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# Fin 448: Fixed-Income Securities Homework 1 Wei Yang Assigned: Apr 3 Due: Apr 10 Each problem counts as 10 points. 1 Bootstrapping In this problem we...

Having trouble with the bootstrapping question. This is the first question in the problem set
Fin 448: Fixed-Income Securities Homework 1 Wei Yang Assigned: Apr 3 Due: Apr 10 Each problem counts as 10 points. 1 Bootstrapping In this problem we extract zero yields from coupon bond prices through a basic and intuitive approach called “bootstrapping”. Table 1 lists the prices of selected U.S. government, or Treasury securities. These secu- rities pay semi-annual coupons (the coupon rate is also annualized). We want to extract semi-annually compounded zero yields. Recall the price of a coupon bond with \$1 face value is given by P = p 0 . 5 ± c 2 ² + p 1 ± c 2 ² + p 1 . 5 ± c 2 ² + ... + p n ± 1 + c 2 ² p k = 1 ( 1 + y k 2 ) 2 k where c is the annualized coupon rate, p k is the zero-coupon bond price, and y k is the annualized zero yield. (1.1) Compute p 0 . 5 and y 0 . 5 , using the quotes for the 6-month Treasury. Table 1: Ex-coupon prices of Treasury securities Coupon rate (%) Time to maturity Price per \$100 0 6 mo 98.16 3.500 1 yr 99.50 6.625 1.5 yr 103.31 4.250 2 yr 99.63 1
(1.2) Use p 0 . 5 and the quotes for the 1-year Treasury to compute p 1 and y 1 . (1.3) Use p 0 . 5 and p 1 , and the quotes for the 1.5-year Treasury to compute p 1 . 5 and y 1 . 5 . (1.4) By now you should have understood how this iterative process works. Find p 2 and y 2 . Now we have all the zero prices or yields. (1.5) Compute the yield of the 2-year Treasury. (1.6) Find the par coupon rate for a 2-year coupon bond. (1.7) What are the zero yields if we switch to continuous compounding? 2 Arbitrage We continue with the previous problem, in which we have computed the term structure of zero yields from coupon bond prices in Table 1. Now consider Treasury note X, a 2-year bond with a 3% coupon rate and a face value of \$100. (2.1) Use the term structure to calculate the price of X. (2.2) The market bid and ask quotes for X are 97.25 – 97.29 per \$100. How is your answer compared with the market quotes? Let us assume that you are able to both buy and sell X at 97 per \$100 face value. We want to explore if there exists an arbitrage strategy that captures a positive proﬁt today, and leaves no obligations in the future. (2.3) How much of each Treasury in Table 1 is needed to replicate the payoﬀs of X? You will not get whole numbers, and that is okay. Assume all bonds have a face value of \$100. (2.4) You have now constructed a basket of bonds to replicate the payoﬀs of bond X. What is the price of this basket? (2.5) You now have one bond X, and a basket of bonds that replicate the payoﬀs of X. Could you earn an arbitrage proﬁt? How much? 2
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