Download Assume that the CAPM holds, the expected return on the market

Assume that the CAPM holds, the expected return on the market
portfolio is 12%, the standard deviation of return on the market
portfolio is 18% and all investors can borrow and lend at the riskless
rate of 4%. Determine the expected return and standard deviation of
return for the following portfolios:
(a) The total investment is $235,000. Of this, $47,000 is invested in
the riskless asset, and the rest is invested in Regis Corporation stock.
Regis stock return has a correlation coefficient of 0.7 with the market
portfolio return. The standard deviation of return on Regis stock is
45%.
Solution:
Expected Return on Regis (Using CAPM)
= Risk free Return + Beta(Market Return-Risk free Return)
=.04+.7(.12-.04) =.096 =9.6%
Expected Return on Portfolio = W1*E(r1) + W2*E(r2)
Where
W1 = weight of First Security
W2 = weight of Second Security
E(r1) = Expected Return on First Security
E(r2) = Expected Return on Second Security
Expected Return on Portfolio
= (47000/235000)*.04 + (188,000/235000)*.096= .0848
=8.48%
Standard Deviation = (188,000/235000)*.45=.36 = 36%
(Since the other asset is a risk free asset Standard deviation
for portfolio will be Weight of risky asset* Standard deviation
of risky asset)
(b) The total investment is $200,000. This portfolio includes a short
position in an asset whose beta is 0.4 and whose standard deviation of
return is 8%, as well as an investment of $400,000 in the market
portfolio. There are no other investments.
Total Portfolio = $200000
Investment in Market Portfolio= $400,000.
Investment in Short Position = -$200,000
Expected Return on Asset(using CAPM) = .04+ .4(.12-.04)
=.072=7.2%
Expected Return on Portfolio = (-200,000/200,000)*.072 +
(400,0000/200,000)*.12=16.8%
Variance for Portfolio = σ² = W1² σ1² + W2² σ2² + 2(W1W2 ρ12 σ1σ2)
σ² = W1² σ1² + W2² σ2² + 2(W1W2 *Covariance (stock versus market
returns))
Where
σ²= Variance of the Portfolio
W1 = weight of First Security
W2 = weight of Second Security
σ1 = Standard Deviation of First Security
σ2 = Standard Deviation of Second Security
ρ12= Correlation between First and Second Security
Variance for Portfolio = (-1)^2*(.08)^2 + (2)^2*(.18)^2 + 2*1*2*.01296 = 0.08416
Standard Deviation = (.08416) ^ (1/2) = .2901 OR 29.01%
Note:
Beta = Covariance (stock versus market returns) / Variance of the
Stock Market
0.4 = Covariance (stock versus market returns) / (.18)^2
Covariance (stock versus market returns) = 0.01296