 Brief introduction to risk theory

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### Risk theory

Risk theory is essentially a branch of probability theory, devoted to decision-making under probabilistic uncertainty. Basic concepts of the theory are: risk, risk measure, risk price, and individual attitude to risk.

The following picture presents a simplified scheme of decision-making. Here S is a set on environment states, D is a set of possible decisions, R is a set of achievable results. Result is influenced by both decision and environment state. Thus, a mathematical model of the object just described is a mapping M: S x D --> R, that for an environment state s and decision d calculates the result r = M(s,d).

Environment state is usually uncertain. In the framework of risk theory the uncertainty is described by a probabilistic model, that is, by a probability distribution on S. Together with the mapping M this distribution for each decision d from D induces a distribution on R. Thus for each decision there is a probability distribution on R, so making the best decision mean choosing the "best" distribution on R among those available.

### Example

Consider the following simple example. A picnic might take place indoors or outdoors (so the set of possible decisions D consists of two elements I and O). Outdoors picnic would be much better unless the weather is rainy. Let environment may fall into one of two states: thunderstorm or dry clean weather (denote them T and C). We do not know in advance if the weather is good, but our experience and weather forecast may allow describing this uncertainty in probabilistic terms, say the pobability of thunderstorm may be P(T) = 0.3, thus the probability of dry weather is equal to P(C) = 0.7. Now let the results set R consist of four elements "excellent", "good", "bad" and "awful" (denote them E, G, B, A), and the mapping M is described as follows:

• M(T,O)=A,
• M(T,I)=B,
• M(C,O)=E,
• M(C,I)=G.

Now the decision to make outdoors picnic induces the following probability distribution on the results set:

 Value Awful Bad Good Excellent Probability 0.3 0 0 0.7

Indoors decision would induce the following distribution:

 Value Awful Bad Good Excellent Probability 0 0.3 0.7 0

Making optimal decision means choosing the better from the two probability distributions. A well known approach consists of assigning an "utility" U(r) to each result r, calculating expected (mean) utility u of result of each decision with subsequent choice of decision that leads to greater expected utility. Now let us assign utilities as described in the following table:

 value Utility Awful 0 Bad 2 Good 5 Excellent 10

Calculating expected utilities brings us

• u(I) = 0.3 * 2 + 0.7 * 5 = 4.1,
• u(O) = 0.3 * 0 + 0.7 * 10 = 7.

Thus expected utility of outdoors picnic in greater than that of indoors picnic; so we are heading to forest.

Let us look if things change when probability distribution changes. Let P(T) = 0.8 and P(C) = 0.2. Calculating expected utilities leads to

• u(I) = 0.8 * 2 + 0.2 * 5 = 2.6,
• u(O) = 0.8 * 0 + 0.2 * 10 = 2,

so now we would better stay indoors.

In the scheme just described decision is influenced not only by distribution on S , but by prescribed utility values as well.

### Illustration

The following picture shows how utilities of decisions depend on the probability of thunderstorm. Here one can find an Excel spreadsheet, that was used to create the above picture. Other parameters values may be set there as well.

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