Risk is not the variance
The finance literature often uses the term ‘risk’ for the variance or spread (standard deviation) of the distribution of the rates of return of investments. This would be an improper use of the term. Suppose that one has a very profitable venture without the possibility of a loss. Suppose that the rate of return of this venture has a large variance, from mildly profitable to highly profitable. Is this a risky venture ? No, not in the usual understanding of the term.
Risk is not the negative of expected revenue
In mathematical statistics, some authors, like Ferguson (1967), define ‘risk’ as ‘expected loss’. However, it appears that they actually regard ‘loss’ as the negative of total returns (i.e. - revenue), so the definition used is -(p profit + (1-p) (-loss)), which is the negated expected value. This use of the term ‘risk’ is inappropriate. My proposal is to use the word “due” to stand for the negative of expected value, so that the standard statistical decision theory (with the game against nature) can be described as minimising due.
Note on Bernstein’s “Against the gods”
I came across Bernstein (1996) “Against the gods”, and found it equally entertaining as his “Capital Ideas”. One comment is that Bernstein indeed emphasises Knight’s and Keynes’s statements on “uncertainty”. My answer to that is, again, that unknown probabilities or even unknown categories indeed are serious cases of uncertainty, so that earlier writers on the subject were right in emphasising that seriousness. However, we should not be tempted to reserve the word “uncertainty” to only those cases. So with all due respect to Knight and Keynes, the definitions provided here are the proper ones.
Note on Wilson & Crouch (2001)
Wilson & Crouch (2001), “Risk-benefit analysis”, adopt the same definition of “risk” as discussed here. I saw this only after the first edition of this book. Since professor Wilson has been teaching on the subject for decades and his book only collects his teaching material I apparently only rediscovered what was already clear to him. Perhaps my presentation is a bit clearer since I use the formal E[.] notation. This chapter remains useful since it clarifies the confusions from the other definitions. Where risk is the product of probability and severity, this book also benefits from the emphasis on this definition, since, where I started to develop this argument after the Fall of the Berlin Wall in 1989, we have to deal with a future where there are huge dangers: though with only a small probability but on balance a relevant risk.