When the Expected Value Is Not Expected: A Conservative Approach

The takeoff point in this paper is a random variable X for which large positive values are desired. When its probability distribution is highly skewed, the possibility of a long fat-tailed distribution can lead to an expected value, μ = E[X], which is unduly optimistic. For the reverse case when small values of X are desired, the ideas in this paper are applied to -X. This issue of over-optimism in the expected value is particularly important when a mission-critical random variable is involved. For example, when considering earthquake intensity or flood levels, reporting an understated expected value to a technically unsophisticated general public would be considered by many as highly undesirable. The Conservative Expected Value (CEV) of X, denoted by CEV (X) is a new definition provided in this paper. It is a metric which we argue is particularly useful when risk aversion must be highly emphasized. At the same time, while being conservative, the CEV is also aimed at avoiding reflection of excessive pessimism. When, then CEV (X), while being conservative, is defined in such a way so as not to be unduly pessimistic. In classical analysis, enhancement of a calculation often includes the variance σ 2 = var(X). However, when large X-values are desired, this may further distort one's overview of the risk at hand; e.g., if X is profit, values above the mean should not be penalized. With these one-sidedness considerations in mind, we work with a new reward-risk pair, the CEV and the so-called Conservative Semi-Variance (CSV) of X, denoted by CSV(X). Whereas the CEV definition is entirely new, the CSV definition is motivated by ideas in the area of finance. This paper also illustrates calculation of the CEV and CSV pair for a number of classical probability distributions and includes description of a number of properties of these metrics which suggest that this new theory is mathematically rich. Finally, we demonstrate the potential for application of the theory via two numerical examples.