Relative risk is defined as r(t) = t - E[x < t] for some target level t. Risk (or absolute risk) takes t = 0, and relative risk would allow for a different target level. [113]
An interesting application is when x is a stochastic rate of return and r the certain rate, so that there is relative risk r(r) = r - E[x < r]. This relative risk answers the question: What is the probable loss with respect to a target return of r ? Here, r - r(r) = E[x < r] gives the weight of underperformance in the total target return (which weight has to be compensated by probable profits to achieve the target).
Conditional (relative) risk is defined as k(t) = t - E[x | x < t] for some target level t. With respect to rates of return, conditional risk k(r) answers the question: What would one expect to lose with respect to r, if earnings actually underperform and fall below r. Indeed, r - k(r) would give your expected return when actually underperforming.
Conditional risk is related to relative risk by the property that E[x | x < t] = E[x < t] / Pr[x < t]. The probable loss thus is corrected for the probability of the loss. Or, the probability measure in the expectation is corrected so that a density is taken that sums to 1. [114]
Example
In everyday parlance, profit and loss are nonnegative concepts. For example, if the difference between revenue and costs is $-10, then your loss is $10. It is only in mathematical economics that profits are defined as a general profit function such that ‘negative profits’ are possible. To understand risk, we however return to the everyday parlance convention.
Let us have a prospect that can give profit with probability p, and loss with probability 1 - p. We denote this as Prospect[profit, -loss, p]. We call profit * p ‘probable profit’ and loss * (1 - p) ‘probable loss’. Then the following definitions apply:
· Expected Value =
