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Unformatted text preview: 84 Chapter 8: Risk and Rates of Return Answers to Endof—Chapter Questions No, it is not riskless. The portfolio would be free of default risk and liquidity risk, but inﬂation
could erode the portfolio's purchasing power. If the actual inﬂation rate is greater than that
expected, interest rates in general will rise to incorporate a larger inflation premium (IP) and—
as we saw in Chapter 7—the value of the portfolio would decline. No, you would be subject to reinvestment rate risk. You might expect to “roll over” the
Treasury bills at a constant (or even increasing) rate of interest, but if interest rates fall, your
investment income will decrease. A U.S. governmentbacked bond that provided interest with constant purchasing power (that is,
an indexed bond) would be close to riskless. The U.S. Treasury currently issues indexed bonds. . The probability distribution for complete certainty is a vertical line. . The probability distribution for total uncertainty is the X—axis from oo to +00. The expected return on a life insurance policy is calculated just as for a common stock. Each
outcome is multiplied by its probability of occurrence, and then these products are summed.
For example, suppose a 1year term policy pays $10,000 at death, and the probability of the
policyholders death in that year is 2%. Then, there is a 98% probability of zero return and a
2% probability of $10,000: Expected return = 0.98($0) + 0.02($10,000) = $200. This expected return could be compared to the premium paid. Generally, the premium will
be larger because of sales and administrative costs, and insurance company proﬁts, indicating a
negative expected rate of return on the investment in the policy. . There is a perfect negative correlation between the returns on the life insurance policy and the returns on the policyholder’s human capital. In fact, these events (death and future lifetime
earnings capacity) are mutually exclusive. The prices of goods and services must cover their
costs. Costs include labor, materials, and capital. Capital costs to a borrower include a return
to the saver who supplied the capital, plus a mark—up (called a “spread") for the ﬁnancial
intermediary that brings the saver and the borrower together. The more efﬁcient the financial
system, the lower the costs of intermediation, the lower the costs to the borrower, and, hence,
the lower the prices of goods and services to consumers. People are generally risk averse. Therefore, they are willing to pay a premium to decrease the
uncertainty of their future cash flows. A life insurance policy guarantees an income (the face
value of the policy) to the policyholder’s beneficiaries when the policyholder’s future earnings
capacity drops to zero. Yes, if the portfolio’s beta is equal to zero. In practice, however, it may be impossible to find
individual stocks that have a nonpositive beta. In this case it would also be impossible to have a
stock portfolio with a zero beta. Even if such a portfolio could be constructed, investors would
probably be better off just purchasing Treasury bills, or other zero beta investments. Integrated Case 1 87 88 2 Security A is less risky if held in a diversiﬁed portfolio because of its negative correlation with other
stocks. In a single—asset portfolio, Security A would be more risky because CA > as and CVA > CVB. No. For a stock to have a negative beta, its returns would have to logically be expected to go up in
the future when other stocks' returns were falling. Just because in one year the stock’s return
increases when the market declined doesn’t mean the stock has a negative beta. A stock in a given
year may move counter to the overall market, even though the stock’s beta is positive. The risk premium on a highbeta stock would increase more than that on a lowbeta stock.
RP, = Risk Premium for Stockj = (rM — rRF)bj. If risk aversion increases, the slope of the SML will increase, and so will the market risk premium
(rM — rRF). The product (rM — rRF)bj is the risk premium of the jth stock. If b, is low (say, 0.5), then
the product will be small; RP, will increase by only half the increase in RPM. However, if b, is large
(say, 2.0), then its risk premium will rise by twice the increase in RPM. According to the Security Market Line (SML) equation, an increase in beta will increase a company’s
expected return by an amount equal to the market risk premium times the change in beta. For
example, assume that the riskfree rate is 6%, and the market risk premium is 5%. If the
company’s beta doubles from 0.8 to 1.6 its expected return increases from 10% to 14%.
Therefore, in general, a company’s expected return will not double when its beta doubles. a. A decrease in risk aversion will decrease the return an investor will require on stocks. Thus,
prices on stocks will increase because the cost of equity will decline. b. With a decline in risk aversion, the risk premium will decline as compared to the historical
difference between returns on stocks and bonds. c. The implication of using the SML equation with historical risk premiums (which would be higher
than the “current” risk premium) is that the CAPM estimated required return would actually be
higher than what would be reflected if the more current risk premium were used. Integrated Case Chapter 8: Risk and Rates of Return Solutions to Endof—Chapter Problems 31 F (0.1)(50%) + (o.2)(5%) + (0.4)(160/0) + (o.2)(25%) + (0.1)(60%) 11.40%. II II 62 = (—50% — 11.40%)2(0.1) + (5% — 11.40%)2(0.2) + (16% — 11.40%)2(0.4)
+ (25% — 11.40%)2(0.2) + (60% — 11.40%)2(0.1)
02 = 712.44; a = 26.69%. 0
CV _ 26.69 /0 — = 2.34.
11.40% 82 Investment Beta
$35,000 0.8
40 000 1.4 Total M
bp 2 ($35,000/$75,000)(0.8) + ($40,000/$75,000)(1.4) = 1.12. 83 r... = 6%; rM = 13%; b = 07; r = ?
r = rRF + “M _ rRF)b
= 6% + (13% — 6%)0.7
= 10.9%.
84 rRF = RPM 2 60/0; rM = rM : 5% +‘(6%)1 = 11%.
rwhen b = 1.2 = ? r = 5% + 6%(l.2) = 12.2%. 3‘5 a r = 11%; rap = 7%; RPM = 4%. r = rRF + (M — rRF)b
11% = 70/0 + 40/ob
4% = 4%b b = 1. Chapter 8: Risk and Rates of Return Integrated Case 3 b. rRF = 7%; RPM 2 6%; b = 1.
r = rm: + (rM — rRF)b = 7% + (6%)1
= 13%. 86 a. F: if;i .
i=1 R, = 0.1(35%) + 0.2(00/0) + 0.4(200/0) + 0.2(250/0) + 0.1(450/0)
= 14% versus 12% for X. b. 0: [ﬁn—WE. of, = (10% — 12%)2(0.1) + (2% — 12%)2(0.2) + (12% — 12%)2(0.4)
+ (20% — 12%)2(o.2) + (38% — 12%)2(0.1) = 148.8%. ox = 12.20% versus 20.35% for Y. CVx = csx/Fx = 12.20%/12% = 1.02, while CVY = 20.35%/14% = 1.45. If Stock Y is less highly correlated with the market than X, then it might have a lower beta than Stock X, and hence be less risky in a portfolio sense. $400,000 $600,000 $1,000,000 $2,000,000
— 1.50 +——— 0.50 +—— 1.25 +———
$4,000,000 ( ) $4,000,000 ( ) $4,000,000 ( ) $4,000,000 bp = (0.1)(15) + (0.15)(0.50) + (0.25)(1.25) + (0.5)(0.75)
= 0.15 — 0.075 + 0.3125 + 0.375 = 0.7625. 87 Porh‘olio beta = (0.75) rp = FR; + (FM  I'RFXbp) = 6% + (14% — 6%)(0.7625) = 12.1%. Alternative solution: First, calculate the return for each stock using the CAPM equation
[rRF + (rM — rRF)b], and then calculate the weighted average of these returns. rRF = 6% and (m — rRF) = 8%. m Investment ﬂ rirﬁﬂﬂigﬁb Weight
A v $ 400,000 1.50 18% 0.10
B 600,000 (0.50) 2 0.15
C 1,000,000 1.25 16 0.25
D 2,000,000 0.75 12 M Total ME Q9 rp = 18%(0.10) + 2%(015) + 16%(0.25) + 12%(050) = 12.1%. 4 Integrated Case Chapter 8: Risk and Rates of Return 8—8 In equilibrium:
r, = F, = 12.5%. r] = rRF + (M — rRF)b
12.5% = 4.5% + (10.5% — 4.5%)b
b = 1.33. 89 We know that bR = 1.50, b5 = 0.75, rM = 13%, rRF = 7%.
r, : FRI: + (rM — rRF)b, : 70/0 + (130/0 — 70/0)bj. rR = 7% + 6%(1.50) = 16.0%
rS = 7% + 6%(0.75) = 11.5
4.5% 810 An index fund will have a beta of 1.0. If rM is 12.0% (given in the problem) and the riskfree rate is
5%, you can calculate the market risk premium (RPM) calculated as rM — rRF as follows: r = rm: ‘1' 12.0% = 5% + (RPM)1.0
7.00/0 = RPM. Now, you can use the RPM, the rm, and the two stocks’ betas to calculate their required returns. Bradford: r3 = I'm: + = 5% + (7.0%)145
= 50/0 + 10.150/0
= 15.19%. Farley: FF = I'm: + = 5% + (7.0%)085
= 5% + 5.95%
=, 10.95%. The difference in their required returns is:
15.15% — 10.95% = 4.2%. 811 rm: = r* + IP = 2.5% + 3.5% = 6%. r5 = 6% + (6.5%)1.7 = 17.05%. Chapter 8: Risk and Rates of Return Integrated Case 5 812 Using Stock X (or any stock):
90/0 = FR}: + (rM — rRF)bx
9% = 5.5% + (rM — rRF)0.8
(rM — rRF) = 4.3750/0.
8'13 3 l'j = rRF + (I‘M " TRF)bi = 90/0 + (140/0 — = 15.50/0. b. 1. rRF increases to 10%:
rM increases by 1 percentage point, from 14% to 15%. ri = rRF + (rM — rRF)b. = 10% + (15% — 10%)13 = 16.5%. 2. rRF decreases to 8%:
no decreases by 1%, from 14% to 13%. r~I = rRF + (rM — rRF)bi = 8% +(13% — 8%)1.3 = 14.5%. c. 1. no increases to 16%:
ri = FR}: + (FM —‘ rRF)bi = 90/0 + (160/0 — 90/0)1.3 = 18.10/0. 2. rM decreases to 13%:
r; = I‘RF + (I’M — I‘RF)bi = 9% + (13% — 9%)13 = 14.2%. . _ $142,500 $7,500
8 14 Old portfolio beta $150,000 (b) + $150,000 (1.00)
1.12 : 0.95b + 0.05
1.07 = 0.9513
1.1263 = b. New portfolio beta = 0.95(1.1263) + 0.05(1.75) = 1.1575 2 1.16. Alternative solutions: 1. Old portfolio beta = 1.12 = (0.05)b1 + (0.05M;2 + + (0.05)b20
1.12 = (Zbi)(0.05)
2b. = 1.12/0.05 = 22.4. New portfolio beta = (22.4 — 1.0 + 1.75)(0.05) = 1.1575 2 1.16. 2. Zbi excluding the stock with the beta equal to 1.0 is 22.4 — 1.0 = 21.4, so the beta of the
portfolio excluding this stock is b = 21.4/ 19 = 1.1263. The beta of the new portfolio is: 1.1263(0.95) + 1.75(0.05) = 1.1575 5 1.16. 6 Integrated Case Chapter 8: Risk and Rates of Return 815 bHRI = 1.8; bLRJ = 0.6. No changes occur.
rRF = 6%. Decreases by 1.5% to 4.5%.
rM = 13%. Falls to 10.5%.
Now SML: r, = rRF + (rM — rRF)b,.
rHRI = 4.5% + (10.5% — 4.5%)1.8 = 4.5% + 6%(1.8) = 15.3% rm, 2 4.5% + (10.5% — 4.5%)06 = 4.5% + 6%(0.6) = 8.1%
Difference 7 0/ 816 Step 1: Determine the market risk premium from the CAPM: 0.12 = '1‘ (FM — rRF)1.25
(FM — TRF) = Step 2: Calculate the beta of the new portfolio:
($500,000/$5,500,000)(0.75) + ($5,000,000/$5,500,000)(1.25) = 1.2045. Step 3: Calculate the required return on the new portfolio:
5.25% + (5.4°/o)(1.2045) = 11.75%. 817 After additional investments are made, for the entire fund to have an expected return of 13%, the portfolio
must have a beta of 1.5455 as shown below: 13% = 4.5% + (5.5%)b
b = 1.5455. Since the fund's beta is a weighted average of the betas of all the individual investments, we can
calculate the required beta on the additional investment as follows: ($20,000,000)(1.5) + $5,000,000X
$25,000,000 $25,000,000 1.5455 = 1.2 + 0.2x
0.3455 = 0.2x
X = 1.7275. 1.5455: 818 3. ($1 million)(0.5) + ($0)(0.5) = $0.5 million.
b. You would probably take the sure $0.5 million.
c. Risk averter.
d. 1. ($1.15 million)(0.5) + ($0)(0.5) = $575,000, or an expected proﬁt of $75,000.
2. $75,000/$500,000 = 15%. 3. This depends on the individual's degree of risk aversion. Chapter 8: Risk and Rates of Return Integrated Case 7 819 8 4. Again, this depends on the individual. 5. The situation would be unchanged if the stocks’ returns were perfectly positively
correlated. Otherwise, the stock portfolio would have the same expected return as the
single stock (15%) but a lower standard deviation. If the correlation coefﬁcient between
each pair of stocks was a negative one, the portfolio would be virtually riskless. Since p for stocks is generally in the range of +0.35, investing in a portfolio of stocks would
deﬁnitely be an improvement over investing in the single stock. Fx = 10%; bx = 0.9; ox = 35%. IA”, = 12.5%; by = 1.2; Uy = 25%. rRF = 6%; RPM = 5%. b. Integrated Case CVX = 35%/10% = 3.5. CVY = 25%/12.5% = 2.0. For diversiﬁed investors the relevant risk is measured by beta. Therefore, the stock with the
higher beta is more risky. Stock Y has the higher beta so it is more risky than Stock X. rx = 6% + 5%(09)
= 10.5%. rY = 6% + 5%(1.2)
= 12%. l’x = 10.5%; PX = 10%.
FY = 12%; FY = 12.5%. Stock Y would be most attractive to a diversiﬁed investor since its expected return of 12.5% is
greater than its required return of 12%. b, = ($7,500/$10,000)0.9 + ($2,500/$10,000)1.2
= 0.6750 + 0.30
= 0.9750. rp = 6% + 5%(0975)
= 10.875%. If RPM increases from 5% to 6%, the stock with the highest beta will have the largest increase
in its required return. Therefore, Stock Y will have the greatest increase. Check:
rx = 6% + 6%(O.9)
= 11.4%. Increase 10.5% to 11.4%. ry = 6% + 6%(1.2) = 13.2%. Increase 12% to 13.2%. Chapter 8: Risk and Rates of Return 8—20 The answers to a, b, c, and d are given below: re r_B Portfolio 2001 (18.00%) (14.50%) (16.25%)
2002 33.00 21.80 27.40
2003 15.00 30.50 22.75
2004 (0.50) (7.60) (4.05)
2005 27.00 26.30 26.65
Mean 11.30 11.30 11.30
Std. Dev. 20.79 20.78 20.13
Coef. Var. 1.84 1.84 1.78 e. A riskaverse investor would choose the portfolio over either Stock A or Stock B alone, since the
portfolio offers the same expected return but with less risk. This result occurs because returns
on A and B are not perfectly positively correlated (rAB = 0.88). 321 a. f... = 0.10%) + 0.2(90/6) + 0.4(110/6) + 0.203%) + 0.1(150/6) = 11%. rRF = 6%. (given) Therefore, the SML equation is:
n = rm: + (I‘M  rRF)bi = 60/0 + (11%  6°/o)bi = 6% + (5°/o)bi. b. First, determine the fund’s beta, bF. The weights are the percentage of funds invested in each
stock: A = $160/$500 = 0.32.
B = $120/$500 = 0.24.
C = $80/$500 = 0.16. ' D = $80/$500 = 0.16.
E = $60/$500 = 0.12. bF = 0.32(o.5) + 0.24(2.0) + 0.16(4.0) + 0.16(1.0) + o.12(3.0)
= 0.16 + 0.48 + 0.64 + 0.16 + 0.36 = 1.8. Next, use bF = 1.8 in the SML determined in Part a:
FF = 6% + (11% — 6%)1.8 = 6% + 9% = 15%.
c. rN = Required rate of return on new stock = 6% + (5%)2.0 = 16%.
An expected return of 15% on the new stock is below the 16% required rate of return on an
investment with a risk of b = 2.0. Since rN = 16% > PM = 15%, the new stock should n_ot be purchased. The expected rate of return that would make the fund indifferent to purchasing the
stock is 16%. Chapter 8: Risk and Rates of Return Integrated Case 9 ...
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 THOMAS
 Interest, Risk premium, rM — rRF

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