Question

Two financial assets with gross return (R1, R2) are jointly normally distributed. E(R1) = 1.06 and E(R2) = 1.12. V ar(R1) = 0.03, V ar(R2) = 0.04, and Cov(R1, R2) = 0.025. Draw the mean-variance frontier formed by the two assets. Mark the minimum variance. Notice that the mean of portfolio formed by these two assets falls in the interval of [1.06, 1.12].

Answer 1

Let us assume the weight as w1 and w2 for the two stocks. Since our portfolio has only 2 stocks w2 = 1-w2

The expected return of the portfolio E(R) = w1*E(R1) + (1-w1) * E(R2)

Variance of portfolio = w1² * V(R2)² + (1-w1)² * V(R2)² + 2* w1* (1-w1) *Cov (R1,R2)

To draw the mean-variance efficient frontier, we will need assume various values of w1 from 0 to 1 ( in increments of 0.1) and plot the expected return and standard deviation on y and x-axis respectively. We will use excel to calculate the values and draw the graph.

The calculation of E(R) and std dev (R) for various values of w1 is copied below from excel

W1	E(R )	V (R )	Std. Dev. (R )
0	1.12	0.0400	0.2000
0.1	1.114	0.0372	0.1929
0.2	1.108	0.0348	0.1865
0.3	1.102	0.0328	0.1811
0.4	1.096	0.0312	0.1766
0.5	1.09	0.0300	0.1732
0.6	1.084	0.0292	0.1709
0.7	1.078	0.0288	0.1697
0.8	1.072	0.0288	0.1697
0.9	1.066	0.0292	0.1709
1	1.06	0.0300	0.1732

Copied below are the formula used to calculate these values

The efficient frontier drawn using E(R) as Y-axis and E(R) as x-axis is copied below:

The minimum variance is slightly lower than 0.1697