Hi loman!!
The market portfolio is assumed to be composed of four securities.
Their covariances with the market and their proportions follow:
Security Covariance with Market Proportion
A 242 .2
B 360 .3
C 155 .2
D 210 .3
Given these data, calculate the market porfolio's standard deviation.
Answer:
Recall that the standard deviation of the market portfolio is equal to
square root of a weighted average of the covariance of all securities
with it, where the weights are equal to the proportions of the
respective securities in the market portfolio, in this case:
STD_M = sqrt(pA*cov_A + pB*cov_B + pC*cov_C + pD*cov_D) =
= sqrt(0.2*242 + 0.3*360 + 0.2*155 + 0.3*210) =
= sqrt(250.4) =
= 15.82
The market porfolio's standard deviation is 15.82
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Based on a three-factor model, consider a portfolio composed of three
securities with the following characteristics:
Sensitivities
Security Factor 1 Factor 2 Factor 3 Proportion
A -.2 3.6 .05 .6
B .5 10.0 .75 .2
C 1.5 2.2 .30 .2
What are the sensitivities of the portfolio to factors 1, 2, and 3?
The sensitivity of the portfolio to each factor Fi (sFi) is equal to
the weighted average of the respective sensitivities of each security
in the market portfolio, in this case:
sFi = pA*sFi_A + pB*sFi_B + pC*sFi_C
For factor 1:
sF1 = pA*sF1_A + pB*sF1_B + pC*sF1_C =
= 0.6*(-0.2) + 0.2*0.5 + 0.2*1.5 =
= 0.28
Using the same method for factors 2 and 3 you will get:
sF2 = 4.60
sF3 = 0.24
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Based on a one-factor model, two portfolios, A and B, have equilibrium
expected returns of 9.8 % and 11.0%, respectively. If the factor
sensitivity of portfolio A is 0.8 and that of portfolio B is 1.0, what
must the risk free rate be?
On a one-factor model at equilibrium the expected return for each portfolio is:
E(rP) =Beta_P*(rM - rF) + rF [eq.1]
where:
rP = expected rate of return of the portfolio
Beta_P = beta or factor sensitivity of the portfolio
rM = expected market return
rF = risk free rate
We can rewrite [eq.1] as follows:
E(rP) = Beta_P*rM + (1-Beta_P)*rF
For portfolio A we know that:
9.8 = 0.8*rM + (1-0.8)*rF = 0.8*rM + 0.2*rF
For portfolio B we know that:
11.0 = 1.0*rM + (1-1.0)*rF = rM + 0*rF = rM
Replacing rM in the portfolio A expected return formula:
9.8 = 0.8*rM + 0.2*rF =
= 0.8*11.0 + 0.2*rF =
= 8.8 + 0.2*rF
Now if we isolate rF:
rF = (9.8-8.8)/0.2 =
= 1.0/0.2 =
= 5
The risk free rate is 5%
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I hope that this helps you. Feel free to request for a clarification
if you need it before rate this answer.
Best regards.
livioflores-ga |
