Question

7) There are only two securities (A and B, no risk free asset) in the market. Expected returns and standard deviations are as follows:

Security	Expected return	standard Deviation
Stock A	25%	20%
Stock B	15%	25%

1. The correlation between stocks A and B is 0.8. Compute the expected return and standard deviation of a portfolio that has 0% of A, 10% of A, 20% of A, etc, until 100% of A. Plot the portfolio frontier formed by these portfolios
2. Repeat the previous question, assuming that the correlation is

−0.8.

1. Explain intuitively why the portfolio frontier is different in the two cases.

Answer 1

For ease of understanding we will calculate in Excel and Excel formulas will be provided in the Screen Shot.

W1 = Weight of Stock 1

W2 = Weight of Stock 2

SD1 = Standard Deviation of Stock 1

SD2 = Standard Deviation of Stock 2

Return =W1*Return of Stock1 + W2*Return of Stock2

SD of Portfolio = ((W1*SD1)^2+(W2*SD2)^2 +2*W1*W2*SD1*SD2* co-relation)^0.5

For -0.8 Correlation

Now portfolio frontier is created by keeping the Standard deviation on X-Axis and Return on Y Axis. The return of portfolio only depends on the weight of individual stocks and the individual stock's return. So the frontier is changing due the change in standard deviation of the portdolio.

Now in the standard deviation formula the co-relation is changed so the standard deviation changes due to co-relation. Now the formula has corelation in last part 2*w1*w2*SD1*SD2*co-relation. The negative co relation changes the sign of that part and as it gets bigger it reduces the standard deviation and if it gets small then the standard deviation increases.

You can see from the screenshot the change in standard deviation.