Question:

Consider a three-asset portfolio consisting of three stocks called A, B, and C. The stocks are weighted, respectively, 20%, 30% and 50% (i.e., weight A = 20%, weight B = 30%, and weight C = 50%). The volatility (standard deviation) of returns for all three assets is 20%. The covariances are as follows: COVAR(A,B) = 0.09 = 9%, COVAR(A,C) = 0.16 = 16%, COVAR(B,C) = 0.01 = 1%. Using matrix math, calculate the three-asset portfolio variance.

Answer:

This is likely a harder question than you would encounter on the exam (not for conceptual difficulty but rather time required to perform the calculation). It is basic matrix application.

Recall that if w is the vector of portfolio weights and Sigma is the covariance matrix, we have a very COMPACT FORMULA for PORTFOLIO VARIANCE:

portfolio variance = (w’)(Sigma)(w)
where w’ is row vector of weights, having been transposed.

In this case, w’ is given by:
[20% 30% 50%]

And the COVARIANCE MATRIX is 3 x 3 and given by:

| 0.04 0.09 0.16 |
| 0.09 0.04 0.01 |
| 0.16 0.01 0.04 |

Note the diagonal of the covariance matrix contains the variances (0.04) which are squared volatilities (0.04 = 20%^2).

The portfolio variance is the product of:
(w’)(Sigma)(w)

Note that matrix multiplication happens to be ‘associative’ so that the order of multiplication does not matter: (AB)C = A(BC).

The worked out answer can viewed in this editgrid spreadsheet.

For reflection:
* The diagonal of the covariance matrix consists of variances. Why? The covariance of a thing with itself is its variance.

* Where is the correlation matrix? It is implicit in the covariance matrix. By the way, what elements are in the diagonal of a correlation matrix? Ones (1). Why? Because the correlation of a thing with itself is 1.0.

rocky420,

the diagonal contains covar(A,A), covar(B,B) and covar(C,C) which = Variance(A), Variance(B) and Variance(C). Since 20% is volatiltiy for each, this is 20%^2 = 4% along the diagonal. Covar(a,b) = 0.09 so that value is found in two elements: 0.09 = [1,2] and [2,1]. David

I am not getting the same numbers that you get when you multiply the matrix.

I looked up some matrix math and it looks like our solution is multipleing the W’s more than once.

What am I missing?