Pricing in General Insurance

357

22

Liability Rating Using Increased Limit Factor Curves

We have seen how exposure curves may help pricing property business. With suitable

adjustments, the same idea can be used to price liability business. The method is based on

the use of the so-called increased limit factor (ILF) reference curves (Miccoli 1977; Palmer

2006; Riegel 2008). The key idea behind ILF curves is that they should help you determine

how the expected losses increase by increasing the limit purchased, and ultimately that

helps you determine the cost of reinsuring a specic layer of reinsurance.

22.1 How ILF Curves Arise

The main difference between liability and property in the construction of reference curves

is that for liability, there is no natural upper limit to the size of the loss such as the sum

insured or the maximum possible loss (MPL) for property. True, liability is usually pur-

chased up to a given limit L, but this limit does not provide in itself information on the

potential loss severity: you can buy £5M of cover for employers’ liability but still get a

£50M claim.

One of the consequences of this difference is that the ILF curves – the reference curves

for liability business – inevitably deal with absolute monetary amounts and not relative

damages – and unless the ILF curve has a very specic shape that is invariant at different

scales (such as a single-parameter Pareto), then the shape of the curve will depend on the

basis value, that is, the lowest amount for which the curve is provided.

22.1.1 Assumptions Underlying ILF Curves

There are two main assumptions behind ILF curves (Palmer 2006):

• The ground-up loss frequency is independent of the limit purchased

• The ground-up severity is independent of the number of losses and of the limit

purchased

These two assumptions are quite reasonable although not unquestionable: the second

one is basically one of the assumptions of the collective risk model, and we have already

discussed in Chapter 6 how this can easily be breached. As usual, they are quite use-

ful hypotheses that should be abandoned only if someone had good empirical evidence

that they are not true – and a more sophisticated model to replace that created on these

assumptions.

If the assumptions above are satised, the shape of an ILF curve depends only on the

underlying severity distribution – in other words, there is a 1:1 correspondence between

severity curves and ILF curves, as was the case for property exposure curves.

358 Pricing in General Insurance

22.1.2 Definition of ILF Curves

Suppose that we know the expected losses to a liability risk with a policy limit b and that

we want to know the expected losses to a policy with a limit u. Using the familiar results

on the expected losses to a layer of insurance (in this case, a ground-up layer with limit u)

from Chapter 20, we can then write

ES EXuEN

EXu

EXb

E

u

() (min(,)) ()

(min(,))

((,))

=×

=×

min

((min(,)) ()

() ()

Xb EN

uES

bb

×

=×ILF

(22.1)

where

ILF

min

b

u

EX

u

EX

b

() 

(min(,))

((,))

=

(22.2)

As we did in Chapter 20, we ask: would it not be good if we could have ILF

b

(u) already

tabulated for us, so that we could simply plug that information into Equation 22.1 and nd

the expected losses for any limit we want? This is what casualty rating with ILF curves is

all about. Indeed, ILF curves are often provided as tables that give the value of the ILF as a

function of a monetary amount (in a specied currency, and at a given point in time), as in

Figure 22.1. An example of an ILF curve in graphical format is given in Figure 22.2.

x ILF(x)

100,000 1.00

200,000 1.37

300,000 1.58

400,000 1.72

500,000 1.83

600,000 1.92

700,000 1.99

800,000 2.05

900,000 2.10

1,000,000 2.15

1,100,000 2.19

1,200,000 2.23

1,300,000 2.26

1,400,000 2.30

1,500,000 2.33

1,600,000 2.35

FIGURE 22.1

Example of ILF curve in tabular format. Note that the currency unit of x must be specied along with the context

to which this curve can be applied and the point in time at which this curve was valid.