Question

1. You are given the following table:

Securities	Weight	Beta	Expected return	Variance of return
1	0.3	0.6	0.7	0.06
2	0.4	0.8	0.9	0.08
3	0.3	1.1	1	0.13

Variance of market portfolio return = 0.07

               Given the assumption of a single factor model, calculate the following:

1. The residual variance of each of the above stocks;

2. The expected return on this 3-stock portfolio;

3. The beta factor of this 3-stock portfolio;

4. The variance of this 3-stock portfolio.

Answer 1

a)

The residual variance of a stock is given by = Total variance of the stock - (Beta^2)*Market variance

Residual variance of security 1 = 0.06-0.6*0.6*0.07 = 0.0348

Residual variance of security 2 = 0.08-0.8*0.8*0.07 = 0.0352

Residual variance of security 3 = 0.13-1.1*1.1*0.07 = 0.0453

b)

Expected return is the average of security return weighted by weights

Expected return on the portfolio = 0.3*0.7 + 0.4*0.9 +0.3*1 = 0.87

c)

Beta factor is the average of individual security beta weighted by weights

Beta factor on the portfolio = 0.3*0.6 + 0.4*0.8 +0.3*1.1 = 0.83

d)

First we need to find the covariance between the stocks. Since this is a single index model,

Covariance (1,2) = Beta1*Beta2*Market variance

Covariance (1,2) = 0.6*0.8*0.07 = 0.0336

Covariance (2,3) = 0.8*1.1*0.07 = 0.0616

Covariance (3,1) = 1.1*0.6*0.07 = 0.0462

The portfolio variance is given by

= (0.3*0.3*0.06) + (0.4*0.4*0.08) + (0.3*0.3*0.13) + 2*0.3*0.4*0.0336 + 2*0.4*0.3*0.0616+ 2*0.3*0.3*0.0462

= 0.061064