Lecture 4
1
Bond valuation
Exercise 1.
A Treasury bond has a coupon rate of 9%, a face value of $1000 and matures 10 years from today. For a
treasury bond the interest on the bond is paid in semi-annual installments. The current riskless interest rate
is 12% (compounded semi-annually).
1. Suppose you purchase the Treasury bond described above and immediately thereafter the riskless interest
rate falls to 8%. (compounded semi-annually). What would be the new market price of the bond?
2. What is your best estimate of what the price would be if the riskless interest rate was 9% (compounded
semi-annually)?
Solution to Exercise 1.
1. If the interest rate is 8%:
1
1
$1000
−
= $1067.95
+
P0 = $45
0.04
0.04 · 1.0420
(1.04)20
2. If the interest rate is 9%: A quick calculation will verify that it is P0 = 1000.0.
1
1
$1000
P0 = $45
−
+
= $1000
0.045
0.045 · 1.04520
(1.045)20
Exercise 2.
Suppose you are trying to determine the interest rate sensitivity of two bonds. Bond 1 is a 12% coupon bond
with a 7-year maturity and a $1000 principal. Bond 2 is a ‘zero-coupon’ bond that pays $1120 after 7 year.
The current interest rate is 12%.
1. Determine the duration of each bond.
2. If the interest rate increases 100 basis points (100 basis points = 1%), what will be the capital loss on
each bond?
Solution to Exercise 2.
1. Duration
Year
1
2
3
4
5
6
7
7
P0
Cash Flow
Bond 1 Bond 2
120
0
120
0
120
0
120
0
120
0
120
0
120
0
1000
1000
PV(r=12%)
Bond 1 Bond 2
107.14
95.66
85.41
76.26
68.09
60.80
54.28
452.34
452.34
1000.00 452.34
Duration bond 1:
[107.14 + 95.66 · 2 + 85.41 · 3 + 76.26 · 4 + 68.09 · 5 + 60.80 · 6 + 507.63 · 7]
= 5.11139
1000
Duration bond 2:
452.34 · 7
=7
452.34
Bond 2 will be more sensitive to interest rate changes.
1
2. If the interest rate increases 100 basis points to 13%, the new prices of each bond will be:
Price Bond 1 =
T
X
$1000
$120
+
= $955.77
t
7
(1.13)
(1.13)
t=1
1120
= 476.07
1.137
Capital Loss Bond 1 = 1000 − 955.77 = 44.23
44.23
Percentage Loss Bond 1 =
= 4.423%
1000
Capital Loss Bond 2 = 506.63 − 476.07 = 30.56
30.56
Percentage Loss Bond 2 =
= 6.032%
506.63
Note: The percentage loss on each bond is approximately equal to
Price Bond 2 =
Percentage Loss ≈
Duration
· ∆r
1+r
5.11139
· 0.01 = 4.563%
1.12
7.0
Percentage Loss Bond 2 ≈
· 0.01 = 6.25%
1.12
Percentage Loss Bond 1 ≈
Exercise 3.
A $100, 10 year bond was issued 7 years ago at a 10% annual interest rate. The current interest rate is 9%.
The current price of the bond is 100.917. Use annual, discrete compounding.
1. Calculate the bonds yield to maturity.
Solution to Exercise 3.
1. YTM: Calculate the internal rate of return on:
t
Ct
=
=
0
−100.917
1
10
2
10
3
110
IRR = 0.096344 = 9.6344%
Exercise 4.
A two-year Treasury bond with a face value of 1000 and an annual coupon payment of 8% sells for 982.50.
A one-year T bill, with a face value of 100, and no coupons, sells for 90. Compounding is discrete, annual.
Given these market prices,
1. Find the prices d(0, 1) and d(0, 2) of one dollar received respectively one and two years from now.
2. Find the corresponding interest rates.
Solution to Exercise 4.
1. Discount factors (prices):
982.50 = d(0, 1)80 + d(0, 2)1080
90 = d(0, 1)100
90
= .90
100
982.50 = 0.90 × 80 + d(0, 2)1080
982.50 − 0.9 × 80
d(0, 2) =
1080
d(0.2) = 0.843055555556
d(0, 1) =
Solving these equations we find prices
d(0, 1) = 0.9
d(0, 2) = 0.84
2
2. and interest rates
r(0, 1) = 11%
r(0, 2) = 9%
>> B=[982.50 90]
B =
982.500
90.000
>> C=[80 1080;100 0]
C =
80
1080
100
0
>> d=inv(C)*B’
d =
0.90000
0.84306
Exercise 5.
Suppose you want to invest $1000 for two years. The current term structure looks like the following:
Year
1
2
Spot rate
6%
7%
1. If you want to be certain of the amount you will have after two years, what is the amount you get in
year 2?
2. Suppose you only invest for one year, and enter into a contract that guarantees the interest rate you
will get one year from now, the forward rate. What must this forward rate be?
Solution to Exercise 5.
If you want to be certain of the amount you will have after two years you will have to lend at the two-year spot rate,
r2 = 7%.
The amount you will have after two years is
F V2
=
$1000(1 + r)2
=
$1000 · 1.072
=
$1145
The rate at which you are implicitly agreeing to lend over the second year is the forward rate for year 2, f2 . This rate
is:
f2
=
=
=
(1 + r2 )2
−1
(1 + r1 )
1.072
−1
1.06
8%
Thus, by buying a two-year pure discount bond, you are implicitly contracting to lend at an 8% interest rate in the
second year.
An alternative way to lend for two years is to buy a one-year pure discount bond yielding r1 = 6% over the first year
and rolling over into another one-year pure discount bond yielding 1 r̃2 over the second year. This strategy is risky ,
however, since you do not know what 1 r̃2 will be until the end of the first year.
The expected amount you will have after two years is:
E[F˜V 2 ]
=
$1000(1 + r(0, 1))(1 + E[r̃(1, 2)])
=
$1000 · 1.06 · (1 + E[r̃(1, 2)])
Relationship:
E[r̃(1, 2)] > f (0, 1, 2) = 8%
E[r̃(1, 2)] = f (0, 1, 2) = 8%
E[r̃(1, 2)] < f (0, 1, 2) = 8%
⇒
E[F˜V 2 ] > $1145
E[F˜V 2 ] = $1145
E[F˜V 2 ] < $1145
3
2
Common Stock Valuation.
Exercise 6.
Expected return = Expected dividend yield + Expected capital gain return.
E[r] =
E[D1 ] E[P1 ] − P0
+
P0
P0
In equilibrium, the price of the stock (P0 ) will adjust so that the expected return E[r] equals the required
return of investors, r. The required return, r, is sometimes called the opportunity cost of capital or market
capitalization rate.
Show that this implies the following expression for the current stock price
P0 =
∞
X
E[Dt ]
(1
+ r)t
t=1
Solution to Exercise 6.
Substituting r for E[r] in
E[r] =
E[D1 ]
E[P1 ] − P0
+
P0
P0
and rearranging yields an equation for today’s price:
P0 =
E[D1 ] + E[P1 ]
1+r
The price one year from now, however, will be equal to
P1 =
E[D2 ] + E[P2 ]
1+r
Substituting this expression into the expression for P0 yields:
P0 =
E[D2 ] + E[P2 ]
E[D1 ]
+
1+r
(1 + r)2
Repeating the process again, this time substituting for P2 , yields:
P0 =
E[D1 ]
E[D2 ]
E[D3 ] + E[P3 ]
+
+
1+r
(1 + r)2
(1 + r)3
Continuing in this fashion over and over produces the following valuation formula
P0 =
∞
X
E[Dt ]
(1
+ r)t
t=1
Exercise 7.
Consider the following valuation formula for stock prices:
P0 =
∞
X
E[Dt ]
(1
+ r)t
t=1
where P0 is todays stock price, Dt the dividend payment on date t, and r the required rate of return on the
stock.
• Under what circumstances does this collapse into the valuation formula
Po =
D1
r−g
4
Solution to Exercise 7.
When the dividend grows at at a rate g per period:
Dt = D0 (1 + g)t
then the
Exercise 8.
The common stock of the Handy Dandy Hardware store chain is currently selling for $30 per share. Last
year’s dividend per share was $4.00. Earnings and dividends per share are expected to grow at a constant
rate of 5% per year for the indefinite future.
1. Estimate the market capitalization rate for Handy Dandy.
2. What is the expected price of the stock one year from now?
3. What are the expected dividend and capital gain returns over the next year?
Solution to Exercise 8.
E[D1 ] = D0 (1 + g) = $4.00 · 1.05 = 4.20
4.20
+ 0.05
30
r = 0.14 + 0.05 = 19%
r=
What is the expected price of the stock one year from now?
There are different ways to think about this. One way is to realize that in expected terms the price should increase
with the capitalization rate r, which gives
E[P1 ] = P0 (1 + r) = 35.7
This is the price before the stock pays dividend. If the dividend is 4.20, the ex-dividend price is expected to be
35.70 − 4.20 = 31.50
We can also find this by direct calculation of the value of the stock going forward
E[P1 ] =
E[D2 ]
r−g
E[D2 ] = D0 (1 + g)2 = 4.00 · (1.05)2 = 4.41
4.41
= 31.50
E[P1 ] =
0.19 − 0.05
What are the expected dividend and capital gain returns over the next year?
E[D1 ]
4.20
=
= 14%
P0
30.00
E[P1 ] − P0
31.50 − 30.00
=
= 5%
P0
30.00
Exercise 9.
The Handy Dandy Hardware Store chain is expected to benefit greatly from the recent interest in ‘do-ityourself’ home repair. Analysts are forecasting that Handy Dandy will experience two years of abnormally
high growth of 20% in earnings and dividends before settling down to a normal growth rate of 5% in year
3 and beyond. Last year’s dividend per share was $4.00. Assume that the appropriate opportunity cost of
capital is 19%.
1. Determine the market price of Handy Dandy’s common stock.
5
Solution to Exercise 9.
P0 =
∞
X
E[Dt ]
(1
+ r)t
t=1
P0 =
E[D1 ]
E[D2 ] + E[P2 ]
+
(1 + r)
(1 + r)2
We need estimates of E[D1 ], E[D2 ] and E[P2 ].
E[D1 ]
=
4.00 · 1.20 = 4.80
E[D2 ]
=
E[P2 ]
=
P0
=
4.00 · 1.202 = 5.76
E[D3 ]
5.76 · 1.05
=
= 43.20
r−g
0.19 − 0.05
4.80
5.76 + 43.20
+
= 38.60
1.19
(1.19)2
6