Pricing in General Insurance

239

17

Aggregate Loss Modelling

In Chapter 6, we introduced the scientic basis for the stochastic modelling of risk and

weexplored the implications of the two main types of aggregate loss models, the individ-

ual risk model and the collective risk model. We then devoted several chapters (Chapters

7 through 16) to creating the building blocks for these aggregate loss models: a frequency

model and a severity model.

Based on the frequency and the severity models that we have built, we already know

how to calculate some simple statistics for the aggregate loss model: for example, the

mean of the aggregate model is simply the mean claim count times the mean loss amount.

However, to make sensible risk-management decisions, we need to know the full aggre-

gate loss distribution so that we can answer questions such as, ‘What is the amount the

company is going to lose in a very bad year, for example, with a probability of 1%?’

This problem is very difcult to solve exactly through analytical formulae. In this chap-

ter, we will explain why this is the case, and how it is possible to get around the issue by

using approximate solutions, the most famous of which is the Monte Carlo simulation.

Figure17.1 puts this step into the context of the risk costing process.

17.1 ‘Exact’ Solution for the Collective Risk Model

Calculating the full distribution of the aggregate loss amount S is a tough problem because,

except in a few cases, it cannot be solved analytically. The cumulative distribution function

of S, F

S

(s), can be expressed as follows:

F

S

(s) = Pr(X

1

+ … + X

N

≤ s) (17.1)

There are two difculties here: rst of all, we have a sum of random variables of which

we know the distribution F

X

(x), and the distribution of the sum of random variables is

a complicated function of the individual distribution, technically called a convolution.

Second, the number N over which the sum must go is itself a random variable, and we

need to take into account all possible values of N in the calculation.

Let us start by addressing the second problem rst: because the different values of N

correspond to mutually exclusive cases, we can write the probability as a weighted sum of

the probability of mutually exclusive events:

Fs XXsNnX

Xs

SN n

n

() Pr()Pr()Pr

()

=+…+ ≤= =+…+ ≤

=

11

0

∞∞

∑

(17.2)

240 Pricing in General Insurance

Now the problem has been reduced to calculating convolutions of a xed number of vari-

ables, and Equation 17.2 can be written as

Fs NnFs

SX

n

n

() ()Pr()

()

==

=

∞

∑

*

0

(17.3)

where (F

X

(s))*

n

is dened recursively as follows (see, for example, Klugman et al. 2008):

• (F

X

(s))*

0

= H(s), where H(s) is the Heaviside step function, dened as 1 for s ≥ 0 and

0 elsewhere

•

((

)) (( ))

()

Fs Fs

sdFs

X

n

X

n

X

**

=−

′′

−

−∞

∞

∫

1

, which can be written as

() (() (Fs Fs

X

n

X

*

=−

∞

∫

0

′′

′

−

sfsds

n

X

)) ()*

1

when the severity distribution is continuous and with zero prob-

ability for negative losses*

*

Note that in the continuous case (F

X

(s))

*n

is dened as the convolution of (F

X

(s))

*n−1

with f

X

(s) rather than with

F

X

(s), so we should not speak of (F

X

(s))

*n

as the n-fold convolution of F

X

(s) as it is sometimes done.

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 mode

l

regate distribution

e. g. Monte Carlo simulation, Fast Fourier

transform, Panjer recursion…

Gross aggregate loss model

Allocate losses between (re)insurer and

(re)insured

Cover

data

Estimate gross agg

Ceded/retained aggregate loss model

FIGURE 17.1

The Monte Carlo simulation (or one of the other methods outlined in this chapter) allows us to produce an

aggregate loss model based on a frequency and severity model; it also allows us to calculate the distribution of

ceded and retained losses based on the policy coverage.