### Investment Management

UNIVERSITY OF TEXAS AT DALLAS SCHOOL OF MANAGEMENT FIN6310: INVESTMENT MANAGEMENT SOLUTIONS TO PROBLEM SET #1 PROF. ARZU OZOGUZ SPRING 2013 1. Calculate the value of the following two bonds. Assume that coupon payments are made semi-annually and that par value is $1,000 for both bonds. Coupon rate Time to maturity Yield-to-maturity Bond A 5% 5 yrs 7. 2% Bond B 5% 25 yrs 7. 2% Recalculate the bonds’ values if the yield to maturity changes to 9. 4%. Which bond is more sensitive to the changes in the yield? Will this always be the case? When the yield-to-maturity is 7. %, the bond prices are, respectively, 1 1 1. 036 0. 036 1 1. 036 0. 036 1 1. 047 0. 047 1 1. 047 0. 047 25 1000 1. 036 1000 1. 036 908. 98 1 25 746. 58 When the yield-to-maturity is 9. 4%, the bond prices are, respectively, 1 25 1000 1. 036 1000 1. 047 827. 62 1 25 579. 01 Price of bond A decreases by 8. 95%, while price of bond B drops by 22. 45%. The longer term bond is more sensitive to a given change in the discount rate. This will always be the case. Mathematically, there are more terms in the equation for the longer-term bond that are influenced by the discount rate.

Practically speaking, your money is tied up longer with a longer term bond and so you will experience greater capital losses and gains when interest rates change. 2. A bond with a coupon rate of 4. 7% is priced to yield 6. 30%. Coupon is paid is semi-annually; the par value is $1,000. The bond has 5 years remaining until maturity. Assuming that market rates stay the same over the next five years, calculate the value of the bond at the beginning of each year and the amount of change in the bond’s value from year to year. Describe the behavior of the bond’s value over time.

At t = 0, at issue the price will be 1 1 1. 0315 0. 0315 1 1. 0315 0. 0315 1 1. 0315 0. 0315 1 1. 0315 0. 0315 1 1. 0315 0. 0315 23. 5 1000 1. 0315 932. 28 At the end of year 1, the price becomes 1 23. 5 1000 1. 0315 1000 1. 0315 1000 1. 0315 1000 1. 0315 944. 20 1 23. 5 956. 88 1 23. 5 970. 37 1 23. 5 1000 984. 73 The price change from year to year is ? ? ? ? ? 11. 92 12. 68 13. 49 14. 36 15. 27 The bond is selling at a discount today; its price will rise to move toward par value at maturity. The change in price increases as it gets closer to maturity. 3.

Suppose that you purchased a 20-year bond that pays an annual coupon of $40 and is selling at par. Calculate the one –year holding period return for each of these three cases. a. The yield-to-maturity is 5. 5% one year from now. If the yield-to-maturity is 5. 5% one year from now, the bond will be selling for 1 1 1000 1. 055 40 825. 89 1. 055 0. 055 Hence, the holding-period-return (HPR) is: 825. 89 40 1000 13. 41% 1000 b. The yield-to-maturity is the same one year from today as it is today. In this case, the bond price will remain at par and therefore the holding period return equals to coupon rate 4% c.

The yield-to-maturity is 2. 5% one year from now. 1 1000 1. 025 40 1224. 68 1. 025 0. 025 Hence, the holding-period-return (HPR) is: 1224. 68 40 1000 26. 47% 1000 1 4. Plot the yield curve implied by the data in the following table. Time to maturity 3 months 6 months 1 year 2 years 5 years 10 years 15 years 20 years Yield-tomaturity 2. 40% 2. 60% 3. 00% 4. 30% 4. 80% 5. 70% 6. 40% 5. 20% Based on the Expectations Hypothesis, what does the yield curve tell us about short-term rates 5 years from now? What does it tell us about short rates 15 years from now and 20 years from now?

Since the yield curve is upward sloping through the fifth year, investors expect that short term rates will be higher during that period than they are today. That is, they expect the 3-month rate to be higher than 2. 4% when five years have passed. They also expect short term rates to be higher than current rates in 15 years. This is reflected in the slope of the yield curve which is positive through year 15. However, the expectation is that after 15 years, short term rates will begin to fall again. The downward slope in the yield curve is a sign of that expectation.

That is, the 3-month rate that prevails 20 years from now is expected to be lower than the 3-month rate that prevails 15 years from now. 5. The current yield curve for default free zero-coupon bonds is as follows: Maturity (years) 1 2 3 Yield-tomaturity 10% 11% 12% a. What are the implied one year forward rates? The one-year forward rate for time 2 solves the following equation: 1. 11 1. 10 1 12. 009%. Similarly, the one-year forward rate for time 3 solves That is, the equation: 1. 12 That is, 14. 0271% 1. 11 1 b. Assume that the expectations hypothesis of the term structure is correct.

If market expectations are accurate, what will the yields to maturity on one year and two year zero coupon bonds be next year? We have already computed the forecast for the one year rate next year. We must now compute the expectation for the 2-years to maturity. This must equate the strategy that consists of investing for 3 years at the current 3-year spot rate with the strategy of investing at the one-year spot rate and then rolling over the profits into a two-year bond one year from now: 1. 10 1 1. 12 13. 0136%. Hence, the forecast for the one-year yield is This implies that 12. 09%, and forecast for the two-year yield is 13. 0136%. c. If you purchase a two year zero coupon bond now, what is the expected total rate of return over the next year? What if you purchase a three year zero coupon bond? You can assume that the par value is $100. We need to compute the forecasted price of the two-year zero-coupon bond at the end of the first year. Notice that by that time this has become a one-year bond. Hence its price is 1000 1. 12009 892. 79 Today the price of this bond is simply 892. 79 811. 62 does not pay any coupons, its return is given by: 1 1 10% . 11. 62. Since this bond Similarly, if you purchase a three-year zero coupon bond today, the forecasted price a year later is 1000 1. 130136 Today, this bond’s price is simply expected holding period return is 78. 295 71. 178 1 78. 295 . 71. 178. Therefore, the 10% 6. Consider the following three bonds. You are investigating how the bonds would react to changes in interest rates. Bond A Face value Years to maturity Coupon rate Yield-to-maturity $1,000 3 5. 5% 4. 80% Zero-coupon bond $1,000 2. 85 0 4. 80% Bond B $1,000 3 8. 75% 4. 80% Assume that coupons are paid once a year. . Find the duration of each bond. Bond A Time 1 2 3 Price ZCB Time 2. 85 Price Bond B Time 1 2 3 Price Cash Flow 87. 5 87. 5 1087. 5 Present value 83. 49 79. 67 944. 81 1107. 97 Weight 0. 075 0. 072 0. 853 Cash Flow 1000 Present value 874. 92 874. 92 Weight 1. 000 Cash Flow 55 55 1055 Present value 52. 48 50. 08 916. 58 1019. 13 Weight 0. 051 0. 049 0. 899 Hence, the durations are: 0. 051 0. 075 1 1 0. 049 0. 072 2 2 0. 899 0. 853 3 3 2. 85 2. 78 2. 85 b. Calculate the modified duration of each bond. The modified durations are ? ? 2. 85 2. 72 1. 048 2. 78 2. 5 1. 048 c. Calculate the estimated percentage change in price of each bond due to a 0. 50% change in yield to maturity. The percentage change in the price of each bond due to a change in the yield? ? ? to-maturity is ? ? ? 2. 72 2. 65 0. 5% 1. 36% 1. 33% 0. 5% d. What can you conclude about the reactions of the bonds? Specifically, compare the percentage price changes of the bonds with similar durations and the bonds with similar maturities. Bonds with equal durations are more alike than bonds with equal maturities in their reactions to changes in yields. 7.

Suppose that your insurance company has issued a Guaranteed Investment Contract (GIC) that matures in three years and promises to pay an interest rate of 23. 36%. The amount invested in GIC today is $150,000. You have decided to immunize your position by purchasing a bond that has a par value of $150,000, a coupon rate of 23. 36%, and four years to maturity. The bond is selling currently at par value. a. What is the future value of your company’s obligation? The future value of the obligation is $150,000 1. 2336 $281,588. 13 b. Assume that the interest rate stays at 23. 36%.

At the date at which each payment is received, compute the accumulated value of reinvested coupons and the proceeds from the bond sale. How close will you come to your meeting your obligation? The bond pays a coupon of $150,000 23. 36% $35,040. If the market rates remain unchanged, at the end of year three it will be possible to sell the bond still at par. With this information, we can construct the following table: Year 1 2 3 3 Total future value Cash flow 35,040 35,040 35,040 150,000 Accumulated value 53,322. 78 43,225. 34 35,040 150,000 281,588. 13 That is, you will be able to repay your obligation in full.