CAPM

Question 1

What must be the beta of a portfolio with E(rP)=18%; rf=6% and E(rM )=14%?

ANSWER

E(rP) = rf + P [E(rM ) – rf ]

18 = 6 + P (14 – 6) P = 12/8 = 1.5

Question 2

Assume that the risk-free rate is 6% and the E(RM) = 16%. A share of stock sells for $50 today.

It will pay a dividend of $6 per share at the end of the year. Its beta is 1.2. What do investors

expect the stock to sell for at the end of the year?

ANSWER

Since the stock’s beta is equal to 1.2, its expected rate of return is:

6 + [1.2 (16 – 6)] = 18%

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Question 3

According to the CAPM, the expected rate of return of a portfolio with a beta of 1.0 and alpha

of 0 is:

a. Between RM and rf

b. The risk-free rate rf

c. (RM – rf)

d. E(RM)

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ANSWER

It is (d) the Expected rate of return on the market portfolio E(RM)

Question 4

Suppose that the rate of return on short-term government bonds (risk-free assets) is 5%, and

that the expected rate of return required by the market for a portfolio with a beta of 1 is 12%.

According to the CAPM:

a. What is expected rate of return on the market portfolio E(RM)

b. What would be the expected rate of return on a stock with beta equal zero?

c. Suppose you consider buying a share of stock at $40. The stock is expected to pay $3

dividends next year and you expect to sell it then for $41. The stock risk has been

evaluated at beta = -0.5. Is the stock overpriced or underpriced?

ANSWER

a. Since the market portfolio, by definition, has a beta of 1, its expected rate of return is 12%.

b. = 0 means no systematic risk. Hence, the stock’s expected rate of return in market

equilibrium is the risk-free rate, 5%.

c. Using the SML, the fair expected rate of return for a stock with = –0.5 is:

E(r) = 5 + [(–0.5)(12 – 5)] = 1.5%

The actually expected rate of return, using the expected price and dividend for next year is:

E(r) = $[(41 -40+3)/40] = 0.10 = 10%

Because the actually expected return exceeds the fair return, the stock is underpriced.

Question 5

Within the CAPM context, assume that the E(RM) = 15%; rf = 8%; the E(R) on GAMMA

security is = 17% and the beta of GAMMA security = 1.25. Which of the following is correct?

a. Gamma is overpriced

b. Gamma is fairly priced

c. Gamma’s alpha is 0.25

ANSWER

c From CAPM, the fair expected return = 8 + 1.25(15 8) = 16.75%

Actually expected return of GAMMA= 17%

= 17 16.75 = 0.25%

Gamma stock is underpriced

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Question 6

Are the following true or false? Explain.

a. Stocks with a beta of zero offer an expected rate of return of zero

b. The CAPM implies that investors require a higher return to hold highly volatile

securities.

c. You can construct a portfolio with beta of .75 by investing .75 of the investment budget in

T-Bills and the remainder in the market portfolio.

ANSWER

a. False. = 0 implies E(r) = rf , not zero.

b. False. Investors require a risk premium only for bearing systematic (undiversifiable or

market) risk. Total volatility includes diversifiable risk.

c. False. Your portfolio should be invested 75% in the market portfolio and 25% in T-bills.

Then:

P = (0.75 1) + (0.25 0) = 0.75

Question 7

Consider the following table, which gives a security analyst expected return on two stocks for

two particular market returns:

Market Return Aggressive stocks Defensive stocks

5% -2% 6%

25% 38% 12%

a. What are the betas of the two stocks?

b. What is the expected rate of return on each stock if the market return is equally likely to

be 5% or 25%?

c. If the T-Bill rate is 6% and the market return is equally likely to be 5% or 25%, draw

the SML for this economy.

d. Plot the two securities on the SML graph. What are the alphas of each security?

ANSWER

a. Call the aggressive stock A and the defensive stock D. 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:

2.00

5 25

2 38

A

4

0.30

5 25

6 12

D

b. With the two scenarios equally likely, the expected return is an average of the two possible

outcomes:

E(rA ) = 0.5 (–2 + 38) = 18%

E(rD ) = 0.5 (6 + 12) = 9%

c. The SML is determined by the market expected return of [0.5(25 + 5)] = 15%, with a beta

of 1, and the T-bill return of 6% with a beta of zero. See the following graph.

The equation for the security market line is:

E(r) = 6 + (15 – 6)

d. Based on its risk, the aggressive stock has a required expected return of:

E(rA ) = 6 + 2.0(15 – 6) = 24%

The analyst’s forecast of expected return is only 18%. Thus the stock’s alpha is:

A = actually expected return – required return (given risk)

= 18% – 24% = –6% (A plots below the SML – overpriced asset)

Similarly, the required return for the defensive stock is:

E(rD) = 6 + 0.3(15 – 6) = 8.7%

The analyst’s forecast of expected return for D is 9%, and hence, the stock has a positive

alpha:

D = actually expected return – required return (given risk)

= 9 – 8.7 = +0.3% (D plots above the SML –underpriced asset)

The points for each stock plot on the graph as indicated above.

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Question 8

Company X has a beta of 0.8 and an expected return of 9.8%. Company Y has an expected

return of 15.8%, but has a beta of 1.8, both lie on the SML. Company Z is floating on the stock

market offering an expected return of 13%, having a beta of 1.2.

a. What do you think will happen to the share price of company Z?

b.If you created a portfolio of just stocks X and Y, what proportion of X would you require

in order to ensure the beta of your portfolio was 1.4?

ANSWER:

a. We could simply plot X and Y on a graph and join the points to give the SML line. If we then

plotted Z on the same graph we would see that it lies above the SML line. The return being

offered is too high so there will be a high demand for the stock, this will push up the price

which in turn will reduce the return until it reaches the SML

Alternatively we could find the SML directly. Given X and Y are on the SML we know

0.098 = rf +0.8 * market premium

0.158 = rf +1.8 * market premium

We could solve these two (simultaneous) equations and show that market premium = 0.06 = 6%

and rf = 0.05 = 5%. Note we could use the graph to find rf and the market premium. The riskfree

asset has beta = 0, so we can use the graph to read off the return when beta = 0 (and show rf =

5%). The market has a beta =1, so we could read of the return when beta = 1, this would be the

return on the market (it turns out to be 11%), so again giving a market premium of (11% – 5%) =

6%. Therefore the expected return for Z is 0.05 + 1.2 * 0.06 = 0.122 = 12.2%. We have shown the

return being offered (13%) is higher than it needs to be (12.2%) and hence as above we would

expect the share price to rise.

Since we expect the share price to rise we would say the stock is UNDER VALUED.

*

Z

*

*

SML

X

Y

Beta

Return

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(REMEMBER: If a stock lies ABOVE the SML line, the PRICE must rise, so the RETURN will fall)

b. We know the beta of Stock X is 0.8 and the beta of Stock Y is 1.8. We also know the beta of a

portfolio is just the weighted average beta of the stocks.

If we call the proportion of Stock X in the portfolio WX and call the proportion of Stock Y in the

portfolio WY then;

The beta of the portfolio is given by:

Portfolio Beta = βP = WX * βX + WY * βY and we want this to be 1.4

Since the portfolio only has stocks X and Y in it we know that:

WX + WY = 1 and hence WY = 1 – WX

So we have: 1.4 = WX * 0.8 + WY * 1.8

1.4 = WX * 0.8 + (1 – WX) * 1.8

Thus 1.4 = 0.8WX +1.8 – 1.8WX

1.4 = 1.8 – WX

Hence WX = 0.4

So WX = 0.4 = 40% and therefore WY = 1 – 0.4 = 0.6 = 60%