博迪第八版投资学第十章课后习题答案

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CHAPTER 10: ARBITRAGE PRICING THEORY

AND MULTIFACTOR MODELS OF RISK AND RETURN

PROBLEM SETS

1. The revised 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 sensitivity coefficient: revised estimate = 12% + [(1 2%) + (0.5 3%)] = 15.5%

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

sources of uncertainty that are of concern to many investors.

Researchers 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.

3. Any pattern of returns can be “explained” if we are free to choose an

indefinitely large number of explanatory factors. If a theory of asset

pricing is to have value, it must explain returns using a reasonably

limited number of explanatory variables (i.e., systematic factors).

4. Equation 10.9 applies here:

E(r p) = r f + P1 [E(r1 ) r f ] + P2 [E(r2) – r f ]

We need to find the risk premium (RP) for each of the two factors: RP1 = [E(r1) r f ] and RP2 = [E(r2) r f ]

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

31 = 6 + (1.5 RP1) + (2.0 RP2)

27 = 6 + (2.2 RP1) + [(–0.2) RP2]

The solution to this set of equations is:

RP1 = 10% and RP2 = 5%

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Thus, the expected return-beta relationship is: E(r P ) = 6% + (P1 10%) + (P2 5%)

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5. 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 G ) = (0.5 12%) + (0.5 6%) = 9%

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 selling an equal amount of Portfolio E. The profit for this

arbitrage will be:

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

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

6. Substituting the portfolio returns and betas in the expected return-

beta relationship, we obtain two equations with two unknowns, the risk-free rate (r f ) and the factor risk premium (RP):

12 = r f + (1.2 RP)

9 = r f + (0.8 RP)

Solving these equations, we obtain:

r f = 3% and RP = 7.5%

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

stocks and 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 M , the expected dollar return is (noting that the

expectation of non-systematic risk, e, is zero):

$1,000,000 [0.02 + (1.0 R M )] $1,000,000 [(–0.02)

+ (1.0 R M )]

= $1,000,000 0.04 = $40,000

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The sensitivity of the payoff of this portfolio to the market factor is zero because the exposures of the positive alpha and negative alpha stocks cancel out. (Notice that the terms involving R M sum to zero.) Thus, the systematic component of total risk is also zero. The variance of the analyst’s profit is not zero, however, since this portfolio is not well diversified.

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可编辑 精品文档，欢迎下载 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 exposure 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)2 ] = 18,000,000,000 The standard deviation of dollar returns is $134,164.

b.

If n = 50 stocks (25 stocks long and 25 stocks short), the investor will have a $40,000 position in each stock, and the variance of dollar returns is: 50 [(40,000 0.30)2 ] = 7,200,000,000 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 position in each stock, and the variance of dollar returns is: 100 [(20,000 0.30)2 ] = 3,600,000,000 The standard deviation of dollar returns is $60,000. Notice that, when the number of stocks increases by a factor of 5 (i.e., from 20 to 100), standard deviation decreases by a factor of 5= 2.23607 (from $134,164 to $60,000).

8. a.

)e (22M 22σ+σβ=σ 88125)208.0(2222A =+?=σ 50010)200.1(2222B =+?=σ 97620)202.1(2222C =+?=σ

b.

If there are an infinite number of assets 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.

c. There is no arbitrage opportunity because the well-diversified

portfolios all plot on the security market line (SML). Because they are fairly priced, there is no arbitrage.

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9. a. A long position in a portfolio (P) comprised of Portfolios A and B

will offer an expected return-beta tradeoff lying on a straight

line between points A and B. Therefore, we can choose weights such

that P = C but with expected 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.

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

coefficient of 2 must be zero in order to preclude arbitrage

opportunities.

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

b.Surprises in the macroeconomic factors will result in surprises in

the return of 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. The APT required (i.e., equilibrium) rate of return on the stock based

on r f and the 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% (because the expected surprises on all factors are zero by definition). Because the actually expected

return based on risk is less than the equilibrium return, we conclude that the stock is overpriced.

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12. The first two factors seem 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 sectors of investors. The third factor,

while important to Pork Products, is a poor choice for a multifactor SML because the price of hogs is of minor 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 confuse 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.

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13. The maximum residual variance is tied to the number of securities (n)

in the portfolio because, as we increase the number of securities, we

are more likely to encounter securities with larger residual variances.

The starting point is to determine the practical limit on the portfolio residual standard deviation, (e P), that still qualifies as a ‘well-

diversified portfolio.’ A reasonable approach is to compare 2(e P) to the market variance, or equivalently, to compare (e P) to the market

standard deviation. Suppose we do not allow (e P) to exceed p M, where p is a small decimal fraction, for example, 0.05; then, the smaller the value we choose for p, the more stringent our criterion for defining

how diversified a ‘well-diversified’ portfolio must be.

Now construct a portfolio of n securities with weights w1, w2,…,w n, so

that w i =1. The portfolio residual variance is: 2(e P ) = w122(e i)

To meet our practical definition of sufficiently diversified, we

require this residual variance to be less than (p M)2. A sure and

simple way to proceed is to assume the worst, that is, assume that the residual variance of each security is the highest possible value

allowed under the assumptions of the problem: 2(e i) = n2M

In that case: 2(e P ) = w i2n M2

Now apply the constraint: w i2 n M2 ≤ (p M)2

This requires that: n w i2≤ p2

Or, equivalently, that: w i2 ≤ p2/n

A relatively easy way to generate a set of well-diversified portfolios is

to use portfolio weights that follow a geometric progression, since the computations then become relatively straightforward. Choose w1 and a

common factor q for the geometric progression such that q < 1. Therefore, the weight on each stock is a fraction q of the weight on the previous stock in the series. Then the sum of n terms is:

w i= w1(1– q n)/(1– q) = 1

or: w1 = (1– q)/(1–q n)

The sum of the n squared weights is similarly obtained from w12 and a

common geometric progression factor of q2. Therefore:

w i2 = w12(1– q2n)/(1– q 2)

Substituting for w1 from above, we obtain:

w i2 = [(1– q)2/(1–q n)2] × [(1– q2n)/(1– q 2)]

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