Why Risk Is Not Variance: An Expository Note

Variance (or standard deviation) of return is widely used as a measure of risk in financial investment risk analysis applications, where mean‐variance analysis is applied to calculate efficient frontiers and undominated portfolios. Why, then, do health, safety, and environmental (HS&E) and reliability engineering risk analysts insist on defining risk more flexibly, as being determined by probabilities and consequences, rather than simply by variances? This note suggests an answer by providing a simple proof that mean‐variance decision making violates the principle that a rational decisionmaker should prefer higher to lower probabilities of receiving a fixed gain, all else being equal. Indeed, simply hypothesizing a continuous increasing indifference curve for mean‐variance combinations at the origin is enough to imply that a decisionmaker must find unacceptable some prospects that offer a positive probability of gain and zero probability of loss. Unlike some previous analyses of limitations of variance as a risk metric, this expository note uses only simple mathematics and does not require the additional framework of von Neumann Morgenstern utility theory.