Capital allocation methods and dynamic risk analysis based on Monte Carlo simulation approach.

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Capital allocation methods and dynamic

risk analysis based on Monte Carlo

simulation approach

Meng Lang

Master's Thesis to obtain the degree in

Actuarial Science and Mathematical Finance

University of Amsterdam

Faculty of Economics and Business Amsterdam School of Economics

Author: Meng Lang Student nr: 10609423

Email: langmeng111@gmail.com Date: August 24, 2014

Supervisor: Prof.dr.R.J.A.Laeven Second reader: Prof.dr.R.Kaas

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Abstract

This study investigates capital allocation methods and the issues related to risk modeling, based on the Monte Carlo simulation approach. A dynamic risk analysis process is resorted to seek a solution of risk choices, in a dynamic setting, using the capital allocation results.

Firstly, the theoretical properties of various capital allocation methods are analyzed. Secondly, practical modeling issues are elaborated, concerning the implementation of the VaR

contribution method, which is the allocation method specifically applied in Achmea Re.. Next, comparisons of the VaR contribution method and two other allocation methods are conducted based on simulation results. Finally, a dynamic risk analysis is carried out, in order to obtain a measure of risk performances, which is invariant to the order of risks.

Keywords

capital allocation, value-at-risk, Euler principle, risk contribution, Monte Carlo simulation, expected utility value

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Preface

This master thesis is the result of my one year study at the University of Amsterdam and five months internship at Achmea Reinsurance Company. I am sincerely thankful to everyone who has given me endless support, both materially and spiritually, in all types.

I would like to, at first, demonstrate my gratitude to Professor dr. R.J.A.Laeven of the University of Amsterdam. As my supervisor, he granted me this opportunity to have this internship at Achmea Re, and diligently advised me in order to conclude this thesis.

Secondly, I extend my gratitude to Achmea Reinsurance ARC team and all other colleagues in Tilburg, for the amazing experience, during my internship working with them. Countless thanks to Joris van Kempen, my supervisor in Achmea Re, who gave me so much inspiration through fruitful discussions and instructions, and in all of his performances. I am also thankful to Rob Vermunt, for all the supports both technically and mentally, along this journey. To my boss Gijs with so much guidance and brilliant ideas, which helped me to keep on the right track, without losing the direction of my research. Colleagues including Ruud, Hélène, Gernot, Roy, Joost should also be remembered and applauded, without whom I would not have had such a great time within Achmea.

In addition, I would like to thank Michiel Janssen of Achmea Group for the enlightening conversations leading me to many discoveries presented in my paper. Tim, Umut and

Zhenzhen Fan from FEB faculty are also thanked for their time and energy to discuss with me about academic issues.

Last but not least, I am really grateful to my parents, for their spiritual support, far from my home China. No words can express how thankful I am for everything they have done for me. They are all the reason I am here in the Netherlands enriching my life and experiences.

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1 Introduction

This paper addresses the issues concerning capital allocation to business divisions, based on the Monte Carlo simulation approach. It attempts to analyze risks in a dynamic circumstance. Capital allocation has been long regarded as a challenge in banking and insurance industry. In order to protect the company against potential and extreme events, financial institutions are required to reserve a certain amount of capital. However, the capital is determined according to the level of the entire company, leading the stock and shareholders to allocate capital to each business divisions.

There are multiple reasons for allocating capital to individual business divisions. Below four major points are mentioned. First, a well-functioning insurance company is established on healthy and profitable business-taking. Business divisions are responsible for their own performances. To assess and compare the performances of divisions, capital allocation is combined with return to calculate return on capital, as a useful performance measurement. Secondly, to support decision-making and optimize portfolio choice, capital allocation provides an effective measure for divisions’ capital consumption. Thirdly, the total cost of capital is redistributed among the various divisions by capital allocation, such that this cost is transferred back to the policyholders by premium. Fourthly, allocated capital serves as a core figure for financial reporting purposes.

In the past decades, countless studies have been conducted by researchers both academically and practically. Denault (2001), at first, proposed several principles of capital allocation, similar to the coherent risk measure properties Artzner et al. (1997) advocated. Various methods for capital allocation have been proposed, from different points of view, such as: Dhaene, Goovaerts and Kaas (2003) and Laeven and Goovaerts (2004) from the viewpoint of mathematical optimization, Denault (2001) and Tsanakas (2004,2008) based on game theory, and Myers and Read (2001) on the basis of results obtained from the default value.

This paper analyzes several capital allocation methods that are widely used throughout the industry including the haircut allocation principle, the covariance allocation principle, the quantile allocation principle, the conditional tail expectation principle and the Euler principle associated with two risk measures. Practical issues in risk modeling, which are rarely explored, are discussed to fill in the gaps between practice and academic fields.

Most insurance and reinsurance companies have discarded the traditional proportional methods since they give imprudent results. More sophisticated method is capable of fully revealing the feature of risk whereas being very sensitive to changes in the portfolio. This makes it difficult for the management to use in a dynamic risk analysis process, as consistencies may need between reporting periods. That is why a new measurement of risk performances, which is indifferent to the choice of risks and their orders, is desired. This paper describes the occasion when the challenge arises. By a numerical example, a unique

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decision is derived, based on the expected utility theory, the aim of which is to obtain a clearer perspective of risk selection for the management.

Our focus dwells on question of how the Euler principle associated with Value-at-Risk, the so-called VaR contribution method, is implemented in practice with simulation approach and to what extent it is possible to trace back to the principle behind it. In addition, how to create a risk measurement that is indifferent to the changes in portfolio in the dynamic risk analysis is stressed.

This investigation on capital allocation is conducted using the risk model applied in Achmea Reinsurance Company. Several risk measures, dependence structures and allocation methods are discussed. Nevertheless, as a special investigation on Achmea internal risk model, the research concentrates on the VaR, Gaussian Copula and Euler principle together with the Monte Carlo simulation approach. Several conclusions are obtained from different aspects that are observed in practice.

Understanding the challenge of capital allocation is not trivial. The paper starts by discussing several other important risk management concepts that is fundamental to the capital allocation theories in Chapter 2. Chapter 3 introduces and compares several allocation methods including the Euler principle that is implemented in the real model. Chapter 4 explains the implementation of capital allocation method using the Monte Carlo simulation approach. In Chapter 5, sensitivity analyses are carried out based on the results of the model and further discussions about the theories behind are provided. Chapter 6 focuses on the process of dynamic risk analysis and calculates the expected utility value of risks that is indifferent to the order of the risks. The issue is addressed by illustrating a numerical example as a sensible starting point for subsequent investigations. Chapters 5 and 6 form the main contribution of this paper. In Chapter 7, conclusions are drew and further suggestions are given on the capital allocation and the dynamic risk analysis.

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2 Capital allocation and related risk concepts

In this chapter, several risk concepts are discussed, which are intimately connected to risk modelling and necessary to understand the capital allocation problem.

2.1 Risk measures

Definition 2.1 A risk measure is a mapping  from a set



of real valued random variables defined on (,F,P) to R : ) ( : : R X  X    ,

where  is the state space, F is a σ-algebra on space  representing the information available to practitioners and P is the physical probability measure on (,F). Risk

measures are used in risk management representing certain amount of capital to quantitatively measure risks. Four properties are proposed by Artzner et.al.(1999) as the requirements of coherence, defined as follows:

1. Sub-additivity: for all X ,Y,we have :(XY)(X)(Y).

Sub-additivity is the most debatable risk property probably because it rules out the Value-at-Risk as a risk measure when the underlying risk distribution is non-elliptical. Nevertheless, it is argued to be a reasonable property mostly for three reasons. Firstly, practitioners are supposed to benefit from risk diversifications. Secondly, financial institutions should have no incentive to split the portfolio. Thirdly, sub-additivity makes the decentralization of risk management possible.

2. Monotonicity: for all X ,Y, such that X Y, almost surely we have (X)(Y). This property naturally holds for the fact that the larger losses may caused by risks the larger amount of capital is preserved.

3. Positive homogeneity: for every X and c0, it is required that(cX)c(X). This property is also controversial since the concentration of risks is supposed to generate diversification benefits according to sub-additivity: (nX)n(X). It is also

counterintuitive when considering the liquidity risk. Relaxing the positive homogeneity property leads to the convex risk measures.

4.Translation Invariance : (X r)(X) holds for everyX, and all real numbers R, where ris the total return on a risk free investment of .

This property implies that the risk free investment of cash will undermine the capital requirement.

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5. Comonotonic Additivity: for allX ,Y, such that X and Y are comonotonic, we have

that: (X Y)(X)(Y).

Comonotonicity is the strongest form of positive dependence between random variables, defined as follows: Two random variables, X ,Y, are called comonotonic if there exists a random variable U and non-decreasing functions d,esuch that X d(U),Ye(U).

Those properties of coherent risk measure rule out a number of conventional measures of risks in insurance industry such as standard deviation criticized for violating the monotonicity property. The most commonly used risk measures: the Value-at-Risk and the Expected Shortfall will be discussed.

2.1.1 Value-at-Risk

Definition 2.2 Value-at-Risk (hereafter referred as VaR) is defined as:

} ) ( | inf{ ) ( ] [X F 1 p x R F x p VaR_p  _X    , (2-1) for any real-valued random variable X with density function F , where probability level

] 1 , 0 [ 

p . VaR represents the worst possible loss given that there is only, for example, a 1 in 200 chance of occurring a loss worse than VaRp.

VaR is prevalently applied in the industry for three noticeable advantages. First of all, VaR successful captures the crucial aspects of risk in a single number enabling it comparatively convenient to be understood and communicated with. Secondly, VaR is relatively easy to estimate accurately in practice using historical simulation approach, Monte Carlo simulation approach or analytically computation. More importantly, as a quantile estimate, VaR is more stable dealing with heavy tailed risks comparing to the Expected shortfall. Further discussions can be referred to Yasuhiro Yamai, Toshinao Yoshiba(2005).

Nevertheless, VaR is broadly criticized for three major drawbacks. First, it violates the sub-additive property when risks are non-elliptical distributed, proved by Aztzner et al. (1997, 1999). This severely diminishes its reliability in most cases encountered in practice, especially when the loss distribution is heavily skewed or the dependences are exotic. Secondly, as a single order statistic withdrew from a probability distribution, VaR fails to reflect the complete information in the tail. Practitioners may hold wrong perceptions about risks by using VaR. Thirdly, VaR should not be the only constraint in optimal investment problem, which may lead to a gambling portfolio.

2.1.2 Expected shortfall

Tail-VaR, also called expected shortfall or conditional tail expectation, is a favorable alternative of the VaR. It is an extended computation of the VaR.

Definition 2.3 For any random variable X , with VaR at confidence level [0,1], the

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

    1 ) ( 1 1 )] ( | [ ) (    X E X X VaR X _ VaR X du ES _u , (2-2)

As a coherent measure, the ES satisfies all four properties previously discussed and incorporates the complete information in the tail.

However, Yasuhiro Yamai and Toshinao Yoshiba discovered that there are large magnitude of estimation errors when estimating the ES in fat-tail distributions, since that the ES is withdrew from the large infrequent losses in the tail. The un-robust estimate of the ES comparing to that of VaR severely undermines its quality as a risk measure. Furthermore, due to the limited information available in practice, the ES sometimes does not even exist. Also, the ES is computationally less attractive than the VaR.

To sum up, the VaR and the ES are both commonly used as risk measure across the insurance and finance industry with their own strengths and weaknesses respectively. Since the ES is the arithmetic average of the VaR of risk X above confidence level, a conspicuous conclusion can be achieved that

ES

_

(

X

)



VaR

_

(

X

)

indicating that the ES is more conservative than the VaR. Performances of two risk measures are tested in Chapter 5 in practical examples.

2.2 Economic Capital

As a financial institution, it is necessary to ensure that the balance sheet is stabilized throughout the business circle. Using the risk measure as a buffer, simply promises a non-profit future. The economic Capital, on the other hand, can make sure the company remains solvent with a sufficient surplus. Economic capital is defined as:

) ( )

(X E X

EC   , (2-3) where ( X)is a risk measure, E( X) is the expected loss of risk X .

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.01 10.01 20.01 30.01 40.01 50.01 E(X) Economic Capital 99,5% Value-at-Risk

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Figure1: Economic capital and value-at-risk

In other words, the risk will only tell us something about the deviations of the portfolio cash flow from its expected value, the economic capital will also take into account the expected value itself. Given that E X( )0, applying the economic capital in risk management provides the reinsurers a relatively conservative capital requirement comparing to the risk measures.

2.3 Performance measurement

Economic capital is an emerging standard in insurance industry to reveal the downside of risks. It also lends itself to the performance evaluation as the denominator of Return on Risk adjusted Capital (referred as RORAC). As a financial institution, with certain risk appetite, management needs to achieve a balance between risk and return in order to make tactical decisions. As a result, performance measurement arises to be a critical standard for decision-making. Various kind of performance measurements have been explored starting from

2 ) , ( M M i i R R Cov 

  under the CAPM framework, thereafter the Sharpe Ratio from the

investment theory etc. However, the ratio of the corresponding return to the allocated capital is praised for being a much more realistic and intact evaluation of risks. This paper use the notion of RORAC instead of the equivalent R@Par used in Achmea Re. Definition of compatibility of RORAC is recalled according to Tasche(2007):

) ( ) ( ] [ ) ( 1 S S S E S RORAC n i i   



   , (2-4) where



  n i i X S 1

is the total risk portfolio comprised of

n

individual risks, (S) is the aggregated economic capital and



_i is the expected return of i th risk. Consequently, the total

expected return equals to the sum of individual expected return. Analogously, RORAC of the

i th risk is defined by:

) | ( ) | ( ] [ ) | ( S X S X X E S X RORAC i i i i i      . (2-5)

Definition 2.3 Risk contribution of

X

i to (S)is defined as



(

X

i

|

S

)

. More specifically, risk contribution is the allocated economic capital conditioning on the aggregated economic capital(S). For later reference, we also define VaR contrbution as the risk contributons of subdivisions with respect to the risk measure VaR. The process of assigning total VaR to subdivisions by calculating VaR contribution is called the VaR contribution method.

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Based on the notion of RORAC, compatibility property is desirable to practitioners from an economic point of view.

Risk contribtuions are RORAC compatible if there are some



_i 0, such that :

)

(

)

(

)

(

)

|

(

X

S

RORAC

S

RORAC

S

hX

RORAC

S

RORAC

_i







_i



for all 0 h



_i.

The compatibility of RORAC is a substantial criterion to explore when optimizing our portfolio. As the denominator of RORAC, allocated capital plays a major role here determining risk performance.

2.4 Dependence structure

Dependence structure is the main factor contributing to the diversification benefits on the aggregated level. It is highly important to model the dependence structure between risks precisely, so that the risk model is able to characterize the true interactions of risk behaviors in the real world. Dependence may exist in various partition of financial models such as risk taxonomy for financial conglomerates including market risk, credit risk, counterparty default risk, operational risk etc. Several dependence measurements are introduced in this section. Gaussian Copula is the most considered one in the discussion for the convenience of illustrating the risk model in Achmea Re.in Chapter 4.

2.4.1 Linear correlation

The most common approach to measure multivariate dependencies is the Pearson correlation, calibrated as: , ) ( ) ( ) , cov( ) , ( Y X Y X Y X     cov(X,Y)E[XY]E[X]E[Y],

Correlation coefficient behaves nicely under linear combinations, which models only symmetric dependence. It is a good indicator of dependence with the class of multivariate Gaussian distributions.

However, the correlation coefficients, is not able to measure the dependencies of non-linear correlated risks. A simple example would be to assumeY X2, where X follows a standard Gaussian distribution. Thus,

cov(

X

,

Y

)



E

[

X

3

]



E

[

X

]

E

[

X

2

]



0

. This is a highly contradicting measurement of the dependence to the reality, where X and Y exhibits the strongest dependence in between.

We should also be careful when applying monotonic transformations to variables which may affect the correlation between variables and in turn may affect our decision. Assuming that a strictly increasing function f , such that for all X and Y , where X < Y , f x( ) f y( ),

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( , )X Y ( ( ), ( ))f x f y

  .

Moreover,  is defined for random variables with finite variances. For variables with heavy tails, it may not exist.

2.4.2 Copula dependence measure

Copulas are an advantageous concept to capture the asymmetric dependencies. It enables us to separate the modeling of the marginal distributions from the modeling of the dependence structure.

2.4.2.1 Copula

The copula of a set of random variables

X

₁

,

X

₂

,

X

₃

,...,

X

_n is a function

C

(

u

₁

,

u

₂

,

u

₃

,...,

u

_n

)

defined on

[

0 ,

1 ]

n such that

)

(

,...,

)

(

)

,...,

,

(

u

₁

u

₂

u

₃

u

_n

P

F

₁

X

₁

u

₁

F

_n

X

_n

u

_n

C





. (2-6)

Therefore, copula can be presented as the conjugated probability distribution function of uniform random variables transformed from different marginal distributions. A bivariate copula is any function

[

0 ,

1 ]

2



[

0 ,

1 ]

that has the following three properties:

1. C(u₁,0)C(0,u₂)0 2. C(u₁,1)u₁,C(1,u₂)u₂ 3. u₁,u₂,v₁,v₂[0,1],u₁ v₁,u₂ v₂, 0 ) , ( ) , ( ) , ( ) , (v₁ v₂ C v₁ u₂ C u₁ v₂ C u₁ u₂  C .

Equivalently, a bivariate copula is a bivariate density function with domain

[0,1]

2and uniform (0, 1) marginals. Every bivariate copula C satisfies the Fréchet–Hoeffding inequality:

( , ) max{0; 1} ( , ) min{ ; } ( , )

W u v  u  v C u v  u v M u v ,

for every

( , ) [0,1]

u v



2. Here, the Fréchet–Hoeffding bounds W and M are themselves bivariate copulas. Contrary to the Pearson linear correlation, copula is invariant under monotone increasing transformations of the random variables.

According to Sklar theorem(1959), let F be a joint distribution function with margins

d

F

F ,...,

₁ . There exists a copula C:

[

0 ,

1 ]

d



[

0 ,

1 ]

such that, for all

x ,...,

₁

x

_din R[,],

))

(

),...,

(

)

,...,

(

x

₁

x

_d

C

F

₁

x

₁

F

_d

x

_d

F



. (2-7) The proof is simple in the sense that assuming all F_j(x_j)are continuous, each has an inverse function F_j1(u)such that F_j[F_`_l1(u)]u, for all 0u1. If U_j F_j(x_j), then U_j ~ U(0,1),

u u F F u F x P u x F P u U P( _j  ) [ _j( _j) ] [ _j  _j1( )] _j[ _j1( )]

)

,...,

,

(

)

,...,

(

x

₁

x

_d

P

X

₁

x

₁

X

₂

x

₂

X

_d

x

_d

F





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))

(

),...,

(

))

(

),...,

(

),

(

U

₁

F

₁

x

₁

U

₂

F

₂

x

₂

U

_d

F

_d

x

_d

C

F

₁

x

₁

F

_d

x

_d

P





The opposite direction can be verified to be true: given copula C and marginal distributions, the joint distribution can always be defined as

F

(

x

₁

,...,

x

_d

)



C

(

F

₁

(

x

₁

),...,

F

_d

(

x

_d

))

. In other words, copula uniquely exists for all possible joint behavior of random variables.

There are two families of copula that are mostly considered in the insurance and finance industry: Archimedean and elliptical copulas. Archimedean copula includes Gaussian copula and t copula. Elliptical copulas deal with Clayton and Gumbel copula. To be compatible with the internal risk model in Achmea Re., this study skips the theoretical illustration of Elliptical families and t copula, but emphasizes on the description of Gaussian Copula. It is interesting to notice that Gaussian Copula is regarded as a special case of the t Copula when the degree of freedom in multivariate t distribution goes to infinity.

2.4.2.2 Gaussian copula

Modeling and simulating the dependent transitions is usually described in terms of Gaussian random variables instead of uniform variables. If

F

(

x

₁

,...,

x

_d

)

is a multivariate normal distribution:

N

_d

(



,



)

, then

C

(

F

₁

(

x

₁

),...,

F

_d

(

x

_d

))

is a Gaussian copula. Since the location and scale of the distribution does not change the form of copula, so the conventional assumption is

0



 and the correlation matrix is R. According to the definition of Gaussian copula, the joint distribution of

U

₁





(

Z

₁

),...,

U

_d





(

Z

_d

)

,

Z

_i

~ N

(

0 ,

1 )

, is equivalently the joint distribution of

N

_d

( R

0 ,

)

.

The representation of Gaussian copula is:

)

),

(

),

(

)

,

(

u

v

₂ 1

u

1

v



C









 ,

where 1is the inverse of the cdf for the normal distributions. ₂is the bivariate normal cumulative distribution function and  is the correlation coefficient. In multivariate normal distribution,  represents the correlation matrix R . Notice that R is necessarily non-negative definite. A curious property of Gaussian Copula is that no tail dependence structure imposed on risks regardless of the correlation matrix R .

2.4.2.3 Rank correlation for Gaussian copula

Rank correlations are simple scalar measures of dependence calculated by only looking at the ranks of the data. Unlike linear correlation, rank correlations are determined only by the copula of a bivariate distribution and not by the marginal distributions. There are two main varieties of rank correlation: Kendall’s tau and Spearman’s rho. We will only introduce Spearman Rank correlation rho in this paper. The definition of the Spearman Rank correlation rho is:

1 2

( , ) ( ( ), ( ))

S X Y F X F Y

  ,

where F₁ and F₂ are the cumulative distribution functions of X and Y respectively. Consequently, Spearman’s rho is simply the linear correlation coefficient of F X₁( ) and

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2( )

F Y which for continuous random variables is the linear correlation of their unique copula. It can be specifically expressed in the terms of copula:

1 1

0 0

( , ) 12 [ ( , ) ]

S X Y C x y xy dxdy

 

 

 .

Moreover, there exists a relation between spearman rank correlation _S and linear correlation  in Gaussian copula model:

1 2 6 1 ( , ) arcsin 2 s X X





 . (2-8) Spearman’s rho can be easily calibrated in real practice and one can form a matrix of pair-wise correlations with more than two variables. In order to apply Gaussian Copula in risk modeling, estimates is computed using the above theoretical relationship between spearman rank correlation and linear correlation.

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3 Capital allocation methods

This chapter describes the problem of capital allocation, elaborates the theories behind several allocation methods and discusses the connections between them. Euler principle is closely examined associated with two different risk measures.

3.1 Description of the capital allocation and allocation principles

A preliminary setting to describe the capital allocation problem is as follows: Considering a portfolio of

n

individual risks

X

₁

,

X

₂

,

X

₃

,...,

X

_n, the random variables have a dependence structure characterized by the joint distribution of the random vector

(

X

₁

,

X

₂

,

X

₃

,...,

X

_n

)

.

Hence, the aggregated risk is



  n i i X S 1

and the aggregated level of capital K is determined

by the risk measure  such that K (S)R. While quantifying the overall risk of the company is important for strategic management, it is of great interest to allocate the overall economic capital back to the individual business divisions enabling the linking of tactical decisions with strategic goals. Denault (1999) first suggested four appealing principles of coherent allocation that are widely adopted in the industry analogous to coherent risk measure.

An allocation principle is coherent if for every allocation problem(N,), the allocation satisfies the following three properties:

1. No undercut:



     M i i M i i X K N M , ( ).

2 .Symmetry: if by joining any subset M N\{i,j}, portfolio i and j both make the

same contribution to the economic capital, thenKi Kj.

3. Riskless allocation: K_n 



(



r_f)



, whererf is price of

th

n riskless portfolio at time T.

In addition to the principles above, the preliminary restraint of capital allocation is the full

allocation principle: K K n i i 



1

, an integral part of coherent allocation. Loosely speaking, the

overall economic capital K (S) needs a mapping to obtain

K

_i as a consistent indicator of individual risk. It is reasonable and straightforward to require the capital to be distributed seamlessly among subdivisions. No-undercut principle implies that individual risk or subsets of risks will not be better off deviated from the risk pool, because the standalone economic capital is lower than the capital allocated from the overall capital. The second principle conforms with our intuition, that the allocated capital should always reflect the contribution of

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individual risk to the total portfolio no matter what combination of risks. However, it is utterly ambiguous and invasive for the practitioner to calculate the risk contributions given that diversification benefits are the results of the integrated influence of all risks. The last allocation principle is not that applicable in this paper, therefore it will not be discussed.

3.2 Diversification benefit

As we mentioned in chapter 2, sub-additive property of coherent risk measure postulates the existence of diversification benefits, which is generated by pooling the risks that are not completely comonotonic with each other. Suppose we fix the number of risks in our portfolio

)

,

(

X

₁

X

₂



X

_n and the probability space(,F,P), each risk is mapped to certain level of economic capital by risk measure. The following inequality holds:



   n i i X S K 1 ) ( ) (   .

Consequently, the challenge in capital allocation that arises is how to allocate the diversification benefits back to individual risks.



         N i i N i i N i i n K X K X X X d ( ₁) ( ₂) ( ) ( ) i i i

X

d

K





(

)



, (3-1) where

d

is the total diversification benefits,

d

_i is the diversification benefits allocated to the standalone risk.

Calculating the total diversification benefits alone can remotely reveal the link between individual risk contributions and total risk pooling effect. The way of attributing diversification benefits to each division appears to be obscure without closed mathematical form of expression.

3.3 Traditional capital allocation methods

There are several traditional capital allocation methods widely being applied in insurance and finance industry. Traditional capital allocation methods include the haircut allocation

principle, the covariance allocation principle, the quantile allocation principle and the conditional tail expectation principle. We set KF_S1(p) as the overall risk capital at confidence level p. Andthe differences between economic capital and risk measure are negligible. Risk measure (S) is chosen to be the VaR in order to be compatible with the discussions in the following chapters.

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13

3.3.1 The haircut allocation principle:

), ( ) ( 1 1 1 p F p F K K i j X n j X i   



 i1,...,n. (3-2)

The haircut allocation principle divides the total capital by the ratio of standalone capital to the overall economic capital. The more risky it is on a standalone basis the more capital will be allocated to the risk. However since the VaR does not satisfy the sub-additive property of risk measure. It may occur that F p K

n j Xj 



  1 1 )

( , which results in the violation of no undercut

allocation principle. Nevertheless, the haircut allocation principle overlooks the dependence structure and distribution feature of the risk such that changing the dependence structure will not, as expected, change the allocation result. Moreover, due to equation (3-1):

K X K X d N i i N i i N i i



       ( ) ( ) ,

If we deviate the allocated capital as:

], ) ( 1 [ ) ( ) ( ) ( ) ( ) ( ) ( ) ( 1 1 1 1 1 1 1 1 1 1 1 1



                    _n j X X X n j X X X n j X n j X i p F d p F p F p F d p F p F p F d p F K j i i j i i j j

As can be seen, diversification benefits are allocated proportionally by the haircut allocation principle. The allocation result is determined by the standalone risk performance independent from its influences on the total portfolio.

3.3.2 The quantile allocation principle:

)), ( ( ) ( 1 K F F K c k i S X i    (3-3) where c

S is defined as the comonotonic sum



   n i X c U F S i 1 1 )

( , U is the uniform random

variable on (0,1). The quantile allocation principle is allocating diversification benefits by loosening the confidence level on a standalone basis. If we rewrite the equation (3-5):



           n i X X S p F p F p F K i i i c 1 1 1 1 ), ( ) ( ) (

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14



        n i X X S q F q F q F i i i c 1 1 1 1 ), ( ) ( ) (

here the inequality holds if and only if assuming that the risk measure satisfies the sub-additive property. In the case of the VaR, underlying risk distribution is restricted to be an elliptical distribution. Consequently, if F 1(p) F 1 (p),

i i c X S      

conclusion comes along that

p

p , herewith it is derived that:

) ( ) ( ) (X K F 1 p F 1 p d i i X X i i i         .

In other words, the diversification benefits are allocated according to how extended the tail of loss distribution is. The quantile allocation method is a special case of the haircut allocation principle.

3.3.3 The covariance allocation principle:

) , ( ) (S Cov X S VaR K K_i  _i , i1,...,n. (3-4)

The covariance allocation principle is one of the most widely used allocation methods. It captures the volatility of the risk itself and dependence structure between risks by the variance and covariance. However, since variance and covariance are above all the dependence estimators of the entire loss distribution, they lose track of the extreme events’ occurrences. A more debatable issue is that covariance only measures the linear correlation between risks. The tail dependence of the joint distribution is what in fact determines the allocated capital. Therefore, the covariance allocation principle is broadly criticized for omitting the tail dependence of the risks. Many studies have signified that the covariance principle is closely related to the Euler principle with the ES as the risk measure, or with the VaR as the risk measure conditioning on the elliptical distribution.

3.3.4 The conditional tail expectation principle:

The conditional tail expectation , also called the ES, is an alternative of the VaR computed by taking the average of losses that are greater or equal than VaR,

)] ( | [ ) ( 1 p F S X E S CTE K K i S p i    , i1,...,n. (3-5)

It has been proved by Landsman and Valdez (2003) that the conditional tail expectation principle and covariance principle coincide with each other while risks are elliptically joint distributed. These two allocation principles are equally sensitive to the dependence between risks.

Proportional allocation methods are advocated for their simplicity and intuitively appealing expression while bearing a number of weaknesses. The haircut allocation principle does not

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15

take the dependence structure between risks into consideration. It is indifferent to the change of the correlation between risks by allocating capital evenly according to the ratio of individual VaR to the total VaR. The quantile allocation principle is analogous to the haircut allocation using the individual losses of quantileF c1(K)

S



as the allocated capital of risks. These point-wise estimates of the risk contributions are imprudent because the risk contributions are presumed to measure risks in a dynamic way, making allowance for the joint loss distribution of the total portfolio as a whole. The covariance allocation principle reveals both the volatility of individual risk itself and the dependence structure in between but ignores the tail structure of risks. Hallerbach (1999) demonstrated that it could be efficiently analyzed from the perspective of OLS regression. The conditional expectation allocation principle also conforms with the results of VaR contribution proved by Chris Marrison (2002).

3.4 Euler principle

Alternative method to allocate capital using Euler theorem was given by Tasche (1999) with rather appealing results of the risk contributions. Before presenting the result of Euler principle, we introduce weight variables

u



(

u

₁

,

u

₂

,...,

u

_n

)

:

1 2 1 ( ) ( , ,..., ) n n i i i S X u X u u u u X    



,

Evidently, the actual loss

1 (1,...,1) n i i S X X   



.

Theorem 3.1 (Euler’s theorem) Let U Rn be an open set and f U: Rbe a

continuously differentiable function. Then f is homogeneous of degree



if and only if it satisfies the following equation:

1 2 1 ( ) ( ) , ( , ,..., ) n i n i i f u f u u u u u u U u         



.

Since f_ is positively homogenous of degree 1, differentiable homogeneous function is represented as a weighted sum of their derivatives in a canonical manner by Euler theorem.

Combined with full allocation principle

1 ( ) ( | ) n i i i S u X S   





, Euler’s theorem suggests a natural way to assign the total capital to bussiness divisions with respect to their weights. Moreover, Tasche(1999) proved that gradient of risk measure (S)is identified to be the only suitable definition of the risk contributions for performance measurement.

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16

Definition 3.1 Let be a risk measure homogeneous of degree 1 and f_ the function defined by f_=



( )S . Assume that f_is continuously differentiable with units

u



(

u

₁

,

u

₂

,...,

u

_n

)

. The risk contributions



(

X

_i

|

S

)

are uniquely determined as:

0 ( _i| ) ( _i) |_h (1,...,1) i f d X S S hX dh u      _   (3-6) The risk contributions computed according to (3-6) are called the Euler contributions. The process of allocating capital to business divisions by calculating Euler contributions is called the Euler allocation. As we discussed in (2-1), the VaR and ES are both positive homogenous risk measures, if assuming that the risks or portfolios are not significant comparing to the whole market and are not sheltered from market liquidity risk.

This section will theoretically illustrate how to obtain a closed-form expression according to Euler principle associated with the VaR and the ES. In addition, the connection between the result of Euler principle and other similar allocation methods will be exploited.

3.4.1 Value-at-Risk

Many literatures argued that the VaR is not differentiable due to its lack of smoothness. However, Tasche(1999) proposed that VaR differentiate can be expressed in forms of

conditional expectation in a general context. Starting from deriving the risk contributions with a heavily restricted assumption of multivariate Gaussian joint distribution to examine the partial differentiate of VaR in a direct manner, a more widely applicable expression of VaR differentiate will be given later in this section.

Multivariate normality

Assumption of multivariate normal distribution simplifies the problem given that under Gaussian distribution with mean



_sand variance



_s2, VaR at confidence level 99.5% can be expressed as:

s

s N

VaR₉₉_,₅_%   1(0,995) ,

Assuming that individual risk

(

X

₁

,

X

₂

,

X

₃

,...,

X

_n

)

follows normal distribution, the portfolio return, continuously compounded over the horizon t, therefore, also follows multivariate normal distribution,

S

~

N

(



_s

,



_s

).

Given that:

2 2 2 1 , 1... n n s i i i j i j i i j u u u i n       







 

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17 c S VaR S VaR S X Cov c S VaR S X Cov u VaR u S S X i i i i i _           ( ) ) ( ) , ( ) ( ) , ( ) ( ) | (  99,5%  , (3-7)

where,

c



N

1

(

0 .

995 )

. By using standard deviation as risk measure,

) ( ) , ( S VaR S X Cov _i can,

therefore, be identified as a portion of the total risk, which happens to be fully consistent with the covariance allocation principle in (3-6). Broadly speaking, conclusion can be obtained that any elliptical distributions and their joint elliptical distribution can be applied with the

covariance allocation method. If we extend the covariance between

X

_iandS ,



   n i j j i i i S Var X X X X Cov( , ) ( ) cov( , ), i1,...,n. (3-8)

Equation (3-8) indicates that Euler principle associated with Standard deviation as risk measure is allocating the total risk capital according to the variance and covariance contribution of individual risk.

Introducing the notion of Marginal risk contribution may shed light on the logic of allocating capital by the Euler principle and the covariance principle. Marginal risk contributions to the total capital are the differences of the total capital amount with risk i and total capital without risk i.

Definition 3.2 Marginal risk contributions of risk i, i1, 2,..,n is defined by:

( | ) ( ) ( )

mar Xi S S S Xi

    .

Assume that we have two risks X and Y in our portfolio, calculating marginal risk

contributions under Gaussian distribution denotes the extra volatility of portfolio by adding risk X . Given risk Y come after X in the portfolio, the marginal risk contributions of adding Y to X is:

( | ) ( ) ( ) ( ) 2cov( , )

mar X S Var X Y Var X Var Y X Y

      .

Therefore, both the Euler principle and the covariance principle allocate the capital by only taking 1 times the covariance between the risk i and the rest of the portfolio. In other words,

the covariance is evenly distributed. The covariance principle allows for the volatility of risk itself: Var(Y) and the dependence between risks: cov(X,Y), however, ignoring the heavy tail impacts on the total portfolio.

Tasche (1999) and Hallerbach (1999) both exhibited that the Euler principle assuming normal distribution is coincidently identical with the results of Markowitz theory in the CAPM. The same results can be derived by averaging the marginal variances of adding new risks using the Shapley value in Game theory, in-depth discussion can be referred to Donald.F.Mango (1997).

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18

Without losing generality, we provide a expression of the VaR differentiate in a general context, since Gaussian assumption cannot be uphold for most non-life risks to reinsurance companies. Computing risk contributions associated with the VaR directly requires several technical assumptions imposed on the joint distribution as follows:

Let d2and (X₁,...,X_d) be an R -valued random vector with a conditional density d  of

1

X given fixed (X₂,...,X_d). We say that  satisfies assumption in an open set 

(u₁,...,u_d)

u U R if the following conditions hold:

(i) For fixed x ,...,₂ x_d the function t(t,x₂,...x_d)is continuous in t .

(ii) The mapping

) , 0 [ ], ) ,..., ), ( ( [ ) , ( ₂ 2 1 1 



_     U R X X X u t u E u t   d_j j j d

is finite-valued and continuous. (iii) For each i2,...,dthe mapping

) , 0 [ ], ) ,..., ), ( ( [ ) , ( ₂ 2 1 1 



_     U R X X X u t u X E u t  i d_j j j d

is finite-valued and continuous.

One of the most important assumptions proposed by Tasche(1999) can be stated as: at least one among the fluctuations X must have a continuous density. The mathematical proof is _i

carried out rigorously by Tasche(1999, Lemma5.3) using implicit function theorem and various conditions. This paper will leave out the technical part. Suppose that

)

(

)

(

S

q

S

VaR

_



_ , it is manifested that the decomposed VaR to individual risks is presented as: )] ( ) ( | [ ) ( ] | [ E X VaR S q S h hX S VaR S X VaR K i _i i i _         , (3-8)

This result is identical with the conclusion of Hallerbach (1999) simply using the property of conditional expectation. Nevertheless, there is no closed-form mathematical expression of the risk contribution

VaR

[

X

_i

|

S

]

available due to the limitation that the VaR is an unsmooth quantile estimate.

3.4.2 Expected shortfall

The expected shortfall (ES), as an alternative risk measure, is homogeneous of degree 1, co-monotonic additive and sub-additive. Under roughly the same assumption as differentiating VaR, the ES can be differentiated as:

h hX S ES S X ES K i i i      _[ | ] ( ), (3-9) ] 1 [ ) 1 ( )] ( | [ _i 1 _i ₍_S _VaR ₍_S₎ i E X S VaR S E X K   _     _ . (3-10)

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19

Section 5.2.2 illustrates the similarities and differences of calculating capital allocation using these two risk measures in a practical example.

Euler principle associated with both the VaR or the ES as risk measure will lead to the similar conclusion that the capital should be allocated according to the conditional expectation of individual losses. However, little efforts have been made to explore the application of the theoretical results in real practice. Therefore, in the following chapter, practical procedures and methods to calculate capital allocation based on Monte Carlo simulation is described.

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20

4 Capital allocation based on Monte Carlo

simulation approach

Analytical approaches to calculate the VaR contribution is usually established on cumbersome assumptions and the outcomes are always constructed in an approximate way with first moment. Nevertheless, without non-linear loss patterns, analytical approaches are much more tractable and efficient computationally than practical simulation approaches. Simulation methods, on the other hand, outperform analytical approaches when coping with non-elliptical multivariate distributions. Simulation methods are exceptionally flexible to construct the joint distribution with non-normal marginal distributions. VaR is the risk measure used in Achmea. This chapter attempts to investigate the technique of using simulation approaches to estimate the VaR and specifies several solutions to achieve highly accurate estimate of the VaR contribution..

4.1 Monte Carlo simulation approach

The Monte Carlo simulation method was first brought up by John von Neumann in 1946 which uses repeated samplings to determine the properties of random variable or the behavior of more than one random variables. The VaR is obtained by withdrawing sample quantile from randomly created scenarios generated from the predetermined loss distribution. This section provides the detail of multi-risk modeling using the Monte Carlo simulation approach.

4.1.1 Marginal loss distribution

Assume that portfolio is constituted with three reinsurance risks

X

₁

,

X

₂

,

X

₃ each of whom follows loss distribution

f

(

x

₁

),

f

(

x

₂

),

f

(

x

₃

)

. Considering the task of estimating VaR at 99.5% confidence level from a Monte Carlo simulation of size 1000, conventionally, VaR is

withdrew as the 995th order statistic L₁(995)from the simulated random samples. Similarly, taking L₂(995) and L₃(995)as the sample quantile of

X

₂

, X

₃ , namely the standalone

) ( % 5 , 99 Xi VaR acquired.

4.1.2 Imposing dependence structure by Gaussian copula

The dependence structure between risks is not only intricate to capture and quantify but also strenuous to implement in the risk model. The dependence structure between risks can be represented by a linear correlation matrix  transformed from the real rank correlation C by _

(2-8). As we discussed in 2.4.2, copula is able to construct dependence structure in a way without limitation on the marginal distributions. This section depicts the method to impose a Gaussian Copula with known marginal distributions in a real example. Gaussian Copula is

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21

chosen here in the view of the fact that it is applied in Achmea Re. risk model. Nonetheless, other types of copula can be inserted in analogous way.

At first, marginal distributions are transformed into uniform distributions )

( ),

( ₁ ₂ ₂

1 F x x F x

x   . Secondly, (u1,u2,...,un) row-vectors are randomly drew from a

n

dimensional uniform distribution. Mapping the uniform distributions back to random

variables from standard normal distribution

Z

~

N

_n

(

0 ,

1 )

using ( 1),..., 1( )

1

1 u Zn un

Z   ,

provides two uncorrelated standard normal distributions. Then Cholesky decomposition is applied by using the linear correlation matrix . The Cholesky decomposition of Hermitian positive-definite matrix A is a decomposition of the formALLT. The multivariate normal distribution

N

_n

(

0 ,



)

can be written as:

Z L Z L Z Z Nn(0,)( 1,..., n) T   T  ,

where Zis the

n

dimensional joint distribution with correlation matrix . As a result, the normalized rank

U

₁





rank

₁

(

Z

₁



),

U

₂





rank

₂

(

Z

₂



),...,

U

_n





rank

_n

(

Z

_n



)

is rank correlated by



C indicated by copula. A pairwise rank correlation dependence structure is formed by the

core trick that rank correlation is indifference to the transformation of distributions. Thus, the rank of each random variable is determined and paired with others in a way that correlation between their ranks follows the correlation matrix. The next step is to trace back to the marginal distribution by taking inverse of F which is predetermined to be correlated with _i C , _

where X₁ F₁1(u₁),X₂ F₂1(u₂),...,X_n F₃1(u₃). To better understand the risk modeling process, a diagram is drew to illuminate the steps:

Table 4.1 Cholesky decomposition matrix

Choleski decomposition I11 1.00000 I21 0.50000 I22 0.86600

Table 4.1 Shuffling the random variables by using Gaussian Copula

U1 U2 Z1 Z2 Z'1 Z'2 RANK1 RANK2 U'1 U'2

0.786 0.177 0.793 -0.928 0.011 -0.01 8 3 0.8 0.3 0.79 0.351 0.805 -0.384 0.011 -0.001 9 5 0.9 0.5 0.925 0.919 1.438 1.395 0.019 0.033 10 10 1 1 0.093 0.165 -1.322 -0.975 -0.018 -0.025 2 1 0.2 0.1 0.76 0.34 0.706 -0.412 0.009 -0.002 7 4 0.7 0.4 0.098 0.861 -1.293 1.083 -0.017 0.009 3 9 0.3 0.9 0.19 0.16 -0.878 -0.993 -0.012 -0.023 4 2 0.4 0.2

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22

0.079 0.867 -1.413 1.112 -0.019 0.009 1 8 0.1 0.8 0.397 0.583 -0.262 0.209 -0.003 0.002 6 7 0.6 0.7 0.268 0.618 -0.62 0.299 -0.008 0.001 5 6 0.5 0.6

Expanding the random sample to100 results, correlation can be easily identified by the scatter plot.

Figure 4.1 Uncorrelated r.v. with rho=0 Figure 4.2 Correlated r.v. with rho=0,5

After the uniform distribution is pair-wisely correlated, the sum of the individual loss distribution simulation results forms into the joint distribution. Examining the joint distribution and marginal distribution enables people to take the order statistics in the loss distribution as VaR. Gaussian Copula provides us more benefits with its flexibility of the choices of loss distribution.

4.2 Calibrating the risk contributions based on simulation

approach

As a generalized expression of the VaR contribution is derived by Tasche(2000), assuming

% 5 . 99





, 1 million scenarios are generated by Monte Carlo simulation, N reinsurance portfolios are estimated and aggregated. Recall in (3-10), the 995,000th scenario out of the ranked 1million scenarios consequently serves as VaR at confidence level 99.5%. VaR₉₉_,₅_%(S)

is the sum of N individual marginal distributions simulation results. The challenge is to take the expectation of individual losses on the condition that only one VaR simulation results on the aggregated level is available. To the best of our knowledge, there is no simple method available to estimate the conditional contribution of VaR by simulation approaches.

-3.000 -2.000 -1.000 0.000 1.000 2.000 3.000 -4.000 Z' -2.000 0.000 2.000 4.000 2 Z'1

Correlation of normal r.v. after Cholesky decomposition with rho=0,5

-3.000 -2.000 -1.000 0.000 1.000 2.000 -4.000 -2.000 0.000 2.000 4.000 Z2 Z1

Correlation of normal r.v before Cholesky decomposition with rho=0

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23

As we mentioned in the previous chapter, the risk contributions defined on the basis of Euler principle measures the change of total capital needed when the individual loss exposure increase by 1%. In order to calibrate the risk contributions, a direct way of generating enough random samples could be to re-estimate VaR whenever the portfolio changes due to new risks. Whereas using computationally intensive simulation approaches to re-estimate VaR is fairly tedious and time consuming, therefore is not plausible in practice. Analytically deriving the conditional distribution of VaR is equally laborious to accomplish.

4.2.1 Conditional expectation

To be consistent with previous assumptions, the VaR of the total portfolio in size of 1 million simulation results at confidence level 99.5% will be the sample quantile, namely the 995,000th order statistic L(995,000). Denote the sample quantile as the order statistic:

)

( 

 LM

VaR  , (4-1) where M is the size of Monte Carlo simulation,  is the confidence level required. The total loss in any particular scenario is given by the sum of individual losses. For instance, the contribution of risk i to the 99.5% VaR is estimated by sum of L(i995,000)(i.e., the loss of risk i that occurs in the 5000 largest portfolio loss). However, one sample of individual loss is far from adequate to derive the expectation of losses conditioning on the total loss.

It brings to our attention that for a 1million times Monte Carlo simulation, a large number of drawing around one confidence level deviate within the scope of 1 in 10,000 times of the absolute number. If VaR_(S)L(k), for fairly wide range of statisticskmin kkmax, most of the simulation results in between are roughly the same with total loss

VaR

_

(S

)

. Hence, the conditional expectation of individual risks is written as:



       max min ) ( min max 1 1 )] ( ) ( | [ ] | [       k k k k i i i i L k k S q S VaR X E S X VaR K , (4-2)

Apparently, increasing the size of simulation enables to obtain a highly accurate estimate of conditional VaR distribution. In other words, the total portfolio loss differs extremely subtle around the confidence level that we can safely ignore the difference between L(995,000) and

) 001 , 995 ( L as a model error.

Hallerbach (1999) also inferred this technique by comparing the results of several other approximating approaches including example OLS regression, rational approximation etc.

4.2.2 Convergence of sample average

Having analyzed that great of number of drawings around

VaR



(S

)

is equivalent to the

Capital allocation methods and dynamic risk analysis based on Monte Carlo simulation approach.