Having discussed the iid and non-iid losses separately, it is now time to compare them. When 1 or 2 business lines are included, the three ways are put together in 1 graph. From 3 lines, this becomes too excessive and the 3 graphs will be placed side by side. The proportional graphs will be compared with each other, and so will the Euler graphs. In Subsection 4.1 and 4.2, the colors were arranged in a specific way. The ‘’dark, normal, color was given to the proportional allo-cation and its lighter version to the Euler alloallo-cation. This division is now no longer possible because more than 2 tones of a particular color must be used and these are not available. Because a maximum of 2 business lines are added together, the arrangement is in this section as follows. Line 1 of each manner gets the ‘dark’, normal, color and line 2 the lighter version of it. These colors only apply when the different distributions are put together in 1 graph. This is only the case when including 1 or 2 business lines. Now that all the necessary information is clear, the comparisons can start.
Figure 15: Proportional and Euler allocation when including 1 business line; all three methods. For both the proportional and the Euler method, the first and second non-iid distributions are the same as the iid distribution.
As already discussed in past subsections and clearly visible in Figure 15, if there is only 1 business line the risk capital is the same in all three ways. This is in both cases; when using the proportional formula and also when using the Eu-ler formula. Both, the proportional and EuEu-ler peak is at the value of Φ−1(0.995).
Compared to the graphs containing 1 business line, Figure 15, these graph in Figure 16 are more cluttered. Of the three distributions, no two lines are the same. When looking carefully, it is clear that in both methods the two colors red and pink are missing. The first non-iid distribution does not exist for the Euler method and is zero for the proportional method.
The left graph shows a clear difference between the iid distributed losses and the second distribution of non-iid losses. The risk capitals of the two lines that are iid are at a higher level than those of the non-iid losses. In addition, the spread of these histograms is much greater and therefore the histograms appear larger. The middle histograms in the right graph belong to business line 1. The left of the two belongs to the proportional allocation. In contrast to the outer
Figure 16: Proportional and Euler allocation when including 2 business lines;
all three methods. For both the proportional and the Euler allocation, the lines of the first non-iid distributions are left out, because of non-existing or being zero.
two histograms where the right one belongs to the proportional allocation. The shape of the four resulting histograms are similar, only the leftmost one, line 2 of the non-iid distributed losses, is a bit smaller.
From now on, the different ways of calculating cannot be put together in 1 graph. The graphs of the proportional method will be placed above the graphs of the Euler method. From left to right, the graph of the iid distributed losses is shown first, then the first distribution of non-iid and lastly the second non-iid distribution.
Figure 17: Proportional and Euler allocation when including 3 business lines; all three methods. For the first non-iid distribution, all three resulting risk capitals are the same for both the proportional as well as the Euler. For the second non-iid distribution, this is only the case for the Euler allocation.
First, the proportional graphs of Figure 17 will be compared and discussed.
When the losses are iid distributed, no business line has the same risk capital.
The first non-iid distribution has three risk capitals which are all the same. The
three independent histograms completely overlap each other which results in only one histogram being displayed in the end. The rightmost graph results in two visible histograms. The risk capital of lines 1 and 3 is the same. Line 2 results in an average capital half as big as that of line 1 and 3. So the three graphs all differ from each other. This is not the case when using the Euler formula. The two graphs of the non-iid distributed losses are exactly the same.
All three business lines overlap each other again and the average risk capital is also the same. Compared to these two graphs, the iid graph is way different.
Here, all three lines have their own histogram. The most remarkable thing here is that line 3 has a lower risk capital than both lines 1 and 2. After comparing both allocation methods, the proportional allocation seems more appropriate when three business lines are included.
Figure 18: Proportional and Euler allocation when including 4 business lines;
all three methods. For the first non-iid distribution, the risk capital does not exist or is zero. For the second non-iid distribution, the proportional allocation has equal values for line 1and 3 and line 2 and 4 and the Euler allocation for line 1 and 3.
When there are 4 business lines included, as shown in Figure 18, the two middle graphs of both methods do not show any results. Reading back through the theoretical framework, this was also what was expected. The losses of the four lines, Z, −Z, Z and −Z, cancel each other out, which results in an error in the Euler formula. The other two distributions do have reasonable results.
Whereas the 4 lines with the iid losses have 4 independent histograms, the second non-iid distribution results in only two histograms. Line 2 and 4 have the same risk capitals and so do line 1 and 3. When the losses are iid, the first line has the lowest risk capital. Adding a second line results in a higher risk capital for line 2. But when a third line is added, it has a lower capital than line 2. The fourth line added has even a lower capital than line 3. This process happens almost the same when the Euler formula is applied. The main difference is that line 3 is not located between line 2 and 4, but that it ends up at a lower risk capital than line 1. In the non-iid case, the lowest risk capital belongs to line 4. For iid losses, the risk capital of line 2 is the highest, but for
non-iid losses, line 1 and 3 end up higher.
Looked at as a whole, the proportional method seems to fit iid losses better, but the Euler seems to distribute the non-iid more fairly.
Figure 19: Proportional and Euler allocation when including 5 business lines; all three methods. For the first non-iid distribution, all three resulting risk capitals are the same for both the proportional as well as the Euler. For the second non-iid distribution, the proportional allocation equals at line 1, 3 and 5 and line 2 and 4 and the Euler allocation for line 1 and 3.
The last comparison is applied to the proportional and Euler allocation when 5 business lines are included and can be seen in Figure 19. The graphs of the first non-iid distribution looks most similar. For the second non-iid distribution, the Euler method results in more histograms and for the iid distribution, this is the case for the proportional one. Looking only at the proportional allocation, the iid losses result in 5 independent histograms. When the second non-iid distribution is implemented, line 1, 3 and 5 and line 2 and 4 end up at the same risk capital. Only 2 different values are found for 5 business lines. Using the Euler formula, this is almost reversed. Now the iid losses result in only 2 different risk capitals and the second non-iid losses result in 4 different ones.
As in the case of 4 business lines, the proportional method seems to fit iid losses better and the Euler seems to distribute the second non-iid more fairly. For the first non-iid distribution, both allocations result in 5 identical risk capitals for the 5 business lines.
5 Conclusion
The goal of this study is to investigate the effect of different distributions and a different number of business lines on the resulting risk capital allocation. It can be claimed that the methods presented in Section 3 are useful to analyze this.
Three different distributions were used in this paper. One where the losses are iid Gaussian distributed and two where they are non-iid Gaussian distributed.
In all three, cases the proportional and the Euler rule are applied. Results are obtained for the Value-at-Risk at α = 99.5%. This level of α is equal to what is used in Solvency II. In this paper, the choice was made to use Value-at-Risk as a risk measure. When using a Gaussian distribution, the risk measures Value-at-Risk and Expected Shortfall should provide the same information (Boonen, 2017).
The results are obtained with a simulation-based approach, which is subject to simulation errors. For financial applications it is important to reduce the simulation errors as much as possible. Therefore, 10.000 repetitions were used.
To make it even more realistic and reduce the simulation error further, a higher number of repetitions can be used.
All resulting graphs were compared to each other in an organized manner.
First for each distribution separately and then the different distributions with each other. The simulations not only resulted in graphs, but the risk capitals were also given in real numbers. From these values it can be quickly concluded that the property subadditivity is satisfied. The capital requirement for all the risks combined is greater than for the risks treated separately. In addition, as described in Section 2, the sum of the independent risk capitals is equal to the risk measure of the whole firm.
An important result for the iid distributed losses is that it is most advanta-geous to merge as much as possible. An example showing that this is true for business lines 1 is presented in Figure 6. This is not always the case for the non-iid distributed losses. Whether it is best to merge depends on the distribution of the different lines. In general, there is no consensus on the best allocation method. For a different number of business lines and different distributions, a different allocation fits best. In some cases, the choice between the proportional and the Euler allocation is indifferent. For example, the first way of the non-iid distributed losses gives for both proportional and Euler allocation the same risk capital to the included lines. The only difference between the methods is the height of the risk capitals.
When further research on this topic is undertaken, for example the Expected Shortfall can be chosen as risk measure and it can be checked whether the re-sults are the same as with the VaR. In doing so, a distribution other than Gaussian can be studied. Yamai and Yoshiba (2002, 2005) state that the esti-mation errors for the Expected Shortfall will increase when the tails are too fat compared to the Value-at-Risk. In addition, if there are fat tails in the underly-ing distribution, fittunderly-ing a Gaussian distribution can substantially underestimate the corresponding Value-at-Risk. It is shown that if data is more tailed than a Gaussian distribution, risk capital allocations are significantly different and
thus differ from Solvency II requirements. Continuing about the fat tails, it is the case that for a more fat-tailed distribution, the Euler rule is more volatile than a more simple allocation method, such as the proportional rule. When the Euler rule is used, the gradient calculated by hand can be used instead of the formula used in this study.
To summarize, it can be said that a different distribution and a different number of business lines both have an impact on the resulting risk capital. When deciding between the proportional and the Euler method, it is recommended for the financial institution to first consider how the losses are distributed and taking into account the number of current business lines. When this information is known, it is easier to make a decision between the different allocation methods.
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