Question

The table below shows standard deviations and correlation coefficients for eight stocks from different countries. Calculate the variance of a portfolio with equal investments in each stock. (Use decimals, not percents, in your calculations. Do not round intermediate calculations. Round your answer to 5 decimal places.)

	BHP	BP	Fiat Chrysler	Heineken	Korea Electric	Nestlè	Sony	Tata Motors
BHP	1.00	.34	.38	.45	.14	.45	.16	.29
BP	.34	1.00	.29	.14	.16	.36	−.07	.12
Fiat	.38	.29	1.00	−.03	−.02	−.10	.24	−.05
Heineken	.45	.14	−.03	1.00	.38	.56	−.12	.46
Korea Electric	.14	.16	−.02	.38	1.00	.15	−.15	.11
Nestlè	.45	.36	−.10	.56	.15	1.00	−.14	−.01
Sony	.16	−.07	.24	−.12	−.15	−.14	1.00	.10
Tata Motors	.29	.12	−.05	.46	.11	−.01	.10	1.00
Standard deviation (%)	30.00	41.80	52.80	28.30	22.00	42.50	50.40	52.50

Portfolio variance            

Answer 1

)Portfolio variance is used to calculate the risk of that portfolio. Lets first understand how is the variance of two stock calculated.

let w_1,w₂ be the weights assigned to the two assets and let their standard deviations be_1,  2 and the correlation coefficient between them be _1,2.

Based on the above information, portfolio variance can be calculated as = w_1^2*1²⁺w_2*²2²+2*w_1*w_{2*1,2* 1* 2}

Similarly, for an 8 stock portfolio, there will n*(n-1)/2 pairs of correlations i.e (8*7)/2= 28 pairs of the correlation coefficient

Hence portfolio variance of an 8 stock portfolio = .

Using this equation and putting the values of w_i = 1/8 =0.125 since all weights are equal.

From the table, the values of standard deviation can be taken from the last row and correlation coefficients can be picked from the other rows.

Hence the formula gives us the value of portfolio variance as 0.040838.

Another method to calculate portfolio variance is by using matrix multiplication.

Let be the variance of the portfolio then

= * correlation matrix given in the question *

Putting the values of w₁........w₈ = 0.125 , ............. as given in the last row of the matrix and using the correlation coefficient between different pairs of the stocks, use the following formula

= MMULT(MMULT(TRANSPOSE(w_ii )_[1,8]*correlation coefficient matrix _[8,8]* (w_ii )_[8,1]

the value obtained is 0.040838