THE PRACTICING MANAGER
As was discussed in Chapter 4, interest-rate forecasts are extremely important to managers of financial institutions because future changes in interest rates have a significant impact on the profitability of their institutions. Furthermore, interest-rate
M05_MISH5006_09_GE_C05.indd 150 13/10/17 1:06 PM
www.rasabourse.com
forecasts are needed when managers of financial institutions have to set interest rates on loans that are promised to customers in the future. Our discussion of the term structure of interest rates has indicated that the slope of the yield curve pro- vides general information about the market’s prediction of the future path of interest rates. For example, a steeply upward-sloping yield curve indicates that short-term interest rates are predicted to rise in the future, and a downward-sloping yield curve indicates that short-term interest rates are predicted to fall. However, a financial institution manager needs much more specific information on interest-rate forecasts than this. Here we show how the manager of a financial institution can generate spe- cific forecasts of interest rates using the term structure.
To see how this is done, let’s start the analysis using the approach we took in developing the pure expectations theory. Recall that because bonds of differ- ent maturities are perfect substitutes, we assumed that the expected return over two periods from investing $1 in a two-period bond, which is (1 + i2t)(1 + i2t) - 1, must equal the expected return from investing $1 in one-period bonds, which is (1 + it)(1 + iet+1) - 1. This is shown graphically as follows:
Today
0 1 1 it
Year 1
Year 1 1 i et 1 1 2
(1 1 i2t ) (1 1 i2t )
In other words,
(1 + it)(1 + iet+1) - 1 = (1 + i2t)(1 + i2t) - 1 Through some tedious algebra we can solve for iet+1:
iet+1 = (1 + i2t)2
1 + it - 1 (4)
This measure of iet+1 is called the forward rate because it is the one-period interest rate that the pure expectations theory of the term structure indicates is expected to prevail one period in the future. To differentiate forward rates derived from the term structure from actual interest rates that are observed at time t, we call these observed interest rates spot rates.
Going back to Example 3, which we used to discuss the pure expectations theo- ry earlier in this chapter, at time t the one-year interest rate is 5% and the two-year rate is 5.5%. Plugging these numbers into Equation 4 yields the following estimate of the forward rate one period in the future:
iet+1 = (1 + 0.055)2
1 + 0.05 - 1 = 0.06 = 6%
Not surprisingly, this 6% forward rate is identical to the expected one-year in- terest rate one year in the future that we used in Example 3. This is exactly what we should find, as our calculation here is just another way of looking at the pure expectations theory.
We can also compare holding the three-year bond against holding a sequence of one-year bonds, which reveals the following relationship:
(1 + it)(1 + iet+1)(1 + iet+2) - 1 = (1 + i3t)(1 + i3t)(1 + i3t) - 1
M05_MISH5006_09_GE_C05.indd 151 13/10/17 1:06 PM
www.rasabourse.com
and plugging in the estimate for iet+1 derived in Equation 4, we can solve for iet+2: iet+2 = (1 + i3t)3
(1 + i2t)2 - 1
Continuing with these calculations, we obtain the general solution for the forward rate n periods into the future:
iet+n = (1 + in+1t)n+1
(1 + int)n - 1 (5) Our discussion indicated that the pure expectations theory is not entirely satisfacto- ry because investors must be compensated with liquidity premiums to induce them to hold longer-term bonds. Hence we need to modify our analysis, as we did when discussing the liquidity premium theory, by allowing for these liquidity premiums in estimating predictions of future interest rates.
Recall from the discussion of those theories that because investors prefer to hold short-term rather than long-term bonds, the n-period interest rate differs from that indicated by the pure expectations theory by a liquidity premium of lnt. So to allow for liquidity premiums, we need merely subtract lnt from int in our formula to derive iet+n:
iet+n = (1 + in+1t - ln+1t)n+1
(1 + int - lnt)n - 1 (6) This measure of iet+n is referred to, naturally enough, as the adjusted forward-rate forecast.
In the case of iet+1, Equation 6 produces the following estimate:
iet+1 = (1 + i2t - l2t)2 1 + it - 1
Using Example 4 in our discussion of the liquidity premium theory, at time t the l2t liquidity premium is 0.25%, l1t= 0, the one-year interest rate is 5%, and the two-year interest rate is 5.75%. Plugging these numbers into our equation yields the following adjusted forward-rate forecast for one period in the future:
iet+1 = (1 + 0.0575 - 0.0025)2
1 + 0.05 - 1 = 0.06 = 6%
which is the same as the expected interest rate used in Example 3, as it should be.
Our analysis of the term structure thus provides managers of financial institu- tions with a fairly straightforward procedure for producing interest-rate forecasts.
First they need to estimate lnt, the values of the liquidity premiums for various n.
Then they need merely apply the formula in Equation 6 to derive the market’s fore- casts of future interest rates.
A customer asks a bank if it would be willing to commit to making the customer a one-year loan at an interest rate of 8% one year from now. To compensate for the costs of making the loan, the bank needs to charge one percentage point more than
Forward Rate
EXAMPLE 5.5
M05_MISH5006_09_GE_C05.indd 152 13/10/17 1:06 PM
www.rasabourse.com
the expected interest rate on a Treasury bond with the same maturity if it is to make a profit. If the bank manager estimates the liquidity premium to be 0.4%, and the one-year Treasury bond rate is 6% and the two-year bond rate is 7%, should the manager be willing to make the commitment?
Solution
The bank manager is unwilling to make the loan because at an interest rate of 8%, the loan is likely to be unprofitable to the bank.
iet+n = (1 + in+1t - ln+1t)n+1 (1 + int - lnt)n - 1 where
in+1t = two-year bond rate = 0.07 ln+1t = liquidity premium = 0.004 int = one-year bond rate = 0.06 l1t = liquidity premium = 0 n = number of years = 1 Thus,
iet+1 = (1 + 0.07 - 0.004)2
1 + 0.06 - 1 = 0.072 = 7.2%
The market’s forecast of the one-year Treasury bond rate one year in the future is therefore 7.2%. Adding the 1% necessary to make a profit on the one-year loan means that the loan is expected to be profitable only if it has an interest rate of 8.2% or higher.
As we will see in Chapter 6, the bond market’s forecasts of interest rates may be the most accurate ones possible. If this is the case, the estimates of the market’s forecasts of future interest rates using the simple procedure outlined here may be the best interest-rate forecasts that a financial institution manager can obtain.
S U M M A R Y
1. Bonds with the same maturity will have different interest rates because of three factors: default risk, liquidity, and tax considerations. The greater a bond’s default risk, the higher its interest rate relative to other bonds; the greater a bond’s liquidity, the lower its interest rate; and bonds with tax-exempt status will have lower interest rates than they otherwise would. The relationship among interest rates on bonds with the same maturity that arise because of these three factors is known as the risk structure of interest rates.
2. Several theories of the term structure provide expla- nations of how interest rates on bonds with differ- ent terms to maturity are related. The expectations theory views long-term interest rates as equaling the average of future short-term interest rates expected to occur over the life of the bond. By contrast, the market segmentation theory treats the determina- tion of interest rates for each bond’s maturity as the outcome of supply and demand in that market only.
Neither of these theories by itself can explain the fact that interest rates on bonds of different maturities
M05_MISH5006_09_GE_C05.indd 153 13/10/17 1:06 PM
www.rasabourse.com
move together over time and that yield curves usually slope upward.
3. The liquidity premium theory combines the features of the other two theories and, by so doing, is able to explain the facts just mentioned. It views long- term interest rates as equaling the average of future short-term interest rates expected to occur over the life of the bond plus a liquidity premium. This theory allows us to infer the market’s expectations
about the movement of future short-term inter- est rates from the yield curve. A steeply upward- sloping curve indicates that future short-term rates are expected to rise; a mildly upward-sloping curve that short-term rates are expected to stay the same;
a flat curve that short-term rates are expected to decline slightly; and an inverted yield curve that a substantial decline in short-term rates is expected in the future.
credit-rating agencies, p. 132 default, p. 130
default-free bonds, p. 131 expectations theory, p. 139 forward rate, p. 151
inverted yield curve, p. 137 junk bonds, p. 133
liquidity premium theory, p. 144 market segmentation theory, p. 143 preferred habitat theory, p. 144
risk premium, p. 131
risk structure of interest rates, p. 129 spot rate, p. 151
term structure of interest rates, p. 129 yield curve, p. 137
K E Y T E R M S
Q U E S T I O N S
1. What is the relationship between a corporate bond rating and a risk premium?
2. What is default risk and risk premium? How can default risk influence interest rates?
3. “Corporate bonds and stocks are a bad combination of investments as both have different characteristics that do not complement each other.” Discuss.
4. Describe the relationship between bond prices and interest rates during a recession.
5. Just before the collapse of the subprime mortgage market in 2007, the most important credit-rating agencies rated mortgage-backed securities with Aaa and AAA ratings. Explain how it was possible that a few months into 2008, the same securities had the lowest possible ratings. Should we always trust credit-rating agencies?
6. If a yield curve looks like the one shown here, what is the market predicting about the movement of future short-term interest rates? What might the yield curve indicate about the market’s predictions concerning the inflation rate in the future?
Yield to Maturity
Term to Maturity
7. If a yield curve looks like the one below, what is the market predicting about the movement of future short-term interest rates? What might the yield curve indicate about the market’s predictions concerning the inflation rate in the future?
Yield to Maturity
Term to Maturity
M05_MISH5006_09_GE_C05.indd 154 13/10/17 1:06 PM
www.rasabourse.com
8. How will a reduction in tax rates affect an individual’s prefernce for municipal bonds compared to Treasury bonds? What conclusion can you draw regarding tax rates and interest rates on securities?
Predicting the Future
9. Predict what will happen to interest rates on corpo- rate bonds if governments adopt a new regulation that guarantees payment of principal in case of default in order to encourage investment in corporate bonds.
Will it affect the demand for Treasury securities?
10. If the current interest rate on one-year bonds is 6%, you may expect an increase in interest rates by 1% in the following year. Predict what interest rate would be suitable now for two-year bonds. Explain your answer and state the theory that justifies it.
11. If the income tax exemption on municipal bonds were abolished, what would happen to the interest rates on these bonds? What effect would it have on interest rates on U.S. Treasury securities?
Q U A N T I T A T I V E P R O B L E M S
1. Assuming that the expectations theory is the correct one of the term structure, calculate the interest rates in the term structure for maturities one to six years:
a. 4%, 4%, 5%, 6%, 6%, 6%
b. 5%, 5%, 4%, 4%, 4%, 4%
Explain what is happening to yield curve.
2. Refer to the previous problem. Assume that instead of the expectations theory, the liquidity premium theory takes place. What will be your answer to parts a and b, if the following liquidity premiums are expected? 0%; 0.25%, 0.5%, 0.75%, 1%, and 1.25%
respectively?
3. How does the after-tax yield on a $1,000,000 munici- pal bond with a coupon rate of 8% paying interest annually compare with that of a $1,000,000 corporate bond with a coupon rate of 10% paying interest annu- ally? Assume that you are in the 25% tax bracket.
4. Consider the decision to purchase either a five-year corporate bond or a five-year municipal bond.
The corporate bond is a 14% annual coupon bond with a par value of $1,000. It is currently yielding 12%. The municipal bond has a 10% annual coupon and a par value of $1,000. It is currently yielding 8%. Which of the two bonds would be more ben- eficial to you? Assume that your marginal tax rate is 25%.
5. Debt issued by Southwest Airways currently yields 24%. A municipal bond of equal risk currently yields 16%. At what marginal tax rate would an investor be indifferent between these two bonds?
6. One-year T-bill rates are expected to steadily increase by 250 basis points per year over the next nine years.
Determine the required interest rate on a five-year T-bond and a nine-year T-bond if the current one-year
interest rate is 15.5%. Assume that the expectations hypothesis for interest rates holds.
7. The one-year interest rate over the next eight years will be 4%, 5.5%, 6%, 8.5%, 10%, 11.5%, 14%, and 15.5%. Using the expectations theory, what will be the interest rates on a four-year bond, a six-year bond, and an eight-year bond?
8. The one-year interest rate over the next 10 years will be 3%, 4.5%, 6%, 7.5%, 9%, 10.5%, 13%, 14.5%, 16%, and 17.5%. Assume that investors prefer hold- ing short-term bonds so that a liquidity premium of 10 basis points is required for each year of a bond’s maturity. What will be the interest rates on a three- year bond, a six-year bond, and a nine-year bond?
9. Suppose that the expectations theory is true and that you can buy a three-year bond with an interest rate of 6% or three consecutive one-year bonds with interest rates of 4%, 5%, and 6%. Which option would you choose to undertake?
10. Suppose you are asked to make a loan for 10% from one year for now; you decided to compare interest rates with Government bonds and make at least 2%
premium over that. One year bond has 5% and two year rate 7%. You have been harmed about liquidity risk, so you evaluated it at 0.5%. Given this data, are you willing to make a loan?
11. One-year T-bill rates are 2% currently. If interest rates are expected to go up after three years by 2%
every year, what should be the required interest rate on a 10-year bond issued today? Assume that the expectations theory holds.
12. One-year T-bill rates over the next five years are expected to be 4%, 5%, 6%, 6.5%, and 8%. If five- year T-bonds are yielding 35.5%, what is the liquidity premium on this bond?
M05_MISH5006_09_GE_C05.indd 155 28/10/17 10:40 AM
www.rasabourse.com
13. At your favorite bond store, Bonds-R-Us, you see the following prices:
One-year $100 zero selling for $90.19
Three-year 10% coupon $1,000 par bond selling for $1,000
Two-year 10% coupon $1,000 par bond selling for
$1,000
Assume that the expectations theory for the term structure of interest rates holds, no liquidity premium exists, and the bonds are equally risky. What is the implied one-year rate two years from now?
14. Assume that you make investment decisions based on the expectations theory. Based on research, you have obtained the following information about certain market interest rates:
You can borrow at 5% and lend at 4.5% for one year You can borrow at 6% and lend for 6.5% for two years
One-year rate one year from now is 7% for bor- rowing and lending
What decision is the most profitable for you? Show your calculations.
15. Predict the one-year interest rate three years from today if interest rates are 3.5%, 4.0%, 4.5%, and 5%
for bonds with one to four years to maturity and liquidity premiums are 0%, 0.1%, 0.25%, and 0.50%.
W E B E X E R C I S E S
The Risk and Term Structures of Interest Rates 1. The amount of additional interest investors receive
due to the various risk premiums changes over time.
Sometimes the risk premiums are much larger than at other times. For example, the default risk premium was very small in the late 1990s, when the economy was so healthy that business failures were rare. This risk premium increases during recessions.
Go to the Federal Reserve Bank of St. Louis FRED database at http://research.stlouisfed.org/fred2/
and find the interest-rate listings for AAA- and Baa- rated bonds at three points in time: the most recent;
August 1, 2007; and October 1, 2008. Prepare a graph that shows these three time periods (see Figure 5.1 for an example). Are the risk premiums stable or do they change over time?
2. Figure 5.7 shows a number of yield curves at various points in time. Go to www.treasury.gov, and in the
“Resource Center” at the top of the page click on
“Data and Charts Center.” Find the Treasury yield curve. Does the current yield curve fall above or below the most recent one listed in Figure 5.7? Is the current yield curve flatter or steeper than the most recent one reported in Figure 5.7?
3. Investment companies attempt to explain to inves- tors the nature of the risk the investor incurs when buying shares in their mutual funds. For example, go to https://personal.vanguard.com/us/funds/
vanguard/all?sort=name&sortorder=asc.
a. Select the bond fund you would recommend to an investor who has a very low tolerance for risk.
Justify your answer.
b. Select the bond fund you would recommend to an investor who has a higher tolerance for risk and a long investment horizon. Justify your answer.
M05_MISH5006_09_GE_C05.indd 156 28/10/17 10:38 AM
www.rasabourse.com
157 over time, we look at the efficient market hypoth- esis. In this chapter we examine the basic reasoning behind the efficient market hypothesis in order to explain some puzzling features of the operation and behavior of financial markets. You will see, for exam- ple, why changes in stock prices are unpredictable and why listening to a stockbroker’s hot tips may not be a good idea.
Theoretically, the efficient market hypothesis should be a powerful tool for analyzing behav- ior in financial markets. But to establish that it is in reality a useful tool, we must compare the theory with the data. Does the empirical evidence support the theory? Though mixed, the avail- able evidence indicates that for many purposes, this theory is a good starting point for analyzing expectations.
PREVIEW
Throughout our discussion of how financial mar- kets work, you may have noticed that the subject of expectations keeps cropping up. Expectations of returns, risk, and liquidity are central elements in the demand for assets; expectations of inflation have a major impact on bond prices and interest rates;
expectations about the likelihood of default are the most important factor that determines the risk structure of interest rates; and expectations of future short-term interest rates play a central role in deter- mining the term structure of interest rates. Not only are expectations critical in understanding behavior in financial markets, but as we will see later in this book, they are also central to our understanding of how financial institutions operate.
To understand how expectations are formed so that we can understand how securities prices move