博迪投资学答案chap010-7thed

1 a.

CHAPTER 10: ARBITRAGE PRICING THEORY

AND MULTIFACTOR MODELS OF RISK AND RETURN

2222



M

(e)

2222

(0.820)25881

A

2222

(1.020)10500

B

2222

(1.220)20976

C

b. If t团建总结 here are an infinite number of asts with identical characteristics, then a

well-diversified portfolio of each type will have only systematic risk since the

non-systematic risk will approach zero with large n. The mean will equal that

of the individual (identical) stocks.

There is no arbitrage opportunity becau the well-diversified portfolios all c.

plot on the curity market line (SML). Becau they are fairly priced, there is

no arbitrage.

2. The expected return for Portfolio F equals the risk-free rate since its beta equals 0.

For Portfolio A, the ratio of risk premium to beta is: (12 6)/1.2 = 5



For Portfolio E, the ratio is lower at: (8 – 6)/0.6 = 3.33

This implies that an arbitrage opportunity exists. For instance, you can create a

Portfolio G with beta equal to 0.6 (the same as E’s) by combining Portfolio A and

Portfolio F in equal weights. The expected return and beta for Portfolio G are then:

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

G





G

= (0.5 1.2) + (0.5 0) = 0.6



Comparing Portfolio G to Portfolio E, G has the same beta and higher return.

Therefore, an arbitrage opportunity exists by buying Portfolio G and lling an

equal amount of Portfolio E. The profit for this arbitrage will be:

r – r =[9% + (0.6 F)]  [8% + (0.6 F)] = 1%

GE



That is, 1% of the funds (long or short) in each portfolio.

10-1

3. Substituting the portf红豆沙馅的做法 olio returns and betas in the expected return-beta relationship,

we obtain two equations with two unknowns, the risk-free rate (r) and the factor

f

risk premium (RP):

12 = r + (1.2 RP)

f



9 = r + (0.8 RP)

f



Solving the equations, we obtain:

r = 3% and RP = 7.5%

f

Equation 10.9 applies here: 4.

E(r) = r +  [E(r)  r] +  [E(r) – r]

p fP11 f P22 f

We need to find the risk premium (RP) for each of the two factors:

RP = [E(r)  r] and RP = [E(r)  r]

11 f 22 f

In order to do so, we solve the following system of two equations with two unknowns:

31 = 6 + (1.5 RP) + (2.0 RP)



1 2

27 = 6 + (2.2 RP) + [(–0.2) RP]



1 2

The solution to this t of e佳能1100d quations is:

RP = 10% and 车子年审 RP = 5%

12

Thus, the expected return-beta relationship is:

E(r) = 6% + ( 10%) + ( 5%)

P P1P2



a. A long position in a portfolio (P) comprid of Portfolios A and B will offer 5.

an expected return-beta tradeoff lying on a straight line between points A and

B. Therefore, we can choo weights such that  =  but with expected

PC

return higher than that of Portfolio C. Hence, combining P with a short

position in C will create an arbitrage portfolio with zero investment, zero beta,

and positive rate of return.

The argument in part (a) leads to the proposition that the coefficient of  b.

2

must be zero in order to preclude arbitrage opportunities.

6. The revid estimate of the expected rate of return on the stock would be the old

estimate plus the sum of the products of the unexpected change in each factor times

the respective nsitivity coefficient:

revid estimate = 12% + [(1 2%) + (0.5 3%)] = 15.5%



10-2

7. a. Shorting an equally-weighted portfolio of the ten negative-alpha stocks and

b. If n = 50 stocks (25 stocks long and 25 stocks short), the investor will have a

8. a. This statement is incorrect. The CAPM requires a mean-variance efficient

b. This statement is incorrect. The CAPM assumes normally distributed curity

c. This statement is correct.

investing the proceeds in an equally-weighted portfolio of the ten positive-

alpha stocks eliminates the market exposure and creates a zero-investment

portfolio. Denoting the systematic market factor as R, the expected dollar

return is (noting that the expectation of non-systematic risk, e, is zero):

M

$1,000,000  [0.02 + (1.0  R)]  $1,000,000  [(–0.02) + (1.0  R)]

M M

= $1,000,000  0.04 = $40,000

The nsitivity of the payoff of this portfolio to the market factokyo怎么读 tor is zero

becau the exposures of the positive alpha and negative alpha stocks cancel

out. (Notice that the terms involving R sum to zero.) Thus, the systematic

component of total risk is also zero. The variance of the analyst’s profit is not

M

zero, however, since this portfolio is not well diversified.

For n = 20 stocks (i.e., long 10 stocks and short 10 stocks) the investor will

have a $100,000 position (either long or short) in each stock. Net market

exposur游戏人生壁纸 e is zero, but firm-specific risk has not been fully diversified. The

variance of dollar returns from the positions in the 20 stocks is:

20  [(100,000  0.30)] = 18,000,000,000

2

The standard deviation of dollar returns is $134,164.

$40,000 position in each stock, and the variance of dollar returns is:

50  [(40,000  0.30)] = 7,200,000,000

2

The standard deviation of dollar returns is $84,853.

Similarly, if n = 100 stocks (50 stocks long and 50 stocks short), the investor

will have a $20,000 positio如何艾灸 n in each stock, and the variance of dollar returns is:

100  [(20,000  0.30)] = 3,600,000,000

2

The standard deviation of dollar returns is $60,000.

Notice that, when the number of stocks increas by a factor of 5 (i.e., from 20

to 100), standard deviation decreas by a factor of = 2.23607 (from

5

$134,164 to $60,000).

market portfolio, but APT does not.

returns, but APT does not.

10-3

9. b. Since Portfolio X has  = 1.0, then X is the market portfolio and E(R) =16%.

10. a. E(r) = 6 + (1.2 6) + (0.5 8) + (0.3 3) = 18.1%

b. Surpris in the macroeconomic factors will result in surpris in the return of

M

Using E(R) = 16% and r = 8%, the expected return for portfolio Y is not

M f

consistent.



the stock:

Unexpected return from macro factors =

[1.2(4 – 5)] + [0.5(6 – 3)] + [0.3(0 – 2)] = –0.3%

E(r) =18.1% − 0.3% = 17.8%

11. d.

12. The APT factors must correlate with major sources of uncertainty, i.e., sources of

uncertainty that are of concern to many investors. Rearchers should investigate

factors that correlate with uncertainty in consumption and investment opportunities.

GDP, the inflation rate, and interest rates are among the factors that can be expected

to determine risk premiums. In particular, industrial production (IP) is a good

indicator of changes in the business cycle. Thus, IP is a candidate for a factor that is

highly correlated with uncertainties that have to do with investment and

consumption opportunities in the economy.

13. The first two factors em promising with respect to the likely impact on the firm’s

cost of capital. Both are macro factors that would elicit hedging demands across

broad ctors of investors. The third factor, while important to Pork Products, is a

poor choice for a multifactor SML becau the price of hogs is of mino狮子卡通图片 r importance to

most investors and is therefore highly unlikely to be a priced risk factor. Better

choices would focus on variables that investors in aggregate might find more

important to their welfare. Examples include: inflation uncertainty, short-term

interest-rate risk, energy price risk, or exchange rate risk. The important point here is

that, in specifying a multifactor SML, we not confu risk factors that are important to

a particular investor with factors that are important to investors in general; only the

latter are likely to command a risk premium in the capital markets.

14. c. Investors will take on as large a position as possible only if the mispricing

opportunity is an arbitrage. Otherwi, considerations of risk and

diversification will limit the position they attempt to take in the mispriced

curity.

10-4

15. d.

16. d.

17. The APT required (i.e., equilibrium) rate of return on the stock bad on r and the

f

factor betas is:

required E(r) = 6 + (1 6) + (0.5 2) + (0.75 4) = 16%



According to the equation for the return on the stock, the actually expected return

on the stock is 15% (becau the expected surpris on all factors are zero by

definition). Becau the actually expected return bad on risk is less than the

equilibrium return, we conclude that the stock is overpriced.

18. Any pattern of returns can be “explained” if we are free to choo an indefinitely

large number of explanatory factors. If a theory of ast pricing is to have value, it

must explain returns using a reasonably limited number of explanatory variables

(i.e., systematic factors).

19. d.

20. c.

10-5