Pricing in General Insurance

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25

Credibility Theory

The experience-based calculation of the ‘risk premium’ (basically, the pure cost of risk

per unit of exposure, without any allowance for expenses or prot) for an insurance policy

is affected by several sources of uncertainty, the most obvious – and perhaps the best

understood – of which is the limited size of the historical database of losses of the client.

Other important uncertainties include data uncertainty and model uncertainty.

To make up for such uncertainties, the analyst may use average, or typical, informa-

tion from the market (the market risk premium) to replace or complement the client risk

premium. The problem with this is that the market experience is not fully relevant to a

particular client. This is usually captured by the spread, or heterogeneity, of the client

risk premiums around the standard market rate. (As an added complication, although the

market rate is typically computed from a larger data set than that of a client, it, too, is based

on a loss database of limited size and is therefore affected by the same type of uncertainty.

However, we will not deal with this complication here.)

The standard way to combine client and market information is through credibility. The

credibility risk premium is a convex combination of the client risk premium and the mar-

ket risk premium:

Credibility risk premium = Z × Client risk premium + (1 − Z) × Market risk premium

where Z is a real number between 0 and 1, reecting the relative weight that we give to the

client’s experience.

25.1 Gentle Start: A Noninsurance Example

To see how credibility works, let us make a gentle start with a simple, noninsurance

example, which – because it is about salaries – may be slightly more memorable. If you are

already familiar with the concept of credibility, this section will perhaps be too gentle for

you – in which case, feel free to skip to Section 25.2.

As you can imagine, the average income of people differs widely depending on loca-

tion. As an example, the town of Greenwich in Connecticut is graced with the highest

income per head in the United States, if not in the world. Something similar is claimed by

Luxembourg City, which is part of the Grand Duchy of Luxembourg in Europe.

Having read about Greenwich, Connecticut, and Luxembourg City, you begin won-

dering how your small town in the South of England, Crickhollow, fares against these

renowned places in terms of average salary. You therefore decide to do a little survey in the

weekend by asking passers-by in the High Street to write their salary on a piece of paper

and put it anonymously in a black bag. The Crickhollowians turn out to be not as keen to

share this type of information as you had hoped, whether anonymously or not, and you

only collect 30 responses.