威廉夏普投资学key for chap10

1. The auto industry's earnings are highly cyclical. Therefore auto company

stocks possess a high sensitivity (in a positive direction) to the trend in economic activity.

Savings and loan companies (whose primary business is home loans) generally have large portfolios of fixed-rate loans. When interest rates rise (fall), their cost of funds rises (falls), while revenues remain relatively stable. As a result, their earnings fall (rise). Thus the stocks of these companies are often responsive (in a negative direction) to movements in real interest rates. For the real estate and air line, it is a negative signal.

Electric utilities operate in regulatory environments. They may have trouble passing on cost increases to consumers, especially in the short run. Thus their stocks are sensitive (in a negative direction) to unexpected inflation.

Crude oil producers and their stocks are sensitive (in a positive direction) to the level of oil prices.

In order to derive the curved Markowitz efficient set, the investor needs to estimate the expected returns, variances, and covariances for all assets. One can show that without a factor model, the investor must estimate (N² + 3N)/2 parameters to derive the efficient set.

On the other hand, based on the assumptions underlying a factor model, the common responsiveness of securities to the factor(s) eliminates the need to estimate directly the covariances between securities. These covariances are captured by the securities' sensitivities to the factor(s) and the factor'(s) variance(s). As a result the number of parameters that must be estimated to derive the efficient set with a factor model is significantly reduced.

Factor model relationships are based on two critical assumptions. The first is that the random error term and the factor are uncorrelated, meaning that the outcome of the factor has no bearing on the outcome of the random error term.

The second assumption is that the random error terms of any two securities are uncorrelated, meaning that the outcome of the random error term of one security has no bearing on the outcome of the random error term of any other security.

As a violation of the first assumption, consider a one-factor model where the factor is growth in GDP. If it were the case that a security had a positive random error term value every time GDP was higher than expected, then the factor model has been misspecified and should be adjusted to take into 2. 4.

account this unexplained sensitivity.

As a violation of the second assumption, suppose that whenever security A

had a positive random error term value, security B also had a positive random error term value, then the factor model has been misspecified. In this case there must be some source of common responsiveness between the two securities that has not been captured by the factor model.

By the term "similar stocks" Cupid presumably means that they display similar sensitivities to various economic and financial factors. If a factor model is correctly specified, then two stocks with similar sensitivities to the model's factors should generate returns that are roughly the same over time. In the short run their returns may differ by the differences in the values of their respective random error terms. Given that the expected value of the random error term is zero, over the long-run one would expect the random error term to equal zero and thus the average return on the two securities to be the same.

a. 22In a one-factor model, a portfolio's factor risk is expressed as b p σF . 5. 7.

b. Since the sensitivity of the portfolio to the factor is the weighted average of the component securities' sensitivities (with their proportions serving as weights), then: Factor risk = (.40 ⨯ .20 + .60 ⨯ 3.50)2 ⨯ 225 = 1,069.3 2Non-factor risk (expressed as σep is the weighted average of the

9.

c. component securities' random error term variances (with the square of the securities' proportions serving as weights), then: Non-factor risk = .40² ⨯ 49 + .60² ⨯ 100 = 43.8 The standard deviation of the portfolio is given by: 2221/2σp =(b p σF +σep ) = (1,069.3 + 43.8)½= 33.4% The covariance between two securities in a one-factor world is given by: 2 σij =bb i j σF In this case, the equation should be solved for F F . That is: = [½ij /b i b j ] = [(-312.50)/(-0.50 ⨯ 1.25)]½

= 22.4%

10. In a one-factor model world, the standard deviation of a security is given by:

221/2σi =(b i 2σF +σei ) For security A : ⨯ (18)² + (25)²]½ A = [(.8)² = 28.9% For security B : ⨯ (18)² + (15)²]½ B = [(1.2)²

= 26.3%

11. The nonfactor risk of a portfolio is given by:

2 σep =∑X i 2σei i =1n

13. In order to calculate the expected return and standard deviation of a

thirty-stock portfolio based on a five-factor model (with uncorrelated factors), the following parameters must be estimated:

Zero-factor for each security 30 Sensitivity of each security to each factor (5 ⨯ 30) 150 Variance of the random error term for each security 30 Variance of each factor 5 Expected value of each factor Total 220 If the factors are correlated, then there will be (N ² - N ) factor covariances to

estimate in addition to the parameters listed above. In this case, the number of additional parameters would be (5² - 5) = 20.

14. Factors thought to pervasively affect security returns are usually viewed as

"macroeconomic" or "microeconomic" in nature. The text discussed several possible macroeconomic factors. Other such factors might include money supply growth, the size of the budget deficit (or surplus), the size of the trade deficit (or surplus), or the level of consumer confidence.

Microeconomic factors (or at least proxies for those factors) that might Assuming that the securities in the portfolio are equal-weighted, the portfolio's nonfactor risk is the average nonfactor risk of the securities divided by the number of portfolio securities. Thus the nonfactor risks of the various portfolios are: 10-security portfolio: 225/10 = 22.5 100-security portfolio: 225/100 = 2.25 1,000-security portfolio: 225/1,000 = 0.225

pervasively influence security returns include dividend yield, earnings growth rate, earnings growth momentum (that is, the rate of change in earnings growth), book value -to-price ratio, market capitalization, and financial leverage.

15. A portfolio's sensitivity to a factor is the weighted average of the component

securities' factor sensitivities. Therefore in this case:

b p1 = (.60 ⨯ -.20) + (.20 ⨯ .50) + (.20 ⨯ 1.50)

= 0.28

b p2 = (.60 ⨯ 3.60) + (.20 ⨯ 10.00) + (.20 ⨯ 2.20)

= 4.60

b p3 = (.60 ⨯ 0.05) + (.20 ⨯ .75) + (.20 ⨯ 0.30)

= 0.24

16. In the context of a factor model, the expected return on securities is a

function of the values expected to be attained by the factor (or factors). Surprises in the actual outcomes for the factor values will determine the actual returns earned on the securities, with the exact nature of those actual returns depending on the structure of the factor model.

Mathematically, the expected return on security based on a single-factor

model can be expressed as:

i = a i + b i where r i and F are the expected return for security i and the expected value of the factor, respectively. Further, realized returns on a security can be expressed as: r i = a i + b i F + e i Substituting (r i - b i F ) for a in the preceding equation gives: r i = r i + b i (F - F ) + e i That is, the actual return on the security is a function of its expected return

and the surprise (or unexpected change) in the value of the factor. The underlying correlations among securities is represented by the sensitivities of the securities to surprises in the factor value, combined with the volatility of the factor value.

18. The time-series approach to factor model estimation begins with the

assumption that the factors are known in advance. Typically, the identification of the factors proceeds from an analysis of the economics of the firms involved. With the factors specified, historical information concerning the values of the factors and security returns are collected from period to period. These data are used to estimate securities' sensitivities to the factors,

the securities' zero factors and unique returns, and the standard deviations of factors and their correlations.

The cross-sectional approach to factor model estimation begins with estimates

of the securities’ sensitivities to certain factors. Then, in a particular time period, the values of the factors are estimated based on the securities' returns and their sensitivities to the factors. By repeating the process over multiple time periods, statistically significant estimates of the factors' standard deviations and correlations can be computed.

The factor analysis approach to factor model estimation begins simply with a set of securities and their corresponding returns. A statistical procedure known as factor analysis is used to identify the number of significant factors and the securities' sensitivities to those factors as well as the standard deviations of the factors and the correlations among the factors.

19. Security prices represent investors’ consensus expectations about the future

prospects for the firms that issue the securities. Past factor values will already be incorporated into security prices. Thus past factor values will have no effect on security price changes and, therefore, security returns. Instead it is what investors expect will be the value of factors in the future that should be related to security price changes and, therefore, security returns.

22. Based on a two-factor model, the variance of a security is:

2222 σi 2=b i 21σF +b i 2σF +2b i 1b i 2COV (F 1, F 2) +σei 12Therefore for the two securities in this problem: 2 = [(1.5)² ⨯ (20)²] + [(2.6)² ⨯ (15)²] + (2 ⨯ 1.5 ⨯ 2.6 ⨯ 225) + 25= 4,201 σA A = (4,201)½ = 64.8% 2 = [(0.7)² ⨯ (20)²] + [(1.2)² ⨯ (15)²] + (2 ⨯ 0.7 ⨯ 1.2 ⨯ 225) + 16= 914 σB B = (914)½ = 30.2% The covariance between two securities in a two-factor world is: 22σij =b i 1b j 1σF +b i 2b j 2σF +(b i 1b j 2+b i 2b j 1) COV (F 1, F 2) 12In this case:

= [(1.5 ⨯ 0.7) ⨯ (20)²] + [(2.6 ⨯ 1.2) ⨯ (15)²+ {[(1.5 ⨯ 1.2) + (2.6 ⨯ 0.7)] ⨯ AB 225} = 1,936.5