Chapter 8
Risk and Return Trade-off Analysis
1.
a.Systematic risk is the part of total risk that results from the tendency of stock prices to move together with the general market. It reflects the fluctuations and change of the general market. Nonsystematic risk is the result of variations peculiar to a firm or industry, for example, a labor strike.
b.The Capital Market Line (CML) describes the relationship between expected return and total risk for efficient portfolios. The Security Market Line (SML)describes the risk-return relationship for all securities held in the efficient market portfolio M found on the CML.
c.Sharpe's Measure for portfolio X = SPX =
and, Jensen's Measure for security X is αx in the following equation:
.
While both measures account for the excess return on the portfolio, Sharpe's measure uses total risk and Jensen's uses beta or systematic risk.
d.The strong-form hypothesis contends that stock prices fully reflect all information regardless of whether it is public, private, or otherwise. The weak-form hypothesis assumes that all current stock prices fully reflect all historical and current stock market information.
e.The market model states that the return on any asset at time t can be expressed as a linear function of the market return at time t plus a random error component. It is given in equation (8.7) on page 291 of the text and is restated as follows:
Ri,t =αi + βiRM,t+ηi,t
The characteristic line is a plot of the market model with alpha as the intercept and beta as the slope.
f.An aggressive stock is one with a beta larger than 1.0 and a defensivestock is one with beta less than 1.0.
2.
The arbitrage pricing theory (APT) demonstrates that the expected rate of return on a security can be explained by k independent factors. The capital asset pricing model (CAPM) in contrast states that the return on a security held in a well diversified portfolio is a function of one variable, the return on the market.
3.
Systematic risk is determined largely by the firm's business and financial risk. Business risk is dependent on a number of factors but some of the more widely recognized ones include demand variability, sales price variability, supplier price variability, and output price flexibility relative to input prices. Financial risk is a function of the amount of fixed charge securities such as debt or preferred stock the company uses. In addition to business risk and financial risk, a firm's beta may be associated with the firm's growth rate, accounting beta, and variance in EBIT.
4.
a.The risk premium on the market = 11% – 5% = 6%.
b.E(Ri) = Rf+ βi[E(RM) – Rf]
= 5% + 2 (11% – 5%) = 17%
Thus, the required rate of return on an investment in XYZ company is 17%.
c.E(Ri)= 5 + 1.5(6) = 14%
Since 11% is less than the required rate of return of 14%, the investment will not yield a positive NPV.
5.
a.
Assuming non-systematic risk is zero.
b.
(30%) = (1.72) (152) +
Thus,
= √.065 + .09 = 39.37%
c.Systematic risk == (1.7)2(15)2 = 6.5%
6.In the original text book, the risk free rate is missing. The question should be in the following:
ABC Company is a manufacturing firm that buys rare raw materials from other countries. However, the supply of these materials is very unstable, causing wide flucuations in ABC’s income. If the expected market rate of return is 10 percent, risk free rate of return is 4 percent, and ABC Company’s beta is 2, what is the company’s expected rate of return?
E(Ri) = Rf+ βi[E(Ri) – Rf]= 4 + (2)[10 –4]
Thus, E(Ri) =16%
7.
Solving for V0,
V0= $88.34
8.
Assume the same risk free rate and return on the market as in problem 8-7.
if V1 is certain.
Thus,
9.
a.Business risk: σroa= √5 = 22.36%
b.Financial risk: σroe– σroa = √7 – √5=4.09%
c.If the firm does not use debt, then σroe= σroa.
Thus, .
10.
a.
b.Required rate of return =
Sellnex:.05 + 1.333(15 – .05) = 18.33%
Kelly:.05 + 0.500(.15 – .05) = 10.00%
Nikita:.05 + 0.833(.15 – .05) = 13.33%
c.Average (market) risk premium = .15 – .05 = .10
d.Risk premium for each company:
Sellnex: / 1.333(.10) = .1333 / .20 – .05 = .15Kelly: / 0.500(.10) = .0500 / .10 – .05 = .05
Nikita: / 0.8330(.10) = .0833 / .13 – .05 = .08
e.Jensen's Performance Measure =α =
Sellnex:α = .15 – .1333= .0127
Kelly:α = .05 – .05 = 0
Nikita:α = .08 – .0833 =–.0033
f.Plot of the Means and Variances are given as follows:
11.
Note that the CML plots the risk-return relationship of efficient portfolios while the SML plots the risk-return relationship of securities held in the efficient portfolio M.
12.
a.
b.Required rate of return = .05 + .375 (.14 – .05) = .0837 or 8.37%.
c.If the expected rate of return is 8%, then Axel is not a good buy.
13.
a.If no debt is used then σroe= 4%.
b.If σroa= 2%, then:
financial risk =σroe – σroa= 4% – 2% = 2%
business risk = σroa = 2%
14.
Required rate of return = 6% + 1.3(14% – 6%) = 16.4%
Then
15.
The SML plots the risk return tradeoff of securities held in a well diversified portfolio. The characteristic line is a plot of the return on one specific security on the SML against the return on the market. The slope of the characteristic line is the beta of the particular security.
16.
a.= (.4)(5%) + (6)(16%) = 11.6%
σp= (.6)(.06) = .036
b.= (1.30)(16%) + (–.30)(5) = 19.30%
σp= (1.30)(.06) = .078
c.We must know the individual investors indifference curves or therefore risk-return preferences to know the investor's optimal portfolio.
17.
If the security’s correlation coefficient with the market portfolio doubles (with all other variables such as variances unchanged), then beta, and therefore the risk premium, will also double. The current risk premium is: 15 – 6 = 9%
The new risk premium would be 18%, and the new discount rate for the security would be: 18 + 6 = 24%
If the stock pays a constant perpetual dividend, then we know from the original data that the dividend (D) must satisfy the equation for the present value of aperpetuity:
Price = Dividend/Discount rate
70 = D/0.15 D = 70 0.15 = $10.50
At the new discount rate of 22%, the stock would be worth: $10.50/0.24 = $43.75
The increase in stock risk has lowered its value by 37.5%.
18.
a.Beta is the sensitivity of the stock’s return to the market return, i.e., the change in the stock return per unit change in the market return.Therefore, we compute each stock’s beta by calculating the difference in its return across the two scenarios divided by the difference in the market return:
- With the two scenarios equally likely, the expected return is an average of the two possible outcomes:
E(rA ) = 0.5 (–5 + 30) = 12.5%
E(rB ) = 0.5 (8 + 14) = 11%
c.
The market expected return is 0.5(7%+26%) =16.5%
Based on its risk, stock A has a required expected return of:
E(rA ) = 5 + 1.8421(16.5 – 5) = 26.18%
The analyst’s forecast of expected return is only 12.5%.Thus the stock’s alpha is:
A = actually expected return – required return (given risk)
= 12.5% – 26.18% = –13.68%
Similarly, the required return for the defensive stock is:
E(rB) = 5 + 0.3158(16.5 – 5) = 8.63%
The analyst’s forecast of expected return for D is 9%, and hence, the stock has a positive alpha:
B = actually expected return – required return (given risk)
= 11 – 8.63 = 2.37%
19.
(a) Possible. If the CAPM is valid, the expected rate of return compensates only for systematic (market) risk, represented by beta, rather than for the standard deviation, which includes nonsystematic risk. Thus, Portfolio A’s lower rate of return can be paired with a higher standard deviation, as long as A’s beta is less than B’s.
(b) Not possible. Portfolio A clearly dominates the market portfolio. Portfolio A has both a lower standard deviation and a higher expected return.
20.
Since the stock’s beta is equal to 1.5, its expected rate of return is:
5 + [1.5 (15 – 5)] = 20%
21.
The series of $1,200 payments is a perpetuity.If beta is 0.8, the cash flow should be discounted at the rate:
6 + [0.8 (16 – 6)] = 14%
PV = $1,200/0.14 = $8571.43
If, however, beta is equal to 1, then the investment should yield 16%, and the price paid for the firm should be:
PV = $1,200/0.16 = $7500
The difference, $1071.43, is the amount you will overpay if you erroneously assume that beta is 0.8 rather than 1.
22.
Using the SML: 7 = 6 + (15 – 6) = 1/9 = 0.11
23.
In the zero-beta CAPM the zero-beta portfolio replaces the risk-free rate, and thus:
E(r) = 10 + 0.8(15 – 10) = 14%
24.
E(rP) = rf + P [E(rM ) – rf ] = 5% + 0.7 (16% − 5%) = 12.7%
= 16% 12.7% = 3.3%
You should invest in this fund because alpha is positive.
25.
The passive portfolio with the same beta as the fund should be invested 70% in the market-index portfolio and 30% in the money market account. For this portfolio:
E(rP) = (0.7 × 16%) + (0.3 × 5%) = 12.7%
26.
r1 = 20%; r2 = 15%; 1 = 1.8; 2 = 1.2
a.To determine which investor was a better selector of individual stocks we look at abnormal return, which is the ex-post alpha; that is, the abnormal return is the difference between the actual return and that predicted by the SML. Without information about the parameters of this equation (risk-free rate and market rate of return) we cannot determine which investor was more accurate.
- If rf = 5% and rM = 15%, then (using the notation alpha for the abnormal return):
1 = 20 – [5 + 1.8(15 – 5)] = 20 – 18 = 2%
2 = 15 – [5 + 1.2(15 – 5)] =15 – 17 = -2%
Here, the first investor has the larger abnormal return and thus appears to be the superior stock selector.
- If rf = 6% and rM = 18%, then:
1 =20 – [6 + 1.8(18 – 6)] = 19 – 21 = –7.6%
2 = 15 – [6+ 1.2(18 – 6)] = 16 – 15 = -5.4%
Here, both investor generate negative alpha meaning neither one of them can produce return other than the general co-movement with the market. However, the second investor appears to be the superior stock selector given its less negative alpha.
27.
a.Since the market portfolio, by definition, has a beta of 1, its expected rate of return is 15%.
- he stock’s expected rate of return in market equilibrium is
E(r) = 6 + [2(15 – 6)] = 24%
- Using the SML, the fair expected rate of return for a stock with = –0.8 is:
E(r) = 6 + [(–0.8)(15 – 6)] = -1.2%
The actually expected rate of return, using the expected price and dividend for next year is:
E(r) = [($53 + $3)/50] – 1 = 12%
Because the actually expected return exceeds the fair return, the stock is underpriced.
28.
(a)The appropriate discount rate for the project is:
rf + [E(rM ) – rf ] = 6 + [1.6 (16 – 6)] = 22%
Using this discount rate:
Annuity factor (22%, 10 years)] = $28.46
(b)The internal rate of return (IRR) for the project is 38%. Recall from your introductory finance class that NPV is positive if IRR > discount rate (or, equivalently, hurdle rate). The highest value that beta can take before the hurdle rate exceeds the IRR is determined by:
38 = 6 +(16 – 6) = 32/10 = 3.2
29.
For these investment proportions, , the portfolio beta is
As the market risk premium is 10%the portfolio risk premium will be