adjustment. (This approach will generate term premiums.) 3.) Generate a numerical illustration which values long-term bonds relative to short-term bonds. (This will give us a description of how the entire term structure evolves through time.) 4.) If time remains, demonstrate the implications of the Vasicek model for potential shifts in the term structure as well as for the expected return for buying a bond and then selling it prior to its maturity. 2.) Summary of Chapter. This chapter values long-term bonds in light of our forecasts about future short-term rates. To complete such a valuation exercise requires us to do three things. First, specify our forecasts about short-term rates. Second, evaluate an intermediate-term bond by determining its risk relative to the possible future scenarios about the shortterm bond. Third, all other long-term bonds are then valued by finding a dynamic synthetic for this long-term bond using just the short-term bond and the intermediateterm bond. All aspects of this methodology will be demonstrated in today's chapter. The dynamic synthetic for the long-term bond is another example of a "dynamic structured strategy." It is a structured strategy because assumptions are made about the evoluation of the short-term interest rate. It is a dynamic strategy because the synthetic requires rebalancing through time -- it is not a buy and hold strategy. You should recall that strategies involving dollar delta were also dynamic structured strategies. Our analysis in this chapter represents a natural extension of what we did in prior chapters. On the proper perspective for this chapter . . . The apparent goal of today's analysis is the valuation of plain vanilla bonds. However, Chapter 14: Vasicek 2: The Term Structure, Preface Page 699 4.) Supplemental Reading. After attending this session, you may find the following reading useful: Richard A. Brealey and Stewart C. Myers. 2000. Principles of Corporate Finance. Sixth Edition. McGraw-Hill: New York.. (This book discusses certainty equivalents, which we will use in deriving the term structure.) Appendix A, "Information about the Finance Museum." Chapter 14: Vasicek 2: The Term Structure, Preface Page 700 Adventures in Debentures Adventures in Debentures Copyright c 2004 by Michael R. Gibbons 5.) New Vocabulary Used in this Chapter. The following buzz words will be used in the lecture notes, the readings, and/or the problem sets: Binomial tree, certainty equivalent, lambda (), risk adjustment factor, term (or "liquidity") premium, and Vasicek model. Announcements and Assignments 6.) Summary of Important Equations. (Expected Return on Bond 1) - (short-term rate) = ... Standard Deviation of Return on Bond 1 = (Expected Return on Bond n) - (short-term rate) = V Standard Deviation of Return on Bond n Chapter 14: Vasicek 2: The Term Structure, Preface Page 701 Chapter 14, Announcements and Assignments Page 702 Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Adventures in Debentures ecture notes L forVasicek 2: The Term Structure A Road Map The Evolution of the Short Term Rate: A Numerical Illustration Assume: T = 2 year and h = 1 year. Assume: = 6%, = 4%, and = .8 . 1.) Provide another numerical illustration of how to generate the evolution of the short-term rate. 2.) Discuss the Vasicek methodology for risk adjustment. (This approach will generate term premiums.) 3.) Generate a numerical illustration which values long-term bonds relative to short-term bonds. (This will give us a description of how the entire term structure evolves through time.) 4.) If time remains, demonstrate the implications of the Vasicek model for potential shifts in the term structure as well as for the expected return for buying a bond and then selling it prior to its maturity. Assume: 0r1 = 7%. With these values: STEP = -2 ln h = (.04) .0267 = -2 ln(.8) 1 and qV,t = 1 + 2 (.06 - trt+h) 1 .04 8 - ln(.8) These numbers generate the following tree diagram . . . Chapter 14: Vasicek 2: The Term Structure, Lecture Notes Page 703 Chapter 14: Vasicek 2: The Term Structure, Lecture Notes Page 704 Adventures in Debentures Adventures in Debentures The Evolution of the Short Term Rate: A Graphical Summary The Evolution of the Short-Term Discount: An Equivalent View .35 .46 . .. .. .. ... ... ... ... ... ... .. ... ... ... .. .. ... ... .. .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. .. 7 + 2.67 = 9.67 7% .65 .57 .54 7 - 2.67 = 4.33 .43 . .. ... ... ... ... ... ... .. ... ... ... .. ... ... ... .. .. ... ... .. .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. . .. ... ... ... ... ... .. ... ... ... .. ... ... ... .. ... ... ... .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. . 7 + (2 2.67) = 12.34% .46 .. .. ... ... ... ... ... ... .. ... ... ... .. ... ... ... . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. . .35 e-.0967 = .9078 7% .9324 .65 .57 .54 e-.0433 = .9576 7 - (2 2.67) = 1.66% Year 2 Year 0 Year 1 .43 . ... ... .. .. ... ... ... ... ... ... .. ... ... ... .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. . . .. .. .. ... ... ... ... ... ... .. ... ... ... .. ... ... ... .. ... ... ... .. ... .. .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. . e-.1234 = .8839 e-.0700 = .9324 e-.0166 = .9835 Year 2 Year 0 Year 1 Chapter 14: Vasicek 2: The Term Structure, Lecture Notes Page 705 Chapter 14: Vasicek 2: The Term Structure, Lecture Notes Page 706 Adventures in Debentures Adventures in Debentures Tracking the Milestones 1.) Provide another numerical illustration of how to generate the evolution of the short-term rate. 2.) Discuss the Vasicek methodology for risk adjustment. (This approach will generate term premiums.) 3.) Generate a numerical illustration which values long-term bonds relative to short-term bonds. (This will give us a description of how the entire term structure evolves through time.) 4.) If time remains, demonstrate the implications of the Vasicek model for potential shifts in the term structure as well as for the expected return for buying a bond and then selling it prior to its maturity. Calculating Risk Adjustments Under Vasicek Model When an investor considers holding a two period bond relative to a one period bond, he may require a term premium to reward him for the extra risk associated with holding the two period bond. The Vasicek model assumes the term premium is proportional to the extra risk (as measured by volatility or standard deviation). That is, the Vasicek model assumes the expected excess return on a bond per unit of standard deviation is a constant across all bonds. This constant we will call V . Formally, the Vasicek model specifies: (Expected Return on Bond 1) - (short-term rate) = ... Standard Deviation of Return on Bond 1 (Expected Return on Bond n) - (short-term rate) = = V Standard Deviation of Return on Bond n where the short-term rate is the yield on the bond that matures in h periods. Keep in mind that the above analysis assumes that h is small. (When h is small, you might want to think of the "shortterm" rate as the overnight rate.) Chapter 14: Vasicek 2: The Term Structure, Lecture Notes Page 707 Chapter 14: Vasicek 2: The Term Structure, Lecture Notes Page 708 Adventures in Debentures Adventures in Debentures Risk Penalty for a 2h Bond To keep the expected excess return per unit of risk constant as specified on the previous page, we need to compute a risk penalty. To compute the risk penalty for the Vasicek model in the case of the bond with maturity of 2h, use: u V - rd h h [e t+h t+2h - e-t+hrt+2h h] , risk penalty = 2 where This risk penalty is used to compute the bond price as of t for a bond that matures in period t + 2h. As long as V is greater than zero, the adjustment factor is positive. However, this risk penalty is only meant to be used for a 2h bond. Bonds with maturities greater than 2h will be analyzed using a different approach. t r t+h .. ... .. .. ... ... ... ... ... ... .. ... ... ... . ... ... ... .. ... ... ... ... ... ... .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. . u t+h r t+2h d t+h r t+2h Period t Period t + h Chapter 14: Vasicek 2: The Term Structure, Lecture Notes Page 709 Chapter 14: Vasicek 2: The Term Structure, Lecture Notes Page 710 Adventures in Debentures Adventures in Debentures Tracking the Milestones 1.) Provide another numerical illustration of how to generate the evolution of the short-term rate. 2.) Discuss the Vasicek methodology for risk adjustment. (This approach will generate term premiums.) 3.) Generate a numerical illustration which values long-term bonds relative to short-term bonds. (This will give us a description of how the entire term structure evolves through time.) 4.) If time remains, demonstrate the implications of the Vasicek model for potential shifts in the term structure as well as for the expected return for buying a bond and then selling it prior to its maturity. Valuing 2h Bonds with Term Premiums: A Numerical Illustration Consider a numerical example. Look at the pricing of the 2 year bond in year 0. In year 0 the bond market might ask what is the smallest certain payoff to be received in year 1 for which the market would exchange the uncertain payoff (or value) of this two year bond. (This is the notion of certainty equivalents.) Assume V equals one third. We need to subtract a "risk penalty" from the expected market value before discounting. The discounted certainty equivalent becomes: e-.071[(.46 e-.09671 + .54 e-.04331) - .0083062] .8639 . = Similarly, we should use a risk penalty of .0080870 for the "up" scenario in year 1 and .0085312 for the "down" scenario in year 1. In this case, the discounted value of the certainty equivalents become: e-.09671[(.35 e-.12341 + .65 e-.071) - .0080870] .8238 = -.04331 -.071 -.01661 .9058 . [(.57 e e + .43 e ) - .0085312] = Chapter 14: Vasicek 2: The Term Structure, Lecture Notes Page 711 Chapter 14: Vasicek 2: The Term Structure, Lecture Notes Page 712 Adventures in Debentures Adventures in Debentures Valuing Longer Term Bonds with Term Premiums: A Numerical Illustration How can we compute the value of the 3 year bond without explicit use of the risk penalty? Consider how to create a synthetic version of a 3 year bond using the 1 year bond and the 2 year bond. We know the 3 year bond becomes a 2 year bond in 1 year. Thus, we want to duplicate the value of a 2 year bond in 1 year in the "up" and "down" states, respectively: "up state:" 1N1 + 0.907808N2 = 0.823822 "down state:" 1N1 + 0.957645N2 = 0.905819 . The solution to these two equations in two unknowns is N1 = -0.6698 and N2 = 1.6453. The cost of this synthetic version of the 3 year bond is: 0.932393N1 + 0.863864N2 = 0.7968 . This example illustrates a dynamic extension of the Lego approach of modern finance. TM Pricing Bonds with Term Premiums A Summary To summarize, we have established for the evolution of the discount function: .35....................... .. .. .. .. ... ... ... ... ... ... 2 d3 = .8839 0 d1 = .9324 0 d2 = .8639 0 d3 = .7968 . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . . .46 d 1 2 1 d3 d 1 2 1 d3 = .9078 = .8238 = .9576 = .9058 .54 . . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . . .. ... ... ... ... ... ... ... ... . . ... ... ... ... . . ... ... ... ... . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . . .65 .57 2 d3 = .9324 .43 2 d3 = .9836 and the evolution of the term structure: .35...................... .. .. .. .. ... ... ... ... ... ... 2r3 = 12.34% 0 r 1 = 7.00% r2 = 7.32% 0 0 r 3 = 7.57% ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . . .46 r 1 2 r3 1 r 1 2 r3 1 = 9.67% = 9.69% = 4.33% = 4.95% .54 . . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . . . ... ... ... ... ... ... ... ... . . ... ... ... ... . . ... ... ... ... . . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . . .65 .57 2r3 = 7.00% .43 2r3 = 1.66% Chapter 14: Vasicek 2: The Term Structure, Lecture Notes Page 713 Chapter 14: Vasicek 2: The Term Structure, Lecture Notes Page 714 Adventures in Debentures Adventures in Debentures Summary of the Steps to Construct a Vasicek Term Structure 1.) Fix values for T , h, , , , and 0rh. (More precisely, set , , and to appropriate values for trt+h. 2.) Draw the relevant tree for trt+h for all t. 3.) Back out the implied rates for trt+2h using term premiums (based on risk penalty and V ) for all t. 4.) For longer maturities, use replication by synthetics to generate prices (and therefore yields). Tracking the Milestones 1.) Provide another numerical illustration of how to generate the evolution of the short-term rate. 2.) Discuss the Vasicek methodology for risk adjustment. (This approach will generate term premiums.) 3.) Generate a numerical illustration which values long-term bonds relative to short-term bonds. (This will give us a description of how the entire term structure evolves through time.) 4.) If time remains, demonstrate the implications of the Vasicek model for potential shifts in the term structure as well as for the expected return for buying a bond and then selling it prior to its maturity. Chapter 14: Vasicek 2: The Term Structure, Lecture Notes Page 715 Chapter 14: Vasicek 2: The Term Structure, Lecture Notes Page 716 Adventures in Debentures Adventures in Debentures Potential Shifts with Vasciek Model 0.1 Continuously Compounded Yield Yield Curve Shifts with the Vasicek Model: A Numerical Illustration Clearly, our numerical example generates non-uniform shifts in the term structure. (Just compare (0r1 - 1r2) versus (0r2 - r3) for either the up or down scenario in period 1.) Also note 1 the yield curve appears more volatile for one year bonds than two year bonds. The following graphs provide a more complete picture of the types of shifts that the Vasicek model can generate . . . 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0 5 10 Maturity (in years) 15 20 Potential Shifts with Uniform Model 0.1 Continuously Compounded Yield 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0 5 10 Maturity (in years) 15 20 Chapter 14: Vasicek 2: The Term Structure, Lecture Notes Page 717 Chapter 14: Vasicek 2: The Term Structure, Lecture Notes Page 718 Adventures in Debentures Adventures in Debentures Expected Returns Versus Bond Maturity Based on the prior numerical illustration, we can calculate the expected return from buying a bond on period 0 and selling it in period 1: Expected Returns Versus Bond Maturity: Some Comments 1.) The annually compounded return on the one year bond is 7.25%. Note that ln(1 + 0.0725) 0.0700; as expected, 0r1 = 7%. 2.) Since V is positive, we expect the bond returns to increase as the risk (or the maturity) increases to compensate investors for holding the riskier bonds. The prior illustration confirms this relation. 3.) In the next session titled "Vasicek 3: More Term Structure," we repeat this numerical illustration except we set V equal to zero. If you reconstruct the expect returns for this case, you will discover the expected returns are all equal across maturities at a point in time. 1 yr: 2 yr: 3 yr: (.46 1.000) + (.54 1.000) - .9324 = 7.25% .9324 (.46 .9078) + (.54 .9576) - .8639 = 8.19% .8639 (.46 .8238) + (.54 .9058) - .7968 = 8.95% . .7968 Chapter 14: Vasicek 2: The Term Structure, Lecture Notes Page 719 Chapter 14: Vasicek 2: The Term Structure, Lecture Notes Page 720 Adventures in Debentures Adventures in Debentures Worksheet Appendix A: Information about the Finance Museum The attached article documents the existence of a finance museum. The article was published in the Baltimore Sun on January 9, 1994. The next page suggests some items that might be found in the gift shop of such a museum. The second page was submitted by Ryan Limaye and Sherman Ma. Please let me know if you have other ideas for the gift shop. Chapter 14: Vasicek 2: The Term Structure, Lecture Notes Page 721 Chapter 14: Vasicek 2: The Term Structure, Appendix A Page 722 Adventures in Debentures Adventures in Debentures Chapter 14: Vasicek 2: The Term Structure, Appendix A Page 723 Chapter 14: Vasicek 2: The Term Structure, Appendix A Page 724 Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Adventures in Debentures Practice Questions with Solutions The degree of difficulty of each question is indicated. The easiest questions are marked by "($)," and the hardest questions are indicated by "($$$$$)." Notice that if there is a term premium, this premium would be built into the price of Bond B, which does affect the price of Bond C already. By just observing the prices of Bonds A and B, it is impossible to say whether or not the prices reflect perfect certainty, the local expectations hypothesis (that is zero term premiums), or positive term premiums 2. ($$$$) 1. ($$$) For both parts of this question, assume that as of today one can purchase a bond (call it bond A) that pays 1,000 dollars in 400 days for 912 dollars. One can also purchase a bond (call it bond B) that pays 1,000 dollars in 850 days for 876 dollars. Assume that some traders in the market can take arbitrarily large long and short positions in these three bonds. Part a. In a world of perfect certainty, what is the value today of a bond (call it bond C) that pays 100 dollars in 400 days and 1,100 dollars in 850 days? The following tree describes the evolution of the yield on the 1 year zero coupon bond. (The yield is quoted an annualized basis with continuous compounding.) 10.00% .. ... ... ... .. .. ... ... ... ... ... ... .. ... ... ... . . ... ... ... ... ... ... ... ... ... ... . . ....... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . 12.00% Part b. Now assume the future interest rates are not certain and that the term premiums are positive -- that is, expected bond returns are larger as the maturity of the underlying bond increases. Would you expect the price of the coupon bond (i.e., Bond C) to be higher, lower, or equal to the price that you calculated in part a? Explain your answer. 8.00% Year 0 Year 1 SOLUTION: Furthermore, the annualized yield (with continuous compounding) on a 2 year zero as of today (i.e., year 0) is 11%. Part a. The purpose of this question is to demonstrate how careful one must be in constructing these tree diagrams in order to avoid arbitrage profits. In fact, the yields for the one year and two year bonds that are given above permit arbitrage. Demonstrate that arbitrage is possible in this example. You should construct a portfolio as of year 0 which will have net value of zero as of year 0. This portfolio should have nonnegative cash flow in year 1 if interest rates go up to 12% and positive cash flow in year 1 if interest rates go down to 8%. Part b. Given the above tree diagram for the yields on one year bonds, determine Part a. From Bond A, we conclude 0 d400 days = 0.912. From Bond B, we conclude that 0 d850 days = 0.876. Therefore to prevent arbitrage Bond C must be worth .912(100) + .876(1,100) = 91.20 + 963.60 = 1,054.80. Part b. Since the problem assumes that the prices of the two zero coupon bonds are the same, the price of Bond C would be the same as well. Any deviation from the 1,054.80 will allow arbitrage. The arbitrage would not depend on future interest rates. Chapter 14: Vasicek 2: The Term Structure, Practice Questions with Solutions Page 725 Chapter 14: Vasicek 2: The Term Structure, Practice Questions with Solutions Page 726 Adventures in Debentures Adventures in Debentures the minimum and maximum values for the yields on the two year zero as of period 0 which do not permit arbitrage. SOLUTION: The following table illustrates the cash flows for this strategy: Strategy Buy 1 unit of a 2-year bond and next year sell this bond. Current revenue e -0 r 02 2 1 year later 1 r 12 e = 8% -.08 1 r 12 e = 12% -.12 Part a. The following table demonstrates an arbitrage is possible: Current revenue e-.112 1 year later 1 r 12 = 12% e-.112 e.10 1 r 12 = 8% Strategy Borrow (e-.112 ) dollars by issuing 1-year zero coupon bonds with yield = 10%. (Each unit is worth e-.10 ; units issued = e-.12 .) Buy 1 unit of a 2-year bond. (Each unit costs e-.112 .) Sell this bond in 1 year. Net e-.112 e.10 Issue a 1-year bond to pay for the 2-year bond. Repay this loan next year. Net e-0 r02 2 (e-0 r02 2 ) e.10 (e-0 r02 2 ) e.10 0 e-.08 - e.10 (e-0 r02 2 ) e-.12 - e.10 (e-0 r02 2 ) e-.112 e-.12 e-.08 To avoid arbitrage, we need: 0 0 e-.08 - e-.12 > 0 e-.08 - e.10 (e-0 r02 2 ) > 0 e-.12 - e.10 e-0 r02 2 < 0 Thus, the strategy in the above table requires no initial capital. In some future scenarios, the strategy generates zero cash flow, and in some future scenarios, a positive cash flow is generated. Since this strategy costs nothing and has a positive probability of a positive cash flow in the future (and zero probability of a negative cash flow), this is arbitrage! e -.18 >e -0 r 02 2 and e-.22 < e-0 r02 2 0 r 02 > .09 0 r 02 < .11 Therefore, to eliminate potential arbitrage, we must require 9% < 0 r02 < 11% Part b. If we have a strategy which requires no initial capital, then there must be a positive probability of a positive future cash flow and a positive probability of a negative future cash flow--otherwise, arbitrage is possible. Consider an investment strategy which buys a 2-year bond and generates the needed capital by issuing a 1-year bond. If 1 r12 = 8%, this is the best possible scenario for this strategy. Conversely, if 1 r12 = 12%, this is the worst possible scenario for this strategy. 3. ($) Assume the Vasicek model is true. The relevant parameters for the Vasicek model are set as follows: = 7.00%, = 2.50%, = 0.65, and V = 0.25. The time between each node in the Vasicek tree is 6 months (i.e., h = 0.5). Finally, the current 6 month continuously compounded rate (i.e., 0.0 r0.5 ) is 5.00%. Chapter 14: Vasicek 2: The Term Structure, Practice Questions with Solutions Page 728 Chapter 14: Vasicek 2: The Term Structure, Practice Questions with Solutions Page 727 Adventures in Debentures Adventures in Debentures The following Vasicek tree is consistent with the above parameters. At each node the discount function is provided where the first number is the price of a 6 month zero coupon bond, the second number is the price of a 12 month zero coupon bond, and the third number (if present) is the price of a 18 month zero coupon bond. All zero coupon bonds have a face value of $1.00. Part b. qV = 1/2 + (.07 - .05) .5 - ln(.65) ( - r) h - ln 1 = /2 + 8 .025 8 = .6312681 , - + Risk Penalty = V /2 h [e-r h - e-r h ] = .25/2 .5 [.9833445 - .9673410] = .0014145 , 0 d1.0 Year 0.0 Year 0.5 + 0.5 d1.0 0.975310 0.0 d1.0 0.0 d1.5 . ... .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. ... ... ... ... ... ... ... ... ... ... ... ... ... . .......... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . . 0.934061 = .975310[(.6312681 .9673410) + (1 - .6312681)(.9833445) - .0014145] = .9478331 . - 0.5 d1.0 Part c. up: down: 1 N0.5 + .9673410 N1.0 = .934061 1 N0.5 + .9833445 N1.0 = .961805 N0.5 = -0.7429414 N1.0 = 1.7336208 0.961805 Part a. Compute Part b. Compute Part c. Compute SOLUTION: + 0.5 d1.0 and - 0.5 d1.0 . To calculate the current cost for 0 d1.5 : .975310 N0.5 + .9478331 N1.0 = 0 d1.5 = .9185849 . 0.0 d1.0 . 0.0 d1.5 . 4. ($$$$) For each part below, indicate if the statement is true, false, or uncertain. Justify your answer by providing a brief explanation. (A few sentences should be sufficient.) Your explanation -- not your selection of true, false, or uncertain -- is the important aspect of your solution. Part a. Assume the Vasicek model for the term structure is true. If the short-term rate is greater than , then the yield curve is downward sloping. Part a. STEP = -2 ln h = .025 -2 ln(.65) .5 = 0.0164085 + .5 r 1.0 = 5.00% + 1.64085% = 6.64085% + -.5.0664085 = .967341 .5 d1.0 = 1 e - .5 r 1.0 - .5 d1.0 = 5.00% - 1.64085% = 3.35915% = 1 e-.5.0335915 = .9833445 . Part b. The Vasicek model of the term structure is not necessarily true. However, you construct a tree for the short term rate by setting qV = .5 for all nodes in the tree and the STEP size to some constant value. In deriving the rest of the term structure, you set the expected holding period return for zero coupon bonds of any maturity equal to the return on the short term zero coupon bond. Under these assumptions, Chapter 14: Vasicek 2: The Term Structure, Practice Questions with Solutions Page 730 Chapter 14: Vasicek 2: The Term Structure, Practice Questions with Solutions Page 729 Adventures in Debentures Adventures in Debentures Copyright c 2004 by Michael R. Gibbons the term structure makes uniform shifts, but it is not flat even though term premiums are zero. Questions Part c. Assume the Vasicek model for the term structure is true. In the absence of arbitrage, the ratio of the expected future spot rate to its standard deviation is equal to V . The degree of difficulty of each question is indicated. The easiest questions are marked by "($)," and the hardest questions are indicated by "($$$$$)." Even if you decide not to submit your answer to this problem set, you should review my solution only after you attempt the questions to know if you could do them or not. Many times my solution is obvious AFTER you see it, but you need to know if the solution was obvious to you BEFORE you are told the solution. If you submit your answers to this problem set, please keep the following in mind: Part a. False. When the current short term yield exceeds the long term forecast of the short term yield, then interest rates are expected to drop. However, the term structure may not slope downward if the term premium is large enough to offset the effect; that is, V is large. Part b. True. Check this out by creating a simple numerical example. Part c. False. V reflects the ratio of expected return in excess of the short-term yield divided by the standard deviation of the bond return. V does not reflect the expected future yield on some short-term bond. Part d. False. Faster mean reversion implies is getting closer to zero. Clearly, the STEP value becomes larger as becomes smaller. Furthermore, the formula for the conditional variance of the short term yield also demonstrates that the volatility increases as becomes smaller. I sometimes fall behind where I expect to be in the lectures and an assignment is due before I get to some of the relevant material. If you believe the lectures have not yet covered the material necessary to answer a particular question and the required reading provides no guide, then you should indicate this on your solution and skip the question. Names of students along with their course and section numbers should be clear. The number of students in a study group should be less than or equal to five. Homework should be stapled. The final numerical answer should be "flagged" in some manner. Boxing, highlighting, and/or underlining the number are appropriate. Pages of spreadsheet printouts should be kept to a minimum. Only the essential information should be incorporated. You should not spend an excessive amount of time trying to solve any particular question. If you cannot complete a question, just describe what you tried to do. Part d. Assume the Vasicek model for the term structure is true. As the speed of mean reversion increases, the volatility of short-term interest rates decreases. SOLUTION: Chapter 14: Vasicek 2: The Term Structure, Practice Questions with Solutions Page 731 Chapter 14: Vasicek 2: The Term Structure, Questions Page 732 Adventures in Debentures Adventures in Debentures 1. ($$$) All parts to this question are based on the tree diagram that follows. The tree describes the evolution of the discount function. The initial node is "today" (t = 0), and the time between the nodes is 3 months (or .25 years). The top number in each node represents the price of a pure discount bond that pays $1 in 3 months. The second number is the price of a zero coupon bond that pays $1 in 6 months. The third number is the price of a zero coupon bond that pays $1 in 9 months. The fourth number is the price of a zero coupon bond that pays $1 in 1 year. so that arbitrage is not available. Further, the tree was generated using the Vasicek methodology. In solving the following parts to this question, please keep the following points in mind. First, you should not try to extend the tree beyond t = .75. All of the following questions can be answered without such an extension. Solutions which are based on extending the tree beyond t = .75 will receive little, if any, credit. Second, you should solve each question in a way that minimizes your computations. If you know the answer using a simple approach, this is the appropriate answer. Solutions requiring excessive computations will receive minimal (if any) credit. Thus, even if you provide the algebraic solution which would give you the correct answer, you may not receive much (if any) credit if the approach is inefficient in terms of the required computations. (Of course, algebraic solutions without a numerical solution will receive most of the credit if the approach is efficient.) Third, be very careful about making careless errors. If you use the wrong numbers from the above tree, this could be treated as a serious error. (For this question, it will be difficult to distinguish careless errors from serious errors.) Part a. What is the value for A .25 d1.25 ? .9851 C 0 d.5 d.75 0 0 d1.0 . .. .. .. .. .. .. .. .. .. . . .. .. . .. .. .. . .. .. .. .. .. .. .. .. . .. . ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . .9832 .9663 .9491 A .25 d1.25 .9870 D .25 d.75 .9592 .9447 . . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . .. .. .. . ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. . . .. .. . .. .. .. . .. .. .. .. .. .. .. .. .. .. .. . ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . .9814 .9627 .9441 .9256 .9851 .9698 .9542 .9383 .9889 .9769 .9643 .9511 . . . .. .. . .. .. .. .. .. .. .. .. . .. .. .. . .. .. .. .. .. .. .. .. . .. . ... . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . .. .. .. . .. .. . . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . .... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . .. .. .. .. .. .. . . .. .. . .. .. .. . .. .. .. .. .. .. .. .. . .. .. .. . .... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . .9795 .9592 G .75 d1.50 d1.75 .75 .9832 .9663 .9491 .75 d1.75 .75 d1.0 .75 d1.25 F .75 d1.50 dF .75 1.75 Part b. In the down scenario at t = .25, use the prices for the 9 month bond or the 12 month bond to determine if term premiums are positive or zero. For Part b, assume that the probability of interest rates increasing from this down scenario at t = .25 is 58.34%. (The above tree may not be based on such an assumption, but use this assumption for this part.) Part c. What is the value for 0 dC ? For Part c, assume that the probability of .5 interest rates increasing after "today" (i.e., t = 0) is 65% and is 0.40. (The above tree may not be based on such assumptions, but use these assumptions for this part.) Part d. What is the value for at t = .25.) Part e. What is the value for D .25 d.75 ? (This is the "2h" bond in the down scenario E .75 d1.0 .75 d1.25 .75 d1.50 .75 d1.75 E .75 d1.0 ? t=0 t = .25 t = .50 t = .75 Part f. What are the values for Part g. What is the value for F .75 d1.5 and F .75 d1.75 ? You should assume that the discount function evolves through time in such a way Chapter 14: Vasicek 2: The Term Structure, Questions Page 733 G .75 d1.5 ? Chapter 14: Vasicek 2: The Term Structure, Questions Page 734 Adventures in Debentures Adventures in Debentures WARNING: In my opinion, part g is a very hard question. Do not waste too much time on this part. Compute 0 r0,2 . 2. ($$$$$) The following tree describes the evolution of the term structure. The initial node is "today" (t = 0), and the time between the nodes is 6 months. The top number in each node represents the continuously compounded yield on a zero coupon bond that pays $1 in 6 months. The second number (when it is present) is the continuously compounded yield on a zero coupon bond that pays $1 in 12 months. The third number (when it is present) is continuously compounded yield on a zero coupon bond that pays $1 in 18 months. The fourth number (when it is present) is continuously compounded yield on a zero coupon bond that pays $1 in 24 months. Unlike most trees we have seen this course, the following tree is not "binomial," but "trinomial" -- that is, at each node there are three possible outcomes over the next year. 3. ($) Assume the Vasicek model with the following parameter values: 0 r0,.25 = 6.5%, h = .25, T = .5, = 8.0%, = 2.5%, = .70, and V = 0.20. The following tree diagram is based on the Vasicek model with the parameter values given above. At each node the first number is the yield on a zero coupon bond with a 3 month maturity; the second number (when present) is the yield on a zero coupon bond with a 6 month maturity; and the third number (when present) is the yield on a zero coupon bond with a 9 month maturity. All the yields are annualized with continuous compounding. 9.9592% 9.8801% 9.8043% 0 r 0,2 . . .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. .. ...................... ....................... . .. .. ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . .. 11.0763% 10.9527% 10.8349% 9.9592% 9.8801% 9.8043% 8.8421% 8.8074% 8.7737% . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . ... .. . .. .......................................... .. .. . .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. . .... .. ... ... .. ... .. ... .. .. .. .. .. .. .. .. .. .. .. .. . .. . ...................... . ....................... .. . . . .. ... . .. .. .. .. .. .. .. .. .. .. . .. . .. .. .. .. .. .. .. ... .... . .. ... .. ... .. ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . .. .. . . .. .......................................... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . 12.1933% 12.0254% 11.0763% 10.9527% 9.9592% 9.8801% 6.5000% 6.6190% C 0 r 0,.75 . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . ... ... ... ... ... ... ... ... . .......... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. . 7.5558% B .25 r .25,.75 5.4442% 5.6104% . . .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . .......... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. . . ... ... ... ... ... ... ... ... ... ... . . ... ... ... ... . . ... ... ... ... ... ... ... ... .. . ... .. .......... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. . 8.6115% 6.5000% A .5 r .5,.75 Part a. Calculate .5 rA . .5,.75 8.8421% 8.8074% 7.7250% 7.7348% Part b. Calculate B .25 r .25,.75 . 0,.75 Part c. Calculate 0 rC . t = 0.0 t = 0.5 t = 1.0 Page 735 Chapter 14: Vasicek 2: The Term Structure, Questions Page 736 Chapter 14: Vasicek 2: The Term Structure, Questions Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Adventures in Debentures Chapter 15 "It is generally agreed that, ceteris paribus, the fertility of a field is roughly proportional to the quantity of manure that has been dumped upon it in the recent past. By this standard, the term structure of interest rates has become in the last dozen years an extraordinarily fertile field indeed." Vasicek 3: More Term Structure -- Edward Kane1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . 739 Announcements and Assignments . . . . . . . . . . . . . . 741 Lecture Notes . . . . . . . . . . . . . . . . . . . . . . . . . 742 A: The Vasicek Model When h Goes to Zero . . . . . . . 758 Practice Questions with Solutions . . . . . . . . . . . . . . 782 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . 786 1 Journal of Finance. May, 1970. Chapter 15: Vasicek 3: More Term Structure Page 737 Chapter 15: Vasicek 3: More Term Structure Page 738 Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Adventures in Debentures 3.) If time remains, discuss the intuition provided by an equation giving the yield of zero coupon bond for any maturity for very small h. Preface to Vasicek 3: More Term Structure 1.) Suggested Preparation Before Attending This Session. Before attending this session, you should review the chapter titled "Vasicek 2: The Term Structure." If you have any questions about this prior chapter, you should bring them to class. 4.) Required Reading. After attending this session, you should read: Appendix A, "The Vasicek Model When h Goes to Zero." (Only sections 1 and 2 are required; section 3 is supplemental reading.) 5.) New Vocabulary Used in this Chapter. 2.) Summary of Chapter. We will develop another example of how the term structure might evolve given forecasts about future short-term rates. We will use the same forecasts about the shortterm rate as we assumed in the prior chapter titled "Vasicek 2: The Term Structure." However, here we will assume term premiums are zero (that is, V = 0), which is a special (but well known) case of the general Vasicek framework. This special case is called "Local Expectations Hypothesis." Comparing the examples in today's chapter with prior chapters will generate some useful insights about interpreting the shape of the yield curve. The following buzz words will be used in the lecture notes, the readings, and/or the problem sets: Implied parameter values and local expectations hypothesis. 6.) Summary of Important Equations. 1 (1 - (t-s) ) (t - s)(- ln ) s rt = rL + (s r - rL ) + 3.) Road Map for Chapter. where t s. The topics in this chapter will be organized as follows: 1.) Price bonds with an arbitrary maturity using the local expectations hypothesis (i.e., set term premiums to zero). 2.) Demonstrate internal consistency of bond pricing across different maturities in the sense that arbitrage is not available. Chapter 15: Vasicek 3: More Term Structure, Preface Page 739 2 (1 - (t-s) )2 2(t - s)(ln )2 rL = + 2 - ln V - 2 - ln Chapter 15: Vasicek 3: More Term Structure, Preface Page 740 Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Announcements and Assignments ecture notes L for Vasicek 3: More Term Structure A Road Map 1.) Price bonds with an arbitrary maturity using the local expectations hypothesis (i.e., set term premiums to zero). 2.) Demonstrate internal consistency of bond pricing across different maturities in the sense that arbitrage is not available. 3.) If time remains, discuss the intuition provided by an equation giving the yield of zero coupon bond for any maturity for very small h. Chapter 15, Announcements and Assignments Page 741 Chapter 15: Vasicek 3: More Term Structure, Lecture Notes Page 742 Adventures in Debentures Adventures in Debentures The Evolution of the Short Term Rate: A Review of a Prior Illustration Assume: T = 2 year and h = 1 year. Assume: = 6%, = 4%, and = .8 . Assume: 0r1 = 7%. With these values: The Evolution of the Short Term Rate: A Graphical Summary .35 .46 . .. .. .. ... ... ... ... ... ... .. ... ... ... .. .. ... ... .. .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. .. 7 + 2.67 = 9.67 7% -2 ln(.8) 1 .65 .57 .54 STEP = -2 ln h = (.04) .0267 = and qV,t = 1 + 2 7 - 2.67 = 4.33 .43 . .. ... ... ... ... ... ... .. ... ... ... .. ... ... ... .. .. ... ... .. .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. . .. ... ... ... ... ... .. ... ... ... .. ... ... ... .. ... ... ... .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. . 7 + (2 2.67) = 12.34% 7% 7 - (2 2.67) = 1.66% Year 2 Year 0 (.06 - trt+h) 1 .04 8 - ln(.8) Year 1 These numbers generate the following tree diagram . . . Chapter 15: Vasicek 3: More Term Structure, Lecture Notes Page 743 Chapter 15: Vasicek 3: More Term Structure, Lecture Notes Page 744 Adventures in Debentures Adventures in Debentures The Evolution of the Short-Term Discount: An Equivalent View The Local Expectations Hypothesis Now consider the case where term premiums are zero (that is, V = 0 so risk adjustment factors are zero as well). This situation is also called the local expectations hypothesis. The way we use the local expectations hypothesis is by fixing the price of a bond today so that it equals the expected value of the bond next period discounted by the 1 period riskless rate. In other words, the price is set today so that the expected holding period return on the bond is just the riskless rate. In other words, .35 .46 .. .. ... ... ... ... ... ... .. ... ... ... .. ... ... ... . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. . e-.0967 = .9078 .9324 .65 .57 .54 e-.0433 = .9576 .43 . ... ... .. .. ... ... ... ... ... ... .. ... ... ... .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. . . .. .. .. ... ... ... ... ... ... .. ... ... ... .. ... ... ... .. ... ... ... .. ... .. .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. . e-.1234 = .8839 e-.0700 = .9324 e-.0166 = .9835 Year 2 Year 0 Year 1 1+ Expected Future Payoff - Vt = (1 + trt,t+h)h Vt 1 [Expected Future Payoff] (1 + trt,t+h)h Vt = = e-h trt,t+h [Expected Future Payoff] . Chapter 15: Vasicek 3: More Term Structure, Lecture Notes Page 745 Chapter 15: Vasicek 3: More Term Structure, Lecture Notes Page 746 Adventures in Debentures Adventures in Debentures Perhaps surprisingly, pricing bonds using the local expectations hypothesis does not imply that forward rates are unbiased predictors of future spot rates. Intuitively, this is like pricing bonds assuming their betas are zero if the CAPM was our model for pricing. Valuing 2h Bonds without Term Premiums: A Numerical Illustration Consider a numerical example. Look at the pricing of the 2 year bond in year 0. Assume V equals zero. Now term premiums are set to zero. Under the local expectations hypothesis, the price of any bond is just the expected value of that bond next period -- discounted at the current 1 period interest rate. From the earlier tree diagram for short term rates, we know the price of a 1 year bond in 1 year is either e-0.9671 with probability of 0.46 or e-0.4331 with a probability of 0.54. Since a 2 year bond becomes a 1 year bond in 1 year, the expected value of the current 2 year bond discounted at the current 1 year rate of 7% is: e-.071(.46 e-.09671 + .54 e-.04331) .8716 . = By the same logic, we can determine the possible prices of a 2 year bond in year 1. In the "up scenario" for year 1: e-.09671(.35 e-.12341 + .65 e-.071) .8312 . = In the "down scenario" for year 1: e-.04331(.57 e-.071 + .43 e-.01661) .9140 . = Chapter 15: Vasicek 3: More Term Structure, Lecture Notes Page 747 Chapter 15: Vasicek 3: More Term Structure, Lecture Notes Page 748 Adventures in Debentures Adventures in Debentures Valuing Longer Term Bonds without Term Premiums: A Numerical Illustration How can we compute the value of the 3 year bond without explicit use of the local expectations hypothesis? Consider how to create a synthetic version of a 3 year bond using the 1 year bond and the 2 year bond. We know the 3 year bond becomes a 2 year bond in 1 year. Thus, we want to duplicate the value of a 2 year bond in 1 year in the "up" and "down" states, respectively: "up state:" "down state:" 1N1 + 0.9078N2 = 0.8312 1N1 + 0.9576N2 = 0.9140 . Pricing Bonds without Term Premiums A Summary To summarize, we have established for the evolution of the discount function: .35....................... .. .. .. .. ... ... ... ... ... ... 2 d3 = .8839 0 d1 = .9324 0 d2 = .8716 0 d3 = .8168 . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . . .46 d 1 2 1 d3 d 1 2 1 d3 = .9078 = .8312 = .9576 = .9140 .54 . . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . . .. ... ... ... ... ... ... ... ... . . ... ... ... ... . . ... ... ... ... . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . . .65 .57 2 d3 = .9324 .43 2 d3 = .9836 and the evolution of the term structure: The solution to these two equations in two unknowns is N1 = -0.6782 and N2 = 1.6627. The cost of this synthetic version of the 3 year bond is: 0.9324N1 + 0.8716N2 = 0.8168 . This numerical example illustrates a dynamic version of the LegoTM approach of modern finance. .35...................... .. .. .. .. ... ... ... ... ... ... 2r3 = 12.34% 0 r 1 = 7.00% r2 = 6.87% 0 0 r 3 = 6.74% ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . . .46 r 1 2 r3 1 r 1 2 r3 1 = 9.67% = 9.25% = 4.33% = 4.50% .54 . . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . . . ... ... ... ... ... ... ... ... . . ... ... ... ... . . ... ... ... ... . . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . . .65 .57 2r3 = 7.00% .43 2r3 = 1.66% Chapter 15: Vasicek 3: More Term Structure, Lecture Notes Page 749 Chapter 15: Vasicek 3: More Term Structure, Lecture Notes Page 750 Adventures in Debentures Adventures in Debentures Pricing Bonds with Term Premiums A Review of the Prior Session To review an illustration from the session titled "Vasicek 2: The Term Structure," we established for the evolution of the discount function: d 1 2 1 d3 d 1 2 1 d3 = .9078 = .8238 = .9576 = .9058 . . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . . ... ... ... ... ... ... ... ... . . ... ... ... ... . . ... ... ... ... . .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . . Tracking the Milestones 1.) Price bonds with an arbitrary maturity using the local expectations hypothesis (i.e., set term premiums to zero). 2.) Demonstrate internal consistency of bond pricing across different maturities in the sense that arbitrage is not available. 3.) If time remains, discuss the intuition provided by an equation giving the yield of zero coupon bond for any maturity for very small h. .35 2 d3 = .8839 0 d1 = .9324 d2 = .8639 0 0 d3 = .7968 .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . . .46 .65 .57 2 d3 = .9324 .54 .43 2 d3 = .9836 and the evolution of the term structure: r 1 2 r3 1 r 1 2 r3 1 = 9.67% = 9.69% = 4.33% = 4.95% . . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . . . ... ... ... ... ... ... ... ... ... ... . . ... ... ... ... .. .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . . .35....................... .. .. .. .. ... ... ... ... ... ... 2r3 = 12.34% 0 r 1 = 7.00% r2 = 7.32% 0 0 r 3 = 7.57% . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . . .46...................... .. .. .. .. ... ... ... ... ... ... .65 .57 2r3 = 7.00% .54 .43 2r3 = 1.66% Chapter 15: Vasicek 3: More Term Structure, Lecture Notes Page 751 Chapter 15: Vasicek 3: More Term Structure, Lecture Notes Page 752 Adventures in Debentures Adventures in Debentures Checking for Internal Consistency and Arbitrage In the previous example, we can "track a security" for years 0 and 1: Checking for Internal Consistency: A Numerical Illustration Consider replicating the one year bond using a synthetic based on the two and three year bonds. To find the composition of the synthetic, we need to solve the following 2 equations with 2 unknowns: "up state:" "down state:" 0.9078N2 + 0.8312N3 = 1 0.9576N2 + 0.9140N3 = 1 . 1 year bond: . .. ... ... ... .. .. ... ... ... ... ... ... .. ... ... ... .. .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. .. 2 year bond: . .. ... ... ... .. .. ... ... ... ... ... ... .. ... ... ... .. .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. .. 3 year bond: . .. ... ... ... .. .. ... ... ... ... ... ... .. ... ... ... .. .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. .. 1 .9078 .8312 .9324 (0d1) 1 .8716 (0d2) (1d2) .9576 .8168 (0d3) (1d3) .9140 The solution to the above system of equations is N2 = 2.4529 and N3 = -1.4759. The cost of this synthetic is: 0.8716N2 + 0.8168N3 = 0.9324 . The cost of this synthetic (0.9324) is the same as the cost of the explicit 1 year zero coupon bond. No arbitrage is possible, and the prices are internally consistent! (1d2) Year 0 Year 1 Year 0 Year 1 Year 0 (1d3) Year 1 Now take any two bonds and try to replicate the third (or create a synthetic version of the third). Note the "1 year bond" matures on year 1; the "2 year bond" matures on year 2; and the "3 year bond" matures on year 3. Chapter 15: Vasicek 3: More Term Structure, Lecture Notes Page 753 Chapter 15: Vasicek 3: More Term Structure, Lecture Notes Page 754 Adventures in Debentures Adventures in Debentures Tracking the Milestones 1.) Price bonds with an arbitrary maturity using the local expectations hypothesis (i.e., set term premiums to zero). 2.) Demonstrate internal consistency of bond pricing across different maturities in the sense that arbitrage is not available. 3.) If time remains, discuss the intuition provided by an equation giving the yield of zero coupon bond for any maturity for very small h. The Vasicek Model as h Becomes Small: Introduction to Appendix A The following result can be established (for t s): srt = r L + (s r - r L ) + 1 (1 - (t-s)) (t - s)(- ln ) 2 (1 - (t-s))2 , 2(t - s)(ln )2 where sr is the intercept of the the continuously compounded yield curve (perhaps you want to think of this intercept as the overnight rate) and rL is the rate on the long-term bond (actually the asymptote of the continuously compounded yield curve): 2 2 rL = + V - . - ln - ln The above equation provides a very useful way to interpret V by relating it to the yield on a very long term zero coupon bond (that is, the asymptote of the term structure). See Appendix A for more details. Chapter 15: Vasicek 3: More Term Structure, Lecture Notes Page 755 Chapter 15: Vasicek 3: More Term Structure, Lecture Notes Page 756 Adventures in Debentures Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Worksheet Appendix A: The Vasicek Model When h Goes to Zero* Sections 1 and 2 of this appendix are required reading while Section 3 is not required reading. The appendix is divided into 3 sections. First, we present an extension of the Vasicek model that describes the evolution of the term structure when time is continuous (that is, as h approaches zero). Second, we discuss the different term structures that the Vasicek model can generate. Third, we propose a method to infer the parameters that generate a particular set of interest rates. The intent is to describe the analytics in a manner that avoids complex calculus. To keep the mathematically-minded content, footnotes are included that outline the basic theory. 1.) When h Becomes Small. In this section, we will illustrate what happens in the Vasicek trees as we let h become smaller. 1.1.) An Illustration. In class, we will apply the Vasicek model to price many types of fixed income securities. We assume that a fixed time interval, T , can be divided up into smaller pieces, and we denote the length of each of these time units by h. Then, we simply determine values for , , and V , and we calculate a binomial tree using the fixed time interval, T , and a continuously compounded yield on a bond with h periods to maturity, 0 rh . For example, when we set T = 3 and h = 1,1 we are implicitly assuming that the shortest maturity on the yield curve (sometimes we call this the intercept of the term * Originally, this note was written by John J. Kraska, III under the supervision of Michael Gibbons, Professor of Finance, The Wharton School, in March 1993. Substantial revisions have been made since that original version. 1 Here, as in the lectures, units of time are measured in years. Therefore, T = 3 years. Chapter 15: Vasicek 3: More Term Structure, Lecture Notes Page 757 Chapter 15: Vasicek 3: More Term Structure, Appendix A Page 758 Adventures in Debentures Adventures in Debentures structure) is the one year continuously compounded rate, 0 r1 , and using the local expectations hypothesis, we calculate the two year and three year rates, 0 r2 and 0 r3 . h = 1: The value we pick for h depends on what time intervals we wish to model. If, for instance, we want to model monthly rates, then setting h = 1 will not do. Instead, 1 we must let h 12 . However, when we decrease h holding everything else constant, a number of things happen. One consequence is that the Vasicek tree becomes quite large and computations become more cumbersome. For instance, for T = 3 and 1 h = 1, there are 6 nodes, but there are over 600 nodes when T = 3 and h = 12 ! You 1 can imagine that calculating 0 r3 can become quite a chore when h = 12 . Another consequence is that we no longer fix 0 r1 , but instead use 0 r1/12 , as our estimate of the shortest maturity on (or intercept of) the yield curve. Then, we have to invoke (say) the local expectations hypothesis to calculate 0 r1 . But, as h decreases beyond a certain point, we would also expect that our estimates for 0 r2 and 0 r3 should not change drastically. In fact, we would hope that as h approaches zero our estimates of 2 s r t for t > s would converge. Let us demonstrate what happens to the Vasicek model as h gets small by making the following assumption. For each h, let T = 3 and hold the one year continuously compounded rate fixed at 8% (i.e., 0 r1 = 8%). When h = 1, assume = 12.25%, and = 6%, = .75, and V = 0.3 Figure 1 displays Vasicek binomial trees for h = 1, 1 1 1 , , , and 1 . As you can see, the tree for h = 1 is substantial compared to that for 2 3 4 8 8 h = 1. How do our estimates of 0 r2 and 0 r3 change for different values of h? Table 1 displays the results of applying the Vasicek model and local expectations hypothesis for h = 1, 1 1 , , and 1 . It appears from Table 1 that the values for 0 r2 and 0 r3 converge as h 2 3 4 approaches zero.4 Figure 1: Vasicek Binomial Trees for Decreasing Values of h 17.10% 12.55% 8.00% 3.45% -1.10% 8.00% h = 1: 2 21.01% 17.65% 14.28% 10.92% 7.56% 4.20% 0.84% -2.53% 7.56% 4.20% 0.84% -5.89% 10.92% 7.56% 14.28% h = 1: 3 24.14% 21.34% 18.54% 15.74% 12.94% 10.15% 7.35% 4.55% 1.75% -1.05% 7.35% 4.55% 1.75% -3.84% -1.05% -6.64% 10.15% 7.35% 4.55% 1.75% -3.84% -9.44% 12.94% 10.15% 7.35% 15.74% 12.94% 18.54% 2 3 4 In fact, one way to determine if h is small enough in any given application of a Vasicek tree is to keep reducing h until the values of the securities do not change by a significant amount. (WARNING: This footnote may confuse you; it is only provided for those who are pathologically curious.) In this illustration, we want to analyze what happens to a certain yield curve as h decreases. Therefore, for each h, we must hold certain underlying assumptions of our yield curve (e.g., our expectation of future short term rates) fixed. This motivates our decision to hold one point on the curve, 0 r1 , constant. At the same time, we are also adding the constraint that, at time 0, our expectation and variance of 1 r2 as well as our expectation of s rt as s goes to infinity remain constant. As h decreases, the parameters , , and 0 rh will change, but expectations of 1 future rates will not. In this example, for h = 2 , = 5.41% and = .68; for h = 1 , = 5.22% 3 and = .65; and for h = 1 , = 5.09% and = .63. 4 The last row of Table 1 is calculated using Equations (15-2) and (15-3). h = 1: 4 26.82% 24.37% 21.92% 19.47% 17.02% 14.57% 12.17% 9.70% 7.23% 4.76% 2.29% 7.23% 4.78% -0.12% 9.68% 7.23% 4.78% 2.33% -2.57% -7.46% -0.12% -5.02% -9.91% 12.12% 9.68% 7.23% 4.78% 2.33% -2.57% Page 760 -7.46% -12.36% 14.57% 12.12% 9.68% 7.23% 17.02% 14.57% 12.12% 19.47% 17.02% 21.92% Chapter 15: Vasicek 3: More Term Structure, Appendix A Page 759 2.33% -0.12% -2.57% Chapter 15: Vasicek 3: More Term Structure, Appendix A -5.02% Figure 1 (continued): Vasicek Binomial Trees for Decreasing Values of h h = 1: 8 35.19% 33.43% 31.67% 29.91% 28.15% 26.39% 24.63% 22.87% 21.11% 19.35% 17.59% 15.83% 14.07% 12.31% 10.54% 8.78% 7.02% 5.26% 3.50% 1.74% -0.02% -1.78% -3.54% -5.30% -7.06% -8.82% -5.30% -7.06% -10.58% -3.54% -3.54% -1.78% -1.78% -0.02% -0.02% -0.02% 1.74% 1.74% 1.74% 1.74% -1.78% -5.30% -8.82% -12.34% -0.02% -3.54% -7.06% -10.58% -14.11% 3.50% 3.50% 3.50% 3.50% 5.26% 5.26% 5.26% 5.26% 5.26% 3.50% 1.74% -1.78% -5.30% -8.82% -12.34% -15.87% -0.02% -3.54% -7.06% -10.58% -14.11% -17.63% 7.02% 7.02% 7.02% 7.02% 7.02% 8.78% 8.78% 8.78% 8.78% 8.78% 7.02% 5.26% 3.50% 1.74% -1.78% -5.30% -8.82% -12.34% -15.87% -19.39% -0.02% -3.54% -7.06% -10.58% -14.11% -17.63% -21.15% 10.54% 10.54% 10.54% 10.54% 10.54% 8.78% 7.02% 5.26% 3.50% 12.31% 12.31% 12.31% 12.31% 14.07% 14.07% 14.07% 14.07% 12.31% 10.54% 8.78% 7.02% 15.83% 15.83% 15.83% 15.83% 14.07% 12.31% 10.54% 17.59% 17.59% 17.59% 19.35% 19.35% 19.35% 17.59% 15.83% 14.07% 21.11% 21.11% 21.11% 19.35% 17.59% 22.87% 22.87% 24.63% 24.63% 22.87% 21.11% 26.39% 28.15% 26.39% 24.63% 29.91% 28.15% 31.67% Adventures in Debentures Chapter 15: Vasicek 3: More Term Structure, Appendix A Page 761 1 4 1 3 1 2 Adventures in Debentures 1 h 0 8.00% 0 r1 8.00% 8.00% 8.00% 8.00% Chapter 15: Vasicek 3: More Term Structure, Appendix A Page 762 0 r2 8.76% 8.73% 8.68% 8.56% 9.35% Table 1: Vasicek Binomial Trees for Decreasing Values of h 8.80% 0 r3 9.31% 9.26% 9.19% 8.98% Adventures in Debentures Adventures in Debentures 1.2.) An Analytical Result for Yields (and Prices). This apparent convergence of 0 r2 and 0 r3 as h goes to zero motivates our next result. Vasicek (1977) established that when h is infinitesimal:5 2 2 V - s rt = + - ln - ln 1 - t-s 2 (1 - t-s )2 2 2 + . 15-1) ( V + + sr - - - ln (t - s)(- ln ) 2(t - s)(ln )2 - ln Equation (15-1)) provide a convenient method for determining values of plain-vanilla bonds when h is very small.7 Imagine the difficulty of generating enough observations to draw a Vasicek performance profile for even a simple zero coupon bond using only a binomial tree.8 Section 3 will also use Equations (15-2) and (15-3) to find the implicit Vasicek parameters consistent with a given term structure. The method outlined in Section 3 could be done without Equations (15-2) and (15-3), but the amount of computations would be enormous. 1.3.) The Vasicek Delta: More Analytical Results. We will discuss in class a method to calculate the interest rate sensitivity of a security (that is, a "delta") based on the Vasicek binomial trees. In this subsection, we will take a different approach based on Equation (15-2). The approach in this subsection will not give the same numerical answer as that in class except when h is very small. First, we can use Equation (15-2) to find an equation for the price of zero coupon -(t-s) bond that pays one dollar on date t (that is, s dt = e 100 s rt , where s rt is a function of s r as given in equation (15-2) and the "100" accounts for rates being quoted in percentage units). Based on this last equation we can study how sensitive the price is to changes in s r. The most obvious way to proceed is to calculate the first derivative of this pricing equation with respect to s r. Such a calculation will reveal (after some simplification): (t-s) - 1) s dt ( . (15-4) 100 ln Equation (15-4) is conceptually the same type of calculation as dollar duration -- that is, it measures the dollar gain in the security from a decrease in interest rates. However, Equation (15-4) assumes the term structure for all maturities makes a shift consistent with the Vasicek model, which is not uniform. Thus, even for the same security, its Vasicek delta need not be equal to its dollar delta. Equation (15-4) will be equal to dollar duration (or delta) as approaches one. Thus, the Vasicek model is more general than the traditional approach of uniform shifts, for the Vasicek model can include the uniform shift model as a special case. 7 Equation (15-1) can be reparameterized by making the following observation. As t becomes large, s rt is the yield on a very long term zero coupon bond. If we let rL be 2 2 the limit of s rt as t approaches infinity, then6 rL = + ln V - - . ln - Substituting rL into Equation (15-1) yields what we will refer to as the Vasicek equation for small h: s rt = rL + (s r - rL ) 2 (1 - t-s )2 1 - t-s + (t - s)(- ln ) 2(t - s)(ln )2 (15-2) where 2 2 V - . rL = + - ln - ln (15-3) The last row of Table 1 shows 0 r2 and 0 r3 using Equations (15-2) and (15-3). Why are Equations (15-2) and (15-3) useful? First, Equation (15-3) provides a convenient way to think about V . If we believe we have reasonable values for , , , and rL , then we can back out V based on Equation (15-3). Second, calculations involving a Vasicek binomial tree can be tedious, and Equations (15-2) and (15-3) (or 5 6 For the mathematical details, see Vasicek, "An Equilibrium Characterization of the Term Structure," in Journal of Financial Economics, Vol. 5, 1977, pp. 177188. This can be shown by applying elementary calculus to Equation (15-1) and noting that: lim t-s = 0 since || < 1. as s - . 8 For questions on exams and problem sets, you should not use Equation (15-2) when you are told h is finite unless the question specifically suggests that you follow such an approach. To generate such a profile, you should calculate an entire tree for every point on the horizontal axis of the profile. That is, think of the horizontal axis measuring 0 rh , as you change the initial 0 r h your entire tree would be different! Chapter 15: Vasicek 3: More Term Structure, Appendix A Page 763 Chapter 15: Vasicek 3: More Term Structure, Appendix A Page 764 Adventures in Debentures Adventures in Debentures As your understanding of the Vasicek model deepens, this connection to dollar duration should become more intuitive. As approaches one, we are essentially saying that all changes in the short-term rate should be viewed as permanent in the sense that the mean reversion is very small. If there is no mean reversion, then investors who buy long-term bonds view today's short-term rate as a good indicator of the short-term rate in the future (when their current long-term bonds become shortterm). Thus, when is near one, the term structure will make uniform shifts, and dollar duration is an appropriate measure of risk. Section 2 will provide some numerical illustrations of Equations (15-2) and (15-3). downward and flat term structures although it does require that all the curves reach a horizontal asymptote. Also, the curve marked C demonstrates the ability of the Vasicek equation to create a curve that has at most one "hump." Term structures that can be described by the Vasicek equation are those which have features similar to the curves in Figure 2.11 Figure 2: Yield Curves Generated by Vasicek Model 2.) Possible Term Structures Using the Vasicek Equation. Now that we have this new equation, we could ask ourselves, where do we get , , and V ? Well, one approach is to estimate the parameters from history. Still another is to rely on divine inspiration.9 A third alternative, and one which we will concentrate on in Section 3, is to try to find , , and V using just the current term structure of interest rates. In other words, we can take s rt in Equation (15-2) as given for a certain set of s and t, and try to back out the , , and V which are consistent with the observed term structure.10 There is a caveat to this third method of determining the Vasicek parameters. Because this approach relies on current market prices, it reflects the market's consensus on expectations of how the term structure will evolve over time. At the same time, it requires that the Vasicek model be an accurate descriptor of the current term structure as well as a reasonable way to characterize the forecasts of future interest rates. Therefore, before we can begin to solve for the Vasicek parameters, we first have to check if the term structure from which we get our interest rates can be generated by the Vasicek model. In this section, we investigate the types of term structures that the Vasicek model can describe. This should gives us some perspective as to whether or not we can solve for a , , , and V just by looking at the properties of a given yield curve. Figure 2 shows different types of yield curves generated by the Vasicek equation. We see that the Vasicek equation is flexible enough to produce a variety of upward, 9 10 There are term structures, though, that no Vasicek model can describe. For instance, curves such as those in Figure 3 have characteristics that Vasicek cannot match exactly. These include curves with two or more "humps" or curves with no horizontal asymptote. To describe these curves exactly, we would have to abandon the Vasicek equation and find another model to describe the term structure. This is usually the best approach. This is similar to the notion of implied standard deviation in the context of option prices using the Black-Scholes model for equity options. 11 Curve A was created using rL = 11%, 0 r = 8%, = .75, and = 6%; Curve B using rL = 7%, 0 r = 10%, = .75, and = 4%; Curve C using r L = 6%, 0 r = 6%, = .9 and = 10%; and, Curve D using rL = 5%, 0 r = 5%, = .3, and = 4%. Chapter 15: Vasicek 3: More Term Structure, Appendix A Page 765 Chapter 15: Vasicek 3: More Term Structure, Appendix A Page 766 Adventures in Debentures Adventures in Debentures Figure 3: Yield Curves Unable to be Generated by Vasicek Model this term structure.13 Figure 4: Example of a Yield Curve Generated by Vasicek Model 3.) Finding the Vasicek Parameters from the Term Structure.12 Suppose we observe a term structure and want to reproduce it using the Vasicek model. How do we know what parameters to plug into the Vasicek equation to generate this term structure? This section presents a method that we can use to solve for the Vasicek parameters, , , and V , given a set of interest rates. We can see that the rates for longer maturities appear to converge to 7.5%. Since we know that Vasicek term structures must reach an asymptote, rL , we can make the assumption that rL = 7.5%. Furthermore, since the intercept of the curve is 8%, we can let 0 r = 8%. Therefore, from Equation (15-2), we only need to solve for and . Once we have these values, we can solve for and V by plugging our estimates for , and rL , into Equation (15-3).14 Solving for and is complicated by the fact that Equation (15-2) is nonlinear in these two variables. In other words, we cannot simply take two of the points from Figure 4 and solve the system of two equations with two unknowns using standard linear algebra. Instead we have to use some other technique. The technique we will employ is known as the "zig-zag" method of solving nonlinear equations.15 It is called 13 3.1.) An Example. An example will be the best way to explain the general approach. Figure 4 displays a yield curve from which we will solve for , , and V as well as certain points on this curve. From just looking at this curve, we know it has the characteristics of a Vasicek model (e.g., an asymptote and only one "hump") and, we have some confidence that we can find a set of Vasicek parameters which will describe 12 14 15 This section is not required reading. There is no guarantee, however, that the solution set of parameters is unique. One could use this methodology without the analytical expressions given in Equations (15-2) and (15-3). These equations could be replaced by the algorithm that generates a Vasicek binomial tree for a finite h. However, using the trees directly would require a large number of computations. There are other ways to solve a system of nonlinear equations. For example, it is possible to transform the system of nonlinear equations into a system of linear equations by using a Taylorseries expansion. In other words, we can take the first partial derivatives of Equation (15-2) with r r ^ ^ respect to and , s t and s t . Then, we can take arbitrary values of and and solve the Chapter 15: Vasicek 3: More Term Structure, Appendix A Page 767 Chapter 15: Vasicek 3: More Term Structure, Appendix A Page 768 Adventures in Debentures Adventures in Debentures "zig-zag" because the method requires us to solve for for given values of , and then solve for using the previously solved values of and then solve for a new with our current value of , etc. This may sound confusing at first, but our example should provide a good illustration. Returning to our example, we have already determined that rL = 7.5% and 0 r = 8%. From Figure 4, we can take some of the points off the graph16 (say, 0 r1 , . . . , 0 r10 ), and plug these and their corresponding values of s and t into Equation (15-2) to get a system of nonlinear equations. This gives us the following ten equations in terms of and : 0 rj - rL - (0 r - rL ) 1 - j 2 (1 - j )2 = j(- ln ) 2j(ln )2 for j = 1, . . . , 10 . (15-5) This system of equations can be solved for 2 by using a simple linear regression with (1-j )2 1-j Yj = 0 rj - .075 - (.005) j(- ln ) as the independent variable, and Xj = 2j(ln )2 as the dependent variable. We can easily organize a spreadsheet to solve for 2 for any 0 < < 1 we choose.17 Given 2 , we take the square root to get . If, in running the regression, we obtain a value for 2 which is negative, we can substitute = 0. Table 2 shows the values of that solve Equation (15-6) when varies from .05 to .95.18 We can create a spreadsheet that allows us to solve for for any value of between 0 and 1. Table 2a lists the values of for each value of in increments of .05. Essentially what we have done is to reduce the problem to the simpler one of solving for one parameter, namely . For each value of , we already know the corresponding value of we should use to solve for our set of interest rates. All that is left to do is find the correct value of . To solve for , we plug in different values for between 0 and 1 and their corresponding values for into Equation (15-2) to generate different sets of interest rates ^ ^ 0 r1 , . . . , 0 r10 . Then, we take the sum of the squares of the differences between our ^ ^ calculated values of 0 r1 , . . . , 0 r10 and the observed values of 0 r1 , . . . 0 r10 from Figure 4. That is: For fixed values of (note that must lie between 0 and 1), Equation (15-5) is linear in terms of 2 . If we take = .05, for instance, Equation (15-5) becomes: 0 rj - .075 - (.005) 1 - .05j (1 - .05j )2 2 = j(- ln .05) 2j(ln .05)2 for j = 1, . . . , 10 . (15-6) following equation for and : s1 r t1 + s1 r t1 s2 rt2 ^ ( - ) + ^ ( - ) + s1 r t1 ^ ( - ) = 0 ^ ( - ) = 0 . ^ ^ , ^ ^ , s2 r t2 + ^ ^ , s2 rt2 ^ ^ , 16 ^ ^ Next, we can take these solution parameters, set = and = and repeat the second step to solve for another set of values for and . This iterative procedure should converge to a solution if our initial guesses are good enough. In particular, the following ten points were selected: 0 r1 0 r2 0 r3 0 r4 0 r5 0 r6 0 r7 0 r9 0 r 10 = 8.12% = 8.18% = 8.21% = 8.22% = 8.21% = 8.19% 17 10 SSR = j=1 (0 rj - 0 rj )2 . ^ = 8.16% 0 r 8 = 8.14% = 8.11% = 8.08% . 18 Note that 2 is being treated as a slope coeficient in a simple linear regression model which excludes an intercept. Any time you run a simple linear regression without an intercept, the Xj Yj formula to calculate the slope coefficient is just . Thus, the spreadsheet just sets 2 equal X2 j to Xj Yj 2 . Xj Analytically, all we have done is to minimize the sum of squares of the residuals in Equation (15-6). Chapter 15: Vasicek 3: More Term Structure, Appendix A Page 769 Chapter 15: Vasicek 3: More Term Structure, Appendix A Page 770 Adventures in Debentures Adventures in Debentures Table 2: , and Their Associated Sum of Squares associated with them. Table 2a shows us that the minimum sum of squares lies somewhere between = .75 and = .85. Table 2b looks at this range more closely and reveals that the optimal solution lies between = .81 and = .82. Table 2c restricts the range even further and reveals to us the optimal solution: = .814 = 6.375% . 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 Table 2a SSR 47.44100 37.68600 32.19000 28.34900 25.36600 22.89500 20.75600 18.84200 17.08710 15.44700 13.89010 12.39360 10.94090 9.52017 8.12473 6.75381 5.41353 4.11626 2.87276 2.17660 2.00760 1.83645 1.66342 1.48906 1.31412 1.13956 0.96659 0.79683 0.63242 0.47633 0.33253 0.20630 0.10425 0.03380 0.00162 0.01036 0.05404 0.11440 0.750 0.755 0.760 0.765 0.770 0.775 0.780 0.785 0.790 0.795 0.800 0.805 0.810 0.815 0.820 0.825 0.830 0.835 0.840 Table 2b SSR 8.12500 7.98600 7.84800 7.71100 7.57300 7.43600 7.29900 7.16200 7.02584 6.88968 6.75381 6.61824 6.48298 6.34804 6.21343 6.07918 5.94528 5.81175 5.67860 0.03380 0.02877 0.02414 0.01989 0.01605 0.01262 0.00959 0.00698 0.00478 0.00299 0.00162 0.00067 0.00013 8.2E-06 0.00030 0.00099 0.00209 0.00358 0.00546 0.810 0.811 0.812 0.813 0.814 0.815 0.816 0.817 0.818 0.819 0.820 0.821 0.822 0.823 0.824 0.825 0.826 0.827 0.828 Table 2c SSR 6.48300 6.45600 6.42900 6.40200 6.37500 6.34800 6.32100 6.29400 6.26724 6.24033 6.21343 6.18655 6.15969 6.13284 6.10600 6.07918 6.05237 6.02557 5.99879 0.00013 7.4E-05 3.3E-05 8.2E-06 0 8.2E-06 3.3E-05 7.4E-05 0.00013 0.00021 0.00030 0.00040 0.00052 0.00066 0.00082 0.00099 0.00118 0.00138 0.00160 Now that we have solved for and , we can plug these values and our estimate of rL into Equation (15-3) to solve for and V . An interesting observation is that there are infinitely many solutions for and V . Does this make sense intuitively? How can ( = 0; V = .95) and ( = 9.475%; V = 0) generate the same term structure? To answer this, we need to understand what and V represent. Recall from the lectures that is an estimate of the future short term interest rate. If > 0 r, then the term structure should be upward sloping and if < 0 r, the term structure should be downward sloping, right? Well, the correct answer is that we cannot tell without knowing what V is. V represents the constant expected excess19 return on a bond per unit of standard deviation. Large values of V increase the values of s rt as t increases. Therefore, it is possible that a large value for V can cause an upward sloping term structure even though < 0 r. In the context of this example, we need to know more about market expectations (e.g., the degree of risk premium investors tack onto bonds) in order to pinpoint the exact values of and V used to generate our particular term structure. To recap, in our example, we have determined the following set of parameters to be our solution for the example: = .814 = 6.375% = (9.475 - 9.937V )% . For each value of , we will get a sum of squares which is associated with it. A sum of squares equal to zero represents the choice of and that generates a set of interest rates which exactly replicates the set of interest rates in Figure 4. Our task is to calculate a sum of squares for each and and to find the minimum. Since, in this example, we know the term structure has been generated by a Vasicek equation, the minimum we find should be zero. Table 2 shows a series of tables for different values of and and the sum of squares 3.2.) Extensions. The previous subsection outlined a method by which to solve for the parameters in the Vasicek equation. In our simplified example, it just so happened that a solution existed because, in fact, the term structure given to us was generated by the Vasicek 19 That is, in excess of the yield on the short-term bond. Chapter 15: Vasicek 3: More Term Structure, Appendix A Page 771 Chapter 15: Vasicek 3: More Term Structure, Appendix A Page 772 Adventures in Debentures Adventures in Debentures equation. However, in the real world, we do not have the luxury of knowing there is a set of parameters that exactly replicates the term structure we want to describe. The advantage of the technique outlined above is that we can always find a solution set that generates a yield curve approximating the term structure. All we have to do is search for the minimum sum of squares. The minimum may not be zero but we can always find one. Consider the yield curve given in Figure 5. The curve has all the characteristics of a term structure created by the Vasicek model (i.e., horizontal asymptote, no "humps"). The optimal solution is = .2209 and = 14.035%. However, the sum of squares associated with this solution is not zero. The solution yield curve is displayed on top of the original curve in Figure 6. As you can see from the plot, the fit is not perfect but it is quite good. We do have to be careful not to abuse the technique outlined in Subsection 3.1. For instance, we can take a curve such as the one shown Figure 7 and try to solve for the Vasicek parameters. Again we will find an optimal solution of = .406 and = 15.774% with a minimum sum of squares different from zero. But, when this solution yield curve is plotted against the original curve in Figure 8, we see that the fit is very poor. This is because the original curve has properties that would lead us to believe that the Vasicek model is an incorrect descriptor of the curve. Figure 5: Yield Curve Not Generated by Vasicek but with Characteristics of Vasicek Figure 6: Optimal Solution "Best Fit" Yield Curve Chapter 15: Vasicek 3: More Term Structure, Appendix A Page 773 Chapter 15: Vasicek 3: More Term Structure, Appendix A Page 774 Adventures in Debentures Adventures in Debentures Figure 7: Yield Curve Not Generated by Vasicek and without Characteristics of Vasicek Model The other advantage of this technique is its flexibility. Suppose we are only concerned with finding the Vasicek equation that replicates a certain range of the term structure (e.g., we are interested in modeling rates of a certain maturity). We can accomplish this by adding a factor to our sum of squares equation that increases the weight of certain points on the term structure when we look for the minimum sum of squares. This method, called weighted least squares, allows us to describe a certain range of the term structure more precisely than other regions. Returning to the curve in Figure 5, we could tailor the technique of Subsection 3.1 to look for the best Vasicek model to describe (say) the beginning of the curve more accurately. We can do this by weighting the sum of squares factor for some of the points on the curve more heavily than the sum of squares for other points. Figure 9 displays the yield curve that generates the best fit for the interest rates of the first few years. As you can see, the fitted curve describes the first few years quite accurately but its ability to describe other maturities is rather poor. Figure 9: Example Using Weighted Least Squares Technique Figure 8: Optimal Solution "Best Fit" Yield Curve Lastly, it is possible to extend the "zig-zag" technique to a setting where the parameters (, , , and V ) of the Vasicek model are time dependent.20 Such an extension will provide more flexibility to fit a given term structure. 20 See Hull and White. "Pricing Interest-Rate-Derivative Securities," Review of Financial Studies 3, 1990, pp. 573592. Chapter 15: Vasicek 3: More Term Structure, Appendix A Page 775 Chapter 15: Vasicek 3: More Term Structure, Appendix A Page 776 Adventures in Debentures Adventures in Debentures 3.3.) Practice Question with Solution. To confirm that you understand the material in this section, you may want to try to answer the following question. A solution is provided below. Question: Suppose you are given a yield curve such as the one in Figure 10. Figure 10: Yield Curve for Practice Question Consider the following 12 points taken from Figure 10: 0 R0.001 0 R1 0 R2 0 R3 0 R4 0 R5 = 9.00% 0 R6 0 R7 0 R8 0 R9 0 R10 0 R1000 = 11.51% = 11.58% = 11.63% = 11.67% = 11.71% = 11.99% = 10.16% = 10.75% = 11.08% = 11.28% = 11.42% What is your best guess for the Vasicek parameters , , , V , r, and rL , which generate a yield curve that replicates this set of interest rates? Chapter 15: Vasicek 3: More Term Structure, Appendix A Page 777 Chapter 15: Vasicek 3: More Term Structure, Appendix A Page 778 Adventures in Debentures Adventures in Debentures Solution: One can infer the variables r and rL from the interest rates given. A reasonable assumption is that r is 9% and rL is 12%. Using the zig-zag technique described above, we can solve for the parameters of and . Table 3 lists the sum of squares for and when ranges from .05 to .95. From this table, we see that the minimum sum of squares lies between = .35 and = .45. table, we see that the minimum sum of squares lies between = .410 and = .420. Table 4 0.350 0.355 0.360 0.365 0.370 0.375 0.380 0.385 0.390 0.395 0.000 1.948 3.251 4.137 4.841 5.436 5.955 6.416 6.832 7.211 Sum of Squares 0.001708024 0.001297705 0.001050832 0.000836751 0.000653090 0.000497518 0.000367749 0.000261547 0.000176723 0.000111145 0.400 0.405 0.410 0.415 0.420 0.425 0.430 0.435 0.440 7.560 7.881 8.180 8.458 8.718 8.962 9.190 9.406 9.609 Sum of Squares 6.27373E-05 2.94863E-05 9.44185E-06 7.23476E-07 1.52378E-06 1.01134E-05 2.48457E-05 4.41624E-05 6.65987E-05 Table 3 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.00 0.00 0.00 0.00 0.00 0.00 0.00 7.56 9.98 11.33 Sum of Squares 2.390928889 1.541449133 0.967868636 0.551329195 0.255805214 0.071417089 0.001708024 6.27373E-05 0.000115476 0.000203277 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 12.11 12.49 12.59 12.45 12.11 11.60 10.95 10.18 9.34 Sum of Squares 6.81996E-05 0.001815766 0.011903924 0.044704628 0.128669378 0.316498439 0.700885222 1.437342999 2.784776274 Table 4 lists the sum of squares for and when ranges from .35 to .45. From this Chapter 15: Vasicek 3: More Term Structure, Appendix A Page 779 Chapter 15: Vasicek 3: More Term Structure, Appendix A Page 780 Adventures in Debentures Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Table 5 lists the sum of squares for and when ranges from .410 to .420. From this table, we see that the minimum sum of squares is zero and that the parameters which generate this minimum are = .417 and = 8.564%. Table 5 0.4100 0.4105 0.4110 0.4115 0.4120 0.4125 0.4130 0.4135 0.4140 0.4145 8.180 8.208 8.237 8.265 8.293 8.321 8.349 8.376 8.404 8.431 Sum of Squares 9.44E-06 8.09E-06 6.85E-06 5.72E-06 4.70E-06 3.78E-06 2.97E-06 2.26E-06 1.65E-06 1.14E-06 0.4150 0.4155 0.4160 0.4165 0.4170 0.4175 0.4180 0.4185 0.4190 8.458 8.485 8.511 8.538 8.564 8.590 8.616 8.642 8.667 Sum of Squares 7.23E-07 4.04E-07 1.79E-07 4.43E-08 0 4.38E-08 1.74E-07 3.89E-07 6.86E-07 Practice Questions with Solutions The degree of difficulty of each question is indicated. The easiest questions are marked by "($)," and the hardest questions are indicated by "($$$$$)." 1. ($) Today a 2 year bond is selling for $845.45. In one year you anticipate that this bond will be worth $904.84 (with a probability of 0.6) or worth $932.39 (with a probability of 0.4). Assume the local expectations hypothesis is true. What is the continuously compounded one year interest rate as of today? SOLUTION: Given these values for r, rL , , and , we can solve for in terms of V using Equation (2b) above. This yields the following equation: = 12.84% - 6.45% V . [(.6 904.84) + (.4 932.39)]e-r = 845.45 r 8.00% = Hence the solution set of parameters is: r = 9.00% rL = 12.00% = 0.417 = 8.564% = 12.84% - 6.45% V 2. ($$$) Part a. Given the following historical data on the one year interest rate ("RATE"), infer the value of , , and in the Vasicek binomial model of the term structure consistent with this information. (Assume the tree is being constructed using h = 1.) You should assume that the values of the Vasicek parameters were stable over the last 11 years. You are to estimate , , and based on this historical data. There are a number of ways to do this -- many of which are correct. You may find it useful to review the appendix titled "Estimating Vasicek Parameters from Historical Data" in the session titled "Vasicek 1: Properties of the Short-Term Rate." Chapter 15: Vasicek 3: More Term Structure, Appendix A Page 781 Chapter 15: Vasicek 3: More Term Structure, Practice Questions with Solutions Page 782 Adventures in Debentures Adventures in Debentures I suggest that you begin by finding the historical1 mean, the historical standard deviation, and the historical correlation between t+1 rt+2 and t rt+1 .2 When calculating , I had to use only 10 observations since we need to use a "lagged" interest rate. For consistency, I calculated the sample mean and sample variance based on the first 10 observations and ignored the last. Sample Mean = Sample Variance = 1 10 10 YEAR RATE 1 5.41% 2 4.03% 3 3.26% 4 4.63% 5 4.23% 6 4.88% 7 6.80% 8 6.79% 9 5.85% 10 5.17% 11 5.85% t r t+1 t=1 10 = 5.11% 1 ( r - 5.11%)2 = .00012 10 t=1 t t+1 Sample Standard Deviation = .00012 = 1.09% (Note: you could also compute the sample variance by dividing by 9, not 10, if you want to make an adjustment for degrees of freedom.) I decided to set equal to the historical sample mean and equal to the historical standard deviation. Thus, Part b. If the annualized interest rate (with continuous compounding) for 100 year zero coupon bonds is 6.00%, can the Vasicek model match this rate using , , and based on part a? If so, what is the implied value for V ? To make this question easier, assume h is very small. HINT: For part b, you need to review some of the equations we know will hold for the Vasicek model when h is small. = 5.11% = 1.09% . To compute , I calculated the slope coefficient in a regression of t+1 rt+2 on t rt+1 . The slope coefficient in a regression is equal to Cov(t+1 rt+2 , t rt+1 ) Var(t rt+1 ) . (If you don't recall this fact, you can just compute the slope coefficient directly in a regression.) Cov(t+1 rt+2 , t rt+1 ) = Var(t rt+1 ) 1 10 10 t=1 (t+1 r t+2 - 5.11%)(t r t+1 - 10 1 2 t=1 (t r t+1 - 5.11%) 10 5.11%) SOLUTION: = .58 Thus, I set equal to this historical sample slope coefficient -- that is, = .58. Part a. My approach to this problem was to estimate the 3 parameters based on this short history. Normally, you would work with many more observations than 11. The question only gave you 11 observations in order to minimize your computational effort. 1 Note: There are a number of ways to use this short history to compute sample statistics. Different ways will produce different numerical answers. Your solution should be similar to my approach, but your numerical answer may differ from mine. 2 If you want to match the numbers in my solution for the mean and the standard deviation, you should calculate these statistics based on the first ten observations and ignore the last observation for year 11. I ignored this last observation because we only have 10 observations to compute the historical correlation between t+1 rt+2 and t rt+1 . Instead of the historical correlation, you could find the slope coefficient in a simple regression model of t+1 rt+2 on t rt+1 for t = 1, 2, . . . , 10. The historical correlation and the slope coefficient in such a regression should be about the same value. Part b. We know from the Vasicek model when h is small that 2 2 V - . rL = + - ln - ln Chapter 15: Vasicek 3: More Term Structure, Practice Questions with Solutions Page 784 Chapter 15: Vasicek 3: More Term Structure, Practice Questions with Solutions Page 783 Adventures in Debentures Adventures in Debentures Copyright c 2004 by Michael R. Gibbons If rL = 6.00%, = 5.11%, = 1.09%, and = .58, then (.0109)2 .0109 2 V - .06 = .0511 + - ln(.58) - ln(.58) V = .4358 Questions The degree of difficulty of each question is indicated. The easiest questions are marked by "($)," and the hardest questions are indicated by "($$$$$)." Even if you decide not to submit your answer to this problem set, you should review my solution only after you attempt the questions to know if you could do them or not. Many times my solution is obvious AFTER you see it, but you need to know if the solution was obvious to you BEFORE you are told the solution. If you submit your answers to this problem set, please keep the following in mind: I sometimes fall behind where I expect to be in the lectures and an assignment is due before I get to some of the relevant material. If you believe the lectures have not yet covered the material necessary to answer a particular question and the required reading provides no guide, then you should indicate this on your solution and skip the question. Names of students along with their course and section numbers should be clear. The number of students in a study group should be less than or equal to five. Homework should be stapled. The final numerical answer should be "flagged" in some manner. Boxing, highlighting, and/or underlining the number are appropriate. Pages of spreadsheet printouts should be kept to a minimum. Only the essential information should be incorporated. You should not spend an excessive amount of time trying to solve any particular question. If you cannot complete a question, just describe what you tried to do. 3. ($) Assume = 0.06, = 0.40, and = .025. Further T is 1 week, and h is 1 week. The current annualized interest rate (with continuous compounding) for a 1 week zero coupon bond is 5%. Assuming local expectations hypothesis, what is the expected annualized return (with annual compounding) from buying a thirty year zero bond today and selling this bond in one week? SOLUTION: Convert the continuously compounded yields to annually compounded values: e.05 - 1 5.13% . = Thus, the expected return from holding a 30-year bond is 5.13%. Chapter 15: Vasicek 3: More Term Structure, Practice Questions with Solutions Page 785 Chapter 15: Vasicek 3: More Term Structure, Questions Page 786 Adventures in Debentures Adventures in Debentures 1. ($$$$) A group of senior bond managers have decided to investigate some of the so-called modern techniques of finance. The group has read many reports on the applications of the Vasicek binomial tree structure for pricing and hedging fixed income securities. They are now interviewing graduates of the Wharton School where the latest techniques in finance are taught. They want to hire someone who can help translate their forecasts about future interest rates into the Vasicek model. You are one of their candidates for a job. As a test of your ability, they ask you to determine the appropriate parameters for the Vasicek model. They provide you with a report which summarizes their forecasts about future interest rates. The bond managers focus on the continuously compounded yields for shortterm zero coupon bonds. They believe that world demand for capital will be unusually high over the next few years because of the need to rebuild Eastern Europe. They also have analyzed the world supply of capital and feel that the supply will not be unusually large. Thus, they are convinced that short-term interest rates are high and will remain high. In fact, they have indicated that their best forecast for the short-term yield in one year is 8.7%. They also estimate that the short-term yield in two years will be 8.43%. This group of bond managers know from experience that forecasting the future is difficult. They believe that this is a particularly difficult time to make forecasts, and they have attempted to quantify their uncertainty about their forecasts with standard deviation. The standard deviation of the short-term interest rate in one year is 2.17945% while the standard deviation of the short-term interest rate in two years is 2.93215%. After reading through the report, you are impressed by their thorough analysis. The group seems to be quite sophisticated in macroeconomics. You grab a copy of the Wall Street Journal and look up the current interest rates. You find the current short-term interest rate is 9%. You also note that the yield on long term zero coupon bonds is about 14.51941%. (Again, treat these yields as annualized continuously compounded rates.) You desperately need this job, so you want to impress these bond managers. Given the above information, what would you select as the values for , , , and V ? This question (as well as its solution) is to demonstrate that a one factor model like the Vasicek term structure is not necessarily too simplistic. Even with such a model, you can incorporate more complex forecasting methods. HINT: In working out this solution, you should assume the "short-term" rate refers to a very short-term bond (e.g., the overnight rate). This assumption allows you to use the equations which describe the expectation and variance of s r given information available prior to time period s. (These equations were given in the session titled "Vasicek 1: The Short-Term Rate.") Also you can use the equation which provides the yield on the long-term bond. (This equation was given in the lecture notes on page 756. It also is equation (15-3) in the appendix titled "The Vasicek Model When h Goes to Zero.") 2. ($$$$) Judge the following statements in the context of the Vasicek model for the term structure. For each part below, indicate if the statement is true, false, or uncertain. Justify your answer by providing a brief explanation. (A few sentences should be sufficient.) Your explanation -- not your selection of true, false, or uncertain -- is the important aspect of your solution. Part a. The local expectations hypothesis implies that the term structure of continuously compounded zero-coupon bond rates makes uniform shifts. Part b. The change (in absolute value) in long-term rates is always less than or equal to the change (in absolute value) in short-term rates. Part c. Under the Local Expectations Hypothesis, the term structure of continuously compounded zero-coupon bond rates is flat. Part d. increase. If the short-term rate is below rL , then interest rates are expected to Part e. If the yield curve is upward-sloping, then the expected returns on long-term bonds are higher than the expected returns on short-term bonds. Part f. Consider a financial institution which borrows money by issuing short-term bonds and invests the money raised by buying long-term bonds. If the term structure is upward sloping, the institution should not anticipate more profit relative to the Chapter 15: Vasicek 3: More Term Structure, Questions Page 787 Chapter 15: Vasicek 3: More Term Structure, Questions Page 788 Adventures in Debentures Adventures in Debentures case where the term structure is downward sloping. The first number at each node is the price of a zero coupon bond that matures on Year 3. This zero coupon bond has a face value of $1000.00. The second number at each node is the price of a coupon-bearing bond. This bond has an annual coupon yield of 13.074%, and it pays the coupon on a semi-annual basis. The face value of the bond is $1000.00. This coupon-bearing bond matures on Year 2. If you own the coupon bond at Year 0 and then you decide to sell the bond at Year 0.5, the buyer of the bond pays the price noted in the tree at Year 0.5. In addition, you would receive the semi-annual coupon from the issuer. (Since the issuer of the bond pays the coupon at each node to the seller of the bond, the full price and the flat price are identical.) The probability of going to the up scenario at Year 0.5 from Year 0.0 is 50%. As of today, what is the continuously compounded yield (annualized) for the zero coupon bond that matures on Year 0.5? (That is, find 0 r0,0.5 .) Part g. As the speed of mean reversion decreases, the more a change in the short-term rate will affect the long-term rate. 3. ($) Assume = 0.16, = 0.45, and = 0.03. Further, assume T = 1 year, and h = 1 year. The current annualized interest rate (with continuous compounding) for a one year zero coupon bond is 14%. Part a. Draw a Vasicek binomial tree for the annualized interest rate (with continuous compounding) for a 1 year zero coupon bond. Part b. Assuming local expectations hypothesis, what is the current price of a two year zero coupon bond with a face value of $1.00? 4. ($$) Today is Year 0. The following tree diagram describes the uncertainty about interest rates. Year 0.0 Year 0.5 + 0.5 d3.0 = 699.55 B + = 979.92 0.0 d3.0 = 676.03 B = 999.92 ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . .......... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . . = 750.54 B = 1026.01 - 0.5 d3.0 - Chapter 15: Vasicek 3: More Term Structure, Questions Page 789 Chapter 15: Vasicek 3: More Term Structure, Questions Page 790 Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Adventures in Debentures Chapter 16 Vasicek 4: The Greeks Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . 793 Announcements and Assignments . . . . . . . . . . . . . . 796 Lecture Notes . . . . . . . . . . . . . . . . . . . . . . . . . 797 Practice Questions with Solutions . . . . . . . . . . . . . . 824 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . 830 Chapter 16: Vasicek 4: The Greeks Page 791 Chapter 16: Vasicek 4: The Greeks Page 792 Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Adventures in Debentures 2.) Illustrate the applications of the delta (and omega) measure. 3.) Discuss theta and gamma measures in the context of the Vasicek binomial model. Preface to Vasicek 4: The Greeks 1.) Suggested Preparation Before Attending This Session. You should review the question and solution for #2a, #2b, and #2c of the chapter titled "Duration and Convexity." The current chapter will refer back to this earlier question. 4.) A Preview of Future Chapters. For the rest of the semester, we will be doing applications involving the Vasicek model. In prior chapters, we used the Vasicek model to value plain vanilla bonds. In today's chapter we study risk management of plain vanilla bonds using the model. While these applications are useful for pedagogical purposes, the model is more interesting in the context of more complicated financial instruments. In several chapters from now, we will start applications of the model involving bond options and bonds with embedded options. Here is where the Vasicek model will offer some important benefits, for traditional methods that we studied in the first part of the course have little, if anything, to say regarding options. If we want to understand these complicated (and very important) securities, it turns out to be essential to have a framework like the one we have developed using the Vasicek model. We also need to study the optimal time to refinance a loan. This is a hard question which is difficult to answer without the help of a framework like that of the Vasicek model. We will turn to this question even later in the course. 2.) Summary of Chapter. The goal of this chapter is to illustrate how we can use the Vasicek model to both measure and manage interest rate risk. This will require that we develop measures like theta, delta, and gamma. The methodology in today's chapter is more general than it may appear. First, the measures really do not require a Vasicek model. Our analysis can be applied to any model that generates a tree diagram like that on page 800. Second, exactly the same measures for theta, delta, and gamma are applicable to more complex fixed income securities -- not just plain vanilla bonds. Today we continue to focus on plain vanilla bonds; however, in future chapters we will broaden our applications to more complex financial instruments. 3.) Road Map for Chapter. The topics in this chapter will be organized as follows: 1.) Develop delta (and omega) measures in the context of the Vasicek binomial model. Chapter 16: Vasicek 4: The Greeks, Preface Page 793 5.) New Vocabulary Used in this Chapter. The following buzz words will be used in the lecture notes, the readings, and/or the problem sets: Horizon analysis, indexed (or inflation indexed) bonds, path dependent, Vasicek delta, Vasicek gamma, and Vasicek omega. Chapter 16: Vasicek 4: The Greeks, Preface Page 794 Adventures in Debentures Adventures in Debentures Copyright c 2004 by Michael R. Gibbons 6.) Summary of Important Equations. V+-V- , r+ - r- V = - Announcements and Assignments V = V , V V net equity = N1 V 1 + N2 V 2 + + Nn V n , V net equity = w1 V 1 + w2 V 2 + . . . + wn V n , = V +- - V , 2h net equity = N1 1 + N2 2 + + Nn n , V = - + - - V V , and r+ - r- V , net equity = N1 V 1 + N2 V 2 + + Nn V n . Chapter 16: Vasicek 4: The Greeks, Preface Page 795 Chapter 16, Announcements and Assignments Page 796 Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Adventures in Debentures ecture notes L for Vasicek 4: The Greeks A Road Map 1.) Develop delta (and omega) measures in the context of the Vasicek binomial model. 2.) Illustrate the applications of the delta (and omega) measure. 3.) Discuss theta and gamma measures in the context of the Vasicek binomial model. Review of Dollar Delta (or Duration) In the first part of the course, we established for a zero coupon bond paying K dollars in T years from now that: T PV(K) T V = , 100 100 where V is the current price for this zero coupon bond. $ = Further, we discussed1 how to approximate this delta by approximating the slope of the performance profile. (This profile was generated assuming uniform shifts in the term structure.) If Va is the price of the security when the short rate equals ra and Vb is the price of the same security when the short rate equals rb, then the approximate dollar delta is: $ - Va - Vb , ra - rb where the entire term structure changes by the uniform amount implied by (ra - rb). 1 For example, see the question on page 535. Page 798 Chapter 16: Vasicek 4: The Greeks, Lecture Notes Page 797 Chapter 16: Vasicek 4: The Greeks, Lecture Notes Adventures in Debentures Adventures in Debentures The Vasicek delta is conceptually similar to the approximation used for dollar delta -- especially if the dollar delta approximation was based on the time-adjusted performance profile. Of course, the numerical values of the dollar delta and Vasicek delta need not be equal because the implicit assumptions about the shift in the term structure are different. r V .. .. .. ... ... ... .. .. ... ... ... ... ... ... .. ... ... ... .. ... ... ... .. ... .. .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. . The Vasicek Delta r+ V+ r- V- .. ... ... ... .. .. ... ... ... ... ... ... .. ... ... ... . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. . . ... ... .. .. ... ... ... ... ... ... .. ... ... ... .. ... ... ... .. .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. . r++ V ++ r+- = r-+ = r V +- = V -+ r-- V -- Period 2h Period 0 Period h Note: r is the continuously compounded rate on the shortterm bond. V is the price of some fixed income security2 The superscripts on r and V are used to indicate the position in the tree. 2 V is the price of any fixed income security, not just plain vanilla bonds. Page 800 Chapter 16: Vasicek 4: The Greeks, Lecture Notes Page 799 Chapter 16: Vasicek 4: The Greeks, Lecture Notes Adventures in Debentures Adventures in Debentures Worksheet For a binomial model, an appropriate way to approximate minus the slope of the time-adjusted performance profile is:3 V = - V+-V- . r+ - r- (If V is to measure the dollar change for a 100 basis point movement in interest rates, be sure to quote r+ and r- in percentage units, not decimal units.) We will use the above idea to measure the Vasicek delta for all securities, not just plain vanilla bonds. 3 Once we know the dollar change (or the delta), it is easy to calculate the percentage change (or the omega) by merely dividing delta by the current price. Thus, V , V = V where V is the current price of the security. Delta measures the dollar change in one unit of a security when the short term rate moves by 100 basis points. Omega measures the percentage change in the value of a security when the short term rate moves by 100 basis points. (Equivalently, omega tells us the dollar change when one dollar is invested in a security.) Page 802 Chapter 16: Vasicek 4: The Greeks, Lecture Notes Page 801 Chapter 16: Vasicek 4: The Greeks, Lecture Notes Adventures in Debentures Adventures in Debentures Vasicek Delta for Portfolios With this definition of V , it is still the case that delta for a portfolio (or balance sheet) with n items is straightforward.4 That is,5 V net equity = N1V 1 + N2V 2 + + NnV n . Tracking the Milestones 1.) Develop delta (and omega) measures in the context of the Vasicek binomial model. 2.) Illustrate the applications of the delta (and omega) measure. 3.) Discuss theta and gamma measures in the context of the Vasicek binomial model. 4 To derive this equation in the text is straightforward. Consider a portfolio formed today with N1 units of asset 1, . . ., and Nn units of asset n. Next period the value of this portfolio in the "up" sce+ + nario is Vnet equity = N1 V1+ + + Nn Vn . Similarly, the value in the - - - "down" scenario is Vnet equity = N1 V1 + + Nn Vn . By definition, V net equity = - + - Vnet equity -Vnet equity r+ - r- + - . Substitute out Vnet equity and Vnet equity , 5 and rearrange to get the equation in the text. We can divide both sides of the equation in the text by the current value of the portfolio (or the value of the net equity of the balance sheet). After some simple algebra, we can establish a similar result for omega: V net equity = w1 V 1 + w2 V 2 + . . . + wn V n , where wi is the proportion of wealth invested in security i. Chapter 16: Vasicek 4: The Greeks, Lecture Notes Page 803 Chapter 16: Vasicek 4: The Greeks, Lecture Notes Page 804 Adventures in Debentures Adventures in Debentures Illustration of Vasicek Delta Let T = 2 years and h = 1 year. Assume = 6%, = 4%, and = .8. Also assume the local expectations hypothesis is true (that is, V = 0). We observe 0r = 7%. These assumptions are the same as in an earlier example from "Vasicek 3: More Term Structure." These assumptions generate the following tree diagrams for the evolution of the term structure and the discount function. Evolution of the Term Structure Recall the evolution of the term structure that we established in the prior session under these same assumptions: 0r1 = 7.00% 0 r 2 = 6.87% 0 r 3 = 6.74% . . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . . .46 r 1 2 1r3 r 1 2 r3 1 = 9.67% = 9.25% = 4.33% = 4.50% .54 . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . .. ... ... ... ... ... ... ... ... ... ... ... . . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . . .35...................... .. .. .. .. ... ... ... ... ... ... 2r3 = 12.34% .65 .57 2r3 = 7.00% .43 2r3 = 1.66% Obviously, the above tree demonstrates that the term structure scenarios generated by the Vasicek model do not make uniform shifts. Thus, dollar delta should not equal the Vasicek delta. Chapter 16: Vasicek 4: The Greeks, Lecture Notes Page 805 Chapter 16: Vasicek 4: The Greeks, Lecture Notes Page 806 Adventures in Debentures Adventures in Debentures Evolution of the Discount Function Questions for the Illustration Consider the following questions: 0 d1 = .9324 0 d2 = .8716 0 d3 = .8168 . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . . .46....................... .. .. .. .. ... ... ... ... ... ... d 1 2 1 d3 = .9078 = .8312 .54 1 d2 = .9576 1 d3 = .9140 . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . .. .. ... ... ... ... ... ... ... ... . . ... ... ... ... . . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . . .35...................... .. .. .. .. ... ... ... ... ... ... 2 d3 = .8839 1.) What are the Vasicek deltas (and omegas) for zero coupon bonds with face values of $1 and which mature in 1, 2, and 3 years? 2.) What is the Vasicek delta (and omega) for a coupon bearing bond that matures in 3 years and pays an annual coupon payment of $10 with a face value of $100? 3.) Finally, consider a balance sheet which consists of 300 units of the coupon bearing bond as a liability, net equity of $60,000, and all the assets are in the form of cash. Using the zero coupon bonds, how should the balance sheet be restructured to achieve a Vasicek delta of zero? (You may not sell off the liability; however, all the cash should be invested.) .65 .57 2 d3 = .9324 .43 2 d3 = .9836 Chapter 16: Vasicek 4: The Greeks, Lecture Notes Page 807 Chapter 16: Vasicek 4: The Greeks, Lecture Notes Page 808 Adventures in Debentures Adventures in Debentures Vasicek Delta for Zeroes: An Example What is the Vasicek delta for zero coupon bonds with face values of $1 and which mature in 1, 2, and 3 years?6 1-1 =0 . 9.67 - 4.33 V 2 = - =- e-.0967 - e-.0433 9.67 - 4.33 .9078 - .9576 = .009325 . 9.67 - 4.33 e-.09252 - e-.04502 9.67 - 4.33 .8312 - .9140 = .015511 . 5.34 V 1 = - V 3 = - =- 6 Similarly, the Vasicek omegas for zero coupon bonds with face values of $1 and which mature in 1, 2, and 3 years are: V 1 = V 1 0.000000 = = 0.00000 .9324 0 d1 V 2 0.009325 = = 0.01070 .8716 0 d2 V 3 0.015511 = = 0.01899 . d3 .8168 0 V 2 = V 3 = Chapter 16: Vasicek 4: The Greeks, Lecture Notes Page 809 Chapter 16: Vasicek 4: The Greeks, Lecture Notes Page 810 Adventures in Debentures Adventures in Debentures Vasicek Delta for Coupon Bonds: An Example What is the Vasicek delta for a coupon bearing bond that matures in 3 years and pays an annual coupon payment of $10 with a face value of $100? A synthetic version of the 3 year coupon bond consists of 10 10 110 1 year zero 2 year zero 3 year zero. Delta of 3 year coupon bond:7 = N1 V 1 + N2 V 2 + N3 V 3 = (10 0) + (10 .009325) + (110 .015511) = 1.79946 . Thus, the value of a 3 year coupon bond (and the value of its synthetic alternative) is: 10 e-.07 + 10 e-.06872 + 110 e-.06743 = 9.3239 + 8.7162 + 89.8625 = 107.9026 . 7 The omega for the coupon bond is straightforward: V ,coup = V ,coup 1.79946 = = 0.0166767 . Vcoup 107.9026 Even though the Vasicek delta for the coupon bond (1.79946) is higher than the Vasicek delta for one unit of the three year zero coupon bond (0.015511), the Vasicek omega provides a different perspective. The omega for the three year coupon bond is less than the omega for the three year zero coupon bond. Chapter 16: Vasicek 4: The Greeks, Lecture Notes Page 811 Chapter 16: Vasicek 4: The Greeks, Lecture Notes Page 812 Adventures in Debentures Adventures in Debentures Vasicek Delta and Risk Management: An Example Finally, consider a balance sheet which consists of 300 units of the coupon bearing bond as a liability, net equity of $60,000, and all the assets are in the form of cash. Using the zero coupon bonds, how should the balance sheet be restructured to achieve a Vasicek delta of zero? (You may not sell off the liability; however, all the cash should be invested.) Assets $92,370.78 "Cash" Liabilities $32,370.78 (= 300107.9) 300 Units of 3 Yr Coup Bond Net Equity To satisfy the above requirements, 60,000 = N1 e-.07 + N2 e-.06872 - 32,370.78 0 = N1 0 + N2(.009325) - 539.838 . That is, 92,370.78 = .9324 N1 + .8716 N2 539.838 = 0 N1 + .009325 N2 . So the second equation implies: N2 = 57,891.475 , and the first equation now requires N1 = 44,950.239 . $60,000 We will use the 1 year and 2 year zero to manage the interest rate risk. Our requirements are: 60,000 = VNE = N1 0d1 + N2 0d2 - 32,370.78 0 = V NE = N1 V 1 + N2 V 2 - (300 1.79946) . Chapter 16: Vasicek 4: The Greeks, Lecture Notes Page 813 Chapter 16: Vasicek 4: The Greeks, Lecture Notes Page 814 Evolution of the Restructured Balance Sheet Assets 44,950.239 1 Adventures in Debentures Liabilities 300 10 300 (10 e-.0967 + 110 e-.09252 ) Total Liabilities: 33,149.926 Net Equity: $64,355.833 57,891.475 e-.0967 Total Assets: 97,505.759 Chapter 16: Vasicek 4: The Greeks, Lecture Notes Assets 44,950.239 e 57,891.475 e-.06872 Total: 92,370.78 Assets 44,950.239 1 57,891.475 e-.0433 Total Assets: $100,388.51 Net Equity: $60,000 -.07 Liabilities 300 107.9026 = 32,370.78 Liabilities 300 10 300 (10 e-.0433 + 110 e-.04502 ) Total Liabilities: 36,032.601 Net Equity: 64,355.909 ... ... .. .. ... ... .. .. ... ... .. .. ... ... .. .. ... ... .. .. ... ... .. .. ... ... .. .. ... ... .. ... ... ... .. .. ... ... .. .. ... ... .. .. ... ... .. .. ... ... .. .. ... ... .. .. ... ... .. .. ... ... .. ... ... ... .. .. ... ... .. .. ... ... .. ... ... .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. . Page 815 8 Chapter 16: Vasicek 4: The Greeks, Lecture Notes Adventures in Debentures 2.) In particular, the Vasicek delta of a bond that matures in h periods is zero. Comments about the Vasicek Delta 3.) As approaches one, it is possible to demonstrate that the traditional dollar duration measure emerges.8. 1.) Unlike dollar duration, the Vasicek delta for a zero is not equal to the maturity of the zero times the price of the zero (and then divided by 100). To develop a Vasicek tree for this case, set qV = 1 and S equal to the 2 standard deviation of the one year change in the short-term rate. Then STEP = h S Page 816 Adventures in Debentures Adventures in Debentures Tracking the Milestones 1.) Develop delta (and omega) measures in the context of the Vasicek binomial model. 2.) Illustrate the applications of the delta (and omega) measure. 3.) Discuss theta and gamma measures in the context of the Vasicek binomial model. Theta and Vasicek Gamma Consider the following tree diagram to establish the relevant notation. Keep in mind that r refers to the continuously compounded short-term rate (quoted in percentage units). Furthermore, V is the price of the same security under different scenarios. r V .. ... ... ... .. ... ... ... .. .. ... ... ... ... ... ... .. ... ... ... .. ... ... ... .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. . r+ V+ r- V- ... ... ... ... ... ... .. .. ... ... ... ... ... ... .. ... ... ... .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. . . .. ... ... ... ... .. ... ... ... .. ... ... ... . .. ... ... .. .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. . r++ V ++ r+- = r-+ = r V +- = V -+ r-- V -- Period 2h Period 0 Period h Chapter 16: Vasicek 4: The Greeks, Lecture Notes Page 817 Chapter 16: Vasicek 4: The Greeks, Lecture Notes Page 818 Adventures in Debentures Adventures in Debentures For a binomial model, an appropriate way to measure the effect from the passage of time: = V +- - V 2h Calculating Theta and Vasicek Gamma: A Numerical Example The following tree describes the evolution of the 1 year rate and the price of a bond which has three years to mature as of year 0: For a binomial model, an appropriate way to approximate the curvature of the performance profile based on Vasicek shifts (or minus the slope of the Vasicek delta profile): + - - V V = - V , + - r- r where + = - V - = - V V ++ - V +- , r++ - r+- V -V , r-+ - r-- -+ -- 7.00% .8168 . . .. .. ... ... ... ... ... ... .. ... ... ... . .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. .. 9.67% .8312 4.33% .9140 . .. ... ... .. .. ... ... ... ... ... ... .. ... ... ... .. ... ... ... .. .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. . .. .. ... ... ... ... .. ... ... ... .. ... ... ... .. ... ... ... .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. .. 12.34% .8839 7.00% .9324 1.66% .9836 Year 2 V +- = V -+ , and r+- = r-+ . Year 0 Year 1 Chapter 16: Vasicek 4: The Greeks, Lecture Notes Page 819 Chapter 16: Vasicek 4: The Greeks, Lecture Notes Page 820 Adventures in Debentures Adventures in Debentures To calculate theta: .9324 - .8168 .1156 = = 0.0578 . 21 2 Comments on Theta and Vasicek Gamma 1.) The theta of a portfolio (or balance sheet) is the weighted sum of the thetas of the components of the portfolio. That is: We also need to calculate the relevant deltas as of year 1: .8839 - .9324 + = - = 0.009082 and V 12.34 - 7.00 - = - V .9324 - .9836 = 0.009588 . 7.00 - 1.66 net equity = N11 + N22 + + Nnn . 2.) The Vasicek gamma of a portfolio (or balance sheet) is the weighted sum of the gamma of the components of the portfolio. That is: V , net equity = N1V 1 +N2V 2 + +NnV n . 3.) In most situations, one can establish that the delta and convexity matching will lead to a solution which only needs to be rebalanced after 2h periods, not h periods. 4.) We will use the same idea to measure the theta and Vasicek gamma for all securities, not just plain vanilla bonds. Now we can calculate the gamma as of year 0: 0.009082 - 0.009588 V = - = 0.000095 . 9.67 - 4.33 Chapter 16: Vasicek 4: The Greeks, Lecture Notes Page 821 Chapter 16: Vasicek 4: The Greeks, Lecture Notes Page 822 Adventures in Debentures Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Worksheet Practice Questions with Solutions The degree of difficulty of each question is indicated. The easiest questions are marked by "($)," and the hardest questions are indicated by "($$$$$)." 1. ($$) Take the Vasicek model as true. Assume h is 2 years, = 12%, = 1%, and = 0.018315. Today the continuously compounded yield on a two year zero coupon bond is two percent on an annualized basis. Today is period 0. Part a. What are the possible values for 2 r2,4 ? Part b. As of today, what is the V for the two year zero coupon bond? Part c. As of today, what is the expected value of 2 r2,4 , and what is the standard deviation of 2 r2,4 ? Part d. Assume V is .25. As of today, what is the V for the four year zero coupon bond? SOLUTION: - 2 ln h = .01 - ln(.018315) 2 2 .04 = + 2 r 2,4 2% - 2 r 2,4 .. .. .. .. ... ... ... ... ... ... .. ... ... ... . . ... ... ... ... ... ... .. ... .. . ....... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... Part a. STEP = 0 r 0,2 .. .. .. .. ... ... ... ... ... ... .. ... ... ... . ... ... ... ... ... ... .. .. ... .. ....... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... 2% + 4% = 6% 2% - 4% = -2% 1-1 Part b. V = - 6.00-(-2.00) = 0 Chapter 16: Vasicek 4: The Greeks, Lecture Notes Page 823 Chapter 16: Vasicek 4: The Greeks, Practice Questions with Solutions Page 824 Adventures in Debentures Adventures in Debentures Part c. qV = 1 + 2 qV > 1, qV = 1. Thus, (-0 r 0,2 ) h - ln 8 = 1 2 + (.12-.02) 2 - ln(.018315) .01 8 = 10.5. Since SOLUTION: 2% .. ... ... ... .. ... ... ... .. .. ... ... ... ... ... ... . ... ... ... ... ... ... . . ....... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... 6% Part a. False. Term premiums have no impact on the h period bond. Since the V of a 2h bond is based on the prices of a h period bond next period, V is not affected for the 2h bond. -2% 3. ($) 1 0 E(2 r2,4 ) = 6% and SD(2 r2,4 ) = 0 All parts to this question are based on the tree diagram that follows. The tree describes the evolution of the discount function. The initial node is "today" (t = 0), and the time between the nodes is 3 months (or .25 years). The top number in each node represents the price of a pure discount bond that pays $1 in 3 months. The second number (when it is present) is the price of a zero coupon bond that pays $1 in 6 months. The third number (when it is present) is the price of a zero coupon bond that pays $1 in 9 months. The fourth number (when it is present) is the price of a zero coupon bond that pays $1 in 1 year. Consider a coupon-bearing bond that has a face value of $100. The quarterly coupons are $5 dollars per quarter. This coupon-bearing bond matures on t = 1. Part a. As of period 0, calculate the Vasicek delta for this instrument. Part d. V = - e -.062 -e+.022 6.00+2.00 = .0192363 Another acceptable answer is to recognize that the probability of "down" moves is zero. Thus V for a 4 year zero is 0. 2. ($$$) Determine whether the following statements are true or false. Provide a brief explanation for your conclusion. Part a. Consider a binomial tree based on the Vasicek model where the time between the nodes is h. As the term premium becomes larger (i.e., as V increases), V for the bond with maturity 2h becomes larger. Part b. As of period 0, calculate the Vasicek omega for this instrument. Part c. As of period 0, calculate the Vasicek gamma for this instrument. Part d. As of period 0, calculate the theta for this instrument. Discuss why this theta calculation may be misleading. Chapter 16: Vasicek 4: The Greeks, Practice Questions with Solutions Page 825 Chapter 16: Vasicek 4: The Greeks, Practice Questions with Solutions Page 826 Adventures in Debentures Adventures in Debentures SOLUTION: . .. . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . .. .. .. . .... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. . . .. .. . .. .. .. .. .. .. .. .. .. . .. .. . . .. . .. ... . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . .. .. .. .. . .. .. . . .. .. . .. .. .. .. .. .. .. .. .. . .. .. . .. .. . ... . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . .9663 Part a. . ..... .... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ... .... ..... ..... ..... ..... ..... ..... . . ..... .... ............... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... .... ..... .. .. ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... . . ..... ..... ..... ..... ..... ..... ..... ..... . . ..... .... ............... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... .9753 .9511 .9273 .9040 . . .. .. .. .. .. .. .. . .. .. . .. .. .. .. .. .. .. .. .. . .. .. . . .. . .. ... . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . .9723 .9454 .9193 .9783 .9567 .9353 . .. .. .. .. .. .. .. .. .. . .. .. . .. .. .. .. .. .. .. .. .. . .. .. . .. .. . ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. . . .. .. . . .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . .. . ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . .9693 .9398 103.53 12.47% .9723 109.19 10.00% .9753 .9511 . ..... .... .... .... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... .... ..... ..... ..... ..... ..... ..... ..... . ............... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... 106.12 11.24% 104.74 10.00% 107.88 8.78% 105.96 7.55% t = .50 .9783 t=0 t = .25 .9813 .9624 The top number at each node represents the price of the coupon-bearing bond. (This price excludes the coupon, if any, that might be paid on that same period; this price reflects just the value of the coupons and face to be received after the particular time period.) For example, at t = 0, .9844 B = 5(.9753 + .9511 + .9273 + .9040) + 100(.9040) 109.19 . = The second number at each node represents the annualized yield (with continuous compounding) of the 3-month bond. For example, at t = 0: t=0 t = .25 t = .50 t = .75 0 d.25 = e-.250 r.25 0 r 25 = -4 ln(0 d.25 ) = -4 ln(.9753) 10.00% . = To calculate the Vasicek delta: V = - 106.12 - 107.88 = 0.7154 . 11.24 - 8.78 You should note that the coupon payment on date t = .25 has been excluded. However, if you included the coupon, it would not change the answer since the coupon is paid in both the up and down states, and it would cancel out. Chapter 16: Vasicek 4: The Greeks, Practice Questions with Solutions Page 827 Chapter 16: Vasicek 4: The Greeks, Practice Questions with Solutions Page 828 Adventures in Debentures Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Part b. V = V /B = .7154/109.19 0.0066 = Part c. + = - V 103.53 - 104.74 12.47 - 10.00 0.4899 = - = - V 104.74 - 105.96 10.00 - 7.55 0.4980 = Questions The degree of difficulty of each question is indicated. The easiest questions are marked by "($)," and the hardest questions are indicated by "($$$$$)." Even if you decide not to submit your answer to this problem set, you should review my solution only after you attempt the questions to know if you could do them or not. Many times my solution is obvious AFTER you see it, but you need to know if the solution was obvious to you BEFORE you are told the solution. If you submit your answers to this problem set, please keep the following in mind: I sometimes fall behind where I expect to be in the lectures and an assignment is due before I get to some of the relevant material. If you believe the lectures have not yet covered the material necessary to answer a particular question and the required reading provides no guide, then you should indicate this on your solution and skip the question. Names of students along with their course and section numbers should be clear. The number of students in a study group should be less than or equal to five. Homework should be stapled. The final numerical answer should be "flagged" in some manner. Boxing, highlighting, and/or underlining the number are appropriate. Pages of spreadsheet printouts should be kept to a minimum. Only the essential information should be incorporated. You should not spend an excessive amount of time trying to solve any particular question. If you cannot complete a question, just describe what you tried to do. V = - + - - 0.4899 - 0.4980 V V =- = .0033 r+ - r- 11.24 - 8.78 As in the case of the delta calculation above, including or excluding the coupon payments has no effect on this numerical answer. Part d. = 104.74 - 109.19 = -8.90 . .5 The theta calculation is very misleading, for it excludes the coupons received on both dates t = .25 and t = .5. Theta is supposed to indicate the increase in one's wealth if interest rates fail to change; clearly, the receipt of coupons should be included in the wealth calculation. The theta calculation would not suffer this problem if the h value is small enough. For example, if we recompute this tree for h < .125, the prior approach would not be misleading for this instrument. Chapter 16: Vasicek 4: The Greeks, Practice Questions with Solutions Page 829 Chapter 16: Vasicek 4: The Greeks, Questions Page 830 Adventures in Debentures Adventures in Debentures 1. ($$) Throughout this question, you should assume that the Vasicek model of the term structure is appropriate. When using this model, set = .10, = .05, = .30, V = .40, T = .5 years, and h = .25. Also assume 0 rh = .15. Part a. Determine the 3 month annualized interest rate (using continuous compounding) for all nodes of the Vasicek tree. year. Based on the tree given below, the following information about zero coupon bonds that pay $1 on maturity could be calculated for the node in period 0. Part b. Determine the annualized interest rate (using continuous compounding) for 6 month and 9 month zero coupon bonds with face value of $1.00. These rates should be specified for time period 0. Also determine the potential 6 month rates for the period occurring 3 months after time period 0. In all cases, also specify the prices as well as the interest rates. . ... ... ... ... .. .. ... ... ... ... ... ... .. ... ... ... .. .. ... ... .. .. ... .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. .. ... ... ... .. ... ... ... .. ... ... ... .. ... ... ... . ... ... ... ... .. .. ... .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. . 12.3444% Part c. Demonstrate whether or not the above formulation of the term structure generates uniform shifts in the term structure. Part d. In time period 0, an investor has $1,000.00. He places 40% of this money in 9 month bonds, 35% in 6 month bonds, and 25% in 3 month bonds. Over the next three months, what is the expected return and the standard deviation of the return from this investment? 7% ... ... .. ... ... ... .. .. ... ... ... ... ... ... .. ... ... ... ... ... ... ... .. .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . . 9.6722% 7% 4.3278% 1.6556% Part e. As of period 0, what do you forecast the 6 month interest rate to be in three months? Compare this forecast with the forward rate, 0 r.25,.75 . . .. ... ... .. ... ... ... .. .. ... ... ... ... ... ... . .. ... ... ... ... ... ... .. .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . . .. ... .. ... ... ... .. ... ... ... .. ... ... ... .. ... ... ... ... ... ... ... .. .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . .. .. .. ... ... ... .. ... ... ... .. ... ... ... . . ... ... ... ... ... ... ... ... .. .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . . 15.0166% 9.6722% 4.3278% -1.0166% Part f. Consider a 6 month bond which pays $25 in 3 months and $125 in 6 months. What is the market value of this bond in period 0? What is the dollar duration of this bond in period 0? What is the Vasicek delta of this bond in period 0? What is the Vasicek delta of this bond in 3 months from period 0? .. .. ... ... ... ... .. ... ... ... .. ... ... ... . .. ... ... .. .. ... ... ... ... .. .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . . ... ... ... ... .. ... ... ... .. ... ... ... .. ... ... ... .. ... ... ... .. ... .. .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . .. ... ... ... .. ... ... ... .. ... ... ... . ... ... ... .. ... ... ... ... ... ... .. .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . . . ... ... .. ... ... ... .. ... ... ... .. ... ... ... .. ... ... ... .. .. ... ... .. .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . . 17.6888% 12.3444% 7% 1.6556% -3.6888% Year 4 Year 0 Year 1 Year 2 Year 3 2. ($$$) For all parts of this question, assume the following tree diagram describes the evolution of the yield on a one year zero coupon bond. (The yield is annualized and quoted on a continuously compounded basis.) For this problem we have set h equal to one Part a. Compute the Vasicek gamma of the three-year zero as of period 0. That is, fill in the following table for V 3 . Chapter 16: Vasicek 4: The Greeks, Questions Page 832 Chapter 16: Vasicek 4: The Greeks, Questions Page 831 Adventures in Debentures Adventures in Debentures Years till Maturity 2 3 4 5 Current Price 0.87160879 0.81680945 0.76713328 0.72181129 Delta 0.00932505 0.01549756 0.01947203 0.02190902 Gamma 0.00000000 V 3 0.00027122 0.00044913 The following tree diagram describes the evolution of the discount function over time. The prices below represent zero coupon bonds with face value of $1.0000. The top number is a one year zero price, the second number is a two year zero price, and the third number is a three year zero price. Today (i.e., year 0) you enter into a forward agreement to borrow $10,000 in year 2. You agree to repay $10,760.03 in year 3. (You may confirm that such a commitment has zero present value as of year 0. ) Part a. What is the implied forward rate on this forward agreement? Part b. In time period 0 a financial institution has assets consisting of $100,000 in cash. Its liabilities are 50,000 units of the three year zero coupon bond. (Each unit pays $1 on maturity.) For a variety of reasons, these 50,000 units of the three year zero cannot be repurchased by the institution. The institution has decided that its net equity should be riskless at the end of two years. That is, the institution wants to implement a financial strategy in year 0 which guarantees a certain level for the net equity in year 2. One further restriction has been imposed by the regulators. This institution may not buy or issue two year zero coupon bonds in year 0. Provide specific numerical values for the desired Vasicek delta and Vasicek gamma of the net equity which guarantees that the value of the net equity will be riskless in year 2. These numerical values should represent characteristics of the net equity as of year 0. Part c. Write out the relevant set of equations that determine the number of three-year, four-year, and five-year zeros that must be bought or sold in order to accomplish your goal in part b. The equations that you write out should specify numeric quantities where possible; the only unknown variables in the equations should be symbols for the number of three-year, four-year, and five-year zeros. Part d. If your strategy in parts b and c are implemented, what will be the value of the net equity in year 2. Part b. Under the scenario that one year zero coupon bonds are selling for $0.84294 in year 2, what is the market value (as of year 2 ) of your forward commitment? Part c. Under the scenario that the one year zero coupon bonds are selling for $0.87772 in year 1, what is the market value (as of year 1 ) of your forward commitment? .91393 .84449 .78484 .. .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . ... ... ... ... . ......... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .87772 u 1 d3 .73036 .95164 .89690 .84165 . .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . .. ... ... ......... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. . . ... ... ... ... ... ... ... ... . ... ... ... ... .. .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. ... ... ......... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. .84294 .73986 .66997 .91393 .84449 .78484 3. ($$$$) Today is year 0. Chapter 16: Vasicek 4: The Greeks, Questions Page 833 (Today) Year 0 Chapter 16: Vasicek 4: The Greeks, Questions .99090 .95216 .90311 Year 1 Year 2 Page 834 Adventures in Debentures Adventures in Debentures Part d. Under the scenario that the one year zero coupon bonds are selling for $0.87772 in year 1, what is the Vasicek delta (as of year 1 ) of your forward commitment? To answer this question, you should use the binomial approach to calculate the delta. in the up scenario on year 1, and the face value is $11,663 at the top scenario on year 2. The coupon payment for this bond is based on the previous period's face value. For example, in the top scenario on year 2, the bondholder would receive a coupon payment of 4% $10, 700 and principal equal to the bond's face value of $11,663. Part a. 4. ($$$$$) Today is year 0. The following tree diagram describes the evolution of the discount function as well as the inflation rate. All zero coupon bonds have a face value of one dollar. The first number is the price of a zero coupon bond with a maturity of 1 year. The second number is the price of a zero coupon bond with a maturity of 2 years. The third number (t-1 it ) at year t is the inflation rate from year t - 1 to year t. This inflation rate is annualized with annual compounding. Thus, if the economy's price index is 100 at t - 1 and 103 at t, then t-1 it is 3%. What is the price of the inflation-indexed bond today (i.e., year 0)? (Assume the issuer of the bond pays the coupon at each node to the seller of the bond. In this case, the full price and the flat price are identical.) Part b. Consider a second bond which is not indexed. Assume this bond matures on year 2. The bond has fixed annual coupon payments of $400 on each year and a face value of $10,000. Using the above tree diagram, the Vasicek omega for this bond is 1.11%. What is the Vasicek omega for this inflation-indexed bond? Compare the omega for the two bonds. Why is the omega negative for the inflationindexed bond? Why is the risk (in absolute value) higher for the bond which is protected against inflation? Part c. For the inflation-indexed bond to sell for par (i.e., $10,000) today, what coupon yield would be necessary? (While you should approximate your answer, the credit you receive will reflect the quality of the methodology that you use in constructing the approximate answer.) HINT: One way to approach part c is to first find the extra value associated with an increase of the coupon by one percent from 4% to 5%. The marginal impact of increasing coupons by 100 basis points can then be used to judge how the coupon should be adjusted to get the bond to sell for par. 2 d3 . .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. ... ... ... ... ... ... ... ... ... ... ... . .......... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. . . .. ... ... ... ... ... ... ... ... . . ... ... ... ... . . ... ... ... ... ... ... ... ... ... ... ... ... . .......... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. . 1 d2 . . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. .. ... ... ... ... ... ... .. ... ... .......... . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. . 0 d1 0 d2 = .8607 = .7432 -1 i0 = 5% = .8509 1 d3 = .7272 0 i1 = 7.0% = .8412 = .7116 1 i2 = 9% 2 d4 2 d3 2 d4 1 d2 = .8706 1 d3 = .7595 0 i1 = 3% = .8607 = .7432 1 i2 = 5% 2 d3 = .8807 2 d4 = .7762 1 i2 = 1% Year 2 Year 0 Year 1 Consider an inflation-indexed bond maturing on year 2. This bond pays an annual coupon at an annual rate of 4%. In addition, the face value of the bond is indexed to inflation. Today (i.e., year 0), the face value of this bond is $10,000. At each future date, the face value is increased from its previous value by the inflation rate experienced over the prior year. For example, the face value of the bond is $10,700 Chapter 16: Vasicek 4: The Greeks, Questions Page 835 Chapter 16: Vasicek 4: The Greeks, Questions Page 836 Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Adventures in Debentures Why It Pays To Be a Broker Randolph: Some of our clients are speculating that the Chapter 17 price of gold will rise in the future, and we have other clients who are speculating that the price of gold is going to fall. They place their orders with us, and we buy or sell their gold for them. Valuation by Monte Carlo Methods Mortimer: Tell him the good part. Randolph: The good part William is that no matter whether our clients make money or lose money Duke and Duke get the commissions. Mortimer: Well, what do you think Valentine? Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . 839 Announcements and Assignments . . . . . . . . . . . . . . 841 Lecture Notes . . . . . . . . . . . . . . . . . . . . . . . . . 842 A: Use of Pseudo Probabilities for Valuation in a Binomial World . . . . . . . . . . . . . . . . . . . . . . Randolph: I told you he would understand. Valentine: Well, it sounds to me like you guys are a couple of bookies. 860 -- from the movie "Trading Places.1 " 1 Eddie Murphy plays William Valentine. Chapter 17: Valuation by Monte Carlo Methods Page 837 Chapter 17: Valuation by Monte Carlo Methods Page 838 Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Adventures in Debentures 4.) New Vocabulary Used in this Chapter. Preface to Valuation by Monte Carlo Methods The following buzz words will be used in the lecture notes, the readings, and/or the problem sets: Equivalent martingale probabilities, Monte Carlo valuation, pseudo probabilities, and risk neutral probabilities. 1.) Summary of Chapter. This chapter provides an alternative method for calculating values in a binomial tree. Monte Carlo valuation takes advantage of the computational power of modern computers. With Monte Carlo valuation practitioners can write very simple algorithms (even in spreadsheets) to analyze even complicated securities. 2.) Road Map for Chapter. The topics in this chapter will be organized as follows: 1.) Explain valuation by Monte Carlo methods under local expectations hypothesis. 2.) Extend valuation method to incorporate risk premiums using pseudo probabilities. 3.) Required Reading. After attending this session, you should read: Appendix A, "Use of Pseudo Probabilities for Valuation in a Binomial World." Chapter 17: Valuation by Monte Carlo Methods, Preface Page 839 Chapter 17: Valuation by Monte Carlo Methods, Preface Page 840 Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Announcements and Assignments ecture notes L for Valuation by Monte Carlo Methods A Road Map 1.) Explain valuation by Monte Carlo methods under local expectations hypothesis. 2.) Extend valuation method to incorporate risk premiums using pseudo probabilities. Chapter 17, Announcements and Assignments Page 841 Chapter 17: Valuation by Monte Carlo Methods, Lecture Notes Page 842 Adventures in Debentures Adventures in Debentures A Fundamental Valuation Equation Until now we have developed valuation techniques which compare values across one period of time. For the Monte Carlo method to be useful, we need to compare values across many periods of time. Assuming risk premiums are not relevant, we need to establish the following result: 0V Worksheet = E0 e-0rhh e-hr2hh e-2hr3hh . . . e-T -hrT h T V , where E0 is the expectation based on information available at time period 0 and VT is the value of the security at time period T . Equivalently, 0V = E0 e-T r T V (i-1)h r ih . ^ , ^ where r 1 T /h T /h i=1 For the case of a zero coupon bond, we demonstrate the valuation equation is reasonable by an algebraic illustration and then a numerical example. Chapter 17: Valuation by Monte Carlo Methods, Lecture Notes Page 843 Chapter 17: Valuation by Monte Carlo Methods, Lecture Notes Page 844 Adventures in Debentures Adventures in Debentures Demonstrating the Valuation Equation: An Algebraic Illustration Continue to assume the local expectations hypothesis where risk premiums play no role in valuation. Assume that today is period 0 and that the time between the periods is 1 year. In period 0, the value of a 3 year zero coupon bond must be set so that the expected return is equal to 0r1 under local expectations hypothesis. That is, -0 r1 E0 {1d3} . 0 d3 = e Demonstrating the Valuation Equation: A Numeric Illustration As in the chapter titled "Vasicek 3: More Term Structure," we assume: 0r1 = 7%, T = 2 year, and h = 1 year. Further, we assume: = 6%, = 4%, and = .8 . Thus, 0 d1 But in period 1, the value of a 2 year zero coupon bond must be set so that the expected return is equal to 1r2 under local expectations hypothesis. That is, 1 d3 = .9324 0 d2 = .8716 0 d3 = .8168 . . ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . . .46 d 1 2 1 d3 d 1 2 1 d3 = .9078 = .8312 = .9576 = .9140 .54 =e -1 r2 E1 e -2 r3 . . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . ... ... ... ... ... ... ... ... ... ... ... ... . . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . . .35....................... .. .. .. ... ... ... ... ... ... ... 2 d3 = .8839 .65 .57 2 d3 = .9324 .43 . r 1 2 r3 1 r 1 2 1r3 = 9.67% = 9.25% = 4.33% = 4.50% .35....................... .. .. 2 d3 = .9836 = 12.34% .. ... ... ... ... ... ... ... 2r3 After algebraic substitution: 0 d3 1 -1 r2 -1 r2 -1 r2 =e -0 r1 E0 e -0 r1 -0 r1 E1 e E1 e e -2 r3 -2 r3 -2 r3 0r1 = E0 e = E0 e 1 e e = 7.00% r2 = 6.87% 0 0 r 3 = 6.74% .. .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... . . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. .. .46 .54 . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . . ... ... ... ... ... ... ... ... ... ... ... ... . . ... ... ... ... .. .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . . .65 .57 2r3 = 7.00% .43 The last equality follows from the "law of iterated expectations." Page 845 Chapter 17: Valuation by Monte Carlo Methods, Lecture Notes 2r3 = 1.66% Page 846 Chapter 17: Valuation by Monte Carlo Methods, Lecture Notes Adventures in Debentures Adventures in Debentures These assumptions imply the following tree structure between period 0 and period 1, which is useful in understanding the price of the 3 year zero coupon bond as of time period 0: As of year 0 the value of a 3 year zero should be (in order to have the expected return equal to 7%): 0 d3 = e-.07[.46e-.0967 .35 e-.1234 + .65 e-.07 + .54e-.0433 .57 e-.07 + .43 e-.0166 ] .46 0 d3 .54 .. ... ... ... ... ... .. ... ... ... .. ... ... ... . ... ... ... .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. . e-.0967 (.35 e-.1234 + + .65 e-.07) = e-.0967 E1 e-2r3 We can rearrange the last equation to state: 0 d3 e-.0433 (.57 e-.07 - + .43 e-.0166) = e-.0433 E1 e-2r3 = (.46)(.35) e-.0700 e-.0967 e-.1234 + (.46)(.65) e-.0700 e-.0967 e-.0700 + (.54)(.57) e-.0700 e-.0433 e-.0700 + (.54)(.43) e-.0700 e-.0433 e-.0166 Year 0 Year 1 Chapter 17: Valuation by Monte Carlo Methods, Lecture Notes Page 847 Chapter 17: Valuation by Monte Carlo Methods, Lecture Notes Page 848 Adventures in Debentures Adventures in Debentures =AVERAGE(D3 . . .) .46 IF(RAND() < B3, STEP, -STEP) .54 .57 1.66% Year 2 e-.07e-.0433e-.0166 step from 0 .43 Note 1: Each row represents a different replication of the simulation. 4.33% e-.07e-.0433e-.0700 IF(RAND() < B4, STEP, -STEP) 7% .65 e-.07e-.0967e-.0700 7% C . . . Year 0 Year 1 Thus, 0 d3 q formula using A3 q formula using A4 of Next Up Probability = E0 e-0r1 e-1r2 e-2r3 Chapter 17: Valuation by Monte Carlo Methods, Lecture Notes Page 849 Chapter 17: Valuation by Monte Carlo Methods, Lecture Notes Page 850 Note 3: The standard error of the average is (STDEVP / B 7.00% 7.00% 0 rh A 1 2 3 4 . . . . . . . . . nbr of replications). .. .. ... ... .. .. ... ... ... ... ... ... .. ... ... ... . .. ... ... .. .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. . 9.67% Note 2: The q formula in column B should be restricted to be between zero and one. .35 ... ... .. ... ... ... .. .. ... ... ... ... ... ... .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. . . .. .. .. ... ... ... ... ... ... .. ... ... ... . .. ... ... ... ... ... ... .. .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. .. 12.34% A Spreadsheet for Monte Carlo Valuation e-.07e-.0967e-.1234 =STDEVP(D3 . . .) =exp(-h*A3) =exp(-h*A4) 0 dh D . . . A Spreadsheet for Monte Carlo Valuation, continued E 1 2 3 4 . . . . . . . . . . . . =A4+C4 q formula using E4 IF(RAND() < F4, STEP, -STEP) =A3+C3 q formula using E3 IF(RAND() < F3, STEP, -STEP) h r 2h of Next Up step from 1h 0 d2h F Probability Adventures in Debentures G H =D3*exp(-h*E3) =D4*exp(-h*E4) . . . =AVERAGE(H3 . . .) =STDEVP(H3 . . .) Chapter 17: Valuation by Monte Carlo Methods, Lecture Notes Note 1: Each row represents a different replication of the simulation. Page 851 Note 2: The q formula in column F should be restricted to be between zero and one. Note 3: The standard error of the average is (STDEVP / nbr of replications). A Spreadsheet for Monte Carlo Valuation, continued I 1 2 3 4 . . . 2h r 3h =E3+G3 =E4+G4 . . . J Probability of Next Up q formula using I3 q formula using I4 . . . step from 2h IF(RAND() < J3, STEP, -STEP) IF(RAND() < J4, STEP, -STEP) . . . 0 d3h Adventures in Debentures K L =H3*exp(-h*I3) =H4*exp(-h*I4) . . . =AVERAGE(L3 . . .) =STDEVP(L3 . . .) Chapter 17: Valuation by Monte Carlo Methods, Lecture Notes Page 852 Note 1: Each row represents a different replication of the simulation. Note 2: The q formula in column J should be restricted to be between zero and one. Note 3: The standard error of the average is (STDEVP / nbr of replications). Adventures in Debentures Adventures in Debentures Some Comments on the Spreadsheet 1.) RAND() generates a random number from a uniform distribution: . . . . .... .... . . 2.) To compute the sampling error of the AVERAGE, use: STDEVP(. . .) number of replications 3.) The prior spreadsheet is only provided as a conceptual illustration. A real application should keep in mind that Excel has a limit on the number of rows and columns. 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.) The good news about the Monte Carlo method is that it handles path dependent securities and non-recombining trees. 5.) The bad news about the Monte Carlo method is that it does not work well for American style options. 6.) To use the spreadsheet to compute the value of a European call option which matures on period 2 where the underlying asset pays one dollar on period 3 is straightforward. If we assume the exercise price of this call option is 0.94, then create a new column M where rows 3 and 4 are: =H3*Max(0,exp(-h*I3) - .94) =H4*Max(0,exp(-h*I4) - .94) then average down all the rows in column M. .. ...... ..... .. . .. 0 ^ q 1 Thus, the probability that a uniform random number is less than q is q . When the uniform ^ ^ random number is less than q , the IF statement ^ is true and returns STEP; otherwise, it returns -STEP. Chapter 17: Valuation by Monte Carlo Methods, Lecture Notes Page 853 Chapter 17: Valuation by Monte Carlo Methods, Lecture Notes Page 854 Adventures in Debentures Adventures in Debentures Tracking the Milestones 1.) Explain valuation by Monte Carlo methods under local expectations hypothesis. 2.) Extend valuation method to incorporate risk premiums using pseudo probabilities. The Basic Idea of Pseudo Probabilities A more complete discussion of pseudo probabilities2 can be found in the appendix titled "Use of Pseudo Probabilities for Valuation in a Binomial World." Within a larger tree consider any binomial node where we know the current value of the h-period bond (i.e., dh), the current value of security b (i.e., Vb), the future values of security b (i.e., Vbu and Vbd), and the future values of security a (i.e., Vau and Vad). But we need to find the current value of security a (i.e., Va). dh .. .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . 1 Vb 1 .. .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . Vbu Va V d b .. .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . Vau V d a 2 Pseudo probabilities are also called "risk neutral probabilities" or "equivalent martingale probabilities." Page 856 Chapter 17: Valuation by Monte Carlo Methods, Lecture Notes Page 855 Chapter 17: Valuation by Monte Carlo Methods, Lecture Notes Adventures in Debentures Adventures in Debentures The following system of three equations in three unknowns (Nh, Nb, and Va) must be satisfied if perfect substitutes sell for the same price to prevent arbitrage: up : Nh1 + NbVbu = Vau down : Nh1 + NbVbd = Vad cost : Nhdh + NbVb = Va Solving the above system for Va: q ^ Va = dh[^Vau + (1 - q )Vad ] , where q= ^ (Vb/dh) - Vbd =q. Vbu - Vbd ^ To determine the relation between qv and qv , recall from the chapter titled "Vasicek 2: The Term Structure:" V u d d u d2h = 0dh qv e-hr + (1 - qv )e-hr - h e-hr - e-hr 0 2 The above expression can be rearranged: V V u d qv + h e-hr + 1 - qv - h e-hr 0 d2h = 0 dh 2 2 So, qv = qv + ^ and v h 2 v h. 2 Since the system of three equations does not contain probabilities, q is not a true probability. However, if arbitrage is ^ not available, then we can show that: 0q1. ^ As a result, q is called a pseudo probability or a risk-neutral ^ probability. Keep in mind that the above analysis works for any security a and/or security b. 1 - qv = 1 - q v - ^ Most importantly, we can use the Monte Carlo methodology for valuation as long as we replace the true probabilities (i.e., qv ) with the pseudo probabilities (i.e. qv ). ^ Chapter 17: Valuation by Monte Carlo Methods, Lecture Notes Page 857 Chapter 17: Valuation by Monte Carlo Methods, Lecture Notes Page 858 Adventures in Debentures Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Worksheet Appendix A: Use of Pseudo Probabilities for Valuation in a Binomial World* If you use the methodology outlined in this appendix in your answers to the exam questions, your exam solution should clearly state that you are relying on pseudo probabilities -- otherwise, you may not receive credit for your solution. 1.) The Basic Idea. In a binomial model of the term structure, there are two possible paths for the evolution of interest rates at each node -- one up and one down. The cash flow from holding for one time period any interest rate dependent security can be replicated with two other interest rate dependent securities. (For example, holding a three-year note for one time period can be replicated with one- and two-year notes, or holding a call option on a ten-year bond for one time period can be replicated with twoand three-year notes.) To achieve this replication relies on a methodology that solves a system of simultaneous equations. As the number of periods increase, the more tedious the calculations can become. Presented here is an alternative framework for simplifying the algebra and the calculations. For any security "a," Va is the value of security a, Vau is the value of security a one period into the future in the up state; Vad is the value of the security one period into the future in the down state. Throughout this document, one time period is of length "h" time units. The following tree illustrates the binomial structure: ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . . ... ... . ......... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. . Vau Va V d a * The initial version of this appendix was prepared by Seth C. Fischoff in March, 1993, under the guidance of Michael Gibbons, Professor of Finance, the Wharton School of the University of Pennsylvania. This version represents a substantial revision of that initial version. Chapter 17: Valuation by Monte Carlo Methods, Lecture Notes Page 859 Chapter 17: Valuation by Monte Carlo Methods, Appendix A Page 860 Adventures in Debentures Adventures in Debentures Section 4 demonstrates that there is a number q which solves the following:1 ^ Va = dh [^Vau + (1 - q )Vad ] , q ^ where the current price of the h-period discount bond is represented by dh . To find Va by solving a system of simultaneous equations, we need to know the possible values of any two securities, b and c. u u . . . . ... ... ... Vb ... Vc ... ... ... ... ... ... Vb . . ... ... ... ... ... ... ... ... ... ... .. ... ... ......... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. . 2.) A Numerical Illustration. Following is a Vasicek tree of one-year rates, one-year discount prices, and two-year discount prices. With the use of those bonds and their prices, q was calculated. q ^ ^ was calculated for each of the nodes where there was a price for a two-year zero. Numerical example: = 10% = 12% = 5% = 70% h=1 V = 0 STEP = 0.04223002 0 rh Vc Vbd . . ... ... ... ... ... ... ... ... ... ... .. ... ... ......... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. . Vcd It will simplify the analysis to use the h-period zero coupon bond as security "c." The h-period zero coupon bond is guaranteed to pay one dollar when it matures h periods from now. That is, . . ... ... 1 ... ... ... ... . dh ........................ .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . . 1 Vasicek Tree of 1-year rates: ... ... ... ... ... ... ... ... ... ... ... ... ... ... . ... ... ... ... . . ......... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . . . . ... ... ... ... ... ... ... . ... ... ... ... . ... ... ... ... .. ... ... ......... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. . With this formulation, Section 4 below demonstrates that: (Vb /dh ) - Vbd q= ^ . Vbu - Vbd 10.0000% In summary, we now have a simple way to find the value of a third security (Va ) given the price of a h-period bond (dh ); the current and future values of a second security (Vb , Vbu , and Vbd ); and the future values of this third security (Vau and Vad ): Va = dh [^Vau + (1 - q )Vad ] , q ^ where q is defined above. ^ Furthermore, if we wish to value a fourth (or more) security, the last two equations would still be applicable. All we would need to do is alter the numerical values for ^ Vau and Vad . The numerical values for dh and q would not change. This achieves a significant computational efficiency!2 1 . .. . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . .. . ......... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. . 18.4460% 14.2230% 10.0000% 5.7770% 1.5540% 2 0 1 As explained above, in the calculation of q for any node, we must know the present ^ and two futures values of two different securities. Based upon the above information, we can calculate the prices of one- and two-period zero coupon bonds. securities are being valued. As you cross to different nodes, dh and q will have different numerical ^ values. 2 It is important to note that q is likely to have different numerical values at each node of the tree. ^ Throughout this discussion we will focus on only one node in the tree, so the changing nature of q may not be obvious to the reader. ^ As already noted, this computational efficiency is only achieved at a single node where many Chapter 17: Valuation by Monte Carlo Methods, Appendix A Page 861 Chapter 17: Valuation by Monte Carlo Methods, Appendix A Page 862 Adventures in Debentures Adventures in Debentures Prices of 1-year zero coupon bonds: .. .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . ... .. ......... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . . .. ... ... ... ... ... ... ... ... ... ... . ... ... ... ... . .. .. ... ... ......... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. . 0.83155317 0.904837418 . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . ... ... . ......... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. . 0.86742171 With the previous calculation of q , we can now begin to value securities. To value a ^ security that pays $100 if the interest rates move up and $20 if interest rates move downward, we would add $100^ and $20(1 - q ) and multiply by the one-year zero q ^ coupon price. This is precisely what is done in the following table. The following is an example of how to apply the pseudo probabilities to the valuation of a security worth 100 in up state and 20 in down state. . . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . ......... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. . 0.90483742 0.94386703 100 0.98458017 ? 2 0 1 20 Prices of 2-year zero coupon bonds: . ... ... ... ... ... ... ... ... ... ... ... ... ... ... . ... ... ... ... . ......... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. . 0.75905908 States 0.87189962 1 up down Value Pseudo prob 0.58446004 0.41553996 Value 100 20 0.81361876 Weighted Value 58.4460043 8.3107991 Discounted Value 52.88413 7.51992 60.40405 0 To calculate q and (1 - q ), we use the values in the above trees, and plug them into ^ ^ the equations that follow. The value of the one-period discount bond is taken from the tree for one-year zero coupon bond prices. For the second security ("b" in the formula), we will use the two-year zero coupon bond. q is the pseudo probability of ^ an up movement, while (1 - q ) is the pseudo probability of a down movement. ^ q= ^ (Vb /dh )-Vbd Vbu -Vbd In the above table, we have multiplied the potential cash flows by their respective pseudo probabilities to get the values in the "Weighted Value" column. Those values are then multiplied by the price of the one-period zero coupon bond to get the value in the "Discounted Value" column. Summing these two values gives the value of this security, $60.40. If we desired to value a fourth (or more) security at time period 0, we could rely on this same framework. All we would do is change the third column in the above table (labeled "Value") and recalculate the fourth column (labeled "Weighted Value.") The discount value (d1 ) used to created the fifth column (labeld "Discounted Value") would not change, but the numbers in the fifth column would change to reflect the new numbers in the fourth column. Let us verify the above calculations by solving for the value of our security by our traditional replication method. We begin by setting up our two simultaneous equations. 1-q = ^ Vbu -(Vb /dh ) Vbu -Vbd Calculation of q ^ ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . . ... ... . ......... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. . Calculation of 1 - q ^ ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . . ... ... . ......... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. . 0.40612257 0.41553996 0.76279752 1 0 0.59387743 0.58446004 0.23720248 1 Page 863 0 Chapter 17: Valuation by Monte Carlo Methods, Appendix A Chapter 17: Valuation by Monte Carlo Methods, Appendix A Page 864 Adventures in Debentures Adventures in Debentures Remember that a two-year bond today becomes a one-year bond one year from now. N1 + N2 du = 100 h N1 + N2 dd = 20 h We then solve these two equations for N1 and N2 . If we then multiply the quantity of each security (N1 and N2 ) by their respective prices, we get the values in the "Weighted Value" column in the following table. We then add the two dollar amounts together to get the value of our security. Bonds 1-year zero 2-year zero Value Number 1,007.75645 -1,046.4996 Prices 0.90483742 0.81361876 Weighted Value 911.85575 -851.45169 60.40405 are mathematical artifacts of the solution to those simultaneous equations. We are simply using algebra to rewrite the solutions to the system of simulataneous equations. If you believe there is some connection between the true probabilities of up versus down states compared with the pseudo probabilites, please review this note. You will notice that formula for pseudo probabilities was derived without including the true probabilities3 -- just like the replicating methodology does not need the true probabilities to find the value of a third security given the information about two other securities. It is important to stress once again that we are not discounting risky cash flows at the riskless rate. We are using some algebra to transform the problem in such a way that it appears to be a simple discounting approach. While this appearance provides some real computational saving, it also creates a dangerous perception. As we see, the valuations obtained under the two methods give exactly the same result, $60.40. 4.) Appendix: Some Mathematical Details. Here we provide the algebra to substantiate the claims made in the prior sections. This section is only included for completeness. It is written for those who are pathologically curious. Throughout this section, we will use the following notation: Va Vau Vad Nh dh Nb Vb Vbu Vbd q ^ 3 3.) A Warning. For many years in teaching this course, I avoided talking about the use of pseudo probabilities. I feared that students would walk away with the impression that the valuation methodology was simply discounting expected, but risky, cash flows, at a riskless discount rate. Although the equations in Section 1 appear to be discounting the expected value of risky cash flows at the riskless rate, this is not the case. First, we are not taking the expected value of the future cash flows. q is not ^ the true probability of state occurring; it is a "pseudo probability." Pseudo probabilities are numbers between 0 and 1 that sum to 1. Because it shares these characteristics with probabilities, these numbers are given the name "pseudo probabilities." Second, by using the pseudo probabilities, we are implicitly solving the simultaneous equations that we use when replicating securities. The pseudo probabilities Chapter 17: Valuation by Monte Carlo Methods, Appendix A Page 865 - - - - - - - - - - Value of Security a today (time 0). Value of Security a, h periods in the future at the up node. Value of Security a, h periods in the future at the down node. Number of h-period bonds to buy/sell to replicate Security a. Price of a h-period zero coupon bond today (time 0). Number of Security b to buy/sell to replicate Security a. Value of Security b today (time 0). Value of Security b, h periods in the future at the up node. Value of Security b, h periods in the future at the down node. pseudo probability. The only exception is in Section 2, where the two period bond was priced from the one period bond. But the pricing of this two period bond is not done by a replicating strategy. Chapter 17: Valuation by Monte Carlo Methods, Appendix A Page 866 Adventures in Debentures Adventures in Debentures h-Period Zero . .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. .. ... ... . ......... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. . Security A . .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. .. ... ... . ......... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. . Security B . .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. .. ... ... . ......... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. . With further rearrangement, we can write the last equation as: Va = dh Now define: (Vb /dh ) - Vbd V u - (Vb /dh ) Vau + dh b u Vad . Vbu - Vbd Vb - Vbd Vb /dh - Vbd , Vbu - Vbd Vbu - Vb /dh . Vbu - Vbd (7) 1 Va 1 Vau Vb Vad Vbu dh Vbd which implies: q= ^ Notice that the value of the h-period bond is guaranteed to be one dollar after one time period (when bond has matured). Claim #1: Given the following equations are true: Nh + Nb Vbu = Vau Nh + Nb Vbd = Vad Nh dh + Nb Vb = Va then we wish to demonstrate that: Va = dh [^Vau + (1 - q )Vad ] . q ^ (1) (2) (3) (1 - q ) = ^ Using the above definitions for q , we can rewrite equation (7) as: ^ q ^ Va = dh [^Vau + (1 - q )Vad ] . Two comments about the first demonstration. 1.) Equation (3) is the formula giving the current cost of a portfolio which provides a synthetic alternative to Security a. This replicating portfolio contains Nh units of the h-period bond and Nb units security b. The current cost of the synthetic depends on the cost of these two components (dh and Vb ). Equation (3) must hold to prevent arbitrage, for the cost of the synthetic must equal the cost of the explicit security (Va ). (4) ^ ^ 2.) Since the variables Vau or Vad do not appear in the definitions of q , q does not depend on the characteristics of the security we are trying to value at time period 0. Thus, we have established a way to price all securities at time period 0 -- we multiply their cash flows in h periods (both up and down states) by q and ^ (1 - q ), respectively, and then multiply this sum by the h-period zero price. ^ While we have determined the value of q , we have not yet demonstrated that q is a ^ ^ pseudo probability. It should be obvious that q +(1-^) is equal to one by construction. ^ q If we demonstrate that q and (1 - q ) are both non-negative, then q and (1 - q ) must ^ ^ ^ ^ be between zero and one since they sum to one. To establish that q and (1 - q ) are both non-negative, we will prove this by contra^ ^ diction. If we assume that q or (1 - q ) is negative and demonstrate that arbitrage is ^ ^ possible, then our assumption must be false -- proving our point. Chapter 17: Valuation by Monte Carlo Methods, Appendix A Page 868 Demonstration of Claim #1: Subtracting (2) from (1): Nb = and Nh = Vau - (Vbu ) Substituting (4) and (5) into (3): Va = dh which rearranges to Va = Vb - dh Vbd u dh Vbu - Vb d V + u V . Vbu - Vbd a Vb - Vbd a (6) Vau - Vu (Vbu ) au Vb - Vad V u - Vad , + Vb au - Vbd Vb - Vbd Vau - Vad Vbu - Vbd Vau - Vad . Vbu - Vbd (5) Chapter 17: Valuation by Monte Carlo Methods, Appendix A Page 867 Adventures in Debentures Adventures in Debentures Claim #2: (1 - q ) is non-negative, where ^ V u - (Vb /dh ) . (1 - q ) = b u ^ Vb - Vbd Demonstration of Claim #2: Contrary to the claim, assume (1 - q ) < 0. This will lead to an ^ arbitrage profit. For (1 - q ) to be negative, either the numerator or the denominator ^ (but not both) must be negative in equation (8). We will examine the case where the denominator is negative;4 that is, Vbu - Vbd < 0 and Vbu - Vb /dh > 0 . We can rearrange the last equation, dh Vbu - Vb > 0 . With these assumptions, we now demonstrate arbitrage is possible. If we now set up portfolio by going long $Vb of Security b and shorting $(dh Vbu ) of a h-period zero, one can reverse positions and arbitrage profits can be made after h periods. (8) Thus, we created a strategy that guarantees a positive cash flow at time zero and a non-negative future cash flow. This is an arbitrage opportunity; therefore, our contra-assumption must be false. We conclude that (1 - q ) 0. ^ Claim #3: q is non-negative, where ^ (Vb /dh ) - Vbd . Vbu - Vbd q= ^ (9) Demonstration of Claim #3: Contrary to the claim, assume q < 0. This will lead to an arbitrage ^ profit. For q to be negative, either the numerator or the denominator (but ^ not both) must be negative in equation (9). We will examine the case where the denominator is negative;5 that is, Vbu - Vbd < 0 and Time h Time 0 Sell 1-period Zero +$dh Vbu Buy security B -$Vb Totals $(dh Vbu - Vb ) > 0 4 (Vb /dh ) - Vbd > 0 . We can rearrange the last equation, Vb - dh Vbd > 0 . With these assumptions, we now demonstrate arbitrage is possible. Up State Down State -$Vbu +$Vbu $0 -$Vbu +$Vbd $(Vbd - Vbu ) > 0 5 If we now set up portfolio by shorting $Vb of Security b and going long $(dh Vbd ) of a h-period zero, one can reverse positions and arbitrage profits can be made after h periods. The reader can extend the proof for the alternative scenario where the numerator is negative. The reader can extend the proof for the alternative scenario where the numerator is negative. Chapter 17: Valuation by Monte Carlo Methods, Appendix A Page 869 Chapter 17: Valuation by Monte Carlo Methods, Appendix A Page 870 Adventures in Debentures Adventures in Debentures Copyright c 2004 by Michael R. Gibbons Time h Time 0 Buy 1-period Zero -$dh Vbd Sell security B +$Vb Totals $(Vb - dh Vbd ) > 0 Up State Down State Part Four +$Vbd -$Vbu $(Vbd - Vbu ) > 0 +$Vbd -$Vbd $0 Options on Bonds Thus, we created a strategy that guarantees a positive cash flow at time zero and a non-negative future cash flow. This is an arbitrage opportunity; therefore, our contra-assumption must be false. We conclude that q 0. ^ 18: Introduction to Bond Options . . . . . . . . . . . 874 19: European Bond Options . . . . . . . . . . . . . . 930 20: American Bond Options . . . . . . . . . . . . . . 971 21: Deja Vu . . . . . . . . . . . . . . . . . . . . . 1024 Chapter 17: Valuation by Monte Carlo Methods, Appendix A Page 871 Part Four: Options on Bonds Page 872 ...
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