The remaining principles of modern quantitative risk management concern risk outside the VaR limit. How do you exploit risk for profit when you don't know the possible outcomes or probabilities? You already know the answer, but to see it you have to go back to Pascal's wager. There are two tricky issues that Pascal finessed.

The first concerns the units in which we measure things. Early probability theorists assumed that all gambles should be evaluated by their expected values, and they unhesitatingly accused anyone who turned down a positive expected value wager of irrationality. *Expected value* has nothing to do with what you expect; it's a mathematical term that means multiply all outcomes by their probabilities and add them up. For example, if you flip a fair coin with which heads you win $100 and tails you pay $50, your expected value is 0.5 × $100 + 0.5 × (−$50) = $50 − $25 = $25.

After a few decades, it dawned on some people that it's not necessarily wise to accept a coin flip with which heads you triple your wealth and tails you lose everything. This has a positive expected value equal to half your wealth, but for most people the pain of losing everything would exceed the pleasure of tripling their wealth.

The fix for this was something called *utility theory*. It was assumed that there was some “utility” you could assign to any outcome, and gambles should be evaluated on the basis of expected utility. If you assign +1 utility to your current state, ...

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