Section: 1 What is Risk?

Sub Section: 1 Variance

Uncertainty surrounding the investment is a function of the magnitude of the possible surprises. To summarize risk with a single number we first define the variance as the expected value of the squared deviation from the mean.

Var(r)  =  σ2   =   Σs p(s) [r(s) – E(r)]2


We first square the deviations because if we did not, negative deviations would offset positive deviations, with the result that the expected deviation from the mean return would necessarily be zero. Squared deviations are necessarily positive. Another result of squaring deviations is that variance has a dimension of percent squared. To give the measure of risk the same dimension as expected return, we use the standard deviation, which is explained, in the next section.


State of the Economy

Scenario, s


HPR (%), r(s)





Normal Growth









Using the data in the table above, we find the variance of expected return as follows. First, we take the difference between the HPR in each scenario and the mean return, then we square that difference, and finally we multiply by the probability of each scenario to find the average of the squared deviations. The result is:


σ2  =  0.25(44 – 14)2 + 0.5(14 – 14)2 + 0.25(–16 – 14)2  =  450