Question

You currently have all your funds invested in a portfolio of shares of Polycorp (PLC). The shares have an expected return of 10%pa and a standard deviation of 20%pa. You are offered the opportunity to invest in two funds managed by the Rearguard Portfolio Management Company. One is a Bond Fund that has an expected return of 5%pa and a standard deviation of 5%pa. The other fund is one that tracks the market index and has an expected return of 15%pa and a standard deviation of 25%pa. The correlation between the index fund and the bond fund is 0.1. Construct a portfolio of the Index and Bond funds that gives the same expected return as a portfolio of Polycorp shares. What is the composition and standard deviation of this portfolio? Use this example to comment on the benefits of diversification.

Answer 1

As the return of a portfolio is the weighted average return of Constituent securities

Let the weight of bond fund be W and that of the Index fund be (1-W), then

W*5%+(1-W)*15% = 10%

=> W =0.5

So, to achieve a portfolio return of 10% , 50% must be invested in Bond fund and 50% in Index fund

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

So, standard deviation of portfolio =sqrt (0.5^2*0.05^2+0.5^2*0.25^2+2*0.5*0.5*0.05*0.25*0.1)

=sqrt(0.016875)

=0.129904 =12.99%

From the above example , we see that the returns of the portfolio of shares of PLC and that of the portfolio of bond and index funds is same. However, the standard deviation of the two portfolios is quite different, PLC portfolio Standard deviation is 20% compard to standard deviation of portfolio of Index and Bond fund's of 12.99%. It is clear that the portfolio created by combining the Index and Bond fund can provide the same returns as that of a portfolio having a single share but at a lesser Standard deviation. This is due to  diversification (investing in more no. of securities with less correlation among them)