FIN 684 Professor Robert Hauswald Fixed-Income Analysis Kogod School of Business, AU Solutions 1 1. For the benchmark maturity sectors in the United States Treasury bill markets, Bloomberg reported the following information. Calculate the missing information. Show all the formulas used and the mechanics of your calculations. Settlement Date was August 31, 2007. Benchmark Maturity Date Discount Yield BEY PRICE Three-month 11/29/2007 4.01%???????? Six-month 02/28/2008???? 4.21%???? 2. For all the benchmark Treasury coupon securities Bloomberg reported the following information. Fill out the missing information. Show all relevant calculations. Settlement
Date was August 31, 2007. Prices are quoted in 32nds. Show the formulas that you have used. Benchmark Coupon Maturity Date YTM Accrued Clean Price Dirty Price Two-year 4.00% 8/31/2009 4.15%???????????? Three-year 4.50% 5/15/2010 4.17%???????????? Five-year 4.125% 8/31/2012???????? 99-12.25???? Ten-year 4.75% 8/15/2017???????? 101-19.5???? Thirty-year 5.00% 5/15/2037 4.84%????????????
3. Refer to problem 2. You need to hedge a long position in $ million par value of the current benchmark 2-year T-note. You decide to use the current benchmark 5-year T-note to perform the hedge. Explain intuitively how you will go about setting up this hedge. What is the implied view taken on the yield curve when the hedge is put in place? Long position in the 2-year note will lose money when the interest rates go up (prices go down), and make money when the interest rates go down (prices go up). Hence in order to hedge, you must short the 5-year note, which will make money when the interest rates go up and vice versa. A $ million par value position (long) in 2-year note will probably require the shorting of less than $ million par of the 5-year note. This is due to the fact that the 5-year note is more sensitive to interest rates than are 2-year notes. Since the hedge is long in 2-year and short in 5-year, the hedged position will make money if the yield curve were to become steeper. 4. An investor buys a face amount $1 million of a six-month (182 days) Treasury bill at a discount yield of 9.25 percent. What is the cost of purchasing these bills? Calculate the bond equivalent yield. Indicate clearly the formula you used and show all the steps in your calculations. Recalculate the bond equivalent yield if the T-bill has a maturity of 275 days. Cost of T-bill: The discount rate per $ face amount is given by the following formula. ( - P) 360 d =, (2.1) n
n = 182, d = 9.25% or 0.0925. Substituting these values and solving for P we get P = 95.323611 or, $953,236.11 invoice price per million. Bond equivalent yield calculated using Equation (2.2) is ( - P) 365 BEY =. (2.2) n Setting n = 182 and P = 9.5323611 in Equation (2.2) we get BEY = 9.38% for 182- day maturity. The BEY for the same bond but with a 275- day maturity is calculated by first computing the price of the T-bill, 275 0.0925 P = 1 = 92.934 360 Using this price we calculate the BEY as 2 2 275 275 2 275 + 2 1 1 365 365 365 92.934 BEY = = 9.9259%. 2 275 1 365 5. On November 18, 1987, a 7 7 /8% T-bond maturing on May 15, 1990 was quoted at 99 29 /33 for settlement on November 20, 1987. The last coupon was paid on November 15, 1987. (a) What is the invoice price of the T-bond? (b) What is the yield on the T-bond? (a) Invoice price: Invoice price is flat price plus accrued interest. We calculate accrued interest as follows.
7.875 5 ai 0.108173. 2 182 = = (2.3) Invoice price = quoted price + accrued; or, in this case, 99.90625 + 0.108173 =.01442. (b) Yield to maturity: Using the price-yield formula, we may find the yield to maturity. In order to calculate the yield to maturity we use the formula in Chapter 2, which is reproduced below: P t = 1+ y 2 N 1+ z x + j= N 1 j= 0 C 2 1+ y 2 j + z x. In the context of this problem, we have the following: The number of days between the last coupon date (11/15/87) and the settlement date (11/20/87) is 5 days. The basis is the number of days between the last coupon date and the next coupon date (5/15/88) and is equal to 182 days. Hence z = 177 days. The accrued interest is 5 /182( 7.875 /2) = 0.10817. The number of coupons remaining is N = 5. Note that z /x = 177 /182 = 0.97253. The full price of the bond is 99 + 29 /32 + 0.10817 =.014. Using the formula we get.014 = y 1 + 2 j= 5 1 + +. 5 1 0.97253 j+ 0.97253 j= 0 7.875 2 y 1 + 2 Solving for the yield to maturity, we get y = 7.9164%. 6. What is the price of a 10-year zero-coupon bond priced to yield 10% under each of the following assumptions? (a) Annual yield.
(b) Semiannual yield. (c) Monthly yield. (d) Daily yield. Explain the differences you found. What is the continuous limit? a) Annual yield P = = 38.5543 (2.4) 10 (1.1) b) Semiannual yield P = = 37.6889 (2.5) 20 (1.05) c) Monthly yield P = 36.9407 120 0.10 = (2.6) 1 + 12 d) Daily yield P = 36.7930 3650 0.10 = (2.7) 1 + 365 For the continuous limit, we apply the formula: P 10 e = = ( 0.1) 36.7879.