Question

Suppose that in this particular economy, there are four assets. Assets 1, 2, and 3 are risky and the fourth asset is risk-free.

The correlations of returns are described in the following table:

Correlation	Stock 1	Stock 2	Stock 3
Stock 1	1	0.6	0.7
Stock 2	0.6	1	0.2
Stock 3	0.7	0.2	1

And the standard deviation of the return of each stock is:

Stock 1	0.3
Stock 2	0.6
Stock 3	0.25

Finally, the number of shares and price of each stock is:

	Price	Number of Shares
Stock 1	$10	100
Stock 2	$15	200
Stock 3	$10	200

1. Construct the variance-covariance matrix for the returns of the three risky assets.

2. Compute the weights of the market portfolio. That is, show that the weight of stock 1 in the market portfolio is 1/6, the weight of stock 2 is 3/6 and the weight of stock 3 is 2/6.

3. If CAPM assumptions hold, compute the investor’s optimal risky portfolio.

Answer 1

The correlations of returns are described in the following table:
Correlation	Stock 1	Stock 2	Stock 3
Stock 1	1	0.6	0.7
Stock 2	0.6	1	0.2
Stock 3	0.7	0.2	1
And the standard deviation of the return of each stock is:
Stock 1 (sd1)	0.3
Stock 2 (sd2)	0.6
Stock 3 (sd3)	0.25

a) Construction of variance -covariance matrix

Variance Covariance Matrix ( value obtained using formula in table below)
	Stock 1	Stock 2	Stock 3
Stock 1	0.09	0.108	0.0525
Stock 2	0.108	0.36	0.03
Stock 3	0.0525	0.03	0.0625

In the table below sd1 = Standard Deviation of stock 1

Corr12 = correlation between stock 1 and 2

Variance Covariance Matrix
	Stock 1	Stock 2	Stock 3
Stock 1	(sd1*sd1*corr11)	(sd1*sd2*corr12)	(sd1*sd3*corr13)
Stock 2	(sd1*sd2*corr12)	(sd2*sd2*corr22)	(sd2*sd3*corr23)
Stock 3	(sd1*sd3*corr13)	(sd2*sd3*corr23)	(sd3*sd3*corr33)

b )Weights of Portfolio

	Price (P)	Number of Shares (N)	Invested Amount (P*N)	Weights (Invested Amount for a stock/ Total Invested Amount)
Stock 1	10	100	1000	0.166666667 (1000/6000)
Stock 2	15	200	3000	0.5 (3000/6000)
Stock 3	10	200	2000	0.333333333 (2000/6000)
		Sum	6000	1

c) Standard Deviation Of Above Portfolio obtained using excel or it can be obtained manually also using the formula SQRT(MMULT(MMULT(TRANSPOSE(weight),varCovarMatrix),weight)) [where MMULT is Matrix Multiplication ,varCovar Matrix is 3*3 variance Covariance matrix , weight is 3*1 matrix ]

Standard Deviation of Portfolio

37%

SQRT(MMULT(MMULT(TRANSPOSE(weight),varCovarMatrix),weight))

In excel to compute above formula one needs to press Ctrl+Shift+Enter button together

Investor optimal portfolio risky portfolio should have minimum Standard deviation

To compute minimum standard deviation (given return is not given ) one can use excel solver function subject to constraint the sum of weight should be1.

Standard Deviation of Portfolio

24.3753593200396%

SQRT(MMULT(MMULT(TRANSPOSE(weight),varCovarMatrix),weight))

Weights

0.080741

0.070054

0.849204