Lecture 1 - Advanced Fixed Income Analytics Backus& Zin...

Info icon This preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Advanced Fixed Income Analytics Backus & Zin April 1, 1999 Vasicek: The Fixed Income Benchmark 1. Prospectus 2. Models and their uses 3. Spot rates and their properties 4. Fundamental theorem of arbitrage-free pricing 5. The Vasicek model: Solution and properties Parameter values  calibration" Hedging: Vasicek v. Duration Where are the bodies buried? 6. Summary and nal thoughts Lecture 1 Advanced Fixed Income Analytics 1-2 1. Prospectus Regard this course as an experiment: no one has ever taught such advanced material at this level What makes this possible? unique approach minimizes technical demands custom software does much of the work steady pace: we refuse to get bogged down in details great teaching! Our approach features discrete time much simpler than stochastic calculus continuous states more accurate than trees" emphasis on numbers a good model with bad parameters is a bad model hands-on experience work in progress, you be the judge Software options: Excel: the old reliable, but awkward for complex problems Matlab: more e ort initially, but much more powerful Our guarantees: you won't understand everything, but you'll learn a lot an honest attempt gets at least a B Advanced Fixed Income Analytics 1-3 2. Uses of Models Valuation: What's this option worth? Given prices of underlying assets, estimate the value of related derivatives. Hedging: How do I o set the risk in these DM swaptions? Find combinations of assets that have relatively little risk. Replication: How can I synthetically reproduce the cash ows of an American swaption? Construct combinations of simple assets to approximate the returns of a more complex asset. Risk management: How bad can things get? Estimate the statistical distribution of the one-day return of a portfolio of assets. Summary and assessment: Di erent objectives demand di erent models. If we thought we knew everything, the perfect model could be used for all four purposes. Instead, models are designed with strengths that suit their uses. They invariably exhibit weaknesses in other uses. Thus complex statistical models are helpful in quantifying risk, but generally ignore the pricing of risk. Our interest lies, instead, in simpler models in which the pricing of risk is explicit. Advanced Fixed Income Analytics 3. Spot Rates and Other De nitions The discount factor bn is the price of an n period zero: the t value at t of one dollar payable n periods in the future. The spot rate ytn is related to the discount factor bn by t n bn = e,nyt t n yt = ,n, log bn t This is based on continuous compounding, an arbitrary convention that helps us down the road. 1 The short rate is rt = yt . 1 The time interval is h years. For periods of one month, h = 1=12 years. To express spot rates as annual percentage rates: Annual Percentage Rate = ytn  100=h For monthly data, we multiply by 1200. 1-4 Advanced Fixed Income Analytics 1-5 4. Statistics Review Data: we have observations yt for t = 1; : : :; T . The sample mean is PT y =  The sample variance is y t=1 t T yt , y   T s is the standard deviation. Some people divide by T , 1, an minor issue we'll ignore. s = 2 PT 2 t=1 The sample autocorrelation an indicator of dynamics PT   t yt , y yt, , y  Corr yt ; yt,  = PT  t yt , y  Values close to one indicate that changes tend to last | they're persistent. 1 =2 1 =1 2 Higher moments for future reference only: PT  j for j  2 t yt , y  mj m T skewness = m  = m kurtosis = m  , 3 =1 3 2 3 2 4 2 2 Skewness measures asymmetry. Kurtosis summarizes the probability of extreme events. Both are zero for normal random variables. Advanced Fixed Income Analytics 1-6 5. Properties of Spot Rates US Treasury spot rates, 1970-95, monthly data Maturity Mean St Dev Skewness Kurtosis Autocorr 1 month 3 months 6 months 9 months 12 months 24 months 36 months 48 months 60 months 84 months 120 months 6.683 7.039 7.297 7.441 7.544 7.819 8.008 8.148 8.253 8.398 8.529 2.699 2.776 2.769 2.721 2.667 2.491 2.370 2.284 2.221 2.138 2.069 1.073 1.007 0.957 0.924 0.903 0.887 0.913 0.944 0.971 1.008 1.035 1.256 0.974 0.821 0.733 0.669 0.484 0.378 0.321 0.288 0.251 0.217 Note Means: increase with maturity picture coming Standard deviations Autocorrelations: close to one very persistent! 0.959 0.971 0.971 0.970 0.970 0.973 0.976 0.977 0.978 0.979 0.981 Advanced Fixed Income Analytics 1-7 5. Properties of Spot Rates continued Spreads Over Short Rate: ytn , yt 1 Maturity Mean St Dev Skewness Kurtosis Autocorr 3 months 6 months 9 months 12 months 24 months 36 months 48 months 60 months 84 months 120 months 0.357 0.614 0.758 0.861 1.136 1.325 1.465 1.570 1.715 1.846 Skip for now 0.350 0.507 0.601 0.683 0.929 1.095 1.213 1.300 1.420 1.526 1.811 1.325 1.117 0.863 0.017 0.424 0.599 0.677 0.728 0.735 5.642 4.538 4.647 4.177 2.197 1.368 0.983 0.765 0.532 0.385 0.403 0.527 0.618 0.680 0.792 0.833 0.855 0.868 0.882 0.892 Advanced Fixed Income Analytics 1-8 5. Properties of Spot Rates continued One-Month Changes: ytn , ytn, 1 Maturity Mean St Dev Skewness Kurtosis Autocorr 1 month 3 months 6 months 9 months 12 months 24 months 36 months 48 months 60 months 84 months 120 months 0.009 0.010 0.010 0.009 0.009 0.009 0.009 0.009 0.008 0.008 0.007 Skip for now 0.764 0.662 0.656 0.648 0.634 0.558 0.499 0.461 0.434 0.401 0.374 1.043 1.747 1.352 1.088 0.926 0.583 0.424 0.345 0.300 0.256 0.220 9.939 11.158 10.691 10.514 10.251 7.902 5.829 4.598 3.845 3.000 2.337 0.003 0.146 0.150 0.153 0.154 0.157 0.157 0.152 0.143 0.120 0.094 Advanced Fixed Income Analytics 1-9 6. The Fundamental Theorem An arbitrage opportunity is a combination of long and short positions that costs nothing but has a positive return a free lunch, in other words. We say prices are arbitrage-free if there are no arbitrage opportunities. Statement of faith in markets, good approximation but not perfect. Notation: let Rt be the gross return between dates t and t + 1. For example, the return on an n-period zero is Rt = bn, =bn. t t +1 +1 1 +1 Theorem. In any arbitrage-free setting, there is a positive random variable m a pricing kernel satisfying 1 = Et mt Rjt for all assets j . Et means the expectation as of time t when the trade is done. Ignore the t's for now, they're a distraction. +1 +1 Risk premiums are governed by covariance with m: Let Rt = 1=bt be the riskfree one-period return Excess returns are xjt = Rjt , Rt . Risk premiums are j j Cov t mt ; xt  Et xt = , E m t t Comment: this is a lot like the CAPM, with m playing the role of the market return. 1 +1 1 +1 +1 1 +1 +1 +1 Needed: a good m +1 +1 Advanced Fixed Income Analytics 1-10 7. Vasicek: The Model Goal: a useful description of the behavior of m The archetype: the Vasicek model , log mt =  =2 + zt + "t zt = 1 , ' + 'zt + "t where " is normal with mean zero and variance one and 0 ' 1. The two-part structure is typical. +1 2 +1 +1 +1 The pricing kernel m controls risk: log guarantees positive m if m is positive log m can be anything, and vice versa If  = 0, m discounts at rate z : mt = e,zt +1  controls risk recall: risk premiums are proportional to covariance with m  =2 is largely irrelevant you'll see 2 Advanced Fixed Income Analytics 1-11 7. Vasicek: The Model continued The state variable" z will turn out to be the short rate. Its properties include: Over one period, the conditional mean and variance are: Etzt = 1 , ' + 'zt Var t zt = Stop if this isn't clear. Over two periods, z looks like this: zt = 1 , ' + 'zt + "t = 1 , '  + ' zt + "t + '"t  The conditional mean and variance are: Etzt = 1 , '  + ' zt Var t zt = 1 + '  Over n periods: n, X j n n zt n = 1 , '  + ' zt + ' "t n,j +1 2 +1 +2 +1 2 +2 2 +2 2 +2 2 +2 +1 2 2 1 + + j =0 The conditional mean and variance are: Etzt n = 1 , 'n + 'nzt Var t zt n = 1 + ' +    + ' n,  As n grows, the mean and variance approach Ez = Var z = 1 + ' + ' +    = 1 , ' We refer to these as the unconditional mean and variance, the theoretical analogs of the mean and variance we might estimate from a sample time series of z . + 2 + 2 2 2 4 2 1 2 2 Advanced Fixed Income Analytics 1-12 8. Vasicek: The Solution Discount factors are log-linear functions of z : , log bn = An + Bn zt t This relatively simple structure is one of the most useful features of the model. All we need to do is nd the coe cients An; Bn. The fundamental theorem gives us An = An +  =2 + Bn 1 , ' ,  + Bn  =2 = An + Bn1 , ' , Bn , Bn  =2 Bn = 1 + Bn' 2 +1 2 2 +1 For future reference, note that Bn = 1 + ' +    + 'n, = 1 , 'n=1 , ' 1 Starting point: bt = 1 a dollar today costs a dollar, so log bt = ,A , B zt = 0  A = B = 0 0 0 0 0 0 0 These formulas are easily computed in a spreadsheet: starting with A = B = 0, we compute An ; Bn  from An; Bn. 0 0 +1 +1 Needed: values for the parameters  ; ; ';  | coming soon. Advanced Fixed Income Analytics 1-13 9. Vasicek: How We Got There Step 1: Fundamental theorem applied to zeros Theorem says returns R satisfy 1 = Et mt Rt  +1 +1 Returns on zeros are Rn = bn =bn t t t ratio of sale price to purchase price +1 +1 +1 +1 Theorem becomes bn = Et mt bn  t t An n + 1-period zero is a claim to an n-period zero one period in the future. +1 +1 +1 Advanced Fixed Income Analytics 1-14 9. Vasicek: How We Got There continued Step 2: Show that z is the short rate Since bt = 1, the short rate satis es: e,yt = bt = Et mt  Stop now if you have any questions about this. 0 1 1 +1 Fact: if log x is normal with mean a and variance b, then E x = ea + b=2 log E x = a + b=2 In words: the log of the mean a + b=2 isn't quite the mean of the log a. There's an extra term due to the variance b. Since " is normal, log mt is normal, too: log mt = , =2 , zt , "t Its conditional mean and variance are Et log mt  = , =2 , zt Var t log mt  =  +1 2 +1 +1 2 +1 +1 2 Apply the fact: Et mt  = e, =2 , zt +  =2 = e,zt ; 2 +1 Result: z is the short rate. 2 Advanced Fixed Income Analytics 1-15 9. Vasicek: How We Got There continued Step 3: Solution for an arbitrary maturity Guess that bond prices are log-linear functions of z : , log bn = An + Bn zt t We know this works for n = 1: A = 0 and B = 1. Given a solution for n, nd n + 1: bn = Et mt bn  t t Work on right-hand-side: log mt = , =2 , zt , "t log bn = ,An , Bn zt t = ,An , Bn 1 , ' + 'zt + "t The sum is log mt bn = , =2 , An , Bn1 , ' , 1 + Bn'zt t , + Bn "t : The conditional mean and variance are Et log mt bn  = , =2 , An , Bn1 , ' , 1 + Bn 'zt t n Var t log mt bt  =  + Bn  Apply the fact: log bn = , =2 , An , Bn 1 , ' , 1 + Bn 'zt t + + Bn  =2 = ,An , Bn zt: Recursions for computing An; Bn one maturity at a time: An = An +  =2 + Bn 1 , ' ,  + Bn  =2 Bn = 1 + Bn' 1 +1 +1 2 +1 1 +1 +1 +1 +1 +1 +1 2 +1 +1 +1 +1 2 +1 +1 +1 2 2 2 +1 +1 +1 2 +1 2 Advanced Fixed Income Analytics 1-16 10. Vasicek: Choosing Parameters Premise: a model with bad parameters is a bad model Question: How do we choose them  ; ; '; ? Answer for now: reproduce broad featues of spot rates see table several pages above Mean and variance of spot rates: ytn = n, An + Bn zt E yn  = n, An + Bn   Bn ! Var z  =  Bn ! n Var y  = n n 1,' n n Corr yt ; yt,  = ' The last follows since linear functions of z inherit z 's autocorrelation. 1 1 2 2 2 2 1 Parameter values: Choose to match the mean short rate: = 6:683 = 0:005569 1200 Choose ' to match autocorrelation of short rate: ' = 0:959 Choose to match variance of short rate:  2:699 !  = 0:0006374 1 , ' = 1200 What about ? 2 2 2 Advanced Fixed Income Analytics 1-17 10. Vasicek: Choosing Parameters continued  controls risk premiums on long bonds Mean spot rates in model and data for di erent choices of : 9 Mean Yield (Annual Percentage) 8.5 8 lambda < 0 7.5 7 lambda = 0 6.5 6 5.5 lambda > 0 5 0 20 40 60 Maturity in Months 80 100 120 Result:  = ,0:1308 matches mean 10-year rate Why? Need negative value to get right sign on risk premium Problem: miss the curvature Advanced Fixed Income Analytics 1-18 11. Hedging Duration-based hedging of a 5-year zero Sensitivity to yield changes: n bn = e,ny n bn  ,ne,ny yn = ,nbnyn n is the duration, measured in months. Arrange positions in n-month zero that o set risk in 60-month: Portfolio consists of v = x b + xnbn 60 60 Change in value: v = x b + xnbn  ,x 60b y  , xn nbnyn  60 60 60 60 60 Hedge ratio results from setting v = 0 the de nition of a hedge and y = yn the assumption underlying duration: xnbn = , 60 Hedge Ratio = x b n The usual: the ratio of the durations. 60 60 60 Advanced Fixed Income Analytics 1-19 11. Hedging continued Vasicek-based hedging of a 5-year zero Sensitivity to z changes: bn = e,An , Bnz bn  ,Bn bnz Arrange positions in n-month zero that o set risk in 60-month: Portfolio consists again of v = x b + xnbn 60 60 Change in value: v = x b + xnbn  ,x B b z  , xnBnbnz  60 60 60 60 60 Hedge ratio results from setting v = 0 the de nition of a hedge: xnbn = , B Hedge Ratio = x b B 60 60 60 Comparison: Recall: Bn = 1 + ' +    + 'n, n 1 When ' = 1, Bn = n and Vasicek- and duration-based hedge ratios are the same When 0 ' 1, Bn n: Vasicek attributes relatively less risk to long zeros than duration Advanced Fixed Income Analytics 1-20 11. Hedging continued Thought experiment: If the world obeyed the Vasicek model, how far wrong could you go by using duration to hedge? Buy $100 of 5-year zeros Construct duration and Vasicek hedges with 24-month zeros Compute pro t loss for given changes in z 3 Profit/Loss on Hedged Position 2 1 Vasicek−based hedge 0 −1 Duration−based hedge −2 −3 −2 −1.5 −1 −0.5 0 0.5 Change in z (x 1200) 1 1.5 2 Advanced Fixed Income Analytics 1-21 11. Hedging continued Comments on the thought experiment Duration hedging: you're overhedged Vasicek hedging: not exactly zero due to convexity Is this reasonable? Qualitatively yes: less volatility for long rates than duration assumes see table for monthly changes Advanced Fixed Income Analytics 12. Where are the Bodies Buried? Strengths of the Vasicek model: relatively simple and transparent ts general features of bond prices Black-like option prices coming soon Weaknesses: insu cient curvature in mean spot rates crude approximation to current spot rates volatility is constant   persistence same for spot rates and spreads allows negative interest rates 1-22 Advanced Fixed Income Analytics 1-23 Summary 1. Modern pricing theory is based on a pricing kernel, which plays a role analogous to the market return in the CAPM 2. Models contain descriptions of interest rate dynamics  ; ; ' assessment of risk  3. The Vasicek model an archetype: many other models have similar structures strengths include simplicity, Black-like option formulas weaknesses include its single factor, constant volatility 4. Coming up: derivatives pricing comparisons with other models 5. Vote on these topics: convexity adjustment for eurocurrency futures Hull and White model Heath-Jarrow-Morton approach volatility smiles for options stochastic volatility two-factor models American options CMT CMS swaps Currency and commodity derivatives ...
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern