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## Uncertainty and risk

February 17, 2012

Often the choices we make lead to uncertain outcomes. When we know or can estimate the likelihood of each possible outcome, we can classify the risk of the outcome occurring. When we have know way of knowing or estimating the likelihood, we are just left with uncertainty.

When we know all the possible outcomes that could occur, then the probabilities of these outcomes will sum to 1.

We use the estimated value and variance to illustrate the expected payoff and amount of risk involved with a project, if we are able to assess value to the different outcomes concerned. This is easiest when we can assess it in financial terms.

Where there are two outcomes, A and B, the estimated value is $EV = Probability(A) Value (A) + Probability(B) Value(B)$. This shows us the expected payoff.

The variance is $Var = Probability(A) [Value (A) - EV]^2 + Probability(B) [Value (B) - EV]^2$. This shows us the degree of risk, the higher the variance, the more the risk.

For instance lets say you are offered a bet, and the choice is either to take the bet or not take the bet. The bet costs £100 to make, and involves tossing a coin. If the coin comes up heads, you lose your £100. If it comes up tails, you get £250.

The estimated value of taking the bet is $0.5(-100) + 0.5(250) = 75$
The variance of taking the bet is $0.5(-100-75)^2 + 0.5(250-75)^2 = 30625$

The estimated value of not taking the bet is $1(0) = 0$
The variance of taking the bet is $1(0-0)^2 = 0$

So here we have a higher estimated value for taking the bet than not taking the bet. That means your expected payoff is better for taking the bet. But there is a large variance involved with taking the best, that means you are taking a lot of risk in taking the bet.

Whether or not you take the bet depends on your aversion to risk.

Someone that is risk-neutral does not care either way about risk, and will accept the bet if it has a higher expected value than not accepting the bet, so here a risk-neutral person would accept the best.

Someone that is risk-averse does not like risk, and will not accept a ‘fair bet’ (a bet with an expected value of 0) or a bet with a higher expected value, if there is risk involved. So here a risk-averse person would not accept the bet. Most people are by nature risk averse to some extent.

At the other end of the spectrum some people are risk-loving and will actively seek risk, so they would accept a fair bet or even some unfair bets, if there was risk involved.