**4. Estimating VaR**

choosing *q*<sup>7</sup> since this loss level has been reached once in 7 years. If the experts judge that the future will be better than the past, they may want to provide a lower assessment for *q*<sup>7</sup> than the largest loss experienced so far. If they foresee deterioration, they may judge that a higher assessment is more appropriate. The other choices of *c* are selected in order to obtain a scenario spread within the range that one can expect reasonable improvement in accuracy from the experts' inputs. Of course, the choice of *c* ¼ 100 may be questionable because judgements on a 1-in-100 years loss level are likely to fall outside many of the experts' experience. In the banking environment, they may also take additional guidance from external data of similar banks which in effect amplifies the number of years for which historical data are available. It is argued that this is an essential input into scenario analysis [12]. Of course requiring that the other banks are similar to the bank in question may be a difficult issue and the scaling of external data in an effort to make it comparable to the bank's own internal data raises further problems (see e.g. [15]). We will not dwell on this issue here and henceforth assume that we do have the 1-in-*c* years scenario assessments for a range of c-values, but have to keep in mind that subjective elements may have affected the reliability of the assessments.

If the annual loss frequency is *Poi*ð Þ*λ* distributed and the true underlying severity distribution is *T*, and if the experts are of oracle quality in the sense of actually

To see this, let *Nc* denote the number of loss events experienced in *c* years and let *Mc* denote the number of these that are actually greater than *qc:* Then *Nc* � *Poi c*ð Þ*λ* and the conditional distribution of *Mc* given *Nc* is binomial with parameters *Nc* and

with *<sup>X</sup>* � *<sup>T</sup>* and *pc* <sup>¼</sup> *T qc*

As illustration of the complexity of the experts' task, take *λ* ¼ 50 then *q*<sup>7</sup> ¼

ð Þ <sup>1</sup> � *<sup>γ</sup>* <sup>≈</sup>*T*�<sup>1</sup>

which implies *c* ¼ 1000, we could ask the oracle the question 'What loss level *q*1000is expected to be exceeded once every 1000 years?'. The oracle will then produce an answer that can be used directly as an approximation for the 99.9% VaR of the aggregate loss distribution. Of course, the experts we are dealing with are not of

In the light of the above arguments one has to take in consideration: (a) the SLA gives only an approximation to the VaR we are trying to estimate, and (b) experts are very unlikely to have the experience or the information at their disposal to assess a 1-in-1000 year event reliably. One can realistically only expect them to assess

Returning to the oracle's answer in (4), the expert has to consider both the true severity distribution and the annual frequency when an assessment is provided. In order to simplify the task of the expert, consider the mixed model in (3) discussed in the previous section. This model will assist us in formulating an easier question for the expert to answer. Note that the oracle's answer to the question in the

<sup>¼</sup> <sup>1</sup> � <sup>1</sup>

the annual frequency. However using the definition of *Tu* and taking *q* ¼ *qb*, *b*<*c*; it

<sup>¼</sup> *<sup>c</sup><sup>λ</sup>* <sup>1</sup> � *T qc*

ð Þ <sup>0</sup>*:*<sup>999</sup> and *<sup>q</sup>*<sup>100</sup> <sup>¼</sup> *<sup>T</sup>*�<sup>1</sup>

*cλ* 

*:* (4)

*cλ*

ð Þ 0*:*9998 which implies that the

ð Þ 1 � *γ=λ* , and by taking *γ* ¼ 0*:*001,

*<sup>c</sup><sup>λ</sup>* (from (4)) and therefore depends on

*<sup>c</sup>* which does not depend on the annual frequency. This

. Therefore

<sup>¼</sup> <sup>1</sup> � <sup>1</sup>

<sup>Þ</sup> . Requiring that *EMc* <sup>¼</sup> 1,

*qc* <sup>¼</sup> *<sup>T</sup>*�<sup>1</sup> <sup>1</sup> � <sup>1</sup>

knowing *λ* and *T*, then the assessments provided should be

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

1 � *pc* ¼ *P* X≥*qc*

yields (4).

oracle quality.

follows that *Tu qc*

**18**

*T*�<sup>1</sup>

<sup>¼</sup> <sup>1</sup> � *T qc*

Returning to the SLA i.e. *CoP*�<sup>1</sup>

previous setting can be stated as *T qc*

<sup>¼</sup> <sup>1</sup> � *<sup>b</sup>*

quantiles that have to be estimated are very extreme.

events occurring more frequently such as once in 30 years.

*EMc* ¼ *EEMc* ð Þ j*Nc* ½ �¼ *E Nc* 1 � *pc*

ð Þ <sup>0</sup>*:*<sup>99714</sup> , *<sup>q</sup>*<sup>20</sup> <sup>¼</sup> *<sup>T</sup>*�<sup>1</sup>

Suppose we have available *a* years of historical loss data *x*1, *x*2, … , *xK* and scenario assessments ~*q*7, ~*q*<sup>20</sup> and ~*q*<sup>100</sup> provided by the experts. In the previous sections two modelling options have been suggested for modelling the true severity distribution *T* and a third will follow below. The estimation of the 99.9% VaR of the aggregate loss distribution is of interest and we will consider three approaches to estimate it, namely the naïve approach, the GPD approach and Venter's approach. The naïve approach will make use of historical data only, the GPD approach (which is based on the mixed model formulation) and Venter's approach will make use of both historical data and scenario assessments. Below we demonstrate that, as far as estimating VaR is concerned, that Venter's approach is preferred to the GPD and naïve approaches.

#### **4.1 Naïve approach**

Assume that we have available only historical data and that we collected the loss severities of a total of *K* loss events spread over *a* years and denote these observed or historical losses by *<sup>x</sup>*1, … , *xK*. Then the annual frequency is estimated by ^*<sup>λ</sup>* <sup>¼</sup> *<sup>K</sup>=a*. Let *F x*ð Þ ; θ denote a suitable family of distributions to model the true loss severity distribution *T*. The fitted distribution is denoted by *F x*; ^*θ* � �, with ^*θ* denoting the (maximum likelihood) estimate of the parameter(s) *θ:* In order to estimate VaR a small adjustment of the Monte Carlo approximation approach, discussed earlier, is necessary.

#### *4.1.1 Naïve VaR estimation algorithm*


#### *4.1.2 Remarks*

The estimation of VaR using the above-mentioned naïve approach has been discussed in several books and papers (see e.g. [11]). [16] stated that heavy-tailed data sets are hard to model and require much caution when interpreting the resulting VaR estimates. For example, a single extreme loss can cause drastic changes in the estimate of the means and variances of severity distributions even if a large amount of loss data is available. Annual aggregate losses will typically be driven by the value of the most extreme losses and the high quantiles of the aggregate annual loss distribution are primarily determined by the high quantiles of the severity distributions containing the extreme losses. Two different severity distributions for modelling the individual losses may both fit the data well in terms of goodness-of-fit statistics yet may provide capital estimates which may differ by billions. Certain deficiencies of the naïve estimation approach, in particular, the estimation of the severity distribution and the subsequent estimation of an extreme VaR of the aggregate loss distribution, are highlighted in [15].

function. This gives us more accentuated view of the tail of the distribution. Then in the bottom panel the Monte Carlo results of the VaR approximations are given by means of a box plot using the 5% and 95% percentiles for the box. As before, one million simulations were used to approximate VaR and the VaR calculations were repeated a 1000 times. In **Figure 2(b)** we assume *λ* ¼ 10, *a* ¼ 10 and generated 100 observations from the T\_Burr(1, 0.6, 2) distribution. The observations generated is plotted in the top panel and in the middle panel the fitted distribution and the maximum likelihood estimates of the parameters are depicted as F\_Burr(1.07, 0.56, 2.2). In the bottom panel the results of the VaR estimates using the naïve approach is provided. Note how the distribution of the VaR estimates differ from those obtained using the true underlying severity distribution. Of course, sampling error is present, and the generation of another sample will result in a different box plot. Let us illustrate this by studying the effect of extreme observations. In order to do this, we moved the maximum value further into the tail of the distribution and repeat the fitting process. The data set is depicted in the top panel of **Figure 2(c)** and the fitted distribution in the middle as F\_Burr(1.01, 0.52, 2.26). Again, the

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

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

resulting VaR estimates are shown in the bottom panel. In this case the

estimates.

**4.2 The GPD approach**

then conditioning it to the interval 0, ~*qb*

Next, *Fu*ð Þ *x* can be modelled by the *GPD x*; *σ*, *ξ*, *qb*

by ~*q*7. Substituting these scenario assessments into *Fu qc*

of the scenario assessments.

*c* ¼ 20, 100 yields two equations.

*Fu* <sup>~</sup>*q*<sup>20</sup> <sup>¼</sup> *GPD* <sup>~</sup>*q*20; *<sup>σ</sup>*, *<sup>ξ</sup>*, <sup>~</sup>*q*<sup>7</sup>

~*q*20�~*q*<sup>7</sup>

exists only if <sup>~</sup>*q*100�~*q*<sup>7</sup>

their assessments.

**21**

introduction of the extreme loss has a profound boosting effect on the resulting VaR

This modelling approach is based on the mixed model formulation (3). As before, we have available *a* years of historical loss data *x*1, *x*2, … , *xK* and scenario assessments ~*q*7, ~*q*<sup>20</sup> and ~*q*100. Then the annual frequency *λ* can again be estimated as ^*<sup>λ</sup>* <sup>¼</sup> *<sup>K</sup>=a*. Next *<sup>b</sup>* and the threshold *<sup>q</sup>* <sup>¼</sup> *qb* must be specified. One possibility is to take *b* as the smallest of the scenario *c*-year multiples and to estimate *qb* as the corresponding smallest of the scenario assessments ~*qb* provided by the experts, in this case ~*q*7. *Te*ð Þ *x* can be estimated by fitting a parametric family *Fe*ð Þ *x*, *θ* (such as the Burr) to the data *x*1, *x*2, … , *xK* or by calculating the empirical distribution and

choice especially if *K* is large and the parametric family is well chosen. Whichever estimate we use, denote it by *<sup>F</sup>*~*e*ð Þ *<sup>x</sup>* . For the sake of future notational consistency, we shall also put tildes on all estimates of distribution functions which involve use

A for the characteristics of this distribution. For ease of explanation, suppose we have actual scenario assessments ~*q*7, ~*q*<sup>20</sup> and ~*q*<sup>100</sup> and thus take *b* ¼ 7 and estimate *qb*

<sup>¼</sup> <sup>0</sup>*:*<sup>65</sup> and *Fu* <sup>~</sup>*q*<sup>100</sup> <sup>¼</sup> *GPD* <sup>~</sup>*q*100; *<sup>σ</sup>*, *<sup>ξ</sup>*, <sup>~</sup>*q*<sup>7</sup>

that can be solved to obtain estimates *σ*~ and ~*ξ* of the parameters *σ* and *ξ* in the GPD that are based on the scenario assessments. Some algebra shows that a solution

>2*:*533. This fact should be borne in mind when the experts do

. Either of these estimates is a reasonable

<sup>¼</sup> <sup>1</sup> � *<sup>b</sup>*

distribution. See Appendix

*c*

; with *b* ¼ 7,

<sup>¼</sup> <sup>0</sup>*:*<sup>93</sup> (5)

In practice, and due to imprecise loss definitions, risk managers may incorrectly group two losses into one extreme loss that has a profound boosting effect on VaR estimates. In the light of this, it is important that the manager is aware of the process generating the data and the importance of clear definitions of loss events.

In **Figure 2** below we used the naïve approach to illustrate the effect of some of the above-mentioned claims. In **Figure 2(a)** we assumed a Burr distribution, i.e. T\_Burr(1, 0.6, 2), as our true underlying severity distribution. In the top panel we show the distribution function and in the middle the log of 1 minus the distribution

#### **Figure 2.**

*Illustration of the effects of VaR estimation using the naïve approach. (a) True Burr distribution,T\_Burr(1, 0.6, 2), (b) simulated observations from the T\_Burr(1, 0.6, 2) distribution with fitted distribution F\_Burr (1.07, 0.56, 2.2), (c) augmented simulated observations with fitted distribution F\_Burr(1.01, 0.52, 2.26).*

*Construction of Forward-Looking Distributions Using Limited Historical Data and Scenario… DOI: http://dx.doi.org/10.5772/intechopen.93722*

function. This gives us more accentuated view of the tail of the distribution. Then in the bottom panel the Monte Carlo results of the VaR approximations are given by means of a box plot using the 5% and 95% percentiles for the box. As before, one million simulations were used to approximate VaR and the VaR calculations were repeated a 1000 times. In **Figure 2(b)** we assume *λ* ¼ 10, *a* ¼ 10 and generated 100 observations from the T\_Burr(1, 0.6, 2) distribution. The observations generated is plotted in the top panel and in the middle panel the fitted distribution and the maximum likelihood estimates of the parameters are depicted as F\_Burr(1.07, 0.56, 2.2). In the bottom panel the results of the VaR estimates using the naïve approach is provided. Note how the distribution of the VaR estimates differ from those obtained using the true underlying severity distribution. Of course, sampling error is present, and the generation of another sample will result in a different box plot. Let us illustrate this by studying the effect of extreme observations. In order to do this, we moved the maximum value further into the tail of the distribution and repeat the fitting process. The data set is depicted in the top panel of **Figure 2(c)** and the fitted distribution in the middle as F\_Burr(1.01, 0.52, 2.26). Again, the resulting VaR estimates are shown in the bottom panel. In this case the introduction of the extreme loss has a profound boosting effect on the resulting VaR estimates.

In practice, and due to imprecise loss definitions, risk managers may incorrectly group two losses into one extreme loss that has a profound boosting effect on VaR estimates. In the light of this, it is important that the manager is aware of the process generating the data and the importance of clear definitions of loss events.

### **4.2 The GPD approach**

data sets are hard to model and require much caution when interpreting the resulting VaR estimates. For example, a single extreme loss can cause drastic changes in the estimate of the means and variances of severity distributions even if a large amount of loss data is available. Annual aggregate losses will typically be driven by the value of the most extreme losses and the high quantiles of the

VaR of the aggregate loss distribution, are highlighted in [15].

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

**Figure 2.**

**20**

aggregate annual loss distribution are primarily determined by the high quantiles of the severity distributions containing the extreme losses. Two different severity distributions for modelling the individual losses may both fit the data well in terms of goodness-of-fit statistics yet may provide capital estimates which may differ by billions. Certain deficiencies of the naïve estimation approach, in particular, the estimation of the severity distribution and the subsequent estimation of an extreme

In **Figure 2** below we used the naïve approach to illustrate the effect of some of the above-mentioned claims. In **Figure 2(a)** we assumed a Burr distribution, i.e. T\_Burr(1, 0.6, 2), as our true underlying severity distribution. In the top panel we show the distribution function and in the middle the log of 1 minus the distribution

*Illustration of the effects of VaR estimation using the naïve approach. (a) True Burr distribution,T\_Burr(1, 0.6, 2), (b) simulated observations from the T\_Burr(1, 0.6, 2) distribution with fitted distribution F\_Burr (1.07, 0.56, 2.2), (c) augmented simulated observations with fitted distribution F\_Burr(1.01, 0.52, 2.26).*

This modelling approach is based on the mixed model formulation (3). As before, we have available *a* years of historical loss data *x*1, *x*2, … , *xK* and scenario assessments ~*q*7, ~*q*<sup>20</sup> and ~*q*100. Then the annual frequency *λ* can again be estimated as ^*<sup>λ</sup>* <sup>¼</sup> *<sup>K</sup>=a*. Next *<sup>b</sup>* and the threshold *<sup>q</sup>* <sup>¼</sup> *qb* must be specified. One possibility is to take *b* as the smallest of the scenario *c*-year multiples and to estimate *qb* as the corresponding smallest of the scenario assessments ~*qb* provided by the experts, in this case ~*q*7. *Te*ð Þ *x* can be estimated by fitting a parametric family *Fe*ð Þ *x*, *θ* (such as the Burr) to the data *x*1, *x*2, … , *xK* or by calculating the empirical distribution and then conditioning it to the interval 0, ~*qb* . Either of these estimates is a reasonable choice especially if *K* is large and the parametric family is well chosen. Whichever estimate we use, denote it by *<sup>F</sup>*~*e*ð Þ *<sup>x</sup>* . For the sake of future notational consistency, we shall also put tildes on all estimates of distribution functions which involve use of the scenario assessments.

Next, *Fu*ð Þ *x* can be modelled by the *GPD x*; *σ*, *ξ*, *qb* distribution. See Appendix A for the characteristics of this distribution. For ease of explanation, suppose we have actual scenario assessments ~*q*7, ~*q*<sup>20</sup> and ~*q*<sup>100</sup> and thus take *b* ¼ 7 and estimate *qb* by ~*q*7. Substituting these scenario assessments into *Fu qc* <sup>¼</sup> <sup>1</sup> � *<sup>b</sup> c* ; with *b* ¼ 7, *c* ¼ 20, 100 yields two equations.

$$F\_{\mathfrak{u}}\left(\check{q}\_{20}\right) = \text{GPD}\left(\check{q}\_{20}; \sigma, \xi, \check{q}\_{7}\right) = 0.6\mathfrak{S} \text{ and } F\_{\mathfrak{u}}\left(\check{q}\_{100}\right) = \text{GPD}\left(\check{q}\_{100}; \sigma, \xi, \check{q}\_{7}\right) = 0.93 \quad \text{(5)}$$

that can be solved to obtain estimates *σ*~ and ~*ξ* of the parameters *σ* and *ξ* in the GPD that are based on the scenario assessments. Some algebra shows that a solution exists only if <sup>~</sup>*q*100�~*q*<sup>7</sup> ~*q*20�~*q*<sup>7</sup> >2*:*533. This fact should be borne in mind when the experts do their assessments.

With more than three scenario assessments, fitting techniques can be based on (5) which links the quantiles of the GPD to the scenario assessments. An example would be to minimise P *<sup>c</sup> GPD* ~*qc*; *σ*, *ξ*, ~*q*<sup>7</sup> � � � ð Þ <sup>1</sup> � *<sup>b</sup>=<sup>c</sup>* � � � �. Other possibilities include a weighted version of the sum of deviations in this expression or deviation measures comparing the GPD quantiles directly to the *qc* assessments. Whichever route we follow, we denote the final estimate of *Fu*ð Þ *<sup>x</sup>* by *<sup>F</sup>*~*u*ð Þ *<sup>x</sup>* . All these ingredients can now be substituted into (3) to yield the estimate *F x* <sup>~</sup>ð Þ of *T x*ð Þ, namely

$$
\hat{\lambda}\tilde{F}(\mathbf{x}) = \left(\hat{\lambda} - \frac{\mathbf{1}}{7}\right)\tilde{F}\_{\epsilon}(\mathbf{x}) + \frac{\mathbf{1}}{7}\tilde{F}\_{\mu}(\mathbf{x}).\tag{6}
$$

are supplied by experts and not oracles the results would differ significantly. This is

*Illustration of VaR estimates obtained from a GPD fit on the oracle quantiles. (a) True Burr distribution, T\_Burr(1, 0.6, 2), (b) fitted distribution F\_Burr(1.07, 0.56, 2.2) on simulated data, (c) fitted distribution*

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

The challenge is therefore to find a way of integrating the historical data and scenario assessments such that both sets of information are adequately utilised in the process. In particular, it would be beneficial to have measures indicating whether the experts'scenario assessments are in line with the observed historical data, and if not, to require them to produce reasons why their assessments are so different. Below we describe Venter's estimation method that will meet these aims.

A colleague, Hennie Venter suggested that, given the quantiles *q*7, *q*20, *q*100; one

� � � � *for q*<sup>7</sup> <sup>&</sup>lt;*<sup>x</sup>* <sup>≤</sup>*q*<sup>20</sup>

*<sup>c</sup><sup>λ</sup>* and it should be clear that the expressions on the right

(7)

*T q*<sup>100</sup> � � � *T q*<sup>20</sup> � � *T x*ð Þ� *T q*<sup>20</sup> � � � � *for q*<sup>20</sup> <sup>&</sup>lt; *<sup>x</sup>*≤*q*<sup>100</sup>

<sup>1</sup> � *T q*<sup>100</sup> � � *T x*ð Þ� *T q*<sup>100</sup> � � � � *for q*<sup>100</sup> <sup>&</sup>lt;*x*<sup>&</sup>lt; <sup>∞</sup>*:*

� � *T x*ð Þ *for x*<sup>≤</sup> *<sup>q</sup>*<sup>7</sup>

� � *T x*ð Þ� *T q*<sup>7</sup>

reduces to *T x*ð Þ. Also, the definition of *T x*ð Þ could easily be extended for more quantiles. Given the previous discussion we can model *T x*ð Þ by *F x*ð Þ , *θ* and estimate it by *F x*, ^*θ* � � using the historical data and maximum likelihood and estimate the

illustrated when we compare the GPD with Venter's approach.

may write the distribution function *T* as follows:

*F\_Burr(1.01, 0.52, 2.26) on augmented simulated data.*

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

*<sup>p</sup>*<sup>7</sup> <sup>þ</sup> *<sup>p</sup>*<sup>20</sup> � *<sup>p</sup>*<sup>7</sup> *T q*<sup>20</sup> � � � *T q*<sup>7</sup>

*<sup>p</sup>*<sup>20</sup> <sup>þ</sup> *<sup>p</sup>*<sup>100</sup> � *<sup>p</sup>*<sup>20</sup>

1 � *p*<sup>100</sup>

*p*7 *T q*<sup>7</sup>

8

>>>>>>>>>>><

>>>>>>>>>>>:

*p*<sup>100</sup> þ

� � <sup>¼</sup> *pc* <sup>¼</sup> <sup>1</sup> � <sup>1</sup>

**4.3 Venter's approach**

**Figure 3.**

*T x*ð Þ¼

Again*T qc*

**23**

Returning now to practical use of Eq. (6), the algorithm below summarises the integration of the historical data with the 1-in-*c* years scenarios following the MC approach.

#### *4.2.1 GPD VaR estimation algorithm*


### *4.2.2 Remarks*

When using the GPD 1-in-*c* years integration approach to model the severity distribution, we realised that the 99.9% VaR of the aggregate distribution is almost exclusively determined by the scenario assessments and their reliability greatly affects the reliability of the VaR estimate. The SLA supports this conclusion. As noted above, the SLA implies that we need to estimate *<sup>q</sup>*<sup>1000</sup> <sup>¼</sup> *<sup>T</sup>*�<sup>1</sup> <sup>1</sup> � <sup>1</sup> 1000*λ* � � and its estimate would be ^*q*<sup>1000</sup> <sup>¼</sup> *GPD*�<sup>1</sup> <sup>1</sup>� <sup>1</sup> <sup>1000</sup>^*<sup>λ</sup>* ð Þ <sup>1</sup>� <sup>1</sup>� <sup>1</sup> 7^*λ* � � , *<sup>σ</sup>*~, <sup>~</sup>*ξ*, <sup>~</sup>*qb* � �. Therefore 99.9% VaR largely depends on the GPD fitted with the scenario assessments. In **Figure 3** below we depict the VaR estimation results by fitting *F*~*<sup>e</sup>* assuming a Burr distribution and *F*~*<sup>u</sup>* assuming a GPD. The top panel in **Figure 3(a)** depicts the tail behaviour of the true severity distribution which is assumed as a Burr and denoted as T\_Burr(1,0.6,2). Using the VaR approximation technique discussed in the second section (Approximating VaR) and assuming *λ* ¼ 10, *I* ¼ 1 000 000 and 1000 repetitions, the VaR approximations are depicted in the bottom panel in the form of a box plot as before. Assuming that we were supplied with quantile assessments by the oracle we use the two samples discussed in **Figure 2** and apply the GDP approach. The results are displayed in **Figure 3(b)** and **(c)** below.

The GPD fit to the oracle quantiles produce similar box plots, which in turn is very similar to the box plot of the VaR approximations. Clearly the fitted Burr has little effect on the VaR estimates. The VaR estimates obtained through the GPD approach is clearly dominated by the oracle quantiles. Of course, if the assessments *Construction of Forward-Looking Distributions Using Limited Historical Data and Scenario… DOI: http://dx.doi.org/10.5772/intechopen.93722*

#### **Figure 3.**

With more than three scenario assessments, fitting techniques can be based on (5) which links the quantiles of the GPD to the scenario assessments. An example

�. Other possibilities include a

*<sup>F</sup>*~*u*ð Þ *<sup>x</sup> :* (6)

1000*λ* � � and its

. Therefore 99.9% VaR largely

� � � ð Þ <sup>1</sup> � *<sup>b</sup>=<sup>c</sup>* � � �

> 7 � �

Returning now to practical use of Eq. (6), the algorithm below summarises the integration of the historical data with the 1-in-*c* years scenarios following the MC

ii. Generate *<sup>X</sup>*1, … , *XNe* � iid *<sup>F</sup>*~*<sup>e</sup>* and *XNe*þ1, … , *XNe*þ*Nu* � iid *<sup>F</sup>*~*<sup>u</sup>* and calculate

estimate the 99.9% VaR by the corresponding empirical quantile of these

*<sup>F</sup>*~*e*ð Þþ *<sup>x</sup>*

1 7

7 � �;

*<sup>n</sup>*¼<sup>1</sup>*Xn* where *<sup>N</sup>* <sup>¼</sup> *Nu* <sup>þ</sup> *Ne*. Using the identity above it easily follows

weighted version of the sum of deviations in this expression or deviation measures comparing the GPD quantiles directly to the *qc* assessments. Whichever route we follow, we denote the final estimate of *Fu*ð Þ *<sup>x</sup>* by *<sup>F</sup>*~*u*ð Þ *<sup>x</sup>* . All these ingredients can now

*<sup>c</sup> GPD* ~*qc*; *σ*, *ξ*, ~*q*<sup>7</sup>

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

be substituted into (3) to yield the estimate *F x* <sup>~</sup>ð Þ of *T x*ð Þ, namely

^*λF x* <sup>~</sup>ð Þ¼ ^*<sup>λ</sup>* � <sup>1</sup>

7

� � and *Nu* � *Poi* <sup>1</sup>

that *A* is distributed as a random sum of *N* i.i.d. losses from *F*~*:*

iii. Repeat i and ii *I* times independently to obtain *Ai*, *i* ¼ 1, 2, … ,*I* and

When using the GPD 1-in-*c* years integration approach to model the severity distribution, we realised that the 99.9% VaR of the aggregate distribution is almost exclusively determined by the scenario assessments and their reliability greatly affects the reliability of the VaR estimate. The SLA supports this conclusion. As noted above, the SLA implies that we need to estimate *<sup>q</sup>*<sup>1000</sup> <sup>¼</sup> *<sup>T</sup>*�<sup>1</sup> <sup>1</sup> � <sup>1</sup>

<sup>1000</sup>^*<sup>λ</sup>* ð Þ <sup>1</sup>� <sup>1</sup>� <sup>1</sup> 7^*λ* � � , *<sup>σ</sup>*~, <sup>~</sup>*ξ*, <sup>~</sup>*qb* � �

depends on the GPD fitted with the scenario assessments. In **Figure 3** below we depict the VaR estimation results by fitting *F*~*<sup>e</sup>* assuming a Burr distribution and *F*~*<sup>u</sup>* assuming a GPD. The top panel in **Figure 3(a)** depicts the tail behaviour of the true severity distribution which is assumed as a Burr and denoted as T\_Burr(1,0.6,2). Using the VaR approximation technique discussed in the second section (Approximating VaR) and assuming *λ* ¼ 10, *I* ¼ 1 000 000 and 1000 repetitions, the VaR approximations are depicted in the bottom panel in the form of a box plot as before. Assuming that we were supplied with quantile assessments by the oracle we use the two samples discussed in **Figure 2** and apply the GDP approach. The results are

The GPD fit to the oracle quantiles produce similar box plots, which in turn is very similar to the box plot of the VaR approximations. Clearly the fitted Burr has little effect on the VaR estimates. The VaR estimates obtained through the GPD approach is clearly dominated by the oracle quantiles. Of course, if the assessments

would be to minimise P

*4.2.1 GPD VaR estimation algorithm*

*<sup>A</sup>* <sup>¼</sup> <sup>P</sup>*<sup>N</sup>*

*Ai*'s as before.

estimate would be ^*q*<sup>1000</sup> <sup>¼</sup> *GPD*�<sup>1</sup> <sup>1</sup>� <sup>1</sup>

displayed in **Figure 3(b)** and **(c)** below.

*4.2.2 Remarks*

**22**

i. Generate *Ne* � *Poi* ^*<sup>λ</sup>* � <sup>1</sup>

approach.

*Illustration of VaR estimates obtained from a GPD fit on the oracle quantiles. (a) True Burr distribution, T\_Burr(1, 0.6, 2), (b) fitted distribution F\_Burr(1.07, 0.56, 2.2) on simulated data, (c) fitted distribution F\_Burr(1.01, 0.52, 2.26) on augmented simulated data.*

are supplied by experts and not oracles the results would differ significantly. This is illustrated when we compare the GPD with Venter's approach.

The challenge is therefore to find a way of integrating the historical data and scenario assessments such that both sets of information are adequately utilised in the process. In particular, it would be beneficial to have measures indicating whether the experts'scenario assessments are in line with the observed historical data, and if not, to require them to produce reasons why their assessments are so different. Below we describe Venter's estimation method that will meet these aims.

#### **4.3 Venter's approach**

A colleague, Hennie Venter suggested that, given the quantiles *q*7, *q*20, *q*100; one may write the distribution function *T* as follows:

$$T(\mathbf{x}) = \begin{cases} \frac{p\_7}{T(q\_7)} T(\mathbf{x}) & \text{for} \quad \mathbf{x} \le q\_7\\ p\_7 + \frac{p\_{20} - p\_7}{T(q\_{20}) - T(q\_7)} \left[ T(\mathbf{x}) - T(q\_7) \right] & \text{for} \quad q\_7 < \mathbf{x} \le q\_{20}\\ p\_{20} + \frac{p\_{100} - p\_{20}}{T(q\_{100}) - T(q\_{20})} \left[ T(\mathbf{x}) - T(q\_{20}) \right] & \text{for} \quad q\_{20} < \mathbf{x} \le q\_{100}\\ p\_{100} + \frac{\mathbf{1} - p\_{100}}{\mathbf{1} - T(q\_{100})} \left[ T(\mathbf{x}) - T(q\_{100}) \right] & \text{for} \quad q\_{100} < \mathbf{x} < \mathbf{ox} \end{cases} \tag{7}$$

Again*T qc* � � <sup>¼</sup> *pc* <sup>¼</sup> <sup>1</sup> � <sup>1</sup> *<sup>c</sup><sup>λ</sup>* and it should be clear that the expressions on the right reduces to *T x*ð Þ. Also, the definition of *T x*ð Þ could easily be extended for more quantiles. Given the previous discussion we can model *T x*ð Þ by *F x*ð Þ , *θ* and estimate it by *F x*, ^*θ* � � using the historical data and maximum likelihood and estimate the

annual frequency by ^*<sup>λ</sup>* <sup>¼</sup> *<sup>K</sup>=a*. Given scenario assessments <sup>~</sup>*q*7, <sup>~</sup>*q*<sup>20</sup> and <sup>~</sup>*q*100, then *T qc* � � can be estimated by *<sup>F</sup>* <sup>~</sup>*qc*, ^*<sup>θ</sup>* � � and *pc* by *<sup>p</sup>*^*<sup>c</sup>* <sup>¼</sup> <sup>1</sup> � <sup>1</sup> *c*^*λ* . The estimated ratios are then defined by

$$R(\7) = \frac{\ddot{p}\_{\mathcal{T}}}{F(\ddot{q}\_{\mathcal{T}}; \hat{\theta})}, R(\text{7, 20}) = \frac{\ddot{p}\_{20} - \hat{p}\_{\mathcal{T}}}{F(\ddot{q}\_{20}; \hat{\theta}) - F(\ddot{q}\_{\mathcal{T}}; \hat{\theta})},$$

$$R(20, 100) = \frac{\hat{p}\_{100} - \hat{p}\_{20}}{F(\ddot{q}\_{100}; \hat{\theta}) - F(\ddot{q}\_{20}; \hat{\theta})} \text{ and } R(100) = \frac{1 - \hat{p}\_{100}}{1 - F(\ddot{q}\_{100}; \hat{\theta})} \tag{8}$$

^*q*<sup>1000</sup> <sup>¼</sup> *<sup>H</sup>*<sup>~</sup> �<sup>1</sup> <sup>1</sup> � <sup>1</sup>

case 2.

**Figure 4.**

**25**

*2 with false quantiles and correct data.*

1000^*λ* <sup>¼</sup> *<sup>H</sup>*<sup>~</sup> �<sup>1</sup>

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

we may expect the same to hold for Venter's approach.

quantiles changes positively the more loss data are supplied.

**4.4 GPD and Venter model comparison**

*<sup>p</sup>*^<sup>1000</sup> . Some algebra shows that the equation

*<sup>F</sup>* ^*q*1000; ^*<sup>θ</sup>* <sup>¼</sup> *<sup>F</sup>* <sup>~</sup>*q*100; ^*<sup>θ</sup>* <sup>þ</sup> *<sup>p</sup>*^<sup>1000</sup> � *<sup>p</sup>*^<sup>100</sup> *<sup>=</sup>R*ð Þ <sup>100</sup> needs to be solved for ^*q*1000.

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

Depending on the choice of the family of distributions *F x*ð Þ , θ , this may be easy (e.g. when we use the Burr family for which we have an explicit expression for its quantile function). This clearly shows that a combination of the historical data and scenario assessments is involved, and not exclusively the latter. In as much as the SLA provides an approximate to the actual VaR of the aggregate loss distribution,

In order to illustrate the properties of this approach we assume that the true underlying severity distribution is the Burr(1.0, 0.6, 2) as before. We then construct a 'false'severity distribution as the fitted distribution to the distorted sample depicted in **Figure 2(c)**, i.e. the Burr(1.00,0.52,2.26). We refer to the true severity distribution as Burr\_1 and the false one Burr\_2. In **Figure 4(a)** the box plots of the VaR approximations of the two distributions are given (using the same input for the MC simulations). We then illustrate the performance of the GPD and Venter approach in two cases. The first case assumes that the correct (oracle) quantiles of Burr\_1 are supplied, but that the loss data are distributed according to the false distribution Burr\_2. In the second case, the quantiles of the false severity distribution are supplied, but the loss data follows the true severity distribution. The box plots of the VaR estimates are given in **Figure 4(b)** for case 1 and **Figure 4(c)** for

The behaviour of the GPD approach is as expected and the box plots corresponds to the quantiles supplied. Clearly the quantiles and not the loss data dictates the results. On the other hand, the Venter approach is affected by both the loss data and quantiles supplied. In the example studied here it seems as if the method is more affected by the quantiles than by the data. This role of the data relative to the

In this section we conduct a simulation study to investigate the effect on the two approaches by perturbing the quantiles of the true underlying severity distributions. We assume the six parameters sets of **Table 1** as the true underlying severity distributions and then perturb the quantiles in the following way. For each simulation run, choose three perturbation factors *u*7, *u*<sup>20</sup> and *u*<sup>100</sup> independently and uniformly distributed over the interval 1½ � � *ϵ*, 1 þ *ϵ* and then take ~*q*<sup>7</sup> ¼ *u*7*q*7, ~*q*<sup>20</sup> ¼ *u*20*q*<sup>20</sup> and ~*q*<sup>100</sup> ¼ *u*100*q*<sup>100</sup> but truncate these so that the final values are increasing,

*Comparison of VaR results for the GPD and Venter approaches. (a) Naïve approach with correct (T\_Burr(1, 0.6, 2)), and false data (F\_Burr(1.01, 0.52, 2.26)), (b) Case 1 with correct quantiles and false data, (c) Case*

Notice that if our estimates were actually exactly equal to what they are estimating, these ratios would all be equal to 1. For example, we would then have *R*ð Þ¼ 7 *p*7*=T q*<sup>7</sup> � � <sup>¼</sup> 1 by (4), and similarly for the others. Our new method is to estimate the true severity distribution function *T* by an adjusted form of *F x*, ^*θ* � �, then Hennie's distribution *H*~ is defined as follows (see de Jongh et al. 2015):

$$\bar{H}(\mathbf{x}) = \begin{cases} R(7)F(\mathbf{x};\hat{\theta}) & \text{for } \mathbf{x} \le \bar{q}\_{7} \\\\ \hat{p}\_{7} + R(7,20) \left[ F(\mathbf{x};\hat{\theta}) - F(\bar{q}\_{7};\hat{\theta}) \right] & \text{for } \bar{q}\_{7} < \mathbf{x} \le \bar{q}\_{20} \\\\ \hat{p}\_{20} + R(20,100) \left[ F(\mathbf{x};\hat{\theta}) - F(\bar{q}\_{20};\hat{\theta}) \right] & \text{for } \bar{q}\_{20} < \mathbf{x} \le \bar{q}\_{100} \\\\ \hat{p}\_{100} + R(100) \left[ F(\mathbf{x};\hat{\theta}) - F(\bar{q}\_{100};\hat{\theta}) \right] & \text{for } \bar{q}\_{100} < \mathbf{x} < \infty. \end{cases} \tag{9}$$

Notice again that this estimate is consistent in the sense that it actually reduces to *T* if all estimators are exactly equal to what they are estimating.

Also note that *<sup>H</sup>*<sup>~</sup> <sup>~</sup>*q*<sup>7</sup> � � <sup>¼</sup> *<sup>p</sup>*^7, *<sup>H</sup>*<sup>~</sup> <sup>~</sup>*q*<sup>20</sup> � � <sup>¼</sup> *<sup>p</sup>*^<sup>20</sup> and *<sup>H</sup>*<sup>~</sup> <sup>~</sup>*q*<sup>100</sup> � � <sup>¼</sup> *<sup>p</sup>*^100, i.e. the equivalents of *T qc* � � <sup>¼</sup> *pc* hold for the scenario assessments when estimates are substituted for the true unknowns. Hence at the estimation level the scenario assessments are consistent with the probability requirements expressed. Thus, this new estimated severity distribution estimate *H*~ 'believes' the scenario quantile information, but follows the distribution fitted on the historical data to the left of, within and to the right of the scenario intervals. The ratios *R*ð Þ7 , *R*ð Þ 7, 20 , *R*ð Þ 20, 100 and *R*ð Þ 100 in (9) can be viewed as measures of agreement between the historical data and the scenario assessments and could be useful for assessing their validities and qualities. The steps required to estimate VaR using this method are as follows:

#### *4.3.1 Venter's VaR estimation algorithm*


#### *4.3.2 Remarks*

The SLA again sheds some light on this method. As noted above the SLA implies that we need to estimate *<sup>q</sup>*<sup>1000</sup> <sup>¼</sup> *<sup>T</sup>*�<sup>1</sup> <sup>1</sup> � <sup>1</sup> 1000*λ* � � and its estimate would be *Construction of Forward-Looking Distributions Using Limited Historical Data and Scenario… DOI: http://dx.doi.org/10.5772/intechopen.93722*

^*q*<sup>1000</sup> <sup>¼</sup> *<sup>H</sup>*<sup>~</sup> �<sup>1</sup> <sup>1</sup> � <sup>1</sup> 1000^*λ* <sup>¼</sup> *<sup>H</sup>*<sup>~</sup> �<sup>1</sup> *<sup>p</sup>*^<sup>1000</sup> . Some algebra shows that the equation *<sup>F</sup>* ^*q*1000; ^*<sup>θ</sup>* <sup>¼</sup> *<sup>F</sup>* <sup>~</sup>*q*100; ^*<sup>θ</sup>* <sup>þ</sup> *<sup>p</sup>*^<sup>1000</sup> � *<sup>p</sup>*^<sup>100</sup> *<sup>=</sup>R*ð Þ <sup>100</sup> needs to be solved for ^*q*1000. Depending on the choice of the family of distributions *F x*ð Þ , θ , this may be easy (e.g. when we use the Burr family for which we have an explicit expression for its quantile function). This clearly shows that a combination of the historical data and scenario assessments is involved, and not exclusively the latter. In as much as the SLA provides an approximate to the actual VaR of the aggregate loss distribution, we may expect the same to hold for Venter's approach.

In order to illustrate the properties of this approach we assume that the true underlying severity distribution is the Burr(1.0, 0.6, 2) as before. We then construct a 'false'severity distribution as the fitted distribution to the distorted sample depicted in **Figure 2(c)**, i.e. the Burr(1.00,0.52,2.26). We refer to the true severity distribution as Burr\_1 and the false one Burr\_2. In **Figure 4(a)** the box plots of the VaR approximations of the two distributions are given (using the same input for the MC simulations). We then illustrate the performance of the GPD and Venter approach in two cases. The first case assumes that the correct (oracle) quantiles of Burr\_1 are supplied, but that the loss data are distributed according to the false distribution Burr\_2. In the second case, the quantiles of the false severity distribution are supplied, but the loss data follows the true severity distribution. The box plots of the VaR estimates are given in **Figure 4(b)** for case 1 and **Figure 4(c)** for case 2.

The behaviour of the GPD approach is as expected and the box plots corresponds to the quantiles supplied. Clearly the quantiles and not the loss data dictates the results. On the other hand, the Venter approach is affected by both the loss data and quantiles supplied. In the example studied here it seems as if the method is more affected by the quantiles than by the data. This role of the data relative to the quantiles changes positively the more loss data are supplied.

#### **4.4 GPD and Venter model comparison**

In this section we conduct a simulation study to investigate the effect on the two approaches by perturbing the quantiles of the true underlying severity distributions. We assume the six parameters sets of **Table 1** as the true underlying severity distributions and then perturb the quantiles in the following way. For each simulation run, choose three perturbation factors *u*7, *u*<sup>20</sup> and *u*<sup>100</sup> independently and uniformly distributed over the interval 1½ � � *ϵ*, 1 þ *ϵ* and then take ~*q*<sup>7</sup> ¼ *u*7*q*7, ~*q*<sup>20</sup> ¼ *u*20*q*<sup>20</sup> and ~*q*<sup>100</sup> ¼ *u*100*q*<sup>100</sup> but truncate these so that the final values are increasing,

#### **Figure 4.**

*Comparison of VaR results for the GPD and Venter approaches. (a) Naïve approach with correct (T\_Burr(1, 0.6, 2)), and false data (F\_Burr(1.01, 0.52, 2.26)), (b) Case 1 with correct quantiles and false data, (c) Case 2 with false quantiles and correct data.*

annual frequency by ^*<sup>λ</sup>* <sup>¼</sup> *<sup>K</sup>=a*. Given scenario assessments <sup>~</sup>*q*7, <sup>~</sup>*q*<sup>20</sup> and <sup>~</sup>*q*100, then

*<sup>F</sup>* <sup>~</sup>*q*7; ^*<sup>θ</sup>* � � , *<sup>R</sup>*ð Þ¼ 7, 20 *<sup>p</sup>*^<sup>20</sup> � *<sup>p</sup>*^<sup>7</sup>

Notice that if our estimates were actually exactly equal to what they are estimating, these ratios would all be equal to 1. For example, we would then have

estimate the true severity distribution function *T* by an adjusted form of *F x*, ^*θ* � �, then Hennie's distribution *H*~ is defined as follows (see de Jongh et al. 2015):

*<sup>R</sup>*ð Þ<sup>7</sup> *F x*; ^*<sup>θ</sup>* � � *for <sup>x</sup>*≤~*q*<sup>7</sup>

*<sup>p</sup>*^<sup>7</sup> <sup>þ</sup> *<sup>R</sup>*ð Þ 7, 20 *F x*; ^*<sup>θ</sup>* � � � *<sup>F</sup>* <sup>~</sup>*q*7; ^*<sup>θ</sup>* � � � � *for* <sup>~</sup>*q*<sup>7</sup> <sup>&</sup>lt;*x*≤~*q*<sup>20</sup>

Notice again that this estimate is consistent in the sense that it actually reduces

� � <sup>¼</sup> *<sup>p</sup>*^<sup>20</sup> and *<sup>H</sup>*<sup>~</sup> <sup>~</sup>*q*<sup>100</sup>

� � <sup>¼</sup> *pc* hold for the scenario assessments when estimates are substituted for

the true unknowns. Hence at the estimation level the scenario assessments are consistent with the probability requirements expressed. Thus, this new estimated severity distribution estimate *H*~ 'believes' the scenario quantile information, but follows the distribution fitted on the historical data to the left of, within and to the right of the scenario intervals. The ratios *R*ð Þ7 , *R*ð Þ 7, 20 , *R*ð Þ 20, 100 and *R*ð Þ 100 in (9) can be viewed as measures of agreement between the historical data and the scenario assessments and could be useful for assessing their validities and qualities.

to *T* if all estimators are exactly equal to what they are estimating.

The steps required to estimate VaR using this method are as follows:

ii. Generate *<sup>X</sup>*1, … , *XN* � iid *<sup>H</sup>*<sup>~</sup> and calculate *<sup>A</sup>* <sup>¼</sup> <sup>P</sup>*<sup>N</sup>*

iii. Repeat i and ii *I* times independently to obtain *Ai*, *i* ¼ 1, 2, … ,*I* and

The SLA again sheds some light on this method. As noted above the SLA

estimate the 99.9% VaR by the corresponding empirical quantile of these

1000*λ*

� � <sup>¼</sup> *<sup>p</sup>*^7, *<sup>H</sup>*<sup>~</sup> <sup>~</sup>*q*<sup>20</sup>

� �;

implies that we need to estimate *<sup>q</sup>*<sup>1000</sup> <sup>¼</sup> *<sup>T</sup>*�<sup>1</sup> <sup>1</sup> � <sup>1</sup>

*<sup>p</sup>*^<sup>20</sup> <sup>þ</sup> *<sup>R</sup>*ð Þ 20, 100 *F x*; ^*<sup>θ</sup>* � � � *<sup>F</sup>* <sup>~</sup>*q*20; ^*<sup>θ</sup>* � � � � *for* <sup>~</sup>*q*<sup>20</sup> <sup>&</sup>lt;*x*≤~*q*<sup>100</sup> *<sup>p</sup>*^<sup>100</sup> <sup>þ</sup> *<sup>R</sup>*ð Þ <sup>100</sup> *F x*; ^*<sup>θ</sup>* � � � *<sup>F</sup>* <sup>~</sup>*q*100; ^*<sup>θ</sup>* � � � � *for* <sup>~</sup>*q*<sup>100</sup> <sup>&</sup>lt; *<sup>x</sup>*<sup>&</sup>lt; <sup>∞</sup>*:*

� � <sup>¼</sup> 1 by (4), and similarly for the others. Our new method is to

*<sup>F</sup>* <sup>~</sup>*q*100; ^*<sup>θ</sup>* � � � *<sup>F</sup>* <sup>~</sup>*q*20; ^*<sup>θ</sup>* � � and *<sup>R</sup>*ð Þ¼ <sup>100</sup>

*c*^*λ*

*<sup>F</sup>* <sup>~</sup>*q*20; ^*<sup>θ</sup>* � � � *<sup>F</sup>* <sup>~</sup>*q*7; ^*<sup>θ</sup>* � � ,

. The estimated ratios are

<sup>1</sup> � *<sup>F</sup>* <sup>~</sup>*q*100; ^*<sup>θ</sup>* � � (8)

� � <sup>¼</sup> *<sup>p</sup>*^100, i.e. the equivalents

*<sup>n</sup>*¼<sup>1</sup>*Xn*;

� � and its estimate would be

(9)

1 � *p*^<sup>100</sup>

� � can be estimated by *<sup>F</sup>* <sup>~</sup>*qc*, ^*<sup>θ</sup>* � � and *pc* by *<sup>p</sup>*^*<sup>c</sup>* <sup>¼</sup> <sup>1</sup> � <sup>1</sup>

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

*<sup>R</sup>*ð Þ¼ <sup>7</sup> *<sup>p</sup>*^<sup>7</sup>

*<sup>R</sup>*ð Þ¼ 20, 100 *<sup>p</sup>*^<sup>100</sup> � *<sup>p</sup>*^<sup>20</sup>

*T qc*

then defined by

*R*ð Þ¼ 7 *p*7*=T q*<sup>7</sup>

*H x* <sup>~</sup> ð Þ¼

8 >>>>>><

>>>>>>:

Also note that *<sup>H</sup>*<sup>~</sup> <sup>~</sup>*q*<sup>7</sup>

*4.3.1 Venter's VaR estimation algorithm*

i. Generate *<sup>N</sup>* � *Poi* ^*<sup>λ</sup>*

*Ai*'s as before.

*4.3.2 Remarks*

**24**

of *T qc*

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 them on the interval 1½ � � *ϵ*, 1 .

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 *ϵ* the following steps are followed:


The above information suggest that provided enough loss data is available the

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

i. Study the loss data carefully with respect to the procedures used to collect the data. Focus should be on the largest losses and one has to establish whether these losses were recorded and classified correctly according to the

ii. Experts should be presented with an estimate of *q*<sup>1</sup> (based on the loss data) and then should answer the question **'**Amongst those losses that are larger than *q*<sup>1</sup> what level is expected to be exceeded only once in *c* years?' where

iii. The assessments by the expert should be checked with the condition

>2*:*533. This bring realism as far as the ratios between the

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:

Venter approach is the best choice to work.

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

*VaR bands for different Burr parameter sets and frequency combinations.*

**Figure 5.**

**5. Implementation recommendations**

definitions used.

*c* ¼ 7, 20, 100.

assessments are concerned.

~*q*100�~*q*<sup>7</sup> ~*q*20�~*q*<sup>7</sup>

**27**

v. Repeat items i–iv 1000 times and then summarise and compare 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 construct 90% VaR bands as before for the 1000 repeated GPD and Venter VaR estimates, i.e. *VaR*ð Þ <sup>51</sup> MedAT � 1, *VaR*ð Þ <sup>951</sup> MedAT � 1 h i. The results are given in **Figure 5**. Note that light grey represents the GPD band and dark grey the Venter band, whilst the overlap between the two bands are even darker.

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

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 the GPD.

*Construction of Forward-Looking Distributions Using Limited Historical Data and Scenario… DOI: http://dx.doi.org/10.5772/intechopen.93722*

**Figure 5.** *VaR bands for different Burr parameter sets and frequency combinations.*

The above information suggest that provided enough loss data is available the Venter approach is the best choice to work.
