A picture of complex risks and their profiles is more useful than knowing expected value is positive or not

Risk profile

A risk profile is a graph showing value - usually expressed in financial terms - and probability.  Looking at the profile of a risk can give a more sophisticated view of it than expected value alone. 

http://spreadsheets.google.com/pub?key=tHk2EpsXiBSmmV6BILRm7IA&output=html


Let's consider a third version of the dice game - version C.  As before, throwing different numbers brings different outcomes.  But in this version, there is the possibility of a severe downside.  Thowing 5 or 6 wins $10; throwing 2, 3, or 4 wins $5; throwing 1 incurs a $10 penalty.

The different outcomes and probabilities are shown in the table above, along with the calculation of expected value for this game.  As before, expected value is calculated by adding together the products of impact and probability for all possible outcomes. 

At first glance, this game looks like the best so far - its expected value is far higher than that of either version A or version B.   ( http://spreadsheets.google.com/pub?key=te9MzyHoIN6EyuoHmfDxMaw&output=html)

But what about the potential downside?  With $5 in our pocket to play with, we could easily incur a debt that we can't pay, and have to declare ourselves bankrupt.  With $20 to play with, we would be a bit safer (the wealth effect). 

The key to this decision is the profile of the risk.  (see the diagram of the risk profile for dice game version C above).    Each vertical black coloured bar represents a possible outcome.  Its position denotes its impact (negative to the left, positive to the right); its height denotes its probability.  The positive side of the graph looks promising, with high probabilities for positive outcomes.  But over on the left, we see the possibility of a serious negative outcome - a potentially fatal downside.  The risk may have an unacceptable profile for us, despite its positive expected value.

More complex risk profiles bring in more and more possible outcomes and probabilities.  They build up a picture of complex risks and their profiles that is more useful than the simple question of whether the expected value is positive or not.

Histograms plot value against probability density, to give a continuous version of the risk's profile.  They are created through advanced risk anlalysis involving techniques such as Monte Carlo simulations, where a large number of probabilities is used to create the risk profile.