**5. Implementation recommendations**

As stated in the introduction to this chapter, Venter's method has been implemented by major international banks and approved by the local regulator. Based on this experience, we can share the following implementation guidelines:


i.e. ~*q*<sup>7</sup> ≤~*q*<sup>20</sup> ≤ ~*q*100. Here the fraction *ϵ* expresses the size or extent of the possible deviations (or mistakes) inherent in the scenario assessments. If *ϵ* ¼ 0 then the assessments are completely correct (within the simulation context) and the experts are in effect oracles. In practice, choosing *ϵ*>0 is more realistic, but how large the choice should be is not clear and we therefore vary *ϵ* over a range of values. 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 ½ � 1 � *ϵ*, 1 þ *ϵ* implies that on average they have the value 1, i.e. the scenario assessments are about unbiased. This may not be realistic and other choices are possible, e.g. we could mimic a pessimistic scenario maker by taking the perturbations to be distributed on the interval 1, 1 ½ � þ *ϵ* and an optimistic scenario maker by taking

*Linear and Non-Linear Financial Econometrics - Theory and Practice*

For each combination of parameters of the assumed true underlying Poisson frequency and Burr severity distributions and for each choice of the perturbation

i. Use the VaR approximation algorithm in the second section to determine the 99.9% VaR for the Burr Type XII with the current choice of parameters. Note that the value obtained here approximately equals the true 99.9% VaR. We refer to this value as the approximately true (AT) VaR.

ii. Generate a data set of historical losses, i.e. generate *K* � *Poi*ð Þ 7*λ* and then generate *x*1, *x*2, … , *xK* � *iid* Burr Type XII with the current choice of parameters. Here the family *F x*ð Þ , θ is chosen as the Burr Type XII but it is refitted to the generated historical data to estimate the parameters as

iii. Add to the historical losses three scenarios ~*q*7, ~*q*20, ~*q*<sup>100</sup> generated by the quantile perturbation scheme explained above. Estimate the 99.9% VaR

iv. Using the historical losses and the three scenarios of item iii), calculate the severity distribution estimate *H*~ and apply Venter's approach to estimate

v. Repeat items i–iv 1000 times and then summarise and compare the

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 construct 90% VaR bands as before for the 1000

given in **Figure 5**. Note that light grey represents the GPD band and dark grey the

For small frequencies (*λ*≤10) the GPD approach outperforms the Venter approach, except for short tailed severity distributions and higher quantile perturbations. When the annual frequency is high (*λ*≥50) and for moderate to high quantile perturbations (*ϵ*≥0*:*2Þ the Venter approach is superior, and more so for higher *λ* and *ϵ*. Even for small quantile perturbations (*ϵ* ¼ 0*:*1) and high annual frequencies (*λ*≥50) the Venter approach performs reasonable when compared to

Venter band, whilst the overlap between the two bands are even darker.

MedAT � 1, *VaR*ð Þ <sup>951</sup>

h i

MedAT � 1

. The results are

them on the interval 1½ � � *ϵ*, 1 .

required.

using the GPD approach.

resulting VaR estimates.

repeated GPD and Venter VaR estimates, i.e. *VaR*ð Þ <sup>51</sup>

From **Figure 5**, we make the following observations:

the 99.9% VaR.

the GPD.

**26**

size parameter *ϵ* the following steps are followed:

iv. The loss data may be fitted by a wide class of severity distributions. We used SAS PROC SEVERITY in order to identify the five best fitting distributions.

**7. Conclusion**

*DOI: http://dx.doi.org/10.5772/intechopen.93722*

**A. Appendix A**

*qb* <sup>þ</sup> *<sup>σ</sup>* ð Þ <sup>1</sup>�*<sup>γ</sup>* �*<sup>ξ</sup>* ð Þ �<sup>1</sup> *ξ*

**29**

**A.2 The Burr distribution**

The GPD given by

In this chapter, we motivated the use of Venter's approach whereby the severity distribution may be estimated using historical data and experts'scenario assessments jointly. The way in which historical data and scenario assessments are integrated incorporates measures of agreement between these data sources, which can be used to evaluate the quality of both. This method has been implemented by major international banks and we included guidelines for its practical implementation. As far as future research is concerned, we are investigating the effectiveness of using the ratios in assisting the experts with their assessments. Also, we are testing

*Construction of Forward-Looking Distributions Using Limited Historical Data and Scenario…*

the effect of replacing *q*<sup>100</sup> with *q*<sup>50</sup> in the assessment process.

� � <sup>¼</sup> <sup>1</sup> � <sup>1</sup> <sup>þ</sup> *<sup>ξ</sup>*

8 < :

with *x*≥ *qb*, thus taking *qb* as the so-called EVT threshold and with *σ* and *ξ* respectively scale and shape parameters. Note the Extreme Value Index (EVI) of the GPD distribution is given by *EVI* ¼ *ξ* and that heavy-tailed distributions have a positive EVI and larger EVI implies heavier tails. This follows (also) from the fact that for positive EVI the GPD distribution belongs to the Pareto-type class of distributions, having a distribution function of the form 1 � *F x*ð Þ¼ *<sup>x</sup>*�1*=ξℓF*ð Þ *<sup>x</sup>* , with *ℓF*ð Þ *x* a slowly varying function at infinity (see e.g. Embrechts et al., 1997). For Pareto-type, when the EVI > 1, the expected value does not exist, and when EVI > 0.5, the variance is infinite. Note also that the GPD distribution is regularly varying with index �1*=ξ* and therefore belongs to the class of sub-exponential distributions. Note that the *<sup>γ</sup>*-th quantile of the GPD is *<sup>q</sup>*ð Þ¼ *<sup>γ</sup> GPD*�<sup>1</sup> *<sup>γ</sup>*, *<sup>σ</sup>*, *<sup>ξ</sup>*, *qb*

*<sup>σ</sup> x* � *qb* � � � � �<sup>1</sup>

> *σ* � � *<sup>ξ</sup>* <sup>¼</sup> 0,

<sup>1</sup> � exp � *<sup>x</sup>* � *qb*

*<sup>ξ</sup> ξ*> 0

� � <sup>¼</sup> *qb* � *<sup>σ</sup>* ln 1ð Þ � *<sup>γ</sup>* when <sup>¼</sup> 0.

, for *x*> 0 (11)

(10)

� � <sup>¼</sup>

**A.1 The generalised Pareto distribution (GPD)**

*GPD x*; *σ*, *ξ*, *qb*

� � when *<sup>ξ</sup>* 6¼ 0 and *GPD*�<sup>1</sup> *<sup>γ</sup>*, *<sup>σ</sup>*, *<sup>ξ</sup>*, *qb*

The three parameter Burr type XII distribution function

*B x*ð Þ¼ ; *<sup>η</sup>*, *<sup>τ</sup>*, *<sup>α</sup>* <sup>1</sup> � <sup>1</sup> <sup>þ</sup> ð Þ *<sup>x</sup>=<sup>η</sup> <sup>τ</sup>* ð Þ�*<sup>α</sup>*

with parameters *η*, *τ*, *α* >0 (see e.g. [10]). Here *η* is a scale parameter and *τ* and *α* shape parameters. Note the EVI of the Burr distribution is given by *EVI* ¼ *ζ* ¼ 1*=τα* and that heavy-tailed distributions have a positive EVI and larger EVI implies heavier tails. This follows (also) from the fact that for positive EVI the Burr distribution belongs to the Pareto-type class of distributions, having a distribution function of the form 1 � *F x*ð Þ¼ *<sup>x</sup>*�1*=<sup>ζ</sup>ℓF*ð Þ *<sup>x</sup>* , with *<sup>ℓ</sup>F*ð Þ *<sup>x</sup>* a slowly varying function at infinity (see e.g. [9]). For Pareto-type, when the EVI > 1, the expected value does

