Question

Suppose that the annual returns on two shares are perfectly negatively correlated and that r_m= 0.07, r_n= 0.20,σ_m=0.12 , and σ_n=0.5.

Assuming that there are no arbitrage opportunities, by using the Goal Seek function (excel) calculate the weight (proportion) of the two assets that produce the lowest portfolio variance? (Use the Goal Seek function)

Answer 1

The return of a portfolio is the weighted return of the two stocks

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

Using Excel Goal Seek. the weights are 0.808 for M and 0.192 for N

Exact weights can be found by minimum variance portfolio weights formula  

The formula for minimum risk weights in a two stock S and Stock B portfolio is

Let S be M and B be N, since correlation coefficient is -1

So, W_S = (0.5^2-0.5*0.12 *(-1)) / ( 0.5^2+0.12^2 -2*0.5*0.12 *(-1))

= 0.806452 = 80.65%

and W_B = 1-   W_S = 1-0.8065 =0.1935=19.35%

So, proportion of M in lowest variance portfolio is 0.8065 or 80.65%

& proportion of N in lowest variance portfolio is 0.1935 or 19.35%