= p profit + (1 - p) (-loss) = probable profit - probable loss
· Risk = risk value = expected value of the risks = probable loss = (1 - p) loss
· Risk Ratio = Risk / (ExpectedValue + Risk) = (1 - p) loss / (p profit)
· Thus: Expected Value = p profit (1 - Risk Ratio)
· Risk Probability = cumulative probability of all losses (in this case 1-p)
Risk is the (absolute value of the) down side of a bet. A venture is judged to be risky if the probable loss is large. Note that this notion still is somewhat vague. A probable loss can be large because of the probability or because of the sum of money involved. This vagueness is unfortunate, in some respects, but here is little to be done about it, since this vagueness is inherent in working with probabilities. In fact, this vagueness is an essentially positive aspect of working with probabilities. For, when we have different prospects, then we can order and evaluate them on risk, neglecting differences in losses and probabilities.
Colignatus (1999, 1999a) further develops these notions for simple binary prospects, multidimensional prospects, joint prospects, and continuous probability densities. An interesting application is the ‘Markowitz efficiency frontier’, but now with risk rather than the spread.
Wrong use in economics 1921-2005
The above definitions are proper in the sense that they conform to every day parlance and the definitions provided by Hornby’s dictionary op. cit.. The definitions provided here however differ from the use within the economics literature. First there are the definitions of Knight (1921) that have been adopted widely in economics, as for example in The New Palgrave (1998:III:358). Or it has become custom in finance to associate risk with the standard deviation. And some mathematical statisticians use another concept of risk. Let us discuss these in turn.