Chapter 4
ANSWERS TO QUESTIONS
1. Would $200, which is to be received in exactly one year, be worth more to you today when
the interest rate is 12% or when it is 17%?
2. What is the formula used to calculate the yield to maturity on a 20-year coupon bond with a
current yield of 12% and $1,000 face value that sells for $2,500?
$2,500 = $120/(1 + i) + C/(1 + i)2 + · · · + C/(1 + i)20 + $1,000/(1 + i)20. Solving for i gives
3. To help pay for college, you have just taken out a $1,000 government loan that makes you
pay $126 per year for 25 years. However, you don’t have to start making these payments
until you graduate from college two years from now. Why is the yield to maturity necessarily
less than 12%? (This is the yield to maturity on a normal $1,000 fixed-payment loan on
which you pay $126 per year for 25 years.)
If the interest rate were 12%, the present discounted value of the payments on the
4. Do bondholders fare better when the yield to maturity increases or when it decreases? Why?
When the yield to maturity increases, this represents a decrease in the price of the bond. If
5. Suppose today you buy a coupon bond that you plan to sell one year later. Which part of the
rate of return formulation incorporates future changes into the bond?
Note: Check Equations 7 and 8 in this chapter.
The rate of capital gain is the part of the rate of return formula that incorporates future
changes in the price of the bond. The other part of the formula, the current yield, is composed
6. If mortgage rates rise from 5% to 10%, but the expected rate of increase in housing prices
rises from 2% to 9%, are people more or less likely to buy houses?
People are more likely to buy houses because the real interest rate when purchasing a house
7. When is the current yield a good approximation of the yield to maturity?
The current yield will be a good approximation to the yield to maturity whenever the bond
8. Why would a government choose to issue a perpetuity, which requires payments forever,
instead of a terminal loan, such as a fixed-payment loan, discount bond, or coupon bond?
The near-term costs to maintaining a given size loan are much smaller for a perpetuity than for
a similar fixed payment loan, discount, or coupon bond. For instance, assuming a 5% interest
rate over 10 years, on a $1000 loan, a perpetuity costs $50 a year (or $500 in payments over
9. Under what conditions will a discount bond have a negative nominal interest rate? Is it
possible for a coupon bond or a perpetuity to have a negative nominal interest rate?
Whenever the current price P is greater than face value F of a discount bond, the yield to
maturity will be negative. It is possible for a coupon bond to have a negative nominal interest
10. True or False: With a discount bond, the return on the bond is equal to the rate of capital
gain.
True. The return on a bond is the current yield iC plus the rate of capital gain, g. A discount
11. If interest rates decline, which would you rather be holding, long-term bonds or short-term
bonds? Why? Which type of bond has the greater interest-rate risk?
You would rather be holding long-term bonds because their price would increase more than
the price of the short-term bonds, giving them a higher return. Longer-term bonds are more
12. Interest rates were lower in the mid-1980s than in the late 1970s, yet many economists have
commented that real interest rates were actually much higher in the mid-1980s than in the
late 1970s. Does this make sense? Do you think that these economists are right?
The economists are right. They reason that nominal interest rates were below expected rates
of inflation in the late 1970s, making real interest rates negative. The expected inflation rate,
13. Retired persons often have much of their wealth placed in savings accounts and other
interest-bearing investments, and complain whenever interest rates are low. Do they have a
valid complaint?
While it would appear to them that their wealth is declining as nominal interest rates fall, as
long as expected inflation falls at the same rate as nominal interest rates, their real return on
ANSWERS TO APPLIED PROBLEMS
14. If the interest rate is 15%, what is the present value of a security that pays you $1,100 next
year, $1,250 the year after, and $1,347 the year after that?
15. Calculate the present value of a $1,300 discount bond with seven years to maturity if the
yield to maturity is 8%.
PV = FV/(1 + i)n, where FV = 1300, i = 0.08, n = 7. Thus, PV = 1300/(1 + 0.08)7 = 758.54.
17. What is the yield to maturity on a $10,000-face-value discount bond, maturing in one year,
which sells for $9,523.81?
18. What is the yield to maturity on a simple loan for $1,500 that requires a repayment of
$15,000 in five years?
58.5%, derived as follows: The present value of the $15,000 payment five years from now is
19. Which $10,000 bond has the higher yield to maturity, a twenty-year bond selling for $8,000
with a current yield of 20% or a one-year bond selling for $8,000 with a current yield of
10%?
If the one-year bond did not have a coupon payment, its yield to maturity would be ($10,000
– $8000)/ $8,000 = $2,000/$8,000 = 0.25, or 25%. Because it does have a coupon payment,
its yield to maturity must be greater than 25%. However, because the current yield is a good
20. Consider a bond with a 6% annual coupon and a face value of $1,000. Complete the
following table. What relationships do you observe between years to maturity, yield to
maturity, and the current price?
Years to Maturity
Yield to Maturity
Current Price
2
4%
2
6%
3
6%
5
4%
5
8%
Years to Maturity
Yield to Maturity
Current Price
2
4%
1037.72
2
6%
1000.00
3
6%
1000.00
5
8%
When yield to maturity is above the coupon rate, the bond’s current price is below its face
21. Consider a coupon bond that has a $900 par value and a coupon rate of 6%. The bond is
currently selling for $860.15 and has two years to maturity. What is the bond’s yield to
maturity?
$860.15 = $54/(1 + i) + $54/(1 + i)2 + $900/(1 + i)2. Solving for i gives a yield to maturity of
0.085, or 8.5%.
22. What is the price of a perpetuity that has a coupon of $70 per year and a yield to maturity of
1.5%? If the yield to maturity doubles, what will happen to the perpetuity’s price?
23. Property taxes in a particular district are 2% of the purchase price of a home every year. If
you just purchased a $150,000 home, what is the present value of all the future property tax
payments? Assume that the house remains worth $150,000 forever, property tax rates never
change, and a 4% interest rate is used for discounting.
The taxes on the $150,000 home are $150,000 × 0.02 = $3,000 per year. The PV of all future
payments = $3,000/0.04 = $75,000 (a perpetuity).
24. A $1,100-face-value bond has a 5% coupon rate, its current price is $1,040, and it is
expected to increase to $1070 next year. Calculate the current yield, the expected rate of
capital gains, and the expected rate of return.
25. Assume you just deposited $1,250 into a bank account. The current real interest rate is 1%,
and the expected rate of inflation over the next year is 5%. What nominal interest rate should
the bank charge you over the next year? How much money will you have at the end of one
year? If you are saving to buy a motorbike that currently sells for $1,300, will you have
enough money to buy it?
The bank will charge you a nominal interest rate equal to, i = rr + πe = 1% + 5% = 6%. At
ANSWERS TO DATA ANALYSIS PROBLEMS
1. Go to the St. Louis Federal Reserve FRED database and find data on the interest rate on a
four-year auto loan (TERMCBAUTO48NS). Assume that you borrow $20,000 to purchase a
new automobile and that you finance it with a four-year loan at the most recent interest rate
given in the database. If you make one payment per year for four years, what will the yearly
payment be? What is the total amount that will be paid out on the $20,000 loan?
For February 2017, the rate on a four-year auto loan is 4.52%. Thus, the yearly payment can
2. The U.S. Treasury issues some bonds as Treasury Inflation Indexed Securities, or TIIS, which
are bonds adjusted for inflation; hence the yields can be roughly interpreted as real interest
rates. Go to the St. Louis Federal Reserve FRED database and find data on the following
TIIS bonds and their nominal counterparts. Then answer the questions below.
• 5 year U.S. treasury (DGS5) and 5-year TIIS (DFII5)
• 7 year U.S. treasury (DGS7) and 7-year TIIS (DFII7)
a. Following the Great Recession of 2008–2009, the 5, 7, 10, and even the 20-year TIIS
yields became negative for a period of time. How is this possible?
Market participants expected the inflation rate on average to be higher than the nominal
interest rate over that same period; thus, the real interest rate, as represented by the
difference between the treasury and TIIS yields, was negative.
The difference between the pairs represents expected inflation over the relevant bond
horizon. Calculations for June 29, 2017, are shown below.
29-Jun-17
Nominal
TIIS
5 Year
1.85
0.23
7 Year
2.10
0.43
10 Year
2.27
0.55
1.72
20 Year
2.59
0.81
1.78
30 Year
2.82
0.96
1.86
c. Based on your answer to part (b), are there significant variations among the differences
in the bond-pair yields? Interpret the magnitude of the variation in differences among the
pairs.