Mishkin • Instructor’s Manual for The Economics of Money, Banking, and Financial Markets, Twelfth Edition 73
Chapter 4
ANSWERS TO QUESTIONS
1. Would a dollar tomorrow be worth more to you today when the interest rate is 20% or when
it is 10%?
It would be worth 1/(1 + 0.20) = $0.83 when the interest rate is 20%, rather than 1/(1 + 0.10)
2. Explain which information you would need to take into consideration when deciding to
receive $5,000 today or $5,500 one year from today.
When comparing amounts of money that are disbursed at different dates, one has to take into
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?
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
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
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
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
rates were above the expected inflation rate and real rates became positive.
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
ANSWERS TO APPLIED PROBLEMS
14. If the interest rate is 10%, what is the present value of a security that pays you $1,100 next
year, $1,210 the year after, and $1,331 the year after that?
15. Calculate the present value of a $1,000 discount bond with five years to maturity if the yield
to maturity is 6%
16. A lottery claims its grand prize is $10 million, payable over 5 years at $2,000,000 per year. If
the first payment is made immediately, what is this grand prize really worth? Use an interest
rate of 6%.
17. Suppose that a commercial bank wants to buy Treasury bills. These instruments pay $5,000 in
one year and are currently selling for $5,012. What is the yield to maturity of these bonds? Is
this a typical situation? Why?
The yield to maturity of these bonds solves the following equation: 5,000/(1+i) = 5,012.
18. What is the yield to maturity on a simple loan for $1 million that requires a repayment of $2
million in five years’ time?
14.9%, derived as follows: The present value of the $2 million payment five years from now
19. Which $1,000 bond has the higher yield to maturity, a twenty-year bond selling for $800 with
a current yield of 15% or a one-year bond selling for $800 with a current yield of 5%?
If the one-year bond did not have a coupon payment, its yield to maturity would be ($1,000 –
$800)/ $800 = $200/$800 = 0.25, or 25%. Because it does have a coupon payment, its yield
20. Consider a bond with a 4% 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 2%
Mishkin • Instructor’s Manual for The Economics of Money, Banking, and Financial Markets, Twelfth Edition 77
Years to Maturity Yield to Maturity Current Price
2 2% 1038.83
2 4% 1000.00
21. Consider a coupon bond that has a $1,000 par value and a coupon rate of 10%. The bond is
currently selling for $1,044.89 and has two years to maturity. What is the bond’s yield to
maturity?
When yield to maturity is above the coupon rate, the bond’s current price is below its face
22. What is the price of a perpetuity that has a coupon of $50 per year and a yield to maturity of
2.5%? If the yield to maturity doubles, what will happen to the perpetuity’s price?
23. Property taxes in a particular district are 4% of the purchase price of a home every year. If
you just purchased a $250,000 home, what is the present value of all the future property tax
payments? Assume that the house remains worth $250,000 forever, property tax rates never
change, and a 6% interest rate is used for discounting.
24. A $1000-face-value bond has a 10% coupon rate, its current price is $960, and its price is
expected to increase to $980 next year. Calculate the current yield, the expected rate of
capital gain, and the expected rate of return.
25. Suppose that you want to take out a loan and that your local bank wants to charge you an
annual real interest rate equal to 3%. Assuming that the annualized expected rate of inflation
over the life of the bond is 1%, determine the nominal interest rate that the bank will charge
you. What happens if, over the life of the loan, actual inflation is 0.5%?
The bank will charge you a nominal interest rate equal to 1% + 3% = 4%. However, if actual
Mishkin • Instructor’s Manual for The Economics of Money, Banking, and Financial Markets, Twelfth Edition 78
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.
x 5 year U.S. treasury (DGS5) and 5-year TIIS (DFII5)
x 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
b. Using the most recent data available, calculate the difference between the yields for each
of the pairs of bonds (DGS5 – DFII5, etc.) listed above. What does this difference
represent?
Mishkin • Instructor’s Manual for The Economics of Money, Banking, and Financial Markets, Twelfth Edition 79
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
Difference (Inflation
Expectations)
5 Year 1.85 0.23 1.62
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.
The difference, which roughly represents inflation expectations, is fairly constant over