University of Amsterdam

MSc Mathematics

Master Thesis

Dynamic Risk Budgeting inInvestment Strategies: The Impact

of Using Expected ShortfallInstead of Value at Risk

Author:Wout Aarts5870151

Supervisors:Bert KramerPeter Spreij

January 10, 2016

Acknowledgements

Writing this Masters thesis in Mathematics has been a challenging endeavorand one which could not have been finished without the incredible support Ireceived from numerous people throughout the process. First of all, I wouldlike to express my gratitude to Ortec Finance for giving me the opportu-nity to write my thesis for the company. More specifically, I would like togive a special thanks to Bert Kramer from Ortec Finance for providing mewith direction in my research and ensuring that it would be useful for oneof Ortec Finance’s clients. Martin van der Schans’ guidance in teachingme programming in Python and his in-depth explanations of how risk man-agement theory and mathematics intersect, has been invaluable throughoutthis research. From the UvA I would like to extend my gratitude to PeterSpreij for the assistance in understanding complex mathematical theory, achallenging yet crucial part for graduating from a Mathematics master’s.Further, I wish to acknowledge the support provided by Aqsa Hussain, par-ticularly in the final stages of my research, and for reading my thesis as asecond viewer before submission. Finally, I am very grateful to all the col-leagues at Ortec Finance; for all the help and patience, explanations aboutbasic theory about risk management and the informal chats.

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Dynamic Risk Budgeting in Investment Strategies: The Impactof Using Expected Shortfall Instead of Value at Risk

Wout Aarts

Abstract

In this thesis we formalize an investment strategy that uses dynamic riskbudgeting for insurance companies. The dynamic component of the invest-ment strategy includes the property that the amount of risk taken by aninsurance company depends on its financial position. This investment strat-egy is applied to examine the effect of using Expected Shortfall (ES) insteadof Value at Risk (VaR) for the risk budgeting component in the investmentstrategy. This effect is measured by observing return-risk characteristics of1000 economic scenarios over a five year time horizon. Further, a detailedproof of the theorem that ES is a coherent risk measure is provided. Wecompare ES-optimal portfolios with VaR-optimal portfolios by looking atthe return-risk ratio, which is the realized return per unit of risk. The firstobservation is that the available risk budget is smaller when ES is usedinstead of VaR to determine shocks. This results in more weight being allo-cated to asset classes which are more risk averse, affecting both the realizedreturn and the risk that the insurer takes. We find that, when using theformalized risk budgeting investment strategy, the return-risk characteris-tics of ES-optimal portfolios are preferable as ES also takes tail risk intoaccount.

Keywords: Expected Shortfall, Value at Risk, Coherent Risk Measure,Solvency II, Investment Strategy, Risk Budgeting

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Contents

1 Introduction 4

2 Background 62.1 Asset Liability Management . . . . . . . . . . . . . . . . . . . 62.2 Solvency II Assumptions . . . . . . . . . . . . . . . . . . . . . 7

3 Properties of the Gaussian Distribution 93.1 Linear Correlation . . . . . . . . . . . . . . . . . . . . . . . . 93.2 Value at Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.3 Aggregation of VaR . . . . . . . . . . . . . . . . . . . . . . . 103.4 Quantile functions . . . . . . . . . . . . . . . . . . . . . . . . 113.5 Log-normal Return Rate . . . . . . . . . . . . . . . . . . . . . 13

4 Dynamic Risk Budgeting Investment Strategy 154.1 General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.2 Stylized Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.3 Scenario Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 21

5 Risk Measures 245.1 Coherent Risk Measures . . . . . . . . . . . . . . . . . . . . . 245.2 Value at Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.3 Expected Shortfall . . . . . . . . . . . . . . . . . . . . . . . . 275.4 Coherence of Expected Shortfall . . . . . . . . . . . . . . . . 295.5 Stylized Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

6 Conclusion 396.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396.2 Concluding Statements . . . . . . . . . . . . . . . . . . . . . . 396.3 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

7 Appendix 427.1 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . 427.2 Python Scripts . . . . . . . . . . . . . . . . . . . . . . . . . . 487.3 Miscellaneous Theorems and Proofs . . . . . . . . . . . . . . 55

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1 Introduction

In the current financial climate, insurance companies are finding it increas-ingly challenging to be financially sound. This is due to low interest rates,declining business and the introduction of the Solvency II Directive - a newprudential regulatory framework which will come into force on 1 January2016 [13]. One of the objectives of Solvency II is to prevent insurance com-panies from becoming insolvent, which happens when the available capitalis insufficient to cover the risks of the liabilities. The liabilities are futurepayments for the insured.

In this new regime, insurers will have to establish technical provisionsto cover expected future claims from policyholders. In addition to this,insurers must have sufficient available resources to cover a Solvency CapitalRequirement (SCR) which is the economic capital needed as a risk budgetfor the investments and liabilities. The SCR is established using the riskmeasure Value at Risk [6].

Value at Risk with confidence level α (VaRα) is defined as follows: fora given risk class (asset or liability), time horizon, and confidence level α,the VaRα is a threshold loss value, such that the probability that the losson the risk class over the given time horizon does not exceed this value is α.The SCR in Solvency II is based on a Value-at-Risk measure calibrated toa 99.5% confidence level over a 1-year time horizon. The SCR supposedlycovers all risks that an insurer faces (e.g. insurance, market, credit andoperational risk) and will take full account of any risk mitigation techniquesapplied by the insurer (e.g. reinsurance and securitisation) [6].

When VaRα is used as risk measure for a certain asset class, an assump-tion can be made about the shock in return rate. In the context of thisresearch, shock is referred to as the maximum assumed loss. In SolvencyII, the shock is used as the risk charge - defined as the amount of moneyinvested in a certain asset class multiplied by the risk charge gives the SCR.

Several of the assumptions the quantitative requirements in SolvencyII are based on are often contested. These assumptions have a significantimpact on the actual aggregated risk [8]. Therefore making it important forinsurance companies to review their current policies and look for alternativeways for estimating risk. For instance, the use of risk measure VaRα iscriticized for not being a coherent risk measure. One implication of workingwith a non-coherent risk measure is that the benefits of diversification onthe overall risk of the portfolio are not reflected properly. This is becauseVaRα does not hold information outside of the confidence level, while somerisk classes might have heavy-tailed distributions.

Further, the correlation in the tail (outside of the confidence interval) ismuch higher than within the confidence level. This means that in a worstcase scenario, there is a higher probability of losses of different asset classesexceeding the confidence level altogether.

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An alternative risk measure is Expected Shortfall with confidence level α(ESα), which is the expected loss of a risk class in the (1−α)-fraction worstcases. Thus, the risk measure ESα represents the expected loss conditionalon the loss exceeding VaRα. Since ESα is in fact a coherent risk measure,many risk managers prefer ESα over VaRα [10]. In this thesis, a detailedproof of the coherence of ESα is outlined. This proof is based on somefundamental results from probability theory.

Several insurance companies prefer a dynamic investment strategy wherethe weight allocation per asset class changes over time. This could meanthe amount of risk an insurer takes depends on its financial position. In thisthesis, the insurers financial position is represented by a value similar to theSolvency ratio, which will be defined later on. An alternative method ofinfluencing the dynamic strategy is by determining the risk appetite. Therisk appetite will be described as an insurers expected solvency ratio aftera maximum shock.

The aim of this research is twofold. Firstly, an optimal dynamic in-vestment strategy using risk budgets is formalized. Therefore, we have tocheck which part of the strategy should be dynamic, what external factorsinfluence the strategy and how the investor’s risk appetite can be taken intoaccount. This will be summarized into an optimization problem.

Secondly, the impact of changing the risk measure from VaRα to ESαwill be investigated. In this comparison, the focus will be on changing therisk measure on the assets side of the insurers balance sheet. We will usethe investment strategy defined in the first part to determine the impact ofusing ESα instead of VaRα, by assessing the return-risk characteristics ofboth. Subsequently, differences between the VaRα-optimal portfolios andESα-optimal portfolios will be highlighted by analyzing the tail risks of therealized return and the corresponding measured risk based on ESα.

Although similar research comparing VaRα with ESα has been con-ducted, this research also involves the additional element of a dynamic riskbudgeting strategy. Dynamic risk budgeting strategies are becoming in-creasingly popular for insurers to implement into their investment policies.Thus, in this thesis, we will evaluate the potential for a Dutch insurancecompany to implement this into their policy.

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2 Background

This section provides a context for the new Solvency II regulation as well asa general background on ALM. We discuss the Economic Scenario Generatorand its role in measuring risk, and subsequently evaluate several mathemat-ical assumptions Solvency II is based on.

2.1 Asset Liability Management

Solvency II is a new supervisory framework that will come into force starting2016 for all insurance companies in Europe. This regulation was initiatedby the European Insurance and Occupational Pension Authority (EIOPA)and has the primary objective to reduce the risks for insurers to becomeinsolvent [3]. Insolvency is a state of being unable to cover liabilities whichare expected future cash flows, given the insurer’s current economic capital.The economic capital that the insurer uses to pay their customers is investedin asset classes like stock, real-estate, sovereign bonds and credits. Theliabilities, as well as the economic capital, are prospective, thus estimatedvalues. For this reason an accurate estimation of what these values will beand how they will evolve in the future is necessary [3].

To obtain insight into the future development of assets and liabilities,often an Economic Scenario Generator (ESG) is used. An ESG uses MonteCarlo simulation and several stochastic models to generate future scenarios[16]. Most insurance companies do not possess an ESG, therefore outsourcetheir Asset Liability Management (ALM) studies to companies like OrtecFinance, who are specialized in this kind of research.

A typical ALM study performed by Ortec Finance is as follows: aninsurance company delivers data concerning their liabilities, expectations -with respect to future new business -, and their current investment strategy.Then, Ortec Finance will model this in a way, such that the ALM model isable to provide results pertaining to the solvency and portfolio value overthe next 5 to 60 years (5 year is more relevant for insurers, whereas 60years is more relevant for pension funds). The ALM model is able to showimportant information concerning a great amount of possible scenarios (ingeneral, for insurers the number of scenarios is 1000 or 2000).

Using such a great number of scenarios results in relevant statisticalvalues, including the 0.5th percentile which the Solvency II Directive usesfor risk measurement as well. The 0.5th percentile of the loss distribution isa risk measure which is used for solvency II and will hereafter be referredto as VaR. The risk measure VaR is probably the most widely used riskmeasure in risk management [10].

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2.2 Solvency II Assumptions

The Solvency II Directive is a new supervisory framework initiated by EIOPA,which puts demands on the required economic capital, risk management andreporting standards of insurance companies [3]. The Solvency II Directiveconsists of three pillars. The first pillar describes the quantitative require-ments every insurer should meet, covering both assets as liabilities. Thesecond pillar describes the standards of risk management and governanceand the third pillar concerns the transparency for supervisors and publicreporting standards [5]. In this thesis, the term Solvency II will only referto the first pillar.

In general, an insurance company has to deal with a great amount ofuncertainties. When assessing liabilities, a proper estimation is important inorder to gain insight into how much technical provision is needed for coveringpayments to its clients. On the asset side, interest rates, credit spreads,equity movements, real estate prices and exchange rates are considered greatrisk factors. This thesis will focus on the risk factors on the asset side.

As mentioned in the introduction, the buffer that an insurer needs to holdaccording to Solvency II is called the SCR, and is based on the VaR withconfidence level 99, 5%, or VaR0,995. The SCR is established by assumingthe VaR0,995-value to be the maximum shock of an asset class, and usingthis as the risk charge.

For example, suppose the maximum shock of the asset class ’stock’ inSolvency II is calculated to be 40%. In the worst case, this assumes thatthis asset class can drop 40% in value within one year. If an insurer hasallocated 1.5 million euro in stock, we have: SCRstock = 1.5 · 0.40 = 0.60million euro. In this case, the risk charge on investments in stock is 40%.

For all asset classes, EIOPA has established the correlation matrix Pfor the corresponding return rates. This correlation is used to determineSCRmarket which is the required capital to cover the market risk. In SolvencyII, the aggregation of risk factors is calculated as if the risk factors arenormally distributed. This means that if for asset class i, the VaR0.995-value si and an insurer invests xi euro in this asset class, we have

SCRmarket =

√∑j

∑i

ρijsixisjxj =

√∑j

∑i

ρijSCRiSCRj ,

where ρij is the correlation between asset class i and j which represent assetclasses like stock, real-estate, credit spread and sovereign bonds.

When SCRmarket is determined, it can be used to calculate SCRtotal

which is the total capital charge to cover the insurer’s risk including liabili-ties. SCRtotal is calculated by the same aggregation formula as SCRmarket.For calculating SCRtotal, instead of summing over the asset classes, the sumis over all risk classes (i.e. market, life, non-life, operational, default). For

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the risk aggregation, EIOPA has established correlation matrix Q = (qkl)k,l.This results in

SCRtotal =

√∑l

∑k

qklSCRkSCRl,

where k, l represent the risk classes.In the Solvency II Directive the term SCRintangibles is also included, pre-

senting the capital requirement for intangible asset risk. This aspect willnot be analyzed in this thesis. The determination of the SCR of all riskfactors can be found in the supplement to the Solvency II Directive of 2015[14].

As aforementioned, the risk aggregation formula is based on multivariatenormal distributions. In reality, the random variables an insurance companycopes with are heavy-tailed and have a much higher dependency in the tail,as opposed to the multivariate normal distribution. If one asset class has ahigh tail dependency with another asset class, and it drops in value morethan the assumed maximum shock, then the correlation in this last 0.5th

percentile is increased enormously. As a result, in a scenario where assetvalues drop outside of the 99, 5% confidence level, the highly correlated tailevents have a much higher probability of occurring at the same time, thanis assumed in Solvency II.

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3 Properties of the Gaussian Distribution

Since Solvency II uses a formula based on a multivariate Gaussian distri-bution for aggregating risk and taking dependency relations into account,in this chapter we summarize the relevant properties of the Gaussian dis-tribution. Firstly, dependency relations and linear correlation are defined.Secondly, upon defining risk measure VaR, we see how risk based on VaRcan be aggregated when a Gaussian distribution is assumed and how thisleads to the aggregation formula used to calculate SCRtotal. Thirdly, we seehow VaR is a quantile when a Gaussian distribution is assumed. Lastly,a return formula based on log-normal return rates is outlined, which willbe revisited in chapter 4 where the optimal risk budgeting strategy will bedefined. In this section, the primary focus is on asset risk.

3.1 Linear Correlation

Using the linear correlation coefficient is a very rudimentary, but also simpleway of describing risk dependencies in a single number. Let the covarianceof random variable X1 and X2 be given by:

Cov(X1, X2) = E[(X1 − E[X1])(X2 − E[X2])

].

For the variance, we write:

V(X1) = E[(X1 − E[X1])2

]= Cov(X1, X1).

The linear correlation coefficient ρ is defined as follows [11]:

ρ(X1, X2) =Cov(X1, X2)√V(X1)× V(X2)

,

where X1, X2 are non-degenerate random variables.Let X and Y be multivariate random variables with finite variance. In

case of multiple dimensions the so-called correlation matrix needs to beapplied. This is a symmetric and positive semi-definite matrix given by

ρ(X,Y ) =

ρ(X1, Y1) · · · ρ(X1, Yn)...

. . ....

ρ(Xn, Y1) · · · ρ(Xn, Yn)

.

This linear correlation coefficient is also known as Pearson’s linear cor-relation. It measures only linear stochastic dependency of two random vari-ables and it holds that −1 ≤ ρ(X,Y ) ≤ 1.

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3.2 Value at Risk

Consider the probability space (Ω,F ,P) on which the following (random)variables are defined:

Let At be the total asset value at time t for a portfolio of n asset classes.Define At =

∑ni=1wt,iAt =

∑ni=1Xt,i, so Xt,i = wt,iAt where Xt,i is the

amount of money invested in asset class i at time t and for the weights wt,iwe have

∑ni=1wt,i = 1. Let Xt = (Xt,1, ..., Xt,n)> represent the portfolio

at time t. In some cases it is convenient to look at the loss distributionL = −X. This depends on the context and will be stated when necessary.Lastly, for asset class i we have return rt,i with mean E[rt,i] = µi.

In this thesis, the investment strategy is defined by the way wt = (wt,1, ..., wt,n)is rebalanced in every step t → t + δt. The rebalancing happens annually,thus the step t→ t+ 1 represents a one year interval.

The measure VaR is based on the maximum possible loss of an invest-ment, which is derived from statistics of historical values. The maximumpossible loss can be defined as infl ∈ R : FL(l) = 1 [10].

However, in most models, the support of FL is unbounded which leadsto the maximum loss being infinity. That is why a quantile of the lossdistribution is taken, and we get the VaR. We extend “maximum loss” to“maximum loss which is not exceeded with a given high probability”, theso-called confidence level, and get the following definition [10]:

Definition 3.1. Let α ∈ (0, 1). The VaR of a portfolio at the confidencelevel α is given by the smallest number l such that the probability that theloss L exceeds l is no larger than (1− α). Formally,

VaRα(L) = infl ∈ R : P(L > l) ≤ 1− α = infl ∈ R : FL(l) ≥ α.

In Solvency II, the confidence interval of α = 0.995 is used. Sometimes,VaR is defined on the basis of a return X instead of loss. In that case wehave VaRα(X) = infx ∈ R : P(X < x) ≥ 1− α which holds an equivalentdefinition. In this thesis, both definitions will be used depending on thecontext. In some cases it is more convenient to have the worst case in theleft tail and in some cases in the right tail. In actuarial science it is commonto use the VaR definition based on expected loss L. Further, throughoutthis thesis, when no α is given, one can assume α = 0.995.

The drawback of VaR0.995 is that it does not contain information aboutthe values outside the confidence interval. For this reason, VaR is not acoherent risk measure. This will be discussed in more detail in the chapter5.

3.3 Aggregation of VaR

The matrix of linear correlations as described in section 3.1, can be used tocharacterize the level of interdependence of marginal losses. For an investor

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it is important, that when the non-diagonal elements are low, the level ofdiversification that can be realized is high, with incremental exposure to arisk component [8].

Let Y1, ..., Yn random variables with standard deviation σi for i = 1, ..., nand correlation matrix P. When Y =

∑ni=1 Yi we have

σ2Y = (σ1, ...σn)P(σ1, ...σn)>

where σ2Y denotes the variance of Y [8]. Now assume a portfolio X is a

multivariate normally distributed random variable. For A =∑

iXi we have

VaR(A) =

√√√√ n∑i=1

n∑j=1

ρi,jVaR(Xi)VaR(Xj),

Note that the random variable Xi in this definition has already taken theweight in asset class i into account. Thus, VaR(Xi) is the amount of moneywhich an investor can possibly loose after a maximum assumed shock si (onthe given confidence interval). Since the shock si defines the risk charge, wehave VaR(Xi) = asiwi = SCRi where a is the initial total asset value of theportfolio. Thus, when we write w s for the component-wise multiplicationof the vectors w and s in matrix form we have

VaR(A) = SCRmarket = a√

(w s)>P(w s),

where P is the correlation matrix of the asset classes.

3.4 Quantile functions

For the next propositions, theory regarding quantile functions will be high-lighted. The cumulative distribution function of the standard normal dis-tribution is given by

Φ(x) =1√2π

∫ x

−∞e−

12t2dt.

In words, this describes the probability that a standard normally dis-tributed random variable X will be found to have a value less than or equalto x. Notation: when we have µ 6= 0 or σ 6= 1 we denote the cdf by Φ(x, µ, σ).

Definition 3.2. (generalized inverse and quantile function)

1. Given some increasing function T : R→ R, the generalized inverse ofT is defined by T←(y) := infx ∈ R : T (x) ≥ y, where the infimumof an empty set is defined to be infinity.

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2. Given some distribution function F , the generalized inverse F← iscalled the quantile function of F . For α ∈ (0, 1) the α-quantile of F isgiven by

qα(F ) := F←(α) = infx ∈ R : F (x) ≥ α

If F is increasing we have qα(F ) = F−1(α), which holds for F = Φ.

Proposition 1. Assume the loss distribution FL is normal with mean µand variance σ2. For α ∈ (0, 1) we can write

VaRα(L) = µ+ σΦ−1(α),

where Φ−1(α) is the α-quantile of Φ.

Proof. By definition of generalized inverse and right continuity of FL weknow that a point x0 ∈ R is the α-quantile of FL if and only if the followingtwo conditions are satisfied: FL(x0) ≥ α;FL(x) < α for all x < x0 [10].Since FL is strictly increasing, we have to show that FL(VaRα) = α. Wehave

FL(VaRα) = P(L ≤ VaRα) = P(L− µ

σ≤ Φ−1(α)

)= Φ(Φ−1(α)) = α.

This implies that the quantile function qa(FL) has the property the ran-dom variable L will exceed µ+ σΦ−1(α) with probability 1−α, and will lieoutside of the interval µ± σΦ−1(α) with probability 2(1− α).

Proposition 2. Suppose we have random variables Y1, ..., Yn with Yi ∼N (0, σ2

i ), covariance matrix Σ, weight vector w = (w1, ..., wn)> and defineY =

∑iwiYi. We have

VaRα(Y ) = Φ−1(α; 0;√w>Σw)

Proof. By proposition 1 we have VaR(Yi) = Φ−1(α)σi.

VaR(Y ) =

√√√√ n∑i=1

n∑j=1

ρi,jwiwjVaR(Yi)VaR(Yj),

=

√√√√ n∑i=1

n∑j=1

ρi,jwiwjσiσjΦ−1(α)Φ−1(α),

= Φ−1(α)

√√√√ n∑i=1

n∑j=1

ρi,jwiwjσiσj ,

= Φ−1(α)√w>Σw,

= Φ−1(α; 0,√w>Σw).

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This proves in the case of µ = 0, VaR(Y ) is a is a percentile of the mixN (0,

√w>Σw).

This shows that when µ = 0, the VaR of a multivariate normally dis-tributed random variable is a quantile of the normal distribution with themean equal to zero and standard deviation equal to

√w>Σw.

3.5 Log-normal Return Rate

In this subsection, we define a formula for the geometric return of a portfolio,based on “Arithmetic and Geometric Mean Rates of Return in DiscreteTime” by Arie ten Cate, 2009 [4].

Let At be the total value of a portfolio at time t. Let rt be the returnon the portfolio using the current investment mix w (which in this subsec-tion will not variate over time). Assume the sequence of return rates rt isindependently distributed. Given that w is constant, by definition we have

At = (1 + rt)At−1.

We assume P(rt < −1) = 0 and E[rt] = µ for all t. At time t = T , byindependence of rs and As−1 for all s we have:

E[AT ] = E[(1 + rT )AT−1] = E[(1 + rT )]E[AT−1] = (1 + µ)E[AT−1].

Repeating this down to t = 0 we get

E[AT ] = (1 + µ)TA0,

and (E[ATA0

])1/T− 1 = µ,

The arithmetic mean is

µar =1

T

T∑t=1

rt =1

T

T∑t=1

( AtAt−1

− 1),

which impliesE[µar] = µ.

Let 1 + µg be the geometric mean of the 1 + ri, we get

1 + µg =( T∏t=1

1 + rt

)1/T=(ATA0

)1/T.

One can show µg ≤ µar. Now assume that all rt are IID and assume that1 + rt is log-normal, we get

log(1 + rt) ∼ N (µ′, σ2).

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It follows that P(rt ≤ −1) = 0, since the logarithm is undefined at zero. Weget

log(AT /A0) = log( T∏t=1

1+rt

)=

T∑t=1

log(1+rt) ∼T∑t=1

N (µ′, σ2) ∼ N (Tµ′, Tσ2),

which leads to

log(1+µg) = log(AT /A0)1/T = (1/T ) log(AT /A0) ∼ (1/T )N (Tµ′, Tσ2) ∼ N (µ′,1

Tσ2)

It follows from the formulas for the expectation and variance of a lognormal variable that

E[1 + rt] = eµ′+ 1

2σ2,

E[1 + µg] = eµ′+ 1

2Tσ2,

V(1 + rt) = (eσ2 − 1)e2µ′+σ2

.

We have

E[1 + µg] = eµ′+ 1

2Tσ2,

= eµ′− 1

2σ2+ 1

2σ2+ 1

2Tσ2,

= (1 + µ)e−12σ2+ 1

2Tσ2.

Thus, for the geometric return, we define

E[µg] = (1 + µ)e−12σ2+ 1

2Tσ2 − 1.

In the next section this formula will be used. Expected return µ will bethe arithmetic mean of total asset value µ =

∑iwiµi with corresponding

marginal weights wi and marginal expected returns µi. The standard de-viation will be given by σ2

w = (w σ)>P(w σ), where σ is the vector ofmarginal standard deviations and w σ component-wise multiplication ofw and σ .

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4 Dynamic Risk Budgeting Investment Strategy

In this section we check how to allocate weights in a portfolio given a cer-tain risk budget, when the objective is to maximize the total return of theportfolio. This allocation depends on the insurers financial state and riskappetite. One of the properties in this investment strategy is when the in-surer has a higher Solvency ratio, they should be able to take more risk. Wewill use the SCR formulas as defined in the Solvency II Directive.

Firstly, the general case will be introduced. Secondly, a four dimensionalcase with realistic numbers will be formalized, which will be used to analyzethe dynamic risk budgeting strategy. Throughout this thesis, this four di-mensional case will be used to examine results about the effect of differentrisk measures using the ESG of Ortec Finance.

4.1 General Case

Assume we have n assets classes. The objective is to find optimal allocationof w when the following parameters are known:

• Initial total assets a

• Initial equity e

• Expected returns µ = (µ1, ..., µn)> of the asset classes 1, ..., n

• Standard deviations σ = (σ1, ..., σn)> of the asset classes 1, ..., n

• Shocks s = (s1, ..., sn)> as a percentage of the initial allocation (inSolvency II the shock of asset class i is determined by VaR(ri) andthis also defines the risk charge for asset class i)

• Correlation matrix P, which is an n × n matrix n asset class returnrates

• Risk appetite parameter γ (which will be defined later on)

We want to find the allocation w that maximizes the expected returnR of the investments. To formalize an optimization problem, we require aformula that represents the expected return. Using the arithmetic returnwµ, which is based solely on the marginal expected returns will directlyresult in all weight being in the asset class with the highest expected returnwhen formalizing this into an optimization problem.

Therefore, the return formula needs to have the property that a highervolatility results in a lower expected return. In section 3, we have seenE[µg] has the volatility in a negative component, hence this representationof expected return suffices this property. Thus, we write

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R(w, µ, σ) = (1 + (w · µ))e−12σ2w+ 1

2Tσ2w − 1,

where σ2w = (w σ)P(w σ)> and the inner product µ · w represents the

arithmetic return rate of the portfolio.By Solvency II regulation, the assumed maximum shock based on weights

w, risk charges s and correlation matrix P can be calculated by

sw =√

(w s)>P(w s).

The objective is to define a dynamic investment strategy. This meansthat the amount of risk the investor will take, depends on the solvency ratioand the risk appetite. Therefore, a risk appetite parameter γ is introduced.The parameter γ represents the solvency ratio after enduring a maximumshock. This means the solvency ratio of the insurer drops to γ when a shocksw is realized.

The risk budget RB induced by Solvency II is given by

RBsolvency = SCRtotal.

The insurers available risk budget RB is defined as

RBavailable := e− γSCRtotal.

By solvency II regulation we have

SCRtotal = asw.

In our optimal investment strategy we want the risk budget to be usedto its maximum. Thus, we set RBsolvency = RBavailable. Combining theseequations results in

sw =e

(1 + γ)a.

which will lead to one of the constraints in the optimization problem. Definesmax = maxsi : i ∈ 1, ..., n. Additional to last constraint we need to boundsw with smax, thus we get

sw = minsmax,

e

(100% + γ)

.

Without this addition, the optimization problem might not have any feasiblesolution, because of the following proposition:

Proposition 3. Let smax = maxsi : i ∈ 1, ..., n and smin = minsi :i ∈ 1, ..., n. If shock sw is calculated by the Solvency II risk aggregationformula, it must hold that smin ≤ sw ≤ smax.

16

Proof.

sw =

√√√√ n∑j=1

n∑i=1

wisiρi,jwjsj

≤√∑

j

∑i

smaxwiρijsmaxwj

= smax

√∑j

wj∑i

wiρij

≤ smax√∑

j

wj∑i

wi

= smax

The proof for smin ≤ sw is analogous to sw ≤ smax, replacing ≤ with ≥ andmax with min.

This shows that for all possible weight allocations w, the shock willalways be bounded by smin and smax.

In the case that e(1+γ)a ≥ smax (resp. ≤ smin), the optimization can

return the weight vector where all weight is allocated to the asset classthat has highest (resp. lowest) shock (and this will be the case when theasset class where the highest shock is assumed also has the highest expectedreturn). Additionally, we have the usual constraints

∑iwi = 1 and wi ≥ 0

for all i. The last constraint is the actual Solvency Capital Requirement,thus

asw ≤ SCRtotal.

For deriving the constraint sw = e(1+γ)a we used that asw = SCRtotal. Since

the factor SCRtotal cancels in this derivation, this constraint is used such thatafter determining the optimal weights, the resulting capital charge does notexceed SCRtotal.

Combining all above constraints and estimation of return, we arrive atthe following optimization problem:

17

Optimization Problem. Given total assets a, equity e, expected returnvector µ, standard deviation vector σ, shock vector s, risk appetite parameterγ and correlation matrix P, the following optimization problem gives anoptimal allocation w based on the log-normally distributed geometric returnrates.

maximizew

R(w, µ, σ),

subject to∑i

wi = 1,

wi ≥ 0,

smin ≤ sw = minsmax,

e

(1 + γ)

,

asw ≤ SCRtotal.

Here we have formulated an optimization problem in terms of the givenvalues a, e, µ, σ, s,P, γ. This optimization will be used to analyze the returnrisk characteristics of a Dutch insurance company.

4.2 Stylized Case

In this subsection, we will analyze a stylized case which involves four assetclasses. This subsection will provide insight on what the effect of the initialvalues a, e,SCRtotal is on the allocation w when the optimization in section4.1 is used. Therefore, the data of a Dutch insurance company will be usedto illustrate the effect of the dynamic risk budgeting strategy.

Since the allocation w is optimized once per time interval, the timeelement t can be disregarded in this subsection. In the second part we willanalyze this optimization using 1000 scenarios over a five year time frame.These scenarios are generated by Ortec Finance’s scenario generator.

Suppose we have four asset classes: stock, real-estate, spread and gov-ernment bonds. We denote the shock by s which is based on VaR0.995, theexpected arithmetic return by µ and the standard deviation by σ. We as-sume the following values which are based on data from Ortec Finance’sALM model:

18

s =

s0

s1

s2

s3

=

0.510.230.12

0

,

µ =

µ0

µ1

µ2

µ3

=

0.0760.0520.0360.022

,

σ =

σ0

σ1

σ2

σ3

=

0.220.120.0950.011

.

In these vectors, stock is indexed by 0, real-estate by 1, spread by 2 andgovernment bonds by 3. Note that the risk charge of government bonds iszero. By letting s3 = 0 ensures there is no risk charge on the governmentbonds.

Further, let total assets a = 40000, equity e = 6000, SCRtotal = 2900(for these values, we have: one unit is one million euro) and γ = 1.3. Theseare realistic values representing the current state of an insurers portfoliothat will lead to the allocation w using the the optimization in section 4.1.Computing this in python gives the following weight vector:

w = (7.9%, 17.8%, 11.1%, 63.2%)>

The script that is used to get these numbers can be found in appendix,section 7.2.

The allocation is determined by two factors, namely the ratio β =(e

(1+γ)a

)2, which equals 0.071 with the current values of a, e, γ and the

other factor is the constraint asw ≤ SCRtotal.

We have β =(

e(1+γ)a

)2=(ea

11+γ

)2. In this ratio, e

a determines the

insurers financial position. With the initial values we used we have ea =

600040000 = 0.15. The factor 1

1+γ determines the risk appetite. With current

values this factor is 12.3 .

The following graphs show the cumulative weights of the four assetclasses. On the left we see the cumulative weight graph that variates inea and the other factor is regarded as the constant 1

1+γ = 12.3 . On the right

we see the cumulative weight graph that variates in risk appetite and thefirst factor is the constant e

a = 0.15:

19

In the left graph, we can see that when ea = 0 all weight will allocated

to government bonds, the risk free asset class (this is not a realistic number,since this would mean the equity equals zero). When e

a increases, moreweight will be on the risky asset classes.

In the right graph, we see that when γ increases, more weight is onthe risk free asset class. When γ increases, the insurer wants to have ahigher solvency ratio after a maximum shock, thus the investment strategybecomes more risk averse. In the appendix, a sensitivity analysis on theinitial parameters can be found.

In the optimization we observe a big allocation in real estate (green). Inpractice, it is common for insurance companies to cap the allocation to acertain percentage due to real estate being a relatively illiquid asset class.An illiquid asset class means shares in this asset class cannot easily be soldor exchanged for money without a substantial loss in value. A realistic capon the weight allocation is 15%. In the next graph we illustrate what theinfluence on the risk budgeting strategy is, when real estate is capped at15%. Again, on the left we see the cumulative weights when e

a variable, andon the right we see the cumulative weights when the risk appetite is variable.

We observe that the w2 increases significantly, especially in the caseswhere more risk budget is available: when e

a high and when γ is low. Thiseffect is due to the optimization using all available risk budget. Since theallocation on real estate is capped at 15%, the allocation to credit spreadwill increase. This increase will be more than only the difference betweenthe initial allocation without the cap, and the 15%, since otherwise the totalrisk budget is not fully used.

20

This is because credit spread is a less riskier asset class than real estate.In this respect, credit spread not only compensates for the weight whichis not allocated to real estate. Additionally, it compensates for the extraavailable risk budget, in redistributing the weight which was also initiallyallocated to sovereign bonds.

4.3 Scenario Analysis

In this subsection, an algorithm using the optimization formalized in section4.1 is provided, modelling an insurers portfolio. For the scenario analysis weuse the following data for 1000 scenarios over a five year time frame: Initialtotal assets a0,ultimo, initial equity e0,ultimo and one is able to calculate initialSCR0,total using the initial values. Note that in this algorithm, for totalassets at, SCRt,total and equity et we distinguish primo year (start of year)and ultimo year (end of year).

In year t, for every scenario we have return rates rt,i for all asset classesi, cash flow rates Ct which is the net income and costs, SCRt,i for i ∈default, life, health, non-life and the primo- and ultimo technical provisionVt,primo, Vt,ultimo which is the expected capital that is required for coveringall liabilities. Note that SCRt,market depends on wt−1. The data for thesescenarios are obtained from the Ortec Finance ALM model. In the scenarioanalysis, we use the Solvency II correlation matrix Q to calculate SCRt,total.This means, if we let

bt =(SCRt,market,SCRt,default, SCRt,life, SCRt,health, SCRt,non−life

)>,

and we have:

Q =

1 0.25 0.25 0.25 0.25

0.25 1 0.25 0.25 0.500.25 0.25 1 0.25 00.25 0.25 0.25 1 00.25 0.50 0 0 1

,

this leads to

SCRt,total =√b>t Qbt,

which agrees with the Solvency II Directive [14].Assume we know at−1,ultimo, et−1,ultimo and SCRt−1,total. For time t =

1, 2, 3, 4, 5, we do the following steps in every scenario:

1. at,primo = at−1,ultimo + Ct,et,primo = et−1,ultimo + Ct.

2. Determine SCRt,total based on SCRt,i, Q, at,primo, ae,primo, whereSCRt,market is based on wt−1, using the the assset classes correlationmatrix P.

21

3. Use the optimization in section 4.1 to find optimal wt, using et,primo,at,primo and SCRt,total.

4. Let a′t,ultimo = at,primo∑

iwt,i(1 + rt,i).

5. Investment return at time t is given by: IRt := a′t,ultimo − at,primo.

6. et,ultimo = et,primo + 0.75(IRt + Vt,primo − Vt,ultimo)− 0.25Ct.

7. at,ultimo = et,ultimo + Vt,ultimo.

This algorithm globally agrees with the ALM model employed by OrtecFinance. Essentially this means: firstly the values ’total assets’ and ’equity’evolve from ultimo year to primo next year by adding the cash flow, secondlythe weights are rebalanced and thirdly these values evolve from primo yearto ultimo year by adding the investment return.

In step 6, the factor of 0.75 is incorporated to assess the tax which ischarged over the net profit. The term 0.25Ct is subtracted, because other-wise the tax over C would be double since it is already taken into accountin step 1.

We are able to check financial stability by looking at the return-risk-ratioin year t. Denote Rt for the return in year t and define

θt =Rtswt

.

Note: The ratio θt is the ratio of two percentages, namely the realized returnand the assumed maximum shock which is dependent on w. The return inyear t is calculated by

at,ultimoat,primo

− 1 and the maximum shock swt is calculated

by swt =√

(wt s)>P(wt s). For each year t we look at the median ofθt of the 1000 scenarios. It is more useful to analyze the median insteadof the mean as there are scenarios which have wt = (0, 0, 0, 1) for some t.This means

√(wt s)>P(wt s) = 0 and we would have θt = ∞, which

increases the mean of θt over 1000 scenarios enormously and therefore doesnot provide representative information.

Median of θt:t 1 2 3 4 5

θt 0.59 1.11 1.05 0.90 0.82

In this table we see that in year 2 and 3 the insurer has a return-risk ratiobigger than one, which means the realized return in these years is biggerthan the assumed maximum shock sw. The shock sw agrees with the riskcharge measured by Solvency II. In year 1, 4 and 5, we see sw is bigger thatthe realized return. In next chapter we will see what the impact of measuringrisk with ES instead of VaR is, by comparing the return-risk ratios.

22

To get insight on how these values of θt are formed, we also check themedians of the returns and the calculated shocks.

Median of return in year t:t 1 2 3 4 5

Rt 0.0308 0.0258 0.0317 0.0307 0.0340

Median of swt:t 1 2 3 4 5

swt 0.0520 0.0279 0.0339 0.0411 0.0487

Note that for the calculation of θt, in every scenario and for all t theratio θt is established and afterwards the median of θt over all scenarios isobserved. This is why dividing the Rt median with the swt median does notlead to the same values as θt.

Since the θt values are higher than the fraction of the median return andmedian swt , one might deduce returns higher (resp. lower) than the medianoccur at the same moments as the swt higher (resp. lower) than the median.The high returns seem to affect the return-risk ratio more than swt , whichresults in the θt being higher when the median is observed after calculatingthe ratios, than when it is the other way around. This implies that higherreturns are realized when a bigger risk is taken. In this table the returnmedians are more constant whereas the medians of swt fluctuate more.

23

5 Risk Measures

In a previous section we have seen what role the risk measure VaR hasin ALM and Solvency II. In this section we investigate what propertiesrisk measures intuitively should have and how this can be translated intomathematics. Firstly, the definition of coherent risk measure is explained.Secondly, we see VaR is not a coherent risk measure, for it does not sufficesubadditivity, a property of a coherent risk measure. Thirdly, we define ESformally and prove that ES is a coherent risk measure. Lastly, we investigatethe effect of using ES instead of VaR on the dynamic investment strategydefined in section 4.

5.1 Coherent Risk Measures

Let L0(Ω,F ,P) be the set of all randoms variables on (Ω,F) which are al-most surely finite. Financial risks are represented by a setM⊂ L0(Ω,F ,P)[10]. The random variables in the set M can be interpreted as portfoliolosses over time horizon ∆. Throughout this thesis, the time horizon is oneyear.

Risk measures are real-valued functions R : M → R, where R(L) rep-resents the amount of capital that should be added to a position with lossgiven by L ∈ M, so that the position becomes acceptable to an externalor internal risk controller. Positions with R(L) ≤ 0 are acceptable withoutinjection of capital; if R(L) < 0, capital might even be withdrawn [10].

The notion of coherent risk measure is an important concept in thiscontext as it is where the economic rationale is translated to mathematics.The definition of coherent risk measure is introduced by Artzner et al., 1999[2]. Note that the original definition is based on random variables in general,whereas in this thesis it is based on the loss distribution, which leads to asign change when compared to the definition in Artzner et al., 1999.

Definition 5.1. Given the random variables L1, L2 ∈ M, a risk measureR whose domain includes M is called a coherent risk measure if it satisfiesthe following conditions:

• Monotonicity : If L1 ≤ L2 almost surely, then R(L1) ≤ R(L2)

• Translation Invariance: For any constant c ∈ R, R(L1+c) = R(L1)+c

• Positive Homogeneity : For any λ > 0, R(λL1) = λR(L1)

• Subadditivity : R(L1 + L2) ≤ R(L1) +R(L2)

Intuitively, we want our risk measure to satisfy monotonicity because ifa portfolio leads to a higher loss, it requires more risk capital. Secondly, wewant our risk measure to be translation invariant, for if a constant quantity

24

of value c is added to (or subtracted from) a portfolio loss L, the risk mustincrease (resp. decrease) by the same amount [2]. Consider a position withloss L and R(L) > 0. Adding the amount of capital R(L) to the positionleads to the adjusted loss L = L − R(L), with R(L) = R(L − R(L)) =R(L) − R(L) = 0, so that the position L is acceptable without furtherinjection of capital [10]. In addition, the risk measure should be positivehomogeneous. This ensures that risk scales linearly with position size [12].

The last property for a risk measure to be coherent is subadditivity. Thisproperty captures the idea that a merger of risk should not create additionalrisk [2]. Suppose investing in asset class 1 leads to loss L1 and investing inasset class 2 leads to loss L2 whilst also assuming that a risk manager wantsto make sure the total risk is less than M . When using subadditive riskmeasure R(·) he can choose bounds M1,M2 such that M1 + M2 ≤ M ,and impose the constraint that R(Li) ≤ Mi. The risk measure R(·) beingsubadditive ensures that R(L) ≤ M1 + M2 ≤ M . If an investor uses anon-subadditive risk measure it would seem more convenient to open twoaccounts and invest in both asset classes separately.

Note that a positive homogeneous, subadditive risk measure is convex.

Definition 5.2. A risk measure R is convex, if for all L1, L2 ∈ M andλ ∈ [0, 1] we have

R(λL1 + (1− λ)L2) ≤ λR(L1) + (1− λ)R(L2).

Convexity agrees with the notion that we need diversification to spreadthe risk. As explained by Follmer, 2010 [7], consider the case where oneinvestment strategy leads to loss L1, and another investment strategy leadsto loss L2. If the investor diversifies, spending the fraction λ using the firststrategy and 1−λ using the second strategy, the investor obtains λL1 +(1−λ)L2. We want this diversification to decrease the risk of investing in theportfolios using these fractions separately.

5.2 Value at Risk

Having defined some relevant properties of risk measures in general, we arenow able to analyze which of these properties VaR carries. In this subsectionwe see that in general VaR satisfies the conditions monotonocity, translationinvariance and positive homogeneity. Moreover, we see an example of VaRthat illustrates VaR is not a coherent risk measure as it does not satisfy thecharacteristic of subadditivity. Recall:

VaRα(L) = infl ∈ R : P(L > l) ≤ 1− α = infl ∈ R : FL(l) ≥ α

Proposition 4. The risk measure VaR satisfies monotonocity, translationinvariance and positive homogeneity.

25

Proof. Let L1, L2 ∈ M as defined above. Firstly, for monotonocity, letL1 ≤ L2. We have VaR(L1) = infl ∈ R : P(L1 > l) ≤ 1− α ≤ infl ∈ R :P(L2 > l) ≤ 1 − α = VaR(L2). For translation invariance let c ∈ R. Wehave VaR(L1 + c) = infl ∈ R : P(L1 + c > l) ≤ 1−α = infl ∈ R : P(L1 >l) ≤ 1− α+ c = VaR(L1) + c.

Lastly, for positive homogeneity let λ > 0. We have

VaR(λL1) = infl ∈ R : P(λL1 > l) ≤ 1− α,= infl ∈ R : P(L1 > l/λ) ≤ 1− α,= infl ∈ R : P(L1 > l′) ≤ 1− α, (where l/λ = l′)

= infλl′ ∈ R : P(L1 > l′) ≤ 1− α, (since l = λl′)

= λinfl′ ∈ R : P(L1 > l′) ≤ 1− α,= λVaR(L1).

The following example by McNeil, Frey, Embrechts [10], shows that VaRα

is not a coherent risk measure in a case where α = 0.95, since the subaddi-tivity property does not hold for VaRα.

Example 5.1. Consider portfolio of 100 defaultable corporate bonds. Thecurrent price of each bond is 100. If there is no default, a bond pays anamount of 105 in one year; otherwise there is no repayment. Hence Li, theloss of bond i, is equal to 100 when the bond defaults and to −5 otherwise.The distribution of the loss of Li, i ∈ 0, . . . , 100 are IID and one unit ofbond i has the following distribution:

Li =

−5 with probability 0.98100 with probability 0.02

Denote Yi = 1 if bond i defaults and Yi = 0 in case it does not default.We can rewrite the distribution of Li to the following loss distribution

Li = 100Yi − 5(1− Yi) = 105Yi − 5.

Now consider two portfolios with a current value equal to 10 000. Inportfolio A the weight is concentrated on bond 1, thus A consists of 100units of bond 1. This implies

LA = 100L1,

so VaR0.95(LA) = 100VaR(L1). Now P(L1 ≤ −5) = 0.98 ≥ 0.95 and forl < −5 we have P (L1 ≤ l) = 0 < 0.95. Hence, VaR0.95(L1) = −5 andtherefore VaR0.95(LA) = −500.

26

This means that the capital charge for this portfolio is -500. Thus, ifthe risk controller has a negative equity of 500, this is still an acceptablestrategy at a 95% confidence level.

In portfolio B the weight is completely diversified and there is one uniton each bond.

LB =100∑i=1

Li = 105100∑i=1

Yi − 500,

and hence:

VaR0.95(LB) = VaR0.95

(105

100∑i=1

Yi − 500)

= 105q0.95

( 100∑i=1

Yi

)− 500.

We know∑100

i=1 Yi ∼ Bin(100, 0.02) and we have P(∑100

i=1 Yi ≤ 5)≈

0.984 < 0.95 and P(∑100

i=1 Yi ≤ 4) ≈ 0.949 < 0.95. So q0.95

(∑100i=1 Yi

)= 5

henceVaR0.95(LB) = 525− 500 = 25.

This means the investor needs an additional risk capital of 25 to satisfy aregulator working with VaR0.95 as a risk measure. Thus, we get

VaR0.95(LA) = −500,

VaR0.95(LB) = 25.

In this example we see the non-subadditivity of VaR:

VaR0.95

( 100∑i=1

Li

)> 100VaR0.95(L1) =

100∑i=1

VaR0.95(Li),

where the last equality is true since the Li are IID implying this risk measureis not coherent.

Intuitively, portfolio B is less risky since it is diversified and thereforeshould have a lower risk. Here we see that for portfolio B, an investor wouldneed an extra risk capital of 25. Meanwhile, for portfolio A, a risk capitalof 500 could be withdrawn and would still be acceptable. This indicates adrawback of non-coherent risk measures.

5.3 Expected Shortfall

Having recognized the consequence of VaR being non-subadditive, we lookat the coherent risk measure ES. In this section, firstly, we provide a formaldefinition for ES and subsequently investigate several properties of ES(L)when L is normally distributed. Secondly, we prove that ES is a coherentrisk measure. Lastly, we calculate shocks of the asset classes in our stylizedcase using ES. This way we can compare the impact of using ES instead ofVaR on the dynamic risk budgeting strategy.

27

The following definitions and theorems can be found in McNeil, Frey,Embrechts [10].

Definition 5.3. For a loss L with E(|L|) <∞ and distribution function FLthe Expected Shortfall at confidence level α ∈ (0, 1) is defined as

ESα(L) =1

1− α

∫ 1

αqu(FL)du,

where qu(FL) = F←L (u) is the quantile function of FL. ES is thus related toVaR by

ESα(L) =1

1− α

∫ 1

αVaRu(L)du.

Instead of fixing a particular confidence level α, VaR is averaged over alllevels u ≥ α and thus some properties of the tail of the loss distribution areregarded. Obviously, we have VaRα ≤ ESα.

For continuous loss distributions an even more intuitive expression canbe derived which shows that ES can be interpreted as the expected loss thatis incurred in the event that VaR is exceeded [10].

Proposition 5. For an integrable loss L with continuous distribution func-tion FL and any α ∈ (0, 1) we have

ESα =E(L;L ≥ qα(L))

1− α= E(L|L ≥ VaRα).

Here we have used the notation E(X;A) := E(X1A) for a generic inte-grable random variable X and a generic set A ∈ F .

Proof. Denote by U a random variable with uniform distribution on theinterval [0, 1]. By a property of the uniform distribution we know that therandom variable F←L (U) has distribution function FL. We have to show that

that E(L;L ≥ qα(L)) =∫ 1α F

←L (u)du. Now

E(L;L ≥ qα(L)) = E(F←L (U);F←L (U) ≥ F←L (α)) = E(F←L (U);U ≥ α);

in the last equality we used the fact that F←L is strictly increasing since FLis continuous. Thus we get E(F←L (U);U ≥ α) =

∫ 1α F

←L (u)du. The second

representation follows since for a continuous loss distribution

This proposition can be used to calculate ES for a Gaussian loss distri-bution.

Proposition 6. If L ∼ N (µ, σ2) and α ∈ (0, 1) then

ESα = µ+ σφ(Φ−1(α))

1− α,

where φ is the density of the standard normal distribution.

28

Proof. We have

ESα = µ+ σE

(L− µσ

∣∣∣L− µσ≥ qα

(L− µσ

))

Hence, it suffices to compute ES for the standard normal random variableL := L−µ

σ . Here we get:

ESα =1

1− α

∫ ∞Φ−1(α)

lφ(l)dl =1

1− α[−φ(l)]∞Φ−1(α) =

φ(Φ−1(α))

1− α

5.4 Coherence of Expected Shortfall

Now, in order to prove that ES is a coherent risk measure, we need to analyzeresults where the following theory is used: Fatou’s Lemma, the Strong Lawof Large Numbers and the Glivenko-Cantelli theorem. These results can befound in the appendix.

The complexity of proving that ES is a coherent risk measure lies with thesubadditivity property. To prove subadditivity, it is convenient to representES as a limit of discrete loss distribution functions using proposition 8.These loss distribution functions are essentially, all the VaRλ-values withλ ∈ (α, 1). Thus, the sum of these loss distribution quantiles multiplied byan averaging scaling factor is exactly the expected loss, given that the lossis outside the confidence interval [0, α].

In proposition 7, the convergence of the discrete case to the continuouscase is proved, which is a result obtained by W.R. van Zwet, 1980 [15]. Toprove this theorem, which is based on order statistics, we look at a theoremon convergence and accompanying definitions which form the base of VanZwet’s theorem.

Having seen Van Zwet’s result and the representation of ES as a limit ofdiscrete loss functions, we are able to prove the subadditivity of ES. Firstly,it is proven for the discrete case which is represented using order statistics,such that Van Zwets theorem can be applied. Secondly, Van Zwet’s theoremconcerning convergence proves subadditivity in the continuous case.

We start by giving some definitions that support lemma 1, which is abasic result in measure theoretic probability regarding convergence which isused by Van Zwet for his theorem about order statistics [15].

Let U1, U2, ... be IID random variables according to the uniform distri-bution on the interval (0, 1). For N = 1, 2, ..., let U1:N < U2:N < ... < UN :N

denote the ordered U1, ..., UN . We have Lebesgue measurable functionsJN : (0, 1)→ R, N = 1, 2, ..., and a Borel measurable function g : (0, 1)→ R.Further, we define gN : (0, 1) → R, N = 1, 2, ..., by gN (t) = g(U[Nt]+1:N ),where [Nt] denotes the integer part of Nt. Additionally, in the case that

29

[Nt] + 1 > N we let gN (t) = g(UN ). In the proof, t is be a number between0 and 1.

Lemma 1. With probability 1, gN converges to g in Lebesgue measure, i.e.P(

limN→∞t : |gN (t)− g(t)| ≥ δ = 0)

= 1 for every δ > 0.

Proof. Choose ε > 0. By Lusin’s theorem [9] (also appendix section 7.3),for a measurable function g : [0, 1] → R there exists a compact E ⊂ [0, 1]such that g restricted to E is continuous and λ(E) > 1 − ε. Now let B =(0, 1) \ E which is Borel. We have B ⊂ (0, 1), λ(B) ≤ ε and let g = g on(0, 1) ∩ Bc. Define gN (t) = gN (U[Nt]+1:N ) and BN = t : U[Nt]+1:N ∈ B,so that gN = gN on (0, 1) ∩ Bc. Since λ(BN ) = λt : U[Nt]+1:N ∈ B =1N

∑Ni=1 1Ui∈B = PN (B), where PN denotes the empirical distribution of

U1, ..., UN , it follows from the strong law that lim supN λ(BN ) ≤ ε withprobability 1. In view of the Glivenko-Cantelli theorem (see appendix) andthe continuity of g, this implies that with probability 1 we have for everyδ > 0: lim supN λt : |gN (t) − g(t)| ≥ δ ≤ λ(B) + lim supN λ(BN ) +lim supN λt : |gN (t) − g(t)| ≥ δ ≤ 2ε. Since ε > 0 is arbitrary, we haveP(

limN→∞t : |gN (t)− g(t)| ≥ δ = 0)

= 1 for every δ > 0, thus the lemmais proved.

We use this lemma to prove Van Zwet’s theorem about order statistics.For this theorem we first need to define the normed space Lp. For 1 ≤ p ≤ ∞,Lp is the Lebesgue space of measurable functions f : (0, 1) → R with finite

norm ||f ||p = ∫ 1

0 |f |pdλ1/p for 1 ≤ p < ∞ and ||f ||∞ = ess sup|f | for

p =∞.The purpose of this note is to show that under integrability assumptions

on JN and g,

MN :=

∫ 1

0JN (gN − g)dλ =

N∑i=1

g(Ui:N )

∫ i/N

(i−1)/NJNdλ−

∫ 1

0JNgdλ

converges to zero for N → ∞ with probability 1, which is Van Zwet’s firstresult. In addition to this, if JN converges in an appropriate sense to afunction J which shares the integrability properties of JN , Van Zwets secondresult states that

MN :=

∫ 1

0JNgNdλ−

∫ 1

0Jgdλ

also converges to zero with probability 1.

Proposition 7 (Van Zwet, 1980 [15]). Let 0 = t0 < t1 < ... < tk = 1and ε > 0. For j = 1, ..., k, let 1 ≤ pj ≤ ∞, p−1

j + q−1j = 1 and define

intervals Aj = (tj−1, tj) and Bj = (tj−1−ε, tj+ε) ∩ (0, 1). Suppose that, forj = 1, ..., k, JN1Aj ∈ Lpj for N = 1, 2, ..., g1Bj ∈ Lqj and either

1. 1 < pj ≤ ∞ and supN ||JN1Aj ||pj <∞, or

30

2. pj = 1 and JN1Aj : N = 1, 2, ... is uniformly integrable.

Then limN→∞MN = 0 with probability 1. If, moreover, there exist a func-tion J with JN1Aj ∈ Lpj for j = 1, ..., k, such that limN→∞

∫ t0 JNdλ =∫ t

0 Jdλ for every t ∈ (0, 1), then also limN→∞ MN = 0 with probability 1.

In this proof, some of the arguments do not go into specific detail. Theextended proof can be found in Van Zwet, 1980 [15].

Proof. Consider an index j with 1 < pj ≤ ∞, so qj < ∞. Choose δ ∈ (0, ε]and define Cj = (tj−1 − δ, tj + δ) ∩ (0, 1). The Glivenko-Cantelli theoremand the strong law of large numbers ensure that with probability one [17]

lim supN

∫Aj

|gN |qjdλ ≤ lim supN

1

N

∑i

|g(Ui)|qj1Cj∪Ui =

∫Cj

|g|qjdλ <∞.

Since δ ∈ (0, ε] is arbitrary, this implies that∫ 1

0 |gN |qj1Ajdλ→

∫ 10 |g|

qj1Ajdλwith probability 1 by Fatou’s lemma.

By Vitali’s convergence theorem (see appendix, section 7.3) this impliesthat

∫ 10 |gN − g|

qj1Ajdλ → 0, and Holder’s inequality yields ||JN ||pj ||gN −g||qj1Aj → 0 with probability 1. Thus, we can conclude that

∫ 10 JN (gN −

g)1Ajdλ→ 0 with probability 1.For an index j with pj = 1 and qj = ∞, the Glivenko-Cantelli theo-

rem ensures that lim supN ||gN1Aj ||∞ ≤ ||g1Bj ||∞ < ∞ with probability 1.Because of the uniform integrability of JN1Aj and Lemma 1, we have withprobability 1,

lim supN

|MN | ≤ δ lim supN

||JN1Aj ||1 + 2||g1Bj ||∞ lim supN

∫|gn−g|1Bj>δ

|JN |

= δ lim supN

||JN1Aj ||1

for every δ > 0. Since supN ||JN1Aj ||pj < ∞, we find that∫ 1

0 JN (gN −g)1Ajdλ→ 0 with probability 1.

This proves the first statement of the theorem. The second statementis apparent, as the assumptions of the theorem imply that

∫ 10 JNg1Ajdλ→∫ 1

0 Jg1Ajdλ for j = 1, ..., k.

This convergence result is the basis of the representation of ESα as thelimit of a sum defined by the order statistics of the loss distribution functions.Intuitively, one might say these loss distribution functions are the functionsVaRλ with λ ∈ (α, 1). The partition 0 = t0 < t1 < ... < tk = 1 in VanZwet’s theorem is used for splitting the range of integration (0, 1) into (0, α]and (α, 1). The next result from Acerbi and Tasche, 2002, gives the desiredrepresentation of ESα.

31

Proposition 8 (Acerbi and Tasche, 2002 [1]). Let α ∈ (0, 1) be fixed. Fora sequence (Li)i∈N of IID random variables, E[Li] <∞ for all i we have

limn→∞

∑[n(1−α)]i=1 Li,n[n(1− α)]

= ESα a.s.,

where L1,n ≥ ... ≥ Ln,n are the order statistics of L1, ..., Ln and where[n(1− α)] denotes the largest integer not exceeding n(1− α).

Proof. The “with probability 1” part of this lemma is essentially a specialcase of proposition 7 with 0 = t0 < (1 − α) = t1 < t2 = 1, J(t) = 1[α,1)(t),JN (t) = 1[ [Nα]−1

N,1)(t), g(t) = F−1(t), and p1 = p2 = ∞. This proves the

’almost sure convergence part’. Concerning the L1-convergence note that

∣∣∣ [n(1−α)]∑i=1

Li:n

∣∣∣ ≤ n∑i=1

|Li|.

By the strong law of large numbers n−1∑n

i=1 |Li| converges to E[L1] in L1.

This implies uniform integrability for n−1∑n

i=1 |Li| and for∣∣∣∑[n(1−α)]

i=1 Li:n

∣∣∣.Together with the already proven almost sure convergence, this implies as-sertion.

In this lemma we see that ESα can be regarded as the limiting averageof the [n(1−α)] upper order statistics from a sample of size n from the lossdistribution. Thus, ESα can be estimated when there is a large sample sizeand [n(1 − α)] is large. With this representation we can prove that ESα isa coherent risk measure, where the proof of ES being subadditive can befound in McNeil, Frey, Embrechts [10].

Theorem 1. Expected shortfall is a coherent measure of risk.

Proof. We have to prove ES satisfies monotonicity, translation invariance,positive homogeneity and subadditivity. The first three properties followdirectly from VaR having these properties, since the operation of integrationis linear and increasing. Thus, we only have to prove subadditivity.

Consider a generic sequence of random variables L1, ..., Ln with associ-ated order statistics L1,n ≥ ... ≥ Ln,n and note that for arbitrary m satisfy-ing 1 ≤ m ≤ n we have

m∑i=1

Li,n = supLi1 + ...+ Lim : 1 ≤ i1 < ... < im ≤ m.

Now consider two random variables L and L with joint distribution func-tion F and a sequence of iid bivariate random vectors (L1, L1), ..., (Ln, Ln)with the same distribution function F . Writing (L + L)i := Li + Li and

32

(L+ L)i,n for an order statistic of (L+ L)1, ..., (L+ L)n we observe that wemust have

m∑i=1

(L+ L)i,n = sup(L+ L)i1 + ...+ (L+ L)im : 1 ≤ i1 < ... < im,

≤ supLi1 + ...+ Lim : 1 ≤ i1 < ... < im+ supLi1 + ...+ Lim : 1 ≤ i1 < ... < im,

=

m∑i=1

Li,n +

m∑i=1

Li,n.

By setting m = [n(1 − α)] and letting n → ∞ we infer from Proposition8 that ESα(L + L) ≤ ESα(L) + ESα(L). Now that we have seen ES issubadditive, the proof of ES being a coherent risk measure is complete.

Since we have seen ES is a coherent risk measure as opposed to VaR, weare able to examine the effect of using ES instead of VaR in Example 5.1regarding defaultable corporate bonds.

Example 5.2. Recall, we have a portfolio with a current value of 10 000.The loss distribtution of the 100 IID random variables is given by

Li =

−5 with probability 0.98100 with probability 0.02

In portfolio A all the weight is concentrated on bond 1, thus A consistsof 100 units of bond 1. In portfolio B the weight is completely diversifiedand there is one unit on each bond.

We haveLA = 100L1,

soES0.95(LA) = 100ES(L1).

Now P(L1 ≤ −5) = 0.98 ≥ 0.95 and for l < −5 we have P (L1 ≤ l) =0 < 0.95. Recal VaR0.95(L1) = −5, hence

ES0.95(L1) =1

1− 0.995

∫ 1

0.995VaRu(L1)du,

=−5(0.98− 0.95) + 100(1− 0.98)

0.05,

= 37

and therefore ES0.95(LA) = 3700.In portfolio B the weight is completely diversified and there is one unit

on each bond.

33

LB =

100∑i=1

Li = 105

100∑i=1

Yi − 500,

therefore we get

ES0.95(LB) = ES0.95

(105

100∑i=1

Yi − 500)

= 105 · ES0.95

( 100∑i=1

Yi

)− 500

= 105 · 1

1− 0.95

∫ 1

0.95VaRu(B)du− 500, (B ∼ Bin(100, 0.02))

Here we need the following numbers,

P(B ≤ 4) ≈ 0.9491,

P(B ≤ 5) ≈ 0.9845,

P(B ≤ 6) ≈ 0.9959,

P(B ≤ 7) ≈ 0.9991,

P(B ≤ 8) ≈ 0.9998,

...

which implies

VaRu(B) = 5 for u ∈ [0.9500, 0.9845)

VaRu(B) = 6 for u ∈ [0.9845, 0.9959)

VaRu(B) = 7 for u ∈ [0.9959, 0.9991)

VaRu(B) = 8 for u ∈ [0.9991, 0.9998)

...

Thus∫ 1

0.95VaRu(B)du = 5× (0.9845− 0.9500) + 6× (0.9959− 0.9845)

+ 7× (0.9991− 0.9959) + 8× (0.9998− 0.9991) + · · · ,≈ 0.28,

which we use to continue the calculation

ES0.95(LB) ≈ 105

0.05× 0.28− 500,

≈ 88.

34

This implies we have

VaR(LA) = −500,

VaR(LB) = 25,

ES(LA) = 3700,

ES(LB) ≈ 88,

As we can see in this example, ES captures our intuitive notion that riskslurk in the tail of a distribution and is readily interpreted as an estimate ofthe average loss that we could experience with a given probability [12]. Inportfolio A, the risk controller is unable to see the possible risk when VaRis used. The expected loss is -500 which means it is expected that the riskcontroller earns 5 on every unit he puts in. Meanwhile the expected loss,conditional on the loss being outside of the confidence level, is very high,better illustrating the tail risk.

In portfolio B we see the values of ES and VaR do not differ much, sincewhen using VaR, the effect of the loss being outside of the confidence levelis already partially observable. When ES is used, the expected loss onlyincreases by (approximately) 63 in comparison to when VaR is used.

5.5 Stylized Case

In this section we analyze the return-risk characteristics when ES is usedas a risk measure instead of VaR. To compare VaR with ES, the same datais used as in the scenario analysis of section 4. Firstly, the behaviour ofthe weights depending on e

a and the risk appetite is illustrated by lookingat the cumulative weight graphs, when shocks are established with ES in-stead of VaR. Secondly, we look at the return-risk ratio when risk (in thedenominator) is measured using ES. Here, we are able to compare ES-basedoptimization with VaR-based optimization, by looking at the return-riskcharacteristics when the risk budgeting investment strategy formulated inchapter 4 is used.

Using ES as risk measure we find the following shock vector, based onthe data of the same Dutch insurance company as in chapter 4,

sES =

s0

s1

s2

s3

=

0.570.260.14

0

We take the same initial values as in the VaR-based scenario analysis. Thus,let total assets a = 40000 and equity e = 6000 and let the SCRtotal = 2900.

35

The optimization gives the following weight vector:

w = (7.2%, 15.4%, 9.1%, 68.3%)>.

Now we compare the optimized allocation when using ES instead of VaR.Again, on the left we see the cumulative weights where e

a is variable, and onthe right we see the cumulative weights when the risk appetite is variable.Recall that the initial values are e

a = 600040000 = 0.15, γ = 1.3.

Which can be compared to the cumulative weight graphs of the optimizationbased on VaR:

Here we see the behaviour of w is quite similar. The ES-based optimizationis more risk-averse, since there is less weight in stock and there is moreweight in sovereign bonds and spread. Now we have established the impacton the strategy, we can compare the return-risk characteristics.

For the comparison of ES with VaR, we use the same scenarios as inchapter 4. Denote RES

t (resp. RVaRt ) as the return in year t with ES-based

(resp. VaR-based) optimization.The return-risk ratio is defined as follows

θESt =

RESt

sESwt

.

Looking at the median of the return-risk ratio θESt , return RES

t and shockssESwt we find that over all scenarios:

36

Median θESt of all scenarios:

t 1 2 3 4 5

θESt 0.551 1.086 1.047 0.881 0.836

Median RESt of all scenarios:

t 1 2 3 4 5

RESt 0.0287 0.0254 0.0312 0.0305 0.0336

Median sESwt of all scenarios:

t 1 2 3 4 5

sESwt 0.0520 0.0270 0.0331 0.0405 0.0468

We can compare this to the medians of the ratio θVaRt , return RVaR

t andshock sES

wt over all scenarios of θ when optimization is executed with VaRand risk is measured with ES. Thus, we denote

θVaRt =

RVaRt

sESwt

.

This leads to the following medians.Median θVaR

t of all scenarios:t 1 2 3 4 5

θVaRt 0.521 0.974 0.923 0.787 0.719

Median RVaRt of all scenarios:

t 1 2 3 4 5

RVaRt 0.0308 0.0258 0.0317 0.0307 0.0340

Median sESwt of all scenarios:

t 1 2 3 4 5

sESwt 0.0592 0.0317 0.0386 0.0467 0.0555

Firstly, we observe that the median of θESt is higher than the median

of θVaRt for all t. Secondly, the returns are slightly higher when the opti-

mization is VaR-based. Thirdly, the measured risk is also higher when theoptimization is VaR-based. Based on these observations, one might deducethe return-risk ratio is favourable when the optimization is ES-based, dueto this strategy being more risk averse. The VaR-based optimization yieldsslightly higher returns, however, this does not compensate for the additionaltail risk taken when considering the return-risk ratio.

To elaborate on this last observation, the actual tail risk is examined.Therefore we check the realized returns of all years in all scenarios focusingon the 0.5th and the 5th percentile. For the realized returns in these tailregions, we also observe the corresponding risk based on ES which is usedto calculate the risk-return ratio θ. In Appendix 7.2, the subsection labelled“Ratio Check”, one is able to see how these results are obtained.

Observing only the lowest 0.5% realized returns, with their corresponding

37

risks and ratios, we take the averages for both strategies resulting in thefollowing table:

0.5% tail VaR-based optimization ES-based optimization

Realized return -0.090 -0.051

Risk (based on ES) 0.160 0.128

θ -0.572 -0.430Comparing these results, we see that in this tail region, the average

realized return of VaR-optimal portfolios is -9.0%, whereas the ES-optimalportfolios have an average of -5.1%. This means almost half of the loss isrealized when using ES-based optimization.

The measured risk for the VaR-based optimization is 16.0%, which ishigher than the measured risk of 12.8%. From this observation, one mightdeduce that when the optimization is ES-based, more weight is allocated toasset classes that are less risky in the tail region as well.

To analyze the sensitivity of the quantile, we check how these values differfrom the 5% lowest values of realized returns with corresponding ES-basedrisk and ratios. This leads to the following averages:

5% tail VaR-based optimization ES-based optimization

Realized return -0.022 -0.014

Risk (based on ES) 0.094 0.071

θ -0.217 -0.192

The averages of the 5% lowest realized returns and the correspondingmeasured risks for ES-optimal portfolios are much closer to the VaR-optimalportfolios, illustrating that the risk lurks in the 0.5% tail.

In conclusion, we see that when ES is used instead of VaR in the riskbudgeting strategy to determine shocks, the available risk budget becomessmaller. This leads to a more risk averse weight allocation, resulting inES-optimal portfolios being better prepared for risk that lurks in the tail.This also illustrates that a risk controller using VaR, should be aware of thetail risk since VaR does not contain information outside of the confidenceinterval.

38

6 Conclusion

In this section, the main findings of this thesis are summarized, followed byconcluding remarks. The limitations of this research will also be outlined,as well as areas for further research in the field.

6.1 Summary

This thesis has introduced an investment strategy for insurance companiesthat optimizes the return-risk ratio, namely maximizing the expected returnwithout exceeding SCRtotal. In the first part of this thesis, an investmentstrategy is proposed where the amount of risk that the insurer takes is pro-portional to their current financial position. This investment strategy isbased on an optimization problem that uses assumptions about statisticalvalues of asset class returns, i.e. expected return, standard deviation, cor-relation and shocks. The outcome of this optimization is an optimal weightallocation.

In the second part of this thesis, firstly, the concept of a coherent riskmeasure is investigated. Secondly, an example of how the benefits of di-versification are not taken into account when risk is measured using VaR,illustrates that VaR is not a coherent risk measure. Thirdly, ES is intro-duced and a detailed proof of the coherence of this risk measure is given.This proof requires the Law of Large Numbers in context of order statistics.An estimator of ES is provided, which is used to show the convergence ofa discrete representation of ES. Further, it is shown that in the extendedexample which illustrates the incoherence of VaR, using ES does in fact leadto a proper valuation of risk that follows the economic rationale.

Lastly, the impact of using ES instead of VaR to measure the shocksis investigated, by examining the return-risk characteristics. The return-risk ratio, where risk is measured using ES, of the ES-optimal portfolio iscompared to the VaR-optimal portfolio. The results are based on the riskbudgeting investment strategy as proposed in the first part of the thesis.The return-risk characteristics are explored by reviewing the tail risk inboth strategies. Therefore, we look at the lowest 0.5% and 5% realizedreturns and the corresponding measured risk based on ES and return-riskratios.

6.2 Concluding Statements

This research concludes that VaR-optimal portfolios face significantly moretail risk than ES-optimal portfolios. When ES is used instead of VaR todetermine the shocks, the available risk budget is smaller. This means moreweight will be allocated to asset classes which are more risk averse, affectingboth the realized return and the risk that the insurer takes.

39

The first part concludes that the optimal weight allocation can be sum-marized in an optimization problem. The objective function to be maxi-mized, is the expected log-normal return rate that depends on weight, ex-pected return and standard deviation. The constraints in the optimizationproblem ensure the available risk budget is fully utilized, without exceedingthe total SCR.

The second part concludes that ES yields a more accurate estimation ofrisk than VaR does. A comparison of two portfolios - one which is diversifiedand the other which is concentrated on one asset class - illustrates howVaR might significantly underestimate the risk when compared to ES. Inthe scenario analysis, we find that an ES-optimal portfolio will have moreweight allocated to less riskier asset classes than a VaR-optimal portfolio,resulting in the decrease of both the realized return and the measured risk.Moreover, this scenario analysis illustrates the limitation of not using acoherent risk measure since it does not contain information outside of theconfidence interval.

Therefore, the overall conclusion of this research is that when using therisk budgeting investment strategy formulated in the first part of this the-sis, ES-optimal portfolios are more risk averse than VaR-optimal portfolios,indicating that VaR-optimal portfolios encounter significantly more tail risk.

6.3 Limitations

This research has been confronted with three noteworthy limitations. Firstly,the stylized case only uses four asset classes, which defines the total assets.In an actual insurers portfolio, besides “stock”, “real-estate”, “sovereignbonds” and “credits” there are typically more asset classes. For example,the Dutch insurance company that is used for the stylized case also has assetclasses like: “mortgages”, “liquid resources”, “currency”, “swap”.

In future research, the stylized case can be extended using a greaterselection of asset classes. In this thesis, this is simplified because the focusis on the impact of risk budgets. Therefore, in order to analyze the effect ofdifferent shocks, the behaviour of all asset classes is assumed to be similar.If one has the objective to research how optimal risk budgets should beestablished per asset class, taking the differences between those asset classesinto account, the stylized case should be extended with modeling differentaspects of the asset classes.

Secondly, the expected geometric return that is used in the formulation ofthe optimization problem, is calculated under the assumption that the returnrates are log-normally distributed. However, in reality insurers deal withheavy tailed distributions. Furthermore, the correlation matrix of SolvencyII is used, since the data set used for this research was too small to deducethe actual correlation values.

The third limitation is that the data used is based on 1000 scenarios

40

with rates of five years, taken from Ortec Finance’s scenario generator.This means that for all return rates the 99.5%-quantile contains informationabout the 5 worst cases per year. The 99.5%-quantile is used for calculatingthe values for VaR and ES. One could argue that five values per year isnot enough to calculate the shock. For future research, it might thereforebe crucial to use a bigger scenario set, where scenarios are spanning over alonger time period, resulting in more accurate values of VaR and ES.

Aside from the given limitations, this research illustrates the importanceof understanding the differences which exist between ES and VaR in a riskbudgeting investment strategy, with ES being a more favorable method asit provides a better estimation of the tail risk.

41

7 Appendix

7.1 Sensitivity Analysis

In this section, we execute a sensitivity analysis on the parameters we usein our optimization, formulated in chapter 4. The parameters we analyzeare expected return and standard deviation. In the optimization we use thefollowing values

µ =

µ0

µ1

µ2

µ3

=

0.0760.0520.0360.022

,

σ =

σ0

σ1

σ2

σ3

=

0.220.120.0950.011

.

µ = (µ0, µ1, µ2, µ3)> = (0.076, 0.052, 0.036, 0.022)>,

σ = (σ0, σ1, σ2, σ3) = (0.22, 0.12, 0.095, 0.011)>.

In the first subsection, we analyze the sensitivity of the expected re-turn. Firstly, we check what happens when we add 0.01 to the marginalexpected returns, and then we investigate the effect of subtracting 0.01 ofthe marginal expected returns. Secondly, we do a similar check for thestandard deviations, only then with adding and subtracting 0.05.

Return rates parameter

Optimization with µ and σ:

Optimization with µ+ (0.01, 0, 0, 0)> and σ:

42

Optimization with µ+ (0, 0.01, 0, 0)> and σ:

Optimization with µ+ (0, 0, 0.01, 0)> and σ:

Optimization with µ+ (0, 0, 0, 0.01)> and σ:

43

Optimization with µ− (0.01, 0, 0, 0)> and σ:

Optimization with µ− (0, 0.01, 0, 0)> and σ:

Optimization with µ− (0, 0, 0.01, 0)> and σ:

Optimization with µ− (0, 0, 0, 0.01)> and σ:

44

Standard deviation parameter

Optimization with µ and σ:

Optimization with µ and σ + (0.05, 0, 0, 0)>:

Optimization with µ and σ + (0, 0.05, 0, 0)>:

45

Optimization with µ and σ + (0, 0, 0.05, 0)>:

Optimization with µ and σ + (0, 0, 0, 0.05)>:

Optimization with µ and σ − (0.05, 0, 0, 0)>:

46

Optimization with µ and σ − (0, 0.05, 0, 0)>:

Optimization with µ and σ − (0, 0, 0.05, 0)>:

Optimization with µ and σ − (0, 0, 0, 0.05)>:

Firstly, we observe that the expected return and standard deviation ofstock are the least sensitive to parameters. This might be because stock hasthe highest values for both parameters, and this will in all cases be the assetclass that gets a high weight allocation in the cases where e

a is high or γ islow.

The second observation is that when real-estate has a higher (resp. lower)expected return or lower (resp. higher) standard deviation, the additional(resp. eliminated) allocated weight is in credit spread, and the other wayaround. For instance, the case where the optimization is executed withµ−(0, 0, 0.01, 0)> = (0.076, 0.052, 0.026, 0.022)> the expected return of real-

47

estate and credit spread are very close. Since the standard deviation ofcredit spread is not much lower, this results in no weight allocated to creditspread. One might conclude, the expected return is not high enough in thiscase considering the risk induced by the extra volatility.

7.2 Python Scripts

Portfolio Algorithm

In this section the algorithm that is used for the scenario analysis is shown.Firstly the initual values are given. This are based on Ortec Finance’s ALMmodel. Secondly, the optimization from section 4.1 is given. Thirdly, the re-quired data is loaded. Lastly, a data frame is generated that holds the initialvalues for all scenarios, and the algorithm of section 4.3 is programmed.

# -*- coding: utf-8 -*-

"""

Created on Tue Jun 21 12:46:10 2015

@author: Wout

"""

from __future__ import division

from __future__ import print_function

from pandas import *

import numpy as np

import pandas as pd

import matplotlib.pyplot as plt

import os

import general_export_reader as ger

import plot_tools as pt

import Formules

import pdb

import scipy as sp

if True: #values gained from ALS/EXCEL

cor_mrkt = np.array([[1, 0 , 0 , 0],

[0, 1 , 0.75, 0],

[0, 0.75, 1 , 0],

[0, 0 , 0 , 1]])

48

cor_life= np.array([[1, 0.25,0.25,0.25,0.25],

[0.25, 1,0.25,0.25,0.50],

[0.25,0.25,1,0.25,0],

[0.25,0.25,0.25,1,0],

[0.25,0.50,0,0,1]])

exp_return = np.array([0.075642,0.052234,0.035625, 0.021967])

stdev = np.array([0.221907, 0.116928, 0.055463, 0.010718])

schokken = np.array([0.5101188, 0.228609, 0.119293, 0])

cvar = np.array([0.567559, 0.2635, 0.13838, 0])

schokken_risk = np.array([0.5101188, 0.228609, 0.119293])

cor_mrkt_risk = np.array([[1, 0 , 0 ],

[0, 1 , 0.75],

[0, 0.75, 1 ]])

################# OPTIMISATION ALGORITHM #######################

def get_weights(rho, s, mu, sigma, b, scrt, ta):

N = mu.shape[0]

con=list()

con.append( ’type’: ’eq’, ’fun’ : lambda x : Formules.VaR(x, s, rho)-b )

con.append( ’type’: ’eq’, ’fun’ : lambda x :np.sum(x)-1 )

con.append( ’type’: ’ineq’, ’fun’ : lambda x :scrt

- np.sqrt(np.dot(x*s, rho.dot(s*x)))*ta )

bounds = [(0,None) for k in range(N)]

x0=np.ones(N)/N

sol=sp.optimize.minimize(lambda x:

-1*Formules.log_georeturn(Formules.r_a(x,mu),

Formules.log_variance(Formules.r_a(x,mu),

Formules.Variance(rho, sigma, x))),

x0, method =’SLSQP’ ,

constraints = con,

bounds=bounds )

return sol[’x’]

####################################################################

def beta(eigenvermogen,assets,gamma):

return np.divide(eigenvermogen,((1+gamma)*assets))

49

if __name__ == ’__main__’:

if True:

sc_set =dict()

if True:

varnames= [’pukkasstroom’,’life’,’nonlife’,’health’,

’operational’, ’market’, ’default’, ’vvpprimo’, ’vvpultimo’,

’aandelen’, ’spread’, ’swap’, ’staats’, ’oghuis’, ’ogwinkel’,

’ogkantoor’]

for varname in varnames:

sc_set[varname]=pd.read_excel(’alsdata2.xlsx’, varname,

index_col=None, na_values=[’NA’]).T

df=pd.read_excel(’alsdata2.xlsx’, varname,

index_col=None, na_values=[’NA’]).T

pass

sc_set = pd.concat(sc_set.values(), axis=1,

keys=sc_set.keys(),names=[’Name’, ’Scenario Nr.’])

simulation_start = sc_set.index[0]#.date()

if True:

sc_nrs = sc_set.columns.get_level_values(’Scenario Nr.’).unique()

scenarios_mix = pd.DataFrame(index=sc_set.index, columns = sc_nrs)

scenarios_mix.iloc[0] = np.ones(len(sc_nrs))/len(sc_nrs)

weights_index=range(2013,2019)

weights_varnames=[’w1’, ’w2’, ’w3’, ’w4’, ’eigenvermogen’,

’belegdvermogen’, ’belegresultaat’, ’beleg’, ’risicovrij’,

’assets’, ’scr_tot’, ’riskratio_asset’,’riskratio_asset_cvar’,

’riskratio_total’, ’asset_shock’,’asset_shock_cvar’,

’betacheck’, ’asset_return’]

n_var=len(weights_varnames)

n_scenarios=1000

scenario_nrs=range(0,n_scenarios)

weights_columns=pd.MultiIndex.from_product([weights_varnames, scenario_nrs],

names=[’Name’, ’Sc. nr.’])

weights = pd.DataFrame(data = 0,index=weights_index, columns = weights_columns)

weights.loc[2013,’eigenvermogen’] = EV_init

weights.loc[2013,’belegdvermogen’] = BV_init

weights.loc[2013,’beleg’] = beleg_init

weights.loc[2013,’risicovrij’] = risicovrij_init

weights.loc[2013,’assets’] = assets_init

weights.loc[2013,’scr_tot’] = 2900

weights.loc[2013,’w1’] = w_init[0]

50

weights.loc[2013,’w2’] = w_init[1]

weights.loc[2013,’w3’] = w_init[2]

weights.loc[2013,’w4’] = w_init[3]

for t in scenarios_mix.index:

if t>2013:

for sc_nr in sc_nrs:

a_t_primo = weights.loc[t-1,(’assets’,sc_nr)]

+ sc_set.pukkasstroom.loc[t,sc_nr]

e_t_primo = weights.loc[t-1,(’eigenvermogen’,sc_nr)]

+ sc_set.pukkasstroom.loc[t,sc_nr]

z=np.array([weights.loc[t-1,(’w1’,sc_nr)] * a_t_primo ,

weights.loc[t-1,(’w2’,sc_nr)] * a_t_primo,

weights.loc[t-1,(’w3’,sc_nr)] * a_t_primo,

weights.loc[t-1,(’w4’,sc_nr)] * a_t_primo])

scr_market=np.sqrt(np.dot(z*schokken,

cor_mrkt.dot(schokken*z)))

x=np.array([scr_market,sc_set.default.loc[t,sc_nr],

sc_set.life.loc[t,sc_nr],

sc_set.health.loc[t,sc_nr],

sc_set.nonlife.loc[t,sc_nr]]) #x is vector met alle SCR

scr_totaal = np.sqrt(np.dot(x, cor_life.dot(x)))

weights.loc[t,(’scr_tot’,sc_nr)] = scr_totaal

bet= beta(e_t_primo,a_t_primo,1.3)

weights.loc[t,(’betacheck’,sc_nr)] = bet

if bet < np.min(schokken):

bet = 0.00001

elif bet > np.max(schokken):

bet = np.max(schokken)

else:

bet=bet

ta = np.max([0,a_t_primo])

wt = get_weights(cor_mrkt, schokken, exp_return,

stdev, bet, scr_totaal, ta)

if t == 2016 and sc_nr==500:

pass

weights.loc[t,(’w1’,sc_nr)]=wt[0]

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weights.loc[t,(’w2’,sc_nr)]=wt[1]

weights.loc[t,(’w3’,sc_nr)]=wt[2]

weights.loc[t,(’w4’,sc_nr)]=wt[3]

X_a= weights.loc[t,(’w1’,sc_nr)] * a_t_primo*

(1+sc_set.aandelen.loc[t,sc_nr])

X_woning = a_t_primo* (1+sc_set.oghuis.loc[t,sc_nr])

* og_vast[0]

X_kantoor = a_t_primo* (1+sc_set.ogkantoor.loc[t,sc_nr])

* og_vast[1]

X_winkel = a_t_primo* (1+sc_set.ogwinkel.loc[t,sc_nr])

* og_vast[2]

X_v = weights.loc[t,(’w2’,sc_nr)]

* (X_woning + X_kantoor + X_winkel)

X_sp = weights.loc[t,(’w3’,sc_nr)] * a_t_primo*

(1+sc_set.spread.loc[t,sc_nr]+sc_set.swap.loc[t,sc_nr])

X_st = weights.loc[t,(’w4’,sc_nr)]

* a_t_primo* (1+sc_set.staats.loc[t,sc_nr])

a_t_tijdelijk=np.sum([X_a,X_v,X_sp, X_st ])

weights.loc[t,(’belegresultaat’,sc_nr)] = a_t_tijdelijk - a_t_primo

weights.loc[t,(’eigenvermogen’,sc_nr)] = np.max([ 0 , e_t_primo +

0.75*(weights.loc[t,(’belegresultaat’,sc_nr)]

+ sc_set.vvpprimo.loc[t,sc_nr]

- sc_set.vvpultimo.loc[t,sc_nr])

- 0.25*sc_set.pukkasstroom.loc[t,sc_nr]])

weights.loc[t,(’assets’,sc_nr)] =

weights.loc[t,(’eigenvermogen’,sc_nr)]

+ sc_set.vvpultimo.loc[t,sc_nr]

Ratio Check

The following script is used in section 5.5 where df1 represents the dataframe that contains the realized return of every scenario every year. Thevariable df2 represents the data frame that contains the calculated shockbased on expected shortfall. The variable q is the examined quantile. Theobjective of this script is to investigate tail risk, by looking at the lowest q%realized returns or highest q% measured risk.

52

def ratio_check(df1,df2,q):

quant = np.percentile(df1,q)

return = []

shock = []

ratio = []

for i in df1.index: #jaren

for j in df1.columns: #scenarios

if df1.loc[i,j]<quant:

return.append(df1.loc[i,j])

shock.append(df2.loc[i,j])

theta = np.divide(df1.loc[i,j],df2.loc[i,j])

ratio.append(theta)

y = pd.DataFrame([return,shock,ratio], index = [’returns’,’shocks’, ’theta’] )

return y

The output of this script, when the optimization is based on VaR andwith q = 0.5% resp. q = 5% is

ratio_check(weights.asset_return, weights.asset_shock_cvar, 0.5)

Out[2]:

0 1 2 3 4 5 6

returns -0.128774 -0.180772 -0.092739 -0.083175 -0.128945 -0.081483 -0.095603

shocks 0.170550 0.170550 0.170550 0.170550 0.170551 0.170550 0.170550

theta -0.755052 -1.059931 -0.543762 -0.487684 -0.756054 -0.477762 -0.560553

7 8 9 ... 20 21 22

returns -0.090343 -0.067693 -0.059687 ... -0.122263 -0.060558 -0.069306

shocks 0.170550 0.170550 0.170550 ... 0.170550 0.170550 0.133192

theta -0.529713 -0.396908 -0.349968 ... -0.716875 -0.355075 -0.520344

23 24 25 26 27 28 29

returns -0.073353 -0.088542 -0.079825 -0.073868 -0.059064 -0.133160 -0.054113

shocks 0.170550 0.127915 0.089021 0.112180 0.170550 0.170550 0.170550

theta -0.430095 -0.692191 -0.896702 -0.658478 -0.346315 -0.780764 -0.317287

[3 rows x 30 columns]

ratio_check(weights.asset_return, weights.asset_shock_cvar, 5)

Out[3]:

0 1 2 3 4 5 6

returns -0.013313 -0.007719 -0.005017 -0.013158 -0.009117 -0.007849 -0.006732

shocks 0.059167 0.059167 0.059167 0.059167 0.059167 0.059167 0.059167

theta -0.225013 -0.130466 -0.084796 -0.222385 -0.154095 -0.132655 -0.113780

53

7 8 9 ... 290 291 292

returns -0.021118 -0.007227 -0.003684 ... -0.005810 -0.024021 -0.014620

shocks 0.059167 0.059167 0.059167 ... 0.170550 0.170550 0.170550

theta -0.356915 -0.122148 -0.062257 ... -0.034067 -0.140847 -0.085722

293 294 295 296 297 298 299

returns -0.041691 -0.054113 -0.016520 -0.005939 -0.008520 -0.022604 -0.004540

shocks 0.098235 0.170550 0.079636 0.071301 0.054480 0.170550 0.059990

theta -0.424406 -0.317287 -0.207438 -0.083296 -0.156395 -0.132537 -0.075676

[3 rows x 300 columns]

Here we see the 30 resp. 300 lowest realized returns (which is the 0.5% resp.5% percentile of 1000 scenarios × 6 years) when based on VaR with thecorresponding shock and return-risk ratio.

The output of this script, when the optimization is based on ES andwith q = 0.5% resp. q = 5% is

ratio_check(weights.asset_return, weights.asset_shock_cvar, 0.5)

Out[2]:

0 1 2 3 4 5 6

returns -0.027608 -0.027713 -0.028129 -0.094455 -0.066574 -0.056968 -0.029034

shocks 0.050177 0.042604 0.152204 0.154389 0.159123 0.149188 0.109645

theta -0.550209 -0.650474 -0.184812 -0.611795 -0.418381 -0.381856 -0.264804

7 8 9 ... 20 21 22

returns -0.046488 -0.029642 -0.090790 ... -0.092640 -0.031341 -0.038260

shocks 0.163243 0.151918 0.149774 ... 0.153986 0.095638 0.074941

theta -0.284776 -0.195118 -0.606180 ... -0.601613 -0.327708 -0.510539

23 24 25 26 27 28 29

returns -0.065291 -0.034631 -0.066862 -0.026814 -0.060457 -0.058749 -0.067784

shocks 0.098932 0.086571 0.076792 0.117449 0.091748 0.151155 0.157126

theta -0.659959 -0.400034 -0.870681 -0.228303 -0.658948 -0.388668 -0.431396

[3 rows x 30 columns]

ratio_check(weights.asset_return, weights.asset_shock_cvar, 5)

Out[3]:

0 1 2 3 4 5 6

returns -0.010411 -0.005202 -0.002998 -0.009006 -0.005932 -0.004143 -0.005090

shocks 0.051965 0.051965 0.051965 0.051965 0.051965 0.051965 0.051965

54

theta -0.200350 -0.100100 -0.057685 -0.173310 -0.114156 -0.079735 -0.097946

7 8 9 ... 219 220 221

returns -0.017110 -0.005704 -0.002606 ... -0.001002 -0.019926 -0.006471

shocks 0.051965 0.051965 0.051965 ... 0.069990 0.066226 0.086018

theta -0.329262 -0.109766 -0.050154 ... -0.014323 -0.300873 -0.075232

222 223 224 225 226 227 228

returns -0.017740 -0.067784 -0.008546 -0.014037 -0.020184 -0.004887 -0.002045

shocks 0.061322 0.157126 0.062812 0.065378 0.087461 0.046079 0.052285

theta -0.289299 -0.431396 -0.136051 -0.214700 -0.230771 -0.106068 -0.039105

[3 rows x 229 columns]

Here we see the 30 resp. 300 lowest realized returns (which is the 0.5%resp. 5% percentile of 1000 scenarios × 6 years) when based on ES with thecorresponding shock and return-risk ratio.

7.3 Miscellaneous Theorems and Proofs

Strong Law of Large Numbers

The strong law of large numbers states that the sample average convergesalmost surely to the expected value, thus X → µ almost surely, when n →∞. That is

P(

limn→∞

Xn = µ)

= 1.

Lusin’s Theorem

For an interval [a, b], let f : [a, b]→ R be a measurable function. Then, forevery ε > 0, there exists a compact E ⊂ [a, b] such that f restricted to E iscontinuous and

λ(E) > b− a− ε.

Note that E inherits the subspace topology from [a, b]; continuity of f re-stricted to E is defined using this topology.

Glivenko-Cantelli

Assume that X1, X2, ... are are independent and identically-distributed ran-dom variables in R with common cumulative distribution function F (x).The empirical distribution function for X1, . . . , Xn is defined by

Fn(x) =1

n

n∑i=1

I(−∞,x](Xi),

55

where IC is the indicator function of the set C. For every (fixed) x, Fn(x) isa sequence of random variables which converge to F (x) almost surely by thestrong law of large numbers, that is, Fn converges to F pointwise. Glivenkoand Cantelli strengthened this result by proving uniform convergence of Fnto F .

Thus the theorem states

‖Fn − F‖∞ = supx∈R|Fn(x)− F (x)|−→0 a.s.

Vitali’s Convergence Theorem

Let (X,F , µ) be a positive measure space. If

1. µ(X) <∞

2. fn is uniformly integrable

3. fn(x)→ f(x) almost everywhere as n→∞ and

4. |f(x)| <∞ almost everywhere

then the following hold:

1. f ∈ L1(µ)

2. limn→∞∫X |fn − f |dµ = 0 (L1-convergence)

Holders Inequality

Let (S,Σ, µ) be a measure space and let p, q ∈ [1,∞] with 1/p + 1/q = 1.Then, for all measurable real- or complex-valued functions f and g on S,

‖fg‖1 ≤ ‖f‖p‖g‖q.

Fatou’s Lemma

Let f1, f2, f3, ... be a sequence of non-negative measurable functions on ameasure space (S,Σ, µ). Define the function f : S → [0,∞] by

f(s) = lim infn→∞

fn(s), s ∈ S.

Then f is measurable and∫Sf dµ ≤ lim inf

n→∞

∫Sfn dµ .

Note: The functions are allowed to attain the value +∞ and the integralsmay also be infinite.

56

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[4] A. ten Cate. Arithmetic and geometric mean rates of return in dis-crete time. Tech. rep. CPB Netherlands Bureau for Economic PolicyAnalysis, 2009.

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[9] N. Lusin and J. Priwaloff. “Sur l’unicite et la multiplicite des fonctionsanalytiques”. In: Annales scientifiques de l’Ecole Normale Superieure.Vol. 42. 1925, pp. 143–191.

[10] A.J. McNeil, R. Frey, and P. Embrechts. Quantitative risk manage-ment: concepts, techniques, and tools. Princeton university press, 2010.

[11] T. Nguyen, R.D. Molinari, et al. “Risk Aggregation by Using Copulasin Internal Models”. In: Journal of Mathematical Finance 1.03 (2011),p. 50.

[12] T. Philips and M. Liu. “Simple and robust risk budgeting with ex-pected shortfall”. In: The Journal of Portfolio Management 38.1 (2011),pp. 78–90.

[13] R. Swain, D. Swallow, et al. “The prudential regulation of insurers un-der Solvency II”. In: Bank of England Quarterly Bulletin 55.2 (2015),pp. 139–153.

[14] European Union. “Supplementing Solvency II Directive”. In: OfficialJournal of the European Union (2015).

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[16] E.M. Varnell. “Economic scenario generators and Solvency II”. In:British Actuarial Journal 16.01 (2011), pp. 121–159.

[17] J.A. Wellner et al. “A Glivenko-Cantelli theorem and strong laws oflarge numbers for functions of order statistics”. In: The Annals ofStatistics 5.3 (1977), pp. 473–480.

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