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Answers to Chapter 3
Questions:
1. The required rate of return is the interest rate an investor should receive on a security given its risk. Required rate
of return is used to calculate the fair present value on a security. The expected rate of return is the interest rate an
investor expects to receive on a security if he or she buys the security at its current market price, receives all
expected payments, and sells the security at the end of his or her investment horizon.
2. Once an expected rate of return, E(r), on a financial security is calculated, the market participant compares this
expected rate of return to its required rate of return (r). If the expected rate of return is greater than the required rate
of return, the projected cash flows on the security are greater than is required to compensate for the risk incurred
3. Most bonds pay a stated coupon rate of interest to the holders of the bonds. These bonds are called coupon bonds.
The interest, or coupon, payments per year are generally constant (are fixed) over the life of the bond. Thus, the
4. a. Premium bond
c. Discount bond
e. Premium bond
5. The valuation process for an equity instrument (such as common stock or a share) involves finding the present
value of an infinite series of cash flows on the equity discounted at an appropriate interest rate. Cash flows from
holding equity come from dividends paid out by the firm over the life of the stock, which in expectation can be
viewed as infinite since a firm (and thus the dividends it pays) has no defined maturity or life. Even if an equity
holder decides not to hold the stock forever, he or she can sell it to someone else who in a fair and efficient market is
6. The present values of the cash flows on bonds decreases as the required rate of return increases. This is the
inverse relationship between present values and interest rates we discussed in Chapter 2. While the examples in the
chapter refer to the relation between fair present values and required rates of returns, the inverse relation also exists
7. All else equal, a long-term bond experiences larger price changes when interest rates change than a short-term
8. The price of the bond with the small coupon will be impacted more by a change in interest rates than the price of
the large coupon bond. For a small coupon bond, the cash flows are weighted much more toward the maturity date
9. Duration is the elasticity, or sensitivity, of the bond’s price to small interest rate (either required rate of return or
yield to maturity) changes, and is represented as follows:
The negative sign in front of the D indicates the inverse relationship between interest rate changes and price
changes. That is, -D describes the percentage value decrease —capital loss—on the security (P/P) for any given
10. The higher the coupon or promised interest payment on the bond, the shorter its duration. This is due to the fact
11. The 8 percent yield to maturity bond has the longer duration. This is because duration decreases as rate of return
increases. This makes intuitive sense since the higher the rate of return on the bond, the lower the present value cost
12. Duration, which measures the weighted-average time to maturity of the asset or liability, also has economic
Problems:
1. 935 = 75{[1 - (1/(1 +
´r
)5)]/
´r
} + 980/(1 +
´r
)5 =>
´r
= 8.83%
´r
3. Vb = 1,000(0.08) {[1 - (1/(1 + 0.09)10)]/0.09} + 1,000(1 + 0.09)10 = $935.82
Or, on a financial calculator: N = 10, I = 9, PMT = 80, FV = 1,000, => PV = $935.82
4. EXCEL Problem: Bond Value = $1,268.27
6. EXCEL Problem: Yield to Maturity = 9.87%
8. $863.73 = 1,000(0.08) {[1 - (1/(1 + 0.10)n)]/0.10} + 1,000/(1 + 0.10)n => n = 12 years
9. a. Vb = 1,000(0.1) {[1 - (1/(1 + 0.06/2)2(10))]/(0.06/2)} + 1,000/(1 + 0.06/2)2(10) = $1,297.55
2
10. a. 985 = 1,000(0.09) {[1 - (1/(1 + ytm/2)2(15))]/ytm/2} + 1,000/(1 + ytm/2)2(15) => ytm = 9.186%
2
c. 1,065 = 1,000(0.11) {[1-(1/(1 + ytm)6)]/ytm} + 1,000/(1 + ytm)6 => ytm = 9.528%
b. Vb = 1,000(0.08) {[1 - (1/(1 + 0.10/2)2(12))]/(0.10/2)} + 1,000/(1 + 0.10/2)2(12) = $862.01
2
b. Vb = 1,000(0.08) {[1 - (1/(1 + 0.08/2)2(12))]/(0.08/2)} + 1,000/(1 + 0.08/2)2(12) = $1,000.00
2
c. Vb = 1,000(0.10) {[1 - (1/(1 + 0.08/2)2(12))]/(0.08/2)} + 1,000/(1 + 0.08/2)2(12) = $1,152.47
2
13. a. Vb = 1,000(0.10) {[1 - (1/(1 + 0.08/2)2(10))]/(0.08/2)} + 1,000/(1 + 0.08/2)2(10) = $1,135.90
2
c. Vb = 1,000(0.10) {[1 - (1/(1 + 0.08/2)2(20))]/(0.08/2)} + 1,000/(1 + 0.08/2)2(20) = $1,197.93
2
14. a. Vb = 1,000(0.10) {[1 - (1/(1 + 0.11/2)2(10))]/(0.11/2)} + 1,000/(1 + 0.11/2)2(10) = $940.25
2
c. Vb = 1,000(0.10) {[1 - (1/(1 + 0.11/2)2(20))]/(0.11/2)} + 1,000/(1 + 0.11/2)2(20) = $919.77 % change 2.28%
2
As interest rates increase the variability in bond prices increases as time to maturity increases but at a decreasing rate.
15. Price before the change in interest rates:
Price after the change in interest rates:
16. Vb = 945.80 = 1,000(0.09) {[1 - (1/(1 + ytm/2)2(7))]/ytm/2} + 1,000/(1 + ytm/2)2(7) => ytm = 10.099%
2
17. P0 = 5/0.10 = $50
19. P0 = 3.50/0.068 = $51.47
0.12 - 0.10
0.145 - 0.125
22. a. Po = 2.50(1 + 0.015) = $24.167
0.12 - 0.015
0.15 - 0.015
23. a. E(rs) = 4.50 + 0.03 = 10.03%
64
64
24. Step 1: Find the present value of dividends during the period of supernormal growth.
Year Dividends (D0(1 + gs) t
) 1/(1 + 0.10) t
Present Value
1 5.5(1 + 0.08)1 = 5.940 0.9091 5.400
2 5.5(1 + 0.08)2 = 6.415 0.8264 5.302
Step 2: Find present value of dividends after period of supernormal growth
a. Find stock value at beginning of constant growth period
rs - g rs - g 0.10 - 0.03
b. Find present value of constant growth dividends
Step 3: Find present value of stock = value during supernormal growth period + value during normal growth period
$30.963 + $72.492 = $103.455
44.12
26. E(rs) = 0.84(1 + 0.15) + 0.15 = 17.41%
40.11
29. a. Year Cash Flows (CF) 1/(1 + 0.12) t PV of CF PV of CF t
1 100 0.8929 89.29 89.29
2 1,100 0.7972 876.91 1,753.83
966.20 1,843.12
c. Duration always will be lower than the maturity for a coupon bond. That is because duration takes into account
If interest rates rise by 1%, the value of the Bank 1’s loan will be
If interest rates rise by 1%, the value of the CD will be
The difference in the changes in the assets ($54,262.43) and liabilities ($58,892.32) is $4,629.89. The asset
decreased in value by less than the liability.
The value of Bank 2’s the zero-coupon bond when rates rise by 1% is $840,074.08 = 1,976,362.88/(1 + 0.13)7
On a financial calculator: N = 7, I = 13, PMT = 0, FV = 1,000,000 => PV = $840,074.08
The value of the Bank 2’s zero-coupon bond was $1,976,362.88 and is now $840,074.08. The value of the Bank 2’s
b. Although the numbers are a bit contrived, the point of the problem is to show that in part a even though Bank 1’s
assets and liabilities had the same face values and maturities, they have different durations and so the changes in
