Page 166 - Contributed Paper Session (CPS) - Volume 7

CPS2051 Mentje G. et al.
3. Result: GPD and Venter model comparison
We conducted a simulation study to investigate the effect of the two
approaches by perturbing the quantiles of the true underlying severity
distributions. We assumed a different extreme value index (EVI) for the true
underlying severity distributions and then perturbed the quantiles in the
following way. For each simulation run, we chose three perturbation
factors , , and 100 independently and uniformly distributed over the
7
20
20
interval [1 − , 1 + ] and then tentatively took = , 20 = and
20 20
7 7
7
= but truncated these so that the final values are increasing, i.e.
100 100 100
≤ 20 ≤ 100 . Here the fraction expresses the size or extent of the
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possible deviations (or mistakes) inherent in the scenario assessments. If =
0 then the assessments are completely correct (within the simulation context)
and the scenario makers are in effect oracles. We chose the values 0, 0.1, 0.2,
0.3 and 0.4 for this purpose in the results below. Choosing the perturbation
factors to be uniformly distributed over the interval [1 − , 1 + ] implies
that on average they have the value 1, i.e. the scenario assessments are about
unbiased.
We assumed a Burr distribution as our true underlying severity distribution.
For each combination of parameters of the assumed true underlying Poisson
frequency and Burr severity distributions and for each choice of the
perturbation size parameter , the following steps were followed:
(a) The VaR approximation algorithm in Section 2.1 was used to determine
the 99.9% VaR. Note that the value obtained here approximately equals the
true 99.9% VaR; the only approximation involved is that it is based on 1 million
repetitions rather than infinitely many. We refer to this value as the
approximately true (AT) VaR.
(b) We generated a data set of historical losses, i.e. generate ~(7)
and then generated , , … , ~ Burr Type XII with choice of parameters.
1

2
Here the family (, ) is chosen as the Burr Type XII but it is refitted to the
generated historical data to estimate the parameters as required.
(c) We added to the historical losses three scenarios , , 100 generated
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20
by the quantile perturbation scheme explained above. We then estimated the
99.9% VaR using the GPD approach.
(d) We used the historical losses and the three scenarios of item (c),
calculated the severity distribution estimate ̃ and applied Venter’s approach
to estimate the 99.9% VaR.
(e) We repeated items (a)-(d) 1000 times and then summarised and
compared the resulting VaR estimates.
Because we are generally dealing with positively skewed data here, we
shall use the median as the principal summary measure. Denote the median
of the 1000 AT values by MedAT. Then we constructed 90% VaR bands for the

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