Page 163 - Contributed Paper Session (CPS) - Volume 7

CPS2051 Mentje G. et al.
As illustration of the complexity of the experts’ task, take = 50 then =
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−1 (0.99714) , 20 = −1 (0.999) and 100 = −1 (0.9998) which implies that the
quantiles that have to be estimated are very extreme.

2.3 Estimating VaR
In the previous section we discussed a way of modelling the true severity
distribution . The estimation of the 99.9% VaR of the aggregate loss
distribution is of interest and de Jongh et al. (2015) discuss three approaches
to estimate it, namely the naïve approach, the generalized Pareto distribution
(GPD) approach and Venter’s approach. The naïve approach make use of
historical data only, whereas the GPD approach (which is based on a mixed
model formulation) and Venter’s approach make use of both historical data
and scenario assessments. Below we focus our discussion on the GPD and
Venter approaches and in Section 3 we demonstrate that, as far as estimating
VaR is concerned, that Venter’s approach is preferred.
2.4 The GPD approach
Select a number and let be the corresponding quantile given by (1),

1
i.e., = −1 (1 − ). We use as a threshold that splice in such a way that

the interval below is the expected part and the interval above the

unexpected part of the severity distribution. Define two distribution functions
() = ()/( ) for ≤ and

() = [() − ( )]/[1 − ( )] for > ,

i.e. () is the conditional distribution function of a random loss ~ given
that ≤ and () is the conditional distribution function given that >

. Note that we then have the identity

() = T( ) () + [1 − ( )] () for all . (2)

This identity represents () as a mixture of the two conditional
distributions. Instead of modelling () with a class of distributions (, ) we
may now consider modelling () with (, ) and () , with (, ) .

Borrowing from extreme value theory, a popular choice for (, θ) could be

the GPD, while a host of choices are available for (, θ), the obvious being

the empirical distribution.
Suppose we have available years of historical loss data , … , and

1
2,
scenario assessments ̃ , ̃ and ̃ 100 . Then the annual frequency can be
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20
̂
estimated as = ⁄ . Next and the threshold must be specified. One

possibility is to take as the smallest of the scenario -year multiples and to
estimate as the corresponding smallest of the scenario assessments ̃

provided by the scenario makers, in this case ̃ . () can be estimated by
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fitting a parametric family (, θ) (such as the Burr) to the data , , … ,
1
2

or by calculating the empirical distribution and then conditioning it to the
̃
interval (0, ̃ ]. We denote it by (). For the sake of future notational

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