Financial Markets and Valuation - Tutorial 2: SOLUTIONS
Bonds, Stock Valuation & Capital Budgeting
(*) denotes those problems to be covered in detail during the tutorial session
Bonds
Problem 1. (Ross, Westerfield & Jaffe) Consider a bond, which pays a $80 coupon
annually and has a face value of $1,000. Calculate the yield to maturity if the bond
has
a. 20 years remaining to maturity and it is sold at $1,200.
b. 10 years remaining to maturity and it is sold at $950.
Solution:
a. Since the bond sells at a premium, its yield is less than the coupon rate of 8%.
PV = 1,200
<=> (80/y) * [1 - 1/((1+y)^20)] + 1,000/[(1+y)^20] = 1,200
<=> y = 6.22%
b. Since the bond sells at a discount, its yield exceeds the coupon rate of 8%.
PV = 950
<=> (80/y) * [1 - 1/((1+y)^10)] + 1,000/[(1+y)^10] = 950
<=> y = 8.77%
(*) Problem 2. (Ross, Westerfield & Jaffe) Available are three zero-coupon, $1,000
face value bonds. All of these bonds are initially priced using an 11-percent interest
rate. Bond A matures one year from today, bond B matures five years from today, and
bond C matures 10 years from today.
a. What is the current price of each bond?
b. If the market rate of interest rises to 14%, what are the prices of these bonds?
c. Which bond experienced the greatest percentage change in prices?
Solution:
a. PV (A, 11%) = 1,000/(1+0.11) = 900.9
PV (B, 11%) = 1,000/[(1+0.11)^5] = 593.45
PV (C, 11%) = 1,000/[(1+0.11)^10] = 352.18
b. PV (A, 14%) = 1,000/(1+0.14) = 877.19
=> % change in A = (877.19 - 900.9)/900.9 = -2.63%
PV (B, 14%) = 1,000/[(1+0.14)^5] = 519.37
=> % change in B = (519.37 - 593.45)/593.45 = -12.48%
PV (C, 14%) = 1,000/[(1+0.14)^10] = 269.74
=> % change in C = (269.74 - 352.18)/352.18 = -23.4%
c. Bond C
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Problem 3. (Brealey and Myers) An 8-percent five-year bond yields 6%.
a. If the yield remains unchanged, what will its price be one year hence? Assume
annual coupon payments.
b. What is the total return to an investor who held the bond over this year?
Solution:
We will find the price in percentages (the same as when assuming that the face value
is $1).
a. Price today = 0.08 * A (5 years, 6%) + 1 / (1.06)^5
= 0.08 * 4.2124 + 0.7473 = 0.337 + 0.7473
= 1.0843 = 108.43%
Price in one year = 0.08 * A(4 years, 6%) + 1/(1.06)^4 = 0.08 * 3.4651 +
0.7921
= 0.2772 + 0.7921
= 1.0693 = 106.93%
b. Total annual return = (coupon plus difference in prices) / price today
= (8% + 106.93% - 108.43%) / 108.43%
= 6.5% / 108.43% = 6%
(*) Problem 4. (Ross, Westerfield & Jaffe) The one-year spot rate equals 10 percent
and the two-year spot rate equals 8 percent.
a. What should a 5-percent coupon two-year bond cost?
b. What is the forward rate expected over year 1?
Solution:
a.
b.
P = $50 / 1.10 + $1,050 / (1.08)2
= $45.45 + $900.21
= $945.66
( 1 + r1 )( 1 + ƒ2 ) = ( 1 + r2 )2
( 1.10 ) ( 1 + ƒ2 ) = ( 1.08 )2
ƒ2 = 0.0604
(*) Problem 5. Your investment bank has just supplied you with the following term
structure of spot rates (all annualized effective rates):
Maturity
3 months
6 months
12 months
18 months
24 months
FMV/Tutorial 2 – Solutions/Sept.-Oct. 2006
Rate
3.5%
4%
5%
5%
5%
2
a. One broker calls to convince you to buy a bond with 18 months to maturity and
face value of €1000 that pays a semiannual coupon (stated rate) of 3%. He says he
will sell you the bond for €1020. Should you buy the bond? (Assume that the bond
has just paid a coupon and ignore taxes).
b. If market expectations about future interest rates are correct, at what price will the
bond sell 1 year from now?
Solution:
a. Coupon = 3% · 1,000 = 30 (annual)
Semiannual Coupon = 30 / 2 = 15 (semiannual)
PV =
15
(1 + r )
0 .5
0,6 m
+
15
+
1015
(1 + r ) (1 + r )
1
0 ,12 m
1 .5
0 ,18 m
=
15
(1 + 0.04)
0 .5
+
15
+
1015
(1 + 0.05) (1 + 0.05)
1
1 .5
= 972.36
The price at which the broker wants to sell is too expensive. You shouldn’t buy the
bond.
NOTE: The result was obvious, because a bond that pays a coupon of 3% when all
market interest rates are above 3% must sell below par. Since the broker was asking a
price above par, you shouldn’t buy the bond.
b. One year from now, the bond will have 6 months to maturity. The only payment
left is the last coupon and the face value.
If market expectations are correct, the 6-month rate prevailing one year from now
equals today’s forward rate between 12 months and 18 months.
(1 + r )
1, 5
0 ,18 m
= ⋅(1 + r0 ,12 m ) ⋅ (1 + f 12 m ,18 m ) ⇔ (1 + 0.05) = ⋅(1 + 0.05) ⋅ (1 + f 12 m ,18 m ) ⇔ f 12 m ,18 m = 0.05
1
0.5
1.5
1
0.5
NOTE: This was again obvious! If the term structure is flat after 12 months and the
market expectations theory holds, the forward term structure is also constant.
PV =
1015
= 990.538
(1 + 0.05)0.5
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Stock Valuation
Problem 6. (Ross, Westerfield & Jaffe) Suppose that a shareholder has just paid $50
per share for XYZ Company Stock. The stock will pay a dividend of $2 per share in
the upcoming year. This dividend is expected to grow at an annual rate of 10% for the
indefinite future. The shareholder felt that she paid the fair price for the stock, given
her assessment of XYZ’s risks. What is the annual required rate of return of this
shareholder?
Solution :
In this problem, you are given the current share price at $50. Dividends at t = 1 is
expected to be $2, and growing indefinitely at 10% p.a. To find the annual required
rate of return, we set up the solution as:
Ρ=
Div
(r − g )
Ù 50 =
2
(r − 0.1)
Ù
50 (r – 0.1) = 2 Ù r =
2
+ 0.10 = 14%
50
(*) Problem 7. (Ross, Westerfield & Jaffe) Brown, Inc. has just paid a $3 dividend
per share of the common stock. The stock is currently being sold at $40. Investors
expect that Brown’s dividend will grow at a constant rate indefinitely. What growth
rate is expected by investors if they require a 8% return on the stock?
Solution :
In this problem, we are given the current dividend (“…..just paid”) at t = 0 of $3 per
share, and we expect the dividend to grow at a constant rate indefinitely. Therefore,
we apply the constant growth model.
The t=1 dividends are DI = DO (1 + g ) = 3 (1 + g )
Po =
3 (1 + g )
Div at t = 1 D0 (1 + g )
Ù 40 =
Ù 40 (0.08 − g ) = 3 (1 + g )
=
(0.08 − g )
(r − g )
(r − g )
and then solving for g, g = 0.47%
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(*) Problem 8. (Ross, Westerfield & Jaffe) Whizzkids, Inc., is experiencing a period
of rapid growth. Earnings and dividends are expected to grow at a rate of 18% during
the next two years, 15% in the third year, and at a constant rate of 6% thereafter.
Whizzkids’ last dividend, which has just been paid, was $1.15. If the required rate of
return on the stock is 12%, what is the price of the stock today?
Solution :
You are asked to compute the share price of a company that is expected to have 2
stages of growth: an initial high growth period of about 2/3 years and a subsequent
constant growth thereafter
$1.15 (1.18)
$1.15 (1.18) 2
$1.15(1.18) 2 (1.15)
ΡO =
+
+
+
3
1.12
(1.12) 2
(1.12)
2
1.15
(1.18) (1.15)(1.06) 



(0.12 − 0.06)


1
(1.12)
3
= $26.95
Problem 9. Allen Inc. is expected to pay an equal amount of dividends at the end of
the first two years. Thereafter the dividend will grow at a constant rate of 4%
indefinitely. The stock is currently traded at $30. What is the expected dividend per
share for the next year if the required rate of return is 12%?
Solution :
D1 = D2 and a growth rate g = 0.04 thereafter
r = 0.12
Po = $30.
Hence, we can set up the following equation:
30 =
D1
1.12
+
D1
(1.12) 2
+
D1 (1.04 )
1
×
(0.12 − 0.04) (1.12)2
Ù 30 = D1 (12.05357)
Ù D1 = $2.49
(*) Problem 10. (Ross, Westerfield & Jaffe) California Electronics, Inc., expects to
earn $100 million per year in perpetuity if it does not undertake any new projects. The
firm has an opportunity that requires an investment of $15 million today and $5
million in one year. The new investment will begin to generate additional annual
earnings of $10 million two years from today in perpetuity. The firm has 20 million
shares of common stock outstanding, and the required rate of return on the common
stock is 15 percent.
(a) What is the price of a share of the stock if the firm does not undertake the new
project?
(b) What is the value of the growth opportunities resulting from the new project?
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(c) What is the price of a share of the stock if the firm undertakes the new project?
Solution
(a) The present value (PV) of the current earnings stream can be obtained using the
perpetuity formula: CF/r. Then simply divide this PV by the number of shares
outstanding to get the price per share.
This implies that PV = $100/(0.15) = $666.67 million
Price per share = 666.67/20 = $33.33
(b) The cash flows from the new project can be depicted as:
Time (Years)
Cash flows
0
1
2 to infinity
-15
-5
10
The NPV of the project is simply:
NPV = −15 −
5
1 
 10
+
×
 = $38.62 million
(1.15)  0.15 1.15 
(c) The new share price can be obtained by adding the NPV of the new project to the
firm’s current PV and then dividing by the shares outstanding. This will be equal
to:
New Share Price = (38.62 + 666.67)/20 = $35.26
Capital Budgeting
(*) Problem 11. (Ross, Westerfield & Jaffe) Dickinson Brothers, Inc., is considering
investing in a machine to produce computer keyboards. The price of the machine is
$400,000 and its economic life is 5 years. The machine is fully depreciated by the
straight-line method. The machine will produce 10,000 units of keyboards each year.
The price of the keyboard is $40 in the first year, and it will increase at 5% per year.
The production costs per unit of the keyboard is $20 in the first year, and it will
increase at 10% per year. Corporate tax rate for the company is 34%. If the
appropriate discount rate is 15%, what is the NPV of the investment?
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Solution :
Straight Line Depreciations (5 yr life) per year =
t
Sales
1
$400,000 − 0
= $80,000
5
Production Costs
10,000 units x $40 each
= $400,000
$400,000 x 1.05
= $420,000
$400,000 x 1.052
= $441,000
$400,000 x 1.053
= 463,050
$400,000 x 1.054
= $486,203
2
3
4
5
0
1
$400,000
($200,000)
($ 80,000)
$120,000
-40,800
Sales Revenues
Production Costs
Depreciation
EBIT
Taxes (34%)
NOPLAT
Add back: Depreciation
CAPEX
($400,000)
Changes in NWC
0
Free Cash Flows
10,000 units x $20 each
= $200,000
$200,000 x 1.10
= $220,000
$200,000 x 1.102
= $242,000
$200,000 x 1.103
= $266,200
$200,000 x 1.104
= $292,820
2
3
$420,000 $441,000
($220,000) ($242,000)
($ 80,000) ($ 80,000)
$120,000 $119,000
-40,800
-40,460
4
$463,050
($266,200)
($ 80,000)
$116,850
-39,389
5
$486,203
($292,820)
($ 80,000)
$113,383
-38,550
$ 79,200
$ 79,200
$ 78,540
$ 77,461
$ 74,833
80,000
0
80,000
0
80,000
0
80,000
0
80,000
0
$159,200
$158,540
$157,461
$154,833
($400,000) $159,200
r = 15%
NPV = − $400,000 +
$159,200
$159,200
$158,540
+
+
+
2
1.15
1.15
1.15 3
$157,461
$154,833
=
+
4
1.15
1.15 5
= - $400,000 + $138,435 + $120,378 + $104,243 + $89,834 + $76,979
= $129,869
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