Measuring model risk of economic capital aggregation techniques

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MASTER IN INTERNATIONAL FINANCE

MASTER THESIS

MEASURING MODEL RISK OF ECONOMIC CAPITAL

AGGREGATION TECHNIQUES

Prepared by

Mr JHG Bisschoff

Date

August 2016

Student number

11081732

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techniques TITLE PAGE

Version: Version 1 Date: August 2016 Page i of ii

Public 11081732 JHG Bisschoff

TITLE PAGE

TITLE: Measuring model risk of economic capital aggregation techniques

THESIS SUPERVISOR: Dr T Yorulmazer

STUDENT NUMBER: 11081732

CLASSIFICATION: Public

SYNOPSIS: Master Thesis

KEYWORDS: Model risk, Economic capital, Copula, Aggregation, Simulation

PREPARED BY: Mr JHG Bisschoff

DOCUMENT VERSION: Version 1

COURSE: Master in International Finance

APPROVED BY:

Dr T Yorulmazer

DATE: August 2016

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techniques TITLE PAGE

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JHG Bisschoff 11081732 Public

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techniques TABLE OF CONTENTS

Version: Version 1 Date: August 2016 Page i of vii

LIST OF FIGURES

Figure 2.1 The three phases of the risk management process as illustrated by Lam ... 5 Figure 2.2 Aggregation process using copula. ... 6 Figure 3.1 Three trees representing the decomposition of a four-dimensional joint density function into bivariate pair-copulas and marginal densities (Schirmacher, 2008). ... 10 Figure 3.2 The error made in estimating some risk measure split into model specification error and sample size error given the right model form was estimated. ... 11 Figure 3.3 Percentage error made in the Combined LOB’s VaR estimates at different percentiles due to sample size. ... 14 Figure 3.4 Proportion of simulations that obtained any fit, split over different copula types simulated from. ... 15 Figure 3.5 The proportion of simulations that obtained a fit, split into how many of the LOB that had the correct underlying distributions fitted and simulated from multiple possible copulas. ... 16 Figure 3.6 Proportion of simulations that obtained a correct fit, split into LOB. Simulations come from multiple possible copulas. ... 17 Figure 3.7 Percentage error made in the 90% VaR estimate for the combined LOB due to fitting either the wrong marginal and/or the wrong copula. ... 18 Figure 3.8 Marginal error for the 90% VaR estimate for the combined business. ... 19 Figure 5.1 A plot of fitted total error vs actual error for the 90% VaR estimate. Actual error represented by the bullets and estimated error by the lines... 24 Figure 5.2 A comparison of before and after adjustment errors made in the 90% VaR estimates. ... 25

Figure A 1 Percentage error made in the Market LOB’s VaR estimates at different percentiles due to sample size. ... A2 Figure A 2 Percentage error made in the Credit LOB’s VaR estimates at different percentiles due to sample size. ... A2 Figure A 3 Percentage error made in the Operational LOB’s VaR estimates at different percentiles due to sample size. ... A3 Figure A 4 The proportion of simulations that obtained a fit, split into how many of the LOB that had the correct underlying distributions fitted and simulated from a Normal copula. ... A3 Figure A 5 The proportion of simulations that obtained a fit, split into how many of the LOB that had the correct underlying distributions fitted and simulated from a T copula. ... A4

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techniques LIST OF FIGURES

Version: Version 1 Date: August 2016 Page iii of vii

Figure A 6 Proportion of simulations that obtained a correct fit, split into LOB. Simulations come from a Normal copula. ... A4 Figure A 7 Percentage error made in the 90% VaR estimate for the combined LOB due to fitting either the wrong marginal and/or the wrong copula, with sample size error removed. A5 Figure A 8 Proportion of simulations that obtained a correct fit, split into LOB. Simulations come from a T copula. ... A5 Figure A 9 Percentage error made in the 95% VaR estimate for the combined LOB due to fitting either the wrong marginal and/or the wrong copula. ... A6 Figure A 10 Percentage error made in the 95% VaR estimate for the combined LOB due to fitting either the wrong marginal and/or the wrong copula, with sample size error removed. A6 Figure A 11 Percentage error made in the 99% VaR estimate for the combined LOB due to fitting either the wrong marginal and/or the wrong copula. ... A7 Figure A 12 Percentage error made in the 99% VaR estimate for the combined LOB due to fitting either the wrong marginal and/or the wrong copula, with sample size error removed. A7 Figure A 13 Marginal error for the 95% VaR estimate for the combined business. ... A8 Figure A 14 Marginal error for the 99% VaR estimate for the combined business. ... A8 Figure A 15 Marginal error for the 99.5% VaR estimate for the combined business. ... A9 Figure A 16 Marginal error for the 90% VaR estimate for the Market LOB. ... A9 Figure A 17 Marginal error for the 95% VaR estimate for the Market LOB. ... A10 Figure A 18 Marginal error for the 99% VaR estimate for the Market LOB. ... A10 Figure A 19 Marginal error for the 99.5% VaR estimate for the Market LOB. ... A11 Figure A 20 Marginal error for the 90% VaR estimate for the Credit LOB. ... A11 Figure A 21 Marginal error for the 95% VaR estimate for the Credit LOB. ... A12 Figure A 22 Marginal error for the 99% VaR estimate for the Credit LOB. ... A12 Figure A 23 Marginal error for the 99.5% VaR estimate for the Credit LOB. ... A13 Figure A 24 Marginal error for the 90% VaR estimate for the Operational LOB. ... A13 Figure A 25 Marginal error for the 95% VaR estimate for the Operational LOB. ... A14 Figure A 26 Marginal error for the 99% VaR estimate for the Operational LOB. ... A14 Figure A 27 Marginal error for the 99.5% VaR estimate for the Operational LOB. ... A15 Figure A 28 Marginal error regression fit for 90th percentile, where marginal error depends on LOB, Percentile and includes an intercept. ... A17 Figure A 29 Marginal error regression fit for 95th percentile, where marginal error depends on LOB, Percentile and includes an intercept. ... A18 Figure A 30 Marginal error regression fit for 99th percentile, where marginal error depends on LOB, Percentile and includes an intercept. ... A18

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LIST OF FIGURES

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Figure A 31 Marginal error regression fit for 99.5th percentile, where marginal error depends on LOB, Percentile and includes an intercept. ... A19 Figure A 32 Copula error regression fit for 90th percentile, where copula error depends on Copula type, Percentile and includes an intercept. ... A21 Figure A 33 Copula error regression fit for 95th percentile, where copula error depends on Copula type, Percentile and includes an intercept. ... A21 Figure A 34 Copula error regression fit for 99th percentile, where copula error depends on Copula type, Percentile and includes an intercept. ... A22 Figure A 35 Copula error regression fit for 99.5th percentile, where copula error depends on Copula type, Percentile and includes an intercept. ... A22 Figure A 36 A plot of fitted total error vs actual error for the 95% VaR estimate. Actual error represented by the bullets and estimated error by the lines... A23 Figure A 37 A plot of fitted total error vs actual error for the 99% VaR estimate. Actual error represented by the bullets and estimated error by the lines... A23 Figure A 38 A plot of fitted total error vs actual error for the 99.5% VaR estimate. Actual error represented by the bullets and estimated error by the lines... A24 Figure A 39 A comparison of before and after adjustment errors made in the 95% VaR estimates. ... A24 Figure A 40 A comparison of before and after adjustment errors made in the 99% VaR estimates. ... A25 Figure A 41 A comparison of before and after adjustment errors made in the 99.5% VaR estimates. ... A25

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techniques LIST OF TABLES

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LIST OF TABLES

Table 2.1 Different risk measures and their formulas ... 8 Table 3.1 Marginal distributions that was fitted to the simulated data. ... 11 Table 3.2 Marginal distributions and parameter values for market, credit and operational risk. ... 12 Table 3.3 Bivariate copulas and parameter values for each node in the vine decomposition of the 3-dimensional copula. ... 13 Table 3.4 Descriptive statistics for the Combined VaR. ... 13

Table A 1 Descriptive statistics for Market VaR. ... A1 Table A 2 Descriptive statistics for Credit VaR. ... A1 Table A 3 Descriptive statistics for Operational VaR. ... A1 Table A 4 Marginal error regression analysis results, where marginal error depends on LOB and Percentile and includes an intercept. ... A16 Table A 5 Marginal error regression analysis results, where marginal error depends on LOB and includes an intercept. ... A17 Table A 6 Marginal error regression analysis results, where marginal error depends on LOB and does not include an intercept... A17 Table A 7 Regression of combined marginal error on the marginal error of the individual LOB. ... A19 Table A 8 Copula error regression analysis results. ... A20

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LIST OF ABBREVIATIONS

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LIST OF ABBREVIATIONS

ERM Enterprise wide risk management

EC Economic capital

VaR Value at risk

ES Expected shortfall

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techniques EXECUTIVE SUMMARY

Version: Version 1 Date: August 2016 Page vii of vii

EXECUTIVE SUMMARY

This report documents the thesis proposal entitled “Measuring model risk of economic capital aggregation techniques”.

Over the past decade there has been a trend to more principle based regulation in Basel II and Basel III. This had led to heavy reliance on quantitative analysis being used to facilitate decision making in banks. Banks constantly use models for various activities, including underwriting risk, valuing exposures to different risks, measuring risk, managing investment portfolios, determining solvency and capital adequacy and many other activities. More recently there has been an effort to determine the total risk the institution faces with enterprise wide risk management (ERM). There has been a drive to create a framework that combines market, credit and operational risk into a single figure, economic capital, that allows for the diversification benefits between these different types of risk. In a world that is increasingly relying on quantitative models and data driven techniques to make business decisions it is important to build models that are free of biases that might cause financial loss or inappropriate decisions to be made. Using VaR to estimate capital has been a growing trend in the last decade, thus a way to measure VaR accurately is required so that capital can be used to fund projects rather than kept as reserves. Using some form of VaR adjustment to remove biases of small sample size and other assumptions that were made is imperative in allocating capital appropriately. By using a methodology as illustrated in this thesis could start to resolve this challenge.

There are many model risks associated with calculating economic capital using quantitative bottom-up models; these include estimation error due to small sample size and selecting the wrong model form; if both errors happen at the same time the error could be exaggerated. In this document a methodology is proposed to determine the size of the model error given a certain sample size, i.e. isolating the effect of choosing the wrong model form and creating a VaR adjustment factor so that biases as a result of sample size and copula error could be removed. From this research it will be clear what are the data requirements to aggregate economic capital using certain bottom up approaches and how such a VaR adjustment factor could be determined.

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techniques INTRODUCTION

Version: Version 1 Date: August 2016 Page 1 of 32

1 INTRODUCTION

Over the past decade there has been a trend to more principle based regulation in Basel II and Basel III. This had led to heavy reliance on quantitative analysis being used to facilitate decision making in banks. Banks constantly use models for various activities, including underwriting risk, valuing exposures to different risks, measuring risk, managing investment portfolios, determining solvency and capital adequacy and many other activities. More recently there has been an effort to determine the total risk the institution faces with enterprise wide risk management (ERM). There has been a drive to create a framework that combines market, credit and operational risk into a single figure that allows for the diversification benefits between these different types of risk. Apart from these consideration, banks have increasingly used data-driven, quantitative decision-making tools for a number of years (Federal Reserve, 2011).

Models are mainly used to inform the decision making process, but these models come at a cost. There are the direct costs of development, implementation and monitoring of these models and also the indirect cost of spurious accuracy, i.e. relying on these models to much, such that there are adverse consequences (including financial loss) of decisions based on models that are misused or incorrect. Those consequences should be addressed by active management and monitoring of model risk. In this document a research method is proposed to determine the error made when selecting the wrong model form. More specifically, in Section 2 some background will be given regarding the definition of a model and also what the main sources of model risk are. After which there will be a brief discussion on economic capital and how it is calculated using copulas. In Section Error! Reference source not found. the background is scoped showing exactly what models are tested and how the data will be simulated and what model risks are evaluated in the thesis. A preliminary data analysis will also be done in Section Error! Reference source not found. so that some features and trends in the simulated data can be identified. In the next section a methodology is proposed that use these factors to estimate a regression that could predict the size of error made in a VaR estimate. This will result in a VaR adjustment factor that could be applied to get a more accurate VaR estimate which is free of sample size and copula error biases. Finally some of the limitations and assumptions of the research will be discussed, proposing future areas of improvement and research followed by some concluding remarks in the final section.

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PROJECT DESCRIPTION AND BACKGROUND

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2 PROJECT DESCRIPTION AND BACKGROUND

In statistics there is a common saying by George Box (1976):

“All models are wrong, but some are useful.”

This expresses that in each model there is some form of risk. To manage model risk, it is important to understand what a model is and what the main sources of model risk are. In this section there will be a short discussion on how models work, highlighting the main sources of model risk. Then a brief discussion regarding economic capital follows and how it is calculated using a bottom-up approach using copulas. Subsequently copulas are defined and Sklar’s theorem is given, but firstly, a discussion on “what is a model?”

2.1 WHAT IS A MODEL?

The Fed (2011) and the OCC (2011) define a model as “a quantitative method, system, or approach that applies statistical, economic, financial, or mathematical theories, techniques and assumptions to process input data into quantitative estimates.” From this definition it is clear that a model is comprised of three parts, namely:

 The input data, hypothesis and assumptions.

 Quantitative theories and techniques that use the input data to produce statistics and summarising figures and forecasts.

 A report that converts the output of the quantitative model into information that can facilitate the business decision making process.

Thus, the definition of a model includes an algorithm that is used to calculate the group wide capital using some form of summarising statistic like value-at-risk (VaR) or expected-shortfall (ES) over multiple business lines. Now that the term “model” has been defined, the main sources of model risk can be explored.

2.2 MAIN SOURCES OF MODEL RISK

In simple terms a model is just a way to express reality in a simplified way so that the main drivers can be determined. Inevitably, by simplification there is the risk that not all drivers are determined, e.g. in “normal” economic circumstances assets prices have a relatively low correlation, but stressed conditions can cause assets with a low correlation to suddenly become more correlated, as was seen during the financial crisis of 2008 (Packham, Kalkbrener, & Overbeck, 2014). However, the simplification cannot be ignored given the complex relationship between different factors that drive reality. According to the Fed (2011),

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techniques PROJECT DESCRIPTION AND BACKGROUND

model risk is defined as potential losses resulting from decisions based on incorrect or misused model output and reports. Inappropriate or inadequate data can also be added to model risk since it can cause the model to provides incorrect results. Model risk thus covers three broad categories, namely: data issues, model selection and estimation and model misuse. Each of these categories can be subdivided into more granular sources of model risk, which will be discussed next.

Issues with data is mainly a result of two sources in the form of quantity and quality. When a model is estimated there is a certain degree of uncertainty around the estimates, i.e. standard error. This standard error is mainly large due to small sample size in the data. Also, if the data do not contain enough historical depth it is difficult to discover reliable trends in certain model factors, which increases the forecasting error. Data also have a lot of quality issues; this could be due to the lack of a critical variable of just plain data input errors. It is also important to consider that the data may contain some biases because of the data gathering and experimental design. An example of this would be banks only capturing operational losses above a certain threshold. This causes a lot of information to be lost in the left hand side of the loss distribution, which could cause estimation error when a distribution is fitted.

Model risk also includes the risk of selecting the wrong model form as well as the estimation of the model parameters. Model selection error is mainly due to oversimplification and approximation, e.g. a variable that has an exponential form could appear to be linear over small changes, thus estimating a linear model when an exponential model is more appropriate. This could cause large errors in the final result if out of sample estimates are required, for example in stress testing or scenario analysis. There is also the risk of inappropriate assumptions, e.g. assuming normal distribution of returns, even though it is known that returns have a more leptokurtic distribution, i.e. heavier tails and a higher peak. Any of these error could occur at any stage of the modelling process, from design to implementation.

Using a model not for its intended purpose could cause inappropriate business decisions to be made, e.g. developing a loan loss provisioning model in the US and implementing the model in one of its subsidiaries in Africa without re-estimating or re-calibrating the model for the specific risk profile of the country.

In this section model risk has been defined and the main sources of model risk have been highlighted. In the next sub-section there will be a discussion on economic capital and risk aggregation using copulas.

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2.3 ECONOMIC CAPITAL AND RISK AGGREGATION

Banks face many types of risks within its different lines of business, these risks are unique to the bank’s business strategy. Lam (2003) describes seven concepts of risk that influence the size of the risk. All other things being equal, the larger any of the first six items are, the greater the risk:

 Exposure: The maximum loss that can be suffered if an event should occur, bearing in mind that harm may not have an immediate monetary value, e.g. damage to a brand name.

 Volatility: Broadly a measure of the variability within the range of possible outcomes. When describing market risk, volatility is usually defined as the standard deviation of returns.

 Probability: Is the likelihood that an event should occur.

 Severity: Is the loss that is likely to be incurred if an event should occur. Severity is generally lower than exposure (except for off balance sheet items), which is the maximum loss. When considered together, severity and probability give a useful assessment of the expected cost of the risk.

 Time horizon: Is the length of time for which an organization is exposed to a risk or the time required to recover from an event.

 Correlation: Is the degree to which different risks behave in a similar manner in response to a common event.

 Capital: A business needs to hold capital to meet the requirements for cash as well as to cover any unexpected losses arising from exposure to risks.

All these factors should be considered when the bank’s risk tolerance and risk limits are determined - the amount of risk the bank is willing to accept is known as its risk appetite. To determine if the bank’s risk profile falls within its risk appetite it is important that the risks the bank faces are understood and managed so that the bank is not overly exposed to risks that fall outside their risk appetite. The type of risks and their exposure to those risks make up the bank’s risk profile. It is important to know that some risks are interrelated and may compound or offset one another. Risk management is thus an important part of a bank’s business to optimise its risk-adjusted returns, given that its risk fall within its risk limits. Lam (2003) illustrates that the risk management process has three stages; first identifying the risks that are faced, then assessing its likelihood and size and lastly deciding how to manage each of the risks faced, as shown in Figure 2.1 below.

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Figure 2.1 The three phases of the risk management process as illustrated by Lam

Risk identification is an important part of the risk management process in which banks identify the risks they face and categorise them. A common risk categorization is made up of six risk types, namely: market risk, credit or default risk, operational risk, underwriting/insurance risk, liquidity risk and reputational risk. This list should not be considered as complete, but is intended to cover major risks to which a financial institution might be exposed (Sweeting, 2011). After identification, risks are measured. There are a number of risk measures that can be used to express the size and likelihood of risks faced, these can give either a ranking of the various risks or an assessment of the absolute levels of risk in order to determine whether those levels are acceptable or not. Good risk measurement practices are essential to ERM and much of the thesis will be devoted to the different methods available to measure and aggregate risks.

Sweeting (2011) defines economic capital (EC) as the amount of capital a bank determines as appropriate to stay solvent at a certain level of confidence over a defined time horizon given their risk profile. In mathematical terms; suppose 𝐿𝑇 is the total financial loss over a time

horizon 𝑇, e.g. 1 year. 𝐿𝑇 is a random variable following some distribution function 𝐹. When a

confidence level 𝛼 is considered, e.g. 𝛼 = 99.9%, EC is the loss level such that:

F(LT) = P(LT ≤ EC) = α (2.1)

In layman’s terms, EC is the capital level such that the bank can survive the worst yearly loss in a 1000 years. It may seem that the EC calculation should be straightforward from the simple equation (2.1), but there are many challenges when calculating EC. The EC calculation should be an all-inclusive calculation that takes all risks into account. These risk are, however, managed and understood at different levels. Credit and market risk have had the most attention in banks and thus have the most developed methodologies, but on the other hand some risks, like operational risk, have less developed methodologies. Some risks are difficult to quantify in terms of losses, e.g. liquidity, reputational and model risk, but should still be included into the EC calculation. Even if all individual risks are well understood and are able to be quantified, there is still difficulty in aggregating these risk into a single loss distribution to finally determine the EC and diversification effects of offsetting risks.

Identifying the risks faced by an organisation

Assessing how likely these risks are to materialise and what their impact could be

Deciding how to deal with each risk

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Banks prefer to do risk-based calculations by risk types, this is because the different risk types require different modelling techniques. Credit risk is usually modelled using a bottom up approach, where each loan is modelled based on its risk parameters such as probability of default (PD), loss given default (LGD) and exposure at default (EAD). These risk parameters are linked both to each other and economic scenarios. Market risk is evaluated by pricing each instrument based on a multi-factor model linked to market factors such as GDP, interest rates, volatilities and credit spreads. Operational risk is usually modelled by fitting loss distributions on both internal and external loss data for each separate sub-level of risk and then aggregated assuming some relationship between them which is facilitated by expert opinion and scenario analysis. After all individual risks have been modelled, combining these risks requires a comprehensive understanding on how these risks are interrelated, not only under normal circumstances but also in stressed conditions.

Risks can be aggregated by using a copula. It allows the risk manager to separate the risk modelling and aggregation processes by first measuring the risk in the different risk categories and then combining them using a copula that explains the relationship between the different risk categories. Using a copula can be a complicated process, from choosing the copula to the actual calibration. Aggregating risk using a copula has two main phases, namely: fitting the underlying distributions for each risk category and fitting a copula that describes the relationship between the risks; as shown in Figure 2.2. This process introduces model risk in each of the two phases, the thesis will show the degree of error introduced in each of the two phases.

Figure 2.2 Aggregation process using copula.

Fit marginal

distributions

• Fit marginal

distribution for risk X,

Y and Z

• Estimate marginal

distribution

parameter values

Fit copula

• Choose between

Gumble, Clayton,

Gaussian and T

-Copula

• Calibrate copula and

estimate parameter

values

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2.4 DEFINITION OF A COPULA

To be able to aggregate risk using a copula it is important to understand what a copula is. A copula is a function 𝐶 that describes the joint distribution function as a function of the marginal distributions of each underlying risk, more formally (Yang, 2013):

A 𝑘-dimensional copula among random variables 𝑋1, … , 𝑋𝑘 is a function 𝐶(𝑢1, … , 𝑢𝑘) from

[0,1]𝑘 _{→ [0,1] such that}

F(𝑥1, … , 𝑥𝑘) = P(X1 ≤ 𝑥1, … , Xk≤ 𝑥𝑘) = C(F1(𝑥1), … , F𝑘(𝑥k)), (2.2)

Where 𝐹𝑖(𝑥𝑖) = 𝑃(𝑋𝑖 ≤ 𝑥𝑖) is the cumulative distribution function for 𝑋𝑖 and 𝐹(𝑥1, … , 𝑥𝑘) =

𝑃(𝑋1≤ 𝑥1, … , 𝑋𝑘 ≤ 𝑥𝑘) is the cumulative joint distribution function for 𝑋1, … , 𝑋𝑘.

A copula is thus a function that describes the relationship between random variables as a function of its marginal distribution, as stated by Sklar’s theorem (1959) below.

2.4.1 SKLAR’S THEOREM (1959)

Let 𝐹 ∈ ℱ(𝐹1, … , 𝐹𝑛) be an 𝑛-dimensional distribution function with marginal 𝐹1, … , 𝐹𝑛. Then

there exist a copula 𝐶 ∈ ℱ(𝒰, … , 𝒰) with uniform marginal such that

F(𝑥1, … , 𝑥𝑛) = C(F1(𝑥1), … , F𝑛(𝑥𝑛)) (2.3)

Now that a copula has been defined some other measures of risk need to be defined for the use of the thesis. There are different ways of quantifying the risk of an organization. In Table 2.1Error! Reference source not found. below some of the different ways of quantifying risk are given. Risk measures should measure rare events, i.e. large losses, which cause fluctuations in the value of the portfolio and thus are related to the variance or a quintile of the portfolio loss distribution. For any real random variable 𝑋 and 𝛼 ∈ [0,1] the quantile of 𝑋 is defined as (Hyndman & Fan, 1996)

𝑞𝛼(𝑋) = min{𝑥: 𝑃[𝑋 ≤ 𝑥] ≥ 𝛼}. (2.4)

If 𝑋 has a strictly increasing and continuous distribution function 𝐹(𝑥) = 𝑃[𝑋 ≤ 𝑥], quantiles of 𝑋 can be expressed by the inverse function of 𝐹, (Steinbrecher & Shaw, 2008)

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For 𝑋 ∈ 𝐿 the following risk measures are defined and assumed it exist finitely (Fisher, 1918) (Nawrocki, 1999) (Acerbi & Tasche, 2002):

Table 2.1 Different risk measures and their formulas

Risk Measure

Formula

Variance

𝜌

_𝑣𝑎𝑟

(𝑋) = 𝑣𝑎𝑟[𝑋]

Standard deviation

_𝜌

_𝑠𝑑

_{(𝑋) = √𝑣𝑎𝑟[𝑋]}

Semi Variance

𝜌

_{𝑠𝑣𝑎𝑟}

(𝑋) = 𝐸[((𝑋 − 𝐸[𝑋])

₋

)

2

_]

Value-at-Risk*

𝜌

_{𝑉𝑎𝑅(𝛼)}

(𝑋) = 𝑞

𝛼

[−𝑋]

Expected Shortfall

𝜌

_{𝐸𝑆(𝛼)}

(𝑋) =

1 1 − 𝛼

∫ 𝑉𝑎𝑅

𝑢

(𝑋)𝑑𝑢

1 𝛼 *In the risk environment a Value-at-Risk measure is used to measure the risk at 𝛼=5%, 1% or 0.5%. (Pearson, 2002)

For the thesis Value-at-Risk at an 𝛼 of 10%, 5%, 1% and 0.5% will be the main risk measure used to measure risk which is in line with standard practice (Pearson, 2002). In this section some background was given regarding what is model risk and what are the main sources of model risk. Also there was a quick discussion regarding EC and how it could be calculated and aggregated using a copula. Then there were some closing remarks on different measures of risk and their formulas. In the next section the specific model risk in the aggregation process will be scoped for the thesis, highlighting how the data will be simulated and what models will be tested. It concludes with some preliminary analysis looking at different features and trends in the data, identifying the main factors that drive model risk and defining how model risk will be decomposed into it constituent parts.

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techniques DATA AND PRELIMINARY ANALYSIS

3 DATA AND PRELIMINARY ANALYSIS

It is clear from the previous section that there is model risk involved in calculating the EC using copula. Also a copula has been defined as a function that describes the joint distribution function as a function of its marginal distribution. Figure 2.2 showed the two phase process of calculating EC using a copula. The main sources of model risk in the two phase process are: fitting the wrong type of marginal distribution and/or copula, the risk that the estimated parameter values differ from the actual parameter values and also the risk that there are insufficient sample data, which exaggerate the previous two sources of risk. These are also the risks that will be focussed on in the thesis. In the rest of this section the simulation method that will be used will be discussed, showing how the size of each of these risks can be isolated and measured. By simulating rather than fitting the distributions and copula to data, one gains control over the different levers that drive the final VaR estimate. The rest of the section will be split into three sub-sections, namely: simulation techniques, fitting of marginal distributions and copulas and simulated data. In the first sub-section the simulation techniques used in the thesis are discussed, explaining how an 𝑛-dimensional copula can be split into multiple 2-dimensional copulas. In the second sub-section the marginal distributions and copulas that are going to be used to simulate from are discussed and also which criteria will be used to select the best fitted distribution and copula. In the third sub-section some preliminary analysis on the simulated data is done, identifying specific features and trends that need to be accounted for in the methodology. The main factors that drive model risk are also identified graphically which will be included and tested formally in subsequent sections. But first a discussion is provided on the simulation techniques that will be used.

3.1 SIMULATION TECHNIQUES

Using simulated data rather than actual data have a couple of advantages, namely: the marginal distributions and parameter values are known, the copula form and parameters are known and the sample size can be chosen to either limit the estimation error in the parameter values or increase the effect of model risk. This is similar to a bank that has limited data in some risk categories, like operational risk, to fit the underlying distribution and copulas. Simulating joint distributions from copulas become increasingly complicated as the dimension of the copula increases, to simplify the simulation process it is possible to split the copula into multiple bivariate copulas using vine decomposition of the joint density function as explained by D. Schirmacher (2008). The basic premise comes from fact that the joint density function 𝑓(𝑥1, 𝑥2) can be written as:

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DATA AND PRELIMINARY ANALYSIS

𝑓(𝑥1, 𝑥2) = 𝑐12(𝐹1(𝑥1), 𝐹2(𝑥2)) × 𝑓1(𝑥1) × 𝑓2(𝑥2). (3.1)

This follows by taking partial derivatives with respect to both arguments in 𝐹(𝑥1, 𝑥2) =

𝐶(𝐹1(𝑥1), 𝐹2(𝑥2)), where 𝐶 is the copula associated with 𝐹 via Sklar’s Theorem (Ruschendor,

2010). Note that from (3.1) it is possible to determine the conditional density of 𝑋2 given 𝑋1,

that is,

𝑓2|1 (𝑥2|𝑥1) =

𝑓(𝑥1, 𝑥2)

𝑓1(𝑥1)

= 𝑐12(𝐹1(𝑥1), 𝐹2(𝑥2)) × 𝑓2(𝑥2). (3.2)

By doing this recursively it is possible to split the 𝑛-dimensional copula into (𝑛₂) combinations of bivariate copulas that can be represented as a tree, as in Figure 3.1 below. The circled nodes represent the four marginal density functions 𝑓1, 𝑓2, 𝑓3, 𝑓4. Each edged labelled with the

pair-copula of the variables that it represents. The edges in level 𝑖 become nodes for level 𝑖 + 1. The edges for tree 1 are labelled as 12,23 and 34. Three 2 has edges labelled 13|2 and 24|3. Finally, tree 3 has one edge labelled 14|23 (Schirmacher, 2008).

To keep the thesis relatively simple the dimension of the copula is limited to 𝑛 = 3, thus only 3 bivariate copulas will have to be simulated simultaneously. These three dimensions represent three risk categories that a bank may have, e.g. market, credit and operational risk, from here on represented by M, C and OP respectively. The three bivariate copulas will be M-C, C-OP and M-OP|M-C, together it forms a tree similar to the one presented in Figure 3.1 above.

3.2 FITTING OF MARGINAL DISTRIBUTIONS AND COPULAS

By simulating the data with known marginal distributions, copulas and parameters it is possible to measure the model risk of not fitting those model forms, estimating the correct parameter values and not having sufficient data. The model risks that will be evaluated are:

Figure 3.1 Three trees representing the decomposition of a four-dimensional joint density function into bivariate pair-copulas and marginal densities (Schirmacher,

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 The risks of inaccurate parameter values because of small sample size.

 The risk of not fitting the correct marginal distribution.

 The risk of fitting an inappropriate copula, i.e. fitting a Gaussian, T-copula or decomposed pair-copula (Multiple copula).

The AIC (Akaike, 1971) was used to choose between different fitted distributions and copulas.

Table 3.1 Marginal distributions that was fitted to the simulated data. RISKCATEGORY ACTUALMARGINAL TESTEDMARGINAL

Market (M) Student-T Normal, Student-T

Credit (C) Log Normal Gamma, Log Normal

Operational (OP) Gamma Log Normal, Gamma

Error was defined as the percentage deviation from the true VaR at the 90%, 95% 99% and 99.5% confidence levels. This error was measured over multiple sample sizes ranging from 30 to 5000 and was also spilt it into different types of errors, namely: Sample size error, which is the estimation error given that the true model form was fitted, Marginal distribution error, which is the estimation error given that the true copula was fitted, but the other marginal distributions were fitted as defined in Table 3.1 above. Finally, Copula error, which is the error made because of fitting the wrong copula form, but the correct marginal distributions. The Marginal distribution error and Copula error can be combined as Model specification error. These errors measured over selected sample sizes can give an indication of the data requirements of building an economic capital model as well as the degree of error made if these data requirements are not adhered to. To illustrate the idea, Error! Reference source

E

rr

or

Sample size

Sample size error Model Specification error

Figure 3.2 The error made in estimating some risk measure split into model specification error and sample size error given the right model form was estimated.

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not found. shows the size of both Sample size error and Model specification error over

different sample sizes. Over small sample sizes there is a larger risk of choosing the wrong model form which exaggerates the error made in estimating the VaR, this decreases as the sample size increases until the correct model was fitted and only sample size error remains. To test when the error is significantly different from zero a T-test can be used. In this sub-section the main sources of risk that will be evaluated have been discussed, giving details on how the “true” data will be simulated using a decomposed 3-dimensional copula. Thereafter the distributions and copulas that will be selected from have been mentioned. In the next sub-section details are given on the simulated data, stating the specific parameter values of both the marginal distributions and the decomposed copulas that specify the relationship between the lines of business.

3.3 SIMULATED DATA

As mentioned earlier, the data that will be used are simulated so that the errors made in estimating the model can be measured against a benchmark. For each line of business an arbitrary underlying loss distribution was chosen that contains specific features of these lines of business.

Table 3.2 Marginal distributions and parameter values for market, credit and operational risk.

RISK CATEGORY

MARGINAL PARAMETER1 PARAMETER2 PARAMETER3

Market (M) Student-T 𝜇 = −6 𝜎 = 20 𝑑𝑓 = 30

Credit (C) Log Normal 𝜇 = 3.5 𝜎 = 0.2

Operational (OP) Gamma 𝛼 = 1.75

𝛽 = 1 15

Table 3.2 above shows the marginal distributions that were simulated from. It is assumed that market risk losses have a T-distribution with 𝜇 = −6, 𝜎 = 20 and 30 degrees of freedom. 𝜇 is negative because the distributions show losses, thus -6 loss is actually a profit of 6 million. Credit risk losses are assumed to follow a Log-normal distribution with 𝜇 = 3.5 and 𝜎 = 0.2. Operational risk losses are assumed to have a gamma distribution with parameters 𝛼 = 1.75 and 𝛽 = 1

15. Next the bivariate copulas that describe the relationships between the marginal

distributions are specified. Table 3.3 show the bivariate copulas that were assumed at each node of the vine decomposition, for example at the M-C node a Gumble copula was assumed with a parameter value of 𝜏 = 0.2, i.e. a Kendal’s Tau correlation of 20%. Similarly, a

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Student-techniques DATA AND PRELIMINARY ANALYSIS

T was assumed for the C-OP node and a Clayton copula for the M-OP|C node. The information contained in Table 3.2 and Table 3.3 is all the information required to calculate the aggregate loss distribution and thus the capital requirements.

Table 3.3 Bivariate copulas and parameter values for each node in the vine decomposition of the 3-dimensional copula.

NODE COPULA PARAMETER 1 PARAMETER 2

M-C Gumble 𝜏 = 0.2

C-OP Student-T 𝜏 = 0.1 𝑣 = 20

M-OP|C Clayton 𝜏 = −0.2

Generating 10000 VaR estimates from the above joint distribution with a sample size of 1000000 resulted in a fairly accurate estimate for the population VaRs at different quintiles for all three lines of business as well as for the combined business. Table 3.4 shows some descriptive statistics for the combined business’s VaR estimates at different percentiles. It shows that the VaR estimates are fairly accurate with an inter quantile range of 0.09 for the 90% VaR estimate and 0.4 for the 99.5% VaR estimate. Thus, for the population VaR estimates the mean VaR estimates were used to measure the degree of error against. Similar tables can be found in Appendix A for the market, credit and operational lines of business.

Table 3.4 Descriptive statistics for the Combined VaR.

Percentile Minimum 1st

Quartile Median Mean

3de Quartile Maximum 90% 98.76 99.01 99.05 99.05 99.10 99.30 95% 114.40 114.80 114.80 114.80 114.90 115.20 99% 147.90 148.60 148.70 148.70 148.80 149.50 99.5% 161.70 162.60 162.80 162.80 163.00 163.90

Now that the “true” VaRs are known the percentage error can be measured in estimating a VaR. This error can be split into marginal and copula error. Another source of error is due to small sample size which needs to be isolated first, because with a large sample size it is more likely to fit the correct copula and marginal distributions and thus the error in the VaR estimates should decrease with an increase in sample size, this can skew the results of both the marginal and copula error. Figure 3.3 shows the percentage error made in the Combined LOB’s VaR estimates at different percentiles due to sample size. It shows that for all the VaR estimates there are large errors made when sample sizes are small, up to 17%, and then the error decreases as the sample size increases. The 90% and 95% VaR estimates are overestimated and the error reaches 0% at sample sizes of 300 and 750 respectively. The 99% and 99.5%

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VaR estimates are first underestimated until sample sizes are large enough to calculate a 1-in-100 and 1-in-200, then overestimated until sample sizes are larger than 2500 and 5000 (not shown in graph) respectively. Similar graphs can be found for the Market, Credit and Operational LOB in Appendix A, they all show the same pattern the sizes of the errors just differ. The largest error comes from Market with an error in the 99.5% VaR of -24.6%, since this is lower than the Combined LOB’s error there seems to be some form of error diversification in combining the LOB with a copula. The remainder of this section is split into Marginal distribution error and Copula error in which the size of these errors are isolated from the sample size error observed in Figure 3.3. Marginal distribution error is the error made due to selecting and fitting the wrong type of marginal distribution, e.g. fitting a normal distribution to the Market data, when a Students-t distribution is the correct distribution and similarly for the credit and operational LOB. This error was analysed in three different ways; first what proportion of the total number of fits fit any distribution, then delving deeper how are these proportions split into one LOB fitting correctly, two LOB fitting correctly and finally all three LOB fit correctly. Thereafter, which LOB causes the trends seen in the data and finally is there a relationship between the number of correct fits and the size of the error made in the final VaR estimates at different percentiles. As can be expected from having small sample sizes the dimensionality of the optimization algorithm might run into problems and not deliver any fit for the marginal distributions. This can be seen in Figure 3.4 which shows how many of the simulations obtained any fit, be that correct or not. From the figure, it shows that sample size

Figure 3.3 Percentage error made in the Combined LOB’s VaR estimates at different percentiles due to sample size.

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below 300 may cause that no fit is obtained and at 300, 94% of the simulations obtained a fit. From the figure it is also clear that the different forms of copulas used to simulate from did not have an impact on proportion of simulations that obtained a fit. This graph only shows the proportion of simulations that obtained a fit and does not contain any information on what proportion of these fits are correct or not. By splitting this up into the number of LOB that had the correct type of distribution fitted, e.g. fitting a Students-T distribution for the Market LOB instead of a Normal distribution, it is possible to see how these fits are comprised. Figure 3.5 shows how these proportions are comprised, in the lower left corner there is a small dark triangle that shows the proportion of simulations that did not have one correct fit over the three LOB, the layer on top of it shows the proportion of simulations that had one LOB where the marginal fitted correctly. The figure also shows the proportion of simulations which had two and three LOB with the correct marginal distributions. At lower sample sizes only a small proportion of the simulations had all three or none of the distributions fit correctly, but as the sample size increases there are fewer simulations with no correct fits. At sample sizes of 300, the majority of simulations had two of the three LOB fit the correct distributions and then at sample sizes of about 5000 the majority of simulations had all three of the LOB fit the correct marginal distribution. Similar graphs were created for the Normal and T copula which can be found in Appendix A, these graphs are similar to Figure 3.5, leading to the conclusion that the

Figure 3.4 Proportion of simulations that obtained any fit, split over different copula types simulated from.

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type of copula simulated from does not affect the proportion of distributions that obtained a fit and not the proportion of correct fits. The next question to ask is which LOB provides the rapid increase in the number of fits at sample sizes of below 300 and what LOB cause the steady improvement for sample sizes larger than 300, i.e. the LOB that takes the largest sample size to fit the correct distribution so that all three LOB fit the correct marginal distribution.

By splitting the simulations by LOB, as in Figure 3.6 below, shows that the operational LOB achieves the fastest improvement in proportion of correct fits obtained and thus the gamma distribution reacts quickly to an increase in sample size, requiring only a sample size of 300 to be fitted. Whereas the Credit LOB is the second fastest to react to the increase in sample size, requiring a sample size of 3500 to be fitted. The Market LOB is the slowest to react to increase in the sample size and thus the T-distribution requires a large sample size, in excess of 5000, to be fitted. In conclusion, the risk of fitting the wrong distribution relies heavily on the sample size. It would be recommended to use sample sizes larger than 300 for the gamma distribution and sample sizes larger than 3500 and 5000+ for the log-normal and student’s-T when using the AIC to choose between distributions. Similar graphs were created for the normal and T copula, in Appendix A, but once again the results are similar and the probability of fitting the correct distribution does not depend on the copula simulated from. Next, how do these misspecifications influence the error made in the final VaR estimates. Figure 3.7 shows the percentage error made in the 90% VaR estimate due to using either the wrong marginal

Figure 3.5 The proportion of simulations that obtained a fit, split into how many of the LOB that had the correct underlying distributions fitted and simulated from multiple

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and/or the wrong copula. In the legend the word before the “-“ describes the type of copula used and the word after the “-“ the marginal distribution used, where multiple stands for fitting a decomposed copula as described in the previous sub-section and fitting the best marginal based on the AIC (Akaike, 1971). Finally the sample size series shows the errors made due to only sample size. When comparing the different series with one another the same colours are compared with one another, where the one is dashed and the other solid, e.g. comparing the “Normal – Multiple” with the “Normal – Actual”. By comparing these two graphs it shows the error made due to only using multiple types of marginal distributions but the copulas used stay the same. By comparing the dashed lines with the solid sample size line it shows the error made due to estimating the wrong type of copulas. From the graph it is clear that all the series still contain sample size error and thus have the same hyperbolic shape as the sample size series. To compare the different series with one another and isolate the marginal distribution and copula error one must first isolate and remove the sample size error, this was done by subtracting the sample size error from the other series that contain the sample size, marginal and copula error as shown in Figure A 7, Figure A 9 and Figure A 12. Figure A 7 show the Percentage error made in the 90% VaR estimate due to fitting either the wrong marginal and/or the wrong copula, with sample size error removed. There seems to be more variance in the error for lower sample sizes, but seems to stabilise as the sample size increases. It seems

Figure 3.6 Proportion of simulations that obtained a correct fit, split into LOB. Simulations come from multiple possible copulas.

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that marginal error is higher than copula error for lower sample sizes but dissipates quickly after sample sizes of 300. This is in line with what was seen in Figure 3.4, Figure 3.5 and Figure 3.6 where the majority of marginal fits were correct at sample sizes above 500, except for the T distribution of the Market LOB. In Figure A 7 it is clear that the different copulas that were fitted are the main driver of error in the 90% VaR estimate. By fitting multiple copulas, the 90% VaR is underestimated by -0.3% and the normal and t copula overestimate the 90% VaR by 1.1% and 0.6% respectively. The average absolute difference between the series with multiple marginal and those with actual marginal distributions show the degree of error relating to fitting the incorrect marginal, but as sample size increases these errors should disappear. Figure A 7 gives an indication that from sample sizes larger than 500 this error should disappear. Similar figures for other percentiles can be found in Appendix A. Plotting the marginal error against sample size show the size of the marginal error made over an increasing sample size, as shown in Figure 3.8 the error is large for small sample sizes but disappears quickly for sample sizes larger than 500. Analysing the figures from Figure A 16 to Figure A 27, the Market LOB has large marginal error which only becomes zero at sample sizes of 5000 which is in line with what was seen earlier in Figure 3.6, where the correct distribution was only fitted for large sample sizes. Also the marginal error is positive for the 90th_{percentile and close to zero for the 95}th_{then negative for 99}th_{and 99.5}th_{. Similar effects}

seem to be true for the credit and operational LOB. There is no significant difference in the

Figure 3.7 Percentage error made in the 90% VaR estimate for the combined LOB due to fitting either the wrong marginal and/or the wrong copula.

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marginal errors for different copulas that were fitted, all two sample T-tests passed at a significance of 5% thus the marginal error does not depend on the type of copula fitted, and this result will be incorporated into the methodology by only regressing the marginal error on LOB and percentile, and not the copula type. The remaining error after sample size and marginal error have been removed is the copula error. Copula error seems to depend on the type of copula used and the percentile being estimated, these will be the factors that will be considered in the regression analysis.

In this section the simulation techniques were discussed showing how the data were simulated and what distributions and copulas were considered in the thesis. After which the simulated data were discussed, highlighting some features of the simulated data and the origins of model risk, describing how the error in the VaR estimate was split into sample size, marginal and copula error. Thereafter, some specific features and trends that were found in a preliminary data analysis were discussed concluding that model risk depends on the sample size, the type of copula, LOB and the percentile being estimated. In the next section a methodology is proposed that combines these factors in a regression to estimate the size of error made in estimating the VaR so that after the error has been estimated the biased estimate of VaR which contains the model error can be adjusted for the sample size and copula used as well

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as the percentile being estimated. Thus obtaining a more accurate estimate for VaR, which does not contain the model risk biases.

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techniques METHODOLOFY AND HYPOTHESIS

4 METHODOLOFY AND HYPOTHESIS

In the previous section sample size, the LOB, copula and percentile were identified as factors that influence the size of error made in a VaR estimate. In this section a methodology is proposed that uses these factors to model the size of the error made using a regression. As stated earlier model risk can be split into three main parts, first sample size error which captures some of the error made due to insufficient sample size in estimating a VaR figure, next marginal distribution error that captures the error made due to estimating the wrong distribution and parameters and finally, copula error that captures the error made in combining the different LOB, i.e. estimating the wrong copula and copula parameters. Thus this can be represented as a regression where the total error made is a function of sample size, marginal and copula error, i.e.

ε_𝑖,𝑗VaR = 𝜀_𝑖,𝑗𝑆𝑎𝑚𝑝𝑙𝑒 𝑆𝑖𝑧𝑒 + 𝜀_𝑖,𝑗𝑀𝑎𝑟𝑔𝑖𝑛𝑎𝑙+ 𝜀_𝑖,𝑗𝐶𝑜𝑝𝑢𝑙𝑎+ 𝜉𝑖,𝑗, j ∈ (1, … , J) (4.1)

where ε_𝑖,𝑗VaR is the error made in the 𝑖th_{VaR estimate of the combined business for the}_𝑗th

percentile, 𝜀_𝑖,𝑗𝑆𝑎𝑚𝑝𝑙𝑒 𝑆𝑖𝑧𝑒 is the error due to sample size conditional on estimating the correct marginal distribution and decomposed copula form, 𝜀_𝑖,𝑗𝑀𝑎𝑟𝑔𝑖𝑛𝑎𝑙 is the error due to fitting the incorrect marginal distributions conditional on estimating the correct copula and 𝜀_𝑖,𝑗𝐶𝑜𝑝𝑢𝑙𝑎 is the error due estimating an incorrect copula conditional on estimating the correct marginal distributions and 𝜉𝑖,𝑗 is the part of the error that is yet to be explained. Each of the underlying

errors, 𝜀_𝑖,𝑗𝑆𝑎𝑚𝑝𝑙𝑒 𝑆𝑖𝑧𝑒, 𝜀_𝑖,𝑗𝑀𝑎𝑟𝑔𝑖𝑛𝑎𝑙 and 𝜀_𝑖,𝑗𝐶𝑜𝑝𝑢𝑙𝑎 are modelled using the factors identified in the previous section. The sample size error will be modelled using an empirical approach as done in the previous section, i.e. the graph in Figure 3.3 will be used directly to “read off” the error made due to sample size. The marginal distribution error will be modelled within the individual lines of business and combined thereafter, i.e. the marginal error of the combined business will be regressed on the marginal error of the other lines of business. The regression will be of the form:

𝜀_𝑖,𝑗𝑀𝑎𝑟𝑔𝑖𝑛𝑎𝑙 = 𝛼 + ∑ 𝛽𝑚𝜀𝑖,𝑗,𝑚 𝑀

𝑚=1

+ 𝜉𝑖,𝑗,𝑚 (4.2)

where 𝜀𝑖,𝑗,𝑚 is the marginal error made in the 𝑖th VaR estimate for the 𝑗th percentile and 𝑚th

LOB and 𝜀_𝑖,𝑗𝑀𝑎𝑟𝑔𝑖𝑛𝑎𝑙 is the marginal error made in the combined business. For each LOB the marginal errors will be regressed on the factors identified in the previous section, i.e. sample

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METHODOLOFY AND HYPOTHESIS

size, LOB and percentile which has been identified to influence the size of error graphically from Error! Reference source not found. to Error! Reference source not found., thus a regression of marginal distribution error on these factors of the form:

𝐿𝑁(𝜀𝑖,𝑗,𝑚) = 𝛼𝑗,𝑚+ 𝛽𝑗,𝑚𝐿𝑁(𝑆𝑖,𝑗,𝑚) + 𝜉𝑖,𝑗,𝑚, (4.3)

where 𝜀_{𝑖,𝑗,𝑚}𝑀𝑎𝑟𝑔𝑖𝑛𝑎𝑙 is the error made in the 𝑖th_{VaR estimate for the 𝑗}th _{percentile and 𝑚}th_LOB.

This thus creates a set of linear equations for each percentile and each LOB, where 𝛼𝑗,𝑚 and

𝛽𝑗,𝑚 need to be estimated for the 𝑗th percentile and 𝑚th LOB. For the thesis, 𝑀 = 3 LOB and

𝐽 = 4 percentile points were considered, thus 12 regressions with 2 parameters each need to be estimated for the marginal error. The natural logarithms are used to convert the hyperbolically shaped curves to straight lines so that a linear regression can be done more easily. Two other models will also be fitted, the first not depending on the percentile being modelled and the second does not include an intercept term, i.e. the first regression will be of the form:

𝐿𝑁(𝜀_{𝑖,𝑗,𝑚}𝑀𝑎𝑟𝑔𝑖𝑛𝑎𝑙) = 𝛼𝑚+ 𝛽𝑚𝐿𝑁(𝑆𝑖,𝑗,𝑚) + 𝜉𝑖,𝑗,𝑚, (4.4)

where in this model the intercept terms, 𝛼𝑚, only depend on the LOB and not on the percentile

as well as in equation (4.3). The third regression that will be tested only depends on the LOB as in the previous regression, but disregards the intercept term, i.e. the regression will be of the form:

𝐿𝑁(𝜀_{𝑖,𝑗,𝑚}𝑀𝑎𝑟𝑔𝑖𝑛𝑎𝑙) = 𝛽𝑚𝐿𝑁(𝑆𝑖,𝑗,𝑚) + 𝜉𝑖,𝑗,𝑚, (4.5)

Finally, the copula error is modelled in a similar manner to the marginal error, i.e. of the form:

𝐿𝑁(𝜀_{𝑖,𝑗,𝑛}𝐶𝑜𝑝𝑢𝑙𝑎) = 𝛼𝑗,𝑛+ 𝛽𝑗,𝑛𝐿𝑁(𝑆𝑖,𝑗,𝑛) + 𝜉𝑖,𝑗,𝑛, (4.6)

where 𝜀_{𝑖,𝑗,𝑛}𝐶𝑜𝑝𝑢𝑙𝑎 is the error made in the 𝑖th_{VaR estimate for the}_𝑗th _{percentile and 𝑛}th_copula

type. Once again this creates a set of linear equations for each percentile and copula type, where 𝛼𝑗,𝑛 and 𝛽𝑗,𝑛 need to be estimated for the 𝑗th percentile and 𝑛th copula type. For the

thesis, 𝑁 = 3 different copula types and 𝐽 = 4 percentile points were considered, thus 12 regressions each with 2 parameters need to be estimated for the copula error. Equation (4.1)

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can be rewritten net of sample size error so that two simple regressions can be done for the marginal and copula errors, i.e.:

ε_𝑖,𝑗VaR− 𝜀_𝑖,𝑗𝑆𝑎𝑚𝑝𝑙𝑒 𝑆𝑖𝑧𝑒 = ∑ 𝜀_{𝑖,𝑗,𝑚}𝑀𝑎𝑟𝑔𝑖𝑛𝑎𝑙 𝑀 𝑚=1 + ∑ 𝜀_{𝑖,𝑗,𝑛}𝐶𝑜𝑝𝑢𝑙𝑎 𝑁 𝑛=1 + 𝜉𝑖,𝑗, (4.7)

where 𝜀_{𝑖,𝑗,𝑚}𝑀𝑎𝑟𝑔𝑖𝑛𝑎𝑙 and 𝜀_{𝑖,𝑗,𝑛}𝐶𝑜𝑝𝑢𝑙𝑎 are the individual LOB’s and copula type’s error that are aggregated to get the total marginal and copula error. If the left hand side of equation (4.7) were plotted against sample size for a 90% VaR it would look similar to Figure A 7. T statistics will be used to test if the individual parameters have an influence on the error being regressed, also what are the size of the influences giving an indication to which of the factors are the most important to model risk. In this section a methodology was proposed to measure the degree of error made in VaR estimates by splitting the error into sample size, marginal and copula error. Where sample size error will be modelled empirically, marginal distribution error will be modelled as a function of the marginal errors within the different lines of business at different percentiles. Copula error will be regressed on the different percentiles and copula types that will be tested. In the next section this methodology was implemented and some regression results are given showing the impact of the factors that were identified in the previous section. After which the biased VaR estimates and the VaR estimates that were adjusted using the methodology above are compared to the population VaR estimates discussed in the previous section.

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techniques RESULTS

5 RESULTS

In the previous section a methodology was proposed to measure the degree of error made in VaR estimates at different percentiles and for different copula types. In this section the methodology was implemented and results are given. By fitting equation (4.3) to the simulated data shows that the natural logarithm of sample size has a significant influence on the logarithm of marginal error for all LOB and percentiles. All these beta parameters are negative indicating that the error decreases with an increase in sample size, the lowest beta coming from the operational LOB indicating that its marginal error declines the fastest and the market LOB has the highest beta parameter indicating that its marginal error has the slowest improvement with increasing sample size. This is in line with what was seen in Figure 3.6, where the gamma distribution used for the operational LOB required only sample sizes in excess of 300 and the t distribution used for the market LOB required sample sizes in excess of 5000. The beta parameters are relatively constant over different percentiles points which might indicate that percentile is not a significant driver of marginal error, on the other hand the intercept terms do increase with percentile points which indicate that these are different levels of error made at different percentiles. The intercept terms increase as the percentile increases, thus there are larger errors made in higher percentiles than at lower ones. Yet only a few of the percentiles are significantly different from zero. Regardless, the operational LOB has the highest intercepts and credit the lowest, indicating that operational risk has a lower level of error and credit the highest level of error. Keeping in mind that the percentile being estimated might not be a significant driver and the intercept terms are not significant, two other regressions were fitted. The first includes an intercept, but the percentile being modelled is removed from the regression, i.e. a regression of the form (4.4). The second does not have an intercept and only depends on LOB and sample size and not the percentile point, i.e. a regression of the form (4.5). The results of the three regressions can be found in Table A 4 to Table A 6. Similar results are obtained for the second and third regression, which does not depend on the percentile and includes an intercept term, as for the first regression. The beta parameters are all significant and of similar magnitude as with the first regression, with the biggest change coming from the credit LOB, where the parameter changed from 0.92 to -1.05. This leads to the belief that percentile is not a significant driver for marginal error. Looking at the intercept terms, it changed from the first to the second regression. The intercept terms are the average of the intercepts for the first regression over the percentile points, but remained insignificantly different from zero. The first regression, which includes an intercept and depends on the percentile, was preferred for both the AIC (Akaike, 1971) and BIC (Wit, van den Heuvel, & Romeyn, 2012) Thus, going forward, only the first regression of form (4.3)