Pricing in General Insurance

221

16

Severity Modelling: Selecting and

Calibrating a Severity Model

The objective of severity analysis is the production of a severity model (Figure 16.1) for

the loss amounts. A severity model is not only essential to calculate – in combination with

a frequency model– the distribution of aggregate losses but also will help us to reply to

questions suchas

• What is the probability that an individual loss is going to be as big as £100,000?

• What level of deductible is adequate for this risk?

• What limit should I purchase for my public liability policy?

All the questions above are not questions about the total losses but about the distribution

of individual loss amounts.

Actuaries have of course always been concerned with the severity of individual loss

amounts, but in traditional practice, both in reserving and pricing, they have often been

happy with considering the average loss amount over a period (see Chapter 12). However,

average loss amount analysis, although a useful tool for exploratory analysis and in con-

texts where the number of claims is very large (such as private motor), is not sufcient to

reply to questions like those listed above.

16.1 Input to Severity Modelling

The input to severity analysis is the database of loss amounts:

• After revaluation (Chapter 10)

• After currency conversion (Chapter 10)

• After correction for IBNER (Chapter 15)

An example of such an input in graphical format is shown in Figure 16.2. The losses

represented there are those of a real-world data set. We have made no distinction here

between open and closed losses.

222 Pricing in General Insurance

Individual

loss data

Assumptions on

– Loss inflation

– Cu

rrency conversion

– …

Exposure

data

Portfolio/market

information

Data preparation

– Data checking

– Data cleansing

– Data transformation

– Claims revaluation and currency conversion

– Data summarisation

– Calculation of simple statistics

Inputs to frequency/severity analysis

Adjust historical claim counts for IBNR

Adjust for exposure/profile changes

Select severity distribution and

calibrate parameters

Select frequency distribution and

calibrate parameters

Adjust loss amounts for IBNER

Severity model

Frequency model

Estimate gross aggregate distribution

e.g. Monte Carlo simulation, Fast Fourier

transform, Panjer recursion…

Gross aggregate loss model

Ceded/retained aggregate loss model

Allocate losses between (re)insurer and

(re)insured

Cover

data

FIGURE 16.1

How the selection of a severity model ts into the overall pricing process.

100

1000

10,000

100,000

1,000,000

10,000,000

0.01%

0.10%1.00%10.00%100.00%

Claim amount (€)

Percentage of losses that exceeds a given amount

CDF of Losses (log X vs. log (1 – p))

FIGURE 16.2

A graphical representation of the loss data base (or data set). Every dot in this graph represents a loss. For each

loss, the value of the x-axis represents the percentage of losses in the database that exceed the value of that loss.

For example, roughly 10% of losses are above £10,000. The graph is in log-log scale, in the sense that although

the values shown are the original ones, the amount represented is actually log X against log (1 − p), where p is

the percentage of losses exceeding a given amount. This log-log representation has many advantages: it allows

showing losses over many orders of magnitude, and the distribution is a straight line if and only if the losses

come from a single-parameter Pareto distribution, which is useful for analysing the behaviour of the tail.