Exercise_3_Week_5_Solutions

CAPM and Market Efficiency

Solutions

1. Ms England

Fund

β

Variability (%) Annual Abnormal

Return

Annual

Actual Return Expected Return T 1.0 36 -19 2 21 R

0.8

24 14 33 19

a) Assuming the CAPM holds the market portfolio has a beta of unity (same as fund T) and hence expected return on the market is 21%. Compute the risk-free rate from CAPM for any of the funds

Expected Return for security P = 23 = r f + 1.2 (21 - r f ) r f =11%

or from security R: 19 = r f + 0.8 (21 - r f ) gives r f = 11%.

(b) New R M =20% and r f =10%. Call this portfolio A. Ms England expects return on security with βj to be ER j = 10 + βj *(20-10)

The beta of Portfolio A is (1.2 + 1.0 + 0.8)/3 = 1 hence the expected return of Portfolio A is 20%.

To obtain an expected return of 25%, she creates a new Portfolio B in which assume she puts an amount x in the risk free asset and 1-x in her Portfolio A then x 10 + 20(1-x) = 25. hence x = -0.5.

This implies that Portfolio B consists of -0.5 (i.e. borrows) in the risk free asset and 1.5 in her Portfolio A or borrows £15,000 and invests £45,000 in Portfolio A of three funds.

(c) Variability of market is 20%.

Note: specific risk, unique risk and diversifiable risk are all the same terms.

Similarly: market risk, systematic risk, and non-diversifiable are all the same terms

We can compute the specific risk of P, T, R from σβσσεi m i 2222=+

Assuming independence of each companies specific risk from the definition of specific risk, we can compute the specific risk in Portfolio A using the standard formula for portfolio risk, but with the assumption that specific risk has ZERO

covariance: Specific Risk in Portfolio A = (13)2 (580+896+320) = 200

We have already computed the beta of Portfolio A as unity, and so systematic risk of Portfolio A = 12 *400 = 400. So total risk of Portfolio A is 600 (stdev = 24.5%)

% Diversifiable risk in Portfolio A with just three securities = 200

400200

+ = 33%

In the case where she borrows and creates Portfolio B, the relative weights in the risk free asset and the three securities are -0.5, 0.5, 0.5, 0.5

Beta of Portfolio B = 0*(-0.5)+ 0.5*1.2 +0.5*1 + 0.5*0.8 (remember beta of risk-free asset is zero)

ie Beta of portfolio B = (1.2 + 1.0 + 0.8)*0.5 = 1.5

The specific risk of Portfolio B is (½)2 (580+896+320) = 449

Total risk of portfolio B =1.52 * 400 + 449 = 1,349 (stdev = 36.7%)

% diversifiable risk in Portfolio B =

%33449

400*5.1449

2

=+.

Result: You do not alter the percentage of diversifiable risk by combining securities with a riskless asset. You do of course alter the total risk in the portfolio.