# VALUATION OF DEBT CONTRACTS AND THEIR PRICE VOLATILITY CHARACTERISTICS QUESTIONS See answers below

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1 VALUATION OF DEBT CONTRACTS AND THEIR PRICE VOLATILITY CHARACTERISTICS QUESTIONS See answers below 1. Determine the value of the following risk-free debt instrument, which promises to make the respective payments when the appropriate annual rates are as shown in the last column. Year Cash Payment Appropriate Annual Rate 1 \$15, % 2 17, , , For each of the following, calculate the price per \$1,000 of par value assuming semiannual coupon payments. Coupon Bond Years to Rate Required Maturity Yield A 8% 9 7% B C D Consider a bond selling at par (\$100) with a coupon rate of 6% and 10 years to maturity. a. What is the price of this bond if the required yield is 15%? b. What is the price of this bond if the required yield increases from 15% to 16%, and by what percentage did the price of this bond change? c. What is the price of this bond if the required yield is 5%? d. What is the price of this bond if the required yield increases from 5% to 6%, and by what percentage did the price of this bond change? e. From your answers to parts (b) and (d) what can you say about the relative price volatility of a bond in high compared to low interest rate environments? 5. Suppose you purchased a debt obligation 3 years ago at its par value of \$100,000.The market price of this debt obligation today is \$90,000. What are some of the reasons why the price of this debt obligation could have declined since you purchased it 3 years ago? 9. A portfolio manager is considering buying two bonds. Bond A matures in 3 years and has a coupon rate of 10% payable semiannually. Bond B, of the same credit quality, matures in 10 years and has a coupon rate of 12% payable semiannually. Both bonds are priced at par. 1

2 a. Suppose the portfolio manager plans to hold the bond that is purchased for 3 years. Which would be the best bond for the portfolio manager to purchase? b. Suppose the portfolio manager plans to hold the bond that is purchased for 6 years instead of 3 years. In this case, which would be the best bond for the portfolio manager to purchase? c. Suppose that the portfolio manager is managing the assets of a life insurance company that issued a 5-year guaranteed investment contract (GIC). The interest rate that the life insurance company agreed to pay is 9% on a semiannual basis. Which of the two bonds should the portfolio manager purchase to assure that the GIC payments will be satisfied and that a profit will be generated by the life insurance company? 10. What is meant by reinvestment risk when purchasing a bond? 11. Can you tell from the following information which of the three bonds will have the greatest price volatility, assuming each is trading to offer the same yield to maturity? Coupon Bond Rate Maturity X 8% 9 years Y Z Calculate the requested measures for bonds A and B (assume each bond pays interest semiannually): A B Coupon 8% 9% Yield to maturity 8% 8% Maturity (in 2 5 years) Par Price a. Calculate the duration for the two bonds by changing the yield up and down 25 basis points. b. Calculate the duration for the two bonds by changing the yield up and down by 10 basis points. c. Compare your answers to parts (a) and (b). d. Calculate the Macaulay duration for the two bonds. e. Calculate the modified duration for the two bonds. f. Compare the modified duration for the two bonds computed in parts (e) and (a). 14. Which of the following bonds will have the larger price change for a 50 basis point change in yield? Bond Duration Price 2

3 E 7 50 F What is the difference between modified duration and effective duration? 16. What is assumed about how the yield curve changes when using duration? 17. An investor is discussing the duration of a highly complex bond with his broker. The broker tells the investor that the duration of this bond is negative 5. The investor is confused about the negative value and tells the broker that the figure must be in error because the duration is always positive because it is some type of weighted average of time of the cash flows. Comment. 18. As a portfolio manager, you present a report to a client. The report indicates the duration of each security in the portfolio. One of the securities has a maturity of 15 years but duration of 25. The client believes that the report contains an error because she believes that the duration cannot be greater than the security's maturity. What would be your response to this client? 19. A strategy called immunization is used by institutional investors to protect a portfolio against an adverse change in interest rates. Basically this strategy seeks to offset interest rate risk and reinvestment risk. Why do these two risks offset each other to a certain extent when interest rates change? 3

4 VALUATION OF DEBT CONTRACTS AND THEIR PRICE VOLATILITY CHARACTERISTICS ANSWERS TO QUESTIONS 1. In terms of the notation used in the chapter: a 1 = \$15,000 r 1 =.080 a 2 = \$17,000 r 2 =.085 a 3 = \$20,000 r 3 =.090 a 4 = \$21,000 r 4 =.095 P 0 = 15, 000 (1.080) 17, (1.080)(1.085) 20, (1.080)(1.085)(1.090) 21, (1.080)(1.085)(1.090)(1.095) = \$13, \$14, \$15, \$15, = \$59, The price of bond A is \$1,065.95; the price of bond B is \$1,000, price of bond C is \$ and price of bond D (zero coupon) is \$ The original bond yield is 6 percent. a. If required yield is 15%, the price of bond will be b. When yield increases to 16%, price of bond will decrease to , and percentage decrease is c. If the required yield is 5%, the price of bond will be d. If required yield increases from 5% to 6%, we are back to the original data, and the selling price will be 100. The decrease in price is ( ) = -7.17%. e. The percentage change observed in prices of bonds in (b) and (d) suggest that relative price volatility is more when initial yield is low (part d) and less when initial level is high (part b). 5. The price of this debt obligation can change over time for any one of the following reasons: (1) a rise in the level of interest rates in the economy over the past three years; (2) an increase in the required yield due to a change in the yield spread between non-treasury and Treasury securities; (3) a decline in the perceived credit quality of the debtor. 9. a. It must be stressed that there is no way to determine which bond will provide the best return. Since these bonds are priced at par, the yield to maturity for bonds A and B is 10% and 12%, respectively. However, yield to maturity tells us absolutely nothing about the dollars that will be generated from holding each bond. The number of dollars that will be received from investing in Bond A depends on the reinvestment rate over the three years. Thus for this bond, there is reinvestment risk. For Bond B, the dollars that will be realized depend on the yield for 7-year bonds three years from now (because Bond B will be a 7-year bond at that time). The portfolio manager is 4

5 exposed to both reinvestment risk and interest rate (or price) risk. b. If the holding period is six years rather than three years, the answer is the same as in (a); it is not possible to determine a priori which bond will provide the best return. The dollars received from investing in Bond A will not only depend on the reinvestment rates over the first three years, but also reinvestment rates from years three to six. c. Once again, it is not possible to lock in a profit or spread with either bond because of reinvestment risk for both bonds and interest rate (or price risk) for Bond B. As an aside, this fundamental principle has been overlooked by many financial institutions who believe that a "yield" indicates the return that will be realized. 10. The concept of yield to maturity assumes that the investor will realize the yield to maturity that is calculated at the time of purchase only if (1) all the coupon payments can be reinvested at the yield to maturity, and (2) the bond is held to maturity. With respect to the first assumption, the risk that an investor faces is that future interest rates at which the coupon can be reinvested will be less than the yield to maturity at the time the bond is purchased. This risk is reinvestment risk. 11. From the properties of price volatility, it is not possible to say which has the greatest price volatility. An unambiguous answer requires that both the coupon rate be lower and the maturity be greater than the other bonds. The purpose of this question is to demonstrate to students the need to quantify price volatility, i.e., the factors that affect price volatility are insufficient to indicate relative price volatility. 12. Note that a 25 bp increase in yield translates into a.125% increase in semiannual yield when calculating prices. Thus, in this case, an increase in 25 bp gives a semiannual yield of 4.125% and a decrease leads to a yield of 3.875%; a 10% bp increase gives a yield of 4.05% and a decrease leads to a yield of 3.95%. To estimate duration using Equation 18.10, however, dy must NOT be adjusted. In the text examples, the values in Table 18-1 are found using semiannual rates, where dy is applied to the annual equivalent -- a 50 basis point increase in rates, from 9%, would give a semiannual yield of 4.75%. Part a For Bond A in part a of the question, the duration is found using Bond A N = 4, I = 8/2 = 4, FV = 100, PMT = 4, cpt PV = 100 N = 4, I = 8.25/2 =4.125, FV = 100, PMT = 4, cpt PV = N = 4, I = 7.75/2 =, FV = 100, PMT = 4, cpt PV = Equation 18.10, where V + is , V - is , dy is , and V 0 is

6 Duration Part b Bond A N = 4, I = 8/2 = 4, FV = 100, PMT = 4, cpt PV = 100 N = 4, I = 8.1/2 =4.05, FV = 100, PMT = 4, cpt PV = N = 4, I = 7.9/2 =3.95, FV = 100, PMT = 4, cpt PV = Equation 18.10, where V + is 99.5, V - is , dy is , and V 0 is 100. Duration Bond A Bond B a. Duration with 25 bp change b. D with 10 bp change c. There was a much larger change in duration for the bond which was trading above par, as should be expected. d. Macaulay duration for Bond A is found using the following equation D (1) \$ 40 (1.04 ) (2 ) \$ 40 (1.04 ) 2 \$ 1, 000 (3 ) \$ 40 (1.04 ) 3 ( 4 ) \$ 1, 040 (1.04 ) semiannual periods This amount would be annual periods. The Macaulay duration for Bond B is on a semiannual basis, or annual periods. e. The modified duration for Bond A is modified duration = Macaulay duration 1 + yield For Bond B, the modified duration is

7 f. Modified duration appears to track closely with the numbers produced in part a of the question. 14. Bond E will have 3.5 percentage change in price, while bond F will have a 2.5 percentage change in its price. However, the dollar amount of price change for bond F is larger than bond E. 15. Modified duration doesn't take into account changes in the bond value due to embedded options. Effective duration is flexible enough to allow analysts to reflect options and their exercise in the value of the bond. 16. When using any measure of duration, there is an implicit assumption that the yield curve shifts in parallel. 17. Duration describes how the value of a security changes when rates change, or what our best guess of that change will be. Measures such as the Macaulay duration are required to be positive because of the way that they are derived, but duration itself can take any value. One example of securities which have a negative duration is interest-only securities derived from mortgage portfolios. The value of these securities falls when rates fall due to the fact that homeowners will prepay their mortgages and the income of the portfolio will fall, lowering the expected cash flows from the pool of mortgages over time. 18. When the client believes that there is an error in your report, he is referring to the concept of Macaulay duration as some measure of the weighted average life of a bond. Thus he is right that a bond with a maturity of 15 years cannot have a Macaulay duration of 25. The duration of the bond, however, can take on any value. In this case, it means that the bond in question will be more sensitive to changes in rates than bonds with a 15- year term. 19. It is possible to create a portfolio of debt securities where the loss in one component of the return (for example, reinvestment loss) is offset by a gain in the other (for example, price gain). Particularly, if a portfolio is constructed such that its duration is equal to the investment horizon, it may be possible to offset interest rate risk (price risk) and reinvestment risk. A rise in market interest rates will lead to a loss in the market value of the portfolio, but a gain on account of higher returns earned on reinvested coupons. A fall in market interest rates will lead to a gain the market value of the portfolio, but a loss on account of lower returns earned on reinvested coupons. Thus the portfolio is "immunized" against changes in interest rates. 7

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