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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.
- If the interest rate is 8%:
P 0 = $
[
]
(1.04)^20
- If the interest rate is 9%: A quick calculation will verify that it is P 0 = 1000. 0.
P 0 = $
[
]
(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.
- Duration Cash Flow PV(r=12%) Year Bond 1 Bond 2 Bond 1 Bond 2 1 120 0 107.14 - 2 120 0 95.66 - 3 120 0 85.41 - 4 120 0 76.26 - 5 120 0 68.09 - 6 120 0 60.80 - 7 120 0 54.28 - 7 1000 1000 452.34 452. P 0 1000.00 452. Duration bond 1: [107.14 + 95. 66 · 2 + 85. 41 · 3 + 76. 26 · 4 + 68. 09 · 5 + 60. 80 · 6 + 507. 63 · 7] 1000
Duration bond 2:
- 34 · 7
- 34
Bond 2 will be more sensitive to interest rate changes.
- If the interest rate increases 100 basis points to 13%, the new prices of each bond will be:
Price Bond 1 =
∑^ T
t=
(1.13)t^
(1.13)^7
Price Bond 2 =
Capital Loss Bond 1 = 1000 − 955 .77 = 44. 23 Percentage Loss Bond 1 =
Capital Loss Bond 2 = 506. 63 − 476 .07 = 30. 56 Percentage Loss Bond 2 =
Note: The percentage loss on each bond is approximately equal to
Percentage Loss ≈ Duration 1 + r · ∆r
Percentage Loss Bond 1 ≈ 5.^11139
- 12
Percentage Loss Bond 2 ≈
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.
- YTM: Calculate the internal rate of return on: t = 0 1 2 3 Ct = − 100. 917 10 10 110
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.
- Discount factors (prices): [ 982 .50 = d(0, 1)80 + d(0, 2) 90 = d(0, 1)
]
d(0, 1) = 90 100
982 .50 = 0. 90 × 80 + d(0, 2) d(0, 2) =
982. 50 − 0. 9 × 80
d(0.2) = 0. 843055555556 Solving these equations we find prices [ d(0, 1) = 0. 9 d(0, 2) = 0. 84
]
2 Common Stock Valuation.
Exercise 6.
Expected return = Expected dividend yield + Expected capital gain return.
E[r] =
E[D 1 ]
P 0
E[P 1 ] − P 0
P 0
In equilibrium, the price of the stock (P 0 ) 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
P 0 =
∑^ ∞
t=
E[Dt]
(1 + r)t
Solution to Exercise 6.
Substituting r for E[r] in
E[r] =
E[D 1 ]
P 0
E[P 1 ] − P 0
P 0
and rearranging yields an equation for today’s price:
P 0 =
E[D 1 ] + E[P 1 ]
1 + r
The price one year from now, however, will be equal to
P 1 = E[D^2 ] +^ E[P^2 ] 1 + r
Substituting this expression into the expression for P 0 yields:
P 0 = E[D^1 ] 1 + r
+ E[D^2 ] +^ E[P^2 ]
(1 + r)^2
Repeating the process again, this time substituting for P 2 , yields:
P 0 =
E[D 1 ]
1 + r
E[D 2 ]
(1 + r)^2
E[D 3 ] + E[P 3 ]
(1 + r)^3
Continuing in this fashion over and over produces the following valuation formula
P 0 =
∑^ ∞
t=
E[Dt] (1 + r)t
Exercise 7.
Consider the following valuation formula for stock prices:
P 0 =
∑^ ∞
t=
E[Dt]
(1 + r)t
where P 0 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 =
D 1
r − g
Solution to Exercise 7.
When the dividend grows at at a rate g per period:
Dt = D 0 (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[D 1 ] = D 0 (1 + g) = $4. 00 · 1 .05 = 4. 20
r =
r = 0.14 + 0.05 = 19%
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[P 1 ] = P 0 (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
- 70 − 4 .20 = 31. 50
We can also find this by direct calculation of the value of the stock going forward
E[P 1 ] = E[D^2 ] r − g
E[D 2 ] = D 0 (1 + g)^2 = 4. 00 · (1.05)^2 = 4. 41
E[P 1 ] =
What are the expected dividend and capital gain returns over the next year?
E[D 1 ] P 0
E[P 1 ] − P 0
P 0
Exercise 9.
The Handy Dandy Hardware Store chain is expected to benefit greatly from the recent interest in ‘do-it-
yourself’ 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.
