Comparing VaR and TVaR : a simulation-based approach

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–A Simulation-Based Approach

Shen, Guiyu

Master’s Thesis to obtain the degree in Actuarial Science and Mathematical Finance University of Amsterdam

Faculty of Economics and Business Amsterdam School of Economics Author: Shen, Guiyu Student nr: 10828435

Email: 1000010532@pku.edu.cn Date: August 11, 2015

Supervisor: dhr. dr. S.U. (Umut) Can Second reader: dhr. prof. dr. R. (Rob) Kaas

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Abstract

Value-at-Risk (VaR) and Tail Value-at-Risk (TVaR) are two very important risk measures in risk management. In this thesis, these two risk measures are studied for an aggregate risk with known marginal distributions and unknown dependence structure. I compare these two measures by testing how well the VaR can predict the corresponding TVaR. Along with some common copula models, I also investigate some special copula models to represent some possible extreme cases. The main conclusion is that the VaR is not good enough to predict TVaR because of the possibility of extreme cases.

Keywords Risk measure, Value-at-Risk, Tail Value-at-Risk, Copula, Thickness of tail, Risk aggregation, Monte-Carlo simulation.

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2.1.1 Definition of VaR. . . 7 2.1.2 Definition of TVaR . . . 7 2.2 Copula . . . 7 2.2.1 Definition of copula . . . 7 2.2.2 Gaussian copula . . . 8 2.2.3 Farlie-Gumbel-Morgenstern copula . . . 8 2.2.4 Monotonic-Countermonotonic copula. . . 8 2.2.5 Countermonotonic-Monotonic copula. . . 9 2.3 Probability distributions . . . 10 2.3.1 Exponential distribution . . . 10 2.3.2 Pareto distribution . . . 10 2.4 Aggregated risk . . . 11 3 Methodology 12 3.1 The choice of marginal distributions . . . 12

3.2 Dependence structure . . . 13

3.3 Confidence level. . . 13

3.4 Implementation with R code . . . 13

3.5 The following steps . . . 14

4 Results and analysis 15 4.1 The presentation of the results after implementing R . . . 15

4.1.1 Results when the marginal distributions are one light-tailed dis-tribution and one heavy-tailed disdis-tribution . . . 15

4.1.2 Results when the marginal distributions are both light-tailed dis-tributions . . . 26

4.1.3 Results when the marginal distributions are both heavy-tailed dis-tributions . . . 36

4.2 Results explanations . . . 46

4.2.1 The choice of ordinary linear regression model . . . 46

4.2.2 Interpreting the results of OLM. . . 46 iv

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4.3 Results analysis . . . 47

4.3.1 The effect of the thickness of tail . . . 47

4.3.2 The effect of the confidence level . . . 48

4.3.3 The effect of copula . . . 48

5 Conclusions 50 Appendix: R code 53 References . . . 57

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Preface

Completing a master’s thesis is always a great challenge, I am really happy that I finally finish it. Here, I would like to thank my supervisor, Dr. Sami Umut Can. I have received a lot of help and learned a lot from him in the whole process of writing this thesis. I also would like to thank all the other professors of the ASMF program. I have had a wonderful experience in this one-year master’s program. Finally, I want to thank my parents and my friends for their great support.

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Introduction

1.1 Managing risks

The financial market is a market full of risks. For the insurance field, the uncertainty of the future stock price, interest rate, mortality rate and inflation rate can bring great profitability as well as risks. Risk is not adverse in the financial market, instead, risk implies the possibility of making a great profit. If the financial institutions manage to hedge all the risks in their portfolios, they will spend a lot of money and also lose their chance of making profit.

However, financial institutions should not neglect the risks in their portfolios. It is not advisable to hedge all the risks, but it is even less appropriate if they decide not to manage the risks at all. A higher probability of bankruptcy or other financial difficulties will cost them a lot. So, managing risks in an appropriate way is quite important for all the financial institutions. And to manage the risks, we first need a good way to measure the risks. Furthermore, it is desirable that we can measure the risk in a way that many people can understand.

1.2 Risk measure

Risk measure is a way to quantify the risk associated with an investment with usually a single number. A good risk measure should be easy to understand, and should describe an important characteristic of the risk. Also, some intuitive meanings of a risk measure will make it more acceptable.

Historically, standard deviation had been the standard risk measure for a long time. However, voluminous criticism of standard deviation as a risk measure has been pub-lished (Kaplanski and Kroll,2002). Especially, it does not take into account the tail of a risk distribution. Two risks, one following the uniform distribution and the other the normal distribution, can have exactly the same standard deviation measure. However, the possible values of the first distribution are limited to a bounded interval, while the normal distribution has two tails and can never be bounded. Moreover, for some dis-tributions with heavy tails, the probability of extremely large losses is too large to be neglected. But the standard deviation measure cannot differentiate these distributions from distributions with light tails, because different distributions can have the same standard deviation regardless of the thickness of the tail.

So, a new risk measure, Value-at-Risk, or VaR, was introduced. This risk measure is now widely used. For example, CEIOPS(2010) describes SCR, the Solvency Capital Requirement under the Solvency II, as follows: “The SCR should correspond to the Value-at-Risk of the basic own funds of an insurance or reinsurance undertaking subject to a confidence level of 99.5% over a one-year period”.

The idea of VaR is quite simple. For a given confidence level, for instance, 99%, VaR(99%) means that the probability that the loss is not bigger than VaR(99%) is

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2 Shen, Guiyu — Comparing VaR and TVaR

99%. It is just the quantile of a distribution, but it avoids the problem of standard deviation. And it is very easy to understand and useful. In the example of SCR, we can conclude that with SCR, the probability that the insurance or reinsurance company will not go bankrupt within one year is 99.5%.

However, VaR also has its own problem. Artzner et al. (1999) set the standard of a coherent risk measure. To be a coherent risk measure, four properties are to be met. Here X and Y are the risks and ρ is the risk measure.

1. Subadditivity. For all X and Y , ρ(X + Y ) ≤ ρ(X) + ρ(Y ).

2. Monotonicity. For all X and Y such that X < Y almost surely, we have ρ(X) < ρ(Y ).

3. Positive homogeneity. For all X and c > 0, cρ(X) = ρ(cX).

4. Translation invariance. For all X and all real numbers α, ρ(X + αr) = ρ(X) − α, where r is the total return on a risk free investment of α.

It turns out that VaR does not satisfy the subadditivity. I will show it in an example. Subadditivity means there should be a benefit of risk diversification, that is, the risk measure of the merged risk should not be larger than the sum of the risk measures of the separate risks. Moreover, VaR does not consider the worst cases beyond a certain confidence level. David Einhorn, an American hedge fund manager, described the VaR as “an airbag that works all the time, except when you have a car accident”.

Some alternatives to VaR have been put forward. One of those is TVaR, Tail Value-at-Risk. It is closely related to VaR, but considers all the scenarios beyond a certain confidence level instead of one. Also, TVaR is subadditive. Yamai and Yoshiba (2005) have discussed the problem and presented some advantages and disadvantages of TVaR.

1.3 Risk aggregation and its effect on risk measures

Financial institutions usually need to deal with multiple sources of risk, so they need to consider the sum of a few risks. Also, they need to measure the riskiness of the aggregated risk, which is usually of more importance for the institutions. However, the VaR or TVaR of the aggregated risk is seldom equal to the sum of the VaRs or TVaRs of the separate risks respectively. Here, some examples of the effect of risk aggregation on VaR and TVaR are given.

1.3.1 Example 1

Assume risk X follows the uniform distribution on [0,1], that is, X follows U[0,1]. Then we define risk Y as,

Y =      X if X < 0.9; 1.9 − X else (1.1)

Obviously, Y has the same distribution as X. Now, we need to aggregate risk X with a risk. There are two choices, the first choice is aggregating X with itself. We define Z := 2X. The second choice is aggregating X with Y . We define W = X + Y .

From some simple computations and the definitions of VaR and TVaR (to be given in the next chapter), we can easily get the following results:

VaR[Z; 0.95] = 1.9 (1.2)

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VaR[W ; 0.95] = 1.9 (1.4) TVaR[W ; 0.95] = 1.9 (1.5) VaR[Z; 0.94] = 1.88 (1.6) TVaR[Z; 0.94] = 1.94 (1.7) VaR[W ; 0.94] = 1.9 (1.8) TVaR[W ; 0.94] = 1.9 (1.9) VaR[Z; 0.9] = 1.8 (1.10) TVaR[Z; 0.9] = 1.9 (1.11) VaR[W ; 0.9] = 1.9 (1.12) TVaR[W ; 0.9] = 1.9 (1.13)

From this example, we can clearly see that the relation between VaR and TVaR depends on how the separate risks depend on each other. Here, aggregated risks Z and W have the components with exactly the same distributions. To be more precise, Z has the components of (the first) X and (the second) X, while W has the components of X and Y . The (first) X of Z and the X of W have the same distribution, as do the (second) X of Z and the Y of W . The only difference between Z and W is the dependence structure between the components. And we can conclude something interesting from the outcomes.

First, assuming everything else the same, the difference of the dependence can result in different VaR or TVaR values. That is, the marginal distributions are not sufficient to determine the nature of the aggregated risks. (See equations (1.6), (1.8) and (1.7), (1.9)).

Second, assuming everything else the same, the difference of the dependence can result in a situation where two aggregated risks have the same VaR but different TVaR, or the same TVaR but different VaR similarly. That is, the knowledge of VaR (or TVaR) and marginal distributions is not enough to determine TVaR (or VaR). (See equations (1.2), (1.3), (1.4), (1.5) and (1.10), (1.11), (1.12), (1.13)).

Third, assuming everything else the same, the difference of the dependence can result in in a situation where one aggregated risk has higher VaR and lower TVaR compared to the other one. That is, we cannot predict a higher TVaR with a higher VaR, or vice versa. (See equations (1.6), (1.7), (1.8), (1.9)).

1.3.2 Example 2

Assume risks X and Y have the following joint distribution: Pr(X = 1, Y = 0) = 0.04, Pr(X = 0, Y = 1) = 0.04,

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Pr(X = 0, Y = 0) = 0.92. From simple computation, we have:

Pr(X = 1) = Pr(Y = 1) = 0.04, Pr(X = 0) = Pr(Y = 0) = 0.96,

Pr(X + Y = 1) = 0.08, Pr(X + Y = 0) = 0.92.

Then, calculating the VaR at 95% level, we have: VaR[X; 0.95] = VaR[Y ; 0.95] = 0 and VaR[X + Y ; 0.95] = 1. So, we have

VaR[X + Y ; 0.95] > VaR[X; 0.95] + VaR[Y ; 0.95]. (1.14) It looks like the aggregation increases the riskiness of two risks. However, the problem is caused by the risk measure VaR, since it does not satisfy the subadditivity. This example shows the possibility that two risks that seems to have very low riskiness aggregate into a risk that seems to have very high riskiness. In fact, risk X and risk Y have a high riskiness. They both have a probability of 0.04 to be 1. However, the risk measure VaR neglects it totally because of the given confidence level 95%. After the aggregation, the risk measure VaR finally senses the riskiness because the probability that X + Y equals to 1 is 0.08, big enough for the 95% level. But for TVaR, this problem never happens because TVaR is subadditive.

Also note that, VaR[X + Y ; 0.9199] = 0, VaR[X + Y ; 0.9201] = 1. It means that a small change in the confidence level may result in a totally different VaR. It makes VaR very dangerous from a practical point of view. The difference between the level 91.99% and 92.01% is not significant at all, while the difference between 0 and 1 can never be ignored. Because VaR is not continuous with respect to the confidence level, VaR may enormously underestimate the risk.

So, it is not sufficient to get more details of the aggregated risk if we only know the marginal distributions and the VaR of the aggregated risk. Their TVaR, one way to describe the tail, may differ from each other on a large scale. VaR does have some advantages over TVaR, so we cannot simply choose TVaR as the better alternative. And similarly, if we use TVaR as the risk measure only, two aggregated risks may have the same TVaR but different VaR.

However, using the one-dimensional risk measure VaR is quite questionable, because it neglects a lot of other important information about a risk. The combination of VaR and TVaR may outperform solely using VaR. For some marginal distributions, no matter how they depend on each other, the VaR and TVaR of the aggregated risk may have a relatively fixed relation, that is, a larger VaR may generally imply a larger TVaR. In this situation, combining with TVaR would not improve the performance of VaR much, so it is acceptable to use VaR only. But for some marginal distributions, the situation is totally diverse, and the use of a combination may be very important, even though the combination requires a lot of extra work.

1.4 Literature study

Not many works on combining VaR and TVaR have been done previously, but there are some works quite related to this topic.

Gorvett and Kinsey (2006) have done some study on a two-dimensional risk mea-sure. They introduced a two-dimensional risk measure which is based on the relationship between both upside and downside conditional expected values. For a particular distri-bution, and with respect to a selected threshold, the conditional expected downside and

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upside values are calculated, and also the difference between the values. This difference is calculated for all possible thresholds. And they find the threshold where the distance between upside and downside is minimized. Then, this threshold value, along with the corresponding difference, represents the two dimensions of the risk measure they in-troduced. In comparing various distributions, it is preferred to have a higher optimal threshold value for a given value of minimum spread, or a smaller spread for a given threshold. They have also introduced the concept of iso-risk curves.

This two-dimensional risk measure is not similar to the combination of VaR and TVaR, however, it is shown that it is feasible to measure the risk with a two-dimensional risk measure.

Another relevant study is done by Yamai and Yoshiba (2005). They focus on com-paring VaR and TVaR. They optimize the utility function of a rational investor with the restriction of VaR or TVaR to get two portfolios. They conclude from the results that the optimal portfolio obtained with the restriction of VaR is vulnerable because of the larger losses under the conditions beyond the VaR level. But at the same time, with the same size of data, VaR approach has a smaller estimation error. Each of the two risk measures has its own advantage and disadvantage and thus, they suggest that a single risk measure cannot dominate the risk management and complementing VaR with TVaR represents an effective way to provide more comprehensive risk monitoring. They suggested the possibility of not using VaR as a single risk measure, but did not mention how to measure the improvement after introducing the TVaR to work with VaR as the risk measure.

1.5 Research question

After observing the potential effects of risk aggregation on VaR and TVaR, and reading the relevant literature, the following research question arises:

What kind of marginal distributions will require the combination of VaR and TVaR other than solely using VaR as the risk measure to measure the aggregated risks, because the combination of VaR and TVaR outperforms VaR only?

This question has some practical implications. Financial institutions usually have a good knowledge of a single risk. They may have a big set of historical data. They always have a good estimation of the distribution of the historical data, and sometimes even fit the historical data with a well known distribution, for instance, normal distribution. However, it is not that easy to estimate the joint distribution of a few risks at the same time, nor the distribution of the aggregated risk, because it is very difficult to measure and model the dependence structures properly. The correlation coefficient is widely used but other dependence measures have also been introduced because some problems of correlation coefficient have been detected. However, the VaR and TVaR of the aggregated risks can be still calculated through the historical data. If we solve the research problem, then the institutions will know whether they need to estimate the TVaR or not for the marginal distributions they are interested in. Note that estimating TVaR in extra might be both time and money consuming. TVaR is the average of the tail of a distribution while VaR is the quantile of the distribution, so, estimating TVaR is usually more costly. So, the institutions can save a lot if we find out there is no need to combine VaR with TVaR for the marginal distributions they are interested in.

A subquestion related to the research question is, how to classify different risk distri-butions? It is not possible to study all the marginal distributions. Financial institutions are worried about bankruptcy, and the cause of the bankruptcy are usually extremely large losses, or in a mathematical way, the tail of the loss distribution. So, classifying the distributions by their thickness of tail is quite reasonable. A thicker tail implies a higher risk of very big losses. So, I will select a light-tailed as well as a thick-tailed distribution as marginal distributions and study the outcomes brought by the combinations of these

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two marginal distributions.

Another problem is how the marginal distributions depend on each other. A few copula models will be selected to model the dependence. Even though it is not possible to include all the copula models, I will try to include some different kinds of copulas with various properties. Besides some copula models that are commonly used, I also select two copula models that are closely connected to the dependence structure of Example 1 in Section 1.3. With these two copula models, I can replicate the problems brought by the special dependence structure in that example, even if the marginal distributions are different from uniform distribution.

Solving the problem with a theoretical approach is very difficult and for most pur-poses not necessary. For a given pair of marginal distributions and their dependence structure, I will use the Monte-Carlo simulation approach to generate enough scenarios, in order to calculate the VaR and TVaR. A fairly accurate result can be obtained as long as I generate enough scenarios of simulation.

To calculate the VaR and TVaR, we should also decide what confidence levels we will consider. The choice should consider the common choice. And I will analyze the different results brought by the different confident levels.

The final step of this thesis is to find a way to compare the performance of VaR and VaR & TVaR combined. Using utility functions is considered. However, VaR is one-dimensional while VaR & TVaR is two-one-dimensional, so it is not convincing to directly compare them through the utility function. So, I will focus on the numerical relation between VaR and TVaR. If VaR and TVaR tend to have a claer relation, the prediction of TVaR by VaR will be relatively precise, then the combination of VaR and TVaR will not outperform solely using VaR too much.

The rest of the thesis is structured as follows. In Chapter 2, some basic definitions will be reviewed. In Chapter 3, the methodology to obtain the results will be introduced. In Chapter 4, the simulation results will be presented and analyzed. In Chapter 5, the final conclusions will be made and some possible improvements will be discussed.

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Definitions

In this chapter, I will introduce all the relevant definitions I will use throughout the process of answering the research question.

2.1 Value-at-Risk and Tail Value-at-Risk

2.1.1 Definition of VaR

The definition of VaR is

VaR[X; p] := F−1(p) := inf{x : F (x) ≥ p}. (2.1) Here F (x) is the cumulative distribution function of risk X. And the Value-at-Risk is a function of confidence level p.

The use of the cumulative distribution function is to manifest the intuitive meaning of the quantile. The infimum here is needed for the cases when no unique value for the inverse distribution function exists for the given p.

2.1.2 Definition of TVaR

The definition of TVaR is

TVaR[X; p] := 1 1 − p

Z 1

p

VaR[X; t]dt (2.2)

In this definition, X is the risk and p is the confidence level. Also, the Tail Value-at-Risk is a function of confidence level p. It can be interpreted as the average of all the VaR beyond the given level p.

Kaas et al. (2008) can be consulted for more details of VaR and TVaR.

2.2 Copula

2.2.1 Definition of copula

The copula of (X, Y ) with marginal cdfs (cumulative distribution functions) F (x) and G(y) is the joint cdf of the ranks F (X) and G(Y ).

Therefore

C(u, v) := Pr[F (X) ≤ u, G(Y ) ≤ v]. (2.3) Furthermore, if the cdf of (X, Y ) is H(x, y), we have:

C(u, v) := H(F−1(u), G−1(y)). (2.4)

Kaas et al. (2008) gives more details and some examples of copula.

In the rest of this section, I will introduce all the copulas I will use in my work. 7

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2.2.2 Gaussian copula

Let (X, Y ) follow a bivariate normal distribution with correlation coefficient r(X, Y ) = r, then for any choice of marginal cdfs F and G, (F−1(Φ(X; µX, σ_X2)), G−1(Φ(Y ; µY, σ2_Y)))

has a Gaussian copula (or normal copula) and marginal distributions F and G.

Note that r is a parameter that we can change in order to obtain a different depen-dence structure between two risks. And −1 ≤ r ≤ 1.

The idea of this copula is to assign the dependence structure of a bivariate normal distribution to any bivariate joint distribution. And bivariate normal distribution is a distribution that almost everybody who works in the financial industry knows. Therefore Gaussian copula is easy to understand and has some intuitive meanings.

2.2.3 Farlie-Gumbel-Morgenstern copula

The FGM copula is:

C(u, v) = uv[1 + α(1 − u)(1 − v)]. (2.5) We have u, v ∈ (0, 1). And −1 ≤ α ≤ 1 is the range of the parameter.

Similarly, we can get a different dependence structure by changing the parameter of the FGM copula.

The advantage of the FGM copula is that we can define the copula with a closed-form formula. Also, there is a parameter that we can change in order to obtain a different dependence structure.

2.2.4 Monotonic-Countermonotonic copula

The copula in this section is based on the distribution of (X, Y ) of Example 1 in Section 1.3.

For risks X and Y with any marginal distributions F (x) and G(y), let U = F (X) and V = G(Y ), then both U and V follow the uniform distribution on [0,1].

If U and V have the following relation:

V =      U if U < a; 1 + a − U else (2.6)

then (X, Y ) has a “monotonic-countermonotonic copula” (MCM copula). Here, a is the parameter of this type of copula. And a ∈ [0, 1].

Let C(u, v) be the cdf of (U, V ). If u ≤ v and u < a, then

Pr(U < u, V < v) = Pr(V < v | U < u)Pr(U < u) = Pr(U < u)Pr(U < v | U < u) = u × 1

= u. Similarly, if v < u and v < a, then

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If a ≤ v ≤ u, then

Pr(U < u, V < v) = Pr(U < a, V < a) + Pr(U < a, a ≤ V < v)

+ Pr(a ≤ U < u, V < a) + Pr(a ≤ U < u, a ≤ V < v) = a + 0 + 0 + Pr(a ≤ U < u)Pr(a ≤ V < v | a ≤ U < u) = a + Pr(a ≤ U < u)Pr(a ≤ 1 + a − U < v | a ≤ U < u) = a + Pr(a ≤ U < u)Pr(1 + a − v < U ≤ 1 | a ≤ U < u) = a + (u − a)max{0,u + v − 1 − a u − a } = max{a, u + v − 1}.

And the same holds for a ≤ u < v. So, we have C(u, v) =      max{a, u + v − 1} if u ≥ a, v ≥ a; min{u, v} else (2.7)

If we assume X and Y both follow the uniform distribution on [0,1], and set a = 0.9, we will observe exactly the situation we found in Example 1. That is, the aggregated risk X + Y may have higher TVaR but lower VaR compared to the aggregated risk X + X, for some confidence levels, or vice versa. I expect that under some conditions, similar situations will happen even if X and Y do not follow the uniform distribution and the parameter changes. This is not a favorable property of a joint distribution for financial institutions if they want to simplify the process of risk measure. So, this copula can be regarded as an extreme example of copula that generates some uncommon situations. These situations are not likely to happen in reality, but considering them will help us to have a more complete perspective of all the possible outcomes. We will be able to observe how severe the potential “problem” is if we include this copula when studying a certain pair of marginal distributions.

And based on the dependence structure of this copula, symmetrically, we can define the following copula.

2.2.5 Countermonotonic-Monotonic copula

For risks X and Y with any marginal distributions F (x) and G(y), let U = F (X) and V = G(Y ), then both U and V follow the uniform distribution on [0,1].

If U and V have the following relation:

V =      U if U > b; b − U else (2.8)

then (X, Y ) has a “countermonotonic-monotonic copula” (CMM copula). Also, b is the parameter. And b ∈ [0, 1].

Let C(u, v) the cdf of (U, V ). If v ≤ b and u ≤ b, then Pr(U < u, V < v) = Pr(U < u, b − U < v) = max{0, u + v − b}. If u ≤ b < v then Pr(U < u, V < v) = Pr(U < u, V < b) + Pr(U < u, V ≥ b) = u + b − b + 0 = u.

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Similarly, if v ≤ b < u then

Pr(U < u, V < v) = v. If b < v ≤ u, then

Pr(U < u, V < v) = Pr(U ≤ b, V ≤ b) + Pr(U ≤ b, b < V < v)

+ Pr(b < U < u, V ≤ b) + Pr(b < U < u, b < V < v) = b + 0 + 0 + Pr(b < V < u, b < V < v) = b + v − b = v. Similarly, if b < u < v then Pr(U < u, V < v) = u. So, we have C(u, v) =      max{0, u + v − b} if u ≤ b, v ≤ b; min{u, v} else (2.9)

The copula introduced in this section is quite similar to the copula in the previous section, so it is reasonable to expect that the VaR and TVaR of the aggregated risk with marginal distributions having a “countermonotonic-monotonic” copula do not have a monotonic relation. Also, this copula is an extreme example of copula that reminds us it is possible for us to face some uncommon situations if the dependence structure is uncommon. Taking it into account also completes our perspective. By comparing this copula and the one in the previous section with the “usual” copulas, we can see how different the situations can be if we do not expect the copula to be extreme.

2.3 Probability distributions

In this section, I will include two probability distributions I will use as the marginal distributions.

2.3.1 Exponential distribution

The probability density function (pdf) of the exponential distribution is f (x; β) = 1

βe

−1

βx _(2.10)

for x ≥ 0. And β > 0.

With some simple calculation, we know that the mean of the exponential distribution is the parameter β.

This distribution is selected because it is a simple and widely used distribution. And it is a representative of light-tailed distributions. Also, the parameter of this distribution is easy to interpret.

2.3.2 Pareto distribution

The pdf of the Pareto distribution is

f (x; A, r) = rA

r

xr+1 (2.11)

for x ≥ A. And r > 0, A > 0.

This distribution is selected because it is a good example of a heavy-tailed distribu-tion. It is easy to understand and the moments (if exist) are easy to calculate.

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2.4 Aggregated risk

Here, I give a clear explanation of “aggregated risk”. Risk Z is the aggregated risk of risk X and risk Y if

Z = X + Y.

In the following chapters, I will assign some distributions to the risk X and risk Y , which are the marginal distributions of aggregated risk Z. And the dependence structure of risk X and risk Y can be interpreted as the dependence structure of the aggregated risk Z.

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Chapter 3

Methodology

In this chapter, I will explain the process to obtain the desired results.

Generally, I need to obtain a series of VaR and TVaR for a given pair of marginal distributions. I can get different VaR and TVar by changing the copula models and their parameters as well. Also, I need to change the marginal distributions, because I want to investigate the relations between the thickness of tail of marginal distributions and the outcomes of VaR and TVaR. Also, I want to change the confidence level of VaR and TVaR.

3.1 The choice of marginal distributions

The goal of this thesis is to classify aggregated risks into two different kinds. I want to test whether it is suitable to measure the risk of a given pair of marginal distributions with VaR only, or it is better to measure together with TVaR.

It is not possible to test all the marginal distributions, but we can classify the marginal distributions. The tail thickness of the distribution is a very important property for a distribution, and TVaR is a risk measure that focuses on the tail of the distribution of a risk. So, it is reasonable to classify the marginal distributions by the thickness of the tail.

Here, I choose the exponential distribution as the representative of light-tailed dis-tributions, and Pareto distribution as the representative of heavy-tailed distributions. All the possible combinations are considered, that is, aggregated risk with marginal dis-tributions light-tailed with light-tailed, heavy-tailed with heavy-tailed and light-tailed with heavy-tailed.

To be more specific, I choose A = 1, r = 3 as the parameters of the Pareto distribu-tion. The choice of A = 1 is mainly based on simplicity. And I choose r = 3 to make sure that the second moment exists, and the tail is thick enough at the same time. Based on the parameters of the Pareto distribution, we know that the mean of the Pareto distri-bution is 1.5. So I choose 1.5 as the parameter of the exponential distridistri-bution to make its mean equal to the mean of the Pareto distribution. To control the mean is equiva-lent to making sure that the two distributions have a comparable scale. Otherwise, for example, if the mean of the exponential distribution is too high, both VaR and TVaR of the exponential distribution of a certain confidence level may be much bigger than those of the Pareto distribution so that the property of the exponential distribution will dominate the property of tail of the aggregated risk, regardless of the dependence structure or other factors. With the control, the influence of the scale is insignificant while the shape of the marginal distributions will count.

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3.2 Dependence structure

To obtain the results of the thesis, we want to test whether the relation between VaR and TVaR of the aggregated risk changes too much as the dependence structure of the marginal distributions changes, for a given pair of marginal distributions. So we need different scenarios of dependence structure.

So, in order to get enough scenarios of VaR and their corresponding TVaR, I will use all the four copula models in Section 2.2. For all these three models, we can change the parameter to get a new dependence structure.

After selecting a pair of marginal distributions, I will do the following. For the first copula model, the Gaussian copula, the parameter is the correlation coefficient of the underlying bi-variate normal distribution, r. The range of this parameter is from -1 to 1. I will select 101 scenarios. The minimum is -1, and the maximum is 1, with the step length of 0.02, that is, -1, -0.98, -0.96...0 ...0.9, 0.98, 1. The selection for the parameter α of the Farlie-Gumbel-Morgenstern copula is exactly the same.

For the MCM copula model, the parameter a ranges from 0 to 1. I will also select 101 scenarios, the minimum is 0 and maximum is 1, with the step length 0.01, that is, 0, 0.01, 0.02... 0.99, 1. And the same applies for the parameter b of the CMM copula model.

So, for a given pair of marginal distributions and a given confidence level, I will obtain 404 scenarios of VaR and TVaR in total. The sample is large enough for the following works. And I give the same weight for all the 404 scenarios.

3.3 Confidence level

The choice of the confidence level of VaR and TVaR is based on what kind of risks the financial institutions are focusing on. If the institutions worry about some very extreme risks, 0.995 is a good choice. However, when they care more about some less severe but more frequent risks, 0.95 is another common choice.

In the thesis, I will calculate the VaR and TVaR of both of these confidence levels. The confidence levels may have an effect on the final results, and we will see if there is any effect after we obtain the results.

3.4 Implementation with R code

I will use Monte-Carlo simulation approach to obtain the VaR and TVaR.

The first step will be drawing copula samples with the R code and convert them into the samples of a given aggregated distribution.

For both Gaussian copula and FGM copula, Hofert et al. (2015) have already pre-pared a copula package of R including functions to draw samples of the copula model. So, I only need to set the parameters of the copula. For each copula model and each parameter, I will draw 100,000 copula samples. The samples we get are the ranks of any distributions. So, we can convert them to the value of the random variable following the corresponding marginal distributions (exponential or Pareto distribution) with R code. Here,Dutang et al. (2008) has offered the package including the function of quantile of Pareto distribution. And I add the value of a pair of the values to get a value of the aggregated risk. So, for a given dependence structure and pair of marginal distributions, I will have 100,000 samples of the aggregated risk.

For the MCM and the CMM copula model, we simply need to draw 100,000 samples following the Uniform distribution on [0,1] for every parameter. These are the ranks of the first marginal distribution of the given pair. Based on these samples and the definitions of the copula models, we can obtain the ranks of the second marginal

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distri-14 Shen, Guiyu — Comparing VaR and TVaR

bution. And similarly, we can convert these 100,000 pairs of ranks into the value of the individual risks and hence compute the aggregated risk.

The second step will be to obtain the VaR and TVaR from the 100,000 samples. According to the definition of VaR, we can simply use R’s “quantile” function to compute the 0.95 and the 0.995 quantile of the set of the 100,000 samples to obtain the VaR of a given dependence structure and a given marginal distributions pair. And according to the definition of TVaR, we need to calculate the average of all the simulated aggregate values that are not smaller than the VaR of the corresponding confidence level to obtain the TVaR.

After finishing these two steps, we will obtain a set of 404 VaR and 404 corresponding TVaR for each marginal distribution pair and each confidence level.

3.5 The following steps

After obtaining the 404 scenarios of each pair of marginal distributions, I will find some appropriate methods to analyze the results and compare the different combinations of marginal distributions.

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Results and analysis

4.1 The presentation of the results after implementing R

In this section, I will present all the simulation results of VaR and TVaR with graphs. With the graphs, I will also present a summary of the ordinary linear regression model TVaR = α + βVaR + ε, where α is the intercept, β is the slope, and ε is the error term following normal distribution with zero mean. The explanation for the linear regression model will be provided in Section 4.2.

4.1.1 Results when the marginal distributions are one light-tailed

dis-tribution and one heavy-tailed disdis-tribution

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16 Shen, Guiyu — Comparing VaR and TVaR 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 VaR T V a R

Figure 4.1: The plot of 95% VaR and TVaR when the marginal distributions are expo-nential distribution and Pareto distribution (Normal copula model).

Results of the ordinary linear regression: Call:

lm(formula = LightTailedHeavyTailedNormalCopulaTvar95 ~ LightTailedHeavyTailedNormalCopulaVar95)

Residuals:

Min 1Q Median 3Q Max

-0.21704 -0.11337 -0.01935 0.11375 0.28798 Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) -2.03074 0.14254 -14.25 <2e-16 LightTailedHeavyTailedNormalCopulaVar95 1.74503 0.02663 65.54 <2e-16 (Intercept) *** LightTailedHeavyTailedNormalCopulaVar95 *** ---Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1 Residual standard error: 0.1287 on 99 degrees of freedom Multiple R-squared: 0.9775, Adjusted R-squared: 0.9772 F-statistic: 4295 on 1 and 99 DF, p-value: < 2.2e-16

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5.0 5.1 5.2 5.3 5.4 5.5 6.6 6.8 7.0 7.2 7.4 VaR T V a R

Figure 4.2: The plot of 95% VaR and TVaR when the marginal distributions are expo-nential distribution and Pareto distribution (FGM copula model).

lm(formula = LightTailedHeavyTailedFGMCopulaTvar95 ~ LightTailedHeavyTailedFGMCopulaVar95)

Residuals:

-0.06241 -0.02098 -0.00178 0.01771 0.08997 Coefficients:

(Intercept) 0.23448 0.09501 2.468 0.0153 LightTailedHeavyTailedFGMCopulaVar95 1.28651 0.01796 71.648 <2e-16 (Intercept) * LightTailedHeavyTailedFGMCopulaVar95 *** ---Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1 Residual standard error: 0.03037 on 99 degrees of freedom Multiple R-squared: 0.9811, Adjusted R-squared: 0.9809 F-statistic: 5133 on 1 and 99 DF, p-value: < 2.2e-16

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18 Shen, Guiyu — Comparing VaR and TVaR 5.0 5.5 6.0 6.5 7.0 7.5 8.0 6.5 7.0 7.5 8.0 8.5 9.0 VaR T V a R

Figure 4.3: The plot of 95% VaR and TVaR when the marginal distributions are expo-nential distribution and Pareto distribution (MCM copula model).

lm(formula = LightTailedHeavyTailed3rdCopulaTvar95 ~ LightTailedHeavyTailed3rdCopulaVar95)

Residuals:

-0.63158 -0.07298 -0.03752 -0.00350 1.19000 Coefficients:

(Intercept) 1.47805 0.22824 6.476 3.66e-09 LightTailedHeavyTailed3rdCopulaVar95 1.03704 0.04264 24.321 < 2e-16 (Intercept) *** LightTailedHeavyTailed3rdCopulaVar95 *** ---Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1 Residual standard error: 0.2854 on 99 degrees of freedom Multiple R-squared: 0.8566, Adjusted R-squared: 0.8552 F-statistic: 591.5 on 1 and 99 DF, p-value: < 2.2e-16

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4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 VaR T V a R

Figure 4.4: The plot of 95% VaR and TVaR when the marginal distributions are expo-nential distribution and Pareto distribution (CMM copula model).

lm(formula = LightTailedHeavyTailed4thCopulaTvar95 ~ LightTailedHeavyTailed4thCopulaVar95)

Residuals:

-1.67309 -0.03526 0.00155 0.03970 1.15457 Coefficients:

(Intercept) 4.77571 0.37225 12.83 <2e-16 LightTailedHeavyTailed4thCopulaVar95 0.69267 0.06077 11.40 <2e-16 (Intercept) *** LightTailedHeavyTailed4thCopulaVar95 *** ---Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1 Residual standard error: 0.2437 on 99 degrees of freedom Multiple R-squared: 0.5675, Adjusted R-squared: 0.5631 F-statistic: 129.9 on 1 and 99 DF, p-value: < 2.2e-16

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20 Shen, Guiyu — Comparing VaR and TVaR 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 6.5 7.0 7.5 8.0 8.5 9.0 VaR T V a R

Figure 4.5: The plot of 95% VaR and TVaR when the marginal distributions are expo-nential distribution and Pareto distribution (All the 4 copula models).

lm(formula = LightTailedHeavyTailedTvar95 ~ LightTailedHeavyTailedVar95) Residuals:

-2.39868 -0.18321 -0.07565 0.30440 3.10629 Coefficients:

Estimate Std. Error t value Pr(>|t|) (Intercept) -1.28699 0.20815 -6.183 1.55e-09 *** LightTailedHeavyTailedVar95 1.60840 0.03756 42.818 < 2e-16 ***

---Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1 Residual standard error: 0.4379 on 402 degrees of freedom Multiple R-squared: 0.8202, Adjusted R-squared: 0.8197 F-statistic: 1833 on 1 and 402 DF, p-value: < 2.2e-16

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9 10 11 12 11 12 13 14 15 16 17 VaR T V a R

Figure 4.6: The plot of 99.55% VaR and TVaR when the marginal distributions are exponential distribution and Pareto distribution (Normal copula model).

lm(formula = LightTailedHeavyTailedNormalCopulaTvar995 ~ LightTailedHeavyTailedNormalCopulaVar995)

Residuals:

-0.58120 -0.14276 -0.04506 0.12200 1.10194 Coefficients:

Estimate Std. Error t value

(Intercept) -1.9282 0.1717 -11.23 LightTailedHeavyTailedNormalCopulaVar995 1.4909 0.0175 85.21 Pr(>|t|) (Intercept) <2e-16 *** LightTailedHeavyTailedNormalCopulaVar995 <2e-16 *** ---Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1 Residual standard error: 0.2384 on 99 degrees of freedom Multiple R-squared: 0.9865, Adjusted R-squared: 0.9864 F-statistic: 7260 on 1 and 99 DF, p-value: < 2.2e-16

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22 Shen, Guiyu — Comparing VaR and TVaR 8.6 8.8 9.0 9.2 9.4 9.6 11.0 11.5 12.0 12.5 VaR T V a R

Figure 4.7: The plot of 99.5% VaR and TVaR when the marginal distributions are exponential distribution and Pareto distribution (FGM copula model).

lm(formula = LightTailedHeavyTailedFGMCopulaTvar995 ~ LightTailedHeavyTailedFGMCopulaVar995)

Residuals:

-0.39254 -0.10663 -0.03077 0.07558 0.75588 Coefficients:

(Intercept) 2.15192 0.56226 3.827 0.000227 LightTailedHeavyTailedFGMCopulaVar995 1.03429 0.06105 16.941 < 2e-16 (Intercept) *** LightTailedHeavyTailedFGMCopulaVar995 *** ---Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1 Residual standard error: 0.1843 on 99 degrees of freedom Multiple R-squared: 0.7435, Adjusted R-squared: 0.7409 F-statistic: 287 on 1 and 99 DF, p-value: < 2.2e-16

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9 10 11 12 13 11 12 13 14 15 16 17 VaR T V a R

Figure 4.8: The plot of 99.5% VaR and TVaR when the marginal distributions are exponential distribution and Pareto distribution (MCM copula model).

lm(formula = LightTailedHeavyTailed3rdCopulaTvar995 ~ LightTailedHeavyTailed3rdCopulaVar995)

Residuals:

-0.9344 -0.1448 -0.0460 0.1039 1.5126 Coefficients:

(Intercept) 1.02062 0.27341 3.733 0.000316 LightTailedHeavyTailed3rdCopulaVar995 1.15453 0.02998 38.507 < 2e-16 (Intercept) *** LightTailedHeavyTailed3rdCopulaVar995 *** ---Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1 Residual standard error: 0.276 on 99 degrees of freedom Multiple R-squared: 0.9374, Adjusted R-squared: 0.9368 F-statistic: 1483 on 1 and 99 DF, p-value: < 2.2e-16

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24 Shen, Guiyu — Comparing VaR and TVaR 9 10 11 12 13 11 12 13 14 15 16 17 18 VaR T V a R

Figure 4.9: The plot of 99.5% VaR and TVaR when the marginal distributions are exponential distribution and Pareto distribution (CMM copula model).

lm(formula = LightTailedHeavyTailed4thCopulaTvar995 ~ LightTailedHeavyTailed4thCopulaVar995)

Residuals:

-0.75365 -0.22613 -0.04254 0.18934 0.73066 Coefficients:

(Intercept) -0.34024 0.76589 -0.444 0.658 LightTailedHeavyTailed4thCopulaVar995 1.37301 0.05988 22.929 <2e-16 (Intercept) LightTailedHeavyTailed4thCopulaVar995 *** ---Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1 Residual standard error: 0.2798 on 99 degrees of freedom Multiple R-squared: 0.8415, Adjusted R-squared: 0.8399 F-statistic: 525.8 on 1 and 99 DF, p-value: < 2.2e-16

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9 10 11 12 13 12 14 16 18 VaR T V a R

Figure 4.10: The plot of 99.5% VaR and TVaR when the marginal distributions are exponential distribution and Pareto distribution (All the 4 copula models).

lm(formula = LightTailedHeavyTailedTvar995 ~ LightTailedHeavyTailedVar995) Residuals:

-2.4875 -0.1595 -0.0206 0.1606 1.1479 Coefficients:

Estimate Std. Error t value Pr(>|t|) (Intercept) -2.046820 0.091649 -22.33 <2e-16 *** LightTailedHeavyTailedVar995 1.498983 0.008862 169.15 <2e-16 ***

---Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1 Residual standard error: 0.3104 on 402 degrees of freedom Multiple R-squared: 0.9861, Adjusted R-squared: 0.9861 F-statistic: 2.861e+04 on 1 and 402 DF, p-value: < 2.2e-16

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4.1.2 Results when the marginal distributions are both light-tailed

distributions 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 7 8 9 10 11 12 VaR T V a R

Figure 4.11: The plot of 95% VaR and TVaR when the marginal distributions are both exponential distributions(Normal copula model).

lm(formula = LightTailedNormalCopulaTvar95 ~ LightTailedNormalCopulaVar95) Residuals:

-0.24073 -0.12505 -0.03542 0.12025 0.30999 Coefficients:

Estimate Std. Error t value Pr(>|t|) (Intercept) -1.46898 0.09908 -14.83 <2e-16 *** LightTailedNormalCopulaVar95 1.47522 0.01372 107.50 <2e-16 ***

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6.6 6.8 7.0 7.2 7.4 7.6 8.0 8.5 9.0 9.5 VaR T V a R

Figure 4.12: The plot of 95% VaR and TVaR when the marginal distributions are both exponential distributions (FGM copula model).

lm(formula = LightTailedFGMCopulaTvar95 ~ LightTailedFGMCopulaVar95) Residuals:

-0.069753 -0.022752 -0.002235 0.027175 0.070242 Coefficients:

Estimate Std. Error t value Pr(>|t|) (Intercept) -0.214915 0.068668 -3.13 0.0023 ** LightTailedFGMCopulaVar95 1.274630 0.009652 132.06 <2e-16 ***

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28 Shen, Guiyu — Comparing VaR and TVaR 6 7 8 9 10 11 7 8 9 10 11 12 VaR T V a R

Figure 4.13: The plot of 95% VaR and TVaR when the marginal distributions are both exponential distributions (MCM copula model).

lm(formula = LightTailed3rdCopulaTvar95 ~ LightTailed3rdCopulaVar95) Residuals:

-0.73170 -0.08467 -0.03411 0.01096 1.44944 Coefficients:

Estimate Std. Error t value Pr(>|t|) (Intercept) 1.23581 0.19615 6.30 8.25e-09 *** LightTailed3rdCopulaVar95 1.03871 0.02739 37.92 < 2e-16 ***

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5 6 7 8 9 7 8 9 10 11 12 VaR T V a R

Figure 4.14: The plot of 95% VaR and TVaR when the marginal distributions are both exponential distributions (CMM copula model).

lm(formula = LightTailed4thCopulaTvar95 ~ LightTailed4thCopulaVar95) Residuals:

-3.02006 -0.01857 0.01298 0.03987 2.48311 Coefficients:

Estimate Std. Error t value Pr(>|t|) (Intercept) 6.95193 0.43564 15.96 <2e-16 *** LightTailed4thCopulaVar95 0.56019 0.04947 11.32 <2e-16 ***

---Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1 Residual standard error: 0.4725 on 99 degrees of freedom Multiple R-squared: 0.5643, Adjusted R-squared: 0.5599 F-statistic: 128.2 on 1 and 99 DF, p-value: < 2.2e-16

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30 Shen, Guiyu — Comparing VaR and TVaR 5 6 7 8 9 10 11 7 8 9 10 11 12 VaR T V a R

Figure 4.15: The plot of 95% VaR and TVaR when the marginal distributions are both exponential distributions (All the 4 copula models).

lm(formula = LightTailedTvar95 ~ LightTailedVar95) Residuals:

-2.3263 -0.2220 -0.1295 0.3918 6.4037 Coefficients:

Estimate Std. Error t value Pr(>|t|) (Intercept) -0.44814 0.20221 -2.216 0.0272 * LightTailedVar95 1.33496 0.02658 50.232 <2e-16 ***

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9 10 11 12 13 14 15 16 12 14 16 18 VaR T V a R

Figure 4.16: The plot of 99.5% VaR and TVaR when the marginal distributions are both exponential distributions (Normal copula model).

lm(formula = LightTailedNormalCopulaTvar995 ~ LightTailedNormalCopulaVar995) Residuals:

-0.33265 -0.10578 0.01116 0.09055 0.30849 Coefficients:

Estimate Std. Error t value Pr(>|t|) (Intercept) -0.662108 0.068990 -9.597 8.28e-16 *** LightTailedNormalCopulaVar995 1.222651 0.005855 208.837 < 2e-16 ***

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32 Shen, Guiyu — Comparing VaR and TVaR 10.0 10.5 11.0 11.5 12.0 11.5 12.0 12.5 13.0 13.5 VaR T V a R

Figure 4.17: The plot of 99.5% VaR and TVaR when the marginal distributions are both exponential distributions (FGM copula model).

lm(formula = LightTailedFGMCopulaTvar995 ~ LightTailedFGMCopulaVar995) Residuals:

-0.199093 -0.046407 0.008225 0.059302 0.201844 Coefficients:

Estimate Std. Error t value Pr(>|t|) (Intercept) 0.75729 0.15934 4.753 6.81e-06 *** LightTailedFGMCopulaVar995 1.08118 0.01437 75.246 < 2e-16 ***

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10 12 14 16 12 14 16 18 VaR T V a R

Figure 4.18: The plot of 99.5% VaR and TVaR when the marginal distributions are both exponential distributions (MCM copula model).

lm(formula = LightTailed3rdCopulaTvar995 ~ LightTailed3rdCopulaVar995) Residuals:

-0.34316 -0.05709 -0.01448 0.04201 1.43009 Coefficients:

Estimate Std. Error t value Pr(>|t|) (Intercept) 1.3027 0.1180 11.04 <2e-16 *** LightTailed3rdCopulaVar995 1.0202 0.0111 91.88 <2e-16 ***

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34 Shen, Guiyu — Comparing VaR and TVaR 10 12 14 16 12 14 16 18 VaR T V a R

Figure 4.19: The plot of 99.5% VaR and TVaR when the marginal distributions are both exponential distributions (CMM copula model).

lm(formula = LightTailed4thCopulaTvar995 ~ LightTailed4thCopulaVar995) Residuals:

-0.34584 -0.08906 -0.02644 0.08629 0.38325 Coefficients:

(Intercept) 0.03365 0.30526 0.11 0.912

LightTailed4thCopulaVar995 1.18653 0.01924 61.67 <2e-16 ***

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10 12 14 16 12 14 16 18 VaR T V a R

Figure 4.20: The plot of 99.5% VaR and TVaR when the marginal distributions are both exponential distributions (All the 4 copula models).

lm(formula = LightTailedTvar995 ~ LightTailedVar995) Residuals:

-1.70213 -0.08906 0.02750 0.12250 0.50913 Coefficients:

Estimate Std. Error t value Pr(>|t|) (Intercept) -0.885581 0.057770 -15.33 <2e-16 *** LightTailedVar995 1.237032 0.004616 267.99 <2e-16 ***

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4.1.3 Results when the marginal distributions are both heavy-tailed

distributions 2.4 2.6 2.8 3.0 3.2 3.4 4.0 4.5 5.0 5.5 6.0 VaR T V a R

Figure 4.21: The plot of 95% VaR and TVaR when the marginal distributions are both Pareto distributions(Normal copula model).

lm(formula = HeavyTailedNormalCopulaTvar95 ~ HeavyTailedNormalCopulaVar95) Residuals:

-0.25704 -0.09535 -0.03750 0.10738 0.31377 Coefficients:

Estimate Std. Error t value Pr(>|t|) (Intercept) -0.56108 0.11882 -4.722 7.7e-06 *** HeavyTailedNormalCopulaVar95 1.86924 0.04102 45.569 < 2e-16 ***

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2.7 2.8 2.9 3.0 4.3 4.4 4.5 4.6 4.7 4.8 4.9 VaR T V a R

Figure 4.22: The plot of 95% VaR and TVaR when the marginal distributions are both Pareto distributions (FGM copula model).

lm(formula = HeavyTailedFGMCopulaTvar95 ~ HeavyTailedFGMCopulaVar95) Residuals:

-0.07412 -0.02716 -0.00370 0.02167 0.10994 Coefficients:

Estimate Std. Error t value Pr(>|t|) (Intercept) 0.92608 0.09595 9.651 6.31e-16 *** HeavyTailedFGMCopulaVar95 1.29592 0.03345 38.741 < 2e-16 ***

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Figure 4.23: The plot of 95% VaR and TVaR when the marginal distributions are both Pareto distributions (MCM copula model).

lm(formula = HeavyTailed3rdCopulaTvar95 ~ HeavyTailed3rdCopulaVar95) Residuals:

-0.49999 -0.06943 -0.03835 0.00322 1.00302 Coefficients:

Estimate Std. Error t value Pr(>|t|) (Intercept) 1.64124 0.14595 11.24 <2e-16 *** HeavyTailed3rdCopulaVar95 1.03274 0.05007 20.62 <2e-16 ***

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2.0 2.5 3.0 3.5 4.5 5.0 5.5 6.0 VaR T V a R

Figure 4.24: The plot of 95% VaR and TVaR when the marginal distributions are both Pareto distributions (CMM copula model).

lm(formula = HeavyTailed4thCopulaTvar95 ~ HeavyTailed4thCopulaVar95) Residuals:

-1.37425 -0.03476 0.00716 0.05081 0.99288 Coefficients:

Estimate Std. Error t value Pr(>|t|) (Intercept) 4.1068 0.2181 18.829 < 2e-16 *** HeavyTailed4thCopulaVar95 0.5958 0.0648 9.193 6.3e-15 ***

---Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1 Residual standard error: 0.2192 on 99 degrees of freedom Multiple R-squared: 0.4605, Adjusted R-squared: 0.4551 F-statistic: 84.51 on 1 and 99 DF, p-value: 6.298e-15

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40 Shen, Guiyu — Comparing VaR and TVaR 2.0 2.5 3.0 3.5 4.0 4.5 4.0 4.5 5.0 5.5 6.0 VaR T V a R

Figure 4.25: The plot of 95% VaR and TVaR when the marginal distributions are both Pareto distributions (All the 4 copula models).

lm(formula = HeavyTailedTvar95 ~ HeavyTailedVar95) Residuals:

-1.9023 -0.2005 -0.1001 0.3204 3.1535 Coefficients:

Estimate Std. Error t value Pr(>|t|) (Intercept) 0.18679 0.15457 1.208 0.228 HeavyTailedVar95 1.62298 0.05121 31.693 <2e-16 ***

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6.5 7.0 7.5 8.0 8.5 9.0 9.5 10 11 12 13 14 15 16 VaR T V a R

Figure 4.26: The plot of 99.5% VaR and TVaR when the marginal distributions are both Pareto distributions (Normal copula model).

lm(formula = HeavyTailedNormalCopulaTvar995 ~ HeavyTailedNormalCopulaVar995) Residuals:

-0.82793 -0.26122 -0.01778 0.18095 1.06783 Coefficients:

(Intercept) 0.10549 0.26596 0.397 0.692

HeavyTailedNormalCopulaVar995 1.54600 0.03581 43.166 <2e-16 ***

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Figure 4.27: The plot of 99.5% VaR and TVaR when the marginal distributions are both Pareto distributions (FGM copula model).

lm(formula = HeavyTailedFGMCopulaTvar995 ~ HeavyTailedFGMCopulaVar995) Residuals:

-0.50558 -0.15751 -0.01055 0.17734 0.84285 Coefficients:

Estimate Std. Error t value Pr(>|t|) (Intercept) 4.3574 0.7714 5.649 1.55e-07 *** HeavyTailedFGMCopulaVar995 0.9064 0.1111 8.156 1.11e-12 ***

---Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1 Residual standard error: 0.2571 on 99 degrees of freedom Multiple R-squared: 0.4019, Adjusted R-squared: 0.3959 F-statistic: 66.53 on 1 and 99 DF, p-value: 1.109e-12

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7 8 9 10 10 11 12 13 14 15 VaR T V a R

Figure 4.28: The plot of 99.5% VaR and TVaR when the marginal distributions are both Pareto distributions (MCM copula model).

lm(formula = HeavyTailed3rdCopulaTvar995 ~ HeavyTailed3rdCopulaVar995) Residuals:

-0.68879 -0.24881 -0.04564 0.17979 1.98233 Coefficients:

Estimate Std. Error t value Pr(>|t|) (Intercept) 3.12706 0.37936 8.243 7.22e-13 *** HeavyTailed3rdCopulaVar995 1.08900 0.05489 19.839 < 2e-16 ***

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44 Shen, Guiyu — Comparing VaR and TVaR 7 8 9 10 11 12 13 14 15 16 17 VaR T V a R

Figure 4.29: The plot of 99.5% VaR and TVaR when the marginal distributions are both Pareto distributions (CMM copula model).

lm(formula = HeavyTailed4thCopulaTvar995 ~ HeavyTailed4thCopulaVar995) Residuals:

-1.18162 -0.33545 -0.08761 0.27232 1.05173 Coefficients:

(Intercept) 1.7644 1.1000 1.604 0.112

HeavyTailed4thCopulaVar995 1.4235 0.1134 12.558 <2e-16 ***

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7 8 9 10 10 12 14 16 VaR T V a R

Figure 4.30: The plot of 99.5% VaR and TVaR when the marginal distributions are both Pareto distributions (All the 4 copula models).

lm(formula = HeavyTailedTvar995 ~ HeavyTailedVar995) Residuals:

-2.87893 -0.24937 -0.04327 0.26071 1.29012 Coefficients:

Estimate Std. Error t value Pr(>|t|) (Intercept) -0.83122 0.13179 -6.307 7.5e-10 *** HeavyTailedVar995 1.67273 0.01683 99.390 < 2e-16 ***

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4.2 Results explanations

In this section, I will explain and analyze the results presented in the previous section in a few different aspects.

4.2.1 The choice of ordinary linear regression model

After obtaining the results of VaR and TVaR, I use the ordinary linear model (OLM) to explain the relation between VaR and TVaR. Here, I list the reasons of choosing OLM. First, after observing the graphs of the results, I found that at least for the FGM copula, it is quite appropriate to choose OLM to explain the relation between VaR and TVaR. For example, we can see a very clear linear relation in Figure 4.2.

Second, for normal copula, it is also appropriate to use OLM for most of the cases, such as Figure4.11, even though in some graphs (such as Figure4.21) there is obviously a curve instead of a straight line.

Third, for the MCM copula model, there is also an obvious linear relation for most of the pairs of VaR and TVaR, except for a few outliers. Figure4.13is a good example. And it is still not easy to take into account these outliers with a more complicated regression model. OLM is good enough.

Fourth, for the CMM copula model, we can see most of the dots concentrate and there are a few outliers. For example, Figure 4.14. So, even though OLM does not give a good explanation, it can be expected that other regression models will not improve a lot because of the special concentration. I do not intend to explain this copula model with any regression models, and presenting the results of OLM is just to make the presentation more complete.

Fifth, in some of the plots of the overall results for all the four copula models, such as Figure4.25we can clearly observe two different (quasi-) straight-line trends. And the slopes of these two lines are both positive. Because of this character, even though OLM does not explain the VaR-TVaR relationship well enough, a more complicated model, such as explaining with a parabola, does not improve too much.

Lastly and most importantly, the purpose of this thesis is to find out whether the VaR is good enough to measure a risk. If we find out that for some marginal distributions, we can predict the TVaR fairly well, then we will conclude that for these marginal distributions, VaR is a very good risk measure because it not only tells us the size of the risk for a certain probability level, but also gives us a good prediction of the average size of the risk for all the levels beyond that probability level. OLM is a very common model used for predicting. More sophisticated regression models may increase the accuracy, but as I have discussed, it will not improve too much in most cases. And these models may be confusing and harder to explain in an intuitive way.

4.2.2 Interpreting the results of OLM

From the summary of the OLM models, we know a lot of information about the re-lation between VaR and TVaR. I will explain these results and try to have a better understanding of the relation.

F -statistics and t-statistics

In the univariate OLM model of VaR and TVaR, both of these two statistics are used to verify that VaR and TVaR are not independent. It is shown that in all the 30 cases, it is statistically significant (99.9%) that VaR and TVaR are not independent. Considering that the coefficients of the variable VaR are always positive, we can conclude that a larger VaR implies a TVaR on average, regardless of the marginal distributions, the copula models or the confidence levels of VaR.

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The results are reasonable. According to the definitions of VaR and TVaR, we know that TVaR is always larger than the corresponding VaR. So, a larger VaR represents a larger lower bound of TVaR.

However, because the statistical significance always holds, we cannot use the F -statistics or the t--statistics to compare different marginal distributions and make further conclusions.

R2

The R2 is a number to indicate how well the data fit the regression model. In other words, the regression model will be a good model of predicting if the R2 is large. To be more specific, the regression model in this thesis is about the dependence of VaR and TVaR, if the R2of one of the cases is high, we will be able to use VaR as a good predictor of TVaR in that case. So we can compare the R2 of different cases to investigate the effect of copula models, marginal distributions or confidence levels on the relation of VaR and TVaR.

Fortunately, we can see that the R2s of different cases differ from each other a lot. So, the height of the R2 _{can be a good criterion in the comparison.}

The maxima and minima of residuals

The R2 measures how well the OLM can predict the TVaR from VaR. However, Figure

4.13 tells us even with a few outliers, the R2 can still be very high.

The maxima and minima of residuals is a good reminder of extreme cases and exposes the problems that are not very likely but very lethal once they happen. Figure 4.22is a very good example to compare with Figure4.13. The R2s of these two cases are quite similar, but they are diverse. In Figure4.22, errors are medium-sized but frequent, while in Figure4.13, most of the errors are small but there are a few infrequent but fairly big ones. The maxima and minima of residuals summarize this difference very well.

So, the maxima and minima of the residuals can be a good supplement of the R2. If we really focus on extreme cases (outliers in the OLM model), it is highly recommended to calculate the TVaR as well even if the R2is high and TVaR can be precisely predicted by VaR for most values of the copula parameter.

4.3 Results analysis

In this section, I will analyze the results in a few aspects.

4.3.1 The effect of the thickness of tail

The thickness of tail of the marginal distribution is expected to have a large influence on the accuracy of predicting TVaR with VaR. I summarize the results of R2 in the following two tables.

Table 4.1: The R2 of all the regression models with 95% confidence level of VaR and TVaR

Gaussian FGM MCM CMM All 4 copulas Light-Light 0.9915 0.9944 0.9356 0.5643 0.8626 Light-Heavy 0.9775 0.9811 0.8566 0.5675 0.8202 Heavy-Heavy 0.9545 0.9381 0.8112 0.4605 0.7142

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Table 4.2: The R2 of all the regression models with 99.5% confidence level of VaR and TVaR

Gaussian FGM MCM CMM All 4 copulas Light-Light 0.9977 0.9828 0.9884 0.9746 0.9944 Light-Heavy 0.9865 0.7435 0.9374 0.8415 0.9861 Heavy-Heavy 0.9495 0.4019 0.799 0.6143 0.9609

From these two tables, we can see the clear trend of the decreasing of R2 _{with the}

increasing of the thickness of tail of the marginal distributions. The only exception is in the CMM column of the Table 4.1. As I have discussed , OLM is not very good to explain the relation between VaR and TVaR when the copula is CMM copula.

4.3.2 The effect of the confidence level

The effect of the confidence level is not clear, if we only consider one single copula model. The increase of the confidence level increases the R2 when the copula is Gaussian but decreases the R2 when the copula is FGM.

Nevertheless, the confidence level seems to have a clear influence on the overall results. From the results of R2_{, we can observe the increase of R}2 _{as the confidence level}

increases. But when we go back to the graphs, we can find out more.

When the confidence level is 95%, we can always distinguish two obvious different trends. For example, consider Figure4.5. In fact, there are three trends in this example. The slopes of Figure 4.1, Figure 4.2 and Figure4.3 are 1.745, 1.287 and 1.037 respec-tively. And these three slopes are also the slopes of the three different trends in Figure

4.5.

However, when the confidence level is 99.5%, things are quite different. Only one trend or two slightly different trends can be observed. For example, the slopes of Figure

4.16, Figure 4.17 and Figure 4.18 are 1.22, 1.08 and 1.02 respectively.The differences among these three are much smaller compared to the differences in the 95% confidence level cases.

4.3.3 The effect of copula

In this part, I will analyze the different results produced by the different copulas.

Gaussian copula

Generally speaking, the R2is very high when the copula is Gaussian copula. Also, there are no obvious outliers. I will then use Gaussian copula as the benchmark when I analyze other copulas.

FGM copula

The R2is still very high when the copula is FGM copula and the confidence level is 95%. However, when the confidence level increases to 99.5%, the R2 becomes very sensitive to the thickness of tail. The increase of the thickness of tail makes the prediction much less reliable. However, we can still observe an increasing trend of TVaR as VaR grows, which is also suggested by the F -statistic, even if the R2 is 0.4019. (Figure 4.27) Also, there are not many obvious outliers.

Comparing VaR and TVaR : a simulation-based approach

–A Simulation-Based Approach

Shen, Guiyu

Contents

Preface

Introduction

1.1

Managing risks

1.2

Risk measure

1.3

Risk aggregation and its effect on risk measures

1.4

Literature study

1.5

Research question

Definitions

2.1

Value-at-Risk and Tail Value-at-Risk

2.2

Copula

2.3

Probability distributions

2.4

Aggregated risk

Chapter 3

Methodology

3.1

The choice of marginal distributions

3.2

Dependence structure

3.3

Confidence level

3.4

Implementation with R code

3.5

The following steps

Results and analysis

4.1

The presentation of the results after implementing R

4.2

Results explanations

4.3

Results analysis