Does risk equal expected value?

Over a series of posts (not consecutive because I realize that not everyone finds risk an intriguing subject) I’m going to sample Misconceptions of Risk by Terje Aven (Wiley, 2010). As traders and investors we’re always told to manage risk (often mistakenly viewed as potential negative consequences; see “Risk management and profit targets”). But how many of us know what we are supposed to manage, let alone how to manage it? Aven analyzes nineteen common views of risk, exposing their strengths, weaknesses, and limitations. My sampling will be much more modest. Let’s start this short series with the first view, that risk equals expected value.

Expected value, as calculated in probability theory, is the sum of each possible outcome with its associated probability. For example, the expected value, or average outcome, of rolling a die some reasonably large number of times would be 3.5 (1 x 1/6 + 2 x 1/6 + 3 x 1/6 + 4 x 1/6 + 5 x 1/6 + 6 x 1/6). Or, one could recast expected value to measure the probability of success or failure; the probability of rolling a 2 is 1/6 and of not rolling it is 5/6.

But does the very simple concept of expected value give us the information we need to make a decision? Aven looks at a financial form of Russian roulette: if the revolver discharges you lose $24 million, if it doesn’t you win $6 million. By the same math as above the expected gain is $1 million (-24 x 1/6 + 6 x 5/6). Sounds like a good deal, doesn’t it? Unfortunately, if you know only the expected gain you don’t understand that there is a 1 in 6 possibility of losing $24 million.

And what happens to the notion of expected value when we apply it outside the world of gambling with its well-controlled probability distributions? Aven shifts to a portfolio perspective, looking at 100 projects (we could easily substitute trades for projects) each with the same risk profile as the Russian roulette example. Assuming that the projects are independent of one another, we can invoke the central limit theorem to calculate the probability distribution of the average gain and plot a Gaussian probability curve. Ah, what a beautifully ordered statistical world!

But there are three major problems with this simple way of viewing and measuring risk—outliers that dominate calculated averages, dependency (that is, are the trades really independent of one another?), and the uncertainty that the future will resemble the past. In brief, once we leave the Las Vegas strip risk becomes a much thornier issue.