Essays on model risk for risk measures

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UNIVERSIDADE FEDERAL DO RIO GRANDE DO SUL

ESCOLA DE ADMINISTRAÇÃO

PROGRAMA DE PÓS-GRADUAÇÃO EM ADMINISTRAÇÃO

Fernanda Maria Müller

Essays on model risk for risk measures

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Fernanda Maria Müller

Essays on model risk for risk measures

Dissertation for Doctoral degree in

Busi-ness Administration at the School of

Ad-ministration of Federal University of Rio

Grande do Sul.

Supervisor: Marcelo Brutti Righi, PhD

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CIP - Catalogação na Publicação

Müller, Fernanda Maria

Essays on model risk for risk measures / Fernanda Maria Müller. -- 2019.

130 f.

Orientador: Marcelo Brutti Righi.

Tese (Doutorado) -- Universidade Federal do Rio Grande do Sul, Escola de Administração, Programa de Pós-Graduação em Administração, Porto Alegre, BR-RS, 2019.

1. Model risk. 2. Risk forecasting. 3. Risk

measures. 4. Capital determination. I. Righi, Marcelo Brutti, orient. II. Título.

Elaborada pelo Sistema de Geração Automática de Ficha Catalográfica da UFRGS com os dados fornecidos pelo(a) autor(a).

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Fernanda Maria Müller

Essays on model risk for risk measures

Dissertation for Doctoral degree in Business Ad-ministration at the School of AdAd-ministration of Federal University of Rio Grande do Sul.

Approved research. Porto Alegre, July 12, 2019:

Marcelo Brutti Righi, PhD Dissertation Advisor

Eduardo Horta, PhD Committee Member

Hudson da Silva Torrent, PhD Committee Member

Márcio Poletti Laurini, PhD Committee Member

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Agradecimentos

Ao meu companheiro Cristian, por todo amor, carinho e incentivo para a conclusão desse trabalho.

Aos meus pais, Nair e Irineu, que sempre me motivaram e me deram apoio.

Ao meu orientador, Professor Dr. Marcelo Brutti Righi, por seu apoio, dedicação, com-petência e atenção nas revisões que foram fundamentais para o desenvolvimento desse trabalho. Ao Programa de Pós-Graduação em Administração e a todos os professores do doutorado que de alguma forma contribuíram para a minha formação.

Aos meus amigos que me incentivaram e torceram pela minha vitória.

À Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) pela bolsa de estudos.

Agradeço também aos membros da banca examinadora, pela disponibilidade em partici-par e pelas contribuições pessoais acerca da tese.

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Essentially, all models are wrong, but some are useful. George Box.

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Abstract

In this dissertation, we present a compilation of three articles discussing model risk for risk measures. In the first one, using Monte Carlo simulation, we analyze the performance of mul-tivariate models in forecasting Value at Risk (VaR), Expected Shortfall (ES), and Expectile Value at Risk (EVaR). In our numerical evaluation, we consider different scenarios, regarding the marginal distributions, correlation and number the assets of the portfolio, and the following models: Historical Simulation (HS), Dynamic Conditional Correlation - Generalized Autore-gressive Conditional Heteroskedastic (DCC - GARCH), Regular copulas, Vine copulas, and Nested Archimedean copulas (NAC). Our results indicate that when the marginal distribution is Gaussian, Regular and Vine copulas demonstrate better performance. On the other hand, for scenarios generated with Student’s t distribution, we observed a better performance of Nested Archimedean copulas. In the second article, we propose a robust risk measurement approach that minimizes the expectation of the sum between costs from overestimation and underestimation. We consider uncertainty by taking the supremum over alternative probability measures. We provide results that guarantee the existence of a solution and explore the properties of minimizer and minimum as risk and deviation measures, respectively. Besides, we explore the use of our loss function as an auxiliary criterion to select risk forecasting models. Additionally, we use our loss function to determine the proportion of model risk that should add to risk measures to cover losses resulting from this risk. Empirical results indicate that our measure leads to more parsimonious capital requirement determination and also that it reduces the mentioned costs. Fur-thermore, the results demonstrate the advantages of our loss function over traditional approaches used in model selection. Finally, in the third article, we review the studies, which propose both model risk measures and alternatives to incorporate model risk in capital determination. The presentation focuses on the procedures, which can be applied in risk forecasting. We observe that the worst case and loss function approach are the main groups of model risk measures. We found two main strategies to incorporate model risk in the capital determination: model risk and backtesting adjusted risk forecasting. Then, we empirically analyze our findings. Moreover, we do also identify some improvements and future directions that can be explored.

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Resumo

Nesta tese, apresentamos uma compilação de três artigos discutindo o risco de modelo para medidas de risco. No primeiro artigo, usando simulação de Monte Carlo, analisamos o desem-penho de modelos multivariados para previsão do Value at Risk (VaR), da Expected Shortfall (ES) e do Expected Value at Risk (EVaR). Em nossa avaliação numérica, consideramos diferentes cenários, quanto às distribuições marginais, correlação e número dos ativos da carteira, e os seguintes modelos: Simulação Histórica (HS), Correlação Condicional Dinâmica - Autorregres-sivo com Heterocedasticidade Condicional Generalizada (DCC -GARCH), cópulas regulares, cópulas Vine e cópulas Nested Archimedean (NAC). Nossos resultados indicam que quando a distribuição marginal é Gaussiana, as cópulas Regular e Vine demonstram melhor desempenho. Por outro lado, para cenários gerados com a distribuição t de Student, observamos melhor desempenho das cópulas Nested Archimedean. No segundo artigo, propomos uma abordagem robusta de mensuração do risco que minimiza o valor esperado da soma entre os custos de superestimação e subestimação do risco. Consideramos a incerteza tomando o supremo sobre medidas de probabilidade alternativas. Fornecemos resultados que garantem a existência de uma solução e analisamos as propriedades do minimizador e do mínimo como medidas de risco e de desvio, respectivamente. Exploramos o uso de nossa função de perda como um critério auxiliar para selecionar modelos de previsão de risco. Além disso, usamos nossa função de perda para determinar a proporção do risco de modelo que deve ser adicionada como penalização às medidas de risco para cobrir as perdas resultantes desse risco. Os resultados empíricos indicam que nossa medida leva a uma determinação do requerimento de capital mais parcimoniosa e reduz os custos mencionados. Além disso, os resultados demonstram vantagens da nossa função de perda em relação às abordagens tradicionais usadas para seleção de modelos de previsão de risco. Finalmente, no terceiro artigo, revisamos a literatura que propõem medidas de risco de modelo e alternativas para incorporar o risco de modelo na determinação de capital. A apresentação centra-se sobre os procedimentos que podem ser aplicados na previsão de risco. Observamos que a abordagem do pior caso e da função de perda são os principais grupos de medidas de risco de modelo. Encontramos duas principais estratégias para incorporar o risco de modelo na determinação de capital: previsões de risco ajustadas ao risco de modelo e previsões de risco ajustadas ao backtesting. Ilustramos empiricamente nossas descobertas. Além disso, apontamos lacunas e direções de trabalhos futuros que podem ser explorados.

Palavras-chaves: Risco de modelo. Previsão de risco. Medidas de risco. Determinação de capital.

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INTRODUCTION

The increasing complexity of financial products has revealed a greater need for financial institutions to use statistical model outputs to assess the risks to which they are exposed. The most common method quantifies the risk, linked to variance, Value at Risk (VaR), and Expected Shortfall (ES) with empirical data distribution. Other approaches often employed are parametric, which includes GARCH (Generalized Autoregressive Conditional Heteroskedasticity) models, and semi-parametric, as Quantile Regression and Filtered Historical Simulation (FHS). The accuracy of the results depends of reliability of the model used. However, models are simplistic representations of reality and inevitably one must acknowledge that there is no perfect model

(seeDanielsson(2008) andFederal Reserve(2011)). The hazard of working with a potentially

incorrect or inadequate model is referred to in the literature as “model risk”. In financial literature,

this risk is investigated in detail by Cont (2006), Kerkhof et al.(2010),Glasserman and Xu

(2014),Barrieu and Scandolo(2015),Bernard and Vanduffel(2015),Danielsson et al.(2016),

andKellner and Rösch(2016).

Studies evaluating model risk are mainly centered around two main streams: i) Empirical evaluation of models for forecasting risk measures, and ii) Approaches to reduce it. Concerning the first stream, and despite the progress made, many studies still focus on the univariate

perspective and use empirical data for their evaluation. We refer toKuester et al.(2006) as an

example of study for the univariate case. However, in the practical sense, the models used for risk management are multivariate. Another limitation of studies is their focus in the investigation of estimation methods for VaR. Although VaR is the most commonly used risk measure, having

good statistical properties, including elicitability1and robustness2, it has a history of criticism

for not being a coherent risk measure and by ignoring potential losses that exceed the quantile of interest.

Given this backdrop, in the first part of this dissertation, we investigate the performance of multivariate models to VaR, ES, and EVaR forecasting. We consider the following models: Historical Simulation, Dynamic Conditional Correlation (DCC) - GARCH, Regular copulas, Vine copulas, and Nested Archimedean copulas (NAC). The evaluation of the performance of the models was done by using Monte Carlo simulations. In each scenario employed, we computed absolute bias (A. Bias), relative bias (R. Bias), and root-mean-square error (RMSE) to identify the model with lower model risk. Pseudo-random samples were generated, and different scenarios considered.

1 _{A functional is elicitable when it is the minimizer of a function score. We suggest}_Ziegel_{(2016) and}_{Acerbi and}

Szekely(2017) for more details.

2 _{A risk measure respects robustness if it is stable at small deviations from the model and in the data theoretical}

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INTRODUCTION 11

Contributions regarding the second stream focus on adjustments to incorporate additional market risk capital into standard risk measures rather than new risk measurement procedure.

Kerkhof et al. (2010), for instance, combine results from a model risk measure belonging

to the worst case approach and a risk measure. Using a different approach, Alexander and

Sarabia(2012) andBoucher et al.(2014) suggest adjustments for VaR estimates using maximum

entropy distribution and backtesting procedures, respectively. Bignozzi and Tsanakas(2016)

andKellner et al.(2016) present other investigations within this context. Despite the progress

made in these proposals, preserving the theoretical properties of the studied risk measures is not guaranteed, especially the fundamental Monotonicity axiom. This contradicts the natural intuition of economic capital regulation, i.e., a position with the highest loss has the highest risk. Additionally, one realizes the absence of approaches that exploit the impacts of risk underestimation and overestimation with the aim of quantifying the amount of capital required for security more accurately.

Based on this perspective, in the second part of this dissertation, we propose a risk measurement procedure, which represents the capital determination for a financial position X that minimizes the expected value of the sum between costs from risk overestimation (opportunity gains) and underestimation (uncovered losses). We measure these two costs in our framework by non-negative random variables G, for gains, and L, for losses, which refer to financial rates traded in the market. We extend our approach to minimization of the sum between costs of risk overestimation and underestimation over a supremum of expectations determined by probability measures. We provide results that guarantee the existence of a solution and explore the properties of minimizer and minimum as risk and deviation measures, respectively. We also suggest the use

of our loss function (X − x)+G+ (X − x)−Las a criterion with practical intuition to compare

risk forecasts. Furthermore, we indicate the use of our loss function to determine additional capital buffer required by regulatory developments to cover the losses from model risk.

Moreover, with the intention of banking institutions to protect themselves from model risk, regulatory developments for the banking sector require that financial institutions quantify and

manage their model risk like other types of risk (seeBasel Committee on Banking Supervision

(2009) andFederal Reserve(2011)). From that moment, there is also a consensus that better

methods for dealing with model risk are pivotal to improving risk management. Understanding and quantifying model risk have become a concern for both banks and regulators. The most

common approach employed to measure model risk is the worst case. We mentionCont(2006),

Kerkhof et al. (2010), and Barrieu and Scandolo (2015) as examples of studies that use it.

Generally, these measures are applied in a set of estimates or forecasts and have a similar structure to deviation measures. Besides this approach, it is also possible to apply risk measures

in a loss or error function, from some estimation or forecasting procedure. We referDetering and

Packham(2016) andMüller and Righi(2018b) as examples of studies that explore this approach.

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INTRODUCTION 12

measures literature. We aim to verify if there is any pattern in the measures used to quantify model risk and unify the procedures for a general approach. We do not explore studies that develop model uncertainty measures in this review since it is not the focus of this study. The only model

uncertainty measure described was proposed byCont(2006) because it can be easily adaptable

to quantify model risk. We also investigate the literature that proposes alternatives to incorporate additional capital buffer required to cover losses from model risk. We consider only works that propose measures and strategies to incorporate model risk in capital determination. Adaptations or extensions are cited where convenient. We conduct an empirical analysis, considering financial data, to illustrate our findings. Furthermore, on the basis of our findings, we highlight some improvements and future directions that can be explored.

Regarding the structure, beyond this initial chapter, which describes an introduction to the subject, as well as the objectives we intend to achieve, this dissertation is organized into three main parts. In summary, we describe three articles that make up each part: Part I, Numerical comparison of multivariate models to forecasting risk measures; Part II, A robust approach for minimization of risk measurement errors; and Part III, Model risk in risk forecasting: Where do we come from and where are we going?. Finally, we summarized the major conclusions and references.

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Part I

NUMERICAL COMPARISON OF

MULTIVARIATE MODELS TO

FORECASTING RISK MEASURES

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INTRODUCTION 14

Abstract

We evaluated the performance of multivariate models for forecasting Value at Risk (VaR), Ex-pected Shortfall (ES), and Expectile Value at Risk (EVaR). We used Historical Simulation (HS), Dynamic Conditional Correlation - Generalized Autoregressive Conditional Heteroskedastic (DCC - GARCH) and copula methods: Regular copulas, Vine copulas, and Nested Archimedean copulas (NAC). We assessed the performance of the models using Monte Carlo simulations, considering different scenarios, regarding the marginal distributions, correlation and number of portfolio assets. Numerical results evidenced the accuracy forecasting risk measures are associ-ated with marginal distributions. For a data generating process where the marginal distribution is Gaussian, Regular and Vine copulas demonstrated better performance. For data generated

with Student0s t distribution, we verified better performance by Nested Archimedean copulas. In

addition, we identified the superiority of copula methods over Historical Simulation and DCC -GARCH, which reduces the model risk.

Keywords: Risk measures. DCC - GARCH. Copulas. Model risk. Monte Carlo simulation.

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15

1 INTRODUCTION

Over the last few years, interest from financial organizations, regulators, and of the academic community in the development of better tools to measure market risk has increased. The main factors contributing to this were the increasing number of collapses and trading activities on the financial markets. In this context, Value at Risk (VaR) has become one of the most popular risk management tools. This measure refers to the maximal loss expected to occur by a financial position for a given period of time and confidence level. For a general review, one

can consult the works ofDuffie and Pan(1997) andJorion(2006). Despite its simplicity and

popularity, VaR has been criticized for presenting theoretical deficiencies as a measure of market

risk. One of the principal weaknesses is it is not a coherent measure of risk1once it does not

meet Sub-additivity. In contrast to the principle of diversification, the VaR of a portfolio is not necessarily smaller than the sum of the VaR for individual assets. This measure ignores potential losses that exceed the quantile of interest.

To circumvent the limitations of VaR,Artzner et al.(1999) presented the concept of Tail

Conditional Expectation (TVaR), known in the literature as Expected Shortfall (ES), a measure

proposed byAcerbi and Tasche(2002). ES is defined as the conditional expectation of a loss,

given this loss exceeds VaR. By its definition, ES considers the losses that exceed the quantile of interest, besides being a coherent risk measure. Another alternative to VaR is to use expectiles,

introduced byNewey and Powell(1987). This measure is referred to as Expectile Value at Risk

(EVaR) and is more sensitive to extreme losses compared to VaR, since the tail probability is determined by the underlying distribution. In addition, it is a coherent risk measure, being the

only example of an elicitable2coherent risk measure beyond the mean. Besides these measures,

in the literature there are other options for quantifying the risk, such as the Entropic, studied by

Föllmer and Schied(2002), and Shortfall Deviation Risk (SDR) proposed byRighi and Ceretta

(2016).

Under the practical relevance of risk measures for financial risk management, there is a need for reliable estimates and forecasts. One of the most commonly used methods is the non-parametric approach, known as Historical Simulation (HS), which is based on the empirical distribution of returns. Parametric methods are also used, such as GARCH (Generalized Autoregressive Conditional Heteroskedastic) models. Regarding the estimation of risk measures for portfolios, difficulties can be observed due to the complexity of modeling the joint distribution of assets. To solve this problem, many authors have proposed the use of copulas. A copula results

1 _{A coherent risk measure is a risk measure that satisfies four axioms: Translation Invariance, Sub-additivity,}

Positive Homogeneity, and Monotonicity (ARTZNER et al.,1999).

2 _{Risk measures are called elicitable when the verification and comparison of competing estimation procedures is}

possible (ZIEGEL,2016). SeeZiegel(2016) andBellini and Di Bernardino(2017) for a detailed discussion

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Chapter 1. INTRODUCTION 16

in a multivariate distribution, combining univariate marginal distributions and dependence

between variables (NELSEN,2006). They allow the aggregation of marginal distributions of two

or more variables and capture important features found in risk management, such as asymmetries

and heavy tails. The mathematical basis of copulas was introduced by Sklar (1959). Their

application in financial practical situations has become clearer more recently. Examples of the

use of bivariate copulas to estimate risk measures are found inMendes and Souza(2004),Junker

and May(2005),Palaro and Hotta(2006), andJäschke(2014).

For the bivariate case, a wide variety of different types of copulas is available, exhibiting flexible and complex dependence patterns. However, for the general multivariate case, the appropriate choice of the copula family is more limited. Standard multivariate copulas do not fit

data well and lack flexibility for modeling the dependence of several assets.Aas et al.(2009)

suggested the use of Vine copulas, originally proposed byJoe(1996), for modeling multivariate

distributions. Vine copulas are a flexible model for describing multivariate relationships, using a

cascade of bivariate copulas, so-called pair-copulas (KUROWICKA,2011). Such pair-copula

constructions (PCCs) decompose a multivariate density into bivariate copula densities, including dependence structures of unconditional bivariate distributions and dependence structures of conditional bivariate distributions. Applications of Vine copula to estimate risk measures are

found in Aas and Berg (2009), Brechmann and Czado(2013), Low et al. (2013), Righi and

Ceretta(2013a),Righi and Ceretta(2013b), andRighi and Ceretta(2015b).

A different structure for building higher-dimensional copula are the Nested Archimedean

copulas (NAC). The hierarchical functional form first appears inJoe(1997). NAC are

generaliza-tions of Archimedean copulas3, allowing asymmetries and providing more flexibility, sharing

properties such as an explicit functional form. Similar to the Vine copulas, NAC are hierarchical

and based on pair-copulas. Applications of NAC to estimate risk measures are found inAas and

Berg(2009).

Amongst a vast range of estimation techniques used in the literature of risk management, a class of studies emerges to perform a comparison among the different estimation methods

of risk measures to determine the most efficient model. We refer to Kuester et al. (2006) as

an example of study for the univariate case. Despite this literature, many studies focus their investigation on estimation methods for VaR. Regarding ES and EVaR, there is not enough research to point out the best estimation method. Most studies use empirical data and focus on the univariate perspective for the comparison of the performance of different models for forecasting risk measures. Multivariate data is a crucial issue in risk management literature and practical situations. An inappropriate model can lead to suboptimal portfolios and inaccurate risk exposure assessments. The Monte Carlo simulation is a computational method utilized to evaluate the performance of competing models. It allows an experiment to be replicated several

3 _{A bivariate distribution function C(u, v) with marginals F}

1and F2is generated by an Archimedean copula if it

can be given by C(u, v) = ϕ−1[ϕ(F1(u)) + ϕ(F2(v))], where ϕ is a function ϕ : I → R∗+, continuous, decreasing,

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times to understand their behaviour. The numerical results obtained from many iterations can be

considered representative of the real situation4.

In this context, the main objective of the current article was to analyse the performance of models for forecasting VaR, ES, and EVaR from the multivariate perspective. We opted for these risk measures because they are the most used in the literature. Regarding multivariate models, we considered the following approaches: HS, Dynamic Conditional Correlation (DCC) - GARCH

(ENGLE,2002), Regular copulas, Vine copulas and NAC5. The performance of the models was

evaluated using Monte Carlo simulations. Pseudo-random samples were generated and different scenarios considered. In each scenario, we computed absolute bias (A. Bias), relative bias (R. Bias), and root-mean-square error (RMSE).

To the best of our knowledge, our paper is the first to perform an evaluation with Monte Carlo simulations for forecasting VaR, ES, and EVaR using multivariate models. Previous studies

using Monte Carlo simulations, such as Manganelli and Engle (2001), have VaR as the first

measure of interest and restrict their analysis to the univariate case.Zhou(2012),Degiannakis

et al. (2013), Tolikas (2014), and Righi and Ceretta (2015a) compared estimators for VaR

and ES; however, they considered empirical data and focus on univariate models; the authors did not provide an extensive focus on VaR and ES forecasting patterns as our study has done.

Regarding EVaR,Bellini and Di Bernardino(2017) performed a review of its known properties,

and financial meaning and compared them with VaR and ES. Their main objective was not to investigate which model offers better performance for computing these risk measures but to

present EVaR as an alternative measure to VaR and ES. In the work ofRighi and Borenstein

(2017), although they considered VaR, ES, and EVaR, their main objective was compare risk

measures regarding performance of optimal portfolio strategies.

Another contribution of our paper is it is the first study to perform a comparison of Historical Simulation, DCC, and copulas to estimate risk measures, in a general multivariate

context. Previous studies, such asWeiß(2013), considered bivariate copula and analyzed the

forecasting of VaR and ES. In addition, they have not used Monte Carlo simulations. As a final

contribution of our paper, it was possible to identify the model that reduces the model risk6of

the risk measures analysed. Appropriate risk-measurement techniques are paramount in finance, once they are used as an input into expensive decisions, such as asset allocation, hedge strategies, and capital requirement. Risk measures estimated with models with a lower model risk are

fundamental to improving risk management and to strengthen global financial stability7.

4 _{We refer to the Law of Large Numbers and Central Limit Theorem (CLT) in which Monte Carlo simulations are}

based for a better understanding.

5 _{This research focused on comparing the performance of traditional multivariate models, such as HS and}

DCC-GARCH, with copulas. We did not use models such as multivariate Quantile Regression and multivariate Extreme Value Theory.

6 _{Model risk refers to errors in modelling assumptions that introduce errors in risk measurement (GLASSERMAN;}

XU,2014).

7 _{Revisions to the Basel II Market Risk Framework require financial institutions quantify model risk (Basel}

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The remainder of this paper is structured as follows: Chapter2presents a background

of the risk measures used in this paper and the estimation methods we considered. In Chapter

3 numerical procedures are described, and in Chapter 4, numerical results are presented and

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2 BACKGROUND

2.1 Risk measures

In this section, we defined VaR, ES, and EVaR. We focused on the continuous distribution case, but generalizations to the discrete case are straightway. X represents the random result of any asset or portfolio, where X ≥ 0 a profit and X < 0 a loss, and α ∈ (0, 1) corresponds to the

significance level. Formally, expression (2.1) defines VaR:

VaRα_{(X ) = −inf{x : F}

X(x) ≥ α} = −F_X−1(α), (2.1)

where FX is the probability function of X and its inverse is F_X−1. Based on formulation (2.1),

we observed VaR does not give information about the severity of losses that occur with low

probability. To outperform VaR drawback,Acerbi and Tasche(2002) suggested the use of ES,

which is defined as:

ESα_{(X ) = −E[X |X ≤ F}−1

X (α)]. (2.2)

This risk measure represents the expected value of a loss, given it is beyond the α-quantile of interest. Nowadays, ES is the most used coherent risk measure. Given the same significance level, ES is higher than VaR. From a risk management point of view, this can lead to higher levels of security. Despite its theoretical advantages ES is estimated with more uncertainty, because it is subject to errors in the estimation of VaR and of the expectation of tail observations. Besides that, ES is not an elicitable risk measure.

Another coherent possibility to measure risk is the EVaR. This risk measure is linked to the concept of expectiles, which is a generalized quantile function employed for obtaining VaR.

Mathematically, EVaR is defined as (2.3):

EVaRα_{(X ) = − arg min}

θ

E[|α − 1X≤θ|(X − θ )2], (2.3)

where 1ais an indicator function with the value 1 if a is true and 0 otherwise.Bellini et al.(2014)

and Bellini and Di Bernardino (2017) argued EVaR is a coherent risk measure for α ≤ 0.5.

Delbaen(2013) described the properties of EVaR as a coherent risk measure. This measure is

also elicitable. Some authors have suggested it as an alternative to both ES and VaR (EMMER et

al.,2015). Although EVaR has good theoretical properties, in literature it is criticized for not

presenting financial interpretation. To circumvent this obstacle,Bellini and Di Bernardino(2017)

discussed its interpretation in relation to acceptance set. In this context, according to EVaR, a position is acceptable when the ratio between the expected value of the gain and of the loss is sufficiently high. According to the authors, the measure presents similarities to VaR and ES, and their numerical results pointed to EVaR as a good alternative to quantify market risk.

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Chapter 2. BACKGROUND 20

2.2 Copulas

To facilitate the notation, we focused on the bivariate case. Extensions to the n-dimensional

case are straightforward. A function C : [0, 1]2→ [0, 1] is a copula, if for 0 ≤ u, v ≤ 1, and

u₁≤ u₂, v₁≤ v₂, (u₁, v₁), (u₂, v₂) ∈ [0, 1]2, it respects the following properties (CHERUBINI et

al.,2004):

1. C(u, 0) = C(0, v) = 0, such that C(u, 1) = u and C(1, v) = v, for every (u, v) of A × B, which represents two non-empty subsets of I = [0, 1] ∈ R. This property means uniformity of the margins.

2. C(u2, v2) −C(u2, v1) −C(u1, v2) +C(u1, v1) ≥ 0, which indicates the function C is called

n- increasing. This property means that P(u1≤ U ≤ u2, v1≤ V ≤ v2) ≥ 0 for (U,V ) with

distribution function C.

Given that C is a copula and F1and F2univariate distribution functions, C is unique if F1

and F2are continuous for each (u, v) ∈ [0, 1]2. A bivariate distribution F with marginals F1and

F₂, is defined by:

F(u, v) = C(F1(u), F2(v)), (u, v) ∈ R2. (2.4)

For a two-dimensional distribution function F with marginal F1and F2, there is a copula

C, which is given by the expression:

C(u, v) = F(F₁−1(u), F₂−1(v)), (2.5)

where F₁−1and F₂−1are the inverse of the marginal distribution functions.

2.2.1 Vine copula

Vine copula represents a flexible and intuitive way of extending bivariate copulas to

higher dimensions (AAS et al.,2009). In the literature, C-vines, R-Vines (BEDFORD; COOKE,

2001;BEDFORD; COOKE,2002), and D-vines (KUROWICKA,2005) have been proposed. In

this work we focused on the D-vine estimation8.

According toAas et al.(2009), D-vine estimation has density c given by:

c(u1, · · · , un) = n

∏

k=1 f_k(u_k) n−1

∏

j=1 n− j

∏

i=1 Ci, j ( F(ui|ui+1, · · · , ui+ j−1), F(ui+ j|ui+1, · · · , ui+ j−1). ) , (2.6)

where, u1, · · · , un = u ∈ [0, 1]n are pseudo-observations; fk is the density function, which is

different for each asset; C(·, ·)i, j is a bivariate copula density, which can be different for each

pair copula.

8 _{To validate the choice of a D-vine, we compared the estimated model with its counterpart through the test}

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The conditional distribution functions are computed with the following equation (JOE,

1996):

F(ui|u) =

∂Cui, uj|u− j{F(ui|u− j), F(uj|u− j)}

∂ F (uj|u− j)

, (2.7)

where i, j ∈ N, i 6= j, Cui, uj|u− j is the dependence structure of uiand ujbivariate conditional

distribution conditioned on u− j (u− jis the vector u excluding the component uj).

2.2.2 Nested Archimedean copulas

We have assumed the reader is familiar with bivariate Archimedean copulas. A good

review can be found inNelsen(2006). Differently from elliptical copulas, Archimedean copulas

can capture different kinds of tail dependences; however, they cannot capture asymmetry. A possibility is to model dependences using a hierarchical structure composed by Archimedean copulas. This hierarchical structure is called Nested Archimedean copulas. Details can be found

inHofert(2008) andMcNeil(2008).

For the n-dimensional case, the corresponding expression becomes:

C(u1, · · · , un) = ϕn−1,1−1 {ϕn−1,1(un) + ϕn−1,1oϕn−2,1−1 {ϕn−2,1(un−1) + ϕn−2,1o· · · o

ϕ₁₁−1{ϕ11(u1) + ϕ11(u2)}}}, (2.8)

Crefers to fully Nested Archimedean copulas, where all bivariate margins are Archimedean

copulas and ϕ corresponding generators.

2.3 DCC - GARCH

This description suggested by Engle (2002) presents a way to model the dynamic

processes of conditional volatilities and correlations simultaneously. Conditional covariance

matrix Ht can be defined as:

Ht= DtRtDt, Dt = diag{

√

σi,t}, (2.9)

where Dt refers to the diagonal matrix of time-varying standard deviations (σ of the asset i

and period t from univariate GARCH models)9. Rt is the correlation matrix containing the

conditional correlations coefficients, which can be defined as:

Rt = Qt∗−1QtQ∗−1t , (2.10)

9 _{The elements of univariate GARCH were obtained through the structure given in (3.1). The model GARCH(P,Q)}

can be described in the following manner: Xt= µt+ σtzt, σt2= ω + ∑Pp=1apεt−p2 + ∑Qq=1bqσt−q2 ; where µt

represents the expectancy, zt is independent and identically distributed (i.i.d.) with zero mean and unit variance,

ω is the constant, a the component ARCH and b represents the component GARCH. One can use other generalizations of the univariate GARCH model.

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where

Q_t= (1 − a + b) ¯Q + aεt−1ε

0

t−1+ bQt−1, (2.11)

where Q_t is the conditional covariance matrix of residuals with unconditional covariance matrix

¯

Q obtained from univariate GARCH models; a and b non-negative scalar parameters satisfying

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23

3 NUMERICAL PROCEDURES

We used Monte Carlo simulations to compare the performance of the models. The number of Monte Carlo replications was set at 10,000. The length of analysed samples was

1,000 observations10. All computational implementations were conducted in the R programming

language (R CORE TEAM,2018). In each replication, we generated one-step-ahead forecast of

risk measures11.

The generating process of the (log)-returns Xi,t, which represent the simulated data, were

drawn from AR(1) - GARCH(1,1) models. This type of specification is commonly used to fit

financial data, because it considers stylized facts of daily returns as inAngelidis et al.(2007) and

other works. The benchmark process utilized to generate the returns is given by12:

Xi,t = 0.10Xi,t−1+ εi,t,

εi,t = σi,tzi,t, zi,t ∼ i.i.d.F(0, 1),

σ_i,t2 = σ2(1 − 0.10 − 0.85) + 0.10ε_i,t−12 + 0.85σ_i,t2, (3.1)

where, for time t and asset i, Xi,t is the return, σ_i,t2 is the conditional variance, εi,t is the innovation

in expectation, zi,t is a white noise process with distribution F. The unconditional volatility is

0.02.

For the analysis, eight scenarios were considered. Marginal distribution was generated

with Normal and Student0s t distribution with six degrees of freedom. The Normal distribution

was considered, because it is often utilized in stock market analysis, and the Student0s t

distri-bution was used to consider the heavy-tailed behaviour of financial assets. We considered high and low association (correlation of 0.20 and 0.80) among assets, obtained from the multivariate

Normal and Student0s t distributions. In addition, we generated portfolios with 4 and 16 assets13.

To forecast the risk measures, we used HS, DCC - GARCH and copula methods14.

Regarding copula methods, after generating returns (considering the eight scenarios described),

we fitted the expectation µ and dispersion components σ with AR(1)-GARCH(1,1) models15

to isolate the marginal behavior. Then, we transformed standardized residuals into pseudo-observations u ∈ [0, 1] by inverting the distribution fitted to each of them. This procedure was

10 _{Studies show that 1,000 observations (four years of daily data) is a good sample size for daily data.}

11 _{We focused on the one-day-ahead forecast, because according to the literature, this is the horizon usually used in}

empirical studies and simulations analysis.

12 _{Many works use similar values for the parameter, such as}_{Christoffersen and Gonçalves}_{(2005) and}_{Righi and}

Ceretta(2015a), because this data-generating process matches the daily returns obtained on the S&P 500 Index.

13 _{We generated portfolios with 2}n_{assets, where n = 2, 3, 4. We have not presented the results of the portfolio with}

8 (23) assets, because the results are similar to those of the portfolios with 4 (22) and 16 (24) assets. Portfolios

with 25or more assets are computationally complicated, due to the copulas approach.

14 _{Regarding the parameters estimation, we used quasi-maximum likelihood (Q-ML) estimate.}

15 _{The distribution used to fit the returns was the same as the scenario considered, to avoid marginal interference}

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Chapter 3. NUMERICAL PROCEDURES 24

needed because of the definition of the copula functions. With this pseudo-information, we could estimate the considered copula. Given the marginal and joint parameters already estimated, we used the following algorithm to forecast the risk measures:

1. Obtain forecasts of the conditional mean µi,t+1and standard deviation σi,t+1of each asset

through marginal models (AR(1)-GARCH(1,1));

2. Simulate N = 10, 000 samples16 u_i,N with the size of 1,000 for each asset i through

estimated Regular, Vine and Nested Archimedean copulas;

3. Convert each set of simulations ui,N to zi,N samples through the inversion of their marginal

probability, according to zi,N= F_i−1(ui,N);

4. For each asset i, determine the returns in accordance to the following specification Xi,N,t+1= µi,N,t+1+ σi,t+1zi,N;

5. Compute portfolio returns as w0XN, where w = {w1, w2, · · · , wn} is the weights vector and

XN= {X1,N,t+1, X2,N,t+1, · · · , Xn,N,t+1} represents the returns. In this study, we assumed

equally weighted portfolios;

6. Forecast VaRα

t+1(w0XN), as the negative of α-quantile of the distribution of the simulated

portfolio returns w0XN;

7. Forecast the ESα

t+1(w0XN), which can be obtained by the negative mean of portfolio returns

w0XN below the α-quantile (-VaRα_t+1(w0XN));

8. Forecast the EVaRα

t+1(w0XN), in accordance with Equation (2.3).

This procedure is an extension of the proposed algorithm byAas et al.(2011) to forecast

VaR and the algorithm utilized byRighi and Ceretta(2013a) to forecast ES. It is similar to the

procedure of Filtered Historical Simulation (FHS) used for the forecasting of risk measures. We considered α equal to 1 % and 5 %. These values are the most common in the literature. In each replicate, we also performed the forecasts of risk measures with HS and DCC - GARCH.

In the approach based on HS, portfolio returns were computed according to steps 5-8of the

algorithm. The only difference was the returns XN used were the returns obtained by the

data-generated process (3.1). For the approach based on DCC - GARCH, the difference in relation

to the algorithm presented was that, in steps2 -3, we obtained zi,t from the DCC - GARCH

model, instead of using the copulas approach. To assess the performance of the forecasts for all scenarios and each replicate, we computed A. Bias, R. Bias, and RMSE from VaR, ES, and

EVaR forecasts17. Then, for each scenario, we calculated the average values of these metrics.

16 _{In each replicate of Monte Carlo, we generated 10,000 samples u}

i,Nwith the size of 1,000 for each asset i. This

procedure was repeated 10,000 times.

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Chapter 3. NUMERICAL PROCEDURES 25

Similar metrics to identify the model with lower model risk were used byYao et al.(2006) and

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26

4 NUMERICAL RESULTS

Based on the numerical procedures, we obtain numerical results of the performance of

VaR, ES, and EVaR for the different scenarios analyzed. Tables1 -6present the A. Bias, R.

Bias, and RMSE for α = 1% and 5% of one-step-ahead forecasts of VaR, ES, and EVaR. Table

7presents the summary of best models at each scenario. Results were analysed in the following

order for a better understanding: first, we analysed the results of the scenarios generated with

Normal distribution, second, we analysed the scenarios generated with Student0s t distribution,

and finally, the behaviour of the absolute bias and relative bias was analysed.

Results for VaR, Tables1-2, showed, for scenarios with Normal distribution, the best

results were presented for Regular and Vine copulas. We noted the values of the criteria are close for these two types of copulas. An interesting result was that, regardless of the value for α, number of assets and correlation, the best performances were presented by the Regular and Vine

copulas. Similar results were reported for estimates of ES (Tables3-4) and EVaR (Tables5-6),

for the scenarios generated with Normal distribution.

Although Normal distribution is one of the most used, there is evidence that financial

data usually distance themselves from this distribution. In their seminal work, Mandelbrot

(1963) stated the Normal distribution is insufficient for modelling financial data and their

heavy-tailed behaviour (we refer to the workCont(2001) as an overview of stylized facts present in

asset returns). It is usually observed that the market returns display negative skewness, excess kurtosis, and structural shifts in the distribution. Improper use of Gaussian models can lead to underestimation of tail risk. This may lead to high losses without proper prevention of investor

portfolios (BROOKS; PERSAND,2002;TOLIKAS,2014). A recent example were the losses

observed in the events of 2007 and 2008, which resulted among other factors of the inappropriate use of Gaussian copulas. During periods of crisis there is an increase in the dependency between the assets, which is not properly captured by Gaussian copulas. In financial data, use of Normal distribution can illustrate a potential model risk, since it is an apparent model specification error

(MILLER; LIU,2006).

Regarding scenarios generated with Student0s t distribution, we noticed for VaR,

con-sidering α = 1%, NAC presents lower values for the metrics considered. This data generation process has more heavy tails and can generate more extreme values than a Normal distribution. The implications of returns with heavy tails can be drastic to a risk manager. This is more

pronounced in high quantil (BRADLEY; TAQQU,2003). Moreover, Angelidis et al. (2004)

showed leptokurtic distributions, especially the Student0s t, are more appropriate than Normal

for VaR forecasting.

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Chapter 4. NUMERICAL RESULTS 27

Table 1 – Monte Carlo simulations results for VaR with 4 assets.

Normal Distribution, Low Correlation

α = 1% A. Bias R. Bias RMSE α = 5% A. Bias R. Bias RMSE

Vine Copula -0.0004 -0.0146 0.0021 Vine Copula -0.0001 -0.0063 0.0013

Nested Copula -0.0039 -0.1463 0.0044 Nested Copula -0.0021 -0.1261 0.0027

Regular Copula -0.0003 -0.0124 0.0020 Regular Copula -0.0001 -0.0051 0.0013

DCC-GARCH -0.0093 -0.2674 0.0119 DCC-GARCH -0.0084 -0.2848 0.0110

HS 0.0052 0.2845 0.0102 HS 0.0032 0.4099 0.0087

Normal Distribution, High Correlation

HS 0.0093 0.3410 0.0175 HS 0.0048 0.3866 0.0135

Studen0s t Distribution, Low Correlation

Vine Copula 0.0246 0.2712 0.0376 Vine Copula 0.0137 0.0798 0.0210

Nested Copula 0.0089 0.0709 0.0302 Nested Copula 0.0119 0.2308 0.0226

Regular Copula 0.0241 0.2649 0.0409 Regular Copula 0.0139 0.1392 0.0208

HS 0.0469 0.8997 0.2722 HS 0.0104 0.4464 0.0901

Studen0s t Distribution, High Correlation

HS 0.1719 0.7657 0.4578 HS 0.0467 0.5908 0.2491

Note: This table shows numerical results of the performance evaluation of multivariate models (Historical Simulation (HS), Dynamic Con-ditional Correlation - Generalized Autoregressive ConCon-ditional Heteroskedastic (DCC - GARCH), Regular copulas, Vine copulas and Nested Archimedean copulas (NAC)) for forecasting VaR of a portfolio with 4 assets. The results are based on 10,000 Monte Carlo replications of length 1,001 (1,000 to consider the larger estimation window plus 1 for out-sample performance). Data generation process of returns corresponds to AR(1)-GARCH(1,1), considering Normal and Student0s t distribution, and low and high correlation between the assets. The performance of the forecasts was analyzed using absolute bias (A. Bias), relative bias (R. Bias), and root-mean-square error (RMSE). The values in bold correspond to the model presented 2 or 3 with lower value criteria.

In the scenario with four assets and low correlation, Vine copulas exhibited the lowest value for criteria. When we considered a portfolio of 16 assets and high correlation, estimates of VaR obtained with the DCC - GARCH model demonstrated better performance, as noted in

Table2. Regarding ES in scenarios with Student0s t distribution, only in the scenario with high

association, the portfolio with 16 assets and a significance level of 1%, the DCC - GARCH model

presents best values for the analysed criteria (Table4). In other cases for ES, NAC also showed a

better performance. In relation to EVaR, we realized, in most scenarios, Nested Archimedean copulas outperformed other models. In general, we realized NAC, among the analysed models,

for the scenarios generated with Student0s t distribution, was the best method for forecasting risk

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Table 2 – Monte Carlo simulations results for VaR with 16 assets.

Regular Copula -0.0001 -0.0035 0.0013 Regular Copula 0.0002 0.0035 0.0008

HS 0.0041 0.2825 0.0073 HS 0.0028 0.6397 0.0067

Regular Copula -0.0006 -0.0184 0.0023 Regular Copula -0.0001 0.0056 0.0015

HS 0.0084 0.3464 0.0168 HS 0.0044 0.4377 0.0136

Nested Copula 0.0048 -0.0012 0.0426 Nested Copula 0.0140 0.4011 0.0249

HS 0.0239 0.6622 0.1376 HS 0.0044 0.8888 0.0864

Nested Copula -0.0120 -0.1188 0.0161 Nested Copula 0.0167 0.2540 0.0245

HS 0.0692 1.1725 0.1729 HS 0.0215 0.8544 0.0720

Note: This table shows numerical results of the performance evaluation of multivariate models (Historical Simulation (HS), Dynamic Con-ditional Correlation - Generalized Autoregressive ConCon-ditional Heteroskedastic (DCC - GARCH), Regular copulas, Vine copulas and Nested Archimedean copulas (NAC)) for forecasting VaR of a portfolio with 16 assets. The results are based on 10,000 Monte Carlo replications of length 1,001 (1,000 to consider the larger estimation window plus 1 for out-sample performance). Data generation process of returns corresponds to AR(1)-GARCH(1,1), considering Normal and Student0s t distribution, and low and high correlation between the assets. The performance of the forecasts was analyzed using absolute bias (A. Bias), relative bias (R. Bias), and root-mean-square error (RMSE). The values in bold correspond to the model presented 2 or 3 with lower value criteria.

We observed the HS did not perform better in any of the analysed scenarios. This result

was more pronounced in scenarios with Student0s t distribution. Problems with HS are presented

in Pritsker(2006), who suggested this method responds slowly to changes in volatility and

large price movements, features present in financial returns. Corroborating with the results,

Christoffersen and Gonçalves (2005) argued the HS model results in bad point estimates for

VaR and ES and poor confidence intervals. This result is troubling. According to the results

identified byPérignon and Smith(2010), about 73 % of institutions that disclose their VaR use

this procedure. Estimation errors of this method accumulate and become large. When employing this method to compose a portfolio, the manager may have difficulties identifying most efficient asset allocation.

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Table 3 – Monte Carlo simulations results for ES with 4 assets.

HS 0.0067 0.2845 0.0115 HS 0.0046 0.3156 0.0096

HS 0.0132 0.3747 0.0211 HS 0.0079 0.3567 0.0159

HS 0.0736 0.9866 0.4244 HS 0.0331 0.8321 0.1940

HS 0.2667 0.8294 0.6621 HS 0.1273 0.7275 0.3748

Note: This table shows numerical results of the performance evaluation of multivariate models (Historical Simulation (HS), Dynamic Con-ditional Correlation - Generalized Autoregressive ConCon-ditional Heteroskedastic (DCC - GARCH), Regular copulas, Vine copulas and Nested Archimedean copulas (NAC)) for forecasting ES of a portfolio with 4 assets. The results are based on 10,000 Monte Carlo replications of length 1,001 (1,000 to consider the larger estimation window plus 1 for out-sample performance). Data generation process of returns corresponds to AR(1)-GARCH(1,1), considering Normal and Student0s t distribution, and low and high correlation between the assets. The performance of the forecasts was analyzed using absolute bias (A. Bias), relative bias (R. Bias), and root-mean-square error (RMSE). The values in bold correspond to the model presented 2 or 3 with lower value criteria.

We realized the absolute bias and the relative bias of the risk measures with HS were

positive. This result was stronger in scenarios with Student0s t distribution. The HS method

overestimates the market risk. This can cause higher capital requirements that could be applied more profitably. A trader may also be required to re-balance his/her portfolio at an inopportune time. Concerning the absolute bias and relative bias of the DCC - GARCH model, in most scenarios, they are negative, especially in scenarios with Normal distribution. In such cases, this model underestimates the risk, which can be dangerous and lead the institution to bankruptcy. The portfolio manager, when using DCC - GARCH to monitor their portfolio, will underestimate the probability of events of tail risk, being sub-optimal for investors that have as their primary objective the minimization of risk. In this case, the portfolio composition may not be appropriate

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Table 4 – Monte Carlo simulations results for ES with 16 assets.

HS 0.0049 0.2616 0.0081 HS 0.0037 0.3292 0.0070

HS 0.0119 0.3729 0.0199 HS 0.0071 0.3704 0.0155

HS 0.0456 0.7419 0.0869 HS 0.0226 0.6335 0.0492

HS 0.1029 1.2898 0.2441 HS 0.0523 1.0416 0.1382

Note: This table shows numerical results of the performance evaluation of multivariate models (Historical Simulation (HS), Dynamic Con-ditional Correlation - Generalized Autoregressive ConCon-ditional Heteroskedastic (DCC - GARCH), Regular copulas, Vine copulas and Nested Archimedean copulas (NAC)) for forecasting ES of a portfolio with 16 assets. The results are based on 10,000 Monte Carlo replications of length 1,001 (1,000 to consider the larger estimation window plus 1 for out-sample performance). Data generation process of returns corresponds to AR(1)-GARCH(1,1), considering Normal and Student0s t distribution, and low and high correlation between the assets. The performance of the forecasts was analyzed using absolute bias (A. Bias), relative bias (R. Bias), and root-mean-square error (RMSE). The values in bold correspond to the model presented 2 or 3 with lower value criteria.

for their risk tolerance, besides compromising their investment strategies. Corroborating,Hwang

and Pereira(2006) andCarnero et al.(2007) identified the estimates of the univariate GARCH

model usually have a negative bias. Given the bias identified in this model0s estimators,Fantazzini

(2009) noted the VaR model estimated with GARCH has a poor performance, especially for

small sample sizes.

For the estimated risk measures through copulas, we noticed a pattern in the sign of bias. Data generated with Normal distribution displayed negative bias. Regarding scenarios

generated with Student0s t distribution, we noted a positive bias for the measures, except for

some scenarios, where the estimates obtained by NAC presented negative relative bias. See, for

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Table 5 – Monte Carlo simulations results for EVaR with 4 assets.

HS 0.0039 0.3699 0.0086 HS 0.0024 0.4759 0.0077

Regular Copula -0.0001 0.0026 0.0023 Regular Copula 0.0001 0.0297 0.0016

DCC-GARCH -0.0072 -0.0978 0.0135 DCC-GARCH -0.0060 0.1262 0.0125

HS 0.0063 0.4131 0.0145 HS 0.0032 0.8001 0.0123

HS 0.0350 1.0684 0.1383 HS 0.0139 1.4188 0.0687

HS 0.0569 1.2837 0.2196 HS 0.0218 1.3672 0.1118

Note: This table shows numerical results of the performance evaluation of multivariate models (Historical Simulation (HS), Dynamic Con-ditional Correlation - Generalized Autoregressive ConCon-ditional Heteroskedastic (DCC - GARCH), Regular copulas, Vine copulas and Nested Archimedean copulas (NAC)) for forecasting EVaR of a portfolio with 4 assets. The results are based on 10,000 Monte Carlo replications of length 1,001 (1,000 to consider the larger estimation window plus 1 for out-sample performance). Data generation process of returns corresponds to AR(1)-GARCH(1,1), considering Normal and Student0s t distribution, and low and high correlation between the assets. The performance of the forecasts was analyzed using absolute bias (A. Bias), relative bias (R. Bias), and root-mean-square error (RMSE). The values in bold correspond to the model presented 2 or 3 with lower value criteria.

distribution), the copulas method results in small risk estimates, resulting in less protection for the portfolios (negative bias), underestimating the market risk. However, in the most turbulent

periods (Student0s t distribution), copulas result in risk measures more parsimonious, reflecting

greater protection. We emphasize that, in general, the relative bias of estimated measures with copulas is small.

We can highlight the results provided evidence of the superiority of copula models over the Historical and DCC - GARCH methods to forecast risk measures. The data generated in

the process simulation present marginals with Normal and Student0s t distribution and linear

dependencies, which should favour the DCC - GARCH model. Some studies have discussed the superiority of copulas over the correlation approach, because they offer more flexibility to model

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Table 6 – Monte Carlo simulations results for EVaR with 16 assets.

Regular Copula -0.0002 -0.0145 0.0011 Regular Copula 0.0001 0.0002 0.0007

HS 0.0031 0.3490 0.0067 HS 0.0023 0.8799 0.0063

HS 0.0054 0.2406 0.0113 HS 0.0027 0.7221 0.0097

HS 0.0196 0.8163 0.0439 HS 0.0085 1.2112 0.0288

HS 0.0468 1.2111 0.1093 HS 0.0208 1.5755 0.0607

Note: This table shows numerical results of the performance evaluation of multivariate models (Historical Simulation (HS), Dynamic Con-ditional Correlation - Generalized Autoregressive ConCon-ditional Heteroskedastic (DCC - GARCH), Regular copulas, Vine copulas and Nested Archimedean copulas (NAC)) for forecasting EVaR of a portfolio with 16 assets. The results are based on 10,000 Monte Carlo replications of length 1,001 (1,000 to consider the larger estimation window plus 1 for out-sample performance). Data generation process of returns corresponds to AR(1)-GARCH(1,1), considering Normal and Student0s t distribution, and low and high correlation between the assets. The performance of the forecasts was analyzed using absolute bias (A. Bias), relative bias (R. Bias), and root-mean-square error (RMSE). The values in bold correspond to the model presented 2 or 3 with lower value criteria.

the dependence between variables (KOLE et al.,2007). Corroborating with our results,Chen and

Tu(2013) demonstrated, under the coverage rates of 95% and 99%, the copula-based VaR model

demonstrates performance superior to both the CCC - GARCH model (Constant Conditional

Correlation - Generalized Autoregressive Conditional Heteroskedastic) (BOLLERSLEV,1990)

and the DCC - GARCH model. However, differently from what we found,Weiß(2013), when

analysing a large number of portfolios, identified that the DCC - GARCH does not lose to any of

the considered copula (elliptical copulas, such as Gaussian and Student0s t, and Archimedean

copulas, such as Clayton, Frank and Gumbel) models in terms of VaR and ES forecasting. One of the possible explanations for this is that the author analysed bivariate portfolios, and in our work, we considered portfolios with 4 and 16 assets. In addition, we point out that using copulas

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will reduce the model risk18, as also noted byChen and Tu(2013) for VaR.

Overall, risk measures are sensitive to the choice of model. By making use of a different model or a different parameterization, the risk manager or regulator is susceptible to different

results. For instance, consider the determination of banks0 capital requirement, which is obtained

directly by the point estimates of risk measure multiplied by portfolio value. Using an inap-propriate model, HS or DCC-GARCH, may cause insufficient quantities of capital, which may be insufficient to absorb losses from unexpected impacts, especially in times of crisis, or can be calculated at a high capital requirement quantity, which could be applied to more profitable investments. Our results can assist the portfolio manager in selecting the most suitable model to manage their portfolio. The choice of model with lower model risk, besides reducing financial losses, can avoid poor strategic decision-making and damaging the reputation of the institution. Another practical implication to use a model with lower model risk is to reduce the problem of

regulatory arbitrage and model misspecification, approached byKellner and Rösch(2016)19.

We noticed similar behaviour in the estimation methods in the two sizes of portfolios. Regarding the α analysed, we realized, in general, there is no pattern for the value of the relative

bias. The results ofChen and Tu(2013) suggested the risk management models should apply a

smaller nominal coverage rate (95% instead of 99%) to reduce model risk. Differently from our work, the author considered only the VaR in their analysis.

According to our results, models with good performance in estimating one of the three measures assessed might provide accurate measurements of the VaR, ES, or EVAR in a

multi-variate perspective. This result is most evident when we analyze the Table7, which summarizes

the results identified. Zhou(2012), Degiannakis et al.(2013) andRighi and Ceretta (2015a)

also identified a similar performance of analysed estimators to measure the VaR and ES. Unlike our research, they restricted their analysis to univariate models. This may be related with the

elicitable. ES is not elicitable, but ES can in practice be jointly elicited with VaRAcerbi and

Szekely(2014). EVaR is elicitable; however, there are no studies that examine the performance

of models to forecast this measure. Another possible reason for the similar performance models is the three measures are tail risk measures, although they present conceptual differences. VaR can be represented as the quantile of the distribution of profit and loss, while ES considers the magnitude of losses beyond the quantile, instead of only the quantile of interest. Differently to VaR and ES, EVaR is associated with expectiles, being more sensitive the extreme values

than VaR. Corroborating with our results, Pfingsten et al. (2004) noted risk measures of the

same group deliver similar risk rankings in their analysis. Differently,Marinelli et al.(2007), to

compare with a backtesting study the performance of univariate models for VaR and ES based

18 _{In this paper, we considered a model presents a greater model risk if the estimate measure with this model}

presents greater relative bias.

19 _{Institutions that present the same portfolio and use different internal models, approved by the regulator, must}

hold the same or at least almost the same amount of regulatory capital, which rarely happens, even if institutions use models that pass in backtesting. This problem is referred to as regulatory arbitrage.

Essays on model risk for risk measures

UNIVERSIDADE FEDERAL DO RIO GRANDE DO SUL

ESCOLA DE ADMINISTRAÇÃO

PROGRAMA DE PÓS-GRADUAÇÃO EM ADMINISTRAÇÃO

Fernanda Maria Müller

Essays on model risk for risk measures

Fernanda Maria Müller

Essays on model risk for risk measures

Dissertation for Doctoral degree in

Busi-ness Administration at the School of

Ad-ministration of Federal University of Rio

Grande do Sul.

Supervisor: Marcelo Brutti Righi, PhD

Fernanda Maria Müller

Essays on model risk for risk measures

Agradecimentos

Abstract

Resumo

Contents

I

NUMERICAL COMPARISON OF MULTIVARIATE

MOD-ELS TO FORECASTING RISK MEASURES

13

II

A ROBUST APPROACH FOR MINIMIZATION OF RISK

MEASUREMENT ERRORS

36

III

MODEL RISK IN RISK FORECASTING: WHERE DO

WE COME FROM AND WHERE ARE WE GOING?

72

INTRODUCTION

Part I

NUMERICAL COMPARISON OF

MULTIVARIATE MODELS TO

FORECASTING RISK MEASURES

Abstract

1 INTRODUCTION

2 BACKGROUND

2.1

Risk measures

2.2

Copulas

2.2.1

Vine copula

∏

∏

∏

2.2.2

Nested Archimedean copulas

2.3

DCC - GARCH

3 NUMERICAL PROCEDURES

4 NUMERICAL RESULTS