chapter 2
Chapter 2:
7.
a. The closing price today is $74.59, which is $0.17 higher than yesterday’s
price. Therefore, yesterday’s closing price was: $74.59 – $0.17 = $74.42
b. You could buy: $5,000/$74.59 = 67.03 shares
c. Your annual dividend income would be 1.20 % of $5,000, or $60.
d. Earnings per share can be derived from the price-earnings (PE) ratio.
Price/Earnings = 16 and Price = $74.59 so that Earnings = $74.59/16 =
$4.66
8. a. At t = 0, the value of the index is: (90 + 50 + 100)/3 = 80
At t = 1, the value of the index is: (95 + 45 + 110)/3 = 83.3333
The rate of return is: (83.3333/80) – 1 = 4.167%
b. In the absence of a split, stock C would sell for 110, and the value of the
index would be: (95 + 45 + 110)/3 = 83.3333
After the split, stock C sells at 55. Therefore, we need to set the divisor
(d) such that:
83.3333 = (95 + 45 + 55)/d…..d = 2.340
c. The rate of return is zero. The index remains unchanged, as it should,
since the return on each stock separately equals zero.
9. a. Total market value at t = 0 is: (9,000 + 10,000 + 20,000) = 39,000
Total market value at t = 1 is: (9,500 + 9,000 + 22,000) = 40,500
Rate of return = (40,500/39,000) – 1 = 3.85%
b. The return on each stock is as follows:
Ra = (95/90) – 1 = 0.0556
Rb = (45/50) – 1 = –0.10
R c = (110/100) – 1 = 0.10
The equally-weighted average is: [0.0556 + (-0.10) + 0.10]/3 =
0.0185 = 1.85%
10. The after-tax yield on the corporate bonds is: [0.09 x (1 – 0.30)] = 0.0630 =
6.30%
Therefore, the municipals must offer at least 6.30% yields.
11. a. The taxable bond. With a zero tax bracket, the after-tax yield for the
taxable bond is the same as the before-tax yield (5%), which is greater
than the yield on the municipal bond.
b. The taxable bond. The after-tax yield for the taxable bond is:
0.05 x (1 – 0.10) = 4.5%
c. You are indifferent. The after-tax yield for the taxable bond is:
0.05 x (1 – 0.20) = 4.0%
The after-tax yield is the same as that of the municipal bond.
d. The municipal bond offers the higher after-tax yield for investors in tax
brackets above 20%.
12. The equivalent taxable yield (r) is: r = rm/(1 – t)
a. 4.00%
b. 4.44%
c. 5.00%
d. 5.71%
13. a. The higher coupon bond
b. The call with the lower exercise price
c. The put on the lower priced stock
14. a. The December maturity futures price is $2.3375 per bushel. If the contract
closes at $2.15 per bushel in December, your profit / loss on each contract
(for delivery of 5,000 bushels of oats) will be: ($2.15 - $2.3375) x 5000 =
$937.50 loss
b. There are 3,907 contracts outstanding, representing 19,535,000 bushels of
oats.
15. a. Yes. As long as the stock price at expiration exceeds the exercise price, it
makes sense to exercise the call.
Gross profit is: $101 - $$ 95 = $6
Net profit = $6 – $ 6.50 = $0.50 loss
Rate of return = -0.50 / 6.50 = - .0769 or 7.69% loss
b. Yes, exercise.
Gross profit is: $101 - $$ 90 = $11
Net profit = $11 – $ 6.50 = $4.50 gain
Rate of return = 4.50 / 6.50 = 0.6923 or 69.23% gain
c. A put with exercise price $95 would expire worthless for any stock price
equal to or greater than $95. An investor in such a put would have a rate
of return over the holding period of –100%.
16. There is always a chance that the option will expire in the money. Investors will
pay something for this chance of a positive payoff.
17.
Value of call Initial Cost Profit at expiration
a. 0 4 -4
b. 0 4 -4
c. 0 4 -4
d. 5 4 1
e. 10 4 6
Value of put Initial Cost Profit at expiration
a. 10 6 4
b. 5 6 -1
c. 0 6 -6
d. 0 6 -6
e. 0 6 -6
CHAPTER 5:
1. V(12/31/2007) = V(1/1/1991) × (1 + g)7 = $100,000 × (1.05) = $140,710.04
2. i and ii. The standard deviation is non-negative.
3. c. Determines most of the portfolio’s return and volatility over time.
4. E(r) = [0.3 × 44%] + [0.4 × 14%] + [0.3 × (–16%)] = 14%
σ^2 = [0.3 × (44 – 14)^2 ] + [0.4 × (14 – 14)^2 ] + [0.3 × (–16 – 14) ^2] = 540
σ2 = 23.24%
The mean is unchanged, but the standard deviation has increased.
5. a. The holding period returns for the three scenarios are:
Boom: (50 – 40 + 2)/40 = 0.30 = 30.00%
Normal: (43 – 40 + 1)/40 = 0.10 = 10.00%
Recession: (34 – 40 + 0.50)/40 = –0.1375 = –13.75%
E(HPR) = [(1/3) × 30%] + [(1/3) × 10%] + [(1/3) × (–13.75%)] = 8.75%
σ^2 (HPR) = [(1/3) × (30 – 8.75)^2 ] + [(1/3) × (10 – 8.75)^2 ] + [(1/3) × (–13.75 – 8.75)^2 ] = 319.79
σ = 319.79 = 17.88%
b. E(r) = (0.5 × 8.75%) + (0.5 × 4%) = 6.375%
σ = 0.5 × 17.88% = 8.94%
2
6. Investment 3. For each portfolio: Utility = E(r) – (0.5 × 4 × σ )
Investment E(r) σ U
1 0.12 0.30 -0.0600
2 0.15 0.50 -0.3500
3 0.21 0.16 0.1588
4 0.24 0.21 0.1518
We choose the portfolio with the highest utility value.
7. Investment 4. When an investor is risk neutral, A = 0 so that the portfolio with the
highest utility is the portfolio with the highest expected return.
9. E(rX) = [0.2 × (–20%)] + [0.5 × 18%] + [0.3 × 50%)] = 20%
E(r Y ) = [0.2 × (–15%)] + [0.5 × 20%] + [0.3 × 10%)] = 10%
10. σ^2X = [0.2 × (–20 – 20) ^2] + [0.5 × (18 – 20)^2 ] + [0.3 × (50 – 20)^2 ] = 592
σX = 24.33%
σ Y = [0.2 × (–15 – 10) ^2] + [0.5 × (20 – 10)^2 ] + [0.3 × (10 – 10)^2 ] = 175
σY = 13.23%
11. E(r) = (0.9 × 20%) + (0.1 × 10%) = 19%
12. The probability is 0.50 that the state of the economy is neutral. Given a neutral
economy, the probability that the performance of the stock will be poor is 0.30,
and the probability of both a neutral economy and poor stock performance is:
0.30 × 0.50 = 0.15
13. E(r) = [0.1 × 15%] + [0.6 × 13%] + [0.3 × 7%)] = 11.4%
15. a. E(r P) – rF = ½Aσ^2 p = ½ × 4 ^2× (0.20) = 0.08 = 8.0%
b. 0.09 = ½Aσ^2P = ½ × A × (0.20)^2 ⇒ A = 0.09/( ½ × 0.04) = 4.5
c. Increased risk tolerance means decreased risk aversion (A), which results
in a decline in risk premiums.
16. For the period 1926 – 2006, the mean annual risk premium for large stocks over
T-bills is 8.42%
E(r) = Risk-free rate + Risk premium = 5% + 8.42% =13.42%
17. In the table below, we use data from Table 5.3 and the approximation: r ≅ R – i:
Large Stocks: r ≅ 12.19% − 3.13% = 9.06%
Small Stocks: r ≅ 18.14% − 3.13% = 15.01%
Long-Term T-Bonds: r ≅ 5.64% − 3.13% = 2.51%
T-Bills: r ≅ 3.77% − 3.13% = 0.64%
Next, we compute real rates using the exact relationship:
r =-1=
Large Stocks: r = 0.0906/1.0313 = 8.79%
Small Stocks: r = 0.1501/1.0313 = 14.55%
Long-Term T-Bonds: r = 0.0251/1.0313 = 2.43%
T-Bills r = 0.0064/1.0313 = 0.62%
18. a. The expected cash flow is: (0.5 × $50,000) + (0.5 × $150,000) = $100,000
With a risk premium of 10%, the required rate of return is 15%.
Therefore, if the value of the portfolio is X, then, in order to earn a 15%
expected return:
X(1.15) = $100,000 ⇒ X = $86,957
b. If the portfolio is purchased at $86,957, and the expected payoff is
$100,000, then the expected rate of return, E(r), is:
The portfolio price is set to equate the expected return with the
required rate of return.
c. If the risk premium over T-bills is now 15%, then the required return is:
5% + 15% = 20%
The value of the portfolio (X) must satisfy:
X(1.20) = $100, 000 ⇒ X = $83,333
d. For a given expected cash flow, portfolios that command greater risk premia
must sell at lower prices. The extra discount from expected value is a penalty
for risk.
19. a. E(rp ) = (0.3 × 7%) + (0.7 × 17%) = 14% per year
σ p= 0.7 × 27% = 18.9% per year
b. Security Investment Proportions
T-Bills 30.0%
Stock A 0.7 × 27% = 18.9%
Stock B 0.7 × 33% = 23.1%
Stock C 0.7 × 40% = 28.0%
c. Your Reward-to-variability ratio = S = (17-7)/27 = 0.3704
Client's Reward-to-variability ratio = (14-7)/18.9 = 0.3704
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