Arbias Hashani

Suppose you are at a football (soccer) match between teams and . You are interested in a gamble – it is assumed you want to earn money in the easiest way whilst adhering to the law), so you are offered a chance to gamble on the situation .

It is known that team is better than team with the additional information that team is very volatile and team is very consistent: stick to the same level most of the time where as can bounce up and down.

Which team would you gamble on to win the match?

What do you expect in return for your gamble?

Both choices poise a risk – is better so you should pick , where as is more consistent and is volatile, so you should pick .

Given an action, the possible outcome being undesirable is precisely the risk of the situation.

What we expect in return for our investment (say ) is the expected return . This is known as the mean of situation .

It is important as this value is what we get in return – a mean that is not desirable implies no gamble hence no investment. The mean is used as a measure of seeing success in an investment in this case.

Assume we have expected returns for team as and for team as .

We know that team is far more likely to achieve the expected return . As is more volatile, it is hard to trust (or perhaps even accept) .

So how do we determine how likely each team is to matching their expected return?

By looking at the variance of . This looks at how far we expect to be away from the expectation as a form of a least square.

Then when we have the variance of each team, clearly the team with the lower variance is more desirable as they are more likely to achieve their expected return, which we would like to be as high as possible.

We translate these problems into mathematics by use of probability theory, to allow the chances of each situation happening and defining the specific functions for expected returns and variance.

We use statistics to find estimates of what we think the expected return and variance are, as they are usually unknown.

We use utility theory to understand the behaviour an investor should exhibit when going through these choices – if he sees two teams who give the same return, so supposing that they are equally as good as each other, then assuming no further information (such as one team being more consistent than the other), he should beat on either team without any worry, as they give the same return.

The behaviour should be consistent. The behaviour is explained by economics and is translated into mathematics by probability theory and statistics.

This is the kind of what actuaries do. Everything explained above can be seen as actuarial science. It is a simple example, but it shows why mean and variance are important and the motivations behind actuarial science.