# «Value-at-Risk: Strengths, Caveats and Considerations for Risk Managers and Regulators Master Thesis by Bogdan Izmaylov Supervisor: Thomas Berngruber ...»

Aarhus University

Business and Social Sciences

**Value-at-Risk:**

Strengths, Caveats and

Considerations for Risk

Managers and Regulators

Master Thesis by Bogdan Izmaylov

Supervisor: Thomas Berngruber

Department of Economics and Business

March 2014

Abstract

In this thesis, critiques and praises of Value-at-Risk measure are studied and compared

to empirical evidence from the literature in order to analyze potential strengths, model

risks and practical application issues of VaR. The study makes use of conceptual literature to define attractive properties and potential pitfalls in using VaR by risk managers and puts it in perspective by analyzing empirical findings. Through discussion, the main strengths and weaknesses of VaR are pointed out as well as the types of decisions which risk managers have to make in order to use the measure effectively.

This research concludes that the VaR is an extremely important, but fragile risk measure.

It also emphasizes the importance of the decision making process for proper implementation and precision of estimates. VaR has proven to be an effective and intuitive risk measure with convenient properties, when calculated and used appropriately to the market conditions and risk management needs. But the empirical evidence and the GFC have shown that there is a need for better understanding of what VaR can and what it cannot measure. It is especially evident in the case of VaR use for regulation purpose, which potentially can undermine the usefulness of VaR as a measure of risk. Expected Shortfall has proven to be less precise in practical applications, but can be used in combination with VaR as a complementary measure.

This thesis proposes further research to be conducted in the areas of VaR model application guidelines and use of risk measure portfolios.

i Acknowledgements I am thankful to my thesis advisor, Thomas Berngruber, for the rewarding discussions that we had and for his guidance throughout the writing process. I am grateful to my family for their support and understanding.

I would like to thank Manuela Markova for proof-reading my thesis. All the errors remaining are my own.

ii Contents Acknowledgements

List of figures

List of abbreviations

1. Introduction

1.1 Problem Formulation

1.2 Delimitations

1.3 Methodology

2. Theory

2.1 Value-at-risk (VaR)

2.2 Expected Shortfall (ES) and Tail Conditional Expectation (TCE)................. 8

2.3 Coherency of VaR

2.4 Precision of VaR

2.5 VaR in Basel Accords

2.6 Backtesting

2.7 Time horizon and the square root of time rule

2.8 Copulas

2.9 Extreme Value Theory (EVT)

3. Literature review

4. Discussion

4.1 VaR as a measure for non-normally distributed returns.

4.2 The quality of high confidence level VaR forecasts – 95%, 99% VaR and higher.

4.5 VaR use for stress testing

4.6 VaR as a widely adopted measure of risk for financial institutions.......... 31

4.7 VaR as an instrument for regulation in Basel Accords.

5. Conclusions and prospects for future research

References

iv List of figures FIGURE 1. 1-DAY VAR FOR S&P500 RETURNS..

** FIGURE 2. EXPECTED SHORTFALL AND VAR.**

FIGURE 3. BACKTESTING ESTIMATION WINDOWS

** FIGURE 4. TYPES OF DISTRIBUTION TAILS.**

** FIGURE 5. DAILY S&P 500 RETURNS 1989-2009 CUMULATIVE PROBABILITY PLOT.**

** FIGURE 6. DAILY S&P 500 RETURNS 1989-2009 DENSITY PLOT.**

** FIGURE 7. THE DISTRIBUTION OF P&L FOR A PROJECT.**

## FIGURE 8. SHIFT IN THE PROBABILITY DISTRIBUTION OF LOSSES CONDITIONAL ON AN ADVERSE

MACROECONOMIC SCENARIO.FIGURE 9. ILLUSTRATION OF THE RELATIVE ERROR (THE RATIO VAR[Η]/(√ΗVAR[1])) FROM THE USE OF THE SQUARE-ROOT OF TIME RULE

** FIGURE 10. P&L DISTRIBUTION BEFORE AND AFTER VAR MANIPULATION.**

FIGURE 11. EXAMPLES OF UNLIKELY, BUT POSSIBLE DISTRIBUTIONS

** FIGURE 12. REAL GDP IN DENMARK, BN.**

DKK, VS. THE LONG-TERM MECHANICAL PROJECTION............. 38

ES – Expected Shortfall ETL – Expected Tail Loss GFC – Global Financial Crisis HS – Historical Simulation iid – Identically and Independently Distributed MC – Monte Carlo MPT – Modern Portfolio Theory niid – Normal Identically And Independently Distributed RMP – Risk Management Professional TCE – Conditional Tail Expectation VaR – Value-at-Risk

Risk is ubiquitous in all areas of human life, but it is convenient to consider a specific definition depending on the area of interest. In the context of financial literature, the term risk refers to the change related to the variability of the future value (Artzner, Delbaen, Eber, & Heath, 1999). Oxford English Dictionary defines

**the word risk as:**

In order to assess variability of the future value, a risk measure (a way of calculating risk) is used to produce a risk measurement or risk metric (a number which quantifies the risk).The most common measure is the volatility of the price or return. The volatility of returns is easier to predict and thus the price volatility is calculated based on the return volatility. Usually, high returns are associated with higher variability in return and thus higher volatility. The famous work of H. Markowitz on portfolio selection has popularized the use of volatility as a measure of risk and introduced the Modern Portfolio Theory (MPT) together with the concept of “efficient frontier”. At its core, the MPT or mean-variance theory presents the benefits of diversification for achieving a minimum variability of outcomes for a given rate of return. One key assumption is that the asset returns are normally distributed, which is almost never true for financial data. The risk is underestimated when the assumption does not hold, thus the focus in the academic research has been on the improvement of volatility forecasts, which in turn would lead to more precise risk estimates and more

risk. Roy (1952) argued that a more realistic situation is when the individuals are not pursuing maximum gains, but reduction of the probability of losses, thus suggesting that a risk measure should reflect the downside variability in returns.

The risk management field has experienced huge growth and revolutionary changes in the past two decades. Many new methods for measuring and managing risks have been developed, with some of them even becoming industry standards and bases for regulation. One of such measures is the Value at Risk (VaR). After being made available to the public by JPMorgan Risk Metrics group in 1993, it was quickly adopted by the industry professionals and Basel II accord has chosen it as a recommended measure for market risk. The first comprehensive work on VaR was provided in the book of P. Jorion “Value at Risk: The New Benchmark for Managing Financial Risk” in 1996. Some of the attractive properties for which VaR owes its popularity are that it is an intuitive measure, it can be used to aggregate different risks (in contrast to volatility, where it makes little sense comparing volatilities of different types of risk), and it is very easy to implement.

VaR as a measure and metric of risk is the central topic of this thesis, which aims to study the current research and evidence, in order to assess the quality of VaR as risk measure for the purposes of regulation and risk management. The motivation for this study comes from the fact, that even though VaR has been criticized since the time it has been introduced for the lack of precision, unrealistic assumptions and inability to predict extreme events, it has become very popular among the Risk Management Professionals (RMPs) both in financial and non-financial companies. The latest financial crisis has shown that we still are not as good at assessing risk as we would like to be, and that there are significant reasons for revision of risk measures we use and the ways we use

the adequacy of the 20-year old risk measure in the modern world.

1.1 Problem Formulation Taking into account the latest developments in risk management and their failure during the Global Financial Crisis (GFC), this thesis aims to review the quality of Value-at-Risk measure and its alternatives, like Expected Shortfall (ES). Many methods for computing VaR exist, and this thesis will attempt to assess the ability of this risk measure to serve its purpose by relying on the characteristics of the most common VaR calculation methods.

The problem of VaR ability to forecast risk can be divided into several research questions, which are connected to the criticism of VaR in “The World According to Nassim Taleb” (1997), “Roundtable: "The Limits of VAR" (1998), “Private Profits and Socialized Risk” (2008), etc. These research areas are formulated as

**follows:**

1.2 Delimitations In the scope of this thesis, only VaR and ES measures are considered, being some of the most ubiquitous downside risk measures. Even though many different methods for estimation of these measures exist, only the most popular and praised in the financial literature will be considered. More detailed information about the calculation of risk measures will be discussed in the Theory Chapter.

VaR, but to assess the benefits and caveats of using it as a risk measure. The precision of different methods is outside the scope of this study, all the assumptions and discussions will be made assuming the best possible precision of the risk measures, based on the current literature. In other words, the model risk of risk measures will be researched.

It is not the ambition of this research to come up with a perfect measure of risk, or at least the one better than the ones currently used. The task is to find out where the VaR excels, where it falls short, and to identify the implications of these qualities for risk management practitioners and regulators.

1.3 Methodology The methodology of the thesis is based on selection, review and discussion of VaR-related conceptual and empirical literature. The main focus are the articles and risk management books published between 1996 and 2013. The methods and theories are discussed and compared in the context of their applicability in financial and non-financial institutions. These theories are compared to the empirical studies of VaR and related risk measures, as well as to regulations implemented by the Basel Committee on Banking Supervision (BCBS). The primary source of articles is Scopus, the largest citation and

**Abstract**

database of peer-reviewed literature. As a primary methodology and methods guide, “Research Methodology: Methods and Techniques” by Kothari (2011) is used.

The quality of VaR measures is assessed by the following criteria, which hare

**based on the research questions:**

The chosen criteria aim to review the statistical properties and the role of the measure in aiding the decision making process. The criterion ease of implementation aims to study the resource and information requirements of a risk measure as well as its applicability in the environment of limited computational resources and sample data.

Through systematic literature research and discussion of challenges in the field of risk management, the thesis aims Value-at-Risk and several alternatives as risk measures and metrics. The ultimate goal, in the light of the GFC, is to find out whether these tools are still adequate in the risk management field.

& ES calculations and provide a review of statistical properties of these risk measures. The aim of this section is to be the reference for some of the main concepts from Literature Review and Discussion sections. Mainly the statistical methods for VaR calculation are reviewed with the exception of Basel Committee’s recommendations on the use and backtesting of VaR models. This section aims to familiarize the reader with the concepts which are related to VaR popularity and critique. Dowd (1998), Jorion (2007; 2009), Danielsson (2011) provide more in-depth description and analysis of VaR methods.

2.1 Value-at-risk (VaR) VaR is calculated as a quantile of the distribution of gains and losses for a target forecast period. This quantile shows the worst possible loss over the chosen time period and confidence level. For example, calculating the 95% 1-day VaR means finding the 5% quantile of the lower tail of the 1-day returns distribution.

** Figure 1. 1-day VaR for S&P500 returns.**

Source: own calculations.

period and estimating the probability distribution. The latter leads to essentially two ways for calculating VaR: non-parametric and parametric.

The non-parametric, also called Historical Simulation (HS), VaR is very easy to implement if the price or return data is readily available. There is no need for any assumptions regarding the shape of the distribution of returns, only that they are identically and independently distributed (iid). The observed returns are arranged by size, and depending on the confidence level α and number of observations n, the observation following the first α% of returns is used in the calculation.

where P0 is the initial value of the position, µ is the mean expected return and r* is the cutoff return for the lowest α*n of returns.

In contrast, the parametric VaR assumes that the returns follow a certain distribution, which simplifies the calculation significantly if this distribution is from the parametric family. For example, for normal independently and identically distributed (niid) returns we need only the mean µ and the volatility