Maastricht University

School of Business and Economics

Bachelor Thesis

Actuarial Modeling

Value-at-Risk, Tail Dependence andCorrelation

Author:Ali Nabbi

Supervisor:Dr. Eric Beutner

July 26, 2015

Abstract

In this thesis, we analyze the interaction of diversification effect in a risk measure with cor-relation and tail dependence of underlying joint distribution, in particular we will focus onthe Value-at-Risk which plays major role in risk calculations in Solvency II and Basel II/III.This thesis shows the underestimation of risk-based capital under misleading assumptions ofValue-at-Risk properties.

Declaration

I, Ali Nabbi, hereby do declare that the thesis entitled Value-at-risk, Tail Dependence andCorrelations is my original work and it has been written by me in its entirety. I have acknowl-edged all the sources of information which have been used in the thesis.

Date:July 26, 2015

Signature:

1

Acknowledgments

I would like to express my deepest gratitude to my supervisor, Dr. Eric Beutner, who hassupported me throughout my thesis with his patience and vast knowledge. I would never havebeen able to finish my thesis without his mentorship and support.

I would also like to thank my parents, my older sister, my younger brother, my uncle Mohsenand my aunt Mechthild. They were always supporting me and encouraging me with their bestwishes.

2

Contents

1 Introduction 41.1 Regulatory frameworks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.1 Basel I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.1.2 Basel II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.1.3 Solvency II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Risks and diversification effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.1 Type of risks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.2 Diversification effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Risk measures 72.1 Insight to risk measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 Loss functions and risks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.2 Value-at-Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Copulas and correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Correlation and diversification 103.1 VaR with discrete risks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2 VaR with continuous risks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4 Tail dependence and diversification 14

5 Conclusion 16

References 17

3

Chapter 1

Introduction

Consider an investor that holds a portfolio of a particular number of stocks or an insurer thatsells policies. Both of them have an asset with uncertain future value. These examples canbe extended to any risky assets e.g. options, bond portfolios, currency forwards, portfolio ofloans. Accurate evaluation of portfolios in risky scenarios is the cornerstone of an effectiverisk management which improves performance against sudden shocks, manages assets in capitalmarkets, increases the value of the firm, reduces taxes etc.

Risk management is defined as a procedure to manage possibility of adverse effects of futureevents (Kloman, 1992). However, the downside of risk is emphasized in the most cases andrarely it caught attention for its upside i.e. opportunity to gain.

Randomness plays a crucial role in quantitative aspect of evaluation of financial products.The concept of risk strongly relates to uncertainty and as a result, the impact of randomness andprobability in the finance and insurance (McNeil et al., 2005). The risk measurement methodsand techniques can be applied to many risk categories e.g. market, credit and operational risk.Although the boundaries of risk categories above are not explicit therefore in the followingchapters we will focus on the risk associated with using a misspecified model —also known asliquidity(model) risk.

Financial institutions need to save capital to insure themselves against possible adversefuture outcomes in order to remain stable economically and prevent from bankruptcy. On agreat scale, risk measurement and risk management lead to stable and sustainable economytherefore need for regulations and frameworks in this field is a necessity i.e. Basel II in bankingsector and Solvency II in insurance.

1.1 Regulatory frameworks

Basel Accords are banking regulations issued by the Basel Committee on Banking Supervisionwhich consists of representatives of central banks of G-20 big economies. In insurance sector,Solvency directives focus on the minimum capital that EU insurances must hold to reduce therisk of insolvency. This section is intended to introduce frameworks more in details from a briefhistory of Basel I to Basel II in banking sector and the simultaneous improvement in SolvencyII in insurance sector.

1.1.1 Basel I

In 1988, the first Basel Accord was the most important step to measure risk (in particular, creditrisk) to specify adequate capital of banks and financial institutions in order to keep them outof bankruptcy. However, the measured risk in Basel I was inaccurate and insufficient. Besides,there was no treatments regarding derivatives. After developing Value-at-Risk, Basel I improved

4

CHAPTER 1. INTRODUCTION 5

their risk valuation by using Value-at-Risk as a standardized market risk by the year 2000 inorder to satisfy the need of immediate and accurate risk measurements.

1.1.2 Basel II

In 2001, the second Basel Accord focused mostly on more risk-sensitive approach to evaluatethe risk of credit portfolios —also known as the three-pillar concept which is the major changewithin Basel II.

- First pillar is to calculate regulatory capital using a quantitative method. In particular,the minimum capital an institution needs to cover its potential financial loss has to beestimated using a particular approach. However, banks may choose different methods tocompute capital charge in different risk classifications.

- Second pillar is designed to review the banks evaluations to be effective and unbiased inseveral risk categories. In other words, the amount of minimum capital(first pillar) shouldbe firmly attached to the organization structure e.g. the board of directors is responsibleto risk overview and forming the future risk structure.

- Third and final pillar is reporting guidelines in order to discipline the market by forcingbanks to report their capitalization accurately and include all necessary information torisk management.

1.1.3 Solvency II

Insurance regulations are codified under the influence of the increasing regulations on bank-ing sector e.g. Solvency I/II directives in EU countries under the influence of Basel Accords.Solvency I was based on minimum guarantee fund and margin of a percentage of non-life pre-miums with part of life premiums. It was straightforward but monitoring this procedure istime-consuming and costly, besides it was value-based rather than risk-based. These issues andneed for an risk-oriented assessment lead to Solvency II (McNeil et al., 2005).

The Solvency II Directive adopts more dynamic risk-based approach by setting new capitalrequirements and valuation tools. Additionally, changes to capital requirements reflects betteroverview of insurer’s risk portfolios which encourage insurance and reinsurance companies tomodel their own Solvency Capital Requirement (hereinafter SCR).

1.2 Risks and diversification effect

1.2.1 Type of risks

In the large scale, risks can be divided in two categories, namely firm-specific(unsystematic)and systematic risks (Berk and DeMarzo, 2014). Macro factors always influence organizationsregardless of their industry. Generally, the risk resulted from external factors cannot be con-trolled, consequently cannot be diversified. On the other hand, firm-specific risks is the effect ofinternal impacts e.g. specific news from firm’s structure, service or organization. Additionally,these impacts can effect related firms. As a result, if a portfolio contains unrelated assets, it istrivial that probability of loss is relatively lower. This technique is known as diversification toreduce unsystematic risk in portfolios.

1.2.2 Diversification effect

Diversification effect is defined as decrement in the value of risk measure of a portfolio containingn risky assets in comparison with sum of n risk measures of assets individually (Hurlimaan,

CHAPTER 1. INTRODUCTION 6

2009a). In other words, merging risky assets in a portfolio does not create extra risk. Otherwise,risk measure has concentration effect which is called a super-additive risk measure. It may leadto violation of reducing risk using diversification, incentive to break up into divisions in orderto reduce the minimum required capital etc. Risk measure’s properties will be discussed in thenext chapter more in details.

Chapter 2

Risk measures

2.1 Insight to risk measures

Since risk analysts are mostly interested in the probability of default on losses, they focused onloss distribution in order to design risk measures. Moreover, the concept of loss distributiongives comparable overview regardless of the size of the portfolio or position of the financialinstitution.

This section focuses on risk measure and their properties based on the loss distribution. Asthe major part of SCR under Solvency II, we will focus on Value-at-Risk and its propertieswhich may lead to diversification or concentration effect.

2.1.1 Loss functions and risks

Let V (t) be the value of a portfolio at time t then loss in value of portfolio in the period [s, s+∆]is defined as follows.

L[s,s+∆] = −(V (s+ ∆)− V (s)

)the loss of a portfolio is more convenient to express in fixed time units. In particular, It can beintroduced a time series process (Θt)t∈N with Θt := Θ(t∆). Loss in the fixed horizon can bewritten as follows.

Lt+1 = L[t∆,(t+1)∆] = −(V ((t+ 1)∆)− V (t∆)

)= −(Vt+1 − Vt) (2.1)

Note that function V (.) maps the vector of risk factors into losses and Vt is a function of timeand risk factors which is observable and known at time t. Moreover, The distribution of Lt isexpressed the loss distribution, denoted by L . Formally, the conditional loss distribution onpast observable history, Ω up to time t is characterized as follows.

FLt+1|Ω(l) = Pr(Lt+1 ≤ l | Ω) for l ∈ R (2.2)

Definition 2.1.1 (Risk measures) let Ψ be a set of random variables on probability space(Ω, E, P ) then risk measure R(.) on latter set is defined as a mapping such that,

Pr(X) = Pr(Y )→ R(X) = R(Y ) for all X,Y ∈ Ψ

If risk measure is negative it means there is no risk of default and capital can be extractedwithout creating positive probability on the loss distribution. Furthermore, a risk measureshould satisfies certain properties in order to fit in appropriate capital requirements. Let X andY be two arbitrary random variables from set Ψ then consider following properties.

1. Dependence on distribution:

Pr(Lt+1 ≤ X | Ω) = Pr(Lt+1 ≤ Y | Ω)⇒ R(X) = R(Y ) for all X,Y ∈ Ψ

7

CHAPTER 2. RISK MEASURES 8

2. Positive homogeneity:

R(cX) = cR(X) for all X and c ≥ 0

3. Translation-invariance:

R(X + c) = R(X) + c for all X and c ≥ 0

4. Monotonicity:R(X) ≤ R(Y ) for all X,Y with X ≤ Y

5. Sub-additivity:

R( n∑i=1

Xi

)≤

n∑i=1

R(Xi) for all non-negative risks Xi for all i and n ∈ N

Note that a proper risk measure satisfies properties 1-4 and coherent risk measure is a properrisk measure that satisfies the sub-additivity property. The interpretation of the propertiesabove, briefly mentioned below in terms of the loss in a portfolio.

• One-on-one dependence on distribution makes sure that each given value of a loss returnsa unique value of risk measure.

• Positive homogeneity insures that if an investor partitions his/her portfolio, each partitioncarries the exact share of risk.

• Additional value to the losses by a fixed amount increases the translation-invariant riskmeasure with the same amount. This property helps institution to control the probabilityof default by injecting capital.

• Monotonic risk measure holds the size effect of portfolio on its risk. To be precise, port-folios contains fewer risky assets carries less risk.

• Diversification is a technique to reduce risk in a portfolio by merging diversified riskyassets. A risk measure that carries diversification effect is called sub-additive risk measure.If a risk measure violates this property, it has concentration effect and it is called super-additive risk measure.

2.1.2 Value-at-Risk

Value-at-Risk(hereinafter VaR) is the risk measure in Basel II/III and Solvency II. It is definedas the probability that the value of a portfolio drops below a certain level in fixed time horizon.

Definition 2.1.2 (Value-at-Risk) Let X be a random variable with loss distribution L andlet x ∈ X then the VaR with coverage δ, denoted by V aRδ(X) is the quantile such that

Pr(x < −V aRδ(X)|Ω

)= δ (2.3)

VaR is the left-tail quantile of the loss distribution at δ level. In other words, worst case scenarioof loss at δ coverage. Formally,

V aRδ(X) = QX(1− δ) for all X and 0 < δ < 1 (2.4)

Where QX(.) denotes the quantile function. In particular, QX(u) = x ∈ R|Pr(X ≤ x) ≥u for 0 < u < 1. In general, VaR is reported in positive values to stay to this pattern, VaRneeds to enter to expression (2.3) with negative sign.

It is evident that to calculate the value of quantile function, distribution of the underlyingvariable has to be known. As a result, estimating VaR is challenging considering the unknownunderlying distribution. At this point Copula can give confident interval in order to observethe coverage of unknown probability distribution.(Embrechts et al., 2005)

CHAPTER 2. RISK MEASURES 9

2.2 Copulas and correlations

Assume two random variables are fully described with their cumulative probability functions.It is explicit that their densities do not provide any information about joint behavior of ran-dom variables. In this point, Copulas are needed to model joint distributions. By using jointprobabilities, dependence structure and correlations can be modeled.

Theorem 2.2.1 (Sklar) Consider F a n-dimensional CDF with marginals F1, ..., Fn wheren ∈ N. Then there exists a copula C with standard uniform marginals such that

H(x1, ..., xn) = C(F1(x1), ..., Fn(xn)

)With following properties:

1. Copula is defined on the unit hypercube.

2. If at least one coordinate of u ∈ Rn is in the unit cube is zero, C(u) = 0.

3. If all the coordinates of u are one except k-th one, C(u) = uk

If the marginal distributions are continuous then Copula is unique and there exists converseof the marginal distributions. As a result, for every (u1, ..., un) in unit hypercube,

C(u) = C(u1, ..., un) = H(F−1

1 (u1), ..., F−1n (un)

)Definition 2.2.1 (Frechet-Hoeffding bounds) Every copula C with domain Rn is boundedwith upper and lower bounds below.

C(u) := Max(u1 + ...+ un − n+ 1, 0) ≤ C(u) ≤ C(u) := Min(u1, ..., un) (2.5)

C(u) and C(u) are called Frechet-Hoeffding upper and lower bound respectively.

Let u be n-dimensional vector where n > 2 then Frechet-Hoeffding lower bound is not a dis-tribution function in contrast to upper bound. If copula C is equal to upper bound it is calledcomonotonic and if it is equal to lower bound it is called countermonotonic. As a result, copulaC in case of n > 2 cannot be countermonotonic.

Proposition 2.2.1 Let F be Copula of a bivariate random vector (X,Y ) with marginal distri-bution FX(x) and FY (y) then the covariance between X and Y will be:

Cov(X,Y ) =

∫∫[−∞,+∞]2

[C(FX(x), FY (y)

)− FX(x)FY (y)

]dxdy (2.6)

Let C be Copula of a bivariate random vector (X,Y ) then instead of joint distribution of Xand Y , corresponding copula can be applied. Note the expression above is concluded fromCov(X,Y ) = E[XY ]− E[X].E[Y ] which implies E[XY ] and first moments of X and Y have tobe finite.

Chapter 3

Correlation and diversification

In this chapter, we will focus on VaR as the major risk measure in Basel II/III and Solvency IIwith an example to investigate diversification or concentration effect. In Solvency II, coverageδ = 0.005 is chosen as standard. The same coverage is selected in the following example.However this simulation can be formulated for every δ ∈ (0, 1).

3.1 VaR with discrete risks

Let X and Y be two random variables with arbitrary values and probabilities. For given randomvariables, negative values represent gains and positive values represent losses. The tail of eachdistribution carries δ mass and so does the tail of the joint distribution. It is explicit that theremaining 1 − δ is carried in the body. Parameter α is applied to distribute 1 − δ − ε mass tomodel joint distribution. The obtained probabilities are given in table (3.1) with ε = 0.001.

x0 80 120 Pr(Y = y) Pr(Y ≤ y)

y-100 α 0.338− α 0.000 0.338 0.338-20 0.656− α α 0.001 0.657 0.995-5 0.000 0.001 0.004 0.005 1.000

Pr(X = x) 0.656 0.339 0.005Pr(X ≤ x) 0.656 0.995 1.000

Table 3.1: Joint distribution of X and Y

The values given in tables (3.1) are valid under 0 ≤ α ≤ 0.338 and range of correlation ofX and Y is between −0.92387 and 0.49619. If the value 0.2199 is assigned to parameter α,the correlation will be zero. Using the given joint probabilities, the total risk of given randomvariables is summarized in table (3.3).

E(.) σ(.)

X 27.72 38.38Y -46.97 39.69

ρ(.)

X,Y −0.92387 + 4.20139α

Table 3.2: Mean, variance and correlation of X and Y

10

CHAPTER 3. CORRELATION AND DIVERSIFICATION 11

s -100 -20 -5 20 60 75 100 115

Pr(S = s) α 0.994− 2α 0.000 0.000 α 0.001 0.001 0.004Pr(S ≤ s) α 0.994− α 0.994− α 0.994− α 0.994 0.995 0.996 1.000

Table 3.3: Distribution of S = X + Y

Given the VaR of X, Y and total risk S from table (3.3), we can observe the concentrationeffect.

V aR0.5%(S) = V aR0.5%(X + Y ) = 75 > 60 = 80 + (−20) = V aR0.5%(X) + V aR0.5%(Y )

which is independent of the parameter α and possible moments and correlations. This exampledoes not show explicit results of risks correlation on VaR and consequently on diversificationeffect. However, if we have a closer look to the joint distributions, we can observe diversificationeffect in higher coverages.

3.2 VaR with continuous risks

Actuaries are mainly interested in finding a loss distribution which can be fitted well into theobserved claims then it can be used to find probabilities of the future claims. First choicesof distributions to fit the data are gamma, log-normal, Pareto, Weibull etc. The main reasonnot to use symmetric distributions is the fact that claims distributions are mostly skewed. Inthe next example, distribution gamma is chosen to investigate the diversification effect of VaR.Copula shuffles of M is chosen to model joint distribution and extreme correlations around theε-neighborhood of VaR coverage. For simplicity sake, the expected value of the variables ischosen to be unity.

The following copula constructed to simulate uncorrelated behavior in the tail which carries1−δ−ε mass and the rest is carried by the copula CL(u, v). The support of constructed copulain (3.1) is illustrated in figure (3.1).

C(u, v) =

CL(u, v) if (u, v) ∈ [0, 1− δ − ε]2u if (u, v) ∈ [0, 1− δ − ε]× [1− δ − ε, 1]v if (u, v) ∈ [1− δ − ε, 1]× [0, 1− δ − ε]Max1− δ − ε, u+ v − 1− (1− δ − ε) if (u, v) ∈ [1− δ − ε, 1]2

(3.1)

Assume joint distribution of X and Y is described by the copula in (3.1) where CL(u, v) ischosen from Archimedean copulas i.e. Gumbel and Frank with parameter λ. Copulas parameterhas to be estimated by fitting the data using multivariate distributions constructed from copulaswith initial copula parameter λ0 = 3. Note that correlation is maximum if among all copulaclasses U and V lay on major diagonal (left support in figure 3.1) and respectively, it will beminimum if U and V lay on minor diagonal (right support in figure 3.1).

Let X and Y be two random variables that follows same Gamma distribution Γ(k, θ) withE(X) = E(Y ) = 1 which results to k = 1

σ2 and θ = σ2 for given σ ∈ 0.1, 0.2, ..., 1. VaR of Xand Y and S = X + Y are illustrated as a function of standard deviation in figure (3.2) withvalues δ = 0.5% and ε = 0.001.

CHAPTER 3. CORRELATION AND DIVERSIFICATION 12

Figure 3.1: The support of constructed copula with maximum correlation condition (left) andminimum correlation condition (right).

The problem of finding the supremum and infimum of VaR (in general, a risk measure) isknown as Frechet problem (McNeil et al., 2005). Define for s ∈ R,

σC,+(Fx, Fy)(s) = Pr(X + Y < s)

τC,+(Fx, Fy)(s) = supxC(FX(x), FY (sup

yx+ y < s)

)In terms of VaR,

V aRδ(X + Y ) ≤ τC,+(Fx, Fy)−1(δ)

Worst case scenario of VaR can occur considering the copula’s lower bound in particular, copulabeing countermonotonic in case of bivariate portfolios. In other words,

τW,+(Fx, Fy)(s) = FX(s

2) + FY (

s

2)− 2 + 1 (3.2)

Note that to calculate correlations, we need to find the value of covariance using proposition2.2.1. In this thesis, numerical procedure is used to calculate the covariance and correspondingcorrelation. Consider following 5 steps,

1. Generate two random variables Gamma-distributed with parameters 1σ2 and σ2 for given

σ ∈ 0.1, 0.2, ..., 1 with sample size 1000.

2. Find u = F (X) and v = F (Y ) where F (.) is the CDF of Gamma distribution.

3. Sort u and v in ascending order.

4. Calculate covariance using Riemann’s left sums with the given copula support.

5. Correlation can be found using the standard deviation of u and v together with covariance.

CHAPTER 3. CORRELATION AND DIVERSIFICATION 13

σ 0.1 0.2 0.3 0.4 0.5V aR(X) + V aR(Y ) 2.5526 3.1796 3.8794 4.6499 5.4887V aR(X + Y ) 2.6074 3.3066 4.0968 4.9766 5.9436ρGumbel 0.8442 0.8412 0.8397 0.8764 0.8405ρFrank 0.4485 0.4469 0.4462 0.4656 0.4465

σ 0.6 0.7 0.8 0.9 1.0V aR(X) + V aR(Y ) 6.3929 7.3592 8.3843 9.4646 10.5966V aR(X + Y ) 6.9953 8.1284 9.3398 10.6258 11.9829ρGumbel 0.8374 0.8666 0.8557 0.8417 0.8476ρFrank 0.4449 0.4604 0.4546 0.4472 0.4503

Table 3.4: Value-at-Risks and correlations

Figure 3.2: Value of V aR(X)+V aR(Y ) (Bottom) and V aR(X+Y ) (Top) of Gamma distributedrandom variables as a function of σ at 0.5% coverage with ε = 0.001.

The discrepancy between aggregated VaR and individual VaRs is getting larger in largerstandard deviation. However, correlations are approximately the same. It is explicit thatdifference in aggregated VaR and sum of VaR is not depend on correlation no matter withwhich copula the risks are coupled.

Chapter 4

Tail dependence and diversification

In this chapter, we consider the continuous example in the previous chapter with positive taildependence in order to investigate whether diversification effect can occur with or withoutpositive tail dependence with fixed marginal distributions.

Following support is used to model tail dependence with γ ∈ (0, δ+ε). The following supportis illustrated in the figure (4.1).

C(u, v) =

CL(u, v) if (u, v) ∈ [0, 1− δ − ε]2u if (u, v) ∈ [0, 1− δ − ε]× [1− δ − ε, 1− γ]v if (u, v) ∈ [1− δ − ε, 1− γ]× [0, 1− δ − ε]Max1− δ − ε, u+ v − (1− γ)− (1− δ − ε) if (u, v) ∈ [1− δ − ε, 1− γ]2

Minu, v elsewhere

(4.1)

Figure 4.1: The support of constructed tail dependent copula with maximum correlation con-dition (left) and minimum correlation condition (right).

In this example, the same generated Gamma distributed random numbers and same nu-merical method are used to find correlations with same values for δ = 0.5% and ε = 0.001,and γ = 0.0005 are chosen. However the approach can be formulated using arbitrary value forγ ∈ (0, δ + ε). Note that CL(u, v) is identical to the copula in [0, 1 − δ − ε]2 in the section(3.2) i.e. Gumbel and Frank from Archimedean copula classes. Moreover, the tail behavior iscountermonotonic if (u, v) ∈ [1− δ − ε, 1− γ]2 and comonotonic elsewhere. In other words, theextreme values of (u, v) is modeled in order to find maximum and minimum correlations in thetail of the joint distribution.

14

CHAPTER 4. TAIL DEPENDENCE AND DIVERSIFICATION 15

Due to the few numbers generated from the tail of both random variables, switching thejoint distribution to the copula constructed in (4.1) does not effect the correlation values. Asa consequence, the size of the sample has to be increased in order to generate more extremeoutcomes to see the effect of tail dependence on the correlations. In other words, using thesample in the section (3.2) with the sample size 1000 with one replication does not generateenough extreme values to alter the correlations by applying tail dependence copula.

Chapter 5

Conclusion

From analysis above, we can conclude that neither correlation pattern in the body of the lossdistribution nor the tail dependence structure has impact on diversification effect. This meansthat the concept of diversification is tightly attached to the joint loss model. In other words,if the purpose of implementing VaR is solely the risk reduction property of a coherent riskmeasure, the underlying aggregated loss distribution has be considered otherwise there is nomathematical proof of sub-additivity property of VaR for all distributions in general. However,further investigation is needed to consider diversification effect under all classes of copulasi.e. Archimedean and Elliptical copulas. Moreover, correlation numerical procedure has to becalculated for different sample sizes e.g. N ∈ 1000, 2000, 5000, 10000 with tremendous numberof replications in order to find accurate outcomes (Monte Carlo approach).

16

References

Janathan Berk and Peter DeMarzo. Corporate Finance. 2014.

Paul Embrechts, Andrea Hoing, and Giovanni Puccetti. Worst var scenarios. 2005.

Werner Hurlimaan. Optimization of the Non-life Insurance Risk Diversification in Solvency II.2009a.

H. Flex Kloman. Rethinking risk management. The Geneva Papers on Risk and Insurance,1992.

Alexander J. McNeil, Rudiger Frey, and Paul Embrechts. Quantitative Risk Management:Concepts, Techniques and Tools. 2005.

Roger B. Nelsen. An introduction to Copulas. 2006.

17