with
Cov(e, )=0. Also assume that stock j has a beta of 1.5, a standard deviation of excess returns of .4, and that the market excess return has a standard deviation of .1.
a. Find E[
b. Find Var[ ]
c. What proportion of the asset j’s variance is systematic?
d. What determines systematic risk? To answer this question use the formulas for the systematic risk.
e. What proportion of the variance is unsystematic?
1. Assume the excess return on stock j can be written as
f. What is the total systematic and unsystematic variance in stock j?
g. What is R‐Squared? To answer this question first answer with words and then using a formula and notation.
h. In the simple regression above (i.e.note it has only one explanatory variable) what is the relation between the correlation between Rj and RM and the R‐square?
i. What is the formula for the beta shown in the model above?
j. What is the formula for the alpha shown above?
k. What is the formula for the security characteristic line for asset j?
E[Rj]= αj + βj E[Rm] ‐‐ note that E(e) = 0.
Var[Rj] = Var[αj + βj Rm + ej] = Var[βj Rm]+Var[ej] = σ2(βjRm) + σ2(ej) = βj2 σ2m + σ2(ej) =systematic risk + firm‐specific risk
This means that the systematic variance in stock j’s excess returns is equal to βj2 σ2m and that the firm‐specific variance is σ2(ej).
This means that the total risk involved is βj2 σ2m + σ2(ej)
This also means that ratio of asset j’s systematic risk to total risk is βj2 σ2m / [ βj2 σ2m + σ2(ej) ] <‐ this is the answer to c
Based on the formulas shown above systematic risk is measured as βj2 σ2m showing that a firm’s overall systematic risk will be determined by both the market volatility and firms’s sensitivity to the market as captured by beta (see book discussion from 168‐172 8ed, 170‐173 9ed)
If the proportion of overall risk that systematic is as follows: βj2 σ2m / [ βj2 σ2m + σ2(ej) ] then 1 minus this proportion will be the proportion of overall risk that is unsystematic.
1
Putting it all together:
Systematic risk in j’s returns = βj2 σ2m = 1.5*1.5*.1*.1 = .0225
Total risk (variance) in j’s returns = σ2j = .4*.4 = .16
Proportion of variance that is systematic= βj2 σ2m / [ βj2 σ2m + σ2(ej) ]
= .0225/.16= .1406
R‐square is the ratio of the explained variance to total variance. In other words it represents the proportion of the variation in the dependent variable explained by the explanatory variable. Using notation from above it would be βj2 σ2m / [ βj2 σ2m
+ σ2(ej) ]
= .0225/.16= .1406
If p is the correlation between Rj and RM then p2 is the R‐square value
The formula for the beta is Cov(Rj , RM )/Var(RM ).
The formula for the intercept is Intercept = Average(Rj ) – average(RM). Note that if
I gave you the average values used in this formula that you could solve for the intercept. The security characteristic line for asset j would be Rj = intercept + beta(Rm)
2. Draw the 3 figures (with labeled axis) that showcase the following terms: CAL, CML, efficient frontier, SML, alpha, minimum variance portfolio, tangent portfolio, security characteristic line, intercept, regression line, beta, R‐square, sharpe ratio, risk free rate, standard deviation. See 3 book figures and related discussion in the text (6.10 + 6.11 +7.1), (6.12+7.4), and
(7.2)
3. What fee structure was common at hedge funds a few years ago?
20% of profits + a flat management fee of 1.5‐2% of assets. 4. Assume the risk‐free rate is 4%. Assume the historical market risk premium is 9%. Assume investors anticipate that stock X with a beta of 0.9 to offer a rate of return of 12 percent. A) What will the CAPM expected return of the stock be, assuming the beta of the stock and the overall expected market return do not change?
