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A pension fund manager is considering three mutual funds. The first is a stock fund, the second is a long-term government and corporate bond fund, and the third is a T-bill money market fund that yields a sure rate of 5.5%. The probability distributions of the risky funds are:

Stock fund (S) Bond fund (B)

Expected Return Standard Deviation 15% 32% 9 23

The correlation between the fund returns is .15. Required: What expected return and standard deviation for the minimum-variance portfolio of the two risky funds? (Round your answers to 2 decimal places. Omit the "%" sign in your response.) Expected return

10.89 ± 1% %

Standard deviation

19.94 ± 1% %

Explanation:

The parameters of the opportunity set are: E(rS) = 15%, E(rB) = 9%, σS = 32%, σB = 23%, ρ = 0.15, rf = 5.5% From the standard deviations and the correlation coefficient we generate the covariance matrix [note that Cov(rS, rB) = ρσSσB]:

Bonds Stocks

Bonds 529.0 110.4

Stocks 110.4 1024.0

The minimum-variance portfolio proportions are: σB2 – Cov(rS,rB) WMin (S) = σS2 + σB2 – 2Cov(rS,rB) 529 – 110.4 =

= 0.3142 1024 + 529 – (2 × 110.4)

w Min(B) = 0.6858 The mean and standard deviation of the minimum variance portfolio are: E(rMin) = (0.3142 × 15%) + (0.6858 × 9%) = 10.89% σMin = [w S2σS2 + w B2σB2 + 2wSw B Cov(rS,rB)]1/2 = [(0.31422 × 1024) + (0.68582 × 529) + (2 × 0.3142 × 0.6858 × 110.4)] 1/2 = 19.94%

A pension fund manager is considering three mutual funds. The first is a stock fund, the second is a long-term government and corporate bond fund, and the third is a T-bill money market fund that yields a sure rate of 5.5%. The probability distributions of the risky funds are: Stock fund (S) Bond fund (B)

Expected Return 15% 9

Standard Deviation 32% 23

The correlation between the fund returns is 0.15.

If you were to use only the two risky funds and still require an expected return of 12%. Required: (a) What would be the investment proportions of your portfolio? (Round your answer to the nearest whole percent. Omit the "%" sign in your response.)

Stocks

50 %

Bonds

50 %

(b)Calculate the standard deviation of the portfolio which yields an expected return of 12%. (Round your answer to 2 decimal places. Omit the "%" sign in your response.) Standard deviation

21.06 ± 1% %

Explanation: (a)

Using only the stock and bond funds to achieve a mean of 12% we solve: 12 = 15w S + 9(1 − wS ) = 9 + 6w S => wS = 0.5 (b)

Investing 50% in stocks and 50% in bonds yields a mean of 12% and standard deviation of: σP = [(0.502 × 1024) + (0.502 × 529) + (2 × 0.50 × 0.50 × 110.4)] 1/2 = 21.06% The efficient portfolio with a mean of 12% has a standard deviation of only 21.06%. Using the CAL reduces the standard deviation by 45 basis points.

A portfolio's expected return is 12%, its standard deviation is 20%, and the risk-free rate is 4%. Required: Which of the following would make for the greatest increase in the portfolio's Sharpe ratio? (Select all that apply.) →

An increase of 1% in expected return.

→

A decrease of 1% in the risk-free rate. A decrease of 1% in its standard deviation.

Check All That Apply

Consider the following table Stock Fund Bond Fund Scenario Probability Rate of Return Rate of Return Severe recession .05 –40% –9 Mild recession .25 –14% 15 Normal growth .40 17% 8 Boom .30 33% –5 Required: (a) Calculate the values of mean return and variance for the stock fund. (Enter your answer in decimals rounded to 4 decimal places.) Mean return Variance

0.112 ± 0.01% 0.0446 ± 0.01%

(b)Calculate the value of the covariance between the stock and bond funds. (Enter your answer in decimals rounded to 4 decimal places. Negative amount should be indicated by a minus sign.) Covariance

-0.0086 ± 0.01%

Explanation:

(a)Calculation of mean return and variance for the stock fund: (A)

Scenario Severe recession Mild recession Normal growth Boom

(B)

(C)

Rate of Probability Return 0.05 –0.4 0.25 –0.14 0.40 0.17 0.30 0.33 Expected Return =

(D) Col. B × Col. C –0.02 –0.035 0.068 0.099 0.112

(E) Deviation from Expected Return –0.512 –0.252 0.058 0.218

(F)

(G)

Squared Deviation 0.2621 0.0635 0.0034 0.0475

Col. B × Col. F 0.0131 0.0159 0.0013 0.0143

Variance = Standard Deviation =

(b)Calculation of covariance: (A)

Scenario Severe recession

(B)

(C) Deviation from Mean Return

Stock Probability Fund 0.05 –0.512

(D)

(E)

(F)

Bond Fund –0.14

Col. C × Col. D 0.0717

Col. B × Col. E 0.0036 –0.0063

Mild recession

0.25

–0.252

0.10

–0.0252

Normal growth

0.40

0.058

0.03

0.0017

Boom

0.30

0.218

–0.10

–0.0218

0.0007 –0.0065

0.0446 0.2112

Covariance =

–0.0086

Suppose that many stocks are traded in the market and that it is possible to borrow at the risk-free rate, rf . The characteristics of two of the stocks are as follows: Stock A B

Expected Return 8% 13

Standard Deviation 40% 60

Correlation = –1 Required: (a) Calculate the expected rate of return on the risk-free portfolio? (Hint: Can a particular stock portfolio be substituted for the risk-free asset?) (Omit the "%" sign in your response.) Rate of return

10 %

(b)Could the equilibrium rf be greater than 10%? No

Explanation: (a)

Since Stock A and Stock B are perfectly negatively correlated, a risk-free portfolio can be created and the rate of return for this portfolio in equilibrium will always be the risk-free rate. To find the proportions of this portfolio [with wA invested in Stock A and w B = (1 – w A) invested in Stock B], set the standard deviation equal to zero. With perfect negative correlation, the portfolio standard deviation reduces to: σP = Abs[wAσA - wBσB] 0 = 40 w A- 60(1 – w A) => wA = 0.60 The expected rate of return on this risk-free portfolio is: E(r) = (0.60 × 8%) + (0.40 × 13%) = 10.0% (b)

E(r) = 10.0% Therefore, the risk-free rate must also be 10.0%.

What must be the beta of a portfolio with E(rP ) = 20%, if rf = 5% and E(rM ) = 15%? (Round your answer to 1 decimal place.) Beta of portfolio

1.5 ± 1%

Explanation: E(rP) = rf + β[E(rM) – rf] 20% = 5% + β(15% – 5%)

β = 15/10 = 1.5

Are the following statements true or false?

(a) Stocks with a beta of zero offer an expected rate of return of zero. False (b)The CAPM implies that investors require a higher return to hold highly volatile securities. False (c) You can construct a portfolio with a beta of .75 by investing .75 of the budget in T-bills and the remainder in the market portfolio. False

Explanation: (a) False. β = 0 implies E(r) = rf , not zero. (b) False. Investors require a risk premium for bearing systematic (i.e., market or undiversifiable) risk. (c) False. You should invest 0.75 of your portfolio in the market portfolio, and the remainder in T-bills. Then: βp = (0.75 × 1) + (0.25 × 0) = 0.75

Consider the following table, which gives a security analyst's expected return on two stocks for two particular expected market returns: Market Return 5% 20

Aggressive Stock 2% 32

Defensive Stock 3.5% 14

Required: (a) What are the betas of the two stocks? (Round your answers to 2 decimal places.)

Beta A

2.00 ± 1%

Beta D

0.70 ± 1%

(b)What is the expected rate of return on each stock if the market return is equally likely to be 5% or 20%? (Round your answers to 2 decimal places. Omit the "%" sign in your response.)

Rate of return on A

17.00 ± 1% %

Rate of return on D

8.75 ± 1% %

(c) What is the expected rate of return for the market, if the T-bill rate is 8%, and the market return is equally likely to be 5% or 20%? (Round your answer to 2 decimal places. Omit the "%" sign in your response.)

Expected rate of return

12.50 ± 1% %

Explanation:

(a) The beta is the sensitivity of the stock's return to the market return. Call the aggressive stock A and the defensive stock D. Then beta is the change in the stock return per unit change in the market return. We compute each stock's beta by calculating the difference in its return across the two scenarios divided by the difference in market return.

2–32 βA =

= 2.00 5–20 3.5–14

βD =

= 0.70 5–20

(b) With the two scenarios equal likely, the expected rate of return is an average of the two possible outcomes: E(rA) = 0.5 × (2% + 32%) = 17.00%

E(rB) = 0.5 ×(3.5% + 14%) = 8.75% (c) The SML is determined by the following: T-bill rate = 8% with a beta equal to zero, beta for the market is 1.0, and the expected rate of return for the market is: 0.5 × (20% + 5%) = 12.50%

If the simple CAPM is valid, state whether the following situation is possible? Portfolio A

Expected Return 20%

B

25

Beta 1.4 1.2

Yes

→

No

Not possible. Portfolio A has a higher beta than Portfolio B, but the expected return for Portfolio A is lower.

If the simple CAPM is valid, state whether the following situation is possible?

Portfolio A B

Expected Return 30% 40 →

Standard Deviation 35% 25

Yes No

Possible. If the CAPM is valid, the expected rate of return compensates only for systematic (market) risk as measured by beta, rather than the standard deviation, which includes nonsystematic risk. Thus, Portfolio A's lower expected rate of return can be paired with a higher standard deviation, as long as Portfolio A's beta is lower than that of Portfolio B.

If the simple CAPM is valid, state whether the following situation is possible?

Portfolio Risk-free Market A

Expected Return 10% 18 16

Standard Deviation 0% 24 12

Yes

→

No

Not possible. The reward-to-variability ratio for Portfolio A is better than that of the market, which is not possible according to the CAPM, since the CAPM predicts that the market portfolio is the most efficient portfolio. Using the numbers supplied: 16 – 10 SA =

= 0.5 12 18 – 10

SM =

= 0.33 24

These figures imply that Portfolio A provides a better risk-reward tradeoff than the market portfolio.

If the simple CAPM is valid, state whether the following situation is possible?

Portfolio Risk-free Market A

Expected Return 10% 18 20

Standard Deviation 0% 24 22

Yes

→

No

Not possible. Portfolio A clearly dominates the market portfolio. It has a lower standard deviation with a higher expected return.

If the simple CAPM is valid, state whether the following situation is possible? Portfolio Risk-free Market A

Expected Return 10% 18 16

Beta 0 1.0 1.5

Yes

→

No

Not possible. Given these data, the SML is: E(r) = 10% + β(18% – 10%) A portfolio with beta of 1.5 should have an expected return of: E(r) = 10% + 1.5 × (18% – 10%) = 22% The expected return for Portfolio A is 16% so that Portfolio A plots below the SML (i.e., has an alpha of –6%), and hence is an overpriced portfolio. This is inconsistent with the CAPM.

If the simple CAPM is valid, state whether the following situation is possible? Portfolio Risk-free Market A

Expected Return 10% 18 16

Beta 0 1.0 .9

Yes

→

No

Not possible. Portfolio A clearly dominates the market portfolio. It has a lower standard deviation with a higher expected return. Here, the required expected return for Portfolio A is: 10% + (0.9 × 8%) = 17.2% This is still higher than 16%. Portfolio A is overpriced, with alpha equal to: –1.2%

If the simple CAPM is valid, state whether the following situation is possible?

Portfolio Risk-free Market A →

Expected Return 10% 18 16

Standard Deviation 0% 24 22

Yes No

Possible. Portfolio A's ratio of risk premium to standard deviation is less attractive than the market's. This situation is consistent with the CAPM. The market portfolio should provide the highest reward-to-variability ratio.

Suppose the yield on short-term government securities (perceived to be risk-free) is about 4%. Suppose also that the expected return required by the market for a portfolio with a beta of 1.0 is 12%. According to the capital asset pricing model: Required: (a) What is the expected return on the market portfolio? (Omit the "%" sign in your response.) Expected rate of return

12 %

(b)What would be the expected return on a zero-beta stock? (Omit the "%" sign in your response.) Expected rate of return

4 %

(c) Suppose you consider buying a share of stock at a price of $40. The stock is expected to pay a dividend of $3 next year and to sell then for $41. The stock risk has been evaluated at β = –0.5. Is the stock overpriced or underpriced? Underpriced

Explanation:

(a) Since the market portfolio, by definition, has a beta of 1.0, its expected rate of return is 12%. (b) β = 0 means the stock has no systematic risk. Hence, the portfolio's expected rate of return is the risk-free rate, 4%. (c) Using the SML, the fair rate of return for a stock with β = –0.5 is: E(r) = 4% + (–0.5)(12% – 4%) = 0.0% The expected rate of return, using the expected price and dividend for next year: E(r) = ($44/$40) – 1 = 0.10 = 10% Because the expected return exceeds the fair return, the stock must be underpriced.

A share of stock is now selling for $100. It will pay a dividend of $9 per share at the end of the year. Its beta is 1.0. Assume the risk-free rate is 8% and the expected rate of return on the market is 18%. Required: What do investors expect the stock to sell for at the end of the year? (Omit the "$" sign in your response.) Expected selling price of the stock

$

109 ± 1%

Explanation:

Since the stock's beta is equal to 1.0, its expected rate of return should be equal to that of the market, that is, 18%. D + P1 – P0 E(r) = P0 9 + P1 – 100 P1 = $109

0.18 = 100

A stock has an expected return of 6%. Assume the risk-free rate is 8% and the expected rate of return on the market is 18%. Required: What is its beta? (Negative amount should be indicated by a minus sign. Round your answer to 2 decimal places.) Beta

-0.20 ± 1%

Explanation: Using the SML: 6% = 8% + β(18% – 8%)

β = –2/10 = –0.2...