Chapter 3
What Do Interest Rates Mean
and What Is Their Role in Valuation?
Measuring Interest Rates
Present Value
Four Types of Credit Market Instruments
Yield to Maturity
The Distinction Between Real and Nominal Interest Rates
Global Box: Negative T-Bill Rates? It Can Happen
The Distinction Between Interest Rates and Returns
Mini-Case Box: With TIPS, Real Interest Rates Have Become Observable in the United States
Maturity and the Volatility of Bond Returns: Interest-Rate Risk
Reinvestment Risk
Summary
Mini-Case Box: Helping Investors to Select Desired Interest-Rate Risk
The Practicing Manager: Calculating Duration to Measure Interest-Rate Risk
Calculating Duration
Duration and Interest-Rate Risk
◼ Overview and Teaching Tips
In my years of teaching financial markets and institutions, I have found that students have trouble with
what I consider to be easy material because they do not understand what an interest rate is—that it is
negatively associated with the price of a bond, that it differs from the return on a bond, and that there is
an important distinction between real and nominal interest rates.
This chapter spends more time on these issues than does any other competing textbook. My experience
has been that giving this material so much attention is well rewarded. After putting more emphasis on this
material in my financial markets and institutions course, I witnessed a dramatic improvement in students’
understanding of portfolio choice and asset and liability management in financial institutions.
An innovative feature of the textbook is the set of over twenty special applications called, “The Practicing
Manager.” These applications introduce students to real-world problems that managers of financial
institutions have to solve and make the course both more relevant and exciting to students. They are not
meant to fully prepare students for jobs in financial institutions—it is up to more specialized courses such
as bank or financial institutions management to do this—but these applications teach them some of the
special analytical tools that they will need when they enter the business world.
10 Mishkin/Eakins • Financial Markets and Institutions, Eighth Edition
This chapter contains the Practicing Manager application on “Calculating Duration to Measure Interest–
Rate Risk.” The application shows how to quantify interest-rate risk using the duration concept and is
a basic tool for managers of financial institutions. For those instructors who do not want a managerial
emphasis in their financial markets and institutions course, this and other Practicing Manager applications
can be skipped without loss of continuity.
◼ Answers to End-of-Chapter Questions
2. 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.
4. People are more likely to buy houses because the real interest rate when purchasing a house has fallen
◼ Quantitative Problems
1. Calculate the present value of a $1,000 zero-coupon bond with 5 years to maturity if the required
annual interest rate is 6%.
2. A lottery claims its grand prize is $10 million, payable over 20 years at $500,000 per year. If the first
payment is made immediately, what is this grand prize really worth? Use a discount rate of 6%.
3. Consider a bond with a 7% annual coupon and a face value of $1,000. Complete the following table:
Years to Maturity
Discount Rate
Current Price
3
5
3
7
6
7
9
7
9
9
Chapter 3: What Do Interest Rates Mean and What Is Their Role in Valuation? 11
What relationship do you observe between yield to maturity and the current market value?
Solution:
Years to Maturity
Yield to Maturity
Current Price
3
5
$1,054.46
3
7
$1,000.00
6
7
$1,000.00
9
5
$1,142.16
9
9
$ 880.10
When yield to maturity is above the coupon rate, the band’s current price is below its face
value. The opposite holds true when yield to maturity is below the coupon rate. For a given
maturity, the bond’s current price falls as yield to maturity rises. For a given yield to
maturity, a bond’s value rises as its maturity increases. When yield to maturity equals the
coupon rate, a bond’s current price equals its face value regardless of years to maturity.
4. Consider a coupon bond that has a $1,000 par value and a coupon rate of 10%. The bond is currently
selling for $1,150 and has 8 years to maturity. What is the bond’s yield to maturity?
5. You are willing to pay $15,625 now to purchase a perpetuity which will pay you and your heirs
$1,250 each year, forever, starting at the end of this year. If your required rate of return does not
change, how much would you be willing to pay if this were a 20-year, annual payment, ordinary
annuity instead of a perpetuity?
7. Property taxes in DeKalb County are roughly 2.66% of the purchase price every year. If you just
bought a $100,000 home, what is the PV of all the future property tax payments? Assume that the
house remains worth $100,000 forever, property tax rates never change, and that a 9% discount rate
is used for discounting.
8. Assume you just deposited $1,000 into a bank account. The current real interest rate is 2% and
inflation is expected to be 6% over the next year. What nominal interest rate would you require from
the bank over the next year? How much money will you have at the end of one year? If you are
saving to buy a stereo that currently sells for $1,050, will you have enough to buy it?
12 Mishkin/Eakins • Financial Markets and Institutions, Eighth Edition
Solution: The required nominal rate would be:
At this rate, you would expect to have $1,000 1.08, or $1,080 at the end of the year.
Can you afford the stereo? In theory, the price of the stereo will increase with the rate of
inflation. So, one year later, the stereo will cost $1,050 1.06, or $1,113. You will be
short by $33.
9. A 10-year, 7% coupon bond with a face value of $1,000 is currently selling for $871.65. Compute
your rate of return if you sell the bond next year for $880.10.
Solution:
170 880.10 871.65 0.09, or 9%.
871.65
tt
t
C P P
RP
+
+− +−
= = =
10. You have paid $980.30 for an 8% coupon bond with a face value of $1,000 that mature in five years.
You plan on holding the bond for one year. If you want to earn a 9% rate of return on this investment,
what price must you sell the bond for? Is this realistic?
Solution: To find the price, solve:
1
11
80 980.30 0.09 for . 988.53.
980.30
t
tt
PPP
+
++
+− ==
Although this appears possible, the yield to maturity when you purchased the bond was
8.5%. At that yield, you only expect the price to be $983.62 next year. In fact, the yield
would have to drop to 8.35% for the price to be $988.53.
11. Calculate the duration of a $1,000 6% coupon bond with three years to maturity. Assume that all
market interest rates are 7%.
Solution:
Year
1
2
3
Sum
Payments
60.00
60.00
1060.00
PV of Payments
56.07
52.41
865.28
973.76
Time Weighted PV of Payments
56.07
104.81
2595.83
Time Weighted PV of Payments
Divided by Price
0.06
0.11
2.67
2.83
This bond has a duration of 2.83 years. Note that the current price of the bond is $973.76,
which is the sum of the individual “PV of payments.”
12. Consider the bond in the previous question. Calculate the expected price change if interest rates
drop to 6.75% using the duration approximation. Calculate the actual price change using discounted
cash flow.
Solution: Using the duration approximation, the price change would be:
0.0025
DUR 2.83 973.76 6.44.
1 1.07
i
PP
i
−
= − = − =
+
Chapter 3: What Do Interest Rates Mean and What Is Their Role in Valuation? 13
Copyright © 2015 Pearson Education, Inc.
The new price would be $980.20. Using a discounted cash flow approach, the price is
980.23—only $.03 different.
Year
1
2
3
Sum
Payments
60.00
60.00
1060.00
PV of payments
56.21
52.65
871.3
980.23
13. The duration of a $100 million portfolio is 10 years. $40 million dollars in new securities are added to
the portfolio, increasing the duration of the portfolio to 12.5 years. What is the duration of the
$40 million in new securities?
Solution: First, note that the portfolio now has $140 million in it. The duration of a portfolio is the
14. A bank has two, 3-year commercial loans with a present value of $70 million. The first is a $30 million
loan that requires a single payment of $37.8 million in 3 years, with no other payments until then.
The second is for $40 million. It requires an annual interest payment of $3.6 million. The principal of
$40 million is due in 3 years.
a. What is the duration of the bank’s commercial loan portfolio?
b. What will happen to the value of its portfolio if the general level of interest rates increased from
8% to 8.5%?
Solution: The duration of the first loan is 3 years since it is a zero-coupon loan. The duration of the
second loan is as follows:
Year
1
2
3
Sum
Payment
3.60
3.60
43.60
PV of Payments
3.33
3.09
34.61
41.03
Time Weighted PV of Payments
3.33
6.18
103.83
Time Weighted PV of Payments
Divided by Price
0.08
0.15
2.53
2.76
The duration of a portfolio is the weighted average duration of its individual securities.
1 1.08
i
+
14 Mishkin/Eakins • Financial Markets and Institutions, Eighth Edition
15. Consider a bond that promises the following cash flows. The required discount rate is 12%.
Year
0
1
2
3
4
Promised Payments
160
170
180
230
You plan to buy this bond, hold it for 2½ years, and then sell the bond.
a. What total cash will you receive from the bond after the 2½ years? Assume that periodic cash
flows are reinvested at 12%.
b. If immediately after buying this bond, all market interest rates drop to 11% (including your
reinvestment rate), what will be the impact on your total cash flow after 2½ years? How does
this compare to part (a)?
c. Assuming all market interest rates are 12%, what is the duration of this bond?
Solution: