Solutions For all the benchmark Treasury securities shown below, compute the PVBP for $1 million

Transcription

1 FIN 684 Professor Robert Hauswald Fixed-Income Analysis Kogod School of Business, AU Solutions 2 1. For all the benchmark Treasury securities shown below, compute the PVBP for $1 million par value. Explain the differences that you found. Show all relevant calculations. Table 7.1: BENCHMARK YIELDS FOR SETTLEMENT ON SEPTEMBER 12, 2007 Maturity Date Coupon Yield to Maturity Clean Price Comments 8/31/ % 3.933%???? Two-year 5/15/ % 3.945%???? Three-year 8/31/ % 4.056%???? Five-year 8/15/ % 4.364%???? Ten-year 5/15/ % 4.646%???? Thirty-year Applying the EXCEL functions as documented in the text, we get the results below:

2 Settlement Date 9/12/2007 Security Coupon Maturity Clean Clean Yield Clean PVBP per Par value $100 Price Price Price $million (32nds) (decs) T-Note 2- year 4.000% 8/31/ % year 4.500% 5/15/ % year 4.125% 8/31/ % year 4.750% 8/15/ % T-Bond 30- year 5.000% 5/15/ % Note that the PVBP increases as we increase the maturity. This suggests that the price risks of longer-term debt securities are in general higher than those of the shorter-term debt securities. For example, the price risk of 10-year T-note is $ dollars per million, which is roughly half of the price risk of the 30-year T-bond, which has a price risk of $1, per million par. 2. For all the benchmark Treasury securities shown in Table 7.1, compute the Modified and McCauley Duration. Explain the differences that you found. Show all relevant calculations. We can compute the Duration and Modified Duration from first principles using the formulas that we derived in class. We can also use directly the EXCEL functions. I do both and show the results below.

3 Settlement Date 9/12/2007 Security Coupon Maturity Clean Clean Yield Clean PVBP per Par value $100 Price Price Price $million (32nds) (decs) T-Note 2- year 4.000% 8/31/ % T-Note 3-year 4.500% 5/15/ % T-Note 5- year 4.125% 8/31/ % T-Note 10-year 4.750% 8/15/ % T-Bond 30- year 5.000% 5/15/ % Days Basis Accrued Dirty Mod Dur Mod Dur Mac Dur Mac Dur Accrued Interest Price (from (from Excel) Excel) Compute the modified duration of a 30-year T-bond in Table 1. Interest-based strip maturing on 5/15/2037 was trading at a yield of 4.677% for settlement on September 12, Compute the modified duration of the strip with a maturity date that is closest to the 30-year T-bond. If you have $1 million par value of each security which is riskier? Why? We know already that the MD of the 30-year T-bond is From first principles, we know that the duration of a strip is simply equal to its maturity, which in our case is 29.7 years. The modified duration is % = Comparison based on par values may be misleading as the 30-year strip will sell at a considerable discount. In fact, the price of this strip can be computed as follows:

4 % 2 = Therefore, $1 million par value of this strip will sell at $253,300 (approximately), and hence the price risk for the same par value is much lower for the strip when compared to the 30- year T-bond. If we equalize the market values strip and the Treasury bond, then the strip is clearly more risky. To see this, let us compute the PVBP of the strip. The price of this strip when the yield goes up by 1 basis point is: % 2 = The PVBP is approximately $734 for $1 million par. Hence on a par value basis strip is less risky. 4. Compute the modified duration of a perpetuity which pays a coupon of 5% and which is currently selling at par. The price of perpetuity is 100 as it sells at par. P C 5 = = = 100. y 0.05 Differentiating price with respect to yield to maturity, we get:

5 dp dy C 5 = = = y 0.05 Modified duration, by definition, is: dp MD = = = 20. dy P 100 Even when the bond has infinite maturity, its duration is only 20 years. This example illustrates that with coupon bonds, buy side institutions cannot hope to get duration of more than 20 to 25 years. Hence, they must resort to strips to achieve higher durations for managing dedicated portfolios to meet pension liabilities. 5. In the following table, fill in the indicated blanks. Show all the steps in your calculations on separate sheets. Duration is denoted by D, coupon by C, and yield by Y. The settlement date is February 15, C Y Mat. Date Price Duration DV01 Yield Value of 32nds 9% 9.00% 5/15/xx xxx xxx 0% 9.00% 2/15/92 xxx xxx xxx xxx 10% 9.00% 2/15/ xxx xxx

6 The solution is as below. Note that there are issues in rounding: the first bond s duration should have been 4.81, the second bond s price 58.97, etc. C Y Mat. Date Price Duration DV01 Yield Value of 32nds 9% 9.00% 5/15/ % 9.00% 2/15/ % 9.00% 2/15/ On November 15, 1986, you bought $10 million (face amount) of a 7.50% T-bond maturing on November 15, 2016, at a yield of 7.60%. All coupons were reinvested at 5% yield. The bond was sold on June 20, 1987, at a yield of 7.50%. Calculate the annualized rate of return from this transaction. Show all the key steps. The bond was held for a total of 217 days, during which time the coupon was paid. We must compute interest on coupon income plus accrued interest at the time of sale. Flat price at a yield of 7.6% = Accrued interest at sale date = /184 = Coupon interest = Interest on interest / = Price at a yield of 7.5% = Total = = 9.663% Further Practice Problems: solutions are straightforward.

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