Question

Stock A has an annual expected return of 8%, a beta of .9, and a firm-specific volatility of 50% Stock B has an annual expected return of 9%, a beta of 1.3, and a firm-specific volatility of 40% The market has a standard deviation of 20%, and the risk-free rate is is 2%.

What is the volatility of stock A? (in %, round to 1 decimal place)

Suppose we construct a portfolio built out of 50% stock A, 30% stock B, and 20% government t-bills.

What is the expected return of this portfolio? (in %, round to 1 decimal place)

What is the beta of this portfolio? (round to 2 decimal places)

What is the non-systematic variance of the portfolio? (round to 3 decimal places)

What is the total volatility of the portfolio? (in %, round to 1 decimal places)

Answer 1

a) Total variance of stock A is given by

Var (RA) = Beta of A ^2 * Var (Market Returns) + Firm Specific variance

=0.9^2*0.20^2+0.5^2

=0.2824

Total Volatility (standard deviation)of stock A = square root of variance of Returns of Stock A

=0.2824^0.5

=0.5314 or 53.1%

Var (RB) = Beta of B ^2 * Var (Market Returns) + Firm Specific variance

=1.3^2*0.20^2+0.4^2

=0.2276

Total Volatility (standard deviation)of stock B = square root of variance of Returns of Stock B

=0.2276^0.5

=0.4771 or 47.71%

The Expected return of a portfolio is the weighted average return of the component stocks

So Expected Return of this portfolio = 0.5 * 8% +0.3 *9% +0.2*2% = 7.1%

Beta of a portfolio is the weighted average beta of the component stocks

Portfolio Beta = 0.5*0.9+0.3*1.3+0.2*0 (Risk free asset has 0 beta)

= 0.84

Covariance between returns of stock A and stock B

= Beta of A * Beta of B * variance of market returns

=0.9*1.3*0.2^2 =0.0468

Similarly Covariance between returns of stock A and Riskfree Asset

= Beta of A * Beta of Riskfree asset * variance of market returns

=0.9*0*0.2^2 =0

and Covariance between returns of stock B and Riskfree Asset

= Beta of B * Beta of Riskfree asset * variance of market returns

=1.3*0*0.2^2 =0

The standard deviation of a portfolio is given by

Where Wi is the weight of the security i,

is the standard deviation of returns of security i.

and is the correlation coefficient between returns of security i and security j

Total volatility or Standard deviation of portfolio = (0.5^2*0.2824+0.3^2*0.2276+0+2*0.5*0.3*0.0468+0+0)^0.5

= (0.105124)^0.5

= 0.324228 or 32.4%

Total Non systematic variance

= Total variance of portfolio - beta of portfolio^2* variance of market returns

=0.105124 - 0.84^2*0.2^2

=0.077