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Risk Capital Allocation in Financial

Conglomerates

Natasha Noskova

Bachelor’s Thesis to obtain the degree in Actuarial Science

University of Amsterdam

Faculty of Economics and Business Amsterdam School of Economics

Author: Natasha Noskova Student nr: 6402445

Email: natasha.noskova@gmail.com Date: June 29, 2016

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1 Introduction 1

2 The Risk Capital Allocation Theory 4

2.1 Academic Literature . . . 4

2.1.1 Approaches to Risk Capital Allocation. . . 4

2.1.2 Capital Allocation Problem and the Concept of a Risk Measure . . . . 5

2.2 Risk Capital Allocation Methods . . . 9

2.2.1 Euler . . . 9

2.2.2 Proportional Allocation Methods. . . 10

3 Implementation and Analysis of Capital Allocation Methods 12 3.1 The Organization of this Research . . . 12

3.1.1 General Information and Assumptions . . . 12

3.1.2 Calculating 99.5% VaR or 99% ES. . . 14

3.1.3 Information on the Implementation of the Proportional Allocation Methods . . . 14

3.1.4 Information on the Implementation of the Euler Method . . . 15

3.2 Practical Results . . . 15

3.2.1 Case 1: The Default Setting . . . 16

3.2.2 Case 2: Volatility of Losses. . . 16

3.2.3 Case 3: Correlation Structure Between Losses. . . 18

3.2.4 Case 4: Heavy-Tailed Distribution (Simulation) . . . 19

3.3 General Analysis of the Results. . . 20

4 Conclusion 24

References

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Introduction

In view of financial crisis 2007-2008, the risk of bankruptcy has been a paramount topic, especially in banking and insurance industries. Changes to financial regulation in the form of the Solvency regime testify to this statement. Solvency directives unify EU regulation on the subject of the amount of capital that insurance companies are required to hold by regulatory bodies in order to reduce the risk of not being able to meet their obligations. This capital is called regulatory. It is assessed at the firm level and is significantly higher under Solvency II Directive, which is in effect since January, 2016. Consequently, the problem of appropriate risk management, which includes an estimation of required capital and its allocation within the firm has never been more to the point. After all, the extra capital held by companies creates costs that need to be properly allocated back to the business units.

The regulatory capital mentioned in the first paragraph requires an amount of risk-free capital added to the firms assets to ensure that the value of the firm is adequate in case of the worst case scenario. On the other hand, the capital that is important for the capital risk allocation process is called economic capital (also known as risk capital). Regulatory capital is slightly different from the economic capital which represents the best estimate of financial institutions in order to internally manage their risk. Ideally, regulatory and economic capitals of a firm should be close to each other and are meant to be a buffer against the effects of unexpected losses in the form of investments held at extremely low risk. From a financial institution’s perspective, holding an amount of money in low risk is burden, because of the low return on this amount, which instead could be invested with much higher return.

To allocate the cost of maintaining regulatory capital among different units within an organization, the calculation of economic capitals of these units is necessary. First economic capitals are calculated at the unit level of an organization. Further, they are aggregated, taking into account correlations between the risks related to these business units. These correlations create a diversification effect, leading to a smaller aggregate economic capital than the sum of all individual parts. (The aggregation of economic capital performed for the purposes of this research is further discussed in Chapter 3.1.1, while the obtained practical results are provided in Chapter 3.2). Finally, the diversification benefits need to be allocated back to the business

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2 Natasha Noskova — RISKCAPITALALLOCATION INFINANCIALCONGLOMERATES

units in a fair manner.

There are a number of reasons why the allocation of risk capital at the unit level is im-portant for companies. The most significant one is performance measurement which is valuable for the overall risk management of the company and its decisions about expansions (reductions) or management compensation. Another reason is related to the requirements of financial re-porting. Further, the proper distribution of capital costs to the business units allows for a better pricing of insurance policies so that these costs are transferred to the policyholders in the form of premiums.

Central in the risk capital allocation process is the concept of a risk measure which is in detail discussed in the next chapter. However, it can be briefly illustrated here since it is essential in the risk allocation process. Take, for example, the most used risk measure at the moment, Value at Risk (VaR). An economic capital that equals daily 99.5% VaR is an amount equal to a 99.5% quantile of the associated distribution of losses. Reserving this economic capital should ensure the satisfactory economic capital for the worst case loss that happens with the probability of 0.5% daily. Therefore, the calculations of economic capitals are risk measure based. In other words, an economic capital is represented by a chosen risk measure, which actually calculates the amount necessary for the company to stay solvent over the measurement period, such as year, given the chosen confidence level and the corresponding distribution of losses.

The idea of a risk measure evolved over many years, and following this evolution can clarify why certain risk measures are still used today. Since Markovitz proposed to measure the risk of investments by variance of outcomes in 1952, this is how risk has been measured for over forty years. Variance as a risk measure is still used by some insurance companies nowa-days. Only starting in 1994, when JP Morgan published their RiskMetrics, did VaR become popular (Jorion, 2001) in financial industry as well as with regulators. VaR is an attractive mea-sure because it is intuitively easy to understand, however, it isn’t always accurate due to the fact that it’s not subadditive, and thus not coherent. The concept of a coherent risk measure and the associated requirement of subadditivity is further discussed in Chapter 2.1.2. The initiative to use coherent risk measures is much debated in academic literature related to the risk capital allocation. Also, there seems to be a recent trend to move to coherent risk measures in prac-tice. Insurance companies pioneer this initiative in financial industry and some regulators are already moving to a coherent risk measure, Expected Shortfall (ES), as evidenced by the Swiss Insurance Supervision Act in January 2006.

Given the recent developments in the measurement of insurance risks, and the influence that the risk measures have on capital risk allocation, the focus of this research is evaluation of the widely used risk measures, coherent and non-coherent, implemented with the most popular allocation methods. The academic literature points to coherent risk measures combined with co-herent allocation methods as most optimal risk capital allocation methods (Denault, 2001) that

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are also suitable for performance measurement. In practice however, non-coherent risk mea-sures and allocation methods are more popular, since they are conceptually easier to understand and are uncomplicated in application. Proportional allocation methods are most direct, and thus, they have been frequently used in quantitative risk management together with the risk measures mentioned above. An example of a coherent allocation method that is gaining popularity is the Euler allocation principle, which ideally used with a coherent risk measure such as Expected Shortfall (ES). However, insurance companies often use the Euler allocation together with VaR, since it is the dominant risk measure as of today. The natural question is then: how signifi-cant is the influence of such liberal use of risk measures and allocation methods on risk capital allocation outcomes? In this thesis performance of proportional allocation methods (haircut, covariance, and CTE) is compared with the performance of the Euler allocation method when using two most popular risk measures: VaR and ES.

The organization of this thesis is as follows. The next chapter provides background re-lated to academic discussion on capital risk allocation. Also the theory on risk measures and risk capital allocation methods is presented in this chapter. Then the research organization is discussed in Chapter 3, as well as the results of the research and their analysis. Finally,

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Chapter 2

The Risk Capital Allocation Theory

2.1

Academic Literature

2.1.1 Approaches to Risk Capital Allocation

The major challenge of risk capital allocation is caused by the fact that there are no common principles for allocating risk capital that suit all. The literature written on this issue reveals a great number of allocation approaches based on different fields of study. The fact that there is no agreement on the risk measure that needs to be used by the allocation complicates the issue even more, since the use of the risk measure has a significant influence on the allocation results. Main approaches to risk based allocation will be discussed in this chapter, focusing on approaches that produced the most popular allocation methods, such as proportional allocation methods and the Euler method.

In academic literature the most known approaches to capital risk allocation stem from the mathematical optimization (Dhaene, Goovaerts and Kaas, 2003; Laeven and Goovaerts, 2004, Dhaene et al., 2011), game theory (e.g., Denault, 2001; Tsanakas and Barnett, 2003) or finance (Tasche, 2000; Myers and Read, 2001). They all offer a different perspective on risk capital allocation as well as reinforce each other when the same allocation method results from two different fields of study, as the Euler method. Each of these points of view on capital risk allocation are reviewed in the paragraphs that follow.

Dhaene et al. (2011) discuss proportional allocation methods as outcomes of optimiza-tion. These allocation methods are simple to use and allocate the aggregate risk capital to a particular business unit based on the ratio of a risk measure of this unit to the sum of all units’ risk measures. (The proportional allocation methods are further discussed in Chapter 2.2 and Chapter 3 of this thesis).

Another perspective on risk capital allocation was proposed by Denault (2001). Using the concepts of the game theory, he justified the Aumann-Shapley method as a coherent allocation method, i.e. the method that is in accordance with a number of desirable theoretical princi-ples. Denault (2001) specified these principles as the properties of a coherent allocation. (The

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concept of a coherent allocation method is further discussed in Chapter 2.1.2). The Aumann-Shapley method assumes a coherent risk measure as described by Artzner et al. (1998). (More background on coherent risk measures is provided in Chapter 2.1.2). Furthermore, this method (also know as the Euler method, which is discussed in Chapter 2.2.1) was corroborated by the papers of Tasche (2000) and Myers and Read (2001) that present the same method, as derived by Denault (2001), from a different perspective.

Myers and Read (2001) developed a model based on option pricing. They noted that every insurance contract has a certain level of default risk. If the insurance company to purchase an option to transfer any liabilities to a third party in the event of a default, than the cost of this put option should be related to the volume of risky insurance liabilities as a cost per unit. Thus, the derivatives are used as a measure of changes in the aggregate risk capital due to very small changes in liabilities of a business unit. This method has an advantage over the other marginal allocation methods because marginal increments in this method add up to the total aggregate capital, just like in the Euler allocation method.

Tasche (2000) also advocates the Euler method. According to him, the RORAC (Return on Risk-Adjusted Capital), compatibility is the most important economic property of a risk capital allocation principle. Tasche (2000) states that the demand for risk capital allocation originates from the practicality and importance of performance measurement. Knowing the risk taken by that unit allows for a better comparison as opposed to knowing only profits of that unit. For example, if a certain business unit generates an average profit, but it hedges some other units, its value for the company as a whole is higher than just its profit. Thus, a properly allocated risk capital should be lower for this hedging unit. The RORAC Ratio demonstrates the real value of such a business unit.

RORAC= expected pro f it risk capital .

Tasche (2000, p.2) points out that one can not calculate RORAC, in the absence of a quantifiable risk capital, and poses it as a motive to derive his risk allocation method. Generally, the issues of measuring performance and risk are often connected in scientific literature. Denault (2001, p. 2) also closely associates performance measurement with the allocation of risk capital. In fact, there are a lot of similarities between the two above mentioned papers and they form a solid theoretical background for the problem of risk capital allocation.

2.1.2 Capital Allocation Problem and the Concept of a Risk Measure

Before specific allocation methods are reviewed in Chapter 2.2, several other theoretical concepts are discussed in this chapter which are necessary for the understanding of the capital risk allocation problem. This discussion is based on the papers of Tasche (2000) and Denault (2001) which were fundamental in the development of the current risk capital allocation theory.

As the first step in developing his capital allocation method, Tasche (2000, p.2) turned to Markowitz portfolio theory, since it offered a risk measure represented by a standard deviation.

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6 Natasha Noskova — RISKCAPITALALLOCATION INFINANCIALCONGLOMERATES

Immediately, he identified (Tasche 2000, p.2) that one has to look for another risk measure. Therefore, Tasche (2000, p.2,3) moved on to the other risk measures. Computing an overall risk capital, ρ(X ), where ρ is a particular risk measure, is in fact the first step in the two step procedure currently used in practice (McNeil et al, 2005):

1) Compute the overall risk capital, ρ(X ), where X = ∑di=1Xi.

2) Allocate the capital ρ(X ) to individual business units according to a chosen allocation method, in such a way that if Ki denotes capital allocation to the unit with a potential loss

Xi, the sum of the allocated amounts corresponds to the overall risk capital ρ(X ) :

ρ (X ) =

d

i=1

Ki. (1)

Risk measure is a function assigning a number to a random variable. Per formulas above, this random variable will be denoted by X, representing a firm’s loss. Formally, risk measure ρ is defined as a function mapping from a space of random variables Γ into R, the real line:

ρ : Γ → R.

In his paper, Tasche (2000, p.3) reviews three popular risk measures: the standard deviation, the VaR and the ES. Since risk measure is such an important idea in risk allocation theory, it is worth looking at a minimum of two most widely known and used ones in this research, the VaR and the ES. Yet before doing so, it is essential to understand the concept of a coherent risk measure, since Denault (2001, p.4) uses coherent risk measure to build his allocation method. According to Denault (2001, p.4) for a risk measure to reasonably quantify the level of risk, it needs to be coherent. It is defined as a risk measure that possesses the following properties:

1. Subadditivity: ρ(X +Y ) ≤ ρ(X ) + ρ(Y );

2. Monotonicity: X ≤ Y ⇒ ρ(X ) ≤ ρ(Y );

3. Positive homogeneity: λ ≥ 0; ρ(λ X ) = λ ρ(X );

4. Translation invariance: α ∈ R; ρ(X + αrf) = ρ(X ) − α. (rf is a riskless investment with

a price of 1).

(For all bounded random variables X and Y).

Subadditivity reflects the requirement that the merger of business units should not create extra risk. This requirement advocates diversification, since a diversified poftfolio is less risky then the sum of the risks related to its components. Monotonicity states that when Y generates more loss than X, it cannot be less risky than X. Therefore, the risk measure should reserve higher economic capital for the asset Y. Positive homogeneity is a controversial property, since accord-ing to subadditivity, there should be a diversificaton benefit. Positive homogeneity, on the other hand, states that doubling the portfolio should double the resulting risk. Denault (2001) calls positive homogeneity requirement a limit case of subadditivity, that’s when the subadditivity equation has an equal sign. Translation invariance comes from the definition of a risk measure. Since rf is a riskless asset, it doesn’t add any risk to the risky portfolio but reduces its overall

risk. Therefore, a risk measure should reduce the loss of the risky portfolio by the amount of the riskless assets.

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Now that the concept of a coherent measure of risk is established, the properties of vari-ance, as a risk measure (Var), VaR, and ES can be reviewed. Variance is the oldest and the best known risk measure that is defined as follows:

Var(X ) = E[(X − E(X ))2] (2)

However, it doesn’t fulfill properties 1 and 2 of a coherent risk measure, and thus fails to be coherent. To demonstrate how variance fails the properties 1 and 2, Example 1 and Example 2 are given accordingly.

Example 1

To illustrate how variance fails the subadditivity property, take two losses, Y1

and Y2. Assume now that X1=12Y1and X2=12Y2. Then according to subadditivity

property, the following should be true:

Var(X1+ X2) ≤ Var(X1) +Var(X2) ⇒ Var(

1 2Y1+ 1 2Y2) ≤ Var( 1 2Y1) +Var( 1 2Y2). Therefore: 1 4Var(Y1) + 1 4Var(Y2) + 2Cov(Y1,Y2) ≤ 1 4Var(Y1) + 1 4Var(Y2).

Taking a positive correlation between Y1and Y2leads to a contradiction. Therefore,

it can be concluded that variance is not subadditive.

Example 2 This example clarifies how variance fails at monotonicity. Take for example a business unit with a loss distribution that generates either a profit or a loss of 1 with equal probabilities. Thus, the mean of the corresponding loss dis-tribution would equal to 0 and the variance to 1. Assume now that the second business unit, created for a comparison, always generates a loss of 1. Thus, its loss distribution has a mean of 1, and a variance of 0. Variance as a risk mea-sure, choses the second asset as less risky, contradicting the monotonicity property: X≤ Y ⇒ ρ(X) ≤ ρ(Y );

It is obvious that variance as a risk measure leads to extremely incorrect conclusions if the loss has a significantly different distribution than the Normal distribution. Since losses in the insurance industry are rarely normally distributed, they are mostly fat-tailed, this risk measure doesn’t seem to be suitable. It is also less widely used than the risk measures that follow, and therefore, it is not investigated further in this thesis.

Value-at-Risk is the risk measure that is often used in financial institutions. At a certain confidence level p ∈ (0, 1) it is a loss that can maximally be suffered with probability p:

VaRp(X ) = FX−1(p). (3)

VaR is then a quantile of the loss distribution. Thus, it is monotone, positive homogeneous and transition invariant. However, VaR is also not subadditive, as proved by Artzner et al. (1998).

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8 Natasha Noskova — RISKCAPITALALLOCATION INFINANCIALCONGLOMERATES

This means that VaR is not a coherent risk measure either. The lack of subadditivity lead to criticizm over the use of VaR for risk capital allocation purposes, since it can cause wrong choices in risk management, such as ignoring the benefits of diversification when adding new business units. Tasche (2000 p.3) also indicates that VaR is not broadly suitable for losses that are not normally distributed. This risk measure and variance can, therefore, be incompatible with some capital allocation methods.

Expected Shortfall (ES), on the other hand, is a coherent risk measure. It is more uni-versal and performs well where VaR fails (i.e. if the loss distributions are heavily tailed). At a confidence level p ∈ (0, 1), it is defined by:

ESp(X ) = 1 1 − p Z 1 p VaRu(X )du. (4)

Thus, it is the average loss in worst (1 − p)% of the cases.

Yet another risk measure associated with both VaR and ES is worth a review, because it can be easily used with proportional allocation methods discussed in Chapter 2.2.2. It is known as conditional tail expectation (CTE). Conceptually, just like ES, this risk measure is built on VaR. It quantifies the loss, given that the event outside the probability, p (as defined in VaR), has occurred. At confidence level p it is defined as:

CT Ep(X ) = E[X |X ≥ VaRp(X )]. (5)

There are no differences between CTE and ES when the distribution of X is continious. In such a case, it is a coherent risk measure.

Example 3 To illustrate how VaR and ES are calculated and the difference in risk estimation that they produce, the following example is offered. Let us define the loss distribution of X as a zero loss with a probability of 0.9, a loss of 1 with a probability of 0.08, and a loss of 10 with a probability of 0.02. Then 95% VaR is calculated as 95% quantile of this loss distribution and 95% ES represents the average of the worst cases beyond that quantile:

VaR95%(X ) = 1;

ES95%(X ) =0.03 ∗ 1 + 0.02 ∗ 10

0.05 = 4.6.

Analogous to coherent risk measure properties, Denault (2001) suggested principles of coherent allocation. According to him, the allocation method can be considered coherent if for every allocation problem (N, ρ) the allocations satisfy these principles:

1) Core compatibility: For all M ⊆ N, ∑i∈MKi≤ ρ(∑i∈MXi).

2) Symmetry: If by joining any subset, M ⊆ N{i, j} portfolio i and j both make the same contribution to the economic capital, then Ki= Kj.

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Also for all these allocations should hold: K = ∑dj=1Ki, the full allocation principle.

Core compatibility implies that individual risk or subsets of risks will not be better off leaving the full risk pool, i.e. allocations are in the core. The second principle, symmetry, states that allocated capital should reflect the contribution of a risk to the total portfolio. The purpose of the riskless allocation requirement is to ensure that addition of risk free assets to the portfolio properly decreases its risk by the same amount. These principles provide a much needed criteria against which allocation methods can be evaluated.

Denault (2001) focuses on coherent risk measures, while Tasche (2000, p.9) focuses on differentiability of common risk measures (the standard deviation, the VaR and the ES). The papers of Denault and Tasche are based on the ideas from different fields of study, yet they strongly reinforce each other’s ideas. Lack of scientific literature on risk capital allocation at the time, forced the authors to search for tools in other fields of study. Consequently, they induced their theories on risk capital allocation by putting restrictions on a general situation to narrow it down to more manageable cases. Denault (2001) did it by restricting himself to the use of coherent risk measures. He followed to establish a set of coherent allocation principles that further confined his work field. These principles produced an Auman-Shapley value as the most attractive allocation method, where per unit cost is an average of the marginal costs of the portfolio (Denault, 2001). Moreover, he noted that the core of the game consisted of a single vector, a gradient of the risk measure function. This is the same result that Tasche (2000) derived. Tasche made assumptions that restricted either the risk measure functions or the underlying distributions of risks to obtain his allocation in the form of weighted sum of the risk measure derivatives. Both papers advocate the Euler method as the one having the best theoretical properties.

2.2

Risk Capital Allocation Methods

2.2.1 Euler

The two papers discussed above laid the ground for Euler allocation method (also known as Auman-Shapley value, when a coherent risk measure, such as ES is used). The methods can be applied to risk measures that are positive homogeneous, see property number three of a coherent risk measure. Thus, this allocation method can be used not only with coherent risk measures but also with VaR.

Before providing the equation of the Euler allocations, Ki, assume the following three

statements. Lambda is a weight variable λ = (λ1, λ2, ...λd). Kirepresents the amount of capital

allocated to one unit of Xi, when overall position of loss is X (λ ), defined as X (λ ) = ∑di=1λiXi.

Also, rρ(λ ) = ρ(∑ d

i=1λiXi) is fully allocated to individual portfolio positions. Then the Euler

principle states: If ρ is a positive homogeneous risk measure function which is differentiable on the set Λ, then the per-unit Euler contributions are defined as:

Ki= dρ dh(X + hXi)|h=0= ∂ rρ ∂ λi (λ )|λ =(1,1,...,1). (6)

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10 Natasha Noskova — RISKCAPITALALLOCATION INFINANCIALCONGLOMERATES

Depending on the ρ used, the expression per above takes different forms.

If ES is used as the risk measure in the Euler contributions expression provided in the previous paragraph, the expression becomes:

Ki=

dESp

dh (X + hXi)|h=0= E[Xi|X ≥ VaRp(X )] = 1

1 − αE[Xi1(X ≥VaRp(X ))]. (7) Note, that this result has a technical proof and requires a number of assumptions on the differen-tiability of the risk measure function. Tasche (2000) offers an extensive proof of a comparable differentiation of VaR, providing these assumptions and advocating the proof. Therefore, the technical part is left out for the purposes of this thesis.

The Euler method is also called allocation by gradient, since the allocations of this method are represented by the sum of the risk measure derivatives as apparent from (6). Tasche (2000) showed that the only vector suitable for performance measurement (RORAC compati-ble) is the gradient of a smooth risk measure function, i.e. the Euler method.

2.2.2 Proportional Allocation Methods

Alternative to the Euler method are the historically used proportional allocation methods, which are much easier to calculate. This calculaton is presented by a simple ratio of a risk measure of a unit divided by the sum of all risk measures. Note that the risk measures used to calculate this ratio are not necessarily the same as the ones used to calculate an aggregate capital in (1). The risk measures displayed in Table 2.1, are specific to the chosen proportional allocation meth-ods. It is generally agreed, for example, that the haircut method uses VaR for the proportional allocation ratio. Thus, the use of the risk measure for the calculation of the allocation ratio is non-discretional and is dependant on the method.

Assume that the amount of the aggregate risk capital is assigned to K. Then Ki denotes

nonnegative amount of risk capital allocated to the unit i. Subsequently, the requirement that full K needs to be allocated to n business units is stated as follows:

K=

d

j=1

Kj; Ki≥ 0; i = 1, ..., d. (8)

Thus, the general proportional capital allocation formula is:

Ki=

ρ (Xi)

∑dj=1ρ (Xj)

· K; i = 1, ..., d. (9)

If one or more risk measures in the formula per above are negative then (in order to sat-isfy the requirement of the full allocation) the capital allocated to corresponding business units has to be set to zero. Note, that the risk measures of the concerned random losses have to be excluded from the sum in the denominator. Finally, if different ρs (which stand for the risk mea-sures) are filled in the last formula per above, it will produce different proportional allocation methods, such as covariance, haircut and CTE allocation methods.

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The specific formulas used to calculate the haircut, the covariance, and the CTE allocation methods are derrived in chapter three of this thesis. The Table 2.1 only lists risk measures that are used in the calculation.

Table 2.1: Proportional Allocation Methods

Allocation Method Risk Measure Haircut ρ (Xi) = FX−1i (p)

Covariance ρ (Xi) = Cov(Xi, X )

CTE ρ (Xi) = E[Xi|X ≥ FX−1i (p)]

The risk measures in this table are used to calculate the allocations for the proportional allocation methods listed per above.

Now that the understanding of the capital allocation theory is established, the next step is practical implementation of the Euler and the proportional allocation methods mentioned above, with VaR and ES as risk measures.

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Chapter 3

Implementation and Analysis of

Capital Allocation Methods

3.1

The Organization of this Research

3.1.1 General Information and Assumptions

The focus of this chapter is the practical application of the theory discussed in the previous chapter. This is in order to evaluate the widely used risk measures, coherent and non-coherent (ES and VaR) and to perform a comparative analysis of the major capital risk allocation methods that are based on these risk measures. The Euler method was compared to proportional allocation methods (haircut, covariance, and CTE).

The following principles were tested:

1. 99.5% VaR with the Euler method;

2. 99% ES with the Euler method;

3. 99.5% VaR with the haircut method;

4. 99% ES with the covariance method;

5. 99% ES with the CTE method.

The economic capitals of the business units used in the examples were calculated based on the risk measure chosen, 99.5% VaR or 99% ES. These confidence levels for the risk mea-sures were chosen because they are most commonly used in practice. For example, Solvency uses 99.5% VaR, while SST uses 99% ES. Further, the economic capitals were aggregated to get the aggregate capital, represented by letter K. The Bottom-Up approach to capital aggregation was used in this research, since it is standard in Solvency.

Before the aggregation process is reviewed, it is important to note that there are two ways to view economic capital. The most obvious one is expressing an economic capital of a business

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unit in a form of the risk measure of the business unit’s losses:

Economic Capital i(ECi) = ρ(Xi).

This is also the definition adopted from McNeil et al. (2005) and used earlier in 2.1.2, see (1). However, this definition of economic capital is incompatible with the Bottom-Up approach of capital aggregation. This leads to the second definition of the economic capital:

Economic Capital i(ECi) = ρ(Xi) − E(Xi).

ρ (X ) is the total balance sheet capital requirement, which includes both the expected loss, E(X ), and the unexpected loss. The financial industry considers the expected loss to be covered by the firm’s reserves, and therefore, only the unexpected portion of the loss needs to be covered by the economic capital. For the purposes of all examples in this thesis, the economic capitals of individual business units were calculated using the second definition. After computing the economic capitals for each of the business units, the Bottom-Up capital aggregation approach can be used to calculate the total economic capital for the firm (EC), which would include a diversification effect from pooling of the business units together. Thus, the EC is calculated as:

EC= v u u t d

i=1 d

j=1 ρi jECiECj.

ρi j denotes correlations between risks i and j.

Further, in order to effectively implement the allocation methods in practice, there are several assumptions that need to be made about the business units involved in the examples as well as the dependency structure between the risks.

For the four risk capital allocation examples in the next chapter, it is assumed that there are a total of three business units in the firm. The firm’s aggregate capital is calculated based on the risk measures and the expected losses for each of the units, as mentioned in the previous paragraph. This aggregate capital, K, needs to be allocated between all three of the business units, which results in allocations Ki. Based on the definitions used earlier in Chapter 2, Xi

represents the loss of the business unit i. X is then the aggregate loss of the firm. In example one through four, Xi0sare assumed to be normally distributed with the finite means and standard

deviations. In example number four, X1is assumed to be Gamma (1, 1) distributed in order to

examine the effect of a fat-tailed distribution of losses on the risk capital allocations. The two parameters of this gamma distribution are chosen this way in order to approximately match the means and the standard deviations of the losses of the other two business units that are identically normally distributed. Therefore, given that the Xi0sare independent, the differences in the risk capital allocations are attributed to the influence of the fat-tailed distribution.

Unlike in example four (where the losses of the three business units were considered to be independent) a linear correlation was assumed between the business units’ losses in examples

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14 Natasha Noskova — RISKCAPITALALLOCATION INFINANCIALCONGLOMERATES

one through three. Presented below is the initial correlation matrix that reflects linear depen-dencies between the losses of the first three examples:

    1 0.3 0.3 0.3 1 0.3 0.3 0.3 1    

Other possible forms of dependence between the losses that need to be modelled via copu-las, such as tail dependence, are not considered in this thesis. The chapters that follow review specifics of the calculations performed.

3.1.2 Calculating 99.5% VaR or 99% ES

In the previous subsection, the role of 99.5% VaR of 99% ES in the context of this research is introduced. The formula for VaR is presented in (3) and for ES in (4) and (5). 99.5% VaR is the 95%-quantile of the loss distribution and is easy to compute theoretically, based on the parameters of the distribution. ES, on the other hand, is not always to compute parametrically for the distributions other than normal, i.e. the closed-form mathematical expression is not always obtainable. For the normal distribution, which is assumed as described in Chapter 3.1.1., Dhaene et al. (2008) derived a formula for the calculation of CTE:

ESp(X ) = µX+ σX

φ (Φ−1(p)) 1 − p

This formula can be used for the calculation of ES. Given that the distribution of X is continu-ous, CTE and ES are the same. For the 99% ES it is further simplified to: ES0.99= 2.6652σX i,

by filling in the known values for standard normal distribution. Since such an expression is not available for the gamma distribution in example four, the ES calculation was accomplished with a simulation. First, 10 000 simulations from Gamma (1, 1) were performed by Matlab. (Note, the parameters of the gamma distribution were chosen in such a way to make all the business units comparable, as explained earlier in this chapter.) Then, the ES was calculated using formula (5) and the ”mean” function of Matlab.

3.1.3 Information on the Implementation of the Proportional Allocation Methods

The proportional allocation methods listed in 3.1.1. can be derived from formula (9) from Chap-ter 2 and the risk measure formulas given in Table 2.1. Thus, to get an expression for haircut allocations, one can substitute (3) as a risk measure in (9), and get:

Ki=

FX−1 i (p) ∑dj=1FX−1j (p)

· K. (10)

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The same procedure is followed to obtain the covariance allocation method, the only difference is the risk measure substituted into (9):

ρ (Xi) = Cov(Xi, X ).

Since ∑di=1Cov(Xi, X ) = Var(X ), the following allocation results:

Ki=

Cov(Xi, X )

Var(X ) · K. (11)

Finally, to calculate the CTE allocations, (5) is substituted into (9) to get:

Ki=

E[Xi|X ≥ FX−1(p)]

∑di=1E[Xi|X ≥ FX−1(p)]

· K. (12)

Dhaene et al. (2008) found a closed-form mathematical expression of the ratio presented above for the normally distributed losses:

Ki= K · [µi+ Cov(Xi, X ) σX ·φ (Φ −1(p)) 1 − p ] · [µX+ σX φ (Φ−1(p)) 1 − p ] −1. (13)

Filling in the known values for standard normal distribution gives:

Ki= K · [µi+

Cov(Xi, X )

σX

· 2.6652] · [µX+ σX· 2.6652]−1. (14)

3.1.4 Information on the Implementation of the Euler Method

As discussed in Chapter 2.3.1. to obtain the Euler method allocations, calculating a derivative of VaR or ES is required. There is not always a closed-form mathematical expression for such a derivative. VaR, for example, does not have one since it is an unsmooth quantile function. There-fore, one can use an approximation of the derivative according to its definition as demonstrated in (6). This approximation reflects the change in aggregate capital due to the small change in the loss of a business unit. The small change h used for this approximation equals 0.001.

Now that the details related to the implementation of the capital allocation methods are reviewed, the next step is the presentation of the practical results.

3.2

Practical Results

Characteristics of a business unit’s losses such as volatility, dependence structure between the losses and the fatness of the distribution’s tail, influence the amount of risk capital allocated to each of the units. Note that since the economic capital is only related to unexpected losses, as provided by the second definition in Chapter 3.1.1, increasing the mean of the loss distribution, would have no influence on either the economic capital or the allocations, because the mean is the expected portion of the loss. Below are descriptions of various cases that test the alloca-tion principles for the sensitivity to these aforemenalloca-tioned changes. The expectaalloca-tion is that when extra risk is introduced via a particular manipulation of a distribution parameter, a well per-forming allocation method would consistently allocate more risk capital to the unit responsible

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16 Natasha Noskova — RISKCAPITALALLOCATION INFINANCIALCONGLOMERATES

for the extra risk. The more sensitive an allocation method is to the introduced risk, the better, assuming the allocations are core compatible as defined in Chapter 2.1.2. Core compatibility is widely accepted as an important property for the evaluation of the allocation methods. Thus, the allocations outside the core are considered to be inferior, due to the fact that they are not ”fair”, as described by Denault (2001). The results of the allocations are arranged in tables for a better overview.

3.2.1 Case 1: The Default Setting

This is a default setting that produces equal allocations. Assumed are the three identical distri-butions of losses, with the following parameters:

Xi∼ N(2, 22).

All the correlations are equal to 0.3. The correlations between the losses are presented in the following matrix:     1 0.3 0.3 0.3 1 0.3 0.3 0.3 1    

Since the risk profiles of the units are identical, as described per above, the contributions of these business units to the overall risk of the firm will also be equal. This leads to the fact that the aggregate capital K is split into the three equal parts.

Table 3.1: The Default Setting

Method Aggregate Capital Allocation 1 Allocation 2 Allocation 3

Euler VaR 11.28 3.76 3.76 3.76

Euler ES 11.67 3.89 3.89 3.89

Haircut 11.28 3.76 3.76 3.76

Covariance 11.67 3.89 3.89 3.89

CTE 11.67 3.89 3.89 3.89

By the same risk profiles all the allocations are equal.

3.2.2 Case 2: Volatility of Losses

Volatility is the classic concept in risk management that determines how risky an asset is. This case tests how the volatility of losses of each of the business units influences the capital allo-cations. To test this, the standard deviations of the losses is varied: they are increased twofold and threefold accordingly, except for the standard deviation of the first unit. Thus, the standard deviations for the first, the second and the third business units’ losses become: 2, 4, and 6. In this case the correlations between the losses as well as the distributions of the losses remain as in Case 1.

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Table 3.2: Increase in Standard Deviations of Losses

Method Aggregate Capital Allocation 1 Allocation 2 Allocation 3

Euler VaR 23.38 2.84 7.26 13.28

Euler ES 24.20 2.94 7.52 13.74

Haircut 23.38 4.53 7.79 11.06

Covariance 24.20 2.94 7.52 13.74

CTE 24.20 2.94 7.52 13.74

The standard deviations for the loss distributions of units 1, 2 and 3 are changed to: 2, 4 and 6.

The total aggregate capital increases significantly due to the increase in standard deviations. This is also expected since out of all the attributes of the above mentioned loss distributions, the standard deviation is the main contributor to the unexpected losses. Therefore, it is only appropriate that all the risk measures translate this additional risk into extra economic capital for the individual units. Take for example VaR, which is a quantile function of the loss distribution. Doubling the standard deviation, per definition doubles the outcomes of the quantile function of the normal distribution. This higher capital calculated by the risk measures per individual units further aggregates to the higher economic capital for the whole firm.

Additionally, the results demonstrate that all the allocation methods respond to the in-crease in the standard deviations, by allocating the least economic capital to the first business unit, while the most capital is allocated to the third unit. The CTE, the covariance and the Euler ES methods demonstrate high sensitivity to the changes in volatility of the losses, by allocating about 5 times more capital to the third business unit compared to the first one, by the threefold increase in standard deviation. The second unit gets about 2.5 times the capital of the first busi-ness unit, although the standard deviation is only doubled. It is obvious that these three methods impose an additional penalty on high volatility. The haircut allocations show much less sensi-tivity to the variation in the standard deviations. The third business unit, for example, gets an allocation of about 2.5 times of the capital allocated to the first business unit.

Noteworthy is the fact that the Euler ES, the CTE and the covariance methods produce the same allocations. This is explained by Dhaene et al. (2011). According to this paper, the CTE method coincides with the covariance method, when the normal distribution of the losses with zero means is assumed. These are assumptions made in this research, since the economic capital is defined in such a way that only the uncertainty of the loss around the mean matters. Further, when continious distribution is assumed, the formula of the CTE method reduces to the sum of weighted derivatives of ES, when the ES is a risk measure. Thus, the CTE method becomes a special case of the Euler method, given the continious distribution of losses (Dhaene et al., 2011).

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18 Natasha Noskova — RISKCAPITALALLOCATION INFINANCIALCONGLOMERATES

Next, when the Euler ES method is compared with the Euler VaR method, no significant differences are noted between the results of these two methods. The allocations are slightly differenct due to the fact that the confidence levels vary as well as the risk measures (ES vs. VaR) that were used. This results in the aggregate capital that is somewhat higher when the ES is used as a risk measure.

Finally, it is important to point out that all the methods in this case produced core com-patible allocations as defined in Chapter 2.1.2.

3.2.3 Case 3: Correlation Structure Between Losses

For the purposes of this case, the influence of the change in the correlation structure between the losses is tested. The correlation matrix changes from:

    1 0.3 0.3 0.3 1 0.3 0.3 0.3 1     To:     1 1 0.3 1 1 0 0.3 0 1    

Note that the rest of the parameters of the loss distribution revert to the default Case 1. When it comes to correlations, the first and the second unit are perfectly positively correlated with the correlation coefficient of 1. Also the losses of the first and the third units are positively correlated with a much lower coefficient of 0.3. The losses of the second and the third business units are not correlated. This case tests how the allocation methods perform in the context of a portfolio.

Table 3.3: Correlation Structure Between Losses

Method Aggregate Capital Allocation 1 Allocation 2 Allocation 3

Euler VaR 12.18 5.00 4.35 2.83

Euler ES 12.61 5.18 4.50 2.93

Haircut 12.18 4.06 4.06 4.06

Covariance 12.61 5.18 4.50 2.93

CTE 12.61 5.18 4.50 2.93

The first and the second business units are perfectly positively correlated. There is also a small correlation between the first and the third units. The losses of the second and the third business units are independent.

The most obvious lack of change to observe is by the haircut method that does not re-spond to the changes in the way losses between the business units are correlated. That is an

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undesirable response, because the dependence structure between risks creates a diversification effect, which drives the risk capital allocation. Thus, the ”fair” response of an alllocation method in the context of a portfolio is important. Further, the allocation that the haircut method gener-ates in this case is core incompatible due to the allocations to units 2 and 3, which are not in accordance with the core compatibility requirement covered in Chapter 2.1.2.

It can also be observed that the Euler method is stable in its allocations with both VaR and ES. The same can be said about CTE and covariance methods that mimic the Euler method, as is expected based on the explanation given in Case 2.

3.2.4 Case 4: Heavy-Tailed Distribution (Simulation)

Modelling losses with heavy-tailed distributions are common in practice. Various recessions over the years and the recent financial crisis of 2007-2008 have shaken the assumption of nor-mally distributed returns. It is generally accepted now that distributions of returns have much fatter tails than the normal distribution. The fat tail of a loss distribution is also one of the main contributors to the unexpected losses. Therefore, it is crucial for allocation methods to be responsive to the risks related to the fat tail of a loss distribution.

This case assumes the independent losses of the three business units:

X1∼ Gamma(1, 1);

X2, X3∼ Normal(1, 1).

The parameters are chosen in such a way that the standard deviations and means of all three distributions are comparable. Note also that independence between the losses is assumed. Such a setup is chosen in order to isolate the influence of the fat-tailed distribution on the allocations.

Table 3.4: Influence of a Heavy-Tailed Distribution

Method Aggregate Capital Allocation 1 Allocation 2 Allocation 3

Euler VaR 5.66 3.33 1.22 1.11

Euler ES 5.93 3.56 1.23 1.14

Haircut 5.66 2.42 1.65 1.59

Covariance 5.93 1.97 1.99 1.97

CTE 5.93 2.05 2.08 1.80

The losses of the first business unit have a fat-tailed gamma distribution.

First of all, it is noticable that the simulation process produces some small discrepancies in the allocation results. Case 1 demonstrated that all the allocation methods produce equal allocations when the risk profile of the business units is the same. Business units 2 and 3 have

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20 Natasha Noskova — RISKCAPITALALLOCATION INFINANCIALCONGLOMERATES

the same risk profiles in the current case, but the allocations are slightly different due to the sampling errors that typically influence simulation results.

Secondly, it is noteworthy that the covariance method’s allocations are the worst given the fat-tailed distribution. These allocations completely fail to recognize the tail risk. The problem lies in the fact that the covariance method responds only to linear correlations and omits the tail behavior of the risks. That is an important weakness of this method due to the practical importance of fat-tailed distributions. The same is true for the CTE method, since both the covariance and the CTE methods produced allocations that are not sensitive to the fat-tail risk and are not core compatible. (The expression used to evaluate the core compatibility is given in Chapter 2.1.2). The poor performance of the CTE method can be explained by the fact that its ratio is based on the risk measure that is only coherent by continious loss distributions. Thus, it can be concluded that the covariance and the CTE methods are not effective in this case.

Thirdly, the haircut allocations are core compatible as defined by the expression of core compatibility in Chapter 2.1.2. However, while the haircut method recognizes the tail risk, it is not as sensitive to it as the Euler method.

It is interesting to note that both the Euler VaR and the Euler ES perform similarly, given the fat-tailed distribution. Due to the lack of subadditivity in VaR, the theoretical expectation for this result was to observe more of a difference in the performance of these two methods.

3.3

General Analysis of the Results

Summarizing the analysis performed on case by case basis in Chapter 3.2, a general conclusion can be reached that the Euler method performs well in all the cases tested. It gives the most sta-ble response and is more sensitive to all the changing parameters compared to the proportional methods, whether it is the change in standard deviations of the losses, the alterations in the cor-relation structure between them or the fat-tailed loss distributions. In fact, it is highly sensitive to the dependence structure between risks, as illustrated in Table 3.3. Note that the Euler and the covariance methods are the same, given the assumptions of Case 3. An attractive property of the covariance method is that this method is specifically focused on the correlations between the risks. The formula (11) reveals that this method allocates more capital to the business units that are more correlated to the whole firm, as oppose to such units that can hedge the risk of the firm. This is a valuable property for a risk capital allocation method, since the sensitivity to the dependencies between risks is crucial in risk modelling. Even intuitively, it appears fair to allo-cate more risk to the highly correlated units than the independent units due to the higher total loss in case of default. However, judging the performance of the method based on the intuition is not always possible or objective. Thus, except for observing how sensitive the methods are to the staged changes, and how stable the allocation results are, one can look to the definition of coherence of the allocation methods defined in Chapter 2. Further in this chapter, all the

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alloca-tion methods are reviewed against the coherence criteria. The Euler method (used with the ES) is an example of a coherent allocation method since it was justified by Denault (2001).

In Chapter 2.1.2. principles of coherent allocation of Denault (2001) are reviewed and ex-plained. According Denault (2001), the full allocation principle needs to hold as a prerequisite. In case of the proportional allocation and the Euler methods, this holds per definition for all of these methods. The core compatibility of all allocations was checked with Matlab against the definition of core compatibility given in Chapter 2.1.2. Thus, the allocations were calculated for one, two, or three element coalitions in search of a coalition where the risk is calculated lower than the capital allocated to this unit or coalition. The allocations in all the four cases in Chapter 3.2 were mostly core compatible. The core incompatible allocations of Case 3 are due to the insensitivity of the haircut method to the dependence structure between the losses. The other two methods that generated incompatible allocations are reported in Case 4, where both the CTE and the covariance methods produced ineffective allocations due to the fat-tailed distribution of the losses. The second principle of symmetry is not easy to evaluate. Moreover, according to Buch and Dorfleitner (2008), symmetry is not a desirable property, since it does not allow for diversification. Therefore, the methods were not evaluated against this property. All the methods were further evaluated against the third principle of a coherent allocation, the riskless allocation. Both of the Euler methods possess the property of riskless allocation, since Buch and Dorfleitner (2008) proved that translation invariant risk measures used with the Euler method lead to a riskless allocation and both VaR and ES are translation invariant. (See Chapter 2.1.2). All of the proportional allocation methods do not comply with the principle of risk-less allocation, since they all ignore risk measures that produce a negative result. (See Chapter 2.2.2.). Moreover, it was established in Chapter 2.1. of this thesis that RORAC-compatibility is an important quality for an allocation method. Tasche(2000) proved that the Euler method is the only method that is RORAC compatible, therefore, it can be concluded that all the proportional allocation methods do not possess this quality. The result is that the Euler is the only allocation method with all the theoretical properties, which also produces the best allocations in practice.

In regards to the proportional allocation methods, it is important to identify the situations which lead to errouneous allocations due to weaknesses of these methods. As opposed to the Euler or the covariance method, the haircut method is a good example of an allocation method that lacks response to correlations between the losses (see Table 3.3). When it comes to the other cases reviewed, the method responds to the rest of the parameter fluctuations, but it is often less sensitive than the other methods under review. It is least sensitive out the five allocation methods when it comes to increasing the standard deviations of the losses, which is also the case with the use of fat-tailed distributions (see Table 3.2 and Table 3.4). However, when the losses are normally distributed and are independent (or at least similarly correlated), the haircut method can be an alternative to consider due to the simplicity of the implementation.

Another proportional allocation method that is simple to implement is the covariance method. It is the most sensitive method to the changes in the variances or the correlation

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struc-22 Natasha Noskova — RISKCAPITALALLOCATION INFINANCIALCONGLOMERATES

ture of the risks. The units with the highest correlations to the whole firm as well as the units with the highest variance are penalised by this method. However, it is obvious from Case 4 that the covariance method does not perform well with the fat-tailed distributions, due to the fact that it only accounts for the linear correlations between the losses. This drawback of the covariance method leads to allocations that are core incompatible (see Table 3.4). Further, the review of the formula for the covariance allocations (11) reveals another weakness of the method that did not come out of the testing in Chapter 3.2. Due to the way the economic capital is defined in this thesis (see Chapter 3.1.1), the means of the loss distributions are disregarded by the allocation methods. As such, it was omitted in the testing that the covariance method does not respond to the changes in an expected loss. (Note that an expected loss is not a part of the formula for the covariance allocations). Given this weakness, it can be inferred that the covariance method is not appropriate for the losses with large differences in in their expected values. Theoretically, if this method is used to analyse the losses that have very different means, it will probably produce core incompatible allocations, as is often the case with methods that completely lack sensitiv-ity to a certain parameter. Therefore, this method can be acceptable for situations, where the means of the losses are comparable and where the dependencies between the losses are linear. Otherwise, it would lead to incorrect allocations, that potentially are not core compatible. On the other hand, given the assumptions that were made for the purposes of the first three cases of Chapter 3.2, it is the best method to use due to both its simplicity and accuracy.

CTE method identifies all the risks and properly translates higher risks to higher alloca-tions for the first three cases in Chapter 3.2, just as the Euler and the covariance methods. This method is based on a coherent risk measure when the underlying loss distribution is continuous. Thus, when the discrete distribution is used in order to simulate in Case 4, the CTE method fails to produce core compatible allocations. The other possible downside that can outweight the better performance of this method is possible difficulties with the estimation of the associated risk measure. As discussed in Chapter 3.1.2, CTE or ES are not always to compute analyti-cally for the distributions other than normal, i.e. there is not always a closed-form mathematical expression available.

Using a simulation, in order to calculate the ES for a gamma distribution in Case 4 of this thesis, lead to an unexpected result. Based on the theory, if the losses are not normally dis-tributed, the lack of subadditivity of VaR should lead to the VaR based Euler method performing worse than the ES based Euler method. The practical results of Table 3.4 do not support this hypothesis. The explanation of this result has to do with an estimation error. Yamai & Yoshiba (2005) compare the estimation error of the ES and VaR in their study, concluding that the es-timation error of ES is especially an issue when the fat-tailed distributions are used. This is to explain with the fact the estimate of ES is more tail dependent than that of VaR, which disre-gards the loss beyond a certain level. Thus, based on the definition in (4), the ES has a greater exposure to infrequent and large tail losses. This exposure leads to more varied estimates due to smaller samples with extreme values. The practical estimation issues that result from the use of ES point to the fact the desirable theoretical properties of a risk measure need to be considered

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against the practical aspects. Yamai & Yoshiba (2005) advocate the use of both coherent and non-coherent risk measures in practice in order to support a better decision making.

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Chapter 4

Conclusion

This thesis handles the practical aspects of the risk capital allocation process. A great number of allocation approaches and the lack of agreement on the risk measures that need to be used, have set the stage of this research. The focus of the research was the evaluation of the most widely used risk measures (ES and VaR) implemented with the most popular allocation methods (the Euler, the haircut and the covariance). The challenge was to distinguish which allocation rules combined with which risk measures ”fairly” share the economic capital.

The performance of the methods was evaluated based on the sensitivity of the methods to the extra risk that was introduced by each of the four cases in Chapter 3.2. More importantly, all the methods were evaluated against the desirable principles of the ”fair” allocation.

Denault (2001) defined the four well known principles of a coherent allocation: full allocation, core compatibility, symmetry and riskless allocation (see Chapter 2.1.2). These principles of-fered a great criteria and helped to define the ”fairness” of an allocation for the purposes of this thesis. However, as mentioned in Chapter 3.3, the principle of symmetry is not always as desirable as the rest of the properties of a coherent allocation. Thus the principle of symmetry was ignored in the evaluation of the allocation methods in this thesis. In place, the RORAC-compatibility was considered as an additional property that is important for the evaluation of the methods.

The full allocation principle, which requires that the sum of all the allocations equals to the risk of the firm, is satisfied by all the proportional allocation methods as well as both of the Euler methods. This requirement is embedded in the definitions of these two groups of methods. Therefore, it is satisfied by all the methods.

The core compatibility principle implies that the allocations of a method fall whithin the core, as defined in Chapter 2.1.2. All the proportional allocation methods failed the core compatibility test at least once in Chapter 3.2, while both of the Euler methods stayed core compatible for all the cases tested.

As to the riskless allocation, this principle is meant to ensure that an addition of risk-free assets to the portfolio reduces the portfolio’s risk by the same amount. Both of the Euler

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methods posses this property, since this property is ensured for the Euler method when the risk measure used with the method is translation invariant. Both VaR and ES are translation invariant as discussed in Chapter 2.1.2. Conversely, the proportional allocation methods do not comply with this property, since the calculation procedure for the proportional allocations (as described in Chapter 2.2.2) ignores the risk measures that produce a negative result.

Finally, it should be mentioned that the performance measurement is a concept that is closely related to the risk capital allocation, as established in Chapter 1 of this thesis. Thus, the RORAC-compatibility principle represents a valuable property for an allocation method. The proportional methods are not RORAC-compatible, since Tasche (2000) proved that the Euler is the only RORAC-compatible allocation method.

Therefore, it can be concluded that the Euler method is the only method that satisfies all the desirable theoretical properties, while also providing the most stable results in the practical testing performed in Chapter 3.2. The risk measures used with the Euler method, VaR and ES, performed equally well in the cases tested. This can be partially explained by the fact that VaR is subadditive when a normal distribution is assumed, which was three out of four cases in Chapter 3.2. However, unexpected was that the VaR based Euler performed as well as the ES based Euler in Case 4, where losses were gamma distributed. This can be explained by the fact that the VaR represents a more robust estimate when the fat-tailed distributions are simulated, as suggested by Yamai & Yoshiba (2005). Whether the subadditivty of VaR presents a practical problem is not tested in this thesis. This can be an interesting topic for a further research.

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26 Natasha Noskova — RISKCAPITALALLOCATION INFINANCIALCONGLOMERATES

References

Artzner, P., Delbaen, F., Eber, J.M. and Heath, D. (1998). ”Coherent measures of risk”, Mathe-matical Finance, 9,203-228.

Buch, A. and Dorfleither, G. (2008). ”Coherent risk measures, coherent capital allocation, and the gradient allocation principle”, Insurance: Mathematics and Economics, 42, 235-242.

Denault, M. (2001). ”Coherent Allocation of Risk Capital”, Journal of Risk, 4, 1-34.

Dhaene, J., Denault, M., Goovaerts, M. J., Kaas, R. (2003). ”Economic capital allocation de-rived from risk measures”, The North American Actuarial Journal, 7, 44-59.

Dhaene, J., Henrard, L., Landsman, Z., Vandendorpe, A., and Vanduffel, S. (2008). ”Some re-sults on the CTE based capital allocation rule”, Insurance: Mathematics and Economics, 42(2), 855-863.

Dhaene, J., Tsanakas, A., Valdez, E., and Vanduffel, E. (2011). ”Optimal capital allocation Principles”, Journal of Risk and Insurance, 78.

Jorion, P. (2001). Value at Risk: The New Benchmark for Managing Financial Risk, 2nd ed. New York, NY: MacGraw-Hill.

Laeven, R. J. and Goovaerts, M.J. (2004). ”An optimization approach to the dynamic allocation of economic capital”, Insurance: Mathematics and Economics, 35, 299-319.

McNeil, A.J., R¨udiger F., Embrechts P. (2005). Quantitative Risk Management: Concepts, Tech-niques and Tools. Princeton, NJ: Princeton University Press.

Myers, S.C. and J.A. Read, Jr. (2001). ”Capital Allocation for Isurance Companies”, Journal of Risk and Insurance, 68,545-580.

Tasche, D. (2000). ”Risk Contributions and Performance Measurement”, Working Paper, Tech-nische Universit¨at M¨unchen.

Yamai, Y. and Yoshiba, T. (2005). ”Value at Risk versus expected shortfall: a practical per-spective”, Journal of Banking and Finance, 29, 997-1015.

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