Links to all tutorial articles (same as those on the Exam pages)Combining expected values and variances
When constructing portfolios we are often concerned with the return (ie the mean, or expected value), and the risk (ie the volatility, or standard deviation) of of combining positions or portfolios. We may also be faced with situations where we need to know the risk and return if position sizes were to be scaled up or down in a linear way. This brief article deals with how mean and variances for two different variables can be combined together, and how they react to being added or multiplied by constants.
If we have a series of two variables A and B with means (or expected value) E(A) and E(B), the expected value of the variable A + B is simply E(A) + E(B). Combining expected values This is fairly intuitive. Now the same logic can be applied if either A or B were to multiplied with a constant, say ‘c’. E(c*A) = c*E(A), and also E(c*A + B) = E(c*A) + E(B) = c*E(A) + E(B) Combining variances
Now if we were to calculate the standard deviation of the summation of two variables, then the relationship would be derived as follows:
Recall that correlation and covariance are related and are almost interchangeable using the following relationship:
What happens to variances when the variable is multiplied by a constant? In other words StDev(c*A) = c*StDev(A) and V(c*A) = c^2*V(A). The reason for this is simple to see. Variance is calculated as
Getting rid of some of the redundant notation (just for the ease of my typing), Another interesting thing to note is V(A + B) = V(A – B) assuming independence between A and B. This is because V(A – B) = V(A + (1)*B) = V(A) + (1)^2V(B) = V(A) + V(B) = V(A + B).
When variables are multiplied together, the relationships that hold are as follows: E(A*B) = E(A) * E(B) + Covar(A,B)
Summary:
