Heavy Tailed risk and Capital Allocation

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Allocation

Nirmal P. Parbhoe

Bachelor’s Thesis to obtain the degree in Actuarial Science

University of Amsterdam

Faculty of Economics and Business Amsterdam School of Economics Author: Nirmal P. Parbhoe Student nr: 11185175

Email: pravesh parbhoe@hotmail.com Date: June 30, 2021

Supervisor: Dr. Tim J. Boonen

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Abstract

Risk capital allocation is an important component of enterprise risk management. Previous research has shown that financial data may present with heavy tails and skewness, implicating

that the Gaussian distribution may not be a reasonable assumption for the empirical distribution of the data. This report provides a simulation study on the effect of heavy tailed losses on capital allocation methods. The Pareto distribution of the second kind will be used to

generate losses with heavy tails and allocations based on the Euler-rule and τ -value method are compared.

Keywords Capital-allocation, Tail-risk, Euler-rule, τ -value method, simulation, Pareto distribution of the second kind

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Statement of originality

This document is written by Student [Nirmal Parbhoe] 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.

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1 Introduction

Capital allocation is an important component of enterprise risk management and refers to distribution of the aggregate capital held by a firm across is its subdivisions, business lines or even individual products in a portfolio. The allocation is based on the risk capital the firm withholds that serves as a buffer to moderate the impact from adverse events or ensure that a company will be able to meet its financial obligation with a specified level of certainty. Capital allocations are used for financial reporting purposes, general investment and also for performance measurement Boonen (2020). However this is a complex process and is dependent on 2 factors. Namely: 1) How the firm measures risk and 2) Which method of capital allocation is used.

In practice the Value at Risk and the Expected shortfall ( TVaR or Tail Conditional Expec- tation ) are the most common used risk measures as these are in accordance with regulatory frameworks such as Basel III and Solvency II. The Solvency II directive prescribes the use of the Value-at-risk with a confidence level of 99.5% to compute the capital requirements for insurance companies, although extensive research has shown that the VaR can under-estimate the risk a company faces when dealing with heavy tailed losses. This calls into question the accuracy of capital allocations based on the Value-at-risk and is the reason for further analysis

The research to establish which risk measure give a better fit or which provides more stable results is abundant. In their paper Artzner et al. (1999) axiomatically define coherent risk measures,Yamai et al. (2002),Yamai and Yoshiba (2005), and Emmer et al.(2015) extensively researched the differences between the two risk measures based on qualities and applicability.

Furthermore,Wang and Zitikis(2021) recently even provided an axiomatic foundation for the expected shortfall after The Basil Committee on Banking Supervision confirmed that the ES will be replacing the VaR as standard risk-measure in the banking sector. Compared to the ES, the VaR is not sub-additive, however it has other favourable properties making it a valid risk measure for certain practical uses.

Additionally there is also a variety of methods for allocating risk capital. Dhaene et al.

(2012) approaches capital allocation by optimisation. and shows that his method requires the capital allocated to the division to be close to the actual risk the division faces, thus making it fair.Dhaene et al.(2008) studies the CTE-based allocation rule. This allocation method can be derived in different contexts, with one being a game-theoretical approach.Denault (2001), Kalkbrener(2005) and Boonen (2019),Boonen (2020) consider capital allocation in a manner of game theory. The Euler allocation rule (Also known as the Auman-Shapley value) gained significant recognition as a solution to the capital allocation problem. The CTE-based allocation mentioned before, is a special case of the Euler allocation rule when using the expected shortfall as risk measure. The Euler rule is favourable in the sense that is hedges the total risk, however Boonen (2019) argued that this method lacks continuity and seems to be very sensitive to measurement error when the underlying risks are normally distributed, which would indicate that the Euler method would be even more volatile when risk present heavy tails. Another

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method of allocation is the τ -value method.Boonen(2020) recently proposed this new method and showed theoretically that the τ -value method to be the only existing allocation rule to satisfy six desirable properties. However not much is known about this method and therefore has to studies further.

The Pareto distribution has been gaining popularity in many insurance and re-insurance related risk management. This distribution is derived from the extreme value theorem and has been used to model heavy-tailed risks. Vernic (2011),Asimit et al. (2013), and Park and Kim (2016) have studied the use of the Pareto distribution and its applications.Vernic(2011) provides an analytic study of the Tail Conditional Expectation ( TCE or Expected shortfall ) when risk have a pareto distribution of the second kind and derives formulas for the TCE- allocation method, which is a special case of the Euler-allocation method. Due to the complex form of the distribution, expressions for theoretical calculation of the TCE-allocation remain challenging. Asimit et al. (2013) further analyzes the effects of heavy-tailed pareto risks with the focus on distortion and weighted risk measures and allocations, whereas Park and Kim (2016) propose a new framework for the estimation of tail risk measures such as the VaR and ES (or Tail-VaR) for heavy tailed-losses.

Throughout the years research has shown that the empirical distribution of financial returns portrays heavy-tails, skewness and inconsistent tail behaviour, volatility clustering and long range dependence. Therefore the standard Gaussian assumption does not suffice. As mentioned above, when dealing with heavy tailed losses the choice of the risk measure is of great importance for capital allocation. Assuming Gaussian losses in this case can also be misleading and result in under- or over-allocation of funds. Furthermore in capital allocation problems one often deals with the co-dependence between divisions and ultimately between the losses. This study will analyze the effects of heavy tails on risk measures such as the VaR and the expected shortfall when losses have a Pareto distribution of the second kind by simulation. Secondarily the influence of dependence between variables on the method of capital allocation will be look into for models where losses have a Pareto distribution of the second kind and by modeling the dependence with the use of copulas.

This report is set out as follows. Section 2 provides the theoretical background and mathematical derivations. Section 3 states the method uses for simulation. Section 4 shows the results and explains the findings. Lastly, section 5 concludes the research.

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2 Theory

2.1 Risk measures

Risk measures are the main tools for calculation or representation of risk when facing uncertainty.

A risk measure is defined as a function ρ : X → R which assigns a value representing the degree of risk associated with random variable X ∈ X within a probability space (Ω,A,P). In terms of risk capital, ρ(X) charactarizes the amount of capital the firm need to withhold when facing risks X. The random variable X can be seen as gains or losses the firm expects at a certain time.

Artzner et al. (1999) presented a set of desirable properties and defines a risk measure as coherent is it satisfies all four of these properties. In his paper the variable X is defines as gains, while on the contrary this study will focus on losses. Because of this the definition of the two properties Translation Invariance and Monotonicity are affected

Definition 2.1 (Coherence). A risk measure is called coherent if it satisfies the following four properties:

1. Monotonicity: For all X, Y ∈ X with X ≥ Y we have ρ(X) ≥ ρ(Y ),

2. Translation invariance: For all X ∈ X and c ∈ R we have ρ(X + c) = ρ(X) + c,

3. Positive homogeneity: For all X ∈ X and c ≥ 0 it holds that ρ(cX) = cρ(X),

4. Subadditivity: For all X, Y ∈ X it holds that ρ(X + Y ) ≤ ρ(X) + ρ(Y ).

To further elaborate, Monotonicity suggests that when the losses of say division X are greater than that of division Y, then the estimated risk of division X is also greater than that of division Y. Translation invariance is a necessary condition for risk measures to be reasonable and the interpretation is simple. If X denotes losses, then by adding an amount which is a loss that will definitely occur the calculated risk will be that of X with the addition of the the amount that will surely be lost. Furthermore positive homogeneity and subadditivity are quite clear. Subadditivity suggests a diversification benefit where as Positive homogeneity in term of portfolios, means that when holding a multiple of the same portfolio there is no benefit, the risk is multiplied by the same factor.

In practise the Value at Risk (VaR) and Expected shortfall ( ES, CVaR, TVaR or TCE ) are the most commonly used risk measures.

The Value at Risk with confidence level α is defined as following:

V aRα(X) = inf(x ∈ R : P[X ≤ x]α), α ∈ (0, 1). (1)

The value at risk is the α-quantile of the loss variable X. Intuitively the value at risk indicates the confidence level with which the firm will be able to meet its obligations. If X ∈ X is a continuously distributed random variable and let F (x) denote its cumulative distribution function, then the Value-at-Risk can be expressed as V aR_α(X) = F_X⁻¹(x)

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Secondly the Expected shortfall for a given α is mathematically expressed as ESα(X) = E(X|X ≥ V aRα(X) = 1

1 − α Z 1

α

V aRτ(X)dτ. (2)

which translates into the average loss, given that losses exceed the VaR level ( Yamai and Yoshiba(2005) ).

The Value-at-Risk has been the most prominent risk measure used in insurance risk management, and setting α = 99.5% we are in line with Solvency II regulations. Nevertheless the Value-at-Risk has been criticized for not being a coherent risk measure, for the reason that it does not satisfy the property of sub-additivity, and underestimating risk due to the fact that it only considers a single quantile of the loss distribution

On the contrary the Expected shortfall, which recently officially replaced the Value-at-Risk as standard risk measure for the banking industryBCBS(2019), is a coherent risk measure and has been shown to be less sensitive to ”tail-risk”

Yamai et al. (2002), Yamai and Yoshiba (2005), Emmer et al. (2015) and Ziegel (2016) extensively studied and compared the Expected shortfall and VaR based on their properties.

When losses are normally distributed, or more generally have an elliptical distribution,Yamai et al. (2002) shows that the VaR is free from tail-risk.Additionally Emmer et al. (2015) says that the VaR is can still be applicable when the underlying risk have a finite variance. However the VaR is subject to tail-risk when this is not the case, whereas the expected shortfall performs better regarding heavy tailed distributions and tail dependence.

Another quality of risk measures is called elicitabilty and concerns the back-testing abilities of risk measures. And while the ES thrives on being coherent and less sensitive to tail-risk, it is not elicitable (Emmer et al.(2015)) When distributions present heavy tails, estimates of the ES shortfall are influenced by large and infrequent observation compared to the VaR, which does not regard observations beyond the VaR level. Nonetheless Emmer et al. (2015) does present feasible approached for back-testing the ES, although to reach the same level of certainty as the VaR, the ES requires more data.

Which of the risk measure is better than the other is a question with many answers, dependant on which of the aforementioned properties is considered. For risk capital allocation purposes tail-risk and dependence are paramount, which are contingent on the distribution of the losses.

2.2 Capital allocation methods

Lets consider the situation where a financial institution comprised of various divisions faces risk.

If this risk is defined as suffering a loss then the aggregate loss the institution faces is defined as S =

n

X

i=1

Xi,

where Xidenotes the loss of division i. The risk capital allocation problem arises when the institution wishes to distribute the total amount of risk capital given by ρ(S) among its subdivisions.

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In the literature a multitude of methods of capital allocation exist, varying from performance measurement approaches, capital allocations by optimisation and even in the context of coop- erative game theory.

An allocation method is called a risk capital allocation when it satisfies:

X

1∈N

Ki= ρ(S),

with Ki being the amount of capital distributed to division i for i = (1...N ) and ρ(S) the amount of risk capital based on the aggregate risk

The Euler allocation method

The Euler allocation method or Auman-Shapley value has been gaining popularity as a solution to the risk capital allocation property.

The Euler allocation method is denoted as:

K_i^E= ∂ρ(P

i∈NλiXi)

∂λ_i |_λ(1...1), (3)

with risk measure ρ for which the partial derivative exists and random loss variable X ∈ X The Euler allocation principle is the gradient of the risk measure ( recall that a risk measure was defined as a mapping 2.1) and translates to the marginal contribution of division i to the aggregate risk of the firm. Therefore taking into account the dependence structure of the divisions and hedging the risk of the firmBoonen(2019)

Furthermore when the loss variable X has a Gaussian distribution X ∼ N (µ, σ) the Value- at-risk can be expressed as

V aRα(X) = µ + Φ⁻¹(α)σ. (4)

Suppose that the losses of each division {Xi}i∈N follow a multivariate Gaussian distribution, it holds thatP

i∈NX_i follows a Gaussian distribution as well, and as shown byBoonen(2019) the Euler rule is reduced to

K_i^E= µi+ Φ⁻¹Cov(P Xi, X_i)

pvar(P Xi) , i ∈ N. (5)

Denault (2001) introduced the notion of coherence for risk capital allocation methods.

Definition 2.2. An allocation principle Π is coherent if for every allocation problem (N, ρ), the allocation Π(N, ρ) satisfies three properties:

1. No Undercut:

X

i∈N

K_i≤ ρ(X

i∈N

X_i),

2. Symmetry if for division i and j it holds that ρ(Xi= ρ(Xj) then

K_i = K_j,

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3. Riskless allocation For Π(N, ρ and i ∈ N such that Xi= c for c ∈ R, it hold that

Ki= c.

Denault (2001) shows that the Euler allocation method is coherent,as it satisfies the three properties given, and also results in the most preferable allocation method. ConverselyVan Gulick et al.(2012) andBoonen(2019) recognise that the Euler allocation principle is not continuous, which is a significant flaw.Boonen (2019) analyzes the volatility of the Euler rule under mild assumptions, such as normally distributed losses, and shows that the Euler rule can be very sensitive to measurement error due to the lack of continuity

The CTE allocation method

The CTE allocation method is a natural method which can be obtained from several theoretical backgrounds. Let the aggregate risk the firm faces be S =P

i∈NXi, the amount of capital Ki

allocated to division i is given by

K_i= E[Xi|S ≥ F_S⁻¹(α)]. (6)

From a game theoretical framework the CTE allocation principle was found to be a special case of the Euler allocation method byDenault(2001), hence it is a coherent allocation method and similarly regards the dependence between the divisions.

Additionally Dhaene et al.(2008) demonstrated that the allocation of risk capital based on the CTE method is justified in the sense of profit maximisation andDhaene et al.(2012) argues that the CTE method is optimal.

Dhaene et al. (2008) considers the CTE method for elliptical distributed risks and provide closed-form expressions.

Consider the case where the random variable X is normally distributed; X ∼ N (µ, σ) the CTE or expected shortfall is denoted as

ES_p(X) = µ + σφ(Φ⁻¹(α))

1 − α . (7)

The CTE allocation method as shown in Dhaene et al.(2008) can be expressed as

K_i= µ_i+φ(Φ⁻¹(α)) 1 − α

Cov(P Xi, X_i)

pvar(P Xi) , (8)

where X:=(X₁...X_n) has a multivariate Normal distribution with parameters (µ, σ, Σ), Lastly the CTE allocation method has also been studied for several heavy-tailed distributions regarding tail-risk. Vernic (2011), and Asimit et al. (2013) consider the Multivariate Pareto Distribution of the Second Kind for heavy tailed risks and capital allocation and Park and Kim (2016) study estimation methods for heavy tailed risk measures with Generalized Pareto distribution. These will be discussed further later on.

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The τ -value method

In a more recent study Boonen (2020) presented a new allocation method called the τ -value method. Originally proposed byTijs et al.(1981) for the class of transferable utility games, the τ -value can be applied to risk capital allocation. As described in Boonen (2020) the τ - value allocation is a convex combination between the utopia allocation and worst-case allocation, with the utopia allocation being the best case allocation a certain division expects or rather the lower bound for the amount of capital allocated to division i and the worst-case allocation being the upper-bound for the amount of capital allocated to division i.

Let the utopia-allocation of division i be given by

Mi(X) = ρ X

j∈N

Xj

− ρ X

j∈N \{i}

Xj

, ∀i ∈ N, (9)

and the worst-case allocation by

mi(X) = min

S⊆N \{i}ρ

X

j∈S∪{i}

Xj

−X

j∈S

Mj(X), ∀i ∈ N. (10)

Then the τ -value allocation is given by

τi(X) = (1 − a)Mi(X) + ami(X), ∀i ∈ N, α ∈ R. (11)

Recall that for any allocation principle to be a risk capital allocation method P

i∈Nτi = ρ(P

i∈NXi) should hold. Hence a =^ρ(

P

i∈NX_i)−P

i∈NM_i(X) P

i∈Nmi_i(X)−M_i(X) .

Moreover Boonen (2020) argues that the τ -vale allocation method is the only allocation method to satisfy the following six properties: Translation Invariance,Scale Invariance, Mono- tonicity, Riskless portfolio, No Diversification, Continuity. Further detail and proof are found inBoonen (2020).

The following example demonstrates how each method of allocation previously discusses is calculated.

Example 1.

Consider a financial institution with 3 sub-divisions where the total amount of risk capital has to be distributed among these divisions. Hence N = {1, 2, 3} and set α = 0.995 when using the VaR and α = 0.99 when using the Expected shortfall or CTE. Let Xi denote the loss variable of division i and assume that X has a multivariate Gaussian distribution with

µ = (0, 0, 0), σ = (1, 1, 1) and Σ =







1 0 0.5

0 1 −0.5

0.5 −0.5 1







(a semi-positive definite matrix)

From 4 it’s clear that V aRα(X1) = V aRα(X2) = V aRα(X3) = Φ⁻¹(α) = 2.575, which are the stand alone risks of the division. The aggregate risk is then computed by: V aRα(X1+ X₂+ X₃). As mentioned earlier, if X_i∼ N (0, 1) then it holds thatP

iX_i∼ N (0, σs), only now

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σ²_s= σ₁²+ σ₂²+ σ₃²+ 2σ1,2+ 2σ1,3+ 2σ2,3. Thus

V aR(X1+ X2+ X3) = µ + Φ⁻¹· σs

= 2.575 ·√ 3

= 4.46.

Likewise, the conditional tail expectation is calculated with 7. CT Eα(X1) = CT Eα(X2) = CT E_α(X₃) = 2.665 and again

CT E(X1+ X2+ X3) = µ +φ⁰(Φ⁻¹(α)) 1 − α · σs

= 2.665 ·√ 3

= 4.62.

with φ the density function of the standard Normal distribution and as before Φ⁻¹ the inverse CDF function of the standard Normal distribution

Evidently, from these calculation the property of sub-additivity can be seen in clear effect.

The aggregate risk is remarkably lower than the sum of all stand-alone risks. Furthermore, the level of the chosen α also impacts the risk evaluation, especially when dealing with heavy tailed risks deviations between the Value-at-Risk and CTE are expected to be larger.

Secondly the Euler-allocation will be derived, with the VaR as risk measure, with α = 0.995.

Hence formula5 is used.

K_i^E = µi+ Φ⁻¹(0.995) · cov(P3

i=1Xi, Xi) pσ_s²

= 2.575 ·cov(P3

i=1Xi, Xi) pσ²_s ,

where Φ⁻¹(0.995) denotes the 99.5-quantile of a standard normal distribution,pσ²_s=√ 3 and the co-variances:

cov(X1+ X2+ X3, X1) = σ²₁+ σ1,2+ σ1,3 = 1.5 cov(X₁+ X₂+ X₃, X₂) = σ²₂+ σ_2,1+ σ_2,3 = 0.5 cov(X1+ X2+ X3, X3) = σ²₃+ σ3,1+ σ3,2 = 1, and results in:

K₁^E = 2.575 ·^1.5^√

3 = 2.23, K₂^E= 2.575 · ^0.5^√

3 = 0.743, K₃^E= 2.575 · ^√¹

3 = 1.487,

For the CTE-allocation method, which can be considered a special case of the Euler- allocation method when using the CTE as risk measure, α is set at 99% and equation 8 is used.

K_i^{CT E}= 2.665 ·cov(P3

i=1X_i, X_i) pσ²_s , and accordingly:

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K₁^{CT E}= 2.665 ·^1.5^√

3 = 2.308, K₂^{CT E}= 2.665 ·^0.5^√

3 = 0.769, K₃^{CT E}= 2.665 ·^√¹

3 = 1.538.

Lastly the τ -value allocation for each division will be calculated step-wise, for the reason that this method consists of components that have to be calculated separately. The τ -value allocation with both the VaR and expected shortfall will be calculated, respectively

1. The Utopia allocation: To calculate the utopia allocation of each division9 will be used.

First the Value-at-Risk is determined for each combination of divisions:

ρ(

3

X

1

X_i) = V aR_α(X₁+ X₂+ X₃) = 4.46,

ρ(X

i6=1

X_i) = V aR_α(X₂+ X₃) = 2.575,

ρ(X

i6=2

Xi) = V aRα(X1+ X3) = 4.46,

ρ(X

i6=3

X_i) = V aR_α(X₁+ X₂) = 3.64.

The corresponding utopia allocations becomes:

M1= 1, 887, M2= 0, M3= 0.82.

2. The worst-case allocation: To calculate the worst case allocation10will be used.

Define S as a subset of N = {1, 2, 3}, thus the sets {∅}, {1},{2},{3},{1, 2},{1, 3},{2, 3}

and {1, 2, 3} are considered. In the next table all combinations of ρ(P

j∈SX_j) −P

j∈SM_j are summarised

i — S {∅} {1} {2} {3} {1, 2} {1, 3} {2, 3} {1, 2, 3}

1 0 2.575 0 0 3.64 3.64 0 3,64

2 0 0 2,575 0 1,753 0 1,775 1,753

3 0 0 0 2,575 0 2,573 2,575 2,573

The worst-case allocation for division i is the minimum value of each row of the table above. Hence

m1= 2.575, m2= 1.753, m3= 2.573.

3. To τ -value allocation: In order to calculate the τ -value allocation, a has to be determined first.

a =ρ(P

i∈NX_i) −P

i∈NM_i(X) P

i∈Nmii(X) − Mi(X) = 0.4179781, and by11the following are obtained:

τ₁= 2.175, τ₂= 0.733, τ₃= 1.553.

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Lastly the τ -value allocation is computed when the conditional tail expectation is used as risk measure. By following the same steps, the amount of capital allocated to each division is:

τ₁^{CT E}= 2.248, τ₂^{CT E}= 0.749, τ₃^{CT E}= 1.600.

For each of the allocation methods illustrated, the sum of the amount allocated to each division equals the aggregate risk capital (P

iXi). This equality is by definition of an allocation method. Secondly the proposed correlation between divisions is also reflected in the allocated amount of capital. Each method shows the same pattern of allocation amongst division 1,2, and 2. Since division 2 is uncorrelated to division 1 and negatively correlated to division 3, it gets rewarded with a low amount op capital for being a good hedge. Division 3 is positively correlated to division 1, but negatively correlated to division 2, due to this relationship its risk is diversified.

Furthermore it has been shown that the Value-at-risk is a coherent risk measure when risks are normally distributed, however this does not specifically hold when the underlying distribution is non-Gaussian and adding to this, the Value-at-risk has also been criticised for underestimating risk when risks present heavy tails. Additionally as mentioned earlier, Boonen (2019) showed that under smooth conditions, such as normally distributed risks, the Euler-allocation method can be quite volatile, which implies that this method could produce even more unstable results in the presence of tail risk. And lastly the τ -value method is still to be studied for extreme situations.

2.3 Tail Risk

As shown earlier, the Value-at-risk is a convenient risk-measure and has favourable traits. How- ever when the underlying risks are fat-tailed or when a division has a high potential for large losses, the Value-at-risk may underestimate the risk and risk manager insufficiently allocate capital among their constituents.

When dealing with heavy-tailed distributions, the Extreme Value Theory (EVT) has emerged as an useful tool in risk management. The EVT has been widely adapted and studied when considering tail-tailed distributions for risk modelling. For instance, Embrechts et al. (2001), Yamai et al.(2002).

For the risk capital allocation problem multivariate distributions are of keen importance, where the tails of the marginal distributions and dependence structure among variable are of significance

Extreme Value Theory

The EVT provides useful information on the distribution of random variable X over a certain threshold.

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Let X be a random variable with distribution function F (x), for fixed u Fu(x) = P[X − u ≤ x|X > u] = F (x + u) − F (u)

1 − F (u) , x ≥ u,

is the distribution function of the variable X over the threshold u. The EVT states the approximation of Fu for high values of u. Nevertheless, Fu is only known if F (x) is known.

Moreover the Pickands-Balkema-De Haan Theorem states that F_uconverges to the Generalised- Pareto distribution as u tends to infinity. Hence,

F_u(x) ≈ G_α,σ(x), G_α,σ(x) = 1 − (1 −ξx σ )⁻¹^ξ.

The Generalised Pareto distribution has two parameters: ξ the tail index and σ the scale parameter. The tail index specifies the fatness of the tail, so for large ξ the distribution has a fat tail. The scale parameter σ indicates how dispersed the values are.

This EVT has proven to be extremely useful in risk management, Yamai et al. (2002) de- scribes in detail how the extreme value theory is used to tackle tail-risk of risk measures such as Value-at-Risk and Expected Shortfall when dealing with fat-tailed distributions.

Copula

In risk capital allocation the correlation between divisions is paramount. When disregarding the correlation, financial institutions may allocate capital to their constituency erroneously or even has an incorrect assessment of the aggregate risk.

When modelling the dependence between random variables, copulas emerged as a mechanism that helps understand the relationships among multivariate outcomes.Frees and Valdez(1998) have done groundbreaking research into copulas and showed their applicability in the finance and insurance industry.

A copula is defined as the distribution function of a vector with standard uniform marginal distributions:

C(u1, u2) = P[U¹≤ u1, U2≤ u2].

Now suppose there are N random variables, in general not assumed to be independant, with distribution functions F₁(x1, F₂(x₂), ..., F_n(x_N) then

C(F₁(x1), F₂(x₂), ..., F_N(x_N)) = F (x₁, x₂, ..., x_N). (12) The copula defines the multivariate distribution function of the random variables.

A well known copula is the Gaussian copula

C(u, v) = Φ_Σ(F⁻¹(u), F⁻¹(v),

with Φ the distribution function of the multivariate standard normal distribution with correlation matrix Σ and F₁⁻¹, F₂⁻¹ the inverse CDF of the random variables X₁and X₂.

The Gaussian copula is easy to use for generating dependant variables in simulation studies.

Ross (2013) has shown how any random variable X with marginal distributions Fi and whose joint distribution is given by the Gaussian copula can be generated by the following 4 steps:

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1. Generate random variable W1, W2, .. from a multivariate Normal distribution with means equal to 0, variances equal to 1 and Cov(W_i, W_j) = Cov(X_i, X_j)

2. Compute Φ(Wi) for all i.

3. Let Fi(Xi) = Φ(Wi)

4. and obtain Xi by solving Xi= F_i⁻¹(Φ(Wi))

The Pareto distribution of the second kind

As shown by the EVT the excess-loss can be approximated by the Generalised Pareto distribution for a large enough boundary. This in turn shows that in real-world scenarios, where losses present heavy-tails, the loss distribution can be approximated by the Pareto distribution of the Second Kind. This distribution has received considerable attention is the recent years in the context of risk management, risk capital allocation, insurance pricing and optimal reinsurance retention by Park and Kim(2016),Vernic(2011),Chiragiev and Landsman(2007),Guillen et al. (2013) and Asimit et al.(2010).

A random variable X has a Pareto distribution of the Second Kind, note X ∼ P II(µ, σ, α) if its probability density function and cumulative distribution function can be written as:

f (x) = a σ(x − µ

σ + 1)^−(a+1), F (x) = 1 − (x − µ

σ + 1)^−a,

with location parameter µ, scale parameter σ, and shape parameter α. The shape parameter indicates the heaviness of the tail. Hence a large α indicates a fat-tail. Furthermore the random vector X has an n-variate Pareto distribution of the Second kind M PⁿII(Θ, α), with α > 0, and Θ = (µ, σ), µi ∈ R, σi> 0 for all i ∈ {1, ..., n} if its density and distribution function are given by:

fX(x) =

ⁿ Y

i=1

a + n − i σi

ⁿ X

i=1

xi− µi

σi

+ 1

−(a+n)

, x ≥ µ,

F_X(x) = 1 −

ⁿ X

i=1

x_i− µ_i σi

+ 1

−a

, x ≥ µ.

For the marginal moments it holds that

E(Xi) = µi+ σi

a − 1, V ar(Xi) = ( σi

a − 1)² a

a − 2, fora > 2. (13) The tail-function F = P[X > x] is then given by

F_X(x)

ⁿ X

i=1

xi− µi

σ_i + 1

−a

, x ≥ µ. (14)

Asimit et al. (2010) noted that the tail function of the multivariate Pareto distribution can be expressed in terms of the Clayton survival copula, from which it follows that the dependence between the marginal risks is signified by the shape parameter α. For the Multivariate Pareto

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distribution the correlation for a > 2 is given by Corr[Xi, Xj] = _α¹. Thus the parameter α indicates the tail-fatness as well as the dependence between the losses of the divisions. If α increases the tails get less fat and the losses are less correlated and if α decreases the tails get heavier and the losses have a stronger correlation. Additionally the Pareto distribution only allows for positively correlated variables

Park and Kim (2016) provide closed form expressions for the Value-at-Risk and ES under the Generalised Pareto distribution. as mentioned before, the second kind Pareto distribution is a special case of the GPD, hence the VaR is given by

V aR_α(X) = σ[(1 − α)^−1/a− 1], (15)

and the expected shortfall by

ES_α(X) = σ (1 − α)^−1/a 1 − 1/a − 1

. (16)

An interesting study by Guillen et al.(2013) has shown that the Pareto distribution of the second kind is a special case of the Beta distribution of the second kind. Guillen et al.(2013) shows that when considering the beta distribution of the second kind, risk measures such as the Value-at-Risk and expected shortfall have a closed form expression.

Consider the random variable X. Then X ∼ B2(p, a, σ) is said to have a second kind beta distribution if its density function is given as

f (x; p, a, σ) = x^p−1

σ^pB(p, a)(1 + x/σ)^p+a,

where p and a are positive shape parameters and σ a scale parameter. Furthermore if p=1, X has a Pareto distribution of the second kind. The density function with p=1 reduced to

f (x; 1, a, σ) = x¹⁻¹

σ¹B(1, a)(1 + x/σ)^p+a B(p,a) is the Betal function and B(1,a)=1/a. Hence

f (x; 1, a, σ) = a

σ(1 + x/σ)^−(1+a)

Guillen et al.(2013) also shows that the VaR and ES have the following form.

Let X ∼ B2(p, a, σ) then the Value-at-risk for level α is given by,

V aRα(X) = σ IB⁻¹(α; p, a)

1 − IB⁻¹(α; p, a) (17)

and the the ES is given by,

ESα(X) = E[X] ·

1 − IB(_1+α/σ^α/σ ; p + 1, a − 1)

1 − IB(_1+α/σ^α/σ ; p, a) (18)

Where IB denotes the Incomplete ratio Beta function, IB⁻¹ denotes the inverse of the In- complete ratio Beta function or also known as the quantile function of the Beta distribution of the first kind and E[X] = _a−1^σp for a > 1

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Figure 1: Probability density of second kind Pareto distribution(red) and second kind Beta distribution (black) from randomly generated values

Additionally these result can be extended to the multivariate case where.Guillen et al.(2013) Shows that the Sum of second kind Beta distributed variables, again have a Beta distribution of the second kind. These results a useful for both analytic and numerical computations of the VaR, ES and even the τ -value allocation.

Vernic (2011) and Chiragiev and Landsman(2007) have been able to provide formulas for the computation of the Euler rule by expressing the n-variate case recursively in terms of the (n-1)-variate case and by using divided differences, respectively.Vernic (2011) shows that the Euler rule can theoretically be calculated with,

K_i^E=E(XiI(S > V aR0.99)) FS(V aR0.99) .

If the underlying loss distribution has a multivariate Pareto distribution of the second kind, then

E(XiI(S > V aR_0.99)) =

ⁿ Y

k=0

(a + n − k)

ⁿ X

k=0

(−1)^k

k!(n − k)!(a + k − 1) × σ^a+k

(V aR0.99+ σ)^a+k−1,

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and

FS(V aR0.99) =

ⁿ Y

k=0

(a + n − k)

ⁿ⁻¹ X

i=0

(−1)ⁱ (a + i)i!(n + i − 1)!

σ

V aR_0.99− σ

^a+i .

These formulas do make it possible to theoretically calculate the amount of capital allocated to each division, nevertheless for large n the computations become technical and require analytical methods to solve.

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3 Methodology

3.1 Capital Allocation and Pareto risks

To analyse the the impact of a heavy tailed distribution on the method of allocation, the τ - value method and the Euler allocation method will be compared based on simulated losses.

The Conditional Tail Expectation with significance level α = 0.99 will be used as risk measure, since it allows for a natural allocation of capital, namely the CTE method, and has become the prescribed risk measure in the banking sector.

Emperically the expected shortfall is calculated as:

ES0.99(Xi) = mean(X1), for X > V aR0.99(Xi)

ES0.99(X1+X2+X3) = mean([X1+X2+X3]) for [X1+X2+X3] > V aR0.99([X1+X2+X3])

Intuitively, the ES is the average of the values larger than the VaR.

For a sample size of 1000, V aR_0.99 equals the 990^th observation of the ordered losses and the ES then equals the average of the 10 highest observations.

A firm with 3 divisions will be considered where the losses Xi of each division will be generated in the following manners:

• Multivariate Pareto distribution of the second kind: Losses will be simulated under the assumption of a multivariate Pareto distribution of the second kind. X∼ M P II((1, 1, 1, ), 3).

Hence Xi∼ P II(1, 3) for i = (1, 2, 3). In this case the divisions are identical and the correlation between the divisions is indicated by the shape parameter a.

• Gaussian copula Marginal losses will be generated from the univariate second kind Pareto distribution with shape parameter 3 and scale parameter 1, under the assumption that their joint distribution is given as a Gaussian copula with correlation matrix as before

in the example Σ =







1 0 0.5

0 1 −0.5

0.5 −0.5 1







To compare the effects of a heavy tailed distribution, losses will also be simulated from a multivariate Normal distribution where the mean and variance will be set equal to those of the marginal distributions second kind Pareto distribution and the correlation matrix will reflect the same dependence structure as both cases given above.

A sample size of 1000 will be generated, which corresponds to losses of approximately 40 years.

To compute the allocation base on the Euler rule of each division first the of the sum losses will be calculated X1+X2+X3. Secondly the 10 largest observations of the sum will be identified and the corresponding 10 losses of the individual divisions that yield these 10 realizations of the

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sum. After this, the amount allocated to each division equals the the mean of the 10 individual losses identified.

In order to compute the τ -value allocation for each division first the Utopia allocation for each division has to be calculated with the ES as risk measure. From9it follows that

M_i= ES_0.99(X

alli

X_i) − ES_0.99(X

j6=i

X_i)

Secondly the worst case allocation is determined for each division. Using equation 10 and the determined utopia allocations. A matrix identical to the table given in the example above 2.2will be defined, where each row contains the result of

ES0.99(X

i∈S

(Xi) − X

i∈S\{i}

Mi

and the worst case allocation for division j equals the minimum value of row j.

To illustrate this consider division 1. The set S indicates the possible combinations of divisions. Thus for division 1, S is each possible combination of division, including division 1.

This gives the following combinations: {1, 2, 3}, {1, 2}, {1, 3}, {1} of the set S. The worst-case allocation is then the minimum value of:

S = {1, 2, 3} →ES0.99

X

i∈S

Xi

− X

i∈S\1

Mi

S = {1, 2} →ES0.99

X

i∈S

Xi

− X

i∈S\1

Mi

S = {1, 3} →ES_0.99

X

i∈S

X_i

− X

i∈S\1

M_i

S = {1} →ES_0.99

X

i∈S

X_i

− X

i∈S\1

M_i

And lastly the τ -value allocation will be computed using11with

a = ES_0.99(X₁+ X₂+ X₃) − M₁− M2− M3

m1+ m2+ m3− M1− M2− M3

The Simulation will be run 10.000 times , yielding 10.000 allocations for each division under each method and each loss generating distribution.

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4 Results and Discussion

In this section the results of the simulated capital allocation proportions are presented. Rather than focusing on the actual amount allocated to each of the division, the proportion of total risk capital allocated to each division is given. In each case the Expected shortfall ( or CTE ) is used as risk measure at a probability level of 99%.

Table 1and Table2 present the descriptive statistics for the compositions of capital allocations based on the Euler allocation method and τ -value method, when the underlying risk, in this case losses, have a second kind Pareto distribution and Normal distribution. Additionally Figure4.1graphically shows the comparison between Normally distributed losses and losses that have multivariate Pareto distribution of the second kind, for each of the allocation methods.

Lastly, Figure 3,4, and 5 exhibit the capital allocation compositions based on the Euler method and for the τ -value, when individual losses have Pareto distribution of the second kind and their joint distribution is given by a Gaussian copula.

4.1 Multivariate second kind Pareto losses vs. Multivariate Normal losses

From the simulation results, it is clear that capital allocation can be quite sensitive to tail risk.

In Table 1 the amount of capital allocated to each division is presented as a fraction of the whole, along with the standard deviations.

Evidently, when dealing with heavy-tailed losses, such as the second kind Pareto distribution, the amount of capital allocated to each division fluctuates strongly compared to the allocated amount of capital when losses follow a normal distribution, which generally does not have a heavy tail. When the underlying risk presents heavy tails, there is a higher probability for large losses. This causes the estimation of risk to be more volatile due to the variability in the losses.

The ES takes the mean of the losses above a certain threshold, and with large and infrequent losses, the estimated risk is also expected to vary. Regardless, as argued byYamai et al.(2002) and many that followed, when a division is prone to heavy-tailed losses the ES gives a better estimation of risk and thus results in more accurate capital allocations compared to using the VaR as risk measure.

It’s clear that both allocation methods are subject to tail-risk. Nonetheless, the τ -value method appears to be less sensitive to tail risk than the Euler allocation method

Congruent to the results presented inBoonen(2019), Euler allocation method appears to be very sensitive to measurement error. Although he compared the Euler method to the propor- tional allocation rule. The τ -value allocation method performs significantly better. The standard deviation decreases faster than that of the Euler allocation method when the thickness of the tail decreases. Figure4.1depicts the spread of the allocated amount of capital, supporting these results graphically.

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ParetoII distribution Normal distribution Division 1 Division 2 Division 3 Division 1 Division 2 Division 3

Euler- mean 0.333 0.334 0.333 0.333 0.333 0.334

allocation std. dev a=3 0.0816 0.0813 0.0816 0.0310 0.0310 0.0310

std.dev a=10 0.7636 0.7572 0.07522

τ -Value mean 0.333 0.334 0.333 0.333 0.333 0.333

std. dev. a=3 0.0661 0.0658 0.0656 0.0195 0.0193 0.0193

std.dev a=10 0.4360 0.04326 0.04321

Table 1: Descriptive statistics of the capital allocation composition based on the Euler-allocation method and τ -value method with the ES as risk measure. The simulation is run 10.000 times and the mean of the portion of risk capital allocated to each division with their standard deviation is shown. a corresponds to the shape parameter of the second kind Pareto distribution

Figure 2: Comparison of allocated portion of risk capital to Division 1 based on the Euler method (left) and τ -value (right) with ES as risk measure under the Assumption that losses have a second kind Pareto distribution( gray) and Normal distribution (green)

The τ -value method allocates capital to each division by taking a convex combination of the utopia allocation and worst-case allocation which in this case basically reduces to a combination of two different values of the ES, therefore the volatility is induced by the chosen risk measure, whereas the Euler rule takes the derivative of the chosen risk measure, meaning that it is directly

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impacted by small changes in the loss distribution. Thus in this case, when dealing with large and infrequent losses, the impact is expected to be even larger.

Another noteworthy result is the composition of the risk capital, which is the same for each method and each distribution. A reasonable explanation for this phenomenon is the uniformity of the divisions. The multivariate Pareto distribution of the second kind only allows for equally correlated random variables induced by the shape parameter a, meaning that the correlation between the risks of each divisions is equivalent.

However this usually not the case in real-world scenarios. In reality the correlation between divisions may differ or even be negative, which impacts the risk capital composition and thus requires different model specifications.

4.2 The Gaussian copula for correlation moddeling

By using the Gaussian copula to specify the joint distribution of the individual losses, the correlation between the divisions can be modeled as desired. Table1presents the capital allocation compositions under the second model as described in section3.1.

These results are similar to the previous ones with respect to the standard deviation of the allocated capital to each division. The standard deviation when using the Euler rule is noticeable larger than the τ -value method.

ParetoII distribution Normal distribution

Division 1 Division 2 Division 3 Division 1 Division 2 Division 3

Euler- mean 0.427 0.192 0.381 0.520 0.146 0.334

rule std. dev. 0.0834 0.1142 0.0930 0.0161 0.0580 0.0482

τ -Value mean 0.411 0.214 0.374 0.516 0.142 0.342

std. dev. 0.063 0.0775 0.0645 0.0123 0.033 0.0266

Table 2: Descriptive statistics of the capital allocation composition based on the Euler-allocation method and τ -value method with the ES as risk measure and the Gaussian copula is used to model the dependence between the divisions. The simulation is run 10.000 times and the mean of the portion of risk capital allocated to each division with their standard deviation is shown.

Looking at the composition of the allocated capital there is a significant difference. The Gaussian copula made it possible to model the joint distribution of the divisions and introduce a different correlation structure.

In this model Division 1 is positively correlated to Division 3, and Division 2 negatively. A known property of the Euler rule is that it considers the correlation structure and can hedge the total risk of the firm. Due to the negative correlation of Division 2, it is considered a good hedge and is rewarded a lower portion of the total risk capital. The opposite is seen in the allocated

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portion to Division 1. Due to its positive correlation to the aggregate risk, it is allocated a larger portion.

Remarkably, this pattern is also seen when the τ -value is used as allocation method. Meaning that the τ -value method also considers the correlation structure, and hedges the risk of the firm.

Furthermore, it can be argued that the τ -value results in more accurate compositions by looking at the standard deviations, which are significantly lower, especially for Division 2. Thus when dealing with heavy-tails, the τ -value method captures the tail-dependence more accurately.

Furthermore, comparing the composition under the ParetoII distribution and normal distribution, a small disparity can be noted. This can be seen as the tail-effect. When losses present heavy tails, using the Normal distribution may over- or under-allocate risk capital to the divisions.

Figure 3: Capital allocation to Division 1 under the Euler method (left) and τ -value method (right) with second kind Pareto losses (gray) and Normal losses (green). The Gaussian copula has been used to model the dependence between divisions. 1000 losses are generated and the simulation is run 10.000 time

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From Figure 3,4, and5 clearly the proportion of risk capital allocated to Division 2 seem to be more unstable compared to the allocations of the other divisions. The standard deviation of Division 2 is also significantly larger, note that this is also the division with the negative

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correlation.

An additional factor that contributes to this volatility is the dependence between the divisions and how this is modeled. Copulas are useful tools for modeling the dependence between variables.

Especially the Gaussian copula, which allows for different correlations between the divisions.

However the Gaussian copula may fail to accurately approximate the dependence in the tails.

Nonetheless Yamai et al.(2002) argues that using the Gaussian copula can still be reasonable for heavy tailed distribution.

When dealing with fat-tailed losses, it is important to have an accurate approximation of the dependence in the tails, since these observation attribute to the estimation of risk and the allocation of risk capital. Hence using the Gaussian copula may fall short. As an alternative the student t copula may be considered. It has been shown that the student t copula performs better in modelling the dependence since it takes into account the dependence in the center and in the tail of the distribution as well.

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5 Conclusion

The Pareto distribution of the second kind is widely used in modeling heavy-tailed risks. This study provides insight into using this distribution to model heavy-tailed losses for the risk capital allocation problem by simulation and shows that additional modelling techniques are required.

When the underlying loss distribution presents heavy tails, the τ -value method results in more stable capital allocations compared to the Euler allocation method. Additionally, the τ - value also takes into account the correlations between divisions, and hedges the risk of the firm

Furthermore, to model the dependence, copula have been proven to be useful. The Gaussian copula especially; This copula allows to model the correlation structure between divisions pair- wise, rather than assuming an uniform dependence structure between the divisions.

Nonetheless, the Gaussian copula may fail to accurately estimate the tail-dependence, and as the tail gets fatter, which may further increase estimation errors.

The simple model used in this study, delivered promising result, although, with an increasing number of divisions and fatter-tailed losses more intricate and advanced modelling techniques are required, which are beyond the scope of this study.

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Heavy Tailed risk and Capital Allocation

Allocation

Nirmal P. Parbhoe

Abstract

Contents

1 Introduction

2 Theory

2.1 Risk measures

2.2 Capital allocation methods

2.3 Tail Risk

3 Methodology

3.1 Capital Allocation and Pareto risks

4 Results and Discussion

4.1 Multivariate second kind Pareto losses vs. Multivariate Normal losses

4.2 The Gaussian copula for correlation moddeling

5 Conclusion

References