Chapter 5: ANSWERS TO "DO YOU UNDERSTAND" TEXT QUESTIONS
DO YOU UNDERSTAND?
1. Why is a dollar today worth more to most people than a dollar received at a future date?
Solution: There are two ways to view this issue. First, people generally have a positive
time preference for consumption; that is, holding the amount of money constant, 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 percent interest compounded
quarterly, how much money would be in the account after three years?
Solution: Use FV = PV(1 + i)n = $100(1.06)3 = $100(1.191) = $119.10.
3. You have just won $60 million in the Powerball lottery! The lottery promises to pay you
$20 million at the end of each year for the next three years. If the market rate of interest is
7 percent, how much money would you accept today in exchange for the three $20 million
payments?
Solution: The bank will only lend you the present value of the lottery winnings. 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?
1. When a bond’s coupon rate is less than the prevailing market rate on interest on similar
bonds, will the bond sell at par, a discount, or a premium? Explain.
Solution: 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 rate of interest 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 alternatives in the
market.
2. Under what conditions will the realized yield on a bond equal the promised yield?
Solution: If the investor receives all cash flows as promised by the issuer and reinvests
them at the bond’s promised yield (the yield to maturity at the time of its purchase), the
realized yield will equal the promised yield. If the issuer defaults on payments of coupon
interest or principal, if the investor consumes the coupon payments rather than reinvesting
them, or if the market interest rate at which the coupon payments can be reinvested
deviates from the promised yield, the realized yield will not necessarily equal the promised
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 percent (annual payments). The bond
currently sells at 98.5 percent of par value.
Solution: 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 percent.
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.
Solution: The current price is $985. The investor expects to sell it after two years for $940
= 0.94($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:
2
985  
t 1
$80
$940

t
(1  i ) (1  i ) 2
The expected yield is 5.90 percent.
DO YOU UNDERSTAND?
1. Consider a four-year bond selling at par with a 7 percent 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.
Solution: 4 year, 7% annual coupon bond selling at par
Duration = 3624.32/1000 = 3.62
Convexity = 15293.33/1000 = 15.29
market rate increase of 50 basis points
2
% change in bond value = -3.62[.005/1.07] + (½)(15.29)(.005 ) = -.0167
The new bond price is $983.25 so the actual change in value is -.0168.
2. Define price risk and reinvestment risk. Explain how the two risks offset each other.
Solution: 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 cause the bond price to fall, but the
reinvestment rate on coupon would increase. 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 coupon.
3. What is the duration of a bond portfolio made up of two bonds: 37 percent of a bond
with duration of 7.7 years and 63 percent of a bond with duration of 16.4 years?
Solution: Estimate the weighted duration of the portfolio or:
7.7(.37) + 16.4(.63) = 13.2 years
4. How can duration be used as a way to rank bonds on their interest rate risk?
Solution: 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.
Solution: 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.