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The expected return of stock A is 20% per year and the stock's annual standard deviation...

The expected return of stock A is 20% per year and the stock's annual standard deviation is 45%. There is also a risk-free asset. When a complete portfolio is formed with a portfolio weight on the risky asset of 35%, the expected return on the complete portfolio is 8.0%.

(a) Compute the risk-free rate of return.

(b) Compute the annual standard deviation of the complete portfolio above

(c) Compute the market price of risk (i.e., the Sharpe ratio)?

Solutions

Expert Solution

Part a:

Expected return on the complete portfolio is calculated as:
Weight on risky asset*Return on risky asset + Weight on risky free asset*Return on risk free asset
Given that, expected return on the complete portfolio =8.0%.

Portfolio weight on the risky asset (stock A)=35%
Portfolio weight on the risk free asset=100%-35%=65%

Expected return of stock A (risky asset)=20%
Assume that the risk-free rate of return=x percent

Expected return on the complete portfolio=Weight on risky asset*Return on risky asset + Weight on risky free asset*Return on risk free asset

Given that expected return on the complete portfolio=8%

So, 35%*20% + 65%*x=8%
=>.35*.2 + .65x=0.08
=>0.07 + .65x=0.08
=>.65x=0.08-0.07
=>.65x=0.01
=>x=0.01/.65=0.015384615 or 1.54% (rounded upto two decimal places)
So, risk-free rate of return=1.54%

Part b:
Standard deviation of portfolio= (Weight on risky asset^2*Standard deviation on risky asset^2 +Weight on risk free asset^2*Standard deviation on risk free asset^2)^1/2
Standard deviation of a risk free asset is zero
So here standard deviation of portfolio = (Weight on risky asset^2*Standard deviation on risky asset^2)^1/2
=Weight on risky asset*Standard deviation on risky asset

Given that, portfolio weight on the risky asset (stock A)=35%
And, the stock's annual standard deviation=45%
Standard deviation of portfolio =35%*45%=15.75%

Part c:
The formula to calculate Sharpe ratio is:
(Expected portfolio return - Risk free rate of return)/Standard deviation of the portfolio

Here,
Given that, expected return on the complete portfolio=8%.
Risk-free rate of return (as we calculated)=1.54%
Standard deviation of portfolio =15.75%
Market price of risk (Sharpe ratio) is calculated as:
(8%-1.54%)/15.75%=6.46%/15.75%=0.41

jeff jeffy answered 2 years ago
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