CHAPTER 8
INTEREST RATES AND BOND
VALUATION
Solutions to Questions and Problems
1.
The price of a pure discount (zero coupon) bond is the present value of the par value. Remember, even
though there are no coupon payments, the periods are semiannual to stay consistent with coupon bond
payments. So, the price of the bond for each YTM is:
a. P = $1,000/(1 + .05/2)30 = $476.74
b. P = $1,000/(1 + .10/2)30 = $231.38
c. P = $1,000/(1 + .15/2)30 = $114.22
2.
The price of any bond is the PV of the interest payment, plus the PV of the par value. Notice this
problem assumes a semiannual coupon. The price of the bond at each YTM will be:
a.
P = $35({1 – [1/(1 + .035)]30 } / .035) + $1,000[1 / (1 + .035)30]
P = $1,000.00
When the YTM and the coupon rate are equal, the bond will sell at par.
b.
P = $35({1 – [1/(1 + .045)]30 } / .045) + $1,000[1 / (1 + .045)30]
P = $837.11
When the YTM is greater than the coupon rate, the bond will sell at a discount.
c.
P = $35({1 – [1/(1 + .025)]30 } / .025) + $1,000[1 / (1 + .025)30]
P = $1,209.30
When the YTM is less than the coupon rate, the bond will sell at a premium.
3.
Here we are finding the YTM of a semiannual coupon bond. The bond price equation is:
P = $1,050 = $32(PVIFAR%,26) + $1,000(PVIFR%,26)
Since we cannot solve the equation directly for R, using a spreadsheet, a financial calculator, or trial
and error, we find:
R = 2.923%
Since the coupon payments are semiannual, this is the semiannual interest rate. The YTM is the APR
of the bond, so:
YTM = 2 × 2.923% = 5.85%
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4.
Here we need to find the coupon rate of the bond. All we need to do is to set up the bond pricing
equation and solve for the coupon payment as follows:
P = $1,060 = C(PVIFA3.8%,23) + $1,000(PVIF3.8%,23)
Solving for the coupon payment, we get:
C = $41.96
Since this is the semiannual payment, the annual coupon payment is:
2 × $41.96 = $83.92
And the coupon rate is the annual coupon payment divided by par value, so:
Coupon rate = $83.92 / $1,000 = .0839, or 8.39%
5.
The price of any bond is the PV of the interest payment, plus the PV of the par value. The fact that the
bond is denominated in euros is irrelevant. Notice this problem assumes an annual coupon. The price
of the bond will be:
P = €45({1 – [1/(1 + .039)]19 } / .039) + €1,000[1 / (1 + .039)19]
P = €1,079.48
6.
Here we are finding the YTM of an annual coupon bond. The fact that the bond is denominated in yen
is irrelevant. The bond price equation is:
P = ¥92,000 = ¥2,800(PVIFAR%,21) + ¥100,000(PVIFR%,21)
Since we cannot solve the equation directly for R, using a spreadsheet, a financial calculator, or trial
and error, we find:
R = 3.34%
Since the coupon payments are annual, this is the yield to maturity.
7. The approximate relationship between nominal interest rates (R), real interest rates (r), and inflation (h)
is:
R=r+h
Approximate r = .045 –.021 =.0240, or 2.40%
The Fisher equation, which shows the exact relationship between nominal interest rates, real interest
rates, and inflation is:
(1 + R) = (1 + r)(1 + h)
(1 + .045) = (1 + r)(1 + .021)
Exact r = [(1 + .045) / (1 + .021)] – 1 = .0235, or 2.35%
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8.
The Fisher equation, which shows the exact relationship between nominal interest rates, real interest
rates, and inflation, is:
(1 + R) = (1 + r)(1 + h)
R = (1 + .024)(1 + .031) – 1 = .0557, or 5.57%
9.
The Fisher equation, which shows the exact relationship between nominal interest rates, real interest
rates, and inflation, is:
(1 + R) = (1 + r)(1 + h)
h = [(1 + .14) / (1 + .10)] – 1 = .0364, or 3.64%
10. The Fisher equation, which shows the exact relationship between nominal interest rates, real interest
rates, and inflation, is:
(1 + R) = (1 + r)(1 + h)
r = [(1 + .125) / (1.053)] – 1 = .0684, or 6.84%
11. The coupon rate, located in the first column of the quote is 4.750%. The bid price is:
Bid price = 109:11 = 109 11/32 = 109.34375% × $1,000 = $1,093.4375
The previous day’s ask price is found by:
Previous day’s asked price = Today’s asked price – Change = 109 13/32 – (–11/32) = 109 24/32
The previous day’s price in dollars was:
Previous day’s dollar price = 109.7500% × $1,000 = $1,097.5000
12. This is a premium bond because it sells for more than 100% of face value. The current yield is:
Current yield = Annual coupon payment / Asked price = $43.75/$1,023.7500 = .0427 or 4.27%
The YTM is located under the “Asked yield” column, so the YTM is 4.2306%.
The bid-ask spread is the difference between the bid price and the ask price, so:
Bid-Ask spread = 102:12 – 102:11 = 1/32
13. Zero coupon bonds are priced with semiannual compounding to correspond with coupon bonds. The
price of the bond when purchased was:
P0 = $1,000 / (1 + .035)50
P0 = $179.05
And the price at the end of one year is:
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P0 = $1,000 / (1 + .035)48
P0 = $191.81
So, the implied interest, which will be taxable as interest income, is:
Implied interest = $191.81 – 179.05
Implied interest = $12.75
Intermediate
14. Here we are finding the YTM of semiannual coupon bonds for various maturity lengths. The bond
price equation is:
P = C(PVIFAR%,t) + $1,000(PVIFR%,t)
Miller Corporation bond:
P0 = $40(PVIFA3%,26) + $1,000(PVIF3%,26)
P1 = $40(PVIFA3%,24) + $1,000(PVIF3%,24)
P3 = $40(PVIFA3%,20) + $1,000(PVIF3%,20)
P8 = $40(PVIFA3%,10) + $1,000(PVIF3%,10)
P12 = $40(PVIFA3%,2) + $1,000(PVIF3%,2)
P13
= $1,178.77
= $1,169.36
= $1,148.77
= $1,085.30
= $1,019.13
= $1,000
Modigliani Company bond:
P0 = $30(PVIFA4%,26) + $1,000(PVIF4%,26)
P1 = $30(PVIFA4%,24) + $1,000(PVIF4%,24)
P3 = $30(PVIFA4%,20) + $1,000(PVIF4%,20)
P8 = $30(PVIFA4%,10) + $1,000(PVIF4%,10)
P12 = $30(PVIFA4%,2) + $1,000(PVIF4%,2)
P13
= $840.17
= $847.53
= $864.10
= $918.89
= $981.14
= $1,000
All else held equal, the premium over par value for a premium bond declines as maturity approaches,
and the discount from par value for a discount bond declines as maturity approaches. This is called
“pull to par.” In both cases, the largest percentage price changes occur at the shortest maturity lengths.
Also, notice that the price of each bond when no time is left to maturity is the par value, even though
the purchaser would receive the par value plus the coupon payment immediately. This is because we
calculate the clean price of the bond.
15. Any bond that sells at par has a YTM equal to the coupon rate. Both bonds sell at par, so the initial
YTM on both bonds is the coupon rate, 7 percent. If the YTM suddenly rises to 9 percent:
PLaurel
= $35(PVIFA4.5%,4) + $1,000(PVIF4.5%,4) = $964.12
PHardy
= $35(PVIFA4.5%,30) + $1,000(PVIF4.5%,30) = $837.11
The percentage change in price is calculated as:
Percentage change in price = (New price – Original price) / Original price
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∆PLaurel% = ($964.12 – 1,000) / $1,000 = –.0359, or –3.59%
∆PHardy% = ($837.11 – 1,000) / $1,000 = –.1629, or –16.29%
If the YTM suddenly falls to 5 percent:
PLaurel
= $35(PVIFA2.5%,4) + $1,000(PVIF2.5%,4) = $1,037.62
PHardy
= $35(PVIFA2.5%,30) + $1,000(PVIF2.5%,30) = $1,209.30
∆PLaurel% = ($1,037.62 – 1,000) / $1,000 = +.0376, or 3.76%
∆PHardy% = ($1,209.30 – 1,000) / $1,000 = +.2093, or 20.93%
All else the same, the longer the maturity of a bond, the greater is its price sensitivity to changes in
interest rates. Notice also that for the same interest rate change, the gain from a decline in interest rates
is larger than the loss from the same magnitude change. For a plain vanilla bond, this is always true.
16. Initially, at a YTM of 10 percent, the prices of the two bonds are:
PFaulk
= $30(PVIFA5%,24) + $1,000(PVIF5%,24)
= $724.03
PGonas
= $70(PVIFA5%,24) + $1,000(PVIF5%,24)
= $1,275.97
If the YTM rises from 10 percent to 12 percent:
PFaulk
= $30(PVIFA6%,24) + $1,000(PVIF6%,24)
= $623.49
PGonas
= $70(PVIFA6%,24) + $1,000(PVIF6%,24)
= $1,125.50
The percentage change in price is calculated as:
Percentage change in price = (New price – Original price) / Original price
∆PFaulk% = ($623.49 – 724.03) / $724.03 = –.1389, or –13.89%
∆PGonas% = ($1,125.50 – 1,275.97) / $1,275.97 = –.1179, or –11.79%
If the YTM declines from 10 percent to 8 percent:
PFaulk
= $30(PVIFA4%,24) + $1,000(PVIF4%,24)
= $847.53
PGonas
= $70(PVIFA4%,24) + $1,000(PVIF4%,24)
= $1,457.41
∆PFaulk% = ($847.53 – 724.03) / $724.03
= +.1706, or 17.06%
∆PGonas% = ($1,457.41 – 1,275.97) / $1,275.97 = +.1422, or 14.22%
All else the same, the lower the coupon rate on a bond, the greater is its price sensitivity to changes in
interest rates.
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17. The bond price equation for this bond is:
P0 = $1,050 = $31(PVIFAR%,18) + $1,000(PVIFR%,18)
Using a spreadsheet, financial calculator, or trial and error we find:
R = 2.744%
This is the semiannual interest rate, so the YTM is:
YTM = 2 × 2.744% = 5.49%
The current yield is:
Current yield = Annual coupon payment / Price = $62 / $1,050 = .0590 or 5.90%
The effective annual yield is the same as the EAR, so using the EAR equation from the previous
chapter:
Effective annual yield = (1 + 0.02744)2 – 1 = .0556, or 5.56%
18. The company should set the coupon rate on its new bonds equal to the required return. The required
return can be observed in the market by finding the YTM on outstanding bonds of the company. So,
the YTM on the bonds currently sold in the market is:
P = $1,063 = $35(PVIFAR%,40) + $1,000(PVIFR%,40)
Using a spreadsheet, financial calculator, or trial and error we find:
R = 3.218%
This is the semiannual interest rate, so the YTM is:
YTM = 2 × 3.218% = 6.44%
19. Accrued interest is the coupon payment for the period times the fraction of the period that has passed
since the last coupon payment. Since we have a semiannual coupon bond, the coupon payment per six
months is one-half of the annual coupon payment. There are two months until the next coupon
payment, so four months have passed since the last coupon payment. The accrued interest for the bond
is:
Accrued interest = $68/2 × 4/6 = $22.67
And we calculate the clean price as:
Clean price = Dirty price – Accrued interest = $950 – 22.67 = $927.33
20. Accrued interest is the coupon payment for the period times the fraction of the period that has passed
since the last coupon payment. Since we have a semiannual coupon bond, the coupon payment per six
months is one-half of the annual coupon payment. There are four months until the next coupon
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payment, so two months have passed since the last coupon payment. The accrued interest for the bond
is:
Accrued interest = $59/2 × 2/6 = $9.83
And we calculate the dirty price as:
Dirty price = Clean price + Accrued interest = $1,053 + 9.83 = $1,062.83
21. To find the number of years to maturity for the bond, we need to find the price of the bond. Since we
already have the coupon rate, we can use the bond price equation, and solve for the number of years to
maturity. We are given the current yield of the bond, so we can calculate the price as:
Current yield = .0842 = $90/P0
P0 = $90/.0842 = $1,068.88
Now that we have the price of the bond, the bond price equation is:
P = $1,068.88 = $90{[(1 – (1/1.0781)t ] / .0781} + $1,000/1.0781t
We can solve this equation for t as follows:
$1,068.88 (1.0781)t = $1,152.37 (1.0781)t – 1,152.37 + 1,000
152.37 = 83.49(1.0781)t
1.8251 = 1.0781t
t = log 1.8251 / log 1.0781 = 8.0004 ≈ 8 years
The bond has 8 years to maturity.
22. The bond has 9 years to maturity, so the bond price equation is:
P = $1,053.12 = $36.20(PVIFAR%,18) + $1,000(PVIFR%,18)
Using a spreadsheet, financial calculator, or trial and error we find:
R = 3.226%
This is the semiannual interest rate, so the YTM is:
YTM = 2 × 3.226% = 6.45%
The current yield is the annual coupon payment divided by the bond price, so:
Current yield = $72.40 / $1,053.12 = .0687, or 6.87%
23. We found the maturity of a bond in Problem 21. However, in this case, the maturity is indeterminate. A
bond selling at par can have any length of maturity. In other words, when we solve the bond pricing
equation as we did in Problem 21, the number of periods can be any positive number.
24. The price of a zero coupon bond is the PV of the par, so:
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a.
P0 = $1,000/1.03550 = $179.05
b.
In one year, the bond will have 24 years to maturity, so the price will be:
P1 = $1,000/1.03548 = $191.81
The interest deduction is the price of the bond at the end of the year, minus the price at the
beginning of the year, so:
Year 1 interest deduction = $191.81 – 179.05 = $12.75
The price of the bond when it has one year left to maturity will be:
P24 = $1,000/1.0352 = $933.51
Year 25 interest deduction = $1,000 – 933.51 = $66.49
c.
Previous IRS regulations required a straight-line calculation of interest. The total interest
received by the bondholder is:
Total interest = $1,000 – 179.05 = $820.95
The annual interest deduction is simply the total interest divided by the maturity of the bond, so
the straight-line deduction is:
Annual interest deduction = $820.95 / 25 = $32.84
d.
The company will prefer straight-line methods when allowed because the valuable interest
deductions occur earlier in the life of the bond.
25. a.
The coupon bonds have a 6 percent coupon which matches the 6 perent required return, so they
will sell at par. The number of bonds that must be sold is the amount needed divided by the bond
price, so:
Number of coupon bonds to sell = $45,000,000 / $1,000 = 45,000
The number of zero coupon bonds to sell would be:
Price of zero coupon bonds = $1,000/1.0360 = $169.73
Number of zero coupon bonds to sell = $45,000,000 / $169.73 = 265,122
b.
The repayment of the coupon bond will be the par value plus the last coupon payment times the
number of bonds issued. So:
Coupon bonds repayment = 45,000($1,030) = $46,350,000
The repayment of the zero coupon bond will be the par value times the number of bonds issued,
so:
Zeroes: repayment = 265,122($1,000) = $265,122,140
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c.
The total coupon payment for the coupon bonds will be the number bonds times the coupon
payment. For the cash flow of the coupon bonds, we need to account for the tax deductibility of
the interest payments. To do this, we will multiply the total coupon payment times one minus the
tax rate. So:
Coupon bonds: (45,000)($60)(1–.35) = $1,755,000 cash outflow
Note that this is cash outflow since the company is making the interest payment.
For the zero coupon bonds, the first year interest payment is the difference in the price of the zero
at the end of the year and the beginning of the year. The price of the zeroes in one year will be:
P1 = $1,000/1.0358 = $180.07
The year 1 interest deduction per bond will be this price minus the price at the beginning of the
year, which we found in part b, so:
Year 1 interest deduction per bond = $180.07 – 169.73 = $10.34
The total cash flow for the zeroes will be the interest deduction for the year times the number of
zeroes sold, times the tax rate. The cash flow for the zeroes in year 1 will be:
Cash flows for zeroes in Year 1 = (265,122)($10.34)(.35) = $959,175.00
Notice the cash flow for the zeroes is a cash inflow. This is because of the tax deductibility of the
imputed interest expense. That is, the company gets to write off the interest expense for the year
even though the company did not have a cash flow for the interest expense. This reduces the
company’s tax liability, which is a cash inflow.
During the life of the bond, the zero generates cash inflows to the firm in the form of the interest
tax shield of debt. We should note an important point here: If you find the PV of the cash flows
from the coupon bond and the zero coupon bond, they will be the same. This is because of the
much larger repayment amount for the zeroes.
26. To find the capital gains yield and the current yield, we need to find the price of the bond. The current
price of Bond P and the price of Bond P in one year is:
P:
P0 = $90(PVIFA7%,10) + $1,000(PVIF7%,10) = $1,140.47
P1 = $90(PVIFA7%,9) + $1,000(PVIF7%,9) = $1,130.30
Current yield = $90 / $1,140.47 = .0789, or 7.89%
The capital gains yield is:
Capital gains yield = (New price – Original price) / Original price
Capital gains yield = ($1,130.30 – 1,140.47) / $1,140.47 = –.0089, or –0.89%
The current price of Bond D and the price of Bond D in one year is:
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D:
P0 = $50(PVIFA7%,10) + $1,000(PVIF7%,10) = $859.53
P1 = $50(PVIFA7%,9) + $1,000(PVIF7%,9) = $869.70
Current yield = $50 / $859.53 = 0.0582 or 5.82%
Capital gains yield = ($869.70 – 859.53) / $859.53 = .0118, or 1.18%
All else held constant, premium bonds pay a high current income while having price depreciation as
maturity nears; discount bonds pay a lower current income but have price appreciation as maturity
nears. For either bond, the total return is still 7%, but this return is distributed differently between
current income and capital gains.
27. a.
The rate of return you expect to earn if you purchase a bond and hold it until maturity is the YTM.
The bond price equation for this bond is:
P0 = $930 = $56(PVIFAR%,10) + $1,000(PVIF R%,10)
Using a spreadsheet, financial calculator, or trial and error we find:
R = YTM = 6.58%
b.
To find our HPY, we need to find the price of the bond in two years. The price of the bond in two
years, at the new interest rate, will be:
P2 = $56(PVIFA5.58%,8) + $1,000(PVIF5.58%,8) = $1,001.44
To calculate the HPY, we need to find the interest rate that equates the price we paid for the bond
with the cash flows we received. The cash flows we received were $90 each year for two years,
and the price of the bond when we sold it. The equation to find our HPY is:
P0 = $930 = $56(PVIFAR%,2) + $1,001.44(PVIFR%,2)
Solving for R, we get:
R = HPY = 9.68%
The realized HPY is greater than the expected YTM when the bond was bought because interest
rates dropped by 1 percent; bond prices rise when yields fall.
28. The price of any bond (or financial instrument) is the PV of the future cash flows. Even though Bond
M makes different coupons payments, to find the price of the bond, we just find the PV of the cash
flows. The PV of the cash flows for Bond M is:
PM = $800(PVIFA4%,16)(PVIF4%,12) + $1,000(PVIFA4%,12)(PVIF4%,28) + $30,000(PVIF4%,40)
PM = $15,200.77
Notice that for the coupon payments of $800, we found the PVA for the coupon payments, and then
discounted the lump sum back to today.
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Bond N is a zero coupon bond with a $30,000 par value; therefore, the price of the bond is the PV of
the par, or:
PN = $30,000(PVIF4%,40) = $6,248.67
29. In general, this is not likely to happen, although it can (and did). The reason this bond has a negative
YTM is that it is a callable U.S. Treasury bond. Market participants know this. Given the high coupon
rate of the bond, it is extremely likely to be called, which means the bondholder will not receive all the
cash flows promised. A better measure of the return on a callable bond is the yield to call (YTC). The
YTC calculation is the basically the same as the YTM calculation, but the number of periods is the
number of periods until the call date. If the YTC were calculated on this bond, it would be positive.
30. To find the present value, we need to find the real weekly interest rate. To find the real return, we need
to use the effective annual rates in the Fisher equation. So, we find the real EAR is:
(1 + R) = (1 + r)(1 + h)
1 + .069 = (1 + r)(1 + .032)
r = .0359, or 3.59%
Now, to find the weekly interest rate, we need to find the APR. Using the equation for discrete
compounding:
EAR = [1 + (APR / m)]m – 1
We can solve for the APR. Doing so, we get:
APR = m[(1 + EAR)1/m – 1]
APR = 52[(1 + .0359)1/52 – 1]
APR = .0352 or 3.52%
So, the weekly interest rate is:
Weekly rate = APR / 52
Weekly rate = .0352 / 52
Weekly rate = .0007 or 0.07%
Now we can find the present value of the cost of the roses. The real cash flows are an ordinary annuity,
discounted at the real interest rate. So, the present value of the cost of the roses is:
PVA = C({1 – [1/(1 + r)]t } / r)
PVA = $8({1 – [1/(1 + .0007)]30(52)} / .0007)
PVA = $7,702.30
31. To answer this question, we need to find the monthly interest rate, which is the APR divided by 12. We
also must be careful to use the real interest rate. The Fisher equation uses the effective annual rate, so,
the real effective annual interest rates, and the monthly interest rates for each account are:
Stock account:
(1 + R) = (1 + r)(1 + h)
1 + .12 = (1 + r)(1 + .04)
r = .0769, or 7.69%
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APR = m[(1 + EAR)1/m – 1]
APR = 12[(1 + .0769)1/12 – 1]
APR = .0743, or 7.43%
Monthly rate = APR / 12
Monthly rate = .0743 / 12
Monthly rate = .0062, or 0.62%
Bond account:
(1 + R) = (1 + r)(1 + h)
1 + .07 = (1 + r)(1 + .04)
r = .0288, or 2.88%
APR = m[(1 + EAR)1/m – 1]
APR = 12[(1 + .0288)1/12 – 1]
APR = .0285, or 2.85%
Monthly rate = APR / 12
Monthly rate = .0285 / 12
Monthly rate = .0024, or 0.24%
Now we can find the future value of the retirement account in real terms. The future value of each
account will be:
Stock account:
FVA = C {(1 + r )t – 1] / r}
FVA = $900{[(1 + .0062)360 – 1] / .0062]}
FVA = $1,196,731.96
Bond account:
FVA = C {(1 + r )t – 1] / r}
FVA = $300{[(1 + .0024)360 – 1] / .0024]}
FVA = $170,316.78
The total future value of the retirement account will be the sum of the two accounts, or:
Account value = $1,196,731.96 + 170,316.78
Account value = $1,367,048.74
Now we need to find the monthly interest rate in retirement. We can use the same procedure that we
used to find the monthly interest rates for the stock and bond accounts, so:
(1 + R) = (1 + r)(1 + h)
1 + .08 = (1 + r)(1 + .04)
r = .0385 or 3.85%
APR = m[(1 + EAR)1/m – 1]
APR = 12[(1 + .0385)1/12 – 1]
APR = .0378, or 3.78%
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Monthly rate = APR / 12
Monthly rate = .0378 / 12
Monthly rate = .0031, or 0.31%
Now we can find the real monthly withdrawal in retirement. Using the present value of an annuity
equation and solving for the payment, we find:
PVA = C({1 – [1/(1 + r)]t } / r )
$1,367,048.74 = C({1 – [1/(1 + .0031)]300 } / .0031)
C = $7,050.75
This is the real dollar amount of the monthly withdrawals. The nominal monthly withdrawals will
increase by the inflation rate each month. To find the nominal dollar amount of the last withdrawal, we
can increase the real dollar withdrawal by the inflation rate. We can increase the real withdrawal by the
effective annual inflation rate since we are only interested in the nominal amount of the last
withdrawal. So, the last withdrawal in nominal terms will be:
FV = PV(1 + r)t
FV = $7,050.75(1 + .04)(30 + 25)
FV = $60,963.34
32. In this problem, we need to calculate the future value of the annual savings after the five years of
operations. The savings are the revenues minus the costs, or:
Savings = Revenue – Costs
Since the annual fee and the number of members are increasing, we need to calculate the effective
growth rate for revenues, which is:
Effective growth rate = (1 + .06)(1 + .03) – 1
Effective growth rate = .0918 or 9.18%
The revenue for the current year is the number of members times the annual fee, or:
Current revenue = 600($500)
Current revenue = $300,000
The revenue will grow at 9.18 percent, and the costs will grow at 2 percent, so the savings each year
for the next five years will be:
Year
1
2
3
4
5
Revenue
$327,540.00
357,608.17
390,436.60
426,278.68
465,411.07
Costs
$127,500.00
130,050.00
132,651.00
135,304.02
138,010.10
Savings
$200,040.00
227,558.17
257,785.60
290,974.66
327,400.96
Now we can find the value of each year’s savings using the future value of a lump sum equation, so:
FV = PV(1 + r)t
Solutions Manual
Year
1
$200,040.00(1 + .09)4 =
2
$227,558.17(1 + .09)3 =
3
$257,785.60(1 + .09)2 =
4
$290,974.66(1 + .09)1 =
5
Total future value of savings =
Future Value
$282,372.79
294,694.43
306,275.07
317,162.38
327,400.96
$1,527,905.64
He will spend $500,000 on a luxury boat, so the value of his account will be:
Value of account = $1,527,905.64 – 500,000
Value of account = $1,027,905.64
Now we can use the present value of an annuity equation to find the payment. Doing so, we find:
PVA = C({1 – [1/(1 + r)]t } / r )
$1,027,905.64 = C({1 – [1/(1 + .09)]25 } / .09)
C = $104,647.22
Calculator Solutions
1.
a.
Enter
30
N
2.5%
I/Y
30
N
5%
I/Y
30
N
7.5%
I/Y
30
N
3.5%
I/Y
30
N
4.5%
I/Y
Solve for
b.
Enter
Solve for
c.
Enter
Solve for
2.
a.
Enter
Solve for
b.
Enter
Solve for
PV
$476.74
PMT
$1,000
FV
PV
$231.38
PMT
$1,000
FV
PV
$114.22
PMT
$1,000
FV
$35
PMT
$1,000
FV
$35
PMT
$1,000
FV
PV
$1,000.00
PV
$837.11
Solutions Manual
c.
Enter
30
N
2.5%
I/Y
Solve for
3.
Enter
26
N
Solve for
2.923% × 2 = 5.85%
4.
Enter
I/Y
2.923%
23
N
3.8%
I/Y
PV
$1,209.30
±$1,050
PV
±$1,060
PV
Solve for
$41.96 × 2 = $83.92
$83.92 / $1,000 = 8.39%
5.
Enter
19
N
3.90%
I/Y
Solve for
6.
Enter
21
N
Solve for
13.
P0
Enter
I/Y
3.34%
50
N
3.5%
I/Y
48
N
3.5%
I/Y
Solve for
P1
Enter
Solve for
$191.81 – 179.05 = $12.75
14. Miller Corporation
P0
Enter
26
N
Solve for
P1
Enter
Solve for
24
N
3%
I/Y
3%
I/Y
$35
PMT
$1,000
FV
$32
PMT
$1,000
FV
PMT
$41.96
$1,000
FV
€45
PMT
€1,000
FV
±¥92,000
PV
¥2,800
PMT
¥100,000
FV
PV
$179.05
PMT
$1,000
FV
PMT
$1,000
FV
$40
PMT
$1,000
FV
$40
PMT
$1,000
FV
PV
€1,079.48
PV
$191.81
PV
$1,178.77
PV
$1,169.36
Solutions Manual
P3
Enter
20
N
3%
I/Y
10
N
3%
I/Y
2
N
3%
I/Y
Solve for
P8
Enter
Solve for
P12
Enter
Solve for
PV
$1,148.77
PV
$1,085.30
PV
$1,019.13
$40
PMT
$1,000
FV
$40
PMT
$1,000
FV
$40
PMT
$1,000
FV
$30
PMT
$1,000
FV
$30
PMT
$1,000
FV
$30
PMT
$1,000
FV
$30
PMT
$1,000
FV
$30
PMT
$1,000
FV
Modigliani Company
P0
Enter
26
N
4%
I/Y
24
N
4%
I/Y
20
N
4%
I/Y
10
N
4%
I/Y
2
N
4%
I/Y
Solve for
P1
Enter
Solve for
P3
Enter
Solve for
P8
Enter
Solve for
P12
Enter
Solve for
PV
$840.17
PV
$847.53
PV
$864.10
PV
$918.89
PV
$981.14
15. If both bonds sell at par, the initial YTM on both bonds is the coupon rate, 7 percent. If the YTM
suddenly rises to 9 percent:
PLaurel
Enter
4
N
4.5%
I/Y
PV
Solve for
$964.12
∆PLaurel% = ($964.12 – 1,000) / $1,000 = –3.59%
PHardy
$35
PMT
$1,000
FV
Solutions Manual
Enter
30
N
4.5%
I/Y
PV
Solve for
$837.11
∆PHardy% = ($837.11 – 1,000) / $1,000 = –16.29%
If the YTM suddenly falls to 5 percent:
PLaurel
Enter
4
2.5%
N
I/Y
PV
Solve for
$1,037.62
∆ PLaurel % = ($1,037.62 – 1,000) / $1,000 = + 3.76%
PHardy
Enter
30
N
2.5%
I/Y
PV
Solve for
$1,209.93
∆ PHardy % = ($1,209.93 – 1,000) / $1,000 = + 20.93%
$35
PMT
$1,000
FV
$35
PMT
$1,000
FV
$35
PMT
$1,000
FV
All else the same, the longer the maturity of a bond, the greater is its price sensitivity to changes in
interest rates.
16. Initially, at a YTM of 10 percent, the prices of the two bonds are:
PFaulk
Enter
24
N
5%
I/Y
24
N
5%
I/Y
Solve for
PGonas
Enter
Solve for
PV
$724.03
PV
$1,275.97
$30
PMT
$1,000
FV
$70
PMT
$1,000
FV
If the YTM rises from 10 percent to 12 percent:
PFaulk
Enter
24
N
6%
I/Y
$30
PMT
$1,000
FV
24
N
6%
I/Y
$70
PMT
$1,000
FV
$30
PMT
$1,000
FV
PV
Solve for
$623.49
∆PFaulk% = ($623.49 – 724.03) / $724.03 = –13.89%
PGonas
Enter
PV
Solve for
$1,125.50
∆PGonas% = ($1,125.50 – 1,275.97) / $1,275.97 = –11.79%
If the YTM declines from 10 percent to 8 percent:
PFaulk
Enter
24
4%
N
I/Y
PV
Solutions Manual
Solve for
$847.53
∆PFaulk% = ($847.53 – 724.03) / $724.03 = +17.06%
PGonas
Enter
24
N
4%
I/Y
PV
Solve for
$1,457.41
∆PGonas% = ($1,457.41 – 1,275.97) / $1,275.97 = +14.22%
$70
PMT
$1,000
FV
All else the same, the lower the coupon rate on a bond, the greater is its price sensitivity to
changes in interest rates.
17.
Enter
18
N
I/Y
Solve for
2.744%
YTM = 2.744% × 2 = 5.49%
±$1,050
PV
$31
PMT
$1,000
FV
18. The company should set the coupon rate on its new bonds equal to the required return; the required
return can be observed in the market by finding the YTM on outstanding bonds of the company.
Enter
40
N
Solve for
3.218% × 2 = 6.44%
I/Y
3.218%
±$1,063
PV
$35
PMT
$1,000
FV
±$1,068.88
PV
$90
PMT
$1,000
FV
±$1,053.12
PV
$36.20
PMT
$1,000
FV
PV
$179.05
PMT
$1,000
FV
PMT
$1,000
FV
21. Current yield = .0842 = $90/P0 ; P0 = $1,068.88
Enter
Solve for
8 years
22.
Enter
N
8.0004
18
N
Solve for
3.226% × 2 = 6.45%
24.
a. Po
Enter
7.81%
I/Y
I/Y
3.226%
50
N
7%/2
I/Y
48
N
7%/2
I/Y
Solve for
b. P1
Enter
PV
Solve for
$191.81
year 1 interest deduction = $191.81 – 179.05 = $12.75
Solutions Manual
P24
Enter
2
N
7%/2%
I/Y
PV
Solve for
$933.51
year 25 interest deduction = $1,000 – 933.51 = $66.49
c.
PMT
$1,000
FV
Total interest = $1,000 – 179.05 = $820.95
Annual interest deduction = $820.95 / 25 = $32.84
The company will prefer straight-line method when allowed because the valuable interest
deductions occur earlier in the life of the bond.
d.
25. a.
The coupon bonds have a 6% coupon rate, which matches the 6% required return, so they will
sell at par; # of bonds = $45,000,000/$1,000 = 45,000.
For the zeroes:
Enter
60
N
6%/2
I/Y
PV
Solve for
$169.73
$45,000,000/$169.73 = 265,122 will be issued.
b.
PMT
$1,000
FV
Coupon bonds: repayment = 45,000($1,030) = $46,350,000
Zeroes: repayment = 265,122($1,000) = $265,122,140
Coupon bonds: (45,000)($60)(1 –.35) = $1,755,000 cash outflow
Zeroes:
c.
Enter
58
N
6%/2
I/Y
10
N
7%
I/Y
9
N
7%
I/Y
$1,000
FV
PV
PMT
$180.07
year 1 interest deduction = $180.07 – 169.73 = $10.34
(265,122)($10.34)(.35) = $959,175.00 cash inflow
During the life of the bond, the zero generates cash inflows to the firm in the form of the
interest tax shield of debt.
Solve for
26.
Bond P
P0
Enter
Solve for
P1
Enter
PV
$1,140.47
$90
PMT
$90
PV
PMT
Solve for
$1,130.30
Current yield = $90 / $1,140.47 = 7.89%
Capital gains yield = ($1,130.30 – 1,140.47) / $1,140.47 = –0.89%
Bond D
P0
Enter
10
7%
$50
$1,000
FV
$1,000
FV
$1,000
Solutions Manual
N
I/Y
9
N
7%
I/Y
Solve for
P1
Enter
PV
$859.53
PV
$869.70
PMT
FV
$50
PMT
$1,000
FV
Solve for
Current yield = $50 / $859.53 = 5.82%
Capital gains yield = ($869.70 – 859.53) / $859.53 = +1.18%
All else held constant, premium bonds pay a higher current income while having price depreciation as
maturity nears; discount bonds pay a lower current income but have price appreciation as maturity
nears. For either bond, the total return is still 7%, but this return is distributed differently between
current income and capital gains.
27.
a.
Enter
10
N
±$930
$56
$1,000
I/Y
PV
PMT
FV
Solve for
6.58%
This is the rate of return you expect to earn on your investment when you purchase the bond.
b.
Enter
8
N
5.58%
I/Y
Solve for
PV
$1,001.44
$56
PMT
$1,000
FV
The HPY is:
Enter
±$930
$56
$1,001.44
I/Y
PV
PMT
FV
Solve for
9.68%
The realized HPY is greater than the expected YTM when the bond was bought because interest
rates dropped by 1 percent; bond prices rise when yields fall.
28.
PM
CFo
$0
$0
C01
12
F01
$800
C02
16
F02
$1,000
C03
11
F03
$31,000
C04
1
F04
I = 4%
NPV CPT
$15,200.77
PN
Enter
2
N
40
4%
$30,000
Solutions Manual
N
I/Y
Solve for
PV
$6,248.67
PMT
FV
31.
Real return for stock account: 1 + .12 = (1 + r)(1 + .04); r = 7.6923%
Enter
7.6923%
12
NOM
EFF
C/Y
Solve for
7.4337%
Real return for bond account: 1 + .07 = (1 + r)(1 + .04); r = 2.8846%
Enter
2.8846%
12
NOM
EFF
C/Y
Solve for
2.8472%
Real return post-retirement: 1 + .08 = (1 + r)(1 + .04); r = 3.8462%
Enter
3.8462%
12
NOM
EFF
C/Y
Solve for
3.7800%
Stock portfolio value:
Enter
12 × 30
N
Solve for
7.4337% / 12
I/Y
PV
$800
PMT
Bond portfolio value:
Enter
12 × 30
N
Solve for
2.8472% / 12
I/Y
PV
$400
PMT
FV
$1,196,731.96
FV
$170,316.78
Retirement value = $1,196,731.96 + 170,316.78 = $1,367,048.74
Retirement withdrawal:
Enter
25 × 12
N
Solve for
3.7800% / 12
I/Y
PMT
$7,050.75
FV
$7,050.75
PV
PMT
FV
$60,963.34
9%
I/Y
$200,040
PV
PMT
FV
$282,372.79
9%
I/Y
$227,558.17
PV
PMT
FV
The last withdrawal in real terms is:
Enter
30 + 25
4%
N
I/Y
Solve for
32.
Future value of savings:
Year 1:
Enter
4
N
Solve for
Year 2:
Enter
3
N
$1,367,048.74
PV
Solutions Manual
Solve for
Year 3:
Enter
$294,694.43
2
N
9%
I/Y
$257,785.60
PV
PMT
FV
$306,275.07
1
N
9%
I/Y
$290,974.66
PV
PMT
FV
$317,162.38
Solve for
Year 4:
Enter
Solve for
Future value = $282,372.79 + 294,694.43 + 306,275.07 + 317,162.38 + 327,400.96
Future value = $1,527,905.64
He will spend $500,000 on a luxury boat, so the value of his account will be:
Value of account = $1,527,905.64 – 500,000
Value of account = $1,027,905.64
Enter
Solve for
25
N
9%
I/Y
$1,027,905.64
PV
PMT
$104,647.22
FV
CHAPTER 9
STOCK VALUATION
Solutions to Questions and Problems
1.
The constant dividend growth model is:
Pt = Dt × (1 + g) / (R – g)
So, the price of the stock today is:
P0 = D0 (1 + g) / (R – g) = $1.90 (1.05) / (.11 – .05) = $37.63
The dividend at year 4 is the dividend today times the FVIF for the growth rate in dividends and four
years, so:
P3 = D3 (1 + g) / (R – g) = D0 (1 + g)4 / (R – g) = $1.90 (1.05)4 / (.11 – .05) = $43.56
We can do the same thing to find the dividend in Year 16, which gives us the price in Year 15, so:
P15 = D15 (1 + g) / (R – g) = D0 (1 + g)16 / (R – g) = $1.90 (1.05)16 / (.11 – .05) = $78.22
There is another feature of the constant dividend growth model: The stock price grows at the dividend
growth rate. So, if we know the stock price today, we can find the future value for any time in the
future we want to calculate the stock price. In this problem, we want to know the stock price in three
years, and we have already calculated the stock price today. The stock price in three years will be:
P3 = P0(1 + g)3 = $37.63(1 + .05)3 = $43.56
And the stock price in 15 years will be:
P15 = P0(1 + g)15 = $37.63(1 + .05)15 = $78.22
2.
We need to find the required return of the stock. Using the constant growth model, we can solve the
equation for R. Doing so, we find:
R = (D1 / P0) + g = ($3.20 / $63.50) + .06 = .1104, or 11.04%
3.
The dividend yield is the dividend next year divided by the current price, so the dividend yield is:
Dividend yield = D1 / P0 = $3.20 / $63.50 = .0504, or 5.04%
The capital gains yield, or percentage increase in the stock price, is the same as the dividend growth
rate, so:
Capital gains yield = 6%
Solutions Manual
4.
Using the constant growth model, we find the price of the stock today is:
P0 = D1 / (R – g) = $2.65 / (.11 – .0475) = $42.40
5.
The required return of a stock is made up of two parts: The dividend yield and the capital gains yield.
So, the required return of this stock is:
R = Dividend yield + Capital gains yield = .043 + .064 = .1070, or 10.70%
6.
We know the stock has a required return of 11.5 percent, and the dividend and capital gains yield are
equal, so:
Dividend yield = 1/2(.115) = .0575 = Capital gains yield
Now we know both the dividend yield and capital gains yield. The dividend is simply the stock price
times the dividend yield, so:
D1 = .0575($72) = $4.14
This is the dividend next year. The question asks for the dividend this year. Using the relationship
between the dividend this year and the dividend next year:
D1 = D0(1 + g)
We can solve for the dividend that was just paid:
$4.14 = D0 (1 + .0575)
D0 = $4.14 / 1.0575 = $3.91
7.
The price of any financial instrument is the PV of the future cash flows. The future dividends of this
stock are an annuity for 12 years, so the price of the stock is the PVA, which will be:
P0 = $9(PVIFA10%,12) = $61.32
8.
The price of a share of preferred stock is the dividend divided by the required return. This is the same
equation as the constant growth model, with a dividend growth rate of zero percent. Remember that
most preferred stock pays a fixed dividend, so the growth rate is zero. Using this equation, we find the
price per share of the preferred stock is:
R = D/P0 = $5.90/$87 = .0678, or 6.78%
9.
The growth rate of earnings is the return on equity times the retention ratio, so:
g = ROE × b
g = .16(.80)
g = .1280, or 12.80%
To find next year’s earnings, we simply multiply the current earnings times one plus the growth rate,
so:
Solutions Manual
Next year’s earnings = Current earnings(1 + g)
Next year’s earnings = $34,000,000(1 + .1280)
Next year’s earnings = $38,352,000
10. Using the equation to calculate the price of a share of stock with the PE ratio:
P = Benchmark PE ratio × EPS
So, with a PE ratio of 18, we find:
P = 18($1.75)
P = $31.50
And with a PE ratio of 21, we find:
P = 21($1.75)
P = $36.75
Intermediate
11. This stock has a constant growth rate of dividends, but the required return changes twice. To find the
value of the stock today, we will begin by finding the price of the stock at Year 6, when both the
dividend growth rate and the required return are stable forever. The price of the stock in Year 6 will be
the dividend in Year 7, divided by the required return minus the growth rate in dividends. So:
P6 = D6 (1 + g) / (R – g) = D0 (1 + g)7 / (R – g) = $3.10(1.06)7 / (.11 – .06) = $93.23
Now we can find the price of the stock in Year 3. We need to find the price here since the required
return changes at that time. The price of the stock in Year 3 is the PV of the dividends in Years 4, 5,
and 6, plus the PV of the stock price in Year 6. The price of the stock in Year 3 is:
P3 = $3.10(1.06)4 / 1.13 + $3.10(1.06)5 / 1.132 + $3.10(1.06)6 / 1.133 + $93.23 / 1.133
P3 = $74.37
Finally, we can find the price of the stock today. The price today will be the PV of the dividends in
Years 1, 2, and 3, plus the PV of the stock in Year 3. The price of the stock today is:
P0
P0
= $3.10(1.06) / 1.15 + $3.10(1.06)2 / (1.15)2 + $3.10(1.06)3 / (1.15)3 + $74.37 / (1.15)3
= $56.82
12. Here we have a stock that pays no dividends for 9 years. Once the stock begins paying dividends, it
will have a constant growth rate of dividends. We can use the constant growth model at that point. It is
important to remember that the general form of the constant dividend growth formula is:
Pt = [Dt × (1 + g)] / (R – g)
This means that since we will use the dividend in Year 10, we will be finding the stock price in Year 9.
The dividend growth model is similar to the PVA and the PV of a perpetuity: The equation gives you
the PV one period before the first payment. So, the price of the stock in Year 9 will be:
P9 = D10 / (R – g) = $15.00 / (.13 – .055) = $200
Solutions Manual
The price of the stock today is simply the PV of the stock price in the future. We simply discount the
future stock price at the required return. The price of the stock today will be:
P0 = $200 / 1.139 = $66.58
13. The price of a stock is the PV of the future dividends. This stock is paying five dividends, so the price
of the stock is the PV of these dividends using the required return. The price of the stock is:
P0 = $15 / 1.12 + $18 / 1.122 + $21 / 1.123 + $24 / 1.124 + $27 / 1.125 = $73.26
14. With differential dividends, we find the price of the stock when the dividends level off at a constant
growth rate, and then find the PV of the future stock price, plus the PV of all dividends during the
differential growth period. The stock begins constant growth in Year 5, so we can find the price of the
stock in Year 4, one year before the constant dividend growth begins, as:
P4 = D4 (1 + g) / (R – g) = $2.75(1.05) / (.13 – .05) = $36.09
The price of the stock today is the PV of the first four dividends, plus the PV of the Year 4 stock price.
So, the price of the stock today will be:
P0 = $10 / 1.13 + $7 / 1.132 + $6 / 1.133 + ($2.75 + 36.09) / 1.134 = $42.31
15. With differential dividends, we find the price of the stock when the dividends level off at a constant
growth rate, and then find the PV of the future stock price, plus the PV of all dividends during the
differential growth period. The stock begins constant growth in Year 4, so we can find the price of the
stock in Year 3, one year before the constant dividend growth begins as:
P3
= D3 (1 + g) / (R – g) = D0 (1 + g1)3 (1 + g2) / (R – g2) = $2.80(1.20)3(1.05) / (.12 – .05) = $72.58
The price of the stock today is the PV of the first three dividends, plus the PV of the Year 3 stock price.
The price of the stock today will be:
P0 = $2.80(1.20) / 1.12 + $2.80(1.20)2 / 1.122 + $2.80(1.20)3 / 1.123 + $72.58 / 1.123
P0 = $61.32
16. Here we need to find the dividend next year for a stock experiencing differential growth. We know the
stock price, the dividend growth rates, and the required return, but not the dividend. First, we need to
realize that the dividend in Year 3 is the current dividend times the FVIF. The dividend in Year 3 will
be:
D3 = D0 (1.30)3
And the dividend in Year 4 will be the dividend in Year 3 times one plus the growth rate, or:
D4 = D0 (1.30)3 (1.18)
The stock begins constant growth after the 4th dividend is paid, so we can find the price of the stock in
Year 4 as the dividend in Year 5, divided by the required return minus the growth rate. The equation
for the price of the stock in Year 4 is:
Solutions Manual
P4
= D4 (1 + g) / (R – g)
Now we can substitute the previous dividend in Year 4 into this equation as follows:
P4 = D0 (1 + g1)3 (1 + g2) (1 + g3) / (R – g3)
P4 = D0 (1.30)3 (1.18) (1.08) / (.11 – .08) = 93.33D0
When we solve this equation, we find that the stock price in Year 4 is 93.33 times as large as the
dividend today. Now we need to find the equation for the stock price today. The stock price today is
the PV of the dividends in Years 1, 2, 3, and 4, plus the PV of the Year 4 price. So:
P0 = D0(1.30)/1.11 + D0(1.30)2/1.112 + D0(1.30)3/1.113+ D0(1.30)3(1.18)/1.114 + 93.33D0/1.114
We can factor out D0 in the equation, and combine the last two terms. Doing so, we get:
P0 = $65.00 = D0{1.30/1.11 + 1.302/1.112 + 1.303/1.113 + [(1.30)3(1.18) + 93.33] / 1.114}
Reducing the equation even further by solving all of the terms in the braces, we get:
$65 = $67.34D0
D0 = $65.00 / $67.34 = $.97
This is the dividend today, so the projected dividend for the next year will be:
D1 = $.97(1.30) = $1.25
17. The constant growth model can be applied even if the dividends are declining by a constant percentage,
just make sure to recognize the negative growth. So, the price of the stock today will be:
P0 = D0 (1 + g) / (R – g) = $9(1 – .04) / [(.11 – (–.04)] = $57.60
18. We are given the stock price, the dividend growth rate, and the required return, and are asked to find
the dividend. Using the constant dividend growth model, we get:
P0 = $58.32 = D0 (1 + g) / (R – g)
Solving this equation for the dividend gives us:
D0 = $58.32(.115 – .05) / (1.05) = $3.61
19. The price of a share of preferred stock is the dividend payment divided by the required return. We
know the dividend payment in Year 5, so we can find the price of the stock in Year 4, one year before
the first dividend payment. Doing so, we get:
P4 = $8.00 / .056 = $142.86
The price of the stock today is the PV of the stock price in the future, so the price today will be:
P0 = $142.86 / (1.056)4 = $114.88
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20. The dividend yield is the annual dividend divided by the stock price, so:
Dividend yield = Dividend / Stock price
.019 = Dividend / $26.18
Dividend = $0.50
The “Net Chg” of the stock shows the stock decreased by $0.13 on this day, so the closing stock price
yesterday was:
Yesterday’s closing price = $26.18 – (–0.13) = $26.31
To find the net income, we need to find the EPS. The stock quote tells us the P/E ratio for the stock is
23. Since we know the stock price as well, we can use the P/E ratio to solve for EPS as follows:
P/E = 23 = Stock price / EPS = $26.18 / EPS
EPS = $26.18 / 23 = $1.138
We know that EPS is just the total net income divided by the number of shares outstanding, so:
EPS = NI / Shares = $1.138 = NI / 25,000,000
NI = $1.138(25,000,000) = $28,456,522
21. To find the number of shares owned, we can divide the amount invested by the stock price. The share
price of any financial asset is the present value of the cash flows, so, to find the price of the stock we
need to find the cash flows. The cash flows are the two dividend payments plus the sale price. We also
need to find the aftertax dividends since the assumption is all dividends are taxed at the same rate for
all investors. The aftertax dividends are the dividends times one minus the tax rate, so:
Year 1 aftertax dividend = $2.25(1 – .28)
Year 1 aftertax dividend = $1.62
Year 2 aftertax dividend = $2.40(1 – .28)
Year 2 aftertax dividend = $1.73
We can now discount all cash flows from the stock at the required return. Doing so, we find the price
of the stock is:
P = $1.62/1.15 + $1.73/(1.15)2 + $65/(1+.15)3
P = $45.45
The number of shares owned is the total investment divided by the stock price, which is:
Shares owned = $100,000 / $45.45
Shares owned = 2,200.03
22. Here we have a stock paying a constant dividend for a fixed period, and an increasing dividend
thereafter. We need to find the present value of the two different cash flows using the appropriate
quarterly interest rate. The constant dividend is an annuity, so the present value of these dividends is:
Solutions Manual
PVA = C(PVIFAR,t)
PVA = $0.80(PVIFA2.5%,12)
PVA = $8.21
Now we can find the present value of the dividends beyond the constant dividend phase. Using the
present value of a growing annuity equation, we find:
P12 = D13 / (R – g)
P12 = $0.80(1 + .01) / (.025 – .01)
P12 = $53.87
This is the price of the stock immediately after it has paid the last constant dividend. So, the present
value of the future price is:
PV = $53.87 / (1 + .025)12
PV = $40.05
The price today is the sum of the present value of the two cash flows, so:
P0 = $8.21 + 40.05
P0 = $48.26
23. Here we need to find the dividend next year for a stock with nonconstant growth. We know the stock
price, the dividend growth rates, and the required return, but not the dividend. First, we need to realize
that the dividend in Year 3 is the constant dividend times the FVIF. The dividend in Year 3 will be:
D3 = D(1.04)
The equation for the stock price will be the present value of the constant dividends, plus the present
value of the future stock price, or:
P0 = D / 1.11 + D / 1.112 + D[(1.04) / (.11 – .04)] / 1.112
$45 = D / 1.11 + D / 1.112 + D[(1.04) / (.11 – .04)] / 1.112
We can factor out D0 in the equation. Doing so, we get:
$45 = D{1 / 1.11 + 1 / 1.112 + [(1.04) / (.11 – .04)] / 1.112}
Reducing the equation even further by solving all of the terms in the braces, we get:
$45 = D(13.7709)
D = $45 / 13.7709 = $3.27
24. The required return of a stock consists of two components, the capital gains yield and the dividend
yield. In the constant dividend growth model (growing perpetuity equation), the capital gains yield is
the same as the dividend growth rate, or algebraically:
R = D1/P0 + g
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We can find the dividend growth rate by the growth rate equation, or:
g = ROE × b
g = .13 × .70
g = .0910, or 9.10%
This is also the growth rate in dividends. To find the current dividend, we can use the information
provided about the net income, shares outstanding, and payout ratio. The total dividends paid is the net
income times the payout ratio. To find the dividend per share, we can divide the total dividends paid by
the number of shares outstanding. So:
Dividend per share = (Net income × Payout ratio) / Shares outstanding
Dividend per share = ($18,000,000 × .30) / 2,000,000
Dividend per share = $2.70
Now we can use the initial equation for the required return. We must remember that the equation uses
the dividend in one year, so:
R = D1/P0 + g
R = $2.70(1 + .0910)/$93 + .0910
R = .1227, or 12.27%
25. First, we need to find the annual dividend growth rate over the past four years. To do this, we can use
the future value of a lump sum equation, and solve for the interest rate. Doing so, we find the dividend
growth rate over the past four years was:
FV = PV(1 + R)t
$1.77 = $1.35(1 + R)4
R = ($1.77 / $1.35)1/4 – 1
R = .0701, or 7.01%
We know the dividend will grow at this rate for five years before slowing to a constant rate indefinitely.
So, the dividend amount in seven years will be:
D7 = D0(1 + g1)5(1 + g2)2
D7 = $1.77(1 + .0701)5(1 + .05)2
D7 = $2.74
26. a.
We can find the price of all the outstanding company stock by using the dividends the same way
we would value an individual share. Since earnings are equal to dividends, and there is no growth,
the value of the company’s stock today is the present value of a perpetuity, so:
P=D/R
P = $950,000 / .12
P = $7,916,666.67
The price-earnings ratio is the stock price divided by the current earnings, so the price-earnings
ratio of each company with no growth is:
P/E = Price / Earnings
P/E = $7,916,666.67 / $950,000
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P/E = 8.33 times
b.
Since the earnings have increased, the price of the stock will increase. The new price of the
outstanding company stock is:
P=D/R
P = ($950,000 + 100,000) / .12
P = $8,750,000
The price-earnings ratio is the stock price divided by the current earnings, so the price-earnings
with the increased earnings is:
P/E = Price / Earnings
P/E = $8,750,000 / $950,000
P/E = 9.21 times
c.
Since the earnings have increased, the price of the stock will increase. The new price of the
outstanding company stock is:
P=D/R
P = ($950,000 + 200,000) / .12
P = $9,583,333.33
The price-earnings ratio is the stock price divided by the current earnings, so the price-earnings
with the increased earnings is:
P/E = Price / Earnings
P/E = $9,583,333.33 / $950,000
P/E = 10.09 times
27. a.
If the company does not make any new investments, the stock price will be the present value of
the constant perpetual dividends. In this case, all earnings are paid as dividends, so, applying the
perpetuity equation, we get:
P = Dividend / R
P = $9.40 / .12
P = $78.33
b.
The investment is a one-time investment that creates an increase in EPS for two years. To
calculate the new stock price, we need the cash cow price plus the NPVGO. In this case, the
NPVGO is simply the present value of the investment plus the present value of the increases in
EPS. So, the NPVGO will be:
NPVGO = C1 / (1 + R) + C2 / (1 + R)2 + C3 / (1 + R)3
NPVGO = –$1.95 / 1.12 + $2.75 / 1.122 + $3.05 / 1.123
NPVGO = $2.62
So, the price of the stock if the company undertakes the investment opportunity will be:
P = $78.33 + 2.62
P = $80.96
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c.
After the project is over, and the earnings increase no longer exists, the price of the stock will
revert back to $78.33, the value of the company as a cash cow.
28. a.
The price of the stock is the present value of the dividends. Since earnings are equal to dividends,
we can find the present value of the earnings to calculate the stock price. Also, since we are
excluding taxes, the earnings will be the revenues minus the costs. We simply need to find the
present value of all future earnings to find the price of the stock. The present value of the
revenues is:
PVRevenue = C1 / (R – g)
PVRevenue = $7,500,000(1 + .05) / (.13 – .05)
PVRevenue = $98,437,500
And the present value of the costs will be:
PVCosts = C1 / (R – g)
PVCosts = $3,400,000(1 + .05) / (.13 – .05)
PVCosts = $44,625,000
Since there are no taxes, the present value of the company’s earnings and dividends will be:
PVDividends = $98,437,500 – 44,625,000
PVDividends = $53,812,500
Note that since revenues and costs increase at the same rate, we could have found the present
value of future dividends as the present value of current dividends. Doing so, we find:
D0 = Revenue0 – Costs0
D0 = $7,500,000 – 3,400,000
D0 = $4,100,000
Now, applying the growing perpetuity equation, we find:
PVDividends = C1 / (R – g)
PVDividends = $4,100,000(1 + .05) / (.13 – .05)
PVDividends = $53,812,500
This is the same answer we found previously. The price per share of stock is the total value of the
company’s stock divided by the shares outstanding, or:
P = Value of all stock / Shares outstanding
P = $53,812,500 / 1,000,000
P = $53.81
b.
The value of a share of stock in a company is the present value of its current operations, plus the
present value of growth opportunities. To find the present value of the growth opportunities, we
need to discount the cash outlay in Year 1 back to the present, and find the value today of the
increase in earnings. The increase in earnings is a perpetuity, which we must discount back to
today. So, the value of the growth opportunity is:
Solutions Manual
NPVGO = C0 + C1 / (1 + R) + (C2 / R) / (1 + R)
NPVGO = –$17,000,000 – $6,000,000 / (1 + .13) + ($4,200,000 / .13) / (1 + .13)
NPVGO = $6,281,143.64
To find the value of the growth opportunity on a per share basis, we must divide this amount by
the number of shares outstanding, which gives us:
NPVGOPer share = $6,821,143.64 / 1,000,000
NPVGOPer share = $6.28
The stock price will increase by $6.28 per share. The new stock price will be:
New stock price = $53.81 + 6.28
New stock price = $60.09
29. a.
If the company continues its current operations, it will not grow, so we can value the company as
a cash cow. The total value of the company as a cash cow is the present value of the future
earnings, which are a perpetuity, so:
Cash cow value of company = C / R
Cash cow value of company = $71,000,000 / .12
Cash cow value of company = $591,666,666.67
The value per share is the total value of the company divided by the shares outstanding, so:
Share price = $591,666,666.67 / 15,000,000
Share price = $39.44
b.
To find the value of the investment, we need to find the NPV of the growth opportunities. The
initial cash flow occurs today, so it does not need to be discounted. The earnings growth is a
perpetuity. Using the present value of a perpetuity equation will give us the value of the earnings
growth one period from today, so we need to discount this back to today. The NPVGO of the
investment opportunity is:
NPVGO = C0 + C1 / (1 + R) + (C2 / R) / (1 + R)
NPVGO = –$16,000,000 – 5,000,000 / (1 + .12) + ($11,000,000 / .12) / (1 + .12)
NPVGO = $61,380,952.38
c.
The price of a share of stock is the cash cow value plus the NPVGO. We have already calculated
the NPVGO for the entire project, so we need to find the NPVGO on a per share basis. The
NPVGO on a per share basis is the NPVGO of the project divided by the shares outstanding,
which is:
NPVGO per share = $61,380,952.38 / 15,000,000
NPVGO per share = $4.09
This means the per share stock price if the company undertakes the project is:
Share price = Cash cow price + NPVGO per share
Share price = $39.44 + 4.09
Share price = $43.54
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30. a.
Using the equation to calculate the price of a share of stock with the PE ratio:
P = Benchmark PE ratio × EPS
So, with a PE ratio of 21, we find:
P = 21($2.35)
P = $49.35
b.
First, we need to find the earnings per share next year, which will be:
EPS1 = EPS0(1 + g)
EPS1 = $2.35(1 + .07)
EPS1 = $2.51
Using the equation to calculate the price of a share of stock with the PE ratio:
P1 = Benchmark PE ratio × EPS1
P1 = 21($2.51)
P1 = $52.80
c.
To find the implied return over the next year, we calculate the return as:
R = (P1 – P0) / P0
R = ($52.80 – 49.35) / $49.35
R = .07, or 7%
Notice that the return is the same as the growth rate in earnings. Assuming a stock pays no
dividends and the PE ratio is constant, this will always be true when using price ratios to evaluate
the price of a share of stock.
31. We need to find the enterprise value of the company. We can calculate EBITDA as sales minus costs,
so:
EBITDA = Sales – Costs
EBITDA = $28,000,000 – 12,000,000
EBITDA = $16,000,000
Solving the EV/EBITDA multiple for enterprise value, we find:
Enterprise value = $16,000,000(7.5)
Enterprise value = $120,000,000
The total value of equity is the enterprise value minus any outstanding debt and cash, so:
Equity value = Enterprise value – Debt – Cash
Equity value = $120,000,000 – 54,000,000 – 18,000,000
Equity value = $48,000,000
So, the price per share is:
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Stock price = $48,000,000 / 950,000
Stock price = $50.53
32. a.
To value the stock today, we first need to calculate the cash flows for the next 6 years. The sales,
costs, and net investment all grow by same rate, namely 14 percent, 12 percent, 10 percent, 8
percent, respectively, for the following 4 years, then 6 percent indefinitely. So, the cash flows for
each year will be:
Year 1
Year 2
Year 3
Year 4
Year 5
Sales
$145,000,000 $165,300,000 $185,136,000 $203,649,600 $219,941,568
Costs
81,000,000
92,340,000 103,420,800 113,762,880 122,863,910
EBT
$64,000,000 $72,960,000 $81,715,200 $89,886,720 $97,077,658
Taxes
25,600,000
29,184,000
32,686,080
35,954,688
38,831,063
Net income $38,400,000 $43,776,000 $49,029,120 $53,932,032 $58,246,595
Investment
15,000,000
17,100,000
19,152,000
21,067,200
22,752,576
Cash flow
$23,400,000 $26,676,000 $29,877,120 $32,864,832 $35,494,019
Year 6
$233,138,062
130,235,745
$102,902,317
41,160,927
$61,741,390
24,117,731
$37,623,660
To find the terminal value of the company in Year 6, we can discount the Year 7 cash flows as a
growing perpetuity, which will be:
Terminal value = $37,623,660(1 + .06) / (.13 – .06)
Terminal value = $569,729,704
So, the value of the company today is:
Company value today = $23,400,000 / 1.13 + $26,676,000 / 1.132 + $29,877,120 / 1.133
+ $32,864,832 / 1.134 + 35,494,019 / 1.135 + ($37,623,660 + 569,729,704) / 1.136
Company value today = $393,449,950
Dividing the company value by the shares outstanding to get the share price we get:
Share price = $393,449,950 / 5,500,000
Share price = $71.54
b.
In this case, we are going to use the PE multiple to find the terminal value. All of the cash flows
from part a will remain the same. So, the terminal value in Year 6 is:
Terminal value = 11($61,741,390)
Terminal value = $679,155,293
Under this assumption for the terminal value, the value of the company today is:
Company value today = $23,400,000 / 1.13 + $26,676,000 / 1.132 + $29,877,120 / 1.133
+ $32,864,832 / 1.134 + 35,494,019 / 1.135 + ($37,623,660 + 679,155,293) / 1.136
Company value today = $446,009,087
Dividing the company value by the shares outstanding to get the share price we get:
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Share price = $446,009,087 / 5,500,000
Share price = $81.09
33. We are asked to find the dividend yield and capital gains yield for each of the stocks. All of the stocks
have a 17 percent required return, which is the sum of the dividend yield and the capital gains yield. To
find the components of the total return, we need to find the stock price for each stock. Using this stock
price and the dividend, we can calculate the dividend yield. The capital gains yield for the stock will be
the total return (required return) minus the dividend yield.
W: P0 = D0(1 + g) / (R – g) = $3.50(1.085) / (.17 – .085) = $44.68
Dividend yield = D1/P0 = 3.50(1.085)/$44.68 = .085, or 8.5%
Capital gains yield = .17 – .085 = .085, or 8.5%
X:
P0 = D0(1 + g) / (R – g) = $3.50 / (.17 – 0) = $20.59
Dividend yield = D1/P0 = $3.50 / $20.59 = .17, or 17%
Capital gains yield = .17 – .17 = 0%
Y:
P0 = D0(1 + g) / (R – g) = $3.50(1 – .05) / (.17 + .05) = $15.11
Dividend yield = D1/P0 = $3.50(0.95) / $15.11 = .22, or 22%
Capital gains yield = .17 – .22 = –.05, or –5%
Z:
P2 = D2(1 + g) / (R – g) = D0(1 + g1)2(1 + g2) / (R – g2) = $3.50(1.30)2(1.08) / (.17 – .08)
P2 = $70.98
P0 = $3.50(1.30) / (1.17) + $3.50 (1.30)2 / (1.17)2 + $70.98 / (1.17)2
P0 = $60.06
Dividend yield = D1/P0 = $3.50(1.30)/$60.06 = .0758, or 7.58%
Capital gains yield = .17 – .0758 = .0942, or 9.42%
In all cases, the required return is 17 percent, but the return is distributed differently between current
income and capital gains. High-growth stocks have an appreciable capital gains component but a
relatively small current income yield; conversely, mature, negative-growth stocks provide a high
current income but also price depreciation over time.
34. a.
Using the constant growth model, the price of the stock paying annual dividends will be:
P0 = D0(1 + g) / (R – g) = $3.20(1.05) / (.11 – .05) = $56.00
b.
If the company pays quarterly dividends instead of annual dividends, the quarterly dividend will
be one-fourth of annual dividend, or:
Quarterly dividend: $3.20(1.05) / 4 = $.84
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To find the equivalent annual dividend, we must assume that the quarterly dividends are
reinvested at the required return. We can then use this interest rate to find the equivalent annual
dividend. In other words, when we receive the quarterly dividend, we reinvest it at the required
return on the stock. So, the effective quarterly rate is:
Effective quarterly rate: 1.11.25 – 1 = .0264, or 2.64%
The effective annual dividend will be the FVA of the quarterly dividend payments at the effective
quarterly required return. In this case, the effective annual dividend will be:
Effective D1 = $.84(FVIFA2.64%,4) = $3.50
Now, we can use the constant growth model to find the current stock price as:
P0 = $3.50/(.11 – .05) = $58.26
Note that we cannot simply find the quarterly effective required return and growth rate to find the
value of the stock. This would assume the dividends increased each quarter, not each year.
35. a.
If the company does not make any new investments, the stock price will be the present value of
the constant perpetual dividends. In this case, all earnings are paid as dividends, so, applying the
perpetuity equation, we get:
P = Dividend / R
P = $8.50 / .12
P = $70.83
b.
The investment occurs every year in the growth opportunity, so the opportunity is a growing
perpetuity. So, we first need to find the growth rate. The growth rate is:
g = Retention Ratio × Return on Retained Earnings
g = 0.20 × 0.10
g = 0.02, or 2.00%
Next, we need to calculate the NPV of the investment. During Year 3, 20 percent of the earnings
will be reinvested. Therefore, $1.70 is invested ($8.50 × .20). One year later, the shareholders
receive a return of 10 percent on the investment, or $0.17 ($1.70 × .10), in perpetuity. The
perpetuity formula values that stream as of Year 3. Since the investment opportunity will
continue indefinitely and grows at 2 percent, apply the growing perpetuity formula to calculate
the NPV of the investment as of Year 2. Discount that value back two years to today.
NPVGO = [(Investment + Return / R) / (R – g)] / (1 + R)2
NPVGO = [(–$1.70 + $0.17 / .12) / (0.12 – 0.02)] / (1.12)2
NPVGO = –$2.26
The value of the stock is the PV of the firm without making the investment plus the NPV of the
investment, or:
P = PV(EPS) + NPVGO
P = $70.83 – 2.26
P = $68.57
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c.
Zero percent! There is no retention ratio which would make the project profitable for the
company. If the company retains more earnings, the growth rate of the earnings on the
investment will increase, but the project will still not be profitable. Since the return of the project
is less than the required return on the company stock, the project is never worthwhile. In fact, the
more the company retains and invests in the project, the less valuable the stock becomes.
36. Here we have a stock with differential growth, but the dividend growth changes every year for the first
four years. We can find the price of the stock in Year 3 since the dividend growth rate is constant after
the third dividend. The price of the stock in Year 3 will be the dividend in Year 4, divided by the
required return minus the constant dividend growth rate. So, the price in Year 3 will be:
P3 = $3.85(1.20)(1.15)(1.10)(1.05) / (.13 – .05) = $76.71
The price of the stock today will be the PV of the first three dividends, plus the PV of the stock price in
Year 3, so:
P0 = $3.85(1.20) / (1.13) + $3.85(1.20)(1.15) / 1.132 + $3.85(1.20)(1.15)(1.10) / 1.133
+ $76.71 / 1.133
P0 = $65.46
37. Here we want to find the required return that makes the PV of the dividends equal to the current stock
price. The equation for the stock price is:
P = $3.85(1.20) / (1 + R) + $3.85(1.20)(1.15) / (1 + R)2 + $3.85(1.20)(1.15)(1.10) / (1 + R)3
+ [$3.85(1.20)(1.15)(1.10)(1.05) / (R – .05)] / (1 + R)3 = $78.43
We need to find the roots of this equation. Using spreadsheet, trial and error, or a calculator with a root
solving function, we find that:
R = .1169, or 11.69%
38. In this problem, growth is occurring from two different sources: The learning curve and the new
project. We need to separately compute the value from the two different sources. First, we will
compute the value from the learning curve, which will increase at 5 percent. All earnings are paid out
as dividends, so we find the earnings per share are:
EPS1 = Earnings/total number of outstanding shares
EPS1 = ($18,000,000 × 1.05) / 7,500,000
EPS1 = $2.52
From the NPVGO model:
P = E / (R – g) + NPVGO
P = $2.52 / (.10 – .05) + NPVGO
P = $50.40 + NPVGO
Now we can compute the NPVGO of the new project to be launched two years from now. The earnings
per share two years from now will be:
EPS2 = $2.52(1 + .05)
Solutions Manual
EPS2 = $2.646
Therefore, the initial investment in the new project will be:
Initial investment = .30($2.646)
Initial investment = $.79
The earnings per share of the new project is a perpetuity, with an annual cash flow of:
Increased EPS from project = $6,500,000 / 7,500,000 shares
Increased EPS from project = $.87
So, the value of all future earnings in Year 2, one year before the company realizes the earnings, is:
PV = $.87 / .10
PV = $8.67
Now, we can find the NPVGO per share of the investment opportunity in Year 2, which will be:
NPVGO2 = –$.79 + 8.67
NPVGO2 = $7.87
The value of the NPVGO today will be:
NPVGO = $7.87 / (1 + .10)2
NPVGO = $6.51
Plugging in the NPVGO model we get;
P = $50.40 + 6.51
P = $56.91
Note that you could also value the company and the project with the values given, and then divide the
final answer by the shares outstanding. The final answer would be the same.