Question

5. Portfolio standard deviation and diversification: You are asked to give financial advice to a mutual fund manager. The fund has had a history of investing in an equally-weighted portfolio of 50 stocks and is considering cutting down the number of stocks to 10 or 20. You assume that all stocks the fund is considering investing in have the same standard deviation of 40% and that the correlation between each pair of stocks is identical and equal to 0.25. a. To start your analysis, you calculate the standard deviation of the equally weighted portfolios with 10, 20, and 50 assets. b. Given your answers to part a. what is the advice that you give the mutual fund manager?

Answer 1

Portfolio standard deviation

Let w1 and w2 be the weights of portfolio consists of pair of stocks A and B with equal standard deviation=0.4.As portfolio is equally weighted ,w1=w2=0.5

Variance of the portfolio=w1²*variance A+w2² *variance B+2*w1*w2Correlation co- efficient*standard deviation of A*standard deviation of B

Variance of portfolio    =0.5²*0.4²+0.5²*0.4²+2*0.5*0.5*0.25*0.4*0.4

= 0.25*0.16+0.25*0.16+2*0.25*0.25*0.16

=0.04+0.04+0.02=0.1

Standard deviation of the portfolio=?variance of the portfolio

                                                                  =?0.1=0.316228(31.62% which is less than 40%)

Thus variance of the portfolio is less than identical individual stocks of the portfolio.A positive correlation of 0.25 means asset returns move together.

Portfolio risk is a measure of diversifiable risk of portfolio, which is less than risk of individual assets.Thus portfolio helps balance the fluctuations of fund. So mutual fund manager is advised to use equally weighted portfolio of 50stocks as used in the past.