# How to calculate the variance of a portfolio?

So I have this question :

Florida Company (FC) and Minnesota Company (MC) are both service companies. Their stock returns for the past three years were as follows: FC: -5 percent, 15 percent, 20 percent; MC: 8 percent, 8 percent, 20 percent.

And I calculate :

Mean return FC=10%, mean return of MC=12%.

Var(FC) = [( -5 - 10)^2 + (15 - 10)^2 + (20 - 10)^2]/3 = 116.67.

Var(MC) = [(8 - 12)^2 + (8 - 12)^2 + (20 - 12)^2]/3 = 32

Standard deviation (FC) = 116.7^0.5 = 10.8%. Standard deviation (MC) = 32^0.5 = 5.7%.

Covariance= : [(-5 - 10)(8 - 12) + (15 - 10)(8 - 12) + (20 - 10)(20 - 12)]/3 = 40.

Correlation coefficient = covariance/[(S.D.(FC)) × (S.D.(MC))] = 40/(10.8 × 5.7) = +0.655.

Now, if FC and MC are combined into a portfolio with 50 percent of the funds invested in each stock, then

Return on portfolio=Rp = (10)(0.5) + (12)(0.5) = 11%.

and finally the variance.

On the answer sheet it states that the variance of this portfolio is:

Var(P) = (0.5^2)(116.7) + (0.5^2)(32) + (2)(0.5)(0.5)(40) = 57.17.

The thing is that I do not see the standard deviations as being part of the formula (at the end).

I thought that the Var(P) was

then,

(0.5^2)(116.7) + (0.5^2)(32) + (2)(0.5)(0.5)(0.108)(0.057)(40)

where the SD's are the ones that we calculated before the portfolio was taken into consideration.

Why is it so that on the answer sheet the variance does not include the old standard deviations? Why is there a mutation of the original formula?

Please explain in the simplest way possible... thanks!

• You have to enclose MathJax in $ signs for the formatting t take effect. – saulspatz Jun 20 '19 at 12:28 • A better place to ask this question is: stats.stackexchange.com - By the way, what is FC and MC? – NoChance Jun 20 '19 at 12:43 • The names of the companies. – Sara Saletti Jun 20 '19 at 12:44 • I'm not well enough versed in this field to give you a good answer (I trust that RRL's is correct), but I do note this: when calculating the mean return, you wisely did not just sum up the 3 yearly returns and divide by$3\$. You knew that rate of return does not work that way. It is multiplicative, not additive. But when you calculated the variances, you calculated them exactly as one would an additive value. – Paul Sinclair Jun 21 '19 at 0:52

$$2\cdot 0.5 \cdot 0.5 \cdot 0.655\cdot 10.8 \cdot 5.7 = 2 w_1w_2\rho_{12}\sigma_1\sigma_2 = 2w_1w_2cov(R_1,R_2) = 2 \cdot 0.5 \cdot0.5\cdot 40\\ = 20$$