Question

1. The riskless interest rate is 2%. You hold a portfolio consisting of short-term safe assets and the market portfolio of risky assets, which has a mean return of 7% and a standard deviation of 20%. You are considering the stock of Microsoft Inc. that you believe to have a beta of 1.2 with the market portfolio, and a standard deviation of return of 30%. a) What is the standard deviation of the idiosyncratic component of Microsoft's return (the residual in the market model regression)? What is the correlation of Microsoft's return with the market's return? b) If the CAPM holds, what must be the mean excess return on Microsoft over the riskfree rate? What must be Microsoft's alpha? What must be its Sharpe ratio? Is the Sharpe ratio on Microsoft higher or lower than the Sharpe ratio on the market? Explain, and generalize to other stocks.

Answer 1

A-1)

Idiosyncratic component or unsystematic risk of Microsoft's return can be calculated by using formula

The difference between variance of security and co variance is the diversifiable or unsystematic risk.

Unsystematic risk. = Variance or Standard deviation of Security - Covariance of Microsoft and market

To calculate Covariance use formula

Beta = Cov_m,m / Var_m

Var_m = 20% (Standard deviation of Market)

Beta of Microsoft 's stock = 1.2

1.2 = Cov_m.m / 20%

Cov_m.m = 1.2 * 0.2

= 24 %

Now,

Variance or Standard deviation of Microsoft = 30%

Unsystematic risk. = Variance or Standard deviation of Security - Covariance of Microsoft and market

= 30% - 24%

Unsystematic risk or Idiosyncratic component = 6%

A-2)

Correlation of Microsoft return with the market return

Another formula for Beta is

Beta = $\sigma$_Microsoft * $\sigma$_market * Cor _m,m / $\sigma$²_m

Beta = 1.2

Variance or Standard deviation of Microsoft ($\sigma$_Microsoft)= 30%

Standard deviation of Market ($\sigma$_market) = 20%

putting values in formula

1.2 = 30 * 20 * Correlation / 20 * 20

Correlation = 1.2 * 20 / 30

Correlation of Microsoft's return with market return = 0.8

B)

As per CAPM

Risk free rate = 2%

Market return (R_m ) = 7%

Beta = 1.2

Expected return on security = R_f + (R_m - R_f) / Beta

= 2 + (7 - 2) / 1.2

= 2 + 5 / 1.2

= 6.16 %

Microsoft's mean excess return over the risk free rate = expected return of microsoft - risk free rate

= 6.16 - 2 => 4.16%

Alpha is the security's excess return over the market return

= Expected Microsoft's return -Market Return

= 6.16 -7 %

= - 0.84 %

Negative Alpha

Sharpe Ratio = [E(Return on security) - Risk free rate ] / Standard deviation of security

Variance or Standard deviation of Microsoft ($\sigma$_Microsoft)= 30%

= (6.16 - 2) / 30

= 4.16 / 30 => 0.138

Standard deviation of Market ($\sigma$_market) = 20%

Return of market = 7%

Risk free rate = 2%

Sharpe Ratio of Market = ( 7 - 2 ) / 20

= 5 / 20 => 0.25

Sharpe ratio of Microsoft is lower than the Sharpe ratio  of market.