Arbias Hashani

Consider the following motivating question:

A person wants to invest some of their wealth into something. How do they know what they will get back?

There are several thoughts to consider:

• How much wealth do they have?
• Why do they want to invest?
• Depending on what the “something” is, should the expectation change?
• How can we be certain we get back what we expect?

To answer this question, we use utility theory and stochastic dominance.

Utility Theory

Suppose you spend some cash on ice cream on a hot day. You eat the ice cream and enjoy the cold taste. This is the utility of your action.

Formally, utility is the satisfaction or welfare gained by an economic agent from the consumption of a good or an investment opportunity.

Some people prefer chocolate ice cream to vanilla ice cream. How can we distinguish different utilities? By assigning a numerical value to a utility, which measures the level of utility derived from a given level of wealth. This is the utility value.

Whether or not we include the raspberries may give a very different utility value…

What about any value of wealth ? Say we have some function of such that this function assigns utility values to given wealth . This is a utility function.

Here are some thoughts that arise:

• We are using wealth, not (necessarily) money. Then the utility gained from a wealth is not measured in money (such as £, $, €) but as a function of wealth .
• I bought ice cream because I enjoy the taste. I expect to enjoy the taste. Investors make decisions based on their expectations. With certainty I expect the ice cream not to taste like spaghetti and cheese. This is a probability: a measure of what is possible in our situation. Buying vanilla ice cream and getting spaghetti and cheese ice cream should have a low probability of happening (although it may be high if are in an ice cream shop that sells the wrong flavours on purpose). So what we expect can be explained by the following theorem.

Expected Utility Theorem

An investor makes decisions based on maximising the expected utility (so I want the best possible taste from the ice cream) under his beliefs about the probability of different outcomes.

We now know how investors think. They want to maximise their expectation of satisfaction. What if two ice creams give the same expected utility value? Then the two ice creams are indifferent.

Stochastic Dominance

The motivating question can be seen as how an investor makes a decision about which assets to purchase when returns are random (we will drop the ice cream language for now). Using utility theory means we have to know the investor’s utility function.

Suppose this function is not completely specified. Then how can we make decisions regarding which assets to pick? By using stochastic dominance. As we have lost some information (by not knowing the utility function) we have to know some other information.

To do this, instead of looking at wealth, we look at the possible returns.

Suppose we have some assets and for each asset there is information on the possible returns. There is also a probability distribution for each asset – the chance of getting each return. If the probability of getting the highest possible return is one (which can be restated as “almost exactly”) then (potentially) this is a great asset to pick – it is not random and we know what we get.

Some thoughts arise:

• How do we know what the returns are? When paying rent, taking loans, mortgages and so on, the returns are specifically stated. But this is not always the case, it is possible for the returns to change depending on how many people invest in that return or for some other reason.
• How do know the probability distribution of each asset? In reality we do not – we estimate it (and how do we do this?).

We know that if you have the choice between having to pay for some same object or receiving it for free, you would receive it or free. Stochastic dominance puts this talk through probability. If (by some manner) the probability of all returns of an asset are lower than the probability of all returns (respectively) of an asset then clearly asset should be favoured: we have a higher chance of getting something back on the returns. The asset is stochastically dominant to asset .

This informal (I have not defined many things properly) post gives us a reasonable answer to our question. This area is known as financial economics and is crucial to the modern world. Consider the following connections:

• Statistics is used to estimate the probability distributions of the returns.
• Economics is used to explain what investors want and how they behave.
• Psychology is used to understand which assets an investor would pick – the risky ones or ones with a safe return. This is important for insurance – companies would like to know what their clients are like.
• Mathematics is the language that formulates all of this – a quick look at this stochastic dominance page shows how all of this can be explained by mathematics.

This is what actuarial science is. It is very basic – yes, but it gives an introduction to how modern day financial problems are tackled.