CHAPTER 5: Answers to “Do You Understand” Questions Do you Understand? (Pg. 110) 1. Why is a dollar today worth more to most people than a dollar received at a future date? There are two ways to view this issue. First, people generally have a positive time preference for consumption; that is, they prefer to consume today to consuming in the future. Second, the dollar today is worth more because if one has it, it can be invested to earn interest. 2. If you were to invest $100 in a savings account offering 6% interest compounded quarterly, how much money would be in the account after three years? 12 FV12 = PV0 (1 + i) 12 = $100 (1.015) = $100 (1.1956) = $119.56 3. Your rich uncle promises you $10,000 when you graduate from college. What is the value of this gift if you plan to graduate in 5 years and interest rates are 10%? PV $10,000 $6209.21 1.105 4. You have just won $60 million in the Power Ball lottery! The lottery promises to pay you $20 million at the end of each year for the next three years. If the market interest rate is 7%, how much money would you accept today in exchange for the three $20 million payments? Calculate the present value of each cash flow and then sum the present values: PV $20,000,000 $20,000,000 $20,000,000 $52,486,321. 1.07 1.07 2 1.07 3 DO YOU UNDERSTAND? (Pg. 117) 1. When a bond’s coupon rate is less than the prevailing market interest rate on similar bonds, will the bond sell at par, a discount, or a premium? Explain. The bond will sell at a discount. If, for example, it were priced at par, the yield to maturity of the bond would be less than the prevailing market interest rate on similar bonds and no one would purchase it. They will only purchase the bond if the price is discounted to provide a yield to maturity equal to that offered by similar bonds in the market. 2 2. Under what conditions will the realized yield on a bond equal the expected yield? If the investor receives all cash flows as promised by the issuer and reinvests them at the bond’s expected yield (the yield-to-maturity at the time of its purchase), the realized yield will equal the expected yield. If the market interest rate at which the coupon payments can be reinvested deviates from the expected yield, the realized yield will not necessarily equal the expected yield. 3. Using the trial-and-error method, find the yield to maturity of a bond with five years to maturity, par value of $1,000, and a coupon rate of 8% (annual payments). The bond currently sells at 98.5 percent of par value. The price of the bond is $985 = 0.985 ($1,000). Set up the known information in the bond pricing formula and try different interest rates until the right hand side equals the left hand side: 5 $80 $1,000 $985 t (1 i )5 t 1 (1 i ) The yield to maturity is 8.38%. 4. An investor purchases a $1,000 par value bond with five years to maturity at $985. The bond pays $80 of interest annually. The investor plans to hold the bond for two years and expects to sell it at the end of the holding period for 94 percent of its face value. What is this investor’s expected yield? Use the trial-and-error method. The current price is $985. The investor expects to sell it after two years for 0.94 ($1,000) = $940. Set up the known information in the bond pricing formula, and try different interest rates until the right hand side equals the left hand side: 2 985 t 1 $80 $940 t (1 i ) (1 i ) 2 The expected yield is 5.90%. DO YOU UNDERSTAND? (Pg. 132) 1. Consider a 4-year bond selling at par with a 7% annual coupon. Suppose that yields on similar bonds increase by 50 basis points. Use duration (Equation 5.8) to estimate the percent change in the bond price. Check your answer by calculating the new bond price. 3 P1 = 1,000 P2 = 70/(1.075) + 70/(1.075)2 + 70/(1.075)3 + 1,070/(1.075)4 = $983.26 % ∆P = [(P2 - P1)/P1] * 100 = [(983.26 - 1,000)/1,000] * 100 = -1.67% i1 = 7% or 0.07 and i2 = 7.5% or 0.075 → ∆i = i2 - i1 = 0.075 - 0.07 = 0.005 D1 = [70 (1)/(1.07) + 70 (2)/(1.07)2 + 70 (3)/(1.07)3 + 1,070 (4)/(1.07)4]/1,000 = 3,624/1,000 = 3.624 years % ∆P = -D1 [∆i/(1 + i1)] * 100 = -3.56 (0.005/1.07) * 100 = -1.66% 2. Define “price risk” and “reinvestment risk.” Explain how the two risks offset each other. Price risk is the variability of return caused by changes in the market price of the bond, while reinvestment risk is the variability in bond return caused by varying reinvestment rates. At the duration point these two risks offset each other. If the bond is held to its duration, the investor will yield the expected YTM. If interest rates were to increase during the holding period, price risk would lower the capital gain on the bond, but rising rates would increase the return from reinvestment of coupons. If interest rates were to fall during the holding period, price risk would increase the capital gain on the bond, but falling rates would reduce the return from reinvestment of coupons. 3. What is the duration of a bond portfolio made up of two bonds: 37% of a bond with duration of 7.7 years and 63% of a bond with duration of 16.4 years? The weighted duration of the portfolio = 7.7 (0.37) + 16.4 (0.63) = 13.2 years 4. How can duration be used as a way to rank bonds on their interest rate risk? Duration is a measure of price variability, given a change in interest rates. The price risk varies directly with the duration of the bond. 5. To eliminate interest rate risk, should you match the maturity or the duration of your bond investment to your holding period? Explain. Duration is a useful measure of interest rate risk because there is a direct relation between bond price volatility and duration. Duration matching is a technique that eliminates interest rate risk. Duration matching matches the duration of the bond to the investor’s holding period. When a bond’s duration matches the holding period, the bond’s price risk directly offsets the bond’s reinvestment rate risk, thus eliminating interest rate risk.