EXAMPLE 3.10
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Examples 3.10 and 3.11 have led the pension fund manager to an important conclusion about the relationship of duration and interest-rate risk: The greater the duration of a security, the greater the percentage change in the mar- ket value of the security for a given change in interest rates. Therefore, the greater the duration of a security, the greater its interest-rate risk.
This reasoning applies equally to a portfolio of securities. So by calculating the duration of the fund’s portfolio of securities using the methods outlined here, a pen- sion fund manager can easily ascertain the amount of interest-rate risk the entire fund is exposed to. As we will see in Chapter 23, duration is a highly useful concept for the management of interest-rate risk that is widely used by managers of banks and other financial institutions.
Now the pension manager has the option to hold a 10-year coupon bond with a coupon rate of 20% instead of 10%. As mentioned earlier, the duration for this 20% coupon bond is 5.98 years when the interest rate is 10%. Find the approximate change in the bond price when the interest rate increases from 10% to 11%.
Solution
This time the approximate change in bond price is –5.4%. This change in bond price is much smaller than for the higher-duration coupon bond.
%∆P ≈ -DUR * ∆i 1 + i where
DUR = duration = 5.98
Δi = change in interest rate = 0.11 - 0.10 = 0.01 i = current interest rate = 0.10
Thus,
%∆P ≈ -5.98 * 0.01 1 + 0.10
%∆P ≈ -0.054 = -5.4%
The pension fund manager realizes that the interest-rate risk on the 20% coupon bond is less than on the 10% coupon bond, so he switches the fund out of the 10% coupon bond and into the 20% coupon bond.
Duration and Interest-Rate
Risk
EXAMPLE 3.11
S U M M A R Y
1. The yield to maturity, which is the measure most accurately reflecting the interest rate, is the inter- est rate that equates the present value of future cash flows of a debt instrument with its value today.
Application of this principle reveals that bond prices and interest rates are negatively related: When the interest rate rises, the price of the bond must fall, and vice versa.
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2. The real interest rate is defined as the nominal inter- est rate minus the expected rate of inflation. It is both a better measure of the incentives to borrow and lend and a more accurate indicator of the tight- ness of credit market conditions than the nominal interest rate.
3. The return on a security, which tells you how well you have done by holding this security over a stated period of time, can differ substantially from the interest rate as measured by the yield to maturity.
Long-term bond prices have substantial fluctuations when interest rates change and thus bear interest- rate risk. The resulting capital gains and losses can be large, which is why long-term bonds are not con- sidered to be safe assets with a sure return. Bonds whose maturity is shorter than the holding period are also subject to reinvestment risk, which occurs because the proceeds from the short-term bond
need to be reinvested at a future interest rate that is uncertain.
4. Duration, the average lifetime of a debt security’s stream of payments, is a measure of effective matu- rity, the term to maturity that accurately measures interest-rate risk. Everything else being equal, the duration of a bond is greater the longer the matu- rity of a bond, when interest rates fall, or when the coupon rate of a coupon bond falls. Duration is addi- tive: The duration of a portfolio of securities is the weighted average of the durations of the individual securities, with the weights reflecting the propor- tion of the portfolio invested in each. The greater the duration of a security, the greater the percent- age change in the market value of the security for a given change in interest rates. Therefore, the greater the duration of a security, the greater its interest-rate risk.
cash flows, p. 78 coupon bond, p. 80 coupon rate, p. 80 current yield, p. 86
discount bond (zero-coupon bond), p. 80
duration, p. 96
face value (par value), p. 80
fixed-payment loan (fully amortized loan), p. 80
indexed bonds, p. 91 interest-rate risk, p. 94 nominal interest rate, p. 88 perpetuity (consol), p. 85
present value (present discounted value), p. 78
rate of capital gain, p. 93 real interest rate, p. 88 real terms, p. 89 reinvestment risk, p. 95 return (rate of return), p. 91 simple loan, p. 78
yield to maturity, p. 81
K E Y T E R M S
Q U E S T I O N S
1. What is the concept of present value? What is discounting?
2. Analyze the risk profiles of short-term bonds and long-term bonds during economic instability and fluctuating interest rates.
3. What is the yield to maturity? Why it is considered as a good measure of interest rates?
4. Property investment constitutes a large and long- term commitment to an individual. Describe the outcome of taking a property loan during fluctuating interest rates.
Q U A N T I T A T I V E P R O B L E M S
1. Calculate the present value of a $1,000 zero-coupon bond with six years to maturity if the yield to matu- rity is 7%.
2. A lottery claims its grand prize is $20 million, payable over 40 years at $500,000 per year. If the first pay- ment is made immediately, what is the grand prize worth? Use an interest rate of 12%.
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3. Consider a bond with an 8% annual coupon and a face value of $1,000. Complete the following table:
Years to Maturity
Yield to Maturity
Current Price
4 6
5 8
7 9
10 11
11 12
4. Consider a coupon bond which has a $1,000 par value and a coupon rate of 20%. The bond is currently sell- ing for $2,300 and has 16 years to maturity. Calculate the bond’s yield to maturity.
5. You are willing to pay $31,250 now to purchase a perpetuity that will pay you and your heirs $2,500 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 40-year, annual payment, ordinary annuity instead of perpetuity?
6. Calculate the yield to maturity on the bond that has a price of $1,000 and pays $50 dividend for the life of the bond. What will happen if the dividend is $25 instead of $50?
7. Suppose you bought a land that costs $500,000 today.
You will need to continue to pay tax on the land, and the rate is 3% of your purchase. Calculate the PV of your payment, using a 10% discount rate. Assume that there are no changes in the land’s price and tax rate.
8. Suppose that you want to take out a loan at a bank that wants to charge you an annual real interest rate equal to 5%. Assuming that the expected rate of infla- tion during the life of the loan is 2%, what will be the nominal interest rate that the bank will charge you? If the real inflation was 3% instead of the expected 2%, what was the actual real interest rate on the loan?
9. Anna bought a bond with a par value of $10,000 and a coupon rate of 8% at par. After a year, she was able to sell her bond for $11,000. Calculate the rate of return on Anna’s investment. What is the current yield and capital gain on her investment?
10. Suppose that you have a bond with a face value of $1,000 and a coupon rate of 8% for one year and that you buy another one after one year. What will be your gain if the interest rate increases up to 10%? How will your answer change if the interest
rate falls to 6%? What conclusion can you draw from these cases?
11. Calculate the duration of a five-year bond with a face value of $1,000 and a coupon rate of 8%. Assume that the current interest rates are 10%. What will your answer be if the current interest rates fall to 7%?
Show all your calculations.
12. Using the data provided in the previous problem, calculate the price difference using the duration formula
13. Financial institutions hold a portfolio comprising the following securities:
10 million bond with duration 4 years 20 million securities with duration 6 years 40 million bonds with duration 5 years What is the duration of the total portfolio?
14. A bank has two 3-year commercial loans with a pres- ent value of $70 million. The first is a $30 million loan that requires a single payment of $37.8 million in three years, with no other payments till then. The second loan is for $40 million. It requires an annual interest payment of $3.6 million. The principal of
$40 million is due in three 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 increases from 8% to 8.5%?
15. Consider a bond that promises the following cash flows. The yield to maturity is 12%.
Year 0 1 2 3 4
Promised
Payments 160 160 170 180 230
You plan to buy this bond, hold it for 2.5 years, and then sell the bond.
a. What total cash will you receive from the bond after the 2.5 years? Assume that periodic cash flows are reinvested at 12%.
b. If immediately after you buy this bond all market interest rates drop to 11% (including your rein- vestment rate), what will be the impact on your total cash flow after 2.5 years? How does this com- pare to part (a)?
c. Assuming all market interest rates are 12%, what is the duration of this bond?
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b. What was the interest rate on the one-month Eurodollar at the end of 2016?
c. What is the most recent interest rate reported for the 10-year Treasury note?
Understanding Interest Rates
1. Investigate the data available from the Federal Reserve Bank of St. Louis FRED database at http://
research.stlouisfed.org/fred2/. Then answer the following questions.
a. What is the difference in the interest rates on com- mercial paper for financial firms versus nonfinan- cial firms?
W E B E X E R C I S E S
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