## The effect of different distributions and a different number of business lines on the

## resulting risk capital allocation

### Sharon Plat

Bachelor’s Thesis to obtain the degree in Actuarial Science

University of Amsterdam

Faculty of Economics and Business Amsterdam School of Economics Author: Sharon Plat Student nr: 12835951

Email: sharonplat2802@gmail.com

Date: June 30, 2022

Academic year: 2021-2022 Supervisor: Tim J. Boonen

### Abstract

The determination of the risk capital allocation is of great importance for all firms. This can be done in several ways. This paper examines the impact of a different number of business lines and different distributions on the resulting risk capital allocation. To compare many different structures, data is simulated in R using iid and non-iid Gaussian losses. An important result is that when losses are iid, it is most advantageous for the firm to merge as much as possible.

Unlike non-iid losses, where this is not always the case.

### Keywords

Risk capital allocation, Euler rule, proportional rule, Value-at-Risk, Expected Shortfall, simulation, business lines

### Statement of originality

This document is written by Student Sharon Plat who declares to take full responsibility for the contents of this document.

I declare that the text and the work presented in this document are original and that no sources other than those mentioned in the text and its references have been used in creating it.

The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.

### Contents

1 Introduction 4

2 Theoretical Framework 6

2.1 Risk measure . . . 6

2.1.1 Value-at-Risk . . . 6

2.1.2 Expected Shortfall . . . 7

2.1.3 Value-at-Risk versus Expected Shortfall . . . 8

2.1.4 Coherence . . . 9

2.2 Allocation methods . . . 10

2.2.1 Proportional allocation . . . 10

2.2.2 τ -value allocation . . . 11

2.2.3 Euler allocation . . . 11

2.3 Example . . . 13

3 Methodology 16 3.1 Risk measures and allocation methods . . . 16

3.2 Iid distributed losses . . . 17

3.3 Non-iid distributed losses . . . 18

3.4 Number of business lines . . . 19

4 Results 20 4.1 Iid distributed losses . . . 20

4.2 Non-iid distributed losses . . . 24

4.3 Comparing iid with non-iid distributed losses . . . 31

5 Conclusion 35

References 37

### 1 Introduction

An important part of risk management is the determination of capital require- ments. Capital allocation is of importance for example the performance mea- surement and pricing. A capital allocation problem arises when an amount associated to the whole has to be distributed among its parts. There are several reasons why the firm may want the total capital allocated over the different lines. Dhaene et al. (2012) mentions four of them. For example, holding cap- ital involves costs. These has to be redistributed across various lines so that this cost is transferred back to the depositors in the form of charges. Thereby, capital allocation is useful for assessing and comparing the performance of the business lines by the return on allocated capital. Zaks and Tsanakas (2014) describe capital allocation as the solution to an optimization problem whereby a quadratic deviation measure between individual losses and allocated capital amounts is minimized.

The amount that has to be invested to compensate for the risk which comes along with assets and liabilities is called the risk capital (Tsanakas, 2004). Dif- ferent frameworks have their own view on the regulation of capital allocation.

There are substantial differences between several regulatory frameworks, such as Solvency II, the Swiss Solvency Test and Basel III requirements for banks (Tasche, 2004). The best known and most widely used of these three is Solvency II. This framework is widely discussed in the article of Boonen (2017) and partly in Zaks & Tsanakas (2014). Solvency II is the new supervisory framework in force for insurers and reinsurers in Europe as of 2016. It puts demands on the required economic capital, risk management and reporting standards of in- surance companies. The risk measure which is in line with Solvency II is the Value-at-Risk.

The underlying meaning of the regulation is to protect the policyholder by regulating the risk that future claims will not be covered. Allocations can be used in the context of investments, business lines and other possible activities that have multiple risk components. In this research, it is assigned to business lines. Risk capital allocations problems emerge when an aggregate capital de- termined by risk measure needs to be allocated to the risk units within a firm (Boonen, Guillen, Santolino, 2019).

Boonen (2019), Dhaene et al. (2012), Denault (2001) and Tsanakas (2004), all four conclude that capital allocation is a non-trivial process. The sum of the risk capitals of each subdivision/business line is usually larger than the risk capital of the company/institution as a whole and there may exist mutual de- pendencies between the performance of the different business lines. The degree of such dependence determines the amount of diversification that a specific line contributes to the aggregate risk. Diversification refers to a strategy of spread- ing risks through a spread across multiple resources. In this paper, the multiple resources are the number of business lines. What is the effect of a different number of business lines on the risk capital allocation and what is the difference between iid and non-idd distributed losses, will be examined in this paper.

There are several methods to allocate risk capital. Some common methods used are the proportional allocation, the Euler allocation and the τ -value allo- cation. These allocation methods will be discussed more in detail in the theory section. In this section, more information will also be given on the different risk measures that can be used. The two most important ones are the Value-at-Risk and the Expected Shortfall. They both will be explained in detail separately and thereafter compared with each other. An important property of risk measures is whether they are coherent. What coherence implies and which properties must be satisfied is made clear in Section 2.1.1. In the end of Section 2 there is given an example to make the theory explained before more clear.

The structure of the rest of the paper is as follows. In ‘Methodology’, Section 3, all the steps that are taken in the research are mentioned. Many simulations will be done with not only a different number of business lines, but also different distributions. First, iid distributed losses are used and then non-iid distributions are considered. In any case, the first simulation includes 1 business line. After that, extra lines are added sequentially. Each additional line can have a different effect on the risk capital. In this way, it should be clear enough to see how the results are achieved. As a logical extension, all the results found are presented in an organized way in the subsequent section. Finally, all conclusions which can be drawn from the study are well identified in Section 5.

### 2 Theoretical Framework

This section outlines the theory of risk capital allocation, covering the most im- portant information from the literature. First, the risk measures are discussed.

It includes an explanation about the two well-known measures the Value-at-Risk and the Expected Shortfall and also about the concept of coherence. Thereafter, the different kind of allocation methods is mentioned. There are a lot of differ- ent methods, but in this paper only three of them are discussed in the theory;

the proportional, the τ -value and the Euler allocation. To finish the section, an example of risk capital allocation using the proportional and Euler method is provided.

### 2.1 Risk measure

All companies have to be able to meet their financial obligations with a high probability as they become due. With these requirements in mind, the amount of the risk capital can be determined (Dhaene, Tsanakas, Valdez, & Vanduffel, 2012). First, all risks of the subdivisions are calculated separately. The required buffer for the financial institution, the risk capital, which has to be kept apart, is determined from the risk-based assessment of the aggregate risk (Boonen, 2019). This is the risk of all subdivisions together. All divisions are of course part of the same company. The allocation, therefore, has to be done as fair as possible. Denault (2001) comes up with the firm-internal risk measure. This is equal to the risk capital of a specific business line minus its allocated share of the diversification advantage. This concept is too complex to cover in this study.

In the literature, ρ(X) is known as the symbol used for indicating the risk capital. The risk measure ρ, a mapping from Γ, which is a subset containing random variables, into real numbers (Γ 7→ R), quantifies the level of risk. The main problem with the risk capital allocation is to allocate the amount of risk (ρ(X)) between the subdivision, business lines or portfolios of the firm. Boo- nen (2019), Tasche (2004) and Dheane et al. (2012) make clear that the two most important risk measures are the Value-at-Risk and the Expected Short- fall. These two measures are in accordance with the Solvency II and Basel III frameworks.

2.1.1 Value-at-Risk

The Value-at-Risk, abbreviated to VaR, is a well-known and often used risk mea- sure. Boonen (2017) describes the Value-at-Risk as the maximum loss within a certain confidence level. Although the VaR is such a well-known risk measure, many academic writers talk about its drawbacks; Asimit, Peng, Wang & Yu (2019), Acerbi & Tasche (2002), Danielsson et al. (2005) and Yamai & Yoshiba (2002, 2005). The two worst, named in all papers are: 1) the VaR is not coherent since it is not subadditive, 2) the VaR disregards any loss beyond the confidence

level; ‘tail risk’. Subadditivity means that the risk of a portfolio can be larger than the sum of the stand-alone risks of its components when measured by VaR.

Danielsen et al. (2005) analyze which asset classes are likely to suffer from violations of subadditivity, and examine the asymptotic and finite sample prop- erties of the VaR risk measure from the point of view of subadditivity. The main result of this study is that for most assets and applications, there is no expectation to see any violations of subadditivity. As a consequence, worries about subadditivity are in general not pertinent for risk management applica- tions relying on VaR. Thereby, Yamai and Yoshiba (2002, 2005) do state that the concept of the VaR is relatively simple compared to other risk measures, it is easy to compute and ready applicable. After considering all the pros and cons together, the VaR is still a standard risk measure for financial risk management.

When there is a confidence level of, for example, 99.5%, there can be expected that the loss in the next year is no more than the VaR with a certainty of 99.5%. The Value-at-Risk with confidence level α ∈ (0, 1) is the α-quantile for all random variables Z, where Z can be interpreted as a loss of a firm or business line (Tasche, 2004). The general formula which is given for the Value-at-Risk is as follows.

V aRα(Z) = inf {z ∈ R : P (Z ≤ z) ≥ α} (1) where Inf means infinimum. The infinimum of a subset can be seen as the largest element that is smaller than or equal to all the elements in this subset. So, the Value-at-Risk for Z with confidence level α is the infinimum from all z in the real numbers whose probability that Z is less than or equal to z is greater than or equal to α.

The standard model of Solvency II assumes a Gaussian distribution. When this is the case, the formula for the Value-at-Risk simplifies a lot.

V aRα(Z) = µ + qασ (2)

The VaR becomes linear in the expectation µ and the standard deviation σ of
Z. Here, qα= Φ^{−1}(α) is the inverse CDF of the standard Gaussian distribution,
in other words ‘the α-quantile’.

2.1.2 Expected Shortfall

Basel III mainly uses the Expected Shortfall (ES) as a risk measure. Also called the TVaR or conditional tail expectation (Panjer, 2002). In contrast to the VaR, the ES is a coherent risk measure. The Expected Shortfall has been proposed as remedy for the deficiencies of the VaR. It is homogeneous of degree 1, co-monotonic additive and sub-additive (Tasche, 2007). A simple and clear definition for the ES is given by Boonen (2017). The Expected Shortfall is equal to the expected value of the loss, given that the loss is larger than the Value-at-Risk. Therefore, the Expected Shortfall also depends on the quantile used.

The Expected Shortfall with a confidence level α ∈ (0, 1) for all random variables Z is given by

ES_{α}(Z) = 1
1 − α

Z 1 α

V aR_{τ}(Z)dτ. (3)

If the integral does not converge, ES equals infinity. Convergence is the property of approaching a limit. When X has a continuous distribution, the Expected Shortfall can be written as conditional VaR.

ESα(Z) = E(Z|Z ≥ V aRα(Z)) (4)

Just as the Value-at-Risk simplified when the losses were Gaussian distributed, this is also the case with the Expected Shortfall.

ESα(Z) = µ +ϕ(qα)

1 − ασ (5)

The ES also becomes linear in the expectation µ and the standard deviation σ.

Here, ϕ stands for the density function for the standard Gaussian distribution.

Similar to the Value-at-Risk, the Expected Shorfall has its benefits and draw- backs too. One of the most important benefits is that the ES is coherent. In addition, it provides a definition of times in which the losses exceed a certain threshold. The Expected Shortfall represents a solid tool for assessing rela- tive risks without limitations on applicability (Acerbi, Nordio, & Sirtori, 2001).

Compared to the VaR, the estimation of ES is more complicated. Furthermore, the first moment needs to exist.

2.1.3 Value-at-Risk versus Expected Shortfall

The Value-at-Risk and Expected Shortfall are related to each other. Both have their pros and cons. When the losses are Gaussian distributed, as in the standard model of Solvency II, the VaR and ES provide similar information. If the losses are non-Gaussian, this will lead to a mismatch.

The Value-at-Risk is popular in insurance regulation, because it is easy to interpret and has a clear connection to the probability of insolvency. In addition, it satisfies the property elicitability which is important to be able to verify and compare competing estimation procedures (Boonen, 2019). Moreover, the Value-at-Risk is less sensitive to model risk compared to the Expected Shortfall.

Lastly, the Expected Shortfall requires a larger sample size than VaR to provide the same level of accuracy.

In some cases, the use of the VaR can cause some problems. When this is the case, the Expected Shortfall can better substitute for it. Examples of such cases are discussed in Yamai and Yoshiba (2005). The Expected Shortfall appears to be a natural choice when the Value-at-Risk is unable to distinguish between portfolios with different riskiness. Also, the Expected Shortfall is aware of the shape of the 5% worst events, while the Value-at-Risk is not. Finally, the VaR

gives insight into only the frequency of the worst-case and the ES gives insight into both the frequency and the size of the worst-case.

Careful considerations have to be made when choosing between the two risk measures. The use of a single risk measure is not recommended. In financial risk management, the most effective way to provide more comprehensive risk monitoring is to supplement the Value-at-risk with the Expected Shortfall.

2.1.4 Coherence

In order to be coherent, the risk measure must meet some important properties.

There is a lot of literature explaining these properties. For example the papers of Artznet et al. (1999), Boonen (2020), Panjer (2002) and Buch & Dorfleitner (2008).

Below, the four properties together with their mathematical meanings are listed.

• Subadditivity: For all Z, Y ∈ Γ; ρ(Z + Y ) ≤ ρ(Z) + ρ(Y ).

• Positive homogeneity: For every Z ∈ Γ and every c ≥ 0, ρ(cZ) = cρ(Z).

• Monotonicity: For all Z, Y ∈ Γ, Z ≥ Y ; ρ(Z) ≥ ρ(Y ).

• Translation invariance: For every Z ∈ Γ and every c ∈ R; ρ(Z + c) = ρ(Z) + c.

When these conditions are satisfied, the risk measure can be called reasonable.

Subadditivity means that the capital requirement for two risks combined will not be greater than for the risks treated separately. When this does not hold, it will be an advantage to dis-aggregate. Positive homogeneity holds when the capital requirement is independent of the currency in which the risk is measured.

Monotonicity implies that the capital requirement should be greater when one risk has always bigger losses than another risk. Because random variables are used, the resulting values of this study must almost surely satisfy this property.

And finally, if there is no additional capital requirement for an additional risk for which there is no uncertainty, then the property of translation invariance applies.

Any measure which does not satisfy some of the axioms will produce para- doxical results of some kind giving a wrong assessment of relative risks (Acerbi, Nordio, & Sirtori, 2001). Because the Value-at-Risk risk measure does not al- ways satisfy the subadditivity property, it is not coherent. In contrast to the Expected Shortfall, which does satisfy all four properties.

Not only the risk measure has properties, also the allocation methods. An allocation principle is called coherent if it satisfies all properties. The six prop- erties which are given in Boonen (2020) are: Translation and Scale invariance, Monotonicity, Riskless Portfolio, No Diversification and Continuity. The (math- ematical) definitions are not of great importance in this study. Since they are rather complicated, they are left out of this paper.

### 2.2 Allocation methods

The capital reserved for the required payments of the company has to be divided between the different business lines. The subdivision of the aggregate capital is called the capital allocation (Dhaene, Tsanakas, Valdez, Vanduffel, 2012).

The properties for a coherent allocation principle are given in the section above.

In addition to these, there are three properties which has to be satisfied by the allocation. The three axioms are: No undercut, Symmetry and Riskless allocation (Denault (2001), Buch Dorfleitner (2008)). No undercut means that a division is not attributed more risk capital than it has on its own. Symmetry implies that the risk capital allocation of a division depends only on the risk that the division contributes to the business. After all, a risk-free portfolio should receive exactly its risk measure allocated; ‘Riskless allocation’. Panjer and Jing (2001) mention, besides these three axioms, a fourth one. All of the capital has to be allocated to the risks. This property is called ‘Full allocation’.

The capital allocation should be done in such a way that for each business line the allocated capital and the loss are sufficiently close to each other. The three most common methods, named below, will be discussed shortly in the common sections.

• The proportional allocation

• The τ -value allocation

• The Euler allocation

2.2.1 Proportional allocation

The literature mentions that none of the proposed capital allocation rules meet all the characteristics (Boonen, 2020). The proportional rule does not take into account the division-specific marginal contributions to the firm’s total risk capital from the aggregation of risks. As a consequence, it does not satisfy the axioms ‘Translation invariance’ and ‘Riskless portfolio’.

The proportional allocation is a general method of several specific ones named in Dhaene et al. (2012). A drawback of this method is not taking into account the division-specific marginal contributions to the total risk capital of the firm by risk aggregation. First, the risk measure ρ is chosen. Thereafter, the capital Ki= αρ(X) is resigned to each business line i, where X is the sum of all Xi for i ∈ (1, ..., n). Xi is assumed to be the loss of business line i. α is chosen in the way that the fourth property, ‘Full allocation’, is satisfied. Ki is assumed to be equal to αρ(X), so K =Pn

i=1Ki can be set equal to ρ(X). The resulting formula for the proportional allocation arises after aggregating this information and values.

K_{i}= ρ(X)
Pn

j=1ρ(Xj)ρ(X_{i}) (6)

The risk measure which is used in the continuation of this study is the Value-

at-Risk. Including this, the formula can be rewritten into
K_{i}^{P} = V aR_{α}(X)

Pn

j=1V aRα(Xj)V aR_{α}(X_{i}). (7)
In this method, all divisions are assigned a standard proportion based on their
standard deviation compared to the standard deviations of the other divisions.

2.2.2 τ -value allocation

The second allocation was made according to the τ -value, described in detail in Boonen (2020). The τ -value capital allocation rule is given by a convex combination of the utopia and the worst-case allocation. The utopia allocation leads to an under-allocation of the risk capital and is given by

Mi(R) = ρ(X) − ρ( X

j∈N\{i}

Xj) (8)

for all i ∈ N. As opposed to the worst-case allocation, which in general leads to an over-allocation. This allocation is given by the following formula.

mi(R) = min

S⊆N\{i}ρ( X

j∈S∪{i}

Xj) −X

j∈S

Mj(R) (9)

for all i ∈ N. Combining these two, the formula for the τ -value capital allocation rule results in

τi(R) = (1 − α)Mi(R) + αmi(R), (10) where α ∈ R is chosen such that

n

X

i=1

[(1 − α)M_{i}(R) + αm_{i}(R)] = ρ(X). (11)

A major advantage of the τ -value allocation is that it satisfies all six prop- erties listed in Section 2.1.4. To the extent known in the literature, there is no known capital allocation rule that also satisfies all of these properties. Tijs (1987) states two more properties that are satisfied by this allocation method.

The τ -value is the unique efficient rule with the minimal right property and the restricted proportionality property.

2.2.3 Euler allocation

The Euler rule has received considerable attention in the academic literature on risk capital allocations. Another well-known name for this method is the Aumann-Shapley value. Boonen (2019) describes this allocation method as the gradient of a properly chosen function, which indicates a marginal contribution of a division’s portfolio to the firm’s aggregate risk. Tasche (2004) defines it in a

different way. It states that the Euler allocation is described as a 1-homogeneous risk measure capital allocation by means of the gradient. Besides the names Euler allocation and Aumann-Shapley value, in Buch, Dortfleitner & Wimmer (2011) it is called the gradient allocation, which is consistent with the definition of Boonen (2019).

The Euler rule takes into account the dependencies between business lines.

The Aumann–Shapley value, or Euler allocation, satisfies some properties such as ‘Translation and Scale invariance’ and ‘No undercut’. However, it does not satisfy ‘Continuity’. The lack of continuity makes risk capital allocation very volatile for small changes in the underlying probability distributions. A small change in the data can have a substantial impact on the allocated capital to a division. Another drawback of the Euler rule is the high volatility. The Euler rule in combination with the Value-at-Risk is highly sensitive to empirical measurement error. The Euler rule with an Expected Shortfall risk measure is already less volatile, but still more volatile than the proportional rule.

Despite all the drawbacks, the Euler allocation is seen as more demanding
than some of the more traditional methods. The Euler rule, K^{E}, is written as:

K_{i}^{E}= ∂

∂λi

ρ(X

i∈N

λiXi)|λ=(1,..,1) (12)

for i ∈ N and λ ∈ R^{N}+. Due to Boonen (2019) and the use of the risk measure
Value-at-Risk, this can be rewritten to:

K_{i}^{E}= ∂

∂λi

V aRα(X

i∈N

λiXi)|λ=(1,..,1), (13)

K_{i}^{E}= E(Xi|X = V aRα(X)). (14)

When the vector of losses of the divisions {Xi}_{i∈N}is multivariate Gaussian dis-
tributed with parameters (µ, σ, Σ), X(λ) has a Gaussian distribution as well and
the formula can be rewritten again. In this case, µ is the vector of expectations,
σ the vector of standard deviations, Σ the correlation matrix, σij the covariance
of (Xi, Xj) and Φ is the CDF of the standard Gaussian distribution.

V aRα(Xi) = µi+ Φ^{−1}(α)σi (15)

V aR_{α}(X(λ)) =X

i∈N

λ_{i}µ_{i}+ Φ^{−1}(α)

sX

i∈N

λ^{2}_{i}σ^{2}_{i} + 2 X

i,j∈N,i<j

λ_{i}λ_{j}σ_{ij} (16)

Taking the partial derivative of this last formula and substituting λ = (1, ..., 1) yield the resulting expression of the Euler rule.

K_{i}^{E} = µi+ Φ^{−1}(α)cov(X, X_{i})

pvar(X) (17)

for i ∈ N.

The main goal of capital allocation is to minimize the sum of the diver- gences between the losses and the allocated capital of the different subdivi- sions/portfolios. As discussed above, there are several alternative allocation methods. Every method has its own benefits and drawbacks. Depending on what is seen as the most important part, the best method can be chosen.

### 2.3 Example

A theoretical example of the risk capital allocation for a simplified firm with 3
business lines is given in this subsection. The losses are multivariate Gaussian
distributed. The actuarial notation for a Gaussian distributed random vari-
able X is X ∼ N (µ, σ^{2}). The most common capital allocation methods are the
proportional allocation and the Euler rule. For these two methods, the cap-
ital allocation will be computed using the risk measure Value-at-Risk with a
confidence level of α = 99.5%. Business line i is denoted by X_{i}, for i = 1, 2, 3.

The distributions of each line is as follows; X_{1}∼ N (µ1, σ_{1}^{2}), X_{2}∼ N (µ2, σ_{2}^{2})
and X_{3} ∼ N (µ_{3}, σ_{3}^{2}). After defining all mu’s and sigma’s, this will result in
X_{1}∼ N (0, 1), X_{2} ∼ N (0, 5) and X_{3} ∼ N (0, 4). It cannot be taken for granted
that the business lines within a firm are all positively correlated. The correla-
tion of the 3 lines in this example are ρ1,2 = 0.6, ρ1,3 = −0.3 and ρ2,3 = −0.5.

The resulting correlation matrix is:

1 0.6 −0.3

0.6 1 −0.5

−0.3 −0.5 1

Because of the correlation between the business lines, the variance of the firm (X) cannot be found by simply adding up the individual variances of the different lines.

var(X) = var(X1) + var(X2) + var(X3)+

2cov(X1, X2) + 2cov(X1, X3) + 2cov(X2, X3) (18) The variance-covariance matrix is equal to:

σ_{1}^{2} ρ_{1,2}σ_{1}σ_{2} ρ_{1,3}σ_{1}σ_{3}
ρ_{1,2}σ_{1}σ_{2} σ^{2}_{2} ρ_{2,3}σ_{2}σ_{3}
ρ_{1,3}σ_{1}σ_{3} ρ_{2,3}σ_{2}σ_{3} σ_{3}^{2}

=

1 0.6√

5 −0.6 0.6√

5 5 −√

5

−0.6 −√

5 4

All values needed for the total variance are known. The final variance is ap- proximately 7.011.

As mentioned earlier, the literature states that the sum of the risk capitals of
each business line is usually larger than the firm’s risk capital as a whole. This
can also be concluded from the example. The sum of the standard deviations
of the three business lines is σ_{1}+ σ_{2}+ σ_{3} = 1 +√

5 + 2 ≈ 5.24 whereas the resulting standard deviation of the firm is lower, namely SD(X) =pvar(X) =

√7.011 ≈ 2.648.

It is assumed that the losses are Gaussian distributed. In this case, the VaR and ES are both linear in the expectation µ and the standard deviation σ.

• ESα(X) = µ +ϕ(qα)

1 − ασ (19)

• V aRα(X) = µ + qασ (20)

Here, q_{α}= Φ^{−1}(α) is the α-quantile and ϕ the density function for a standard
Gaussian distribution. For this example, only the formula for the Value-at-Risk
is needed. α is set equal to 99.5%, so q0.995 equals Φ^{−1}(0.995) = 2.807.

• V aR_{0.995}(X_{1}) = 0 + 2.807 ·√

1 = 2.807 (21)

• V aR0.995(X2) = 0 + 2.807 ·√

5 = 2.807 ·√

5 ≈ 6.277 (22)

• V aR0.995(X3) = 0 + 2.807 ·√

4 = 2.807 · 2 = 5.614 (23)

• V aR_{0.995}(X) = 0 + 2.807 ·√

7.011 = 2.807 ·√

7.011 ≈ 7.432 (24) The sum of the three independent Value-at-Risks is higher than the Value-at- Risk of the total firm. This phenomenon had occurred before with the standard deviations.

σ_{X}_{1}+ σ_{X}_{2}+ σ_{X}_{3} > σ_{X} and V aR(X_{1}) + V aR(X_{2}) + V aR(X_{3}) > V aR(X).

This corresponds to the definition of subadditivity.

The capital which is resigned to each business line, in case of the proportional allocation, can be calculated with the following formula:

K_{i}^{P} = ρ(X)
Pn

j=1ρ(Xj)ρ(X_{i}). (25)

For the 3 business lines used in this example, the following proportional alloca- tion is given.

• K_{1}^{P} = 7.432

2.807 + 6.277 + 5.614· 2.807 ≈ 1.419 (26)

• K_{2}^{P} = 7.432

2.807 + 6.277 + 5.614· 6.277 ≈ 3.174 (27)

• K_{3}^{P} = 7.432

2.807 + 6.277 + 5.614· 5.614 ≈ 2.839 (28) The allocations add up to 7.432.

The formula for the capital allocation of the Euler rule is:

K_{i}^{E} = ∂

∂λi

V aR_{α}(X

i∈N

λ_{i}X_{i}). (29)

By making use of the definition of the VaR for the Guassian distribution, this formula can be rewritten to:

K_{i}^{E} = µi+ Φ^{−1}(α)cov(X, Xi)

pvar(X). (30)

Using the formula for the cov(X, Xi) (cov(P3

j=1Xj, Xi) =P3

j=1cov(Xi, Xj)), the results for the capital allocation is as follows:

• K_{1}^{E}= 0 + 2.807 · 1.742

√7.011 ≈ 1.847 (31)

• K_{2}^{E}= 0 + 2.807 · 4.106

√7.011 ≈ 4.353 (32)

• K_{3}^{E}= 0 + 2.807 · 1.164

√7.011 ≈ 1.234 (33)

All resulting values from both allocations is shown in a clear table below.

Proportional allocation Euler allocation

Line 1 1.419 1.847

Line 2 3.174 4.353

Line 3 2.839 1.234

Total firm 7.432 7.432

The allocations of the individual business lines add up to the one of the total firm. The small difference in the Euler allocation is due to the intermediate rounding off. Line 2 is positively correlated and has the highest variance. As a result, the risk capital for this line is the highest. Business line 3 is negatively correlated with line 1 and 2. The correlation is not taken into account in the proportional allocation. Because it is part of the calculation of the Euler one, the risk capital for line 3 is the lowest. The reason why line 1 receives the lowest risk capital in the proportional allocation is because X1has the lowest variance.

### 3 Methodology

When simulating losses to get eventually to the risk capital allocation, there must first be a decision whether the losses are iid distributed or not. In this section, both cases are covered. Not only in the most straightforward way, but also with different numbers of business lines and different ordering. The Value- at-Risk risk measure is considered for the application to find the risk capitals.

The chosen allocation methods are the ones based on the proportional and Euler allocation rule. Results were obtained in the programming language R.

### 3.1 Risk measures and allocation methods

The formulas to determine the risk capitals with the proportional and Euler allocation are already shown and elaborated in the theoretical framework. Be- cause, in this subsection, there is assumed that the losses are iid and Gaussian distributed, the formulas which can be used are:

K_{i}^{P} = V aRα(X)
Pn

j=1V aRα(Xj)V aR_{α}(X_{i}), (34)
K_{i}^{E} = µi+ Φ^{−1}(α)cov(X, Xi)

pvar(X). (35)

As mentions in the begin of Section 3, the chosen risk measure to work with is the Value-at-Risk.

V aRα(X) = µ + qασ (36)

where µ is the mean of X, σ the standard deviation and qα= Φ^{−1}(α). α is set
equal to 99.5% and so ρ(X) = V aR0.995(X). Thereby, Φ^{−1}(0.995) = 2.807. The
standard deviation of X, which is needed for the Euler method, results from
pvar(X). In the case of three lines, the variance of X can be calculated by

var(X) = var(X1) + var(X2) + var(X3)+

2cov(X_{1}, X_{2}) + 2cov(X_{1}, X_{3}) + 2cov(X_{2}, X_{3}). (37)
The covariance of, for example, X and X_{1} can be found by making use of

cov(X, X_{1}) = cov(X_{1}, X_{2}) + cov(X_{1}, X_{3}) + var(X_{1}). (38)
In R, there are built-in functions to find the Value-at-Risk, the variance
and the covariance. For the Value-at-Risk this is the command ‘quantile’. The
variance and covariance can be found with the simple commands ‘var(...)’ and

‘cov(...,...)’. In this function, the matrix of which the quantile has to be taken and the percentage can be filled in. The simulation to find the quantile for each business line is done 10.000 times. So the resulting Value-at-Risk matrix, which includes, say, 3 business lines, is 10.000 rows by 4 columns. This matrix can now be used to determine the risk capitals using the two formulas given in the beginning of this subsection.

### 3.2 Iid distributed losses

To start the different simulation processes, first a general simulation is done that can be used as a comparison with processes mentioned later in this section.

This general simulation contains 3 business lines and is multivariate-Gaussian distributed. That means that every individual line is Gaussian/normally dis- tributed. This first simulation corresponds to the one in Section 2.3. The only difference is that it is now generated in R and not calculated by hand. The preliminary information is given as follows:

(X_{1}, X_{2}, X_{3}) ∼ N (0, Σ_{3}),

with

Σ3=

1 0.6√

5 −0.6 0.6√

5 5 −√

5

−0.6 −√

5 4

. The corresponding correlation matrix is equal to

1 0.6 −0.3

0.6 1 −0.5

−0.3 −0.5 1

.

With this information available, the simulation can run to find values for Xi,1, Xi,2 and Xi,3for i from 1 to 1.000. Xi is again assumed to be the loss of busi- ness line i.

There is most likely a difference in capital allocation when the number of business lines differs. Therefore, the simulation from above is done again, but then with only 1 and 2 lines and extended to 4 and 5 lines. In the case of 1 line, there is no correlation with different lines. The resulting values for this line are automatically the values for the whole firm.

X1∼ N (0, 1)

When there are 2 lines involved, they are in this case positively correlated.

(X1, X2) ∼ N (0, Σ2),

with ρ1,2 = 0.6 and

Σ_{2}=

1 0.6√

5 0.6√

5 5

.

When increasing the number of business lines to 4 or 5, the variance-covariance and correlation matrix will be more extensive too.

(X1, X2, X3, X4, X5) ∼ N (0, Σ5),

with

Σ5=

1 0.6√

5 −0.6 0.9√

3 −1.1√ 2 0.6√

5 5 −√

5 0.6√

15 0.1√ 10

−0.6 −√

5 4 −1.2√

3 2√

2 0.9√

3 0.6√

15 −1.2√

3 3 −1.3√

6

−1.1√

2 0.1√

10 2√

2 −1.3√

6 8

.

Thereby, the corresponding correlation matrix turns into:

1 0.6 −0.3 0.9 −0.55

0.6 1 −0.5 0.6 0.05

−0.3 −0.5 1 −0.6 0.5

0.9 0.6 −0.6 1 −0.65

−0.55 0.05 0.5 −0.65 1

.

These matrices corresponds obviously to a firm with 5 lines. When it is the case that there are only 4 lines involved, the last row and column of both matrices will no longer be needed and can be removed.

According to the theory, the risk capital allocation will improve when there are more business lines involved. This capital will be calculated in two ways;

with the proportional and the Euler allocation. In the previous subsection, these two methods are discussed. In the next section, in contrast with this section, the losses are non-iid distributed.

### 3.3 Non-iid distributed losses

As mentioned in the start of Section 3, there is a difference between iid dis- tributed losses and non-iid distributed losses. When the losses are iid, it is most advantageous/best to merge as much as possible. How the risk capitals respond to losses which are non-iid distributed, will be examined in this subsection.

There are two different ways in which the X and Z variables are related. In
both ways, there is assumed that Z is normally distributed with mean 0 and
standard deviation 1. This means that the Z of different business lines are all
identical. Just as in the earlier (sub)sections, X_{i} is assumed to be the loss of
business line i.

The first way which is investigated is as follows:

X1= Z (39)

X2= −Z (40)

X_{3}= Z (41)

X4= −Z (42)

X_{5}= Z (43)

It is not necessary to use all five business lines at once. To start the simu- lations, it was chosen to use 1 business line. Hereafter, business lines will be

added sequentially. In this way, it can become clear whether more lines is more profitable or not.

The last column of the resulting X-matrix is the sum of all Xi’s. When using, for example, 2 or 4 business lines, it is clear that the Z cancel each other out. Looking at the general formulas for the proportional and Euler allocation, it can be expected that in the Euler allocation something will go wrong with the calculation of the risk capital.

To avoid this problem, a second distribution is set up.

X_{1}= Z (44)

X_{2}= −1

2Z (45)

X_{3}= Z (46)

X_{4}= 1

2Z (47)

X_{5}= −Z (48)

In this set up, the losses cannot cancel each other out. When the firm consist of
2 lines, the sum of all X_{i} is equal to ^{1}_{2}Z. 3 business lines results in ^{3}_{2}Z, 4 lines
in 2Z and finally 5 lines in Z again. In this second distribution, the business
lines will also be added sequentially.

In both ways, with various amounts of business lines, the risk capital alloca- tion is determined with the proportional and the Euler method. In the end, all resulting values and graphs can be compared and so a conclusion can be drawn about the best setup and the most advantageous number of business lines.

### 3.4 Number of business lines

In the previous subsections, three different ways of simulations have been exe- cuted; iid distributed losses and two ways of non-iid losses. Each way contains 5 individual simulations. In all three ways, it starts with 1 business line. There- after, a second line is added. In this way, an additional business line is added sequentially.

In the theoretical framework there is already stated from the literature that a different number of business lines affect the risk capital allocation. In the case of iid and Gaussian distributed losses, it is most advantageous for the firm to merge as much as possible. In other words, it is best to have as many business lines as possible. For non-iid losses, there has not yet been a clear statement.

This is why adding the lines sequentially is important for this study. For the iid losses, it is possible to check whether the results are consistent with the literature. In doing so, it can be examined whether the same applies to non-iid losses. Will it also be most advantageous for the non-iid losses to merge as much as possible or does this principle not hold? The confirmation/rejection of these statements and answers to these questions will be found in Section 4,

‘Results’, after comparing the resulting graphs of the simulations and discussing the findings.

### 4 Results

As the name of this section already suggests, all the results of the simulations explained in the ‘Methodology’ are presented here. The structure will be as follows. The resulting graphs of each simulation will be displayed in an ordered manner. The graphs being compared are placed together in one graph or side by side so that the differences will be visually clear. When the graphs are displayed side by side, the values are less easy to compare, but 12 or 15 business lines together in 1 graph becomes too much and confusing.

First, the simulations of iid and non-iid losses are shown independently.

After that, the different distributions will be compared with each other.

Although there is stated in the methodology that the first simulation is done with three lines and thereafter with 1, 2, 4 and 5 lines, the representation of the graphs will start with 1 line and than add lines sequentially. This gives a better overview.

### 4.1 Iid distributed losses

This subsection will cover the resulting graphs of the simulations with iid dis- tributed losses. As already mentioned in earlier sections, this starts with the findings containing 1 business line and sequentially moves to 5 business lines.

Figure 1: Proportional and Euler allocation when including 1 business lines; iid distributed losses. With 1 business line both methods should result in the same risk capital.

In the merged graphs, I have tried as much as possible to give the Euler allocation the light version of the corresponding color of the proportional allo- cation. In Figure 1 it is clear for business line 1 (the only one in this case); blue is the proportional allocation and light blue the Euler allocation. It is clear to see that the peak of the Euler method is at the same level as the one of the proportional. However, the spread of the proportional is bigger. The propor- tional is spread over the range of approximately 2.3 to 3.3 and the Euler over

2.6 to 3. The values of the Euler method are all very close to each other. The
reason for this is because the covariance is taken of X1and X1. This is the same
as the variance of X1. The term by which the inverse Gaussian is multiplied
becomes in this case equal to 1. Therefore, the peak of the Euler allocation is
at the value Φ^{−1}(0.995) = 2.807. The small difference in the resulting values is
because the simulation error. Filling in the formula of the proportional method,
this results eventually in Pn^{σ}^{X}

j=1σX jΦ^{−1}(0.995)σ_{X i}. When only 1 business lines
is included, σX, Pn

j=1σX j and σX i are all equal to 1. So, the resulting risk
capital when using the proportional method is equal to the one of the Euler
method; Φ^{−1}(0.995) = 2.807.

Figure 2: Proportional and Euler allocation when including 2 business lines; iid distributed losses. The risk capital for line 1 is the same, for line 2 it differs.

Figure 2 resembles Figure 1 in some cases, the histogram of the Euler allo- cation are more peaked than the proportional one. For line 1, there is almost no difference. The peak is approximately at the same value; 2.2. This is lower than the 2.807 which was appropriate for 1 business line. So for line 1 it is more advantageous to go together with an additional line. The average value for line 2 differs between the proportional and Euler method. The Euler allocation gives an average that is in general 1 higher than the proportional average. The lowest value of both methods differs 1.5, while the highest value differs only 0.4.

Where the additional line in the previous case, Figure 2, had a higher average allocation than the already existing line, this is not the case in Figure 3. At the proportional allocation, line 3 ends within line 1 and 2. With the Euler allocation, it is even much lower than line 1. The capital for line 1 is again decreased. This decrease is bigger for the proportional method than for the Euler method. The difference between the averages of the two methods for line 2 has increased. The most remarkable difference can be found at line 3.

With the proportional allocation it is almost equal to line 2, but with the Euler allocation it is even lower than line 1. The gap between line 2 and 3 with the

Figure 3: Proportional and Euler allocation when including 3 business lines; iid distributed losses. The risk capital for each line is different for the two methods.

Euler method is very large; approximately 3. Looking at the specific lines on itself; for line 1 and 2 the Euler method has more risk capital, but for line 3 it is actually less than the proportional.

Compared to the case with 2 business lines, all risk capitals have become lower. For the two existing business lines it is therefore more advantageous to merge with a third one. But also for this extra line, it is profitable.

Figure 4: Proportional and Euler allocation when including 4 business lines; iid distributed losses. The risk capital for each line is different for the two methods.

As can be seen in Figure 4, there is quite a big difference between the two methods. The first thing to notice is that there is much more overlap between the lines at the proportional version. The histograms of the Euler allocation are much more peaked. Moreover, the histogram of line 4, the yellow one, is the largest with the proportional and the smallest with the Euler. This line ended in both cases between line 1 and 2.

With the transition from 3 to 4 lines, the capital allocation for lines 1, 2 and 3 remained approximately the same with the proportional method. At

the Euler allocation, line 1 and 2 are both increased with a half. In contrast with line 3, which has dropped with 2 to a negative value; from 1.2 to -0.8.

This is not usual, but it is possible if the risk is negative. The overall spread of the proportional method, 1 - 4, is smaller than of the Euler method, -1.5 - 5.5.

Figure 5: Proportional and Euler allocation when including 5 business lines; iid distributed losses. The risk capital for each line is different for the proportional method. For the Euler allocation, line 1, 3 and 4 and line 2 and 5 are approxi- mately the same.

Just as could be seen in the last graphs, there are again some changes with the Euler allocation in Figure 5. The proportional ensures an almost equally spread over the range from 0.8 till 4. Whereas the lines from the Euler are almost split in two. Line 1, 3 and 4 and line 2 and 5 are almost equal to each other. Another big difference is the size and average value of line 5. With the proportional allocation, line 5 has the highest capital value and also the highest frequency. Unlike the Euler allocation where line 5 has a slightly lower value than line 2 and the lowest frequency of all 5 lines.

With the proportional allocation, the average values for the first four lines decreases when adding a fifth line. In the Euler case, the value of line 3 has become positive again, so it increased a bit. Line 1, 2 and 4 have been assigned a lower value just like in the proportional method.

When 2, 3 or 4 business lines are considered, in all three cases the risk cap- ital for line 2 is the highest. Even when a fifth line is added, line 2 is still the highest using the Euler method. Using the proportional method, the fifth line does get the highest capital.

As already mentioned in the theoretical framework and in the beginning of this section, it turns out to be most advantageous for firms with iid and Gaussian distributed losses to merge as much as possible. This will provide the most beneficial risk capital allocations with both the proportional method and the Euler method. To make this more clear, a graph is shown which shows only line 1, but for n = 1 until n = 5. As can be clearly seen in Figure 6, for the proportional allocation, the risk capital decreases with every additional line.

Figure 6: Proportional and Euler allocation when only the 5 first lines are included; iid distributed losses.

### 4.2 Non-iid distributed losses

In the previous subsection, the iid distributed losses are discussed. Now it is time for the non-iid lossses. First, the two different distributions are discussed separately. Thereafter they will be compared. Again, the process starts with including 1 business line. Then add sequential lines to end at 5 business lines.

The initial distribution that is being investigated is

X1= Z (49)

X2= −Z (50)

X3= Z (51)

X4= −Z (52)

X_{5}= Z (53)

Figure 7: Proportional and Euler allocation when including 1 business line; first non-iid distributed losses. With 1 business line both methods should result in the same risk capital.

Figure 7 looks very familiar. This is because it is exactly the same as the graph for 1 business line and iid distributed losses, Figure 1. The peak of the

Euler method is located at 2.807, which is equal to Φ^{−1}(0.995). As expected,
the peak of the proportional method is found at the same level.

Including 2 business lines results in a very weird graph. The graph has not
been added because it does not contribute to a more clear picture of the allo-
cation. There are together 4 lines; 2 of the proportional method and 2 of the
Euler method. Yet only 1 purple block was visible. This is because X_{1}= Z and
X_{2}= −Z cancel each other out. The sum of the business lines, X = X_{1}+ X_{2},
equals 0. The var(0) and the partial derivative do not exist, so the Euler rule
does not operate. Because X = 0, the resulting values of the proportional
method all turn out to be zeros. V aR(0) = 0 and dividing zero by any number
is zero again. The purple block is a combination of the blue of line 1 and the
red of line 2, both containing only zeros.

Figure 8: Proportional and Euler allocation when including 3 business lines;

first non-iid distributed losses. In both methods, line 2 and 3 have the same risk capital as line 1.

Figure 8 is another remarkable graph. All lines of the proportional method result in the same values. This is also the case with the Euler method. The his- tograms of the two methods have the same figure but than mirrored. The aver- age value of the Euler method, 2.8, is higher than that of the proportional, 0.85.

The 2.8 comes from the Φ^{−1}(0.995) = 2.807. X is equal to Z − Z + Z = Z. The
covariance between Z and Z is the same as between Z and −Z. So, the last term
will in all three cases be the same and the resulting average will end at 2.807.

Looking at the proportional formula, the V aR(−Z) is the same as the V aR(Z).

The formula turns into K_{i}^{P} = ^{V aR(X}_{3V aR(X}^{1}^{+X}^{2}^{+X}^{3}^{)}

i) V aR(X_{i}) = ^{V aR(X}_{2V aR(X}^{1}^{+X}^{2}^{+X}^{3}^{)}

i) . When computing i = 1, 2, 3, they will all result in approximately the same val- ues. So the average of all three lines will be equal to 0.85. On average, the Value-at-Risk of the whole firm is 1.7 times as big as the Value-at-Risk of one business line.

In the case with 4 business lines, the same problems arise as when there are

2 business lines. That is why this graph is also omitted. The resulted block for the proportional allocation with the greenish color is a combination of the blue of line 1, the red of line 2, the green of line 3 and the yellow of line 4. These colors are combined, because they all have the same risk capital, namely 0. The values resulting from the Euler method are NaN (Not a Number), so not visible in the graph. All values of the proportional method are again zero, resulting in one block at zero.

Figure 9: Proportional and Euler allocation when including 5 business lines;

first non-iid distributed losses. In both methods, line 2, 3, 4 and 5 have the same risk capital as line 1.

Above graphs in Figure 9 are similar to the graph of 3 business lines, Figure 8. The only difference is that the average value from the proportional method is decreased from 0.85 to 0.5. The average of the Euler method is still 2.8.

While the peak of the proportional value with 3 lines reaches 3000, with 5 lines it reaches only 2000. The graph of the Euler part is the same with 5 lines as with 3 lines. Focused on the two methods only applied to 5 business lines, the frequency between them differs. The highest frequency which is reached by the proportional method is 2000. The average value of the Euler method occurs 1.5 times as often, which results in a frequency of 3000. In value terms, the difference in spread between the methods is very small, only 0.2. But compared with each other, the spread of the Euler is almost two times as big as of the proportional method. Only looked at the shape of the resulting histograms they resemble each other very much.

Looking at all 5 graphs, the average of the Euler method stays the same at the value 2.807. This is because the factor with which this value has to be multiplied is equal to 1. This of course only applies to an odd number of busi- ness lines. As predicted in Section 3, for an even number of business lines, the losses cancel each other out and there are no resulting values. The values for the proportional method decrease when going from 1 to 3 to 5 business lines. For an even number of business lines it is equal to 0. Therefore, looking at the whole, it is also most advantageous with this distribution to merge as much as possible.

Given the problems which arise with an even number of business lines, it would be best to eventually work with an odd number of business lines in the company.

The next distribution that is being investigated is

X1= Z (54)

X2= −1

2Z (55)

X3= Z (56)

X4= 1

2Z (57)

X5= −Z (58)

Figure 10: Proportional and Euler allocation when including 1 business line;

second non-iid distributed losses. With 1 business line both methods should result in the same risk capital.

It is not remarkable that Figure 10 is again the same as the previous two which contain 1 business line. In all three cases, X1 is set equal to Z. In the firm with iid losses, different business lines have different variances. For line 1, the variance of Z is chosen so that, as for the two ways with non-iid losses, it is standard normally distributed. This implies that the mean is equal to 0 and the standard deviation equal to 1. All X1 are standard normally distributed, resulting in three times the same graph.

To make the combination graph more clear, Figure 11 also shows the graphs for the proportional allocation only and the Euler allocation only. Now it is easier to see which histogram belongs to which business line and which allocation method, so they can be discussed.

It is obvious that the Euler results are more extreme than the proportional
values. The peaks of the proportional method are around 0.75, line 1, and 0.4,
line 2, and those of the Euler method at 2.8, line 1, and -1.4, line 2. The dif-
ference between the Euler values is explainable. X1 is set equal to Z. Z is
standard normally distributed, so the risk capital equals Φ^{−1}(0.995). X2 is set
equal to −^{1}_{2}Z. Multiplying the 2.8 of line 1 with −^{1}_{2} results in -1.4. For the
proportional value, the minus-sign does not count. Line 2 has approximately

Figure 11: Proportional and Euler allocation when including 2 business lines;

second non-iid distributed losses. With both methods, the risk captial of line 1 is higher than of line 2. Euler method is more peaked and extreme.

half the risk capital of line 1. Comparing these values with the previous graph, the average value of the Euler line 1 is stayed the same. The additional line even has a negative risk capital. The proportional line 1 has decreased and again the additional line ends at a lower value. So adding an extra line was in general a beneficial decision.

The main difference between Figure 11 and Figure 12 is that the negative
value of one of the Euler lines, line 2, has become positive again. The three
resulting histograms of the three business lines, are all the same. The resulting
average is again settled at 2.8. The risk capital of line 1 of the proportional
method is increased from 0.75 to 1.5. Line 2 is now located at 0.75 and the
additional line is ended the same as line 1. Given the distributions of X_{1}, X_{2}
and X3, it is again explainable that line 2 lies on the half of line 1 and 3. Namely,
X1 and X3 are assumed to be equal to Z and X2at −^{1}_{2}Z.

Compared to 2 lines, the risk capitals of line 1 and 2 only increased or re- mained the same. So adding a third line has not been beneficial to all the already existing lines.

In Figure 13, the graph of the proportional method looks more clear than

Figure 12: Proportional and Euler allocation when including 3 business lines;

second non-iid distributed losses. With the proportional method, line 3 is the same as line 1. For the Euler method, all three lines have the same risk capital.

Figure 13: Proportional and Euler allocation when including 4 business lines;

second non-iid distributed losses. Using the proportional method, line 1 and 3 and line 2 and 4 are the same. With the Euler method this only applies to lines 1 and 3

the one of the Euler method. This is because the 4 lines from the proportional results in values between 0.5 and 2.5. The Euler results in more divided values.

In this case, values as -2 or 3 are almost taken. The fourth line which is added
is assumed to be distributed as ^{1}_{2}Z. Yet the resulting histogram is the same as
line 2, which is distributed as −^{1}_{2}Z. This makes it clear that the formula for the
proportional allocation does not take into account the minus-sign. In contrast
to the formula for the Euler allocation, which does take it into account. For
line 1 and 3, which are assumed to be equal to −Z, the risk capital is set to 2.8.

Line 2, −^{1}_{2}Z, has a risk capital of −^{1}_{2}· 2.8 = −1.4 and line 4 of ^{1}_{2}· 2.8 = 1.4.

Adding a fourth line to the firm makes line 2 return to a negative value. Both risk capitals for the proportional allocation found in the case with 3 lines are increased when a fourth line is added. Adding a fourth line was therefore not advantageous for the proportional allocation to the already existing lines.

The same pattern found in Figure 13 is continued when a fifth line is added,

Figure 14: Proportional and Euler allocation when including 5 business lines;

second non-iid distributed losses. Using the proportional method, line 1, 3 and 5 and line 2 and 4 are the same. With the Euler method this only applies to lines 1 and 3

so in Figure 14. Line 5, X_{5}, is distributed as −Z. Looking at the 4 line case,
this would mean that the risk capital for line 5, using the Euler formula, would
be equal to -2.8. The graph shows that this is indeed the case. Since the pro-
portional formula does not take into account the minus-sign, line 5 would end
up with the same risk capital as line 1 and 3. This is also clearly visible in the
graph shown above. When adding a third or a fourth line, using the propor-
tional allocation, the risk capitals for the already existing lines went up. This
is not the case when a fifth line is added. The average risk capitals are now
dropping from 1.7 to 0.6 and from 0.8 to 0.3. The added line with the Euler
allocation is also lower than the capitals of the other 4 lines. The transition
from 4 to 5 business lines is therefore advantageous for the firm.

Discussed in Section 4.1, were the iid distributed losses. In this case, it is most advantageous for the firm to merge as much as possible. The more business lines, the more beneficial it is for an individual business line. This is not in line with when the losses are non-iid. Considering the first distribution of this section, it is not possible to have an even number of business lines. When going from an even number to an uneven number of lines, the risk capitals will always increase. The second distribution ensures that at every number of business lines the risk capital exists. Nevertheless, not every additional business line ensures lower risk capitals. From 1 to 2 lines, it does decrease. However, from 2 to 3 lines, the risk capital either stays the same or increases. For the proportional allocation this is also the case when adding a fourth line. Unlike the Euler allocation, where risk capital remains the same or even decreases.

Adding the last and fifth business line, both allocation methods provide the same of lower risk capital.

It can be concluded that whether the risk capital decreases or increases, in case of non-iid distributed losses, depends on the kind of distribution.

### 4.3 Comparing iid with non-iid distributed losses

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 allocation. 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 Euler 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