By using the Gaussian copula to specify the joint distribution of the individual losses, the cor-relation between the divisions can be modeled as desired. Table1presents the capital allocation compositions under the second model as described in section3.1.
These results are similar to the previous ones with respect to the standard deviation of the allocated capital to each division. The standard deviation when using the Euler rule is noticeable larger than the τ -value method.
ParetoII distribution Normal distribution
Division 1 Division 2 Division 3 Division 1 Division 2 Division 3
Euler- mean 0.427 0.192 0.381 0.520 0.146 0.334
rule std. dev. 0.0834 0.1142 0.0930 0.0161 0.0580 0.0482
τ -Value mean 0.411 0.214 0.374 0.516 0.142 0.342
std. dev. 0.063 0.0775 0.0645 0.0123 0.033 0.0266
Table 2: Descriptive statistics of the capital allocation composition based on the Euler-allocation method and τ -value method with the ES as risk measure and the Gaussian copula is used to model the dependence between the divisions. The simulation is run 10.000 times and the mean of the portion of risk capital allocated to each division with their standard deviation is shown.
Looking at the composition of the allocated capital there is a significant difference. The Gaussian copula made it possible to model the joint distribution of the divisions and introduce a different correlation structure.
In this model Division 1 is positively correlated to Division 3, and Division 2 negatively. A known property of the Euler rule is that it considers the correlation structure and can hedge the total risk of the firm. Due to the negative correlation of Division 2, it is considered a good hedge and is rewarded a lower portion of the total risk capital. The opposite is seen in the allocated
portion to Division 1. Due to its positive correlation to the aggregate risk, it is allocated a larger portion.
Remarkably, this pattern is also seen when the τ -value is used as allocation method. Meaning that the τ -value method also considers the correlation structure, and hedges the risk of the firm.
Furthermore, it can be argued that the τ -value results in more accurate compositions by looking at the standard deviations, which are significantly lower, especially for Division 2. Thus when dealing with heavy-tails, the τ -value method captures the tail-dependence more accurately.
Furthermore, comparing the composition under the ParetoII distribution and normal distri-bution, a small disparity can be noted. This can be seen as the tail-effect. When losses present heavy tails, using the Normal distribution may over- or under-allocate risk capital to the divi-sions.
Figure 3: Capital allocation to Division 1 under the Euler method (left) and τ -value method (right) with second kind Pareto losses (gray) and Normal losses (green). The Gaussian copula has been used to model the dependence between divisions. 1000 losses are generated and the simulation is run 10.000 time
Figure 4: Capital allocation to Division 2 under the Euler method (left) and τ -value method (right) with second kind Pareto losses (gray) and Normal losses (green). The Gaussian copula has been used to model the dependence between divisions. 1000 losses are generated and the simulation is run 10.000 time
Figure 5: Capital allocation to Division 3 under the Euler method (left) and τ -value method (right) with second kind Pareto losses (gray) and Normal losses (green). The Gaussian copula has been used to model the dependence between divisions. 1000 losses are generated and the simulation is run 10.000 time
From Figure 3,4, and5 clearly the proportion of risk capital allocated to Division 2 seem to be more unstable compared to the allocations of the other divisions. The standard deviation of Division 2 is also significantly larger, note that this is also the division with the negative
correlation.
An additional factor that contributes to this volatility is the dependence between the divisions and how this is modeled. Copulas are useful tools for modeling the dependence between variables.
Especially the Gaussian copula, which allows for different correlations between the divisions.
However the Gaussian copula may fail to accurately approximate the dependence in the tails.
Nonetheless Yamai et al.(2002) argues that using the Gaussian copula can still be reasonable for heavy tailed distribution.
When dealing with fat-tailed losses, it is important to have an accurate approximation of the dependence in the tails, since these observation attribute to the estimation of risk and the allocation of risk capital. Hence using the Gaussian copula may fall short. As an alternative the student t copula may be considered. It has been shown that the student t copula performs better in modelling the dependence since it takes into account the dependence in the center and in the tail of the distribution as well.
5 Conclusion
The Pareto distribution of the second kind is widely used in modeling heavy-tailed risks. This study provides insight into using this distribution to model heavy-tailed losses for the risk capital allocation problem by simulation and shows that additional modelling techniques are required.
When the underlying loss distribution presents heavy tails, the τ -value method results in more stable capital allocations compared to the Euler allocation method. Additionally, the τ -value also takes into account the correlations between divisions, and hedges the risk of the firm
Furthermore, to model the dependence, copula have been proven to be useful. The Gaussian copula especially; This copula allows to model the correlation structure between divisions pair-wise, rather than assuming an uniform dependence structure between the divisions.
Nonetheless, the Gaussian copula may fail to accurately estimate the tail-dependence, and as the tail gets fatter, which may further increase estimation errors.
The simple model used in this study, delivered promising result, although, with an increasing number of divisions and fatter-tailed losses more intricate and advanced modelling techniques are required, which are beyond the scope of this study.
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