CHAPTER 8

QUESTIONS

8-1 The statement is correct. Less risk means less variability of returns, which means a more peaked probability distribution than for a riskier investment.

8-2 a. The probability distribution for complete certainty is a vertical line.

b. The probability distribution for total uncertainty is the X-axis from -∞ to +∞.

8-3 Events that affect a single firm or a group of firms are defined as the components of unsystematic risk. Such events include changes in manufacturing methods, changes in management, labor strikes at the firm, and so forth. These events generally influence the stock prices of individual firms, not all firms. On the other hand, events that affect everyone and every business are defined as the components of systematic risk. Such factors include changes in inflation expectations, increases in consumer and producer prices, unemployment factors, and so forth. These events affect the stock prices of all firms, although not equally.

8-4 Systematic risk is the “relevant” risk because it cannot be diversified away. Unsystematic, or firm-specific, risk can be significantly reduced or eliminated through diversification, but systematic risk cannot. Because investors should not be rewarded for risk they can eliminate—that is, unsystematic risk—the returns earned on investments should be based on the systematic risk associated with the investment. In other words, investors should not be rewarded for risk they do not have to take; rather they should be rewarded only for the risk that they must take because it cannot be eliminated through diversification—that is, systematic risk.

8-5 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 riskier because σA > σB and CVA = 5 > CVB = 0.83.

8-6 a. 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 showed in Chapter 5—the value of the portfolio would decline.

b. No, you would be subject to interest rate 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, and vice versa.

c. A U.S. government-backed bond that provided interest with constant purchasing power (that is, an indexed bond) would be close to riskless (risk free). No such bond exists in the United States, however.

8-7 a. 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 one-year term policy pays $10,000 at death, and the probability of the policyholder’s death in that year is 2 percent. Then, there is a 98 percent probability of zero return and a 2 percent probability of $10,000:

Expected return = 0.98($0) + 0.02($10,000) = $200.

1

Chapter 8

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.

b. 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, because a person has no future earnings when he or she dies.

c. 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.

8-8 The risk premium on a high beta stock would increase more.

RPj = Risk Premium for Stock j = (rM – rRF)ßj.

If risk aversion increases, the slope of the SML will increase, and so will the market risk premium (rM – rRF). The product (rM – rRF)ßj is the risk premium of the j th stock. If ßj is low (say, 0.5), then the product will be small because RP j will increase by only half the increase in RPM. However, if ßj is large (say, 2.0), then its risk premium will rise by twice the increase in RPM.

8-9 At least theoretically, it is possible to combine two stocks that are perfectly negatively correlated to produce a portfolio that has no risk. For example, see Figure 8-5 in the chapter. However, it would be very difficult, if not impossible, to find two stocks that are perfectly negatively correlated. There are other investment instruments that can be used to construct risk-free portfolios, but discussion of these instruments is beyond the scope of this book. It is important to recognize that to construct a risk-free portfolio, investments that are negatively correlated must be available.

8-10 False. The amount of the risk premium changes by the amount of the beta coefficient. For example, when ß = 1.0 and ß = 2.0, the respective risk premiums for the investment are:

RP1 = (rM – rRF)ß1 = (rM – rRF)1.0 = RPM

RP2 = (rM – rRF)ß2 = (rM – rRF)2.0 = 2RPM

Thus, the returns on the investment with these beta coefficients would be:

r1 = rRF + RPM

r2 = rRF + 2RPM

Example: Suppose that rRF = 4% and rM = 11%. Using this information, we have

r1 = rRF + RPM = 4% + (11% - 4%)1.0 = 4% + 7% = 11%

r2 = rRF + 2RPM = 4% + (11% - 4%)2.0 = 4% + 14% = 18%

2

Chapter 8As you can see, the risk premium, not the total return, is twice as large when ß = 2.0 than when ß = 1.0.

____________________________________________________________

PROBLEMS

8-1 = 0.2(-5%) + 0.4(10%) + 0.4(30%) = 15.0%

8-2 = 0.3(30%) + 0.2(10%) + 0.5(-2%) = 10.0%

8-3

8-4 r = 5% + (12% - 5%)1.5 = 15.5%

8-5 r = 4% + (5%)2.0 = 14%

8-6 CVD = 8.0%/10.0% = 0.8

CVE = 24.0%/36% = 0.6667

Stock E has the lower coefficient of variation, so it has the lower relative risk.

8-7 rZR = rRF + (rRF – rM)β

16% = rRF + (10% – rRF)1.8

16% = rRF + 18% – 1.8rRF

rRF = 2%/0.8 = 2.5%

8-8 Portfolio beta:

Investment Weight Beta Portfolio beta (1) (2) (3) (4) = (2) x (3)

$ 400,000 0.40 1.5 0.6500,000 0.50 2.0 1.0

100,000 0.10 4.0 0 .4 $1,000,000 1.00 2.0

rp = rRF + (rM – rRF)(ßp) = 3% + (10% – 3%)2.0 = 17%.

Alternative solution: First compute the return for each stock using the CAPM equation [rRF + (rM – rRF)ßj], and then compute the weighted average of these returns.

3

Chapter 8

rRF = 3% and rM – rRF = 7%.

Investment Beta rj = rRF + (rM – rRF)ßj Weight

$ 400,000 1.5 13.5% 0.40 500,000 2.0 17.0 0.50 100,000 4.0 31.0 0 .10

Total $1,000,000 1 .00

rp = 13.5%(0.40) + 17.0%(0.50) + 31.0%(0.10) = 17%.

8-9 a. = (0.3)(15%) + (0.4)(9%) + (0.3)(18%) = 13.5%.

= (0.3)(20%) + (0.4)(5%) + (0.3)(12%) = 11.6%.

b.

c.

8-10 Expected Amount

Investment Return, Invested Weight Portfolio Return (1) (2) (3) (4) (5) = (2) x (4) ABC 30% $ 10,000 0.1 3.0%EFG 16 50,000 0.5 8.0QRP 20 40,000 0.4 8 .0

$100,000 19.0%

8-11 a. = 0.1(─35%) + 0.2(0%) + 0.4(20%) + 0.2(25%) + 0.1(45%) = 14%

= 12%

b. = (–10% – 12%)2(0.1) + (2% – 12%)2(0.2) + (12% – 12%)2(0.4)

+ (20% – 12%)2(0.2) + (38% – 12%)2(0.1) = 148.8

4

Chapter 8

CVX = 12.20%/12% = 1.02

CVY = 20.35%/14% = 1.45.

If Stock Y is less highly correlated with the market than Stock X, then it might have a lower beta than Stock X, and hence be less risky in a portfolio sense.

8-12 rK = rRF + (rM – rRF)β

17.0% = 3.0% + (10.0% – 3%) β RPM = 10% – 3% = 7%

β = (17.0% – 3.0%)/7% = 2.0

Because RPM should equal 8 percent rather than 7 percent,

rK = 3% + 8%(2.0) = 19.0%

8-13 βold = 1.2; there are five stocks in the portfolio

If one stock with β = 1.0 is sold, then the portfolio’s beta would change

βportfolio = 1.2 = 0.2(β1) + 0.2(β2) + 0.2(β3) + 0.2(β4) + 0.2(1.0) = 0.2(∑β of four stocks) + 0.2(1.0)

0.2(∑β of four stocks) + 0.2 = 1.2

(∑β of four stocks) = 1.0/0.2 = 5.0

βnew = 0.2(∑β of four stocks) + 0.2(2.0) = 0.2(5.0) + 0.2(2.0) = 1.4

8-14 βold = 1.5; portfolio value = $400,000

βnew = 1.8 = 1.5 – ($100,000/$400,000)(βstock)

βstock = 0.3/0.25 = 1.2

The four original stocks must have an average β equal to 1.6 = 1.2/0.75. This can be confirmed by computing the portfolio’s beta using this information:

βold = ($300,000/$400,000)1.6 + ($100,000/$400,000)1.2 = 1.5

8-15 a. rB = rRF + (rM – rRF)ßB

14% = 8% + (11% – 8%)ßB

14% = 8% + 3%(ßB)

6% = 3%(ßB)

5

Chapter 8

2 = ßB.

b. rB = 8% + 3%(ßB)

rB = 8% + 3%(1.5)

rB = 12.5%.

8-16 a. rX = rRF + (rM – rRF)ßX = 9% + (14% – 9%)1.3 = 15.5%.

b. RPM = rM – rRF = 14% - 9% = 5%

(1) rRF increases to 10%; RPM does not change:

rX = rRF + (RPM)ßX = 10% + (5%)1.3 = 16.5%.

rM = rRF + (RPM)ßM = 10% + (5%)1.0 = 15.0%.

(2) rRF decreases to 8%; RPM does not change:

rX = rRF + (RPM)ßX = 8% + (5%)1.3 = 14.5%.

rM = rRF + (RPM)ßM = 8% + (5%)1.0 = 13.0%.

c. (1) rM increases to 16%; RPM = 16% – 9% = 7%

rX = rRF + (RPM)ßX = 9% + (7%)1.3 = 18.1%.

(2) rM decreases to 13%; RPM = 13% – 9% = 4%

rX = rRF + (RPM)ßX = 9% + (4%)1.3 = 14.2%.

8-17 Alternative solutions:

1. Old portfolio beta = 1.12 = (0.05)ß1 + (0.05)ß2 +...+ (0.05)ß20

1.12 = (Σßj)(0.05)Σßj = 1.12/0.05 = 22.4

New portfolio beta = (22.4 - 1.0 + 1.75)(0.05) = 1.1575 = 1.16

2. Σßj 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 ß = 21.4/19 = 1.1263. The beta of the new portfolio is:

1.1263(0.95) + 1.75(0.05) = 1.1575 = 1.16

8-18 We know that ßR = 1.50, ßS = 0.75, rM = 15%, rRF = 9%.

6

Chapter 8rj = rRF + (rM – rRF)ßj = 9% + (15% - 9%)ßj.

rR = 9% + 6%(1.50) = 18.0%rS = 9% + 6%(0.75) = 13 .5 4 .5%

8-19 Portfolio beta:

Stock Investment Weight Beta Portfolio beta (1) (2) (3) (4) (5) = (3) x (4)

A $ 400,000 0.10 1.50 0.1500B 600,000 0.15 -0.50 -0.0750C 1,000,000 0.25 1.25 0.3125D 2,000,000 0 .50 0.75 0 .3750

$4,000,000 1.00 0.7625

rp = rRF + (rM – rRF)(ßp) = 6% + (14% – 6%)(0.7625) = 12.1%.

Alternative solution: First compute the return for each stock using the CAPM equation [rRF + (rM – rRF)ßj], and then compute the weighted average of these returns.

rRF = 6% and rM – rRF = 8%.

Stock Investment Beta rj = 6% + (8%)ßj Weight

A $ 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 0 .50

Total $4,000,000 1 .00

rp = 18%(0.10) + 2%(0.15) + 16%(0.25) + 12%(0.50) = 12.1%.

8-20 Following is information about Investment A, Investment B, and Investment C:

Return on Investment: Economic Condition Probability A B C Boom 0.5 25.0% 40.0% 5.0%Normal 0.4 15.0 20.0 10.0Recession 0.1 -5.0 -40.0 15.0

18.0% 24.0%

23.3% 3.3%

a. = 0.5(5%) + 0.4(10%) + 0.1 (15%) = 8.0%

b. = 0.5(25% - 18%)2 + 0.4(15% - 18%)2 + 0.1(-5% - 18%)2 = 81

7

Chapter 8

c. CVA = 9%/18% = 0.50

CVB = 23.3%/24% = 0.97

CVC = 3.3%/8% = 0.41

8-21 a. ($1 million)(0.5) + ($0)(0.5) = $0.5 million.

b. You would probably take the sure $0.5 million.

c. Risk averter. The expected payoff of both alternatives is the same, but Alternative (1) offers a sure $0.5 million.

d. (1) ($1,150,000)(0.5) + ($0)(0.5) = $575,000, or an expected profit of $75,000.

(2) $75,000/$500,000 = 15%.

(3) This depends on your degree of risk aversion.

(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 percent) but a lower standard deviation. If the correlation coefficient (ρ) between each pair of stocks was negative one, the portfolio would be virtually riskless. Because ρ for stocks is generally in the range of +0.4 to +0.6, investing in a portfolio of stocks would definitely be an improvement over investing in the single stock.

8-22 a. = 0.1(10%) + 0.2(12%) + 0.4(13%) + 0.2(16%) + 0.1(17%) = 13.5%.

b. To determine the fund's beta, ßF, the weight for the amount invested in each stock needs to be computed.

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

ßF = 0.32(0.5) + 0.24(2.0) + 0.16(4.0) + 0.16(1.0) + 0.12(3.0) = 0.16 + 0.48 + 0.64 + 0.16 + 0.36 = 1.8.

c. rRF = 8% (given)

Therefore, the SML equation is

rF = rRF + (rM - rRF)ßF = 8% + (13.5% - 8%)ßF = 8% + (5.5%)ßF.

8

Chapter 8d. Use ßF = 1.8 in the SML determined in Part b:

= 8% + (13.5% - 8%)1.8 = 8% + 9.90% = 17.90%.

e. rNew = Required rate of return on new stock = 8% + (5.5%)2.0 = 19%.

An expected return of 18 percent on the new stock is below the 19 percent required rate of return on an investment with a risk of ß = 2.0. Because rNew = 19% > 18%, the new stock should not be purchased. The expected rate of return that would make McAlhany indifferent to purchasing the stock is 19 percent.

8-23 The answers to a, b, c, and d are given below:

50/50 Portfolio

2003 -18.0% -14.5% -16.25% 2004 33.0 21.8 27.40 2005 15.0 30.5 22.75 2006 - 0.5 - 7.6 - 4.05 2007 27.0 26.3 26.65

Mean 11.30 11.30 11.30 Std. Dev. 20.79 20.78 20.13 Coeff. of Var. 1.84 1.84 1.78

The computations for the average return, standard deviation, and coefficient of variation for Stock A are:

- ( - )2

-18.0% 11.3% -29.3% 858.4933.0 11.3 21.7 470.8915.0 11.3 3.7 13.69-0.5 11.3 -11.8 139.2427 .0 11.3 15.7 246 .49 56.5% 1,728.80

e. A risk-averse investor would choose the portfolio over either Stock A or Stock B alone, because 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 (ρAB = 0.88).

9

Chapter 8

INTEGRATIVE PROBLEM

A. (1) WHY IS THE T-BILL'S RETURN INDEPENDENT OF THE STATE OF THE ECONOMY? DO T-BILLS PROMISE A COMPLETELY RISK-FREE RETURN?

10

8-24 ASSUME THAT YOU RECENTLY GRADUATED WITH A MAJOR IN FINANCE, AND YOU JUST LANDED A JOB IN THE TRUST DEPARTMENT OF A LARGE REGIONAL BANK. YOUR FIRST ASSIGNMENT IS TO INVEST $100,000 FROM AN ESTATE FOR WHICH THE BANK IS TRUSTEE. BECAUSE THE ESTATE IS EXPECTED TO BE DISTRIBUTED TO THE HEIRS IN APPROXIMATELY ONE YEAR, YOU HAVE BEEN INSTRUCTED TO PLAN FOR A ONE-YEAR HOLDING PERIOD. FURTHERMORE, YOUR BOSS HAS RESTRICTED YOU TO THE FOLLOWING INVESTMENT ALTERNATIVES, SHOWN WITH THEIR PROBABILITIES AND ASSOCIATED OUTCOMES. (FOR NOW, DISREGARD THE BLANK SPACES IN THE TABLE; YOU WILL FILL IN THE BLANKS LATER.)

ESTIMATED RETURNS ON ALTERNATIVE INVESTMENTS

STATE OF THE PROB- T- HIGH COLLEC- U.S. MARKET TWO-STOCK ECONOMY   ABILITY BILLS TECH TIONS RUBBER PORTFOLIO PORTFOLIORECESSION 0.1 8.0% -22.0% 28.0% 10.0% -13.0%BELOW AVERAGE 0.2 8.0 - 2.0 14.7 -10.0 1.0AVERAGE 0.4 8.0 20.0 0.0 7.0 15.0ABOVE AVERAGE 0.2 8.0 35.0 -10.0 45.0 29.0BOOM 0.1 8.0 50.0 -20.0 30.0 43.0

σCV

THE BANK'S ECONOMIC FORECASTING STAFF HAS DEVELOPED PROBABILITY ESTIMATES FOR THE STATE OF THE ECONOMY, AND THE TRUST DEPARTMENT HAS A SOPHISTICATED COMPUTER PROGRAM THAT WAS USED TO ESTIMATE THE RATE OF RETURN ON EACH ALTERNATIVE UNDER EACH STATE OF THE ECONOMY. HIGH TECH INC. IS AN ELECTRONICS FIRM, COLLECTIONS INC. COLLECTS PAST-DUE DEBTS, AND U.S. RUBBER MANUFACTURES TIRES AND VARIOUS OTHER RUBBER AND PLASTICS PRODUCTS. THE BANK ALSO MAINTAINS AN “INDEX FUND” THAT INCLUDES A MARKET-WEIGHTED FRACTION OF ALL PUBLICLY TRADED STOCKS; BY INVESTING IN THAT FUND, YOU CAN OBTAIN AVERAGE STOCK MARKET RESULTS. GIVEN THE SITUATION AS DESCRIBED, ANSWER THE FOLLOWING QUESTIONS:

Chapter 8ANSWER: The 8 percent T-bill return does not depend on the state of the economy because the Treasury must (and will) redeem the bills at par regardless of the state of the economy.

The T-bills are risk-free in the default risk sense because the 8 percent return will be realized in all possible economic states. However, remember that this return is composed of the real risk-free rate, say, 3 percent, plus an inflation premium, say 5 percent. Because there is uncertainty about inflation, it is unlikely that the realized real rate of return would equal the expected 3 percent. For example, if inflation averaged 6 percent over the year, then the realized real return would be only 8% ─ 6% = 2%, not the expected 3 percent. Thus, in terms of purchasing power, T-bills are not riskless.

Also, if you invested in a portfolio of T-bills, and rates then declined, your nominal income would fall—that is, T-bills are exposed to reinvestment rate risk. So, we conclude that there are no truly risk-free securities in the United States. If the Treasury sold inflation-indexed, tax-exempt bonds, they would be truly riskless, but all actual securities are exposed to some type of risk.

ANSWER: High Tech’s returns move with, and thus are positively correlated with, the economy, because the firm’s sales, and hence profits, generally will experience the same type of ups and downs as the economy. If the economy is booming, so will High Tech. On the other hand, Collections is considered by many investors to be a hedge against both bad times and high inflation; so if the stock market crashes, investors in this stock should do relatively well. Stocks such as Collections are thus negatively correlated with (move counter to) the economy. [Note: In actuality, it is almost impossible to find stocks that are expected to move counter to the economy. Even Collections shares have positive (but low) correlation with the market.]

ANSWER: The expected rate of return, , is expressed as follows:

Here Pri is the probability of occurrence of the ith state, r i is the estimated rate of return for that state, and n is the number of states. The calculation for High Tech is:

= 0.1(-22.0%) + 0.2(-2.0%) + 0.4(20.0%) + 0.2(35.0%) + 0.1(50.0%) = 17.40%.

We can now add the 17.4% to the bottom of the table, and use the same formula to calculate for the other alternatives. Here they are:

= 8.00%

= 1.74%

= 13.80%

= 15.00%11

A. (2) WHY ARE HIGH TECH'S RETURNS EXPECTED TO MOVE WITH THE ECONOMY WHEREAS COLLECTIONS' ARE EXPECTED TO MOVE COUNTER TO THE ECONOMY?

B. CALCULATE THE EXPECTED RATE OF RETURN ON EACH ALTERNATIVE

AND FILL IN THE ROW FOR IN THE TABLE.

(2) WHAT TYPE OF RISK DOES THE STANDARD DEVIATION MEASURE?

Chapter 8

C. YOU SHOULD RECOGNIZE THAT BASING A DECISION SOLELY ON EXPECTED RETURNS IS ONLY APPROPRIATE FOR RISK-NEUTRAL INDIVIDUALS. BECAUSE THE BENEFICIARIES OF THE TRUST, LIKE VIRTUALLY EVERYONE, ARE RISK AVERSE, THE RISKINESS OF EACH ALTERNATIVE IS AN IMPORTANT ASPECT OF THE DECISION. ONE POSSIBLE MEASURE OF RISK IS THE STANDARD DEVIATION OF RETURNS, σ. (1) CALCULATE THIS VALUE FOR EACH ALTERNATIVE, AND FILL IN THE ROW FOR σ IN THE TABLE.

ANSWER: The standard deviation is calculated as follows:

Here are the standard deviations for the other alternatives:

σT-Bills = 0.00%.σCollections = 13.37%.

σU.S.Rubber = 18.82%. σM = 15.34%.

ANSWER: The standard deviation is a measure of a security's (or a portfolio’s) total, or stand-alone, risk. The larger the standard deviation, the higher the probability that actual realized returns will fall far below the expected return, and that losses rather than profits will be incurred.

ANSWER: Probability distribution curves for High Tech, U.S. Rubber, and T-bills are shown here:

12

(3) DRAW A GRAPH THAT SHOWS ROUGHLY THE SHAPE OF THE PROBABILITY DISTRIBUTIONS FOR HIGH TECH, U.S. RUBBER, AND T-BILLS.

Chapter 8

D. SUPPOSE YOU SUDDENLY REMEMBERED THAT THE COEFFICIENT OF VARIATION (CV) IS GENERALLY REGARDED AS BEING A BETTER MEASURE OF TOTAL RISK THAN THE STANDARD DEVIATION WHEN THE ALTERNATIVES BEING CONSIDERED HAVE WIDELY DIFFERING EXPECTED RETURNS AND RISKS. CALCULATE THE CVs FOR THE DIFFERENT SECURITIES, AND FILL IN THE ROW FOR CV IN THE TABLE. DOES THE CV MEASURE PRODUCE THE SAME RISK RANKINGS AS THE STANDARD DEVIATION?

ANSWER: The coefficient of variation (CV) is a standardized measure of dispersion about the expected value; it shows the amount of risk per unit of return.

CVT-bills = 0.00%/8.00% = 0.00

CVHigh Tech = 20.04%/17.40% = 1.15

CVCollections = 13.37%/1.74% = 7.68

CVU.S. Rubber = 18.82%/13.80% = 1.36 CVM = 15.34%/15.00% = 1.02

When we measure risk per unit of return, Collections, with its low expected return, becomes the riskiest stock. The CV is a better measure of an asset’s total, or stand-alone, risk than σ because CV considers both the expected value and the dispersion of a distribution. A security with a low expected return and a low standard deviation could have a higher chance of a loss than one with a high σ but a high .

13

Chapter 8

E. (1) SUPPOSE YOU CREATED A TWO-STOCK PORTFOLIO BY INVESTING $50,000 IN HIGH TECH AND $50,000 IN COLLECTIONS. (1) CALCULATE THE EXPECTED RETURN ( ,) THE STANDARD DEVIATION (σp), AND THE COEFFICIENT OF VARIATION (CVp) FOR THIS PORTFOLIO AND FILL IN THE APPROPRIATE ROWS IN THE TABLE.

ANSWER: To find the expected rate of return on the two-stock portfolio, we first calculate the rate of return on the portfolio in each state of the economy. Because we have half of our money in each stock, the portfolio’s return will be a weighted average in each type of economy. For a recession, we have: r p = 0.5(-22%) + 0.5(28%) = 3%. We would do similar calculations for the other states of the economy, and get these results:

State Portfolio Recession 3.00% Below average 6.35 Average 10.00 Above average 12.50 Boom 15.00

Add these to the table to complete the last column.

Now multiply the probabilities times outcomes in each state to get the expected return on this two-stock portfolio—that is, 3.0%(0.1) + 6.35%(0.2) + 10.0%(0.4) + 12.5%(0.2) + 15.0%(0.1) = 9.57%.

Alternatively, we could apply this formula:

= wj x = 0.5(17.40%) + 0.5(1.74%) = 9.57%,

which finds as the weighted average of the expected returns of the individual securities in the portfolio.

The standard deviation of the portfolio is:

CVP = 3.336%/9.57% = 0.349

(2) HOW DOES THE RISKINESS OF THIS TWO-STOCK PORTFOLIO COMPARE TO THE RISKINESS OF THE INDIVIDUAL STOCKS IF THEY WERE HELD IN ISOLATION?

ANSWER: Using either σ or CV as our total risk measure, the total risk of the portfolio is significantly less than the total risk of the individual stocks. This is because the two stocks are negatively correlated—when High Tech is doing poorly, Collections is doing well, and vice versa. Combining the two stocks

14

Chapter 8diversifies away some of the risk inherent in each stock if it were held in isolation—that is, in a single-stock portfolio.

ANSWER:

This graph shows the probability distributions for a one-stock portfolio and a portfolio of many similar stocks. The graph shows that the standard deviation gets smaller as more stocks are combined in the portfolio, while rp (the portfolio’s return) remains constant. Thus, by adding stocks to your portfolio, which initially started as a single-stock portfolio, risk has been reduced.

In the real world, stocks are positively correlated with one another—that is, the economy does well, so do stocks in general, and vice versa. Correlation coefficients between pairs of stocks generally range from +0.5 to +0.7. The graph below shows the relationship between portfolio size and risk.

15

F. SUPPOSE AN INVESTOR STARTS WITH A PORTFOLIO CONSISTING OF ONE RANDOMLY SELECTED STOCK. WHAT WOULD HAPPEN (1) TO THE RISKINESS AND (2) TO THE EXPECTED RETURN OF THE PORTFOLIO AS MORE AND MORE RANDOMLY SELECTED STOCKS WERE ADDED? WHAT IS THE IMPLICATION FOR INVESTORS? DRAW TWO GRAPHS TO ILLUSTRATE YOUR ANSWER.

Chapter 8

A single stock selected at random would on average have a standard deviation of approximately 30 percent. As additional stocks are added to the portfolio, the portfolio’s standard deviation decreases because the added stocks are not perfectly positively correlated. However, as more and more stocks are added, each new stock has less of a risk-reducing impact, and eventually adding additional stocks has virtually no effect on the portfolio’s risk as measured by σ. In fact, σ stabilizes at about 15 percent when 40 or more randomly selected stocks are added. Thus, by combining stocks into well -diversified portfolios, investors can eliminate almost one-half the riskiness of holding individual stocks. (Note: It is not completely costless to diversify, so even the largest institutional investors hold less than all stocks. Even index funds generally hold a smaller portfolio that is highly correlated with an index such as the S&P 500 rather than hold all the stocks in the index.)

The implication is clear: Investors should hold well-diversified portfolios of stocks rather than individual stocks. (In fact, individuals can hold diversified portfolios through mutual fund investments.) By doing so, they can eliminate about half of the riskiness inherent in individual stocks.

ANSWER: portfolio diversification does affect investors’ views of risk. A stock’s total, or stand-alone, risk as measured by its σ or CV, might be important to an undiversified investor, but it is not relevant to a well-diversified investor. A rational, risk-averse investor is more interested in the impact that the stock has on the riskiness of his or her portfolio than on the stock’s stand-alone, or total, risk. Stand-alone risk is composed of diversifiable, or company-specific, risk, which can be eliminated by holding the stock in a well-diversified portfolio, and the risk that remains, which is called market risk because it is present even when the entire market portfolio is held.

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G. (1) SHOULD PORTFOLIO EFFECTS INFLUENCE THE WAY INVESTORS THINK ABOUT THE RISKINESS OF INDIVIDUAL STOCKS?

G. (2) IF YOU CHOSE TO HOLD A ONE-STOCK PORTFOLIO AND CONSEQUENTLY WERE EXPOSED TO MORE RISK THAN DIVERSIFIED INVESTORS, COULD YOU EXPECT TO BE COMPENSATED FOR ALL OF YOUR RISK? THAT IS, COULD YOU EARN A RISK PREMIUM ON THAT PART OF YOUR RISK THAT YOU COULD HAVE ELIMINATED BY DIVERSIFYING?

Chapter 8ANSWER: If you hold a one-stock portfolio, you will be exposed to a high degree of risk, but you won't be compensated for it. If the return were high enough to compensate you for your high risk, it would be a bargain for more rational, diversified investors. They would start buying it, and these buy orders would drive the price up and the return down. Thus, you simply could not find stocks in the market with returns high enough to compensate you for the stock's diversifiable risk.

H. THE EXPECTED RATES OF RETURN AND THE BETA COEFFICIENTS OF THE ALTERNATIVES AS SUPPLIED BY THE BANK'S COMPUTER PROGRAM ARE AS FOLLOWS:

SECURITY RETURN ( ) RISK (BETA) HIGH TECH 17.40% 1.29 MARKET 15.00 1.00 U.S. RUBBER 13.80 0.68 T-BILLS 8.00 0.00 COLLECTIONS 1.74 -0.86

(1) WHAT IS A BETA COEFFICIENT, AND HOW ARE BETAS USED IN RISK ANALYSIS?

ANSWER: Draw the framework of the graph, put up the data, plot the points for the market (45° line) and connect them, and then get the slope as δY/δX = 1.0.) State that an average stock, by definition, moves with the market. Then do the same with High Tech and U.S. Rubber. Beta coefficients measure the relative volatility of a given stock vis-a-vis an average stock. The average stock’s beta is 1.0. Most stocks have betas in the range of 0.5 to 1.5. Theoretically, betas can be negative, but in the real world they generally are positive.

Betas are calculated as the slope of the “characteristic” line, which is the regression line showing the relationship between a given stock and the general stock market. The characteristic line for each investment is given in part (4).

(2) DO THE EXPECTED RETURNS APPEAR TO BE RELATED TO EACH ALTERNATIVE'S MARKET RISK?

ANSWER: The expected returns are related to each alternative’s market risk—that is, the higher the alternative’s rate of return the higher its beta. Also, note that T-bills have 0 risk.

(3) IS IT POSSIBLE TO CHOOSE AMONG THE ALTERNATIVES ON THE BASIS OF THE INFORMATION DEVELOPED THUS FAR?

ANSWER: We do not yet have enough information to choose among the various alternatives. We need to know the required rates of return on these alternatives and compare them with their expected returns.

(4) USE THE DATA GIVEN AT THE START OF THE PROBLEM TO CONSTRUCT A GRAPH THAT SHOWS HOW THE T-BILL’S, HIGH TECH’S, AND COLLECTIONS’ BETA COEFFICIENTS ARE CALCULATED. DISCUSS WHAT BETAS MEASURE AND HOW THEY ARE USED IN RISK ANALYSIS.

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Chapter 8

I. (1) WRITE OUT THE SML EQUATION, USE IT TO CALCULATE THE REQUIRED RATE OF RETURN ON EACH ALTERNATIVE, AND THEN GRAPH THE RELATIONSHIP BETWEEN THE EXPECTED AND REQUIRED RATES OF RETURN.

ANSWER: Here is the SML equation:rj = rRF + (rM ─ rRF)ßj.

If we use the T-bill yield as a proxy the risk-free rate, then rRF = 8%. Further, our estimate of rM = is 15%. Thus, the SML is drawn as follows:

10

6

2

r (%)

Beta (β)

18

14

0 1 2

SML

rT-bill = 8

rM = 15

18

Chapter 8

(2) HOW DO THE EXPECTED RATES OF RETURN ( ) COMPARE WITH THE

REQUIRED RATES OF RETURN (rj)?

ANSWER: Using the SML equation, we have the following relationships:

Expected Required Return Return

Security ( ) (r) Condition

High Tech 17.4% 17.0% undervalued: > rMarket 15.0 15.0 fairly valued (market equilibrium)U.S. Rubber 13.8 12.8 undervalued: > rT-bills 8.0 8.0 fairly valued Collections 1.7 2.0 overvalued: r >

These returns are plotted on the SML graph next.

The T-bills and market portfolio plot on the SML, High Tech and U.S. Rubber plot above it, and Collections plots below it. Thus, the T-bills and the market portfolio promise a fair return, High Tech and U.S. Rubber are good deals because they have expected returns above their required returns, and Collections has an expected return below its required return.

(3) DOES THE FACT THAT COLLECTIONS HAS A NEGATIVE BETA MAKE ANY SENSE? WHAT IS THE IMPLICATION OF THE NEGATIVE BETA?

ANSWER: Collections is an interesting stock. Its negative beta indicates negative market risk—including it in a portfolio of “normal” stocks will lower the portfolio’s risk. Therefore, its required rate of return is below the risk-free rate. Basically, this means that Collections is a valuable security to rational, well-diversified investors. To see why, consider this question: Would any rational investor ever make an investment that has a negative expected return? The answer is “yes”—just think of the purchase of a life or fire insurance policy. The fire insurance policy has a negative expected return because of commissions and insurance company profits, but businesses buy fire insurance because they pay off at a time when normal operations are in bad shape. Life insurance is similar—it has a high return when work income

10

6

2

r (%)

Beta (β)

18

14

-1 0 1 2

SML

• Collections

High Tech •

U.S.Rubber •

T-bill •

Market •

19

Chapter 8ceases. A negative beta stock is conceptually similar to an insurance policy.

(4) WHAT WOULD BE THE MARKET RISK AND THE REQUIRED RETURN OF A 50-50 PORTFOLIO OF HIGH TECH AND COLLECTIONS? OF HIGH TECH AND U.S. RUBBER?

ANSWER: Note that the beta of a portfolio is simply the weighted average of the betas of the stocks in the portfolio. Thus, the beta of a portfolio with 50 percent High Tech and 50 percent Collections is:

ßp = 0.5(ßHigh Tech) + 0.5(ßCollections) = 0.5(1.29) + 0.5(─0.86) = 0.215,

and the portfolio’s required return is 9.51 percent:

rp= rRF + (rM ─ rRF)ßp

= 8.0% + (15.0% ─ 8.0%)(0.215) = 8.0% + 7%(0.215) = 9.51%

For a portfolio consisting of 50 percent High Tech plus 50 percent U.S. Rubber, the required return would be 14.90 percent:

ßp = 0.5(1.29) + 0.5(0.68) = 0.985. rp = 8.0% + 7%(0.985) = 14.90%

J. (1) SUPPOSE INVESTORS RAISED THEIR INFLATION EXPECTATIONS BY 3 PERCENTAGE POINTS OVER CURRENT ESTIMATES AS REFLECTED IN THE 8 PERCENT T-BILL RATE. WHAT EFFECT WOULD HIGHER INFLATION HAVE ON THE SML AND ON THE RETURNS REQUIRED ON HIGH- AND LOW-RISK SECURITIES?

ANSWER: This effect is graphed next.

Here we have plotted the SML for betas ranging from 0 to 2.0. The base case SML is based on r RF = 8% and rM = 15%. If inflation expectations increase by 3 percent, with no change in risk aversion, then the entire SML is shifted upward (parallel to the base case SML) by 3 percentage points. Now, r RF = 11%, rM

10

6

2

r (%)

Beta (β)

18

14

-1 0 1 2

Original Situation

Increased InflationIncreased Risk

Aversion

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Chapter 8= 18%, and all securities’ required returns rise by 3 percentage points. Note that the market risk premium, rM – rRF, remains at 7 percentage points.

(2) SUPPOSE INSTEAD THAT INVESTORS’ RISK AVERSION INCREASED ENOUGH TO CAUSE THE MARKET RISK PREMIUM TO INCREASE BY 3 PERCENTAGE POINTS. (INFLATION REMAINS CONSTANT.) WHAT EFFECT WOULD THIS HAVE ON THE SML AND ON RETURNS OF HIGH- AND LOW-RISK SECURITIES?

ANSWER: When investors’ risk aversion increases, the SML is rotated upward about the Y-intercept (rRF). rRF remains at 8 percent, but now rM increases to 18 percent, so the market risk premium increases to 10 percent. The required rate of return will rise sharply on high risk (high beta) stocks, but not much on low beta securities.

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Chapter 88-25 Computer-Related Problem

a. INPUT DATA: KEY OUTPUT:

Probability: Stock A Stock B Stock C Portfolio 0.1 33.00% 60.00% 50.00% 46.50%0.2 20.00% 30.00% 10.00% 25.00%0.4 15.00% 5.00% 25.00% 10.00%0 .3 0.00% -20.00% -6.00% -10.00%1.0 ^

r = 13.30% 8.00% 15.20% 10.65%Std Dev = 10.10% 24.62% 17.31% 17.26% CV = 0.76 3.08 1.14 1.62

Percent in: Stock A Stock B Stock C Portfolio50.00% 50.00% 0.00% 100.00%

b. INPUT DATA: KEY OUTPUT:

Probability: Stock A Stock B Stock C Portfolio 0.1 33.00% 60.00% 50.00% 47.67%0.2 20.00% 30.00% 10.00% 20.00%0.4 15.00% 5.00% 25.00% 15.00%0.3 0.00% -20.00% -6.00% -8.67%1.0

^ r = 13.30% 8.00% 15.20% 12.17%

Std Dev = 10.10% 24.62% 17.31% 16.48% CV = 0.76 3.08 1.14 1.35

Percent in: Stock A Stock B Stock C Portfolio33.33% 33.33% 33.34% 100.00

c. – d. The correlation among the stocks is positive, but not perfect. As a result, it would be a good idea to include all three stocks in a portfolio so as to achieve diversification.

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Chapter 8

ETHICAL DILEMMA

RIP—RETIRE IN PEACE

Ethical dilemma:

Howard, a friend of yours, would like you to recommend an investment called PAIDs be purchased from the company for which he works, Suncoast Investments, and sold as part of the retirement packages offered by Retirement Investment Products (RIP). The commission Howard would earn from such a deal certainly would bail him and his family out of the financial crisis they currently are experiencing. The problem is that PAIDs are complex investment instruments consisting of combinations of many other investments, and you have not been able to determine the degree of risk associated with them, either through Howard or other sources. The return expected from PAIDs is very attractive, and there is no doubt that offering higher returns on its products will help RIP attract more customers. Should you recommend the PAIDs without more complete knowledge about their riskiness? Howard is anxious to get the deal completed—he even offered to reward you if RIP invests in PAIDs through Suncoast Investments. What should you do?

Discussion questions:

● What is the ethical dilemma?

RIP has built its reputation on the fact that the service agents are required to fully inform clients of the risks associated with any investment position. The PAIDs offer a very attractive expected return, but, primarily because they are complex instruments, little is know about the riskiness of these investments. If PAIDs are recommended and a high return is earned, RIP and its customers will be happy. But, if the return for PAIDs actually turns out to be poor, then RIP's customers will be very unhappy and its reputation will be tarnished.

To generate discussion on this topic, talk about risk in general. It will help to define risk as the total variability of returns, which includes both downside variability and upside variability. Investors are primarily concerned with the downside variability. But, when they see an investment with potential for a high return, generally only the upside risk seems to matter. In fact, many "naive" investors actually believe that as long as their investments are doing well (prices are moving up), there is no risk associated with such investments. Discuss the fallacy of such logic with the students.

● Should RIP be more concerned with return than risk when making its decision about the PAIDs?

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Chapter 8This question follows the above discussion. The short, simple answer is

"absolutely not." There are many examples where people have invested considerable amounts in securities that promised significant returns, and much if not all of their investment was lost. It should be emphasized that, when it comes to any investment, the concepts of expected return and risk never should be separated--when evaluating a potential investment, risk and return always should be considered simultaneously.

● If the PAIDs are recommended, what should RIP tell its customers?

RIP could find itself in a great deal of trouble, both financially and legally, if it doesn't fully disclose any information concerning the PAIDs, including what is not known about the risk. It should be made quite clear that the riskiness of the PAIDs is unknown at this time, but there probably are great risks associated with such investments--while investors can gain considerable amounts, they also can lose considerable amounts.

● Would you recommend the PAIDs?

Ask the students whether they would make such an investment. Discuss the pitfalls of making investments that "look too good to be true."

References:

The scenario presented here parallels the well-publicized cases of (1) Orange County, California that came to light in 1994 and (2) Long-Term Capital Management L.P. Orange County lost billions of dollars with its investment fund, apparently because the managers of the fund did not fully understand the risk ramifications of some of the investments in the portfolio, especially derivatives. Long-Term Capital Management L.P., which employed complex arbitrage strategies to construct investment positions that were suppose to generate positive returns in any type of market, was “bailed out” of bankruptcy only after large financial institutions provided nearly $4 billion.

The following articles offer interesting insights into what caused Orange County's problems:

"Untangling the Derivative Mess," Fortune, March 20, 1995, p. 50+.

"Orange County is Looking Green Around the Gills," Business Week, December 26, 1994, p. 66+.

"Derivatives Lead to a Huge Loss in Public Fund," The Wall Street Journal, December 2, 1994, p. A3+.

"Bitter Fruit in Orange County," Business Week, May 30, 1994, p. 44+.

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Chapter 8The following articles describe some of the complexities and the reasons for the

trouble at Long-Term Capital Management L.P.:

“Failed Wizards of Wall Street: Can You Devise Surefire Ways to Beat the Markets? The Rocket Scientists Thought They Could. Boy Were They Wrong,” Business Week, September 21, 1998, p. 114.

25

Chapter 8

“Bailout Blues: How a Big Hedge Funds Marketed Its Expertise and Shrouded Its Risk; Regulators and Lenders Knew Little About the Gambles At Long-Term Capital; ‘Stardust in Investors’ Eyes,” The Wall Street Journal, September 25, 1998, p. A1+.

“A House Built on Sand. John Meriwether’s Once-Mighty Long-Term Capital Has All But Crumbled. So Why Did Warren Buffett Offer to Buy It?,” Fortune, October 26, 1998, p. 110+.

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