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Unformatted text preview: FIXED INCOME SECURITIES
OCTOBER 7, 2010 REVIEW FOR EXAM 1 Fall 2010 FNCE 235/725 Prof. Stephan Dieckmann 1 NAME OF INITIATIVE OR GROUP Chapters 1 - 5 Description of Coupon Bonds / Zero Coupon Bonds Interest Rate Compounding Term Structure of Interest Rates What is a Replicating Portfolio? Arbitrage Concept Yield to Maturity Holding Period Returns 2 Question 1, Exam 2010
TRUE or FALSE? No explanation is required. A correct answer is awarded 3 points, a false answer with 0 points. a. Assume the term structure is downward sloping and stable. The yield to maturity on a zero coupon bond overstates the holding period return if the zero coupon bond is bought and then sold prior to its maturity. 3 Question 2, Exam 2010
Consider the following trading universe which consists of a zero coupon bond and two coupon-bearing bonds. The time till maturity in years, face value, coupon, and today's price for each bond in the trading universe are given in the following table. The coupon-bearing bonds make annual coupon payments; the first coupon payment is one year from today. Bond #A #B #C Maturity Face Value 1 2 2 1 1000 5000 Coupon Price Zero 100 250 0.9524 1073.33 4921.92 Part a. (5 points) Arbitrage appears to be possible within the trading universe, please explain why. 4 Question 2, Exam 2010
Suppose you replicate bond C, by buying a portfolio containing bonds A and C, then the price of the replicating portfolio is different than the Bond C price in the marketplace. There are other ways, of course, to express the same. A quantitative answer was not needed required. However, there are many ways to show the existence of arbitrage here. For example: For bond B, no arbitrage requires 1073.33 = 100d1 + 1110d2, Bond A equals d1, hence, d2= 0.8892. For bond C, 250 d1 + 5250 d2 equals 4906.4, which is not equal to 4921.92. Part b. (5 points) Please set up an arbitrage strategy for which a profit occurs as of today, but the portfolio payoff is zero in year 1 and zero in year 2. What is the arbitrage profit? 5 Question 2, Exam 2010
For example, sell 1 unit of bond C, buy 4.773 units of bond B, and sell 227.27 units of Bond A. This yields no cash flows in year 1 and 2, and a profit of 15.52 as of today. Part c. (5 points) Can you find an alternative strategy, for which the arbitrage profit occurs in year 1 instead of today? In other words, based on the same set of bonds, please create a portfolio, for which the portfolio payoff is zero today and in year 2, and strictly positive in year 1. What is the arbitrage profit? 6 Chapters 6 - 7 Accrued Interest Theta / Passage of Time Time Profiles for Coupon Bearing Bonds What is a Forward Contract? Forward Borrowing / Forward Lending Why are Forward Contracts Risky? Borrowing Rates greater than Lending Rates 7 Question 1, Exam 2010
TRUE or FALSE? No explanation is required. A correct answer is awarded 3 points, a false answer with 0 points. b. Consider a forward contract that was agreed to in the past. The value of this forward contract gets closer and closer to zero as the forward contract approaches its maturity date. c. Suppose there is a forward contract that matures in five years and delivers a two year zero coupon bond upon maturity. You can replicate a short position in this forward contract by buying a two year zero coupon bond and shorting a five year zero coupon bond. e. Regardless of the shape of the term structure, the yield to maturity of a coupon bond is equal to the holding period return of that bond if the intermediate coupons can be reinvested at today's forward rates. 8 Question 3, Exam 2010
Currently, the general level of interest rates is low. At the end of 2009, we observed the following term structure: (i) 3 year zero coupon bonds were trading at 1.65% yield to maturity (assuming continuous compounding) (ii) 10 year zero coupon bonds were trading at 3.8% yield to maturity (assuming continuous compounding) Please assume that the face value for 1 bond is 1,000 dollars. Part a. (5 points) Suppose you set up a trading strategy purchasing 10-year zero coupon bonds (assets), being fully financed by 3-year zero coupon bonds (liabilities). The value of equity is zero. If the total asset value is 1,000,000 dollars, then how many 10 year zeros do you need to purchase, and how many 3 year zeros do you need to sell? Part b. (5 points) Do you think the trading strategy above has positive or negative theta, and why?
9 Question 3, Exam 2010 ...
Price of one 3 year zero, 1000*exp(-.0165*3)=951.71 Price of one 10 year zero, 1000*exp(-.038*10)=683.86 Liabilities: 1,000,000/951.71 = 1050.75 units Assets: 1,000,000/683.86 = 1462.28 units This trading strategy will have positive theta, the assets will accrue at a larger rate than the liabilities. If you would approximate the theta as we did in class, this would be a positive theta of 21,500 (not needed to answer this question). Part c. (5 points) The current expectation of many market participants appears to be that interest rates are going to rise in the near future. Based on the dollar delta of this trading strategy, by how much should the value of your trading strategy change if interest rates increase uniformly by 100 basis points? 10 Question 3, Exam 2010 ...
Part d. (5 points) The sensitivity of your strategy to changes in interest rates identified in part c worries you, so you decide to eliminate this risk. Specifically, you have heard about the possibility of forward borrowing, and decide to look in future financing for your asset position today. At what yield to maturity (continuous compounding) could you obtain financing for 7 years starting 3 years from now? Please assume forward borrowing also takes place in form of a zero coupon bond. Part e. (5 points) Right after entering the contract to obtain forward borrowing to eliminate the risk, does the new total position (consisting of your original assets and liabilities, plus the contract for forward borrowing) have a positive, negative, or zero theta, and why? No need to calculate the actual value. 11 Chapters 8 11 Price Sensitivity of Bonds to Changes in Interest Rates What is Delta? Risk Management Delta Based Strategies Convexity What is Gamma? Geometric Interpretations Is Positive Convexity always Desirable? Risk Management Delta and Gamma Based Strategies Relationship among Delta, Gamma and Theta 12 Question 1, Exam 2010
TRUE or FALSE? No explanation is required. A correct answer is awarded 3 points, a false answer with 0 points. d. It is possible to construct a trading strategy that has an initial value of zero and appreciates when interest rates shift in either a downward or upward direction, without generating arbitrage. 13 Question 4, Exam 2010
You start working for a hedge fund, tailoring fixed income portfolios to satisfy specific investor demands. This question asks you to design such a fixed income portfolio in a simplified way. Specifically, the trading universe consists of 3 zero coupon bonds, maturing in 2, 8 and 14 years, respectively. Assuming continuous compounding, the yield to maturity of those three zeros is 2%, 4%, and 5%, respectively, as of today. Please assume the face value of 1 bond is 1,000 dollars. Part a. (5 points) What is the price, the dollar delta, and the dollar gamma of each of the three zero coupon bonds? 14 Question 4, Exam 2010 ...
Part b. (5 points) The portfolio you are asked to design for an investor should have a value of 10,000 dollars today. But your investor has a strong view about the future change in interest rates: She thinks interest rates are going to change within the next few days by 50 basis points, but she does not know in which direction. Although this might not be the view the overall market has, she wants to bet on interest rates being very volatile. Her idea is to set up a portfolio that is delta-neutral but has positive convexity. Specifically, she wants you to implement a portfolio gamma such that after an immediate 50 basis point increase or decrease in interest rates, her portfolio would be worth approximately 10,500 dollars. What is the gamma of the portfolio (in dollars) that would satisfy such an objective? 15 Question 4, Exam 2010 ...
Part c. (5 points) Based on your results in part b, what are the equations that the portfolio weights in the 3 zero coupon bonds need to satisfy. Part d. (5 points) Please solve the set of equations in part c, and show the portfolio weights. Part e. (5 points) So far, the effect of the passage of time has been ignored in this case, since your customer thinks the change in interest rates is almost immediate. However, you recall the properties of theta from your fixed income class, and would like to advise your customer. Without actually calculating theta, do you think the portfolio theta will be positive or negative, and why? 16 Rules in class closed book may bring one 8 x 11 piece of paper of notes, both sides please bring a calculator, but not a computer 17 ...
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