Department of Business Administration Master Thesis School of Economics Lund University Sweden Spring 2000

Value-at-Risk as a Risk Measurement Tool for Swedish Equity Portfolios

Tutor: Authors: Hossein Asgharian, Department of Economics Mikael Kärrsten Fredrik Olsson

Abstract Title: Value-at-Risk as a Risk Measurement Tool for Swedish Equity

Portfolios Seminar date: 2000-06-08 Subject: Master thesis 10 credits, Finance Authors: Mikael Kärrsten & Fredrik Olsson Tutor: Hossein Asgharian Purpose: The purpose of this Master thesis is to examine the applicability of

different VaR methods for Swedish equity portfolios. In addition, we will analyse if equity market cap has any impact on how well-functioning and reliable the VaR methods are. Based on these results we will discuss the implications of VaR for asset managers.

Method: To assess whether VaR can be considered as a reliable and stable

risk measurement tool for Swedish equity portfolios, we have performed a quantitative study. The study covers three different VaR approaches and seven methods for both the 95% and the 99% confidence levels. Further on, all results are evaluated using nine different performance criteria as well as statis tical significance and normality tests.

Conclusions: We can conclude that most VaR methods work well at the 95%

confidence level, while at the 99% level the results are more ambiguous. The methods based on the assumption about normally distributed returns produce attractive results for the OMX portfolio, but for the small-cap and mixed portfolios these methods tend to underestimate the VaR. Further on, our study shows that the portfolio returns are not normally distributed. Due to this fact we recommend the historical simulation approach, which does not rest on the assumption about normality. In addition, the historical simulation with a window size of 250 trading days produces the most attractive results for the small-cap and mixed portfolios. However, none of the VaR methods seem to produce totally perfect and reliable results. Therefore VaR can be questioned as a useful tool for asset managers managing Swedish equity portfolios.

Key words: VaR, distribution of financial returns, historical simulation,

equally weighted moving average, exponentially weighted moving average.

Table of contents 1. Introduction 1 1.1 Background…………………………………………………………………………... 1 1.2 Problem Discussion…………………………………………………………………... 2 1.3 Purpose………………………………………………………………………………. 3 1.4 Target Group ………………………………………………………………………… 4 1.5 Disposition…………………………………………………………………………… 4 2. Methodology 5 2.1 General methodology………………………………………………………………… 5 2.1.1 Choice of Subject…………………………………………………………………... 5 2.1.2 Perspective………………………………………………………………………5 2.1.3 Scientific Approach…………………………………………………………….. 6 2.1.4 Theory and Object……………………………………………………………….7 2.2 Practical Methodology……………………………………………………………….. 7 2.2.1 Primary Data……………………………………………………………………. 7 2.2.2 Secondary Data…………………………………………………………………. 8 2.2.3 Criticism of the Sources…………………………………………………………8 2.3 Validity……………………………………………………………………………….. 9 2.4 Reliability…………………………………………………………………………….. 9 2.5 Empirical Study………………………………………………………………………. 9 2.6 Criticism of Chosen Methodology…………………………………………………… 10 2.7 Alternative Methodology…………………………………………………………….. 10 3. VaR Theory 11 3.1 Risks………………………………………………………………………………….. 11 3.1.1 Value at Risk (VaR)……………………………………………………………..13 3.2 The Implications of VaR for Asset Managers………………………………………... 14 3.3 Normal Distribution of Financial Returns……………………………………………. 16 3.3.1 Skewness and Kurtosis…………………………………………………………. 18 3.4 VaR Approaches………………………………………………………………………19 3.4.1 The Historical Simulation Approach…………………………………………… 19 3.4.1.1 Advantages and Disadvantages of the HS Approach…………………….. 20 3.4.2 The Equally Weighted Average Approach……………………………………... 21 3.4.2.1 Advantages and Disadvantages of the EqWMA Approach………………. 21 3.4.3 The Exponentially Weighted Average Approach………………………………. 22 3.4.3.1 What Value of λ Should Be Used?……………………………………….. 24 3.4.2.2 Advantages and Disadvantages of the ExpWMA Approach……………... 24 3.4.4 The New Improved VaR Methodology………………………………………… 25 3.4.4.1 Advantages and Disadvantages of the Improved VaR Methodology…….. 26 3.4.5 Monte Carlo Simulation…………………………………………………………26 3.4.5.1 Advantages and Disadvantages with the MCS Approach………………... 27 3.4.6 Semi-parametric VaR Approach……………………………………………….. 27 3.4.6.1 Advantages and Disadvantages with the Semi-parametric VaR Approach. 28 3.4.7 The Stress Testing Approach…………………………………………………… 29 3.4.7.1 Advantages and Disadvantages with Stress Testing……………………… 29 3.5 Multi-day VaR Prediction……………………………………………………………. 30

4. Statistical Methodology 31 4.1 Working Process………………………………………………………………………31 4.2 VaR Methods Used in the Study……………………………………………………... 32 4.3 Calculation Procedure for the HS Approach…………………………………………. 33 4.4 Calculation Procedure for the EqWMA Approach…………………………………... 34 4.5 Calculation Procedure for the ExpWMA Approach…………………………………. 34 4.6 Performance Evaluation Criteria……………………………………………………... 35 4.6.1 Mean Relative Bias……………………………………………………………... 35 4.6.1.1 Calculation Procedure…………………………………………………….. 35 4.6.2 Root Mean Squared Relative Bias……………………………………………… 36 4.6.2.1 Calculation Procedure…………………………………………………….. 36 4.6.3 Annualized Percentage Volatility………………………………………………. 36 4.6.3.1 Calculation Procedure…………………………………………………….. 36 4.6.4 Fraction of Outcomes Covered…………………………………………………. 37 4.6.4.1 Calculation Procedure…………………………………………………….. 37 4.6.5 Multiple Needed to Attain Desired Coverage………………………………….. 37 4.6.5.1 Calculation Procedure…………………………………………………….. 38 4.6.6 Average Multiple of Tail Event to Risk Measure………………………………. 38 4.6.6.1 Calculation Procedure…………………………………………………….. 38 4.6.7 Maximum Multiple of Tail Event to Risk Measure……………………………..38 4.6.7.1 Calculation Procedure…………………………………………………….. 39 4.6.8 Correlation between Risk Measure and Absolute Value of Outcome………….. 39 4.6.8.1 Calculation Procedure…………………………………………………….. 39 4.6.9 Mean Relative Bias for Risk Measures Scaled to Desired Level of Coverage…. 40 4.6.9.1 Calculation procedure…………………………………………………….. 40 4.7 Hypothesis Testing…………………………………………………………………… 40 4.7.1 Actual Portion of Fraction of Outcomes Covered……………………………… 41 4.7.2 Difference in FoOC between OMX and Small-cap Portfolios…………………. 42 4.7.3 Significance Test of the Correlation Coefficients……………………………….42 4.8 Normality Tests………………………………………………………………………. 43 4.9 Criticism of Primary Data……………………………………………………………. 44 5. Results 45 5.1 Mean Relative Bias…………………………………………………………………... 45 5.2 Root Mean Squared Relative Bias……………………………………………………. 46 5.3 Annualized Percentage Volatility…………………………………………………….. 47 5.4 Fraction of Outcomes Covered………………………………………………………..47 5.4.1 Significance Testing……………………………………………………………..48 5.4.1.1 Fraction of Outcomes Covered………………………………………….... 49 5.4.1.2 Difference between the OMX and Small-cap Portfolios…………………. 49 5.5 Multiple Needed to Attain Desired Coverage………………………………………... 50 5.6 Average Multiple of Tail Events to Risk Measure…………………………………… 50 5.7 Maximum Multiple of Tail Event to Risk Measure………………………………….. 51 5.8 Correlation between Risk Measure and Absolute Value of Outcome………………...52 5.8.1 Significance Testing……………………………………………………………. 52 5.9 Mean Relative Bias for Risk Measures Scaled to Desired Level of Coverage………. 52 5.10 Normality Tests……………………………………………………………………... 53 5.10.1 Results from the Normality Tests……………………………………………... 53 6. Conclusions 57 6.1 Evaluation of VaR Methods………………………………………………………….. 57 6.2 The Distribution of Financial Returns………………………………………………... 58 6.3 Implications of VaR for Asset Managers…………………………………………….. 59 6.4 Suggestions for further Research……………………………………………………...61

List of references Appendices List of Tables Table 1. The number of historical observations used by the ExpWMA approach…………... 23 Table 2. Results from normality tests…………………………………………………………54 List of Figures Figure 1. Normal vs leptokurtic distribution…………………………………………………. 17 Figure 2. Normal distributions with different variances……………………………………... 25 Figure 3. A random normal distribution plotted against the OMX portfolio returns…………54 Figure 4. A random normal distribution plotted against the small-cap portfolio returns……..55 Figure 5. A random normal distribution plotted against the mixed portfolio returns………... 55

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Chapter 1 - Introduction

1.1 Background

While finance is about risk/return and risk management, the specialized study of risk is a rather recent phenomenon1. It has become a critical issue over the last decade since organizations have suffered great losses, often from risks they never should have taken in the first place2. The most well known example of this is probably the collapse of Barings Bank in 1995, that was caused by the Singapore based derivatives trader Nick Leeson, who took large positions in futures and options on Asian Stock Exchanges3. Other internationally well known companies that have been seriously hurt by insufficient risk management techniques are the German commodity trading firm Metallgesellschaft in 1993 and Summito Corp. in 1996, that lost more than 1.8 billion USD through unauthorized copper trades4. In Sweden, Electrolux lost 250 million SEK on currency trading and Meritanordbanken lost 290 million SEK taking short positions in stocks. Although these losses to a large extent can be labelled as fraud, they are also results of an unsatisfactory risk communication system.5 Today the financial system is very vulnerable, since the solidity in the banking industry is as low as a few percentage points. One way to solve this problem would be to raise the capital base in the financial sector, so banks more easily could cope with unanticipated disturbances and falls on the financial markets. On the other hand, to keep excess capital is costly and it has to be paid by someone, most certainly the clients of the banks. The solidity has become a less important measure however, since many risks are outside the balance sheet. Therefore a risk measure of the total risk exposure is needed.6 The number one tool in this respect has become Value-at-Risk (from now on mentioned as VaR), which today is used by all US Commercial Banks to monitor trading portfolios on a daily basis 7. The VaR method is basically a statistical estimation which measures, at a certain confidence level, the amount that may be lost within a certain time period, due to potential changes in the market prices of the underlying assets8.

1 Danielsson, J., “Class notes Corporate Finance & Financial Markets”, (1998-1999), p. 1. 2 Thornberg, J., “Derivative users lack refined controls of risk”, (1998), p. 2. 3 Koupparis, P., “Barings – A Random Walk to Self-Destruction”, (1995), p. 3. 4 Danielsson, J., “Class notes Corporate Finance & Financial Markets”, (1998-1999), p. 4-5. 5 Björklund, M., “Bristande kontroll möjliggör svindlerier”, (2000), p. 7. 6 Bäckström, U., “Betydelsen av riskhantering”, (2000), p. 1-2. 7 Jorion, P., “In Defense of VAR”, (1997), p.1. 8 Yiehmin Lui, R., “VaR and VaR derivatives”, (1996), p. 2.

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1.2 Problem Discussion The VaR measure is based on a number of assumptions to facilitate the calculations, where the most important assumption is how the distribution of returns is viewed. Most VaR models use a normal distribution to characterise the distribution of returns, and historical returns are mainly used to make predictions about the future. However, there has been criticism that the VaR is based on assumptions that do not hold in times when the financial markets are experiencing stress, and that the normal distribution does not make a good job in predicting the distribution of outcomes.9 Research has found that financial returns experience fat tails, which implies that the normal distribution works well in predicting frequent outcomes but is not a good estimator to predict extreme events10. In 1997 Alan Greenspan expressed his concern for this problem by stating “…as you well know, the biggest problems we now have with the whole evolution of risk is the fat-tail problem, which is really creating large conceptual difficulties.”11. As an effect of the fat-tail problem new VaR models have been proposed to use as estimators for the distribution of outcomes, but these are complex and have not been fully accepted yet. To further complicate the VaR calculations, the volatility on the stock market is not constant over time. Evidence show that financial returns experience clusters of high volatility, i.e. a day with a large absolute outcome is followed by another day with a large absolute outcome.12 In addition, the volatility on the Stockholm stock exchange has increased during the second half of the 90s and today there is a larger portion of “glamour” stocks, i.e. stocks with low book to market and high p/e-ratios13,14. These stocks have a more volatile share price development, since the time horizon for their expected profits are longer than for other companies and they are more dependent on future expectations. In addition, there might be a difference in how applicable the VaR models are for equities with different market capitalisation (market cap). Smaller companies’ distribution of returns might differ from larger companies, since they might not be as frequently traded, and new information regarding these stocks can have a greater impact on the share price.

9 Bäckström, U., “Betydelsen av riskhantering”, (2000), p. 3. 10 See for instance JPMorgan/Reuters, RiskMetrics – Technical Document, (1996), p. 64., or Dowd, K., “A Value at Risk Approach to Risk -Return Analysis”, (1999), p. 66. 11 Danielsson, J., “Class notes Corporate Finance and Financial Markets”,(1998-1999), p. 11. 12 JPMorgan/ Reuters, RiskMetrics - Monitor (1996), p. 9. 13 See appendix 1. 14 Affärsvärlden, “Riktningen och värde avgör svängningarna”, (1998), p. 28.

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Out of an asset manager perspective the portfolio risk is one of the most decisive parameters to have perfect control over. A well- functioning VaR measurement method could therefore be a superior way to supervise the portfolio risk and quantify potential losses. However, as stated before there are several potential problems that can make VaR an unstable and perhaps unreliable method, where the risks most crucial to capture, i.e. extreme events, are the most difficult to cover. A well- functioning VaR measurement method could also serve as a communication tool between customers/management and the asset manager. Even if the VaR calculations are quite complex for the general public, the results per se are easy to understand. For an asset manager the discreteness regarding the portfolio holdings is important, since the holdings determine the level of competitiveness. VaR could therefore be a way of comparing portfolio risks between asset managers without unmasking too much of the holdings. Some previous studies have been made in this area, but these have mostly been focused on the American financial markets. In addition, none of the studies we have seen have tested different asset characteristics. We have tried to take one step further by looking at possible impacts of equity market cap on the reliability of VaR. From the discussion above we ask the following questions: § Which VaR method is the most applicable as a risk measurement tool for

Swedish equity portfolios? § Are there any differences in how reliable and useful different VaR methods

are with respect to portfolios consisting of shares with different market caps? § Is VaR a useful tool for asset managers to monitor risk?

1.3 Purpose The purpose of this Master thesis is to examine the applicability of different VaR methods for Swedish equity portfolios. In addition, we will analyse if equity market cap has any impact on how well- functioning and reliable the VaR methods are. Based on these results we will discuss the implications of VaR for asset managers.

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1.4 Target Group This Master thesis is intended for people with at least a basic knowledge in finance. The target group is mainly people working with asset management and others with an interest in financial economics.

1.5 Disposition In chapter 2 the methodology, both in a general and a practical perspective, is outlined. We will further discuss the primary and secondary sources that have been used. In the initial stage of chapter 3 different types of risk are presented. Then we will describe the usefulness of VaR for asset managers and present theories regarding the distribution of financial returns. Finally, we will outline seven different VaR approaches, with a particular focus on the three we use in our study. Chapter 4 describes how the study was conducted. We will in detail present our procedure for collection of data, construction of portfolios, choice of VaR methods and calculations. In addition, the performance criteria, normality and statistical significance tests that have been used in the study are described. In chapter 5 the results of the study are presented. Based on the performance criteria, the normality and significance tests presented in the previous chapter we will analyse and comment on our results. In chapter 6 we will present the conclusions of our study. The applicability of the VaR methods as well as the implications for asset managers are discussed.

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Chapter 2 - Methodology The methodology chapter is one of the elementary parts of an academic paper and what is written should be possible to be evaluated and replicated15. This means that the content of the paper should be open for questions and by repeating the same investigation the same result should be reached. This chapter describes our approach to reach the purpose and goal of this Master thesis and how we have tackled the subject. We discuss the validity as well as the reliability of the sources and round off by giving proposals of other ways in which the subject could have been addressed. A detailed description of the methodology and data used in the VaR tests is presented in chapter 4.

2.1 General methodology 2.1.1 Choice of subject VaR is a risk measurement approach that since its breakthrough in the beginning of the 90s has become increasingly popular, especially in the banking industry. Several examinations have been done over the VaR concept with many different perspectives, but we have not identified any research of VaR with a focus on the Swedish equity market. In addition, the topics stock market risk and equity risk management are very timely, both due to recent large losses and the increasing volatility on the Swedish stock market during the last five years. We find the combination of VaR for Swedish equities and the risk topics per se to be very appealing, which explains the choice of subject for this Master thesis. 2.1.2 Perspective We have chosen to write this Master thesis from the perspective of an asset manager managing equity portfolios. This is because we find this perspective to be the most interesting one, since asset managers are likely the actors on the stock market that will benefit the most from a risk measurement tool when stock markets are becoming increasingly more volatile.

15 Backman, J., Att skriva och läsa vetenskapliga rapporter, (1985), p. 27.

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2.1.3 Scientific approach There are two main scientific ways to view a problem, the positivistic and the hermeneutic 16. The positivistic is basically a rationalistic view that has its roots in the growing scientific society in the 18th century. The central point is that there is a reality that we can get knowledge about by observation. The knowledge is neutral and totally objective, i.e. without personal conception. 17 By experiment, quantitative measurement and logical reasoning, these theories are built and can then be converted into hypotheses that can be tested. Statements should be presented with clear definitions and a logical as well as an analytical approach. 18 The hermeneutic approach on the other hand is based on a view that, on the contrary to nature laws, there are no laws for human behaviour or for society that are constant 19. The dialogue between people plays a central role and the hermeneutic approach means that the scientist should try to understand other scient ists’ actions and get a general picture of the subject20. Both of these approaches could be criticized and many scientific studies contain both of these approaches linked together. The positivistic approach could be regarded as too simplistic, but can in some cases be a complement to the hermeneutic approach, for which it is hard to control the reliability. 21 In this Master thesis we will base our research on a quantitative study and therefore we will take a positivistic approach. Hence, we will try to logically analyze our results from our study of the VaR for stocks located on the Stockholm stock exchange. In the interpretation of the study we will try to bring up our findings to a more general level and discuss how they can be of assistance to asset managers. However, asset managers are also influenced by many other variables, for example risk attitude and how they view the stock market. Therefore, we should have a more hermeneutic approach in our discussion of the implications of VaR for asset managers.

16 Svenning, C., Metodboken , (1996), p. 25. 17 Halvorsen, K., Samhällsvetenskaplig metod, (1992), p. 14. 18 Wiedersheim-Paul F. & Eriksson L., Att utreda forska och rapportera , (1991), p. 150. 19 Halvorsen, K., Samhällsvetenskaplig metod, (1992), p. 14. 20 Wiedersheim-Paul F. & Eriksson L., Att utreda forska och rapportera , (1991), p. 151. 21 Ibid, p. 151.

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2.1.4 Theory and object To simplify the reality it is common to use different kinds of theories and models. These theories and models also facilitate how to evaluate our findings. Theories are often more general than models, since theories can incorporate more variables and refer to longer periods of time. Models on the other hand are a development of the theories and aim at specifying what the theories have built up, so that the models can be used in practice. Models can also be built without any support from theories, especially in undeveloped areas.22 In chapter 3, we have chosen to present theories regarding different kinds of risk, VaR as a concept, the implications of VaR for asset managers, the distribution of financial returns and different approaches for estimating VaR. The theories we present reflect the perspective we have selected for this Master thesis. The object we have chosen to study is, as mentioned before, the Stockholm stock exchange. Stocks on the most traded list and the A-list as well as smaller companies traded on the over-the-counter (OTC) list and the O-list are also included in the study.

2.2 Practical methodology Practical methodology involves how data is collected and how it is evaluated and analyzed. This data can be of two kinds, primary and secondary data23. 2.2.1 Primary data The primary data that has been used in this Master thesis is exclusively the processed historical time series stock data that are used to compute different VaR measures. Put in its context this data gives information on how applicable different VaR methods are for equities listed on the Stockholm stock exchange. The data has been collected from the Bloomberg database and was then processed to get data files that could be handled in Excel. A more thorough description on how this research was done is found in chapter 4.

22 Halvorsen, K., Samhällsvetenskaplig metod, (1992), p. 44. 23 Wiedershiem-Paul F. & Eriksson L., Att utreda forska och rapportera , (1991), p. 76.

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2.2.2 Secondary data Articles in economic journals, articles published on the Internet and financial as well as statistical textbooks have been used as a base for this Master thesis. To find the articles we have searched the databases EconLit and AffärsData. Especially Internet proved to be a useful source where many researchers publish their articles. 2.2.3 Criticism of the sources Primary data – The criticism of the primary sources will be discussed in chapter 4, section 4.9. Secondary data – To evaluate the secondary sources three criteria could be used24. The first criterion is to analyze how current the sources are. Since VaR is a relatively new topic that has not been around for long it makes it even more important that the sources are up to date. In our case, we believe that this criterion is met because most of the articles and other sources we have used were published in the mid or late 90s. The second criterion is to evaluate if the authors of the sources have any interests of their own in the subject they are writing about. We have mainly used scientific research articles and hence, we believe that we meet the criterion on this point. However, some of our sources are published by the RiskMetrics group and since this is a profit making company it is possible that they try to promote their way of viewing VaR models. Therefore, these sources have to be interpreted carefully. The third criterion is to investigate if the sources have any relation with one another. Some of our articles are written by the same person or organization. In addition, the sources have made references to each other and hence, they are not totally independent. However, to try to minimize this problem we have used numerous different sources and authors that are well know and accepted. A large part of the secondary data is collected from American studies, which is natural since VaR is a frequently used method in the US. This can imply that these studies are difficult to translate to Swedish circumstances. However, there are no proper alternatives since VaR is a relatively new phenomenon and the American financial markets are normally in the forefront regarding new issues.

24 Wiedersheim-Paul F. & Eriksson L., Att utreda forska och rapportera , (1991), p. 82.

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2.3 Validity In an evaluation of the sources other terms that should be discussed are validity and reliability. Validity can be defined as a measurement tool’s ability to measure what it is intended to measure.25 In our case it means that we have to think about if the theory we have used to build models and the data we have collected for the study actually make us reach the purpose of this Master thesis. VaR research is developing rapidly and hence, there are no absolute truths about which theories and models are the correct ones. However, we have tried to use the latest findings in the area to incorporate what is known about VaR so far. In addition, the theories and models that our study is based on are to a large extent well known and very well accepted in finance.

2.4 Reliability The term reliability means that the measurement tools should give trustworthy and stable results26. The methodology we have used should be able to be used by others and the same result should be reached. We believe that the data we have used is reliable and the fact that it had to be somewhat processed, e.g. filling in missing values on days when no shares changed hands and correcting for dividends, to get the data series does not change this opinion. Another thing that could be questioned is the randomness of the companies chosen for the mixed and small-cap portfolios. We have used Excel to construct the portfolios and hence, no bias should enter these portfolios.

2.5 Empirical study We have put together three different stock portfolios, each containing ten stocks. The VaR for these portfolios is estimated using three different approaches, historical simulation, equally weighted moving average and exponentially weighted moving average. In addition, we have used different window lengths, i.e. the number of observations, to calculate the volatility. One approach in combination with a specific window is referred to as a method. The portfolio outcomes are evaluated from 1995-01-01 to 1999-12-31. However, to estimate the volatility for 1995 historical data for 1994 has been used. 25 Wiedersheim-Paul F. & Eriksson L., Att utreda forska och rapportera , (1991), p. 28. 26 Ibid, p. 29.

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Hence, the total time series of historical stock data ranges from 1994-01-01 to 1999-12-31. To evaluate our findings nine performance criteria are used27. A detailed description on how the study was performed and evaluated can be found in chapter 4.

2.6 Criticism of chosen methodology By using numbers and statistical methods there is a chance to both get precise information, but also to get distorted and therefore inappropriate information28. To be able to use the findings in the right way the data of course have to be collected and treated in an appropriate manner since mistakes can lead to misinterpretations that affect the whole study.

2.7 Alternative methodology To reach the purpose of this Master thesis we believe that there are no proper alternatives for using a quantitative study. A qualitative study would never give us the results we are looking for. However, it is possible that other approaches should have been included in calculating the VaR. We have used the approaches that are most established in financial theory today and that have been used in studies with similar purposes. Regarding the discussion whether VaR is a useful measure for an asset manager, perhaps an opinion poll could be of interest to get a direct viewpoint on the usefulness of VaR in practice. Instead we chose to put the focus on whether VaR calculations are trustworthy in the first place, rather than examine the interest from the market before we know if VaR per se is a useful tool.

27 These are found in Hendricks, D., “Evaluation of Value-at-Risk Models Using Historical Data”, (1996). 28 Wiedersheim-Paul F. & Eriksson L., Att utreda forska och rapportera , (1991), p. 69.

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Chapter 3 - VaR Theory In this chapter the theories which the study is built on are outlined. At first a broad view on different types of risk is given. Next the chapter focuses on market risk, in particular the term VaR is presented, and the implications of VaR for asset managers are described. Further, the distribution of financial returns is discussed, since this is a fundamental point for VaR. Finally, different approaches for measuring VaR are presented in detail.

3.1 Risks Risk per se can be defined as the vola tility of unexpected outcomes, generally for values of assets and liabilities29. Specialized studies of risk are a rather new phenomenon, but recent large losses such as Orange County, Barings and Metallgesellschaft have motivated a rapid development in specialized techniques for risk management 30. This Master thesis is specializing on a financial risk called market risk, which involves the uncertainty of earnings resulting from changes in market conditions such as asset prices, interest rates, volatility, and market liquidity31. However, market risk is just one form of risk to which participants in the financial market are subject to. The major types of risk can briefly be defined as:32,33

• Business risk – a firm or industry specific risk, e.g. technological innovations, product design, and marketing. Firms are mostly specializing in this type of risk. • Strategic risk – risks resulting from fundamental shifts in the economy or the political environment, e.g. the rapid disappearance of the threat of the Soviet Union, which led to a world-wide gradual build-down of defence spendings directly affecting the defence industries. • Financial risk – relates to possible losses in financial markets arising from, for example, movements in interest rates and exchange rates. The volatility of financial variables is the single most important reason for the development of a risk management industry. Financial risk can be divided into the following five types of risk.

• Market risk – arises from changes in the prices of financial assets and liabilities and can be defined as the risk of losses due to adverse market

29 Jorion, P., Value at Risk , (1997), p. 3. 30 Danielsson, J., “Class notes Corporate Finance & Financial Markets”, (1998-1999), p. 1. 31 JP Morgan/RiskMetrics group, Introduction to RiskMetrics, (1995), p. 2. 32 Danielsson, J., “Class notes Corporate Finance & Financial Markets”, (1998-1999), p. 1-2. 33 Jorion, P., Value at Risk , (1997), p. 3-18.

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conditions. Market risk includes basis risk and gamma risk34. Furthermore, market risk can be absolute, the loss measured in dollar terms, or relative, the loss relative to a benchmark index. • Credit risk – is defined as the risk of a loss due to the inability of a counterparty to meet its obligations. Credit risk can also lead to losses when debtors are downgraded by credit agencies, usually leading to a fall in the market value of its obligations. Further on, credit risk also includes sovereign risk and settlement risk. The former can for instance be if a country imposes a foreign-exchange control system, which will limit counterparties obligations. The latter refers to the risk that a counterpart cannot fulfil its obligations after one party has already made payment. • Liquidity risk – can take two forms: market/product liquidity and cash flow/funding. The former type of risk arises when a transaction cannot be conducted at prevailing market prices due to insufficient market activity and poor depth and resiliency in the market. The latter type of risk is associated with the inability of a firm to fund illiquid assets or to meet cash flow obligations, which may force early liquidation. • Operational risk – the risk from the failure of internal systems such as management failure, fraud, and errors made in instructing payments or settling transactions. • Legal risk – risk of changes in regulations or when a counterparty does not have the legal or regulatory authority to engage in a transaction.

The first proposal on market risk was constructed in 1993 and called the “Basle on market risk”. It was a building block approach and a start by the authorities to set up rules and regulate market risks. The proposal was extended in April 1995 to become the “1995 Basle proposal on market risk”.35 This proposal is a reflection of the authorities willingness to prevent systemic risk and it contains a recommendation how to calculate VaR, which states that:36,37 • VaR should be calculated on a daily basis. • VaR should be based on a 10 trading day holding period. • A 99% confidence interval should be used, i.e. the chance of experiencing a

larger loss than the VaR should be 1 in 100 or less. • A historical observation period of at least one year (250 days), with at least

quarterly updates, should be incorporated.

34 Basis risk = the risk of a change or breakdown of a relationship between products used to hedge each other. Gamma risk = the risk due to nonlinear relationships. 35 Styblo Beder, T., “VaR: Seductive but Dangerous”, (1995), p. 17. 36 Maymin, Z., “VaR variations: is multiplication factor still too high?”, (1998), p. 1. 37 Danielsson, J., “Class notes Corporate Finance & Financial Markets”, (1998-1999), p. 7.

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3.1.1 Value at Risk (VaR) The primary tool for market risk is VaR, which is a method of assessing risk through standard statistical techniques. Philippe Jorion defines VaR as a measure for the worst expected loss over a given time interval under normal market conditions at a given confidence level38. Formally VaR is defined as:

∫∞−

=VaR

dxxf )(α or [ ] α=≤ VaRxPr (1)

where x stands for the change in the market value of a given portfolio over a given time horizon with the probability α. Either of the equations states that a loss equal to, or larger than the specific VaR occurs with probability α.39 The inputs used to calculate VaR for a certain asset are the volatility, time horizon and a choice of confidence level. The volatility is estimated implicitly from option pricing or through statistical models. In practice, past observations are often used to estimate the future volatility. The time period chosen affects both the measured volatility and therefore also the VaR, where a longer time period gives a higher volatility measure and hence, a higher VaR. The chosen confidence interval states how often the loss on the specific asset will be greater than the VaR. The most commonly used confidence intervals are 95% and 99%.40 The formula to calculate VaR for one asset is:

MVVCIVaR **= (2) where CI is the confidence interval, V is the volatility and MV is the market value.41 The formula above gives an absolute amount in a certain currency, stating the maximum loss for one asset at a given confidence level. Further, for a portfolio of multiple assets the correlation between the portfolio assets has to be taken into cons ideration. To calculate the portfolio VaR the formula (3) below is used:

2,1212

22

1 ***2)()( ncorrelatioVaRVaRVaRVaRVaRport ++= (3)

38 Jorion, P., Value at Risk , (1997), p. xiii. 39 Danielsson, J. & de Vries, G. C., “Value-at-Risk and Extreme Returns”, (1997), p. 9. 40 Söderlind, L., Att mäta ränterisker, (1996), p. 70-75. 41 Ibid, p. 77.

14

where VaR1 is the VaR for the first asset and VaR2 is the VaR for the second asset. To calculate VaR for a portfolio of more than two assets a row vector (4), which is transposed to a column vector (6), and a correlation matrix (5) are used in formula (7). This can be illustrated with the formulas below: [ ]321 VaRVaRVaRR = (4)

=Ω1

11

2,31,3

3,21,2

3,12,1

CorrCorrCorrCorrCorrCorr

(5)

=

3

2

1

'VaRVaR

VaR

R (6)

The formulas above help us to calculate the VaR of the portfolio as:

'** RRVaRport Ω= (7)

To calculate the VaR for a portfolio of more than three assets these are simply added to the vectors and matrices. 42

3.2 The implications of VaR for Asset Managers Regarding the perspective on this Master thesis, i.e. if the VaR concept is a useful tool to forecast and measure portfolio risk for asset managers managing Sweden based equity portfolios, we have used Culp’s, Mensink’s and Neves’ article “VaR for Asset Managers” as a guideline. VaR is not nearly as well accepted in the institutional investment area as it is elsewhere. The main reason is that asset managers are typically in the business of taking risks, which is directly linked to the aim and effo rt to create excess returns. Therefore asset managers often view risk management in general and VaR in particular as inherently at odds with their primary business mandate. 43 The increased volatility in financial markets and the vast amount of large losses during

42 Söderlind, L., Att mäta ränterisker, (1996), p. 81. 43 Culp, L. C., Mensink, R. and Neves, M. P. A., “VaR for Asset Managers”, (1999), p. 1.

15

the 1990s have further motivated and increased the interest for the understanding of risk44. VaR will never tell an asset manager how much risk to take, it will only tell how much risk is being taken. VaR can be a useful tool for helping asset managers determine whether the risks they are exposed to are the risks to which they think they are and want to be exposed to. Out of an investor perspective VaR is a concept that is easy to understand and thereby a way to monitor the level of risk exposure the asset managers are undertaking. Culp, Mensink and Neves outline four concrete applications of VaR for asset management. These applications involve: Monitoring45 - VaR facilitates a consistent and regular monitoring of market risk both at the aggregate fund level, as well as by asset class, and by issuer/counterparty. VaR also facilitates the comparison of risks of one asset manager’s holdings with other asset manager’s holdings. Since asset managers’ portfolio holdings are not transparently available to investors at all times, the VaR reported to investors can help assuage any investors’ concerns about market risk without necessitating disclosure of portfolio holdings. Especially to ensure investors that market risk is within the specified risk tolerance level of the investment pool. Elimination of ex ante transactional approval requirements46 – VaR can be beneficial for asset managers that wish to eliminate transactional scrutiny by senior management. When the inspection process is removed asset managers reach a higher level of autonomy. Risk Targets and Thresholds47 – this application of VaR involves measurement and monitoring of market risk using a formal system of pre-defined risk targets and thresholds. A system of risk thresholds surrounds all potential investments w.r.t. the asset managers’ investment policy and risk tolerance. The thresholds act as a tripwire defined in terms of the maximum tolerable VaR. The VaR is monitored by regularly comparing actual VaR figures to the pre-defined targets and when a tripwire is hit a reallocation of the portfolio is called for. Risk Limits and Risk Budgets48 – risk limits, also known as a risk budget, is an extreme version of risk targets and risk thresholds. In a risk budget, the portfolios’ total VaR is calculated and then allocated to asset classes. Asset managers are not 44 Danielsson, J., “Class notes Corporate Finance & Financial Markets”, (1998-1999), p. 1. 45 Culp, L. C., Mensink, R. & Neves, M. P. A., “VaR for Asset Managers”, (1999), p. 14-19. 46 Ibid, p. 19-20. 47 Ibid, p. 20-21. 48 Ibid, p. 21-22.

16

allowed to exceed this risk budget as long as the risk dimensions not have been altered. However, the greatest benefit of VaR for an asset manager, according to Philippe Jorion, probably lies in the imposition of a structured methodology for critically thinking about risk. Institutions applying VaR are forced to confront their exposure to financial risk. A well- functioning supervision of VaR should logically also imply less risk of unexpected and uncontrolled losses. Jorion also states that “there is no doubt that VaR is here to stay”, but at the same time highlights that the process and methodology of calculating VaR may be as important as the number itself.49

3.3 Normal distribution of financial returns In most theoretical and empirical work regarding financial returns a normal distribution is assumed, since by assuming that the returns are normally distributed it simplifies all calculations. In addition, it produces tractable results and all moments of positive order exist.50 Further on, the normal distribution is characterised by its mean and variance and by only knowing these two variables you know the entire distribution. Mathematically this is, for a random variable rt, given by the density function below:

−−=

2

2

2 2

)(exp

2

1)(

σ

µ

πσt

t

rrf (8)

where µ is the mean and σ2 is the variance of rt.51 However, these advantages have to be weighed against research showing that the distribution of returns in financial markets experience fat tails52. Financial returns generally exhibit a leptokurtic behaviour and extreme price movements occur more frequently than what is given by the normal distribution53. A leptokurtic

49 Jorion, P., “Value at Risk”, (1997), p. xv. 50 Lucas A. & Klaassen P., “Extreme Returns, Downside Risk and Optimal Asset Allocation”, (1998), p. 71. 51 Hill C., Griffiths W. & Judge G., Undergraduate Econometrics, (1997), p. 30. 52 Hendricks, D., “Evaluation of Value-at-Risk Models Using Historical Data”, (1996), p. 41. 53 JPMorgan/Reuters, RiskMetrics – Technical Document, (1996), p. 64.

17

distribution implies that the distribution has a high peak, the sides are low and the tails are fat, see further 3.3.1 54. This is illustrated in figure 1 below:

Figure 1. Normal vs leptokurtic distribution

Since VaR is concerned with unusual outcomes, e.g. five or one percent, the fact that tails are fat pose a problem. More outcomes than predicted by the normal distribution will fall into the category that exceed the VaR measures generated with normal distribution, i.e. the assumption of normal distribution underestimates the VaR. This is proved by Lucas and Klaassen who show that the normal distribution underestimates VaR by more than 30 percent at the 99% level55. However, it is not necessary that the fat tails lead to a higher VaR for all confidence intervals, because there are two effects working in opposite directions. Firstly, the probability of increasing tail events lead to more careful asset allocations. Secondly, the increase in the precision of the distribution gives a higher certainty of the spread of outcomes and therefore lead to a more aggressive strategy. Lucas and Klaassen show that the latter effect dominates on the 95% level, while the first effect dominates on the 99% level. This means that by using a leptokurtic function the asset allocation becomes more risky when using the 95% level, but more careful when using the 99% level. 56

54 Danielsson, J., “Class notes Corporate Finance & Financial Markets”, (1998-1999), p. 4. 55 Lucas A. & Klaassen P., “Extreme Returns, Downside Risk and Optimal Asset Allocation”, (1998), p. 72. 56 Ibid, p. 75.

Distributional functions

Standard deviations

NormalLeptokurtic

18

3.3.1 Skewness and Kurtosis The normal distribution is symmetric with the mean equal to the median. The distribution is also called mesokurtic. Departure from symmetry usually implies a skewed or kurtosis distribution. Skewness is a measure of the degree of asymmetry of a frequency distribution. Positive skewness, or right-skewed, is an indication of a distribution with an asymmetric side that is expanding towards more positive numbers. Negative skewness, or left-skewed, implies the opposite, i.e. a distribution that stretches asymmetrically to the left.57 The formula for skewness is:

∑=

−

−

−=

n

i x

i

sxx

nnn

xSk1

3

11

2)( (9)

where s is the standard deviation, n is the number of observations, xi is the observed variable at time i and x is the mean of all observations 58. Kurtosis is a measure of the flatness versus peakedness of a frequency distribution. In statistics flat is called platykurtic and peaked is called leptokurtic. A positive kurtosis indicates a relatively leptokurtic distribution, while a negative kurtosis indicates a relatively platykurtic distribution. 59 The formula to calculate the kurtosis is the following:60

∑=

−

−

−−

+=

n

i x

i

sxx

nnnnn

xKur1

4

11

)3)(2()1(

)( (10)

Data is symmetrical when there are no repeated extreme values in a particular direction, i.e. low and high values balance each other out61. If the data is normally or symmetrically distributed, the computed skewness will be close to zero and the kurtosis close to three62.

57 Aczel, A. D., Complete Business Statistics, (1993), p. 19. 58 Kleinbaum, D. G., Kupper, L. L. & Muller, K. E., Applied Regression Analysis and Other Multivariable Methods, (1988), p. 188. 59 Aczel, A. D., Complete Business Statistics, (1993), p. 20. 60 Kleinbaum, D. G., Kupper, L. L. & Muller, K. E., Applied Regression Analysis and Other Multivariable Methods, (1988), p. 188. 61 Afifi, A. A. & Clark, V., Computer-Aided Multivariate Analysis, (1990), p. 66. 62 Benerson, M. L. & Levine, D. M., Basic Business Statistics, (1992), p. 73.

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3.4 VaR approaches In the following sections we will present different approaches for VaR estimation. Firstly, the three approaches performed in our study are outlined in detail and secondly, other approaches that can be used for VaR estimation are presented more briefly. 3.4.1 The Historical Simulation approach Historical simulation (HS) is a non-parametric VaR-method which rests on the assumption that historical returns are a good guide for future returns, i.e. one use observed asset returns as a proxy for future returns. Hence, the HS does not rest on the assumption about normally distributed returns or serial independence, but instead on an empirical distribution of returns. In addition, the distribution of the returns in the portfolio should be constant over the sample period.63 A sample length, a window, for the estimation is chosen and for each day t in the sample the portfolio return t∆Π is evaluated w.r.t. historical prices and portfolio

weights w according to:

∑=

=∆ΠN

itiit yw

1, (11)

where N is the number of assets in the portfolio and tiy , is the return on asset i at

time t. Then the portfolio returns t∆Π should be sorted in ascending order. A level

of confidence and a given probability π is chosen and π=<∆Π )Pr( VaRt states

that a loss equal to, or larger than the specific VaR occurs with probability π . For example, if there were 100 observations, the 5th lowest observation value would be the VaR for a 95% confidence interval. 64,65 There is a trade-off regarding the length of the observation period chosen. Clearly the choice of for example 125 days is motivated by the desire to capture short-term movements in the underlying risk of the portfolio and in contrast the choice of 1250 days may be driven by the desire to estimate the historical percentiles as accurately as possible66. While longer intervals increase the accuracy of estimates

63 Danielsson, J. & de Vries, G., C., “Value-at-Risk and Extreme Returns”, (1997), p. 10. 64 Danielsson, J., “Class notes Corporate Finance & Financial Markets”, (1998-1999), p. 18. 65 Danielsson, J. & de Vries, G., C., “Value-at-Risk and Extreme Returns”, (1997), p. 9. 66 Henricks, D., “Evaluation of Value-at-Risk Models Using Historical Data”, (1996), p. 43.

20

it could use irrelevant data, thereby missing important changes in the underlying process67. HS expresses the distribution of portfolio returns as a bar cha rt or histogram of hypothetical returns. Each hypothetical return is calculated as what would be earned on today’s portfolio if a day in the history of market prices were to repeat itself. The VaR can then be read from this histogram. 68 3.4.1.1 Advantages and disadvantages with the HS approach The advantages of HS are mainly that it is intuitive, easy to implement and that it does not rely on specific assumptions about valuation models or the underlying stochastic structure of the market. Further, the method is relatively easy to perform in a spreadsheet program and that same data can be stored and reused for later estimations of VaR. Furthermore, HS forms the basis for the Basle 1995 proposals on market risk.69,70 Disadvantages are that past extreme returns can be a poor predictor of extreme events, for example Danielsson and de Vries show that HS is unable to address losses which are outside the sample. This drawback is linked to the problem that HS, due to the discreteness of extreme returns that also will make the VaR being discrete, can over- or underestimate observations in the tails and over- or underpredict VaR. 71 The sample size or window length is another decisive aspect to consider, where the inclusion or exclusion of only one or two days at the beginning of the sample can cause large swings in the VaR estimate.72 As mentioned in 3.1, the Basel Committee proposes a window of at least one year of past returns 73. Another criticism is that HS is based on the assumption that past returns represent the immediate future fairly, but risk contains significant and predictable time variation that make the HS approach miss situations with temporarily elevated volatility. Finally, for large portfolios with numerous assets and exposures the historical approach quickly becomes cumbersome. 74,75

67 Jorion, P., Value at risk , (1997), p. 195. 68 Schachter, B., “Value at Risk Resources – An Irreverent Guide to Value at Risk ”, (1997), p. 2. 69 Jorion, P., Value at risk , (1997), p. 195. 70 Danielsson, J., “Class notes Corporate Finance & Financial Markets”, (1998-1999). 71 Danielsson, J. & de Vries, G., C., “Value-at-Risk and Extreme Returns”, (1997), p. 11. 72 Ibid, p. 11. 73 Danielsson, J., Hartmann, P. & de Vries, C.G. “The Cost of Conservatism” (1998), p. 2. 74 Smithson C., “Class notes of CIBC School of Financial Products”, (1998), p. 3. 75 Jorion, P., Value at Risk , (1997), p. 196.

21

3.4.2 The Equally Weighted Moving Average Approach The equally weighted moving average (EqWMA) approach assumes that the distribution of outcomes follow a normal distribution and uses a fixed amount of historical data to calculate the standard deviation. There are different opinions on how wide the data window should be. On the one hand only very recent data, e.g. 50 observations, should be used to incorporate changes in the volatility over time. On the other hand, to be able to estimate potential movements accurately and to estimate the variance with precision a much wider data window should be used.76 The calculation of the standard deviation is shown below:

∑−

−=

−−

=1

2)()1(

1 t

ktsst x

kµσ (12)

where σt is the estimated standard deviation at time t, and k specifies the number of observations included in the moving average. xs is the change in the value of the asset on day s and µ is the mean change in asset value during the estimated period.77 By using shorter periods of time the standard deviation gets more irregular and reacts faster to changes in asset price movements. Other parameters that have to be set are the confidence interval and the covariance of the asset returns. The most commonly used confidence intervals are the 95th and the 99th percentile although VaR is calculated with confidence intervals from the 90th to the 99.9th 78. 3.4.2.1 Advantages and disadvantages with the EqWMA approach An advantage with the EqWMA approach is that it is easy to use, since the normal distribution is only characterised by its mean and variance. The VaR estimation can be derived directly from the portfolio standard deviation by using a multiplicative factor that depends on the confidence level. 79 In addition, many statistical formulas are based on a normal distribution assumption and these facilitate the analysis of the results. Finally, all moments of positive order exist and the normal distribution can model VaR estimations outside the sample range.80

76 Jorion, P., Value at Risk , (1997), p.99. 77 Hendricks, D., “Evaluation of Value-at-Risk Models Using Historical Data”, (1996), p. 41. 78 Ibid, p. 40. 79 Jorion, P., Value at Risk (1997), p.88. 80 Lucas, A. & Klaassen, P., “ Extreme Returns, Downside Risk, and Optimal Asset Allocation”, (1998), p. 71.

22

The most obvious disadvantage with the EqWMA approach, as mentioned above, is that financial returns experience fat tails. Therefore using a normal distribution underestimates the true VaR, which of course is a very serious drawback81. Another disadvantage is that the quality of the VaR estimate calculated with the EqWMA approach degrades if non- linear instruments, like options, are included in the portfolio82. Moreover, low correlations between assets reduce the portfolio risk. However, evidence show that correlations increase in periods of instability on the financial markets, and therefore the normal distribution may underestimate the true VaR measure.83 3.4.3 The Exponentially Weighted Moving Average Approach In contrast to the EqWMA approach the exponentially weighted moving average (ExpWMA) approach attaches different weights to past observations in the observation period84. The weights decline exponentially and therefore, the most recent observations get much higher weight than earlier observations. The formula for the standard deviation under the ExpWMA is shown below:

∑−

−=

−− −−=1

21 )()1(t

ktss

stt x µλλσ (13)

The parameter λ (lambda) determines at which rate past observations decline in value as they become more distant. 85 The formula (13) above can be rewritten as below, which gives a more intuitive understanding of the calculation:

211

2 ))(1( µλλσσ −−+= −− ttt x (14)

Formula (14) shows that on any given day the standard deviation, calculated as an exponentially moving average, is made up of two components. Firstly, the weighted average on the previous day and secondly, yesterday’s squared deviation, which is given a weight of (1-λ). This means that a lower value on λ makes the importance of observations decline at a more rapid speed. 86

81 Danielsson, J., “Value-at-Risk and Extreme Returns”, (1997), p. 14. 82 Schachter, B., “Value at Risk Resources - An Irreverent guide to Value-at-Risk ”, (1997), p. 2. 83 Jorion, P., Value at Risk , (1997), p. 178. 84 Ibid, p. 177. 85 Hendricks, D., “Evaluation of Value-at-Risk Models Using Historical Data”, (1996), p. 42. 86 Ibid, p. 42.

23

The parameter λ is often referred to as the “decay factor” and determines how fast the decline in observation weights should be. For the weights to sum up to one an infinitely number of observations should be used, but in practice a limited number of observations can be used since the sum of weights will converge to one87. By setting a tolerance level, i.e. how close to one the sum has to be, the number of observations that has to be used in the calculation of the standard deviation can be calculated88. The number of observations that have to be used at different tolerance levels can be seen in the table below: Days of historical data at tolerance level:

Decay factor 0.001% 0.01% 0.1% 1% 0.85 71 57 43 28 0.90 109 87 66 44 0.91 122 98 73 49 0.92 138 110 83 55 0.93 159 127 95 63 0.94 186 149 112 74 0.95 224 180 135 90 0.96 282 226 169 113 0.97 378 302 227 151 0.98 570 456 342 228 0.99 1146 916 687 458

Table 1: The number of historical observations used by the exponentially weighted moving average approach. Source: JPMorgan/Reuters, RiskMetrics – Technical Document, (1996), p. 94.

The purpose of using the ExpWMA approach is to capture short-term movements in the volatility89. RiskMetrics uses exponentially moving averages to forecast future volatility in order to make sure that the model is responsive to market shocks and an eventual decline in the volatility forecast90. Further, research show that financial market volatility experience clusters of high and low volatility. The RiskMetrics group examined returns on the S&P 500 among other assets and found that autocorrelation is not present in the distribution of financial returns. However, although returns are uncorrelated they may not be independent. This can be evaluated by looking at the autocorrelation of squared returns. On this matter the RiskMetrics group found that financial returns experience autocorrelation. 91

87 Hendricks, D., “Evaluation of Value-at-Risk Models Using Historical Data”, (1996), p. 42. 88 JPMorgan/Reuters, RiskMetrics – Technical Document, (1996), p. 93. 89 Hendricks, D., “Evaluation of Value-at-Risk Models Using Historical Data”, (1996), p. 42. 90 JPMorgan/RiskMetrics group, Introduction to RiskMetrics, (1995), p. 2. 91 JPMorgan/Reuters, RiskMetrics – Technical Document, (1996), p. 59.

24

3.4.3.1 What value of λ should be used? By using a low decay factor, e.g. 0.94, it implies that almost the entire VaR measure is derived from the most recent observations. This means that the VaR measure becomes very volatile over time. On the one hand, relying on the most recent observations is important for capturing short-term movements in volatility. On the other hand, a smaller sample size increases the possibility of measurement error.92 The decay factor may not only be used to estimate the volatility of a single asset, but also to calculate the covariance matrix, which is shown below:

∑

=

)()()()(

222

321

312112

λσλσλσλσ

(15)

As can be seen above, the covariance matrix is a function of three decay factors. 93 Although it is possible in theory to estimate all possible decay factors, it is too complex in practice to calculate them all. Therefore, it has become necessary to get some form of structure on the value of the decay factors and the most practical thing to do is to use one lambda for the entire matrix. Still, different values of the decay factors can be used for different time periods. RiskMetrics has found 0.94 to be the optimal value for daily returns and 0.97 for monthly returns.94 3.4.3.2 Advantages and disadvantages with the ExpWMA approach The advantages are very much the same as with the EqWMA approach. However, the volatility is much more receptive to variations over time. By using an exponential moving average the standard deviation is responsive to market shocks and the following gradual decline in the forecast of volatility. A simple moving average does not react fast enough to changes in the volatility. 95 In addition, the ExpWMA approach smoothens out the standard deviation over time. In the EqWMA approach the standard deviation varies more, since the estimation is more affected when an observation falls out of the estimation window. 96

92 Hendricks, D., “Evaluation of Value-at-Risk Models Using Historical Data”, (1996), p. 43. 93 JPMorgan/Reuters, RiskMetrics - Technical Document, (1996), p. 97. 94 Ibid, p. 97. 95 JPMorgan, “Introduktion to RiskMetrics”, (1995), p. 4. 96 Ibid, p. 7.

25

The disadvantages, in addition to what was mentioned in section 3.4.2.1, are that the computations are somewhat more difficult and that the volatility over time is more unstable than with the EqWMA approach. 97 3.4.4 The New Improved VaR Methodology98 This methodology is a product from the RiskMetrics group, which has been developed to overcome the problems with financial returns experiencing fat tails. The new approach, which is still under development, tries to estimate the volatility by assuming that returns are generated from a mixture of two different normal distributions. This is shown in the formula below: PDF = p1 * N1 (µ1, σ1) + p2 * N2 (µ2, σ2) (16) where p1+p2=1. PDF stands for the probability density function and p1 is the probability that the return is generated from the normal distribution N1, which is characterised by its mean µ1 and variance σ1

2. Similarly p2 is the probability that the return is generated from the normal distribution N2, which has mean µ2 and variance σ2

2. This model makes it possible to assign large returns a higher probability than the normal distribution. To generate the PDF one can then assign p1 a value close to 1, with µ1=0 and σ1

2=1. The mean for N2 is also set to zero, but the variance is assigned a value higher than 1. The mixture of the two normal distributions then has fatter tails compared to N1. This can be illustrated graphically as below:

Figure 2. Normal distributions with different variances.

97 JPMorgan/Reuters, RiskMetrics – Technical Document (1996), p. 80. 98 JPMorgan/Reuters, RiskMetrics – Monitor, (1996), p. 7-19.

Normal distributions

Standard deviations

N1

N2

26

3.4.5.1 Advantages and disadvantages of the improved VaR methodology The advantage with the improved VaR methodology is that it describes reality more accurately than the traditional VaR models, since it is able to deal with the fat tails of financial returns. This gives a more precise VaR estimation. The disadvantages are firstly that the calculations become more complex and the VaR methodology loses its intuitive appeal, since very sophisticated statistical techniques are used to calculate the VaR with this model. Secondly, this methodology has not yet been thoroughly tested and hence, it is uncertain how well it really works. 3.4.5 Monte Carlo Simulation

Monte Carlo simulation (MCS) is mainly a method used by risk specialists and risk analysts to value complex derivatives such as exotics, but nowadays MCS is also a very useful tool for VaR calculation99. The MCS-method approximates the behaviour of financial prices by using computer simulations to generate random price paths. Before the generation of random prices is started one has to select the distribution the prices should be generated from, as well as the volatility of prices and the correlation between assets100. The first step in the simulation is to choose a stochastic model for the behaviour of prices, and in line with the Black and Scholes option pricing model a geometric Brownian motion can be used101,102. To generate random numbers Jorion propose either to use a two step process or a method called bootstrap. The former implies selecting a uniform distribution for the random number generator over the interval (0, 1), which produces a random variable. The next step is to transform the uniform random numbers into the desired distribution, for example a normal, which can be done by inverting the cumulative probability distribution function. The bootstrap method briefly implies that random numbers are generated by sampling from historical data with replacement.103 When the random numbers are generated the VaR is calculated with (11).

99 Söderlind, L., Att mäta ränterisker, (1996), p. 102. 100 Ibid, p. 101. 101 See for example Hull, J. C., Options, Futures, and Other Derivatives, (1997). 102 Jorion, P., Value at Risk , (1997), p. 232. 103 Ibid, p. 236.

27

The MCS-method is similar to the HS-method, except that the hypothetical changes in prices for assets are created by random draws from a stochastic process in the MCS-method, while HS is directly based on actual historical price changes104. 3.4.6.1 Advantages and disadvantages with the MCS approach An advantage with the MCS approach is that it can account for a wide range of risks, including price, volatility and credit risk, and by using different models it can even account for model risk. The MCS-method is considered to be the most powerful method to compute VaR. 105, 106 Disadvantages are mainly its complexity and that the method involves costly investments in intellectual and systems development. For example if 1000 sample paths are generated with a portfolio of 10 assets, the total number of valuations amounts to 10,000. Another disadvantage of the model is that it relies on a specific stochastic model for the underlying risk factors and pricing models for securities. Hence, there is a sensitivity to model risk. If, for instance the stochastic process chosen for the price is unrealistic the estimated VaR will also be misleading, and therefore there is a risk incorporated that the model is wrong. To conclude, the MCS-method is likely the most comprehensive approach to measure market risk if modelling is done correctly, but it is easy to lose the intuitive appeal with VaR by using this complex method.107 3.4.6 Semi-parametric VaR method108 This approach sheds light on the importance of an accurate prediction of the frequency of extreme events in VaR analysis, where the most frequent risks are modelled parametrically and infrequent risks are captured by a non-parametric empirical distribution function. The common VaR methods fall into two main classes: parametric prediction of conditional volatilities such as the RiskMetrics method, and non-parametric prediction of unconditional volatilities such as HS or stress testing methods. The semi-parametric method, also called the extreme value theory, is a mixture of these two approaches, where the non-parametric HS is

104 Söderlind, L., Att mäta ränterisker, (1996), p. 100. 105 Jorion, P., Value at Risk , (1997), p. 231. 106 Söderlind, L., Att mäta ränterisker, (1996), p. 106. 107 Jorion, P., Value at Risk , (1997), p. 201. 108 Danielsson, J. & de Vries, G. C., “Value-at-Risk and Extreme Returns”,(1997), p. 19-25.

28

combined with a parametric estimation of the tails of the return distribution109. Parametric methods, which are based on cond itional normality, are not well suited for analysing large risks, where the normality assumption probably leads to an underestimation of the risk of heavy losses. The main purpose of this mixed method is to accurately estimate the tails of a distribution, and therefore overcome problems that methods based on normality assumptions underpredict infrequent tail events. The results from a research by Danielsson and de Vries regarding the extreme value approach, shows that this method performs better than both RiskMetrics and HS far out in the tails. For the 5th percentile, RiskMetrics is superior but longer out in the tails the method consistently underpredicts the tail. For HS the opposite problem is the case, i.e. it consistently overpredicts the tails. Further on, HS is unable to address losses that are outside the sample. This approach is relatively more sensitive for a small sample than for example RiskMetrics and HS, and Danielsson and de Vries propose that a sample such as one year is not appropriate to use for the extreme value approach. 3.4.6.1 Advantages and disadvantages with the semi-parametric approach There are several advantages in using the estimated tail distribution for VaR estimation. For example, the extreme value method has a smoothing out effect of events like the ’87 crash compared with for example HS. For HS an event like the ’87 crash will cause a too large VaR estimate and hence impose too conservative capital provisions, while the extreme value method will give a better VaR estimate. Furthermore, one can easily obtain the lowest return that occurs with a given probability by sampling from the tail of the distribution, which facilitates sensitivity experiments. This is not possible by using the HS. In addition, no restrictive assumptions are needed, and the method can be used for large portfolios without endless computation time. However, in accordance with the MCS method the extreme value method is complex and relatively less intuitive, even if there are many advantages.

109 To view how to estimate the tails see Danielsson, J., & de Vries, G. C., “Value-at-Risk and Extreme Returns”, (1997) and Danielsson, J., & de Vries, G.C., “Beyond the Sample: Extreme Quantile and Probability Estimation”, (1997).

29

3.4.7 The Stress testing approach Stress testing is more or less a scenario analysis, where the effects from different events and movements are assessed for an asset or a portfolio of assets110. For an asset manager an example of a scenario could be what happens to the equity portfolio if the interest rate fluctuates heavily or if a currency suddenly devalues by 30 percent. When scenarios are selected the portfolio is revalued according to:

∑=

=N

isitisp RwR

1,,, (17)

where the portfolio return is derived from the hypothetical component siR , under

the new scenario s. Hence, various portfolio returns are generated with the consideration to pre-specified probabilities ),( sp for each scenario. VaR can then

be measured out of this generated distribution of portfolio returns. 111 3.4.7.1 Advantages and disadvantages with Stress testing The advantages with stress testing are mainly that it may cover situations that are completely absent from the historical data. For instance the EMS-breakdown can serve as an example of an event that would have been beneficial to make a scenario analysis on in advance. Hence, stress testing is a way to force management to consider events that they otherwise might ignore. Stress testing is also relatively easy to implement and communicate112.

One serious drawback with stress testing, compared to the methods mentioned before, is the sensitivity to the choice or creation of scenarios. The method is completely subjective, where an untenable scenario will lead to an incorrect measure of VaR. According to Jorion, stress testing does not account for correlation in-between assets, which is a crucial component for risk diversification. The example mentioned above about the EMS breakdown is a good illustration of the ignoration of correlations. Furthermore, stress testing does not either specify any trustworthy probabilities for, or the likelihood of, worst-case situations to actually occur.113

110 Jorion, P., Value at Risk , (1997), p. 196. 111 Ibid, p. 197. 112 Ibid, p. 203. 113 Ibid, p. 196-199.

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To conclude one can say that stress testing should be considered as a complement rather than a single VaR approach. Stress testing as a complement tries to capture what is going on in the tails, but as stated before stress values are subjectively defined without specified likelihood114.

3.5 Multi-day VaR prediction Most financia l firms use one day VaR for internal risk management, but regulators require that VaR also is calculated for longer time periods. There are two ways this can be calculated. One can look at past t-day returns or extrapolate the one-day VaR to t days.115 RiskMetrics uses the square root of t rule, where the one-day VaR is multiplied by the square root of t to obtain the VaR for t days116. However, this might produce an overestimation of the multi-day VaR since the multi-day extreme outcomes are smaller for fat tailed distributed returns than normally distributed returns. Danielsson found that a scaling factor of around 1.7 should be used for a ten day VaR estimate, which is significantly less than the square root of ten (√10≈3.7).117

114 Longin, F. M., “Stress Testing: A Method based on Extreme Value Theory”, (1999), p. 3. 115 Danielsson, J., “Value-at-Risk and Extreme Returns”, (1997), p. 22. 116 JPMorgan/Reuters, RiskMetrics – Technical Document, (1996), p. 84. 117 Danielsson, J., “Value-at-Risk and Extreme Returns”, (1997), p. 8.

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Chapter 4 - Statistical Methodology

4.1 Working process

The very first step in the statistical process was to compare which small- and mid-cap stocks that were listed in the beginning of the test period, i.e. 1994-01-01, and still listed in the end of the test period, i.e. 1999-12-30. Out of this sample stocks with extremely poor volume, which implies a low frequency of pricing and poor data, were excluded. As a rule of thumb stocks changing hands less than two thirds of the total sample of trading days were excluded. Finally the sample was 122 stocks, which were divided into three groups w.r.t. market cap criteria, see appendix 2. To set the market cap limits the Carnegie Small-Cap and Mid-Cap index ranges were used. In the end of the test period the Carnegie Small-Cap range was 0-5.2 billion SEK and the Mid-Cap range was 5.2-18.1 billion SEK118. The small- and mid-cap index values were used to adjust these ranges backward in the test period. After the stocks were divided into three groups, each of them was assigned a number, to be used in the random number generation later on. Three different kinds of equity portfolios, each containing ten stocks, were constructed. The first portfolio is an OMX-portfolio, which implies a portfolio consisting of the ten largest stocks listed on the Stockholm stock exchange over the test period119. Every six-month during the test period the portfolio is reallocated as to maintain the criterion of consisting of the ten largest stocks120. The reallocation dates are the 1st of January and 1st of July between 1995 and 1999. The second portfolio is a small-cap portfolio, consisting of ten stocks with market cap in accordance with the Carnegie Small-Cap index range. The stocks in the portfolio are randomly selected from the total sample of small-cap stocks. Every six-month new small-cap stocks are randomly selected so the reallocation of the portfolio is performed in a statistically correct way. The third portfolio is a mixed portfolio consisting of the five largest OMX-stocks, two randomly selected small-cap stocks, and three randomly selected mid-cap stocks. The procedure is identical as for the first two portfolios. In the portfolios every stock get a weight of ten percent at every day in the sample, i.e. the portfolios are equally weighted. The first two portfolios, the OMX and the small-cap portfolio, are motivated by the purpose to examine the applicability of VaR methods w.r.t. differences in equity market cap. The mixed portfolio is constructed since it will give a more realistic reflection of a portfolio held by an asset manager. The portfolios can be viewed in appendix 3. 118 Segerström, T., Carnegie Asset Management, (00-04-11). 119 The size is measured as market capitalisation, i.e. the share price multiplied by the total number of outstanding shares. 120 OM-Gruppen, Shares in OMX, (1995-1999).

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Last price paid data was collected from the Bloomberg database. Especially for the small-cap stocks the time-series were not totally complete, for example all stocks were not traded every day in the test period. Hence, the data is corrected so the time-series are consistent over the sample, i.e. if a stock was not traded on a specific date the price when the stock last changed hands was used. Out of the last price paid time series the formulas below are applied, to get the percentage returns on a daily basis for each stock in all portfolios.

1

1

−

−−=

t

ttt P

PPR (18)

where Rt is the percentage return at time t and Pt-1 is the price at time t-1. 121 Equation (19) is used for dividend replacement, on the ex dates for dividends122.

1

1))((

−

−−+=

t

tttt P

PDPR (19)

4.2 VaR methods used in the study All in all seven different methods have been used in the study. These are divided into three subgroups, which are the Historical Simulation approach (HS), the Equally Weighted Moving Average approach (EqWMA) and the Exponentially Weighted Moving Average approach (ExpWMA). These three approaches have been chosen, since they are the methods most widely used in empirical finance of today123. No semi-parametric approaches are performed, since none of these are fully developed and accepted yet, and also because they require an estimation window longer than one year. Regarding the Monte Carlo simulation it is well accepted, but too complex, demanding and costly to perform. Stress testing is easy to perform, but scenario analysis is not included in the purpose of this Master thesis. The most commonly used window sizes for the HS approach are six months, one year, two years and five years, i.e. window sizes of 125, 250, 500 and 1250 observations124. In our study windows of 125 and 250 trading days have been used, since the data quality for small-cap stocks were very poor with many

121 JPMorgan/Reuters, RiskMetrics - Technical Document, (1996), p. 46. 122 Ross, S.A., Westerfield, R.W. & Jaffe, J., Corporate Finance, (1996), p. 224. 123 See for instance Danielsson, J. & de Vries, C.G.,“Value-at-Risk and Extreme Returns”, (1997) or Hendricks D., “Evaluation of Value-at-Risk Models Using Historical Returns”, (1996). 124 See for instance Hendricks, D., “Evaluation of Value-at-Risk Models Using Historical Returns”, (1996).

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shares not changing hands frequently in the beginning of the 90s, and in addition few small-cap shares were listed. Regarding the EqWMA approach the most commonly used window sizes are the same as for the HS approach, with the only difference that 50 days is also often used 125. Therefore, we have chosen window sizes of 50, 125 and 250 days for this approach. For the ExpWMA the most widely accepted decay factors are 0.94 and 0.97, and these are the decay factors applied in this study126. For all of the approaches mentioned the calculations are performed at confidence levels of both 95% and 99%. These levels are by far the most used in statistical testing. In summary, the following 14 combinations of methods and confidence levels are performed in the study: § HS with a window size of 250 trading days at both the 95% and 99%

confidence interval (CI). § HS with a window size of 125 trading days at 95% and 99% CI. § EqWMA with a window size of 50 trading days at 95% and 99% CI. § EqWMA with a window size of 125 trading days at 95% and 99% CI. § EqWMA with a window size of 250 trading days at 95% and 99% CI. § ExpWMA with a decay factor of 0.94 at 95% and 99% CI. § ExpWMA with a decay factor of 0.97 at 95% and 99% CI.

4.3 Calculation procedure for the HS approach First, the portfolio return is calculated for each day according to formula (11) viewed in chapter 3. As stated previously windows of 250 and 125 days are used both for the 95% and 99% confidence interval. Thus, the 1st percentile, for the 99% confidence interval, and the 5th percentile, for the 95% confidence interval are calculated using windows of the latest 125 or 250 trading days. For example, to calculate the 95% confidence interval using the 250-day window at time t the 5th percentile of the window ranging from t-250 to t-1 is calculated. To calculate the confidence interval for the next day at time t+1, the window from t-249 to t0 is used. The results, i.e. the daily VaR measures, are then compared to the actual outcomes of the portfolio. The results are then evaluated using the performance evaluation criteria presented in 4.6.

125 See for instance Hendricks, D., “Evaluation of Value-at-Risk Models Using Historical Returns”, (1996). 126 For instance RiskMetrics uses 0.94 for daily observations, and 0.97 for monthly observations.

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4.4 Calculation procedure for the EqWMA approach The EqWMA approach is tested over 50, 125 and 250 days both at the 95% and the 99% confidence interval, as stated in 4.2. For each day the variance for each portfolio is calculated. In the case of a portfolio with ten assets the variance is obtained by using the following formula:

∑ ∑= ≠=

+==N

i

N

jijijijiiipprVar

1 ,1,,

222 2)()( σγγσγσ (20)

where γ is the weight, σi

2 variance, and σ1,2 the covariance127. The square root is taken of formula (20) to obtain the daily standard deviation of the portfolio. We have used formula (12) to calculate the standard deviation for each asset in the portfolio for each day, and the covariance according to formula (21) 128:

N

rrrrrr

N

ttt∑

=

−−== 1

2,21,1

2,121

))((),cov( σ (21)

where

∑=

=N

ttii r

Nr

1,

1 (22)

The daily VaR measures are finally obtained by multiplying the standard deviation by -1.645 for the 95% confidence level and -2.327 for the 99% level129.

4.5 Calculation procedure for the ExpWMA approach The decay factors 0.94 and 0.97 were chosen, and according to table 1 in chapter three 74 and 151 observations are used respectively at the 1% tolerance level. Thus, two time-series are created ranging from 1 to 74 and 1 to 151. When these lambda weights are obtained we use formula (13), presented in chapter 3, to calculate the standard deviation. The results are then multiplied with the left-hand-side standard deviation from a normal distribution at the 95% level, which

127 Benninga, S., Financial Modeling, (1998), p. 74. 128 Gustavsson, M., and Svernlöv, M., Ekonomi & Kalkyler, (1994), p. 706. 129 Körner, S., Tabeller och formler för statistiska beräkningar, (1986), p. 14.

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is –1.645, and at the 99% level, which is –2.327. The results, the daily VaR measures, are then evaluated against the performance criteria as for the other two approaches.

4.6 Performance evaluation criteria Darryll Hendricks uses nine performance criteria to evaluate the quality and performance of the VaR-approaches EqWMA, ExpWMA, and HS 130. We have selected to use all of these criteria to evaluate our VaR examination. Every criterion is calculated for each of VaR method, portfolio and confidence level, i.e. 42 calculations are performed for each criterion (7 VaR methods*3 portfolios*2 confidence levels). The criteria are in order:

4.6.1 Mean Relative Bias131 This criterion estimates whether each VaR method produces risk measures of similar average size. The VaR measures are here compared to each other and not to the actual portfolio outcomes. The mean relative bias is measured in percentage terms, where for example a value of 0.15 implies that a given VaR method on average is 15 percent larger than the average of all methods for the same portfolio and confidence level. 4.6.1.1 Calculation Procedure For each VaR method, portfolio and confidence level an average over the VaR measures is calculated over all observations in the sample. An average is then calculated over all averages, to obtain a “total average” over all methods for each portfolio and confidence level separately. Then we divide the individual average with the total average and subtract one, to get the percentage difference. For example, an average is calculated for the OMX-HS250d at the 95% confidence level over the whole sample period from 1995-1999. This average is then divided by the average of all seven methods for the OMX portfolio using a 95% confidence interval. From this number, one is subtracted and the mean relative bias is obtained.

130 Hendricks, D. “Evaluation of Value-at-Risk Models Using Historical Data” (1996), p. 46. 131 Ibid, p. 46.

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4.6.2 Root Mean Squared Relative Bias132 This criterion assesses to what extent the VaR measures tend to vary around the average VaR measures for a given date. This measure can be compared to a standard deviation for the mean relative bias. 4.6.2.1 Calculation procedure A daily mean relative bias is calculated by taking the daily VaR figure and divide it by the average over all methods using the same confidence level and portfolio for that day, and subtract one. Then we obtain a mean relative bias figure for each day, which is squared, and an average is calculated over the entire sample. To get the root mean squared relative bias we simply take the square root of this squared mean.

4.6.3 Annualized Percentage Volatility133 This criterion evaluates the tendency of the VaR measures to fluctuate over time. An annualized percentage volatility is calculated for each portfolio and VaR method at both confidence levels. 4.6.3.1 Calculation procedure A new column of numbers has been calculated for each day in the sample period according to:

1

1%−

−−=∆

t

ttt VaR

VaRVaRVaR (23)

Hence, a new time-series for each method, portfolio and confidence level is created, which views the percentage change in the VaR measures. For these time-series the standard deviation is calculated. To obtain the annualized percentage volatility we multiply the standard deviation by the square root of 250134. 132 Hendricks, D., “Evaluation of Value-at-Risk Models Using Historical Data”, (1996), p. 47. 133 Ibid, p. 48. 134 In financial theory 250 days is considered to be a measure of the amount of trading days on an annual basis.

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4.6.4 Fraction of Outcomes Covered135 This criterion is more of a fundamental test, where the VaR methods are examined whether they cover the portfolio outcomes that they are intended to capture. For example, to achieve the desired level the coverage should be 95 percent for the methods using a 95% confidence interval. 4.6.4.1 Calculation procedure The fraction of outcomes covered is calculated as the percentage of results where the loss in portfolio value is less than the risk measure. A command is constructed, which states that if the VaR measure exceeds the portfolio return a violation of VaR is present, and vice versa. This test is performed for each day, and to calculate the fraction of outcomes covered (FoOC) we have used the following formula:

−=∑

=

N

violationsFoOC

N

t 1

)0;1(1 (24)

where violations can take either of (1) or (0) depending on if a violation is present or not, i.e. no violation implies a (0) and a violation a (1). For example, if the VaR measure exceeds the portfolio outcome on five trading days out of a hundred the fraction of outcomes covered would be 0.95 (1-5/100). 4.6.5 Multiple Needed to Attain Desired Coverage136 The multiple needed to attain desired coverage is simply a figure on what value the multiple should be for each VaR measure to attain the desired level of coverage. This criterion focuses on the size of the adjustments in the risk measurement required to achieve this perfect coverage. This measure is important because shortcomings in VaR measures that seem small in probability terms may be much more significant when considered in terms of the changes required to remedy them. 135 Hendricks, D., “Evaluation of Value-at-Risk Models Using Historical Data”, (1996), p. 49-50.

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4.6.5.1 Calculation procedure

First, we divide the daily portfolio returns with the VaR measure for each day. Out of these generated numbers the percentiles for a 95% and a 99% confidence interval are calculated, i.e. the 5th and 1st percentile. These percentile values are the multiples that would have been required for each VaR measure to attain the desired level of coverage. A value below one implies that the VaR method overstates the risk and hence a value above one understates the risk. Thus, a multiple of exactly one is to be preferred. 4.6.6 Average Multiple of Tail Event to Risk Measure137 This evaluation method relates to the median size of outcomes not covered by the VaR measures. The average multiple of tail events is calculated and compared with the VaR measure, where tail events are defined as the largest percentage of losses measured relative to the confidence level chosen.

4.6.6.1 Calculation procedure

To simplify matters this method is calculated in a similar way as the multiple needed to attain desired coverage in 4.6.5. The daily portfolio return is divided with the VaR measure for each day in the sample. When the percentiles are calculated an average of one plus the confidence interval is used, i.e. 97.5 for the 95% confidence interval and 99.5 for the 99% confidence interval. The percentile values are the average multiples of tail event to risk measure. To analyze the results we calculate a benchmark from the normal distribution, see further section 5.6.

4.6.7 Maximum Multiple of Tail Event to Risk Measure138 This performance criterion assesses the size of the maximum portfolio loss. The maximum multiples are likely to be highly dependent on the length of the sample period, e.g. for shorter periods the maximum multiple are likely to be lower.

136 Hendricks, D., “Evaluation of Value-at-Risk Models Using Historical Data”, (1996), p. 50. 137 Ibid, p. 51. 138 Ibid, p. 52.

39

4.6.7.1 Calculation procedure Out of the generated multiple series in 4.6.5 the maximum value is found, which is the maximum multiple of tail event to risk measure.

4.6.8 Correlation between Risk Measure and Absolute Value of Outcome

This criterion examines to what extent the VaR measures adjust to changes in the portfolio risks over time139. Briefly the correlation coefficient can be considered as an index showing the degree of linear co-variation between two variables. A correlation coefficient equal to (–1) implies perfect negative correlation, i.e. the variables are moving exactly in the opposite direction. A coefficient of (+1) implies perfect positive correlation and hence the variables are moving in exactly the same direction. 140 According to Hendricks, even a perfect VaR measure cannot guarantee a correlation of (1) between the risk measure and the portfolio outcome, which is an important statement to bear in mind141. Despite this, a value close to one is desired. 4.6.8.1 Calculation procedure First the absolute value of the portfolio returns are calculated. These values are then compared with the generated VaR measures using the following correlation formula:

21

2,12,1 σσ

σρ = (25)

where the covariance is calculated as in (21) and the standard deviation is calculated according to following formula (26):142

1

)(1

2

−

−=

∑=

N

rrN

ii

iσ (26)

where N is the total number of observations, and r is the mean of all observations 143.

139 Hendricks, D., “Evaluation of Value-at-Risk Models Using Historical Data”, (1996), p. 53. 140 Söderlind, L., Att mäta ränterisker, (1996), p. 79. 141 Hendricks, D., “Evaluation of Value-at-Risk Models Using Historical Data”, (1996), p. 53. 142 Ross, S. A., Westerfield, R. W. & Jaffe, J., Corporate Finance, (1996), p. 253.

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4.6.9 Mean Relative Bias for Risk Measures Scaled to Desired Level of Coverage144

Here the mean relative bias that results when VaR measures are scaled to either 95 percent or 99 percent coverage is assessed. The purpose is to determine which approach that could provide the desired level of coverage with the smallest average VaR measures. 4.6.9.1 Calculation Procedure The scaling is performed by multiplying the VaR measures for each method by the multiple needed to attain desired coverage. Then an average of the VaR measures scaled to desired level of coverage is calculated for each portfolio and confidence level. The VaR measure scaled to desired level of coverage for one specific method, portfolio and confidence level is divided by the mean of all scaled VaR measures using the same portfolio and confidence level. From this number, one is subtracted to get the mean relative bias for risk measures scaled to desired level of coverage.

4.7 Hypothesis testing To be able to evaluate our findings in a proper way a number of significance tests have been performed. Tests are carried out for the criteria fraction of outcomes covered, both as an actual portion and as a comparison between portfolios with different market caps, and for the correlation between risk measure and absolute value of outcome. Hypothesis testing on these specific criteria has been done since these are the most fundamental for the evaluation of the VaR methods. A significance level of 5% is denoted with (*), a significance level of 1% is denoted with (**) and a significance level of 0.1% is denoted with (***).

143 Gustavsson, M. & Svernlöv, M., Ekonomi & Kalkyler, (1994), p. 706. 144 Hendricks, D., “Evaluation of Value-at-Risk Models Using Historical Data”, (1996), p. 54.

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4.7.1 Actual portion of Fraction of Outcomes Covered To test if the VaR methods cover the portfolio outcomes as is intended, i.e. if the VaR measures at the 95% level actually cover 95 percent of the outcomes and the VaR measures at the 99% level cover 99 percent of the outcomes, we set up the following hypotheses: H0: VaR95% =95% for Fraction of Outcomes Covered H1: VaR95% ≠95% for Fraction of Outcomes Covered The hypotheses for the 99% confidence interval is set up accordingly. Hence, the null hypothesis, H0, states that the VaR measures cover the portion of outcomes that they are intended to. The alternative hypothesis, H1, states that the VaR measures differ significantly from the portions of outcomes they are supposed to cover. These tests are then performed for all seven VaR methods at both confidence levels and for all three portfolios, which sum up to 42 tests. The formula used for the calculation is shown below:

n

PZ

)1( ππ

π

−

−= (27)

where P is the observed portion of the measured variable in the sample, π is the portion that the test is performed against, i.e. 95% for tests on VaR95% and 99% for tests on VaR99%, and n is the number of observations. The formula is approximately normally distributed if (nπ*(1-π)>5), with zero mean and a standard deviation of one. 145 If the absolute value of Z exceeds 1.96 H0 is rejected at the 5%-level. Similarly if the absolute value of Z exceeds 2.57 H0 is rejected at the 1%-level, and if the value exceeds 3.3 H0 is rejected at the 0.1%-level. For absolute values of Z below 1.96 H0 is accepted. These are the Z-values used, since they correspond to the different significance levels.146

145 Körner, S., Statistisk dataanalys, (1987), p. 283. 146 Körner, S., Tabeller och formler för statistiska beräkningar, (1986), p. 14.

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4.7.2 Difference in FoOC between OMX and small-cap shares To test if there is a difference in the fraction of outcomes covered between OMX and small-cap shares the following hypotheses are set up: H0: VaROMX=VaRSMALL H1: VaROMX>VaRSMALL To get an explanation of why this test is one-sided, see section 5.4.1.2. This test is then performed for all seven methods at both confidence levels. The formula below is used for the calculation of the significance test:

)

11)(1(

21

21

nnPP

PPZ

+−

−= (28)

where P1 is the portion of the variable measured in the first sample, i.e. the fraction of outcomes covered for the OMX portfolios, and P2 is the portion of the variable measured in the second sample, i.e. the fraction of outcomes covered for the small-cap portfolios. n1 is the number of observations in the OMX sample and n2 is the number of observations in the small-cap sample. P is the weighted average of the two samples and Z is approximately normally distributed with zero mean and a standard deviation of one.147 Since H1 is one-sided the critical Z values become 1.645 for the 5%-level, 2.33 at the 1%-level and 3.1 at the 0.1%-level. For all Z values below 1.645 H0 is accepted.148 4.7.3 Significance test of the correlation coefficients To test if the correlation coefficients differ significantly from zero the following hypotheses are set up: H0: correlation=0 H1: correlation≠0

147 Körner, S., Statistisk dataanalys, (1987), p. 285. 148 Körner, S., Tabeller och formler för statistiska beräkningar, (1986), p. 14.

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The test is performed for all seven methods at both confidence levels and for all three portfolios. The formula below is used for the calculation:

21

2

r

nrt

−

−= (29)

where r is the correlation coefficient and n is the number of observations. The formula is t-distributed with (n-2) degrees of freedom.149 The critical values for t are 1.960 at the 5%-level, 2.576 at the 1%-level and 3.291 at the 0.1%-level. For all t-values below 1.960 H0 is accepted.150

4.8 Normality tests According to the formulas (9) and (10) in chapter 3, the skewness and kurtosis for the portfolio returns for each portfolio are computed. When the skewness and kurtosis are obtained, we use them in the Jarque-Bera formula (30) below to test whether our data follows a normal distribution or not. Jarque-Bera is a test statistic for testing if series are normally distributed. The formula (30) follows a Chi-squared distribution (χ2), which is an asymmetric distribution, with two degrees of freedom:

( )

−+

−=− 22 )3

41

6KurSk

kNBeraJarque (30)

where N is the total number of observations and k the number of estimated coefficients used to create the series.151 Tests were made for all portfolios to see if they were normally distributed. The following hypotheses were set up:

:0H The portfolio returns are normally distributed

:1H The portfolio returns are not normally distributed

The results from our tests can be viewed in chapter 5.10. The χ2 generated probability values are the probabilities that a Jarque-Bera statistic exceeds the observed value under the null, where a small value leads to the rejection of the null. 152

149 Körner, S., Statistisk dataanalys, (1987), p. 287. 150 Körner, S., Tabeller och formler för statistiska beräkningar, (1986), p. 17. 151 Eviews 3, User’s Guide, (1998), p. 165. 152 Ibid, p. 165.

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4.9 Criticism of primary data The most decisive point in our study is that the collected data is the correct one and that no price information is missing. Two things that could have an adverse affect on our study is that the data was not corrected for dividends and that companies have merged or been acquired during the period of our study. To cope with these problems the data has been corrected for dividends pay out. When firms have merged and one stock has been changed for another we have corrected the data for this as well. Another thing that is of great importance is that the sample we have used for the mid-cap and the small-cap portfolios are random. To make sure of this we have used Excel to generate random numbers that represent different companies. Some of the small-cap shares have not changed hands on every trading day. To handle this problem we have used the last price paid on the day when the stock was last traded. Companies that have been trading on less than two thirds of the trading days have been excluded in the study to avoid any distorted results.

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Chapter 5 - Results In this chapter the results from the study of VaR over three different approaches, seven different methods and three different kinds of portfolios are presented for both the 95% and the 99% confidence levels. We regard the performance criteria fraction of outcomes covered and correlation between risk measure and absolute value of outcome as most intuitive when analysing the results, and hence we are giving these criteria most attention. In addition, the results from the significance and normality tests are presented, see appendix 10-11 and 16. Tables and diagrams over the results can be found in appendix 4-9, 12-15 and 17-18. In each section the expectations are firstly discussed, followed by the results per se. Finally comments are made on how the results correspond with our expectations.

5.1 Mean Relative Bias It is hard to have any qualified expectations for this criterion. However, previous research has found that longer windows normally give higher VaR measures, i.e. methods with long windows have a mean relative bias above one 1. When interpreting the mean relative bias it is important to keep in mind that this criterion only measure how the size of the VaR measures are to the mean of all VaR measures for the specific category. For instance, a VaR measure of one method for the OMX portfolio is compared to other methods’ VaR measures for the same portfolio and level of confidence. The mean relative bias is measured in percentage terms, so a value of 0.10 implies that the VaR measure is 10 percent larger than the average. For the vast majority of portfolios the mean relative bias is between –0.1 and +0.1, indicating that most VaR measures vary up to 10 percent from the mean. The results are presented in appendix 6, and for the 95% confidence level the HS approach seems to give the lowest VaR measures and the EqWMA approach the highest. At the 99% confidence interval HS tends to give the highest values, especially with a window of 250 trading days. Further, there is a distinct indication that longer windows give higher VaR measures. One possible explanation for this is Jensen’s inequality theorem, which states that if the true conditional variance is changing frequently, then the average of a concave function, i.e. the VaR measure, will tend to be less than the same concave function of the average variance. Briefly the implication of the theorem for VaR is that

1 Hendricks, D., “Evaluating Value-at-Risk Models Using Historical Data”, (1996), p. 46.

46

VaR measures with short windows should on average be smaller than VaR measures with longer windows.2,3 For this criterion we can conclude that the results are in line with our expectations. In addition, the average VaR of different methods deviates as much as ten percent from the mean in both directions, indicating that two methods on average can differ 20 percent from each other.

5.2 Root Mean Squared Relative Bias As with the mean relative bias it is hard to make predictions about the results of the root mean squared relative bias. However, this criterion is sensitive to window size and therefore the VaR methods using windows of average sizes are expected to differ the least from the mean, i.e. give the lowest root mean squared relative bias. As can be seen in appendix 7, for both confidence levels and all three portfolios the EqWMA approach with a window size of 125 trading days and the ExpWMA approach with a decay factor of 0.97, i.e. 151 days – see table 1, give the lowest results. It is also interesting to note that on any given day differences in the range of 30-40 percent between two VaR methods are not uncommon. Many portfolios have values of 0.15-0.20, which means that one method can produce a VaR measure 15-20 percent above the average, while another produces values 15-20 percent below the average. Clearly, it is a serious drawback for VaR as a concept that different methods give remarkably different VaR measures. The methods that give the lowest values are methods with windows of average length, i.e. 125 and 151 observations. The HS with a window size of 125 trading days does not give a low value, which could be due to the fact that the HS approach is very dependent on a few observations. Since the HS approach only relies on five or one percent of the observations, depending on the confidence level, the rest of the observations are irrelevant for the calculations. Therefore it can be questioned whether HS125d produces reliable results.

2 Hendricks, D., “Evaluating Value-at-Risk Models Using Historical Data”, (1996), p. 66. 3 http://mathworld.wolfram.com/JensensInequality.html

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5.3 Annualized Percentage Volatility We expect methods using short windows to have high volatility measures, since they depend on fewer observations that leave the window more quickly. The ExpWMA methods should have higher volatility measures than the other approaches, since they put higher weights on recent observations through the decay factor. In addition, the small-cap portfolio could probably have a higher volatility measure compared to the other portfolios, since small-cap stocks generally have a lower liquidity measured as depth. Poor liquidity can sometimes imply higher volatility since even a small amount of trading can affect prices heavily. It is important to note that the values for approaches using the normal distribution are exactly the same for both confidence levels, since these are multiples of the same standard deviation. From the diagrams, see appendix 8, it is clear that longer windows give lower volatility and vice versa. In addition, the ExpWMA approach, especially with a decay factor of 0.94, gives the highest volatility measures. This is not surprising due to the fact that the VaR measure varies as new observations enter the window. There seems to be a tendency for the small-cap portfolios to have a slightly higher volatility than the OMX shares even though the differences are small. Another conclusion that can be drawn is that, for the same window length, the HS approach tends to have a higher volatility than the EqWMA approach. This is probably due to the fact that the HS to a high degree only depends on a couple of observations. The conclusions from this criterion are very much what we expected. Short windows give high volatility measures, in particular the ExpWMA. The small-cap portfolios give higher volatility measures in 13 cases out of 14, so even though the differences are small there is a tendency for these portfolios to have a higher volatility measure.

5.4 Fraction of Outcomes Covered This criterion is the most important and intuitive for evaluating VaR methods. To be able to use the VaR approach in practice the VaR methods using confidence levels of 95% and 99% should also give a 95 percent and 99 percent coverage respectively. However, financial returns experience fat tails and previous research has shown that VaR models find it hard to cover these confidence levels,

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especially the 99% level4. The RiskMetrics group regards the ExpWMA approach to be superior to the others, especially with a decay factor of 0.94 since this is used for daily VaR estimation and is receptive for short-term volatility swings. Furthermore, RiskMetrics considers the HS125d to produce unsatisfactory results, since the window is too short. 5 The results in appendix 9 show that HS125d sharply underestimates the VaR for all portfolios at both confidence levels. At the 95% confidence interval the EqWMA approach, for which seven out of nine portfolios overestimate the VaR, tends to produce VaR measures that cover more than 95 percent of the outcomes. For the ExpWMA approach five out of six portfolios overestimate the VaR, while only one out of six portfolios for the HS approach produces too high VaR measures. There is no apparent difference in the VaR measures for the small-cap and OMX portfolios. Furthermore, at the 99% confidence level only two out of 21 portfolios actually cover 99 percent of the outcomes. Generally the OMX portfolios have a higher fraction of outcomes covered than both the small-cap and mixed portfolios. At the 95% confidence level four out of seven methods produce higher VaR measures for the OMX portfolios, while two methods have exactly the same coverage. At the 99% level six out of seven methods give higher results for the OMX portfolios compared to the small-cap. To some extent the results for this criterion are in line with our expectations. It seems the VaR methods have a hard time covering 99 percent of the outcomes and distinctly the HS125d produces unsatisfying results. In addition, the fraction of outcomes covered for the OMX portfolios seem to be higher than the small-cap portfolios. However, there is no evidence indicating that the ExpWMA approach produces superior results compared to the other methods. 5.4.1 Significance Testing To further evaluate the fraction of outcomes covered significance testing has been performed both to check if the portfolios cover the portion of outcomes that they are supposed to, and if there is any difference between small-cap and OMX portfolios. The tests are described in section 4.7, and the results are presented in appendix 10-11.

4 Hendricks, D., “Evaluating Value-at-Risk Models Using Historical Data”, (1996), p. 49. 5 JPMorgan/Reuters, RiskMetrics – Technical Document, (1996), p. 78.

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5.4.1.1 Fraction of Outcomes Covered For the OMX portfolio at the 95% confidence level all H0 are accepted, i.e. all methods give coverage of 95 percent. On the other hand, at the 99% level both HS methods underestimate the VaR and especially the HS125d produces too low VaR measures. The small-cap portfolios all produce attractive results at the 95% confidence interval. However, at the 99% level the picture is quite the opposite. Six out of seven portfolios significantly underestimate the VaR and only H0 for the HS250d is accepted. This is especially interesting, since the Basel 1995 proposal for market risk recommends to use HS with a 99% confidence level. In addition, five out of the six portfolios that underestimate the VaR, do so on the lowest, i.e. 0.1%, significance level. Apparently this is a very serious drawback of the VaR concept. For the mixed portfolios no method underestimates the VaR at the 95% confidence level. A surprising result is that one method, the EqWMA250d, overestimates the VaR. At the 99% level only two methods produce results for which H0 is accepted, the HS and the EqWMA with windows of 250 trading days. All of the other methods underestimate the VaR at the 1% significance level or less. 5.4.1.2 Difference between the OMX and Small-cap Portfolios This test is set up to evaluate if there is a difference in the fraction of outcomes covered between the small-cap and OMX portfolios. Unfortunately the mixed portfolio cannot be tested against the OMX portfolio, since the mixed portfolio contains five stocks that are also included in the OMX portfolio and therefore the observations are not independent. The diagrams presented in appendix 9 show that there is a tendency for the OMX portfolios to have higher fraction of outcomes covered than the small-cap portfolios and therefore the test is one-sided. On the 95% confidence level none of the methods signals that there is a difference between the small-cap and the OMX portfolios. On the other hand, for the 99% confidence interval all methods using a normal distribution get their H0 rejected, i.e. the methods produce significantly higher fraction of outcomes covered for the OMX portfolios compared to the small-cap portfolios. For the HS methods no difference could be detected and hence H0 for these methods are accepted.

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5.5 Multiple Needed to Attain Desired Coverage The multiple needed to attain desired coverage is more or less a reflected image of fraction of outcomes covered. Hence, the methods that tend to overestimate VaR measured as fraction of outcomes covered will get a multiple less than one in this test, and vice versa. The larger the deviation from one, the greater the risk measure has to adjust to achieve a perfect coverage. Since we expect a poor coverage for the HS125d in the previous test, the multiple is consequently expected to be relatively high. Similarly we expect values above one for the 99% confidence level, and the ExpWMA to produce values close to one. The diagrams in appendix 12 view that the multiples are a reflection of the fraction of outcomes covered, both at the 95% and the 99% confidence level. Worth noting is the very poor result by the HS125d at the 99% level, which underestimates the VaR by approximately 40 percent. Another interesting notation is the sharp overestimation of VaR by the mixed portfolio for the EqWMA250d. Further on, at the 95% level no method seems to be superior to the other methods, but for the 99% level HS250d and EqWMA250d appear to have somewhat more attractive values. The small-cap portfolios at the 99% level are the worst performers, while its more difficult to distinguish among the portfolios for the 95% level. For all seven methods and both confidence levels the multiple stretches in a range of 0.9 – 1.45.

5.6 Average Multiple of Tail Event to Risk Measure This performance criterion views the size of outcomes not covered by the VaR measure. To evaluate the test we use the normal distribution as a benchmark. In calculating the benchmark we use the table-value for the average percentile, i.e. 97.5 for the 95% confidence level and 99.5 for the 99% level, which is 1.96 and 2.327 respectively. These are then divided by the table-values for both the 95% and the 99% confidence level. Hence, the benchmarks for the 95% level is (1.96/1.645)=1.19 and for the 99% level (2.575/2.2327)=1.11. This benchmark is not applicable for the HS, since the method does not rely on normal distributed returns. Thus, the EqWMA and the ExpWMA approaches should present values close to the benchmark if the portfolio returns should be regarded to follow a normal distribution. At the same time, we are aware of the fact that financial theory states that returns hardly follow a normal distribution.

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The diagrams in appendix 13 indicate that the OMX portfolios, for all methods and both confidence levels, have multiples close to the benchmarks relative the other portfolios. This result imply that the normal distribution is relatively more well- functioning for OMX stocks. Later on, in section 5.10, we will see that this is an ambiguous result. At the same time the small-cap portfolio constantly deviates the most from the benchmarks. Otherwise we cannot conclude that one method is superior to the other methods. The results are somewhat in line with the expectations, since the OMX portfolio produce values close to the benchmarks. On the other hand, the small-cap and mixed portfolios have values significantly above the benchmarks. Even if the results for this criterion should produce values close to the benchmarks financial theory states that returns experience fat tails, and hence the normal distribution is a poor approximation.

5.7 Maximum Multiple of Tail Event to Risk Measure Basically the maximum multiple value is the largest multiple for each method over all observations. Regarding the ExpWMA approach, which is known as a method able to cover short-term volatility well, we expect a relatively low max multiple. The results in appendix 14 show that for 12 out of the 14 combinations of methods and confidence levels the OMX portfolio got the lowest max multiple. One explanation for this characteristic can be higher performance stability and organization quality for the large cap stocks, that are included in the OMX-index, compared to the small- and mid-cap stocks. In addition, the HS approach tends to produce the highest max multiples, while the ExpWMA approach gives the lowest. Worth noting is that it is important not to view VaR as a strict upper bound on the portfolio losses that can occur. This means that even if it seems that VaR at the 99% level captures essentially all of the relevant events the last one percent can entail substantial losses. The results confirm our expectations regarding that the ExpWMA method properly covers the short-term volatility, which implies a relatively low max multiple. Hence, the ExpWMA tends to be the superior method in this respect. However, this conclusion can definitely be too hasty, because the performance criterion is only based on one single observation over the total sample of 1254 observations

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5.8 Correlation between Risk Measure and Absolute Value of Outcome

We expect the methods using short windows to have the highest correlation coefficients, because these are superior in capturing volatility fluctuations. In addition, the ExpWMA approach should produce high values, since the strength of this approach is that it should be more receptive for changes in the portfolio volatility. As can be seen in appendix 15 for methods using the same approach, for example by comparing methods using the EqWMA approach with other methods using the same approach but with other window sizes, it is apparent that a shorter window gives a higher correlation coefficient. Furthermore, the ExpWMA approach produces the highest correlation coefficient. It is also interesting to notice that for all 14 methods, the OMX portfolios have higher correlation coefficients than the small-cap portfolios. As we expected the methods with short windows and methods using the ExpWMA approach produce the highest correlation coefficients. 5.8.1 Significance Testing To test if the correlation coefficients differ significantly from zero we set up the hypotheses described in section 4.7.3. The test results are presented in appendix 16. As can be seen in the appendix all correlation coefficients differ from zero at the lowest significance level, i.e. the 0.1% level. Hence, all VaR methods are receptive to changes in the market volatility.

5.9 Mean Relative Bias for Risk Measure Scaled to Desired Level of Coverage

This performance criterion assesses which method that can provide the desired level of coverage with the smallest average VaR measure. Previous research indicates that the ExpWMA approach should have a relatively low mean relative bias for risk measure scaled to desired level of coverage, and vice versa for the

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HS approach6. In general, as stated before in section 5.1, the mean relative bias criteria are difficult to make predictions for prior to the investigation. The diagrams in appendix 17 indicate that the HS method gives a poor value for the 99% confidence level, especially for the small-cap portfolios. The ExpWMA approach appears to be superior for both confidence levels, except for the mixed portfolio with decay factor 0.94 at the 95% level. Moreover, for the EqWMA approach the small-cap portfolios produce lowest figures two out of three times at the 99% level, but highest values two out of three times at the 95% level. The results follow the expectations, based on previous research, that the ExpWMA has a relatively low mean relative bias for risk measure scaled to desired level of coverage.

5.10 Normality Tests As discussed in section 3.3, the distribution of financial returns can take many different shapes. Since both the EqWMA and the ExpWMA approach rest on the assumption of normally distributed returns we find it interesting to examine whether our data is normally distributed or not. In a first stage we test our data for skewness and kurtosis, and secondly we perform a Jarque-Bera test for normality, see section 4.8. It is important to stress that the normal distribution is a well-functioning approximation for smaller samples, i.e. 30-150 observations, but can be an insufficient approximation for larger samples such as ours of 1254 observations 7. 5.10.1 Results from the Normality Tests If the portfolio returns are perfectly normally distributed the computed skewness should be close to zero and the kurtosis close to three. That will produce a Chi-square probability value of one and hence the null can be accepted, i.e. the portfolio returns are normally distributed. As can be seen in table 2 below, the test values for skewness indicate that the returns from the OMX portfolio are right-skewed, since 0.488 is a relatively high and positive number. Further on, for the small-cap portfolio the test indicates left-skewed returns, while the mixed portfolio got results close to zero.

6 See for instance Hendricks, D., “Evaluation of Value-at-Risk Models Using Historical Data”, (1996). 7 Hagnell, M., Lecturer in Econometrics at Lund University, (00-17-05).

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A distribution is considered to be leptokurtic if the kurtosis coefficient is higher than three and platykurtic below three. The kurtosis calculations show that all three portfolio returns are significantly leptokurtic, especially the returns for the OMX portfolio with a value of 9.86, but the small and mixed portfolio returns are also significantly leptokurtic.

Skewness

Kurtosis

Jarque-Bera

prob-χ2

OMX

0,488

9,860

2504,635

0,000

SMALL

-0,222

7,637

1131,960

0,000

MIX

-0,032

7,656

1131,100

0,000

Table 2. Results from normality tests.

The Chi-squared probability values in table 2 clearly indicates that the null hypotheses set up in section 4.8 are rejected for all portfolios, which implies that the portfolio returns cannot be regarded as normally distributed. To find answers for this result we plotted the portfolio returns for each portfolio against normally distributed random numbers for the same sample size, i.e. 1254 observations. The results are viewed in figures 3-5 below, which demonstrate the difference in distribution between the portfolio returns for all three portfolios and a normal distribution. It is difficult to view the kurtosis and skewness, even for the OMX portfolio with a very high kurtosis value, but compared to the leptokurtic distribution viewed in section 3.3 all figures are similarly leptokurtic distributed.

Figure 3. A random normal distribution plotted against the OMX portfolio returns.

05

10152025303540455055

Standard deviation

Frequency

OMX

NORMAL

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In accordance with the negative skewness coefficients in table 2 more values should be striving towards negative numbers for the small-cap portfolio, and the opposite for the OMX portfolio. The mixed portfolio has a skewness coefficient close to zero, and hence the distribution can be considered as symmetric.

Figure 4. A random normal distribution plotted against the small-cap portfolio returns.

Figure 5. A random normal distribution plotted against the mixed portfolio returns.

Out of the diagrams it is difficult to view the fat tails of the portfolio returns, and therefore the frequency tables are attached in appendix 18. The kurtosis is significant and an answer to this result can be found in the heart of portfolio theory. The figures indicate that too many return observations, as compared with the generated normal distribution, are clustered around relatively small changes of the mean for all three portfolios. We believe that this is an effect from diversification. However, it is important to stress that the clusters around the mean for the small-cap and to some extent the mixed portfolios can be explained by the filling in of missing prices explained in sections 2.4 and 4.1. The main reason to hold a portfolio in the first place is to lower the investment risk via distributing the capital over different business lines and sectors. If the diversification works it simply implies that the fluctuation in returns decreases and hence more observations are clustered around the mean. In addition, there is a tendency for the

05

10152025303540455055606570

Standard deviaton

Frequency

SMALL

NORMAL

05

101520253035404550556065

Standard deviation

Frequency

MIX

NORMAL

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correlation between stocks to increase in periods of global turbulence and that might be the reason why financial returns experience fat tails. These two effects can explain why the portfolio outcomes are not normally distributed. Due to the fact that the normal distribution appears to be a poor approximation for the portfolio returns in our study, the use of another approximation would be beneficial. Here the t-distribution could be an option, since it captures fat tails well. The t-distribution depends on the number of degrees of freedom, which is an insecure estimation and might make the VaR process both unreliable and complicated. Hence, the VaR with an approximated t-distribution could be a less appealing tool in practice. Another interesting conclusion that can be drawn from the normality test is linked to the results from the performance criterion average multiple of tail event to risk measure. To evaluate the results we used the normal distribution as a benchmark, see 5.6, where a value close to the benchmark implies that the portfolio returns should be regarded to follow a normal distribution. The results indicated that the OMX portfolio followed a normal distribution relatively better than the other portfolios, which appears to be very ambiguous w.r.t. the results from the normality tests. However, the average multiple of tail event to risk measure only compares the 95 and 97.5 respectively 99 and 99.5 percentiles, and does not take the whole normal distribution into consideration.

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Chapter 6 - Conclusions

6.1 Evaluation of VaR Methods We regard methods producing results that do not differ significantly from the intended fraction of outcomes covered at the 95% significance level as satisfying. However, it can be questioned whether estimations that come close to having their null hypothesis rejected, such as the OMX-HS125d at the 95% confidence level, can be considered as being a reliable and trustworthy VaR method. Most of our VaR methods work well on the 95% confidence level, while on the 99% level only portfolios consisting of OMX stocks produce satisfying results. For the small-cap and mixed portfolios 11 out of 14 VaR methods significantly underestimate the fraction of outcomes covered at the 99% confidence level. This gives indications of serious shortcomings of the VaR concept when differences in market cap affect the applicability of VaR. These results might suggest that the VaR methods are mostly applicable to large-cap shares and cannot be used with small-cap or mixed portfolios, at least not at the 99% significance level. In our study we can conclude that financial returns do not follow a normal distribution. Therefore, we argue that VaR approaches based on the normal distribution should not be used for VaR estimations, even if the EqWMA and the ExpWMA approaches for the OMX portfolio do well measured as fraction of outcomes covered. Hence, by assuming a normal distribution despite its imperfections, cannot be recommended. The HS approach, which does not rest on the assumption about normal distribution, indeed produces very unsatisfying results with a window of 125 trading days. However, by using a longer window, i.e. 250 days, the results are much more promising. For HS the estimations are acceptable at the 95% level, even though the OMX and small-cap portfolio do not cover exactly 95 percent of the outcomes. For the mixed portfolio on the other hand, the HS250d slightly overestimates the VaR. All differences from the 95 percent of outcomes covered are within the margin of error. At the 99% level the HS250d method clearly produces the best results for the small-cap and mixed portfolios. However, it does significantly underestimate the VaR at the 99% level for the OMX portfolio. Still, we argue that the HS with a window size of 250 trading days is the VaR method producing the most attractive results in our study. In addition, there is no significant difference in the fraction of outcomes covered between the small-cap and OMX portfolio for the HS methods.

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The approaches based on the normal distribution, i.e. the EqWMA and the ExpWMA approach, seem to do very well at the 95% level. No method comes close to having their null hypothesis rejected except for the MIX-EqWMA250d that overestimates the VaR. On the other hand, at the 99% level the results are not as good. These results are in line with Lucas & Klaassen’s findings, i.e. VaR methods based on the normal distribution produce attractive VaR measures at the 95% level but poor at the 99% level, see section 3.3. The approaches work well with the OMX portfolio, but the results with the small-cap and mixed portfolios are not acceptable. All methods except for the MIX-EqWMA250d underestimate the VaR with probability values close to zero. We were expecting the ExpWMA to be superior to the EqWMA approach, since this approach is favoured by RiskMetrics and is known to capture volatility fluctuations in a superior way. However, there is no evidence pointing in this direction in our findings. On the other hand, the ExpWMA approach generates superior results regarding the mean relative bias for risk measures scaled to desired level of coverage and the correlation between risk measure and absolute value of outcome. We can conclude that a high correlation does not automatically give attractive results, and even if the correlation coefficient would have been one (1) the VaR method cannot be regarded as totally perfect. There is a tendency for the VaR methods with long windows to produce superior results compared to the methods using shorter windows. It is possible that by using longer windows than 250 trading days the VaR measures would produce more attractive results. It might be the case that a one-year window is too short to capture the risk of infrequent market crashes. By using a longer window this risk would be captured more accurately. Perhaps a semi-parametric method or a Monte-Carlo simulation would have produced superior results, but as stated in chapter 3 these methods are significantly more complex to perform than the three approaches we selected. Furthermore, since one of the purposes with this research is to discuss whether VaR is a useful tool for asset managers, we found it much more likely that any of the chosen approaches are relevant to be used in practice.

6.2 The Distribution of Financial Returns As we showed in section 5.10 the portfolio returns used in our study do not follow a normal distribution. The returns for all three portfolios experience a leptokurtic behaviour with high peaks, thin sides and fat tails. This might be due to the diversification effect, i.e. portfolio outcomes cluster around the mean, and increasing correlations between portfolio stocks when markets experience times of

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high volatility create fat tails. Finally, the normal distribution is not a well-functioning approximation for large samples, and that the normal distribution cannot handle large samples properly favour the HS even more, where an increasing window size only makes VaR to make progress. In summary, it is obvious that financial returns are not normally distributed and the VaR approached should not be based on assumptions that do not hold. Therefore, we recommend the HS approach to be used for VaR estimations.

6.3 Implications of VaR for Asset Managers The stock markets of today are increasingly more volatile and risky than only a couple of years ago. One of the main reasons is the high paced intrusion of the “new economy” at the expense of the “old economy”. The “new economy” is reflected in the stock market, which to a large extent now consists of a relatively new and untested sector, telecom and IT. The uncertainty regarding the sector per se and the companies’ ability to live up to the expectations are significant and contribute to the volatility increase. The companies in this sector are also relatively more difficult to analyse, which has the effect of a significant dispersion in motivated valuations among market actors. Contradictory opinions regarding the valuation of stocks and a high valuation of the stock market per se are both driving forces for volatility increases. The high valuation and the booming telecom and IT sectors and the launching of the EMU, which force asset managers to reallocate their portfolio holding in the region, are all factors that make higher volatility a natural thing. Asset managers are highly exposed to these increased risks, and are more or less forced to invest in the telecom and IT sector to get and maintain the exposure of the “new economy”. Briefly this implies an increased risk taking. With respect to the enhanced portfolio risk a way to capture and measure these risks is called for, even if the asset managers typically are in the business of taking risk and not avoiding it. The question is therefore if we can recommend VaR as a risk measurement tool for asset managers focused on the Swedish equity market. The primary goals with VaR are to show how much risk that is being taken and to determine whether asset managers are exposed to the risks intended to. Even if the need for a risk management tool is great, especially in the stock market of today, the reliability and stableness are the two first characteristics that have to be commented on. A VaR method that is not reliable and does not give stable results is not a risk tool to consider for an asset manager. Even if a certain VaR method never will tell an asset manager how much risk to undertake, it must perfectly

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reflect how much risk that is being taken. Further on, if a specific VaR method produces measures that underestimate the risk, an asset manager can for instance accept additional risk to be undertaken. This gives serious implications for the portfolio, which consists of more risk than the threshold actually is intended for. Hence, the threshold or risk target defined in terms of maximum tolerable VaR is not consistent and reliable. On the other hand, a well- functioning and reliable VaR method can be a superior way to avoid unexpected and uncontrolled losses. As stated in chapter 3, Philippe Jorion is of the opinion that there is no doubt that VaR is here to stay, but he also stresses that the process and methodology of calculating VaR is equally important as the VaR number itself. On the whole, this goes hand in hand with the results from this study. As long as a VaR method is reliable and stable it is probably the superior risk measurement tool for asset managers managing Swedish equity portfolios. At the same time it is definitely too much to conclude that any of the VaR methods in this study can be regarded as totally reliable and stable. Previously we concluded that the normal distribution is incorrect as an underlying assumption for a VaR approach, which automatically exclude the EqWMA and the ExpWMA approach. The HS seems to work well with a window of 250 trading days and it would have been interesting to view the results from a similar research with longer windows. One of the purposes with our study was to analyse if the market cap of stocks has any impact on how well- functioning and reliable the VaR models are. From a practical viewpoint, both the OMX and the small-cap portfolios can be considered as extremes because they only consist of stocks from a certain range of market caps. Of course specialised equity funds can have a concentration similar to our OMX and small-cap portfolios, such as small-cap and index funds, but these concentrations are no likely alternatives/hold ings for an asset manager. On the contrary, the mixed portfolio appears to be more of a natural kind of diversification among stocks w.r.t. market caps for an asset manager. Therefore, the VaR results for the mixed portfolios for each method are perhaps the most interesting, where the HS250d and the EqWMA50d at the 95% level are the superior methods. At the 99% level the HS250d and the EqWMA250d seem to achieve the most attractive results measured as fraction of outcomes covered. Hence, the HS250d method can be regarded as the most useful tool for risk management of Swedish equity portfolios.

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6.4 Suggestions for further Research We believe that there are a number of VaR areas that have not received enough attention yet. Our study concludes that the normal distribution is an unsatisfactory approximation of portfolio returns and therefore it would be interesting to view how applicable the new VaR approaches are to stock portfolios, for example the semi-parametric approach, the improved VaR methodology and the Monte Carlo simulation approach. One distribution that could be useful for estimation of portfolio returns is the t-distribution and hence, to examine what impact it has on VaR estimation would be an interesting topic. In addition, a more qualitative research of the usefulness of VaR for asset managers could shed light on the subject from another point of view.

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Maymin, Zak (1998) “VaR variations: is multiplication factor still too high?”. Http://www.gloriamundi.org/var/. Schachter, Barry (1997), “Value-at-Risk Resources – An Irreverent Guide to Value at Risk”. Financial Engineering News, Vol. 1 No. 1, August 1997. Smithson, Charles (1996) “Class notes of CIBC School of Financial Products”. CIBC School of Financial Products. Styblo Beder, Tanya (1995) “VaR: Seductive but Dangerous”. Financial Analysts Journal, September/October 1995. Thornberg, John (1998), “Derivative users lack refined controls of risk”. Working Paper University of Paisley, December 1998. Yiehmin, Lui, R. (1996), “VaR and VaR derivatives”. Applied Derivatives Trading, December 1996. Textbooks Aczel, D. Amir (1993), Complete Business Statistics. IRWIN. Afifi, A.A & Clark, Virginia (1990), Computer-Aided Multivariate Analysis. Van Nostrand Reinhold Company. Backman, Jarl (1985), Att skriva och läsa vetenskapliga rapporter. Studentlitteratur, Lund. Benerson, L. Mark & Levine, M. David (1992), Basic Business Statistics. Prentice-Hall Int. Editions. Benninga, Simon (1998), Financial Modeling. Massachusetts Institute of Technology. Eviews 3 (1998), User’s Guide. Eviews Ltd. Gustavsson, Michael & Svernlöv, Magnus (1994), Ekonomi & Kalkyler. Liber-Hermods. Halvorsen Knut (1992), Samhällsvetenskaplig metod. Studentlitteratur, Lund. Hill, Carter, Griffiths, William & Judge, George (1997) Undergraduate Econometrics. John Wiley & Sons, Inc. Hull, John (1997), Options, Futures, and Other Derivatives. Prentice-Hall Int. Jorion, Philippe (1997), Value at Risk. McGraw-Hill. JPMorgan/RiskMetrics Group (1995), Introduction to RiskMetrics. JPMorgan.

JPMorgan/Reuters (1996), RiskMetrics-Technical Document. JPMorgan/Reuters. JP Morgan/Reuters (1996), RiskMetrics - Monitor. JPMorgan/Reuters. Kleinbaum, G. David, Kupper, L. Lawrence & Muller, E. Muller (1988), Applied Regression Analysis and Other Multivariate Methods. PWS-KENT Publishing Company. Körner, Svante (1986), Tabeller och formler för statistiska beräkningar. Studentlitteratur, Lund. Körner, Svante (1987), Statistisk Dataanalys. Studentlitteratur, Lund. Pettersson, Gertrud (1997), Att skriva rapporter. Ekonomihögskolan vid Lunds Universitet. Ross, Stephen.A, Westerfield, Randolph.W. & Jaffe, Jeffrey. (1996), Corporate Finance. McGraw-Hill. Svenning, Conny (1996), Metodboken. Eslöv Lorentz. Söderlind, Lars (1996), Att mäta ränterisker. SNS Förlag. Wiedersheim-Paul, Finn & Eriksson, Lars, Torsten (1991), Att utreda, forska och rapportera. Liber Ekonomi. Collection of Quantitative Data Affärsvärlden (1995), ”Placeringsindikatorn”. Affärsvärlden, 18 January, No. 3, 1995. Affärsvärlden (1995), ”Placeringsindikatorn”. Affärsvärlden, 17 May, No. 20, 1995. Affärsvärlden (1996), ”Placeringsindikatorn”. Affärsvärlden, 10 January, No. 1-2, 1996. Affärsvärlden (1996), ”Placeringsindikatorn”. Affärsvärlden, 22 May, No. 21, 1996. Affärsvärlden (1997), ”Placeringsindikatorn”. Affärsvärlden, 15 January, No. 1-3, 1997. Affärsvärlden (1997), ”Placeringsindikatorn”. Affärsvärlden, 21 May, No. 20, 1997.

Affärsvärlden (1998), ”Placeringsindikatorn”. Affärsvärlden, 14 January, No. 1-3, 1998. Affärsvärlden (1998), ”Placeringsindikatorn”. Affärsvärlden, 13 May, No. 20, 1998. Affärsvärlden (1999), ”Placeringsindikatorn”. Affärsvärlden, 13 January, No. 1-2, 1999. Affärsvärlden (1999), ”Placeringsindikatorn”. Affärsvärlden, 2 June, No. 22, 1999. Bloomberg Database of Financial Information. OM-Group, Shares in OMX, First halfyear 1995 - second halfyear 1999. Electronic Sources AffärsData EconLit Eric Weisstein's World of Mathematics, Http://mathworld.wolfram.com/JensensInequality.html Personal contacts Hagnell, Mats, Lecturer in Econometrics at Lund University. Personal contact May 17th 2000. Segerström, Thomas, Carnegie Asset Management. Telephone contact April 11th 2000.

Appendix 2 – Total sample of stocks

Order Small-Cap stocks Order Small-Cap stocks Order Mid-Cap Stocks

OMX-stocks

1 Bulten b 42 Lap Power b 1 Allgon b ABB a

2 Celsius b 43 Scribona 2 Avesta Sheffield ABB Ltd. a

3 Consilium 44 Lindab 3 Höganäs b Asea a

4 Finnveden b 45 Nea b 4 SSAB b Astra b

5 Gunnebo 46 Active b 5 Seco-Tools b AstraZeneca

6 HL Display b 47 Sintercast a 6 Svedala Electrolux b

7 Haldex 48 Havsfrun 7 Assidomän Ericsson b

8 Itab b 49 Skanditek 8 JM b FSPB a

9 KMT 50 Nordifa 9 Lundbergs b H&M b

10 Kabe b 51 Hexagon b 10 NCC b Investor b

11 Kalmar industrier 52 Midway b 11 Tieto-Enator Nokia

12 Nolato 53 Tivox b 12 Invik b Nordbanken Holding

13 Klippan 54 Westergyllen b 13 Kinnevik b Pharmacia a

14 Munksjö 55 Borås Wäfveri b 14 Perstorp b Pharmacia & Upjohn

15 Rottneros 56 Brio b 15 Trelleborg Sandvik b

16 Rörvik Timber 57 Cloetta b 16 Graninge SEB a

17 Bergman &Beving 58 Spendrups b 17 SAS SHB a

18 Bilia b 59 TV4 b 18 Atle Skandia b

19 Doro 60 VLT b 19 Bure Skanska b

20 Elgruppen b 61 Artema b 20 Custos a Sparbanken a

21 Fjällräven b 62 Elekta b 21 Latour Volvo b

22 Folkebolagen 63 Getinge 22 Ratos 23 OEM b 64 Nobel Biocare 23 OM-gruppen 24 Diös 65 J&W 25 Fastpartner 66 KM b 26 Heba b 67 Scandiaconsult 27 Hufvudstaden 68 Ångpanneföreningen b 28 Ljungbergruppen b 69 Senea 29 Norrporten 70 Bong Ljundahl b 30 Peab b 71 Elanders b 31 Piren b 72 Esselte b 32 Platzer 73 Graphium 33 Realia b 74 Strålfors b 34 Wallenstam b 75 Tryckindustri b 35 Wihlborg 76 Geveko 36 B&N b 77 Svolder b 37 Concordia b 78 Öresund 38 Stena-Line b 79 H&Q 39 IMS 80 Matteus 40 Måldata b 81 NH Nordiska 41 IBS b

Appendix 3 – Equity Portfolios

Portfolio 1 – Swedish equity OMX portfolio (large-cap) Reallocation period / Stocks

95:1 95:2 96:1 96:2 97:1

Astra b Ericsson b Astra b Astra b Ericsson b

Ericsson b Astra b Ericsson b Ericsson b Astra b

Volvo b Volvo b Volvo b Volvo b ABB a

Asea a Asea a Asea a ABB a Volvo b

Electrolux b Sandvik b SHB a Sandvik b SHB a

Sandvik b Stora a SEB a P&U Sandvik b

Stora a Pharmacia a Skanska b SHB a Skanska b

SEB a Electrolux b Sandvik b Investor b H&M b

SHB a SHB a Stora a Skanska b SEB a

Skanska b Skanska b Electrolux b SEB a Investor b

97:2 98:1 98:2 99:1 99:2

Ericsson b Ericsson b Ericsson b Ericsson b Ericsson b Astra b Astra b Astra b AstraZeneca AstraZeneca ABB a Volvo b H&M b H&M b H&M b Volvo b FSPB a FSPB a SHB a ABB Ltd. a Sparbanken a H&M b SHB a FSPB a Skandia H&M b SHB a SEB a NB Holding Volvo b SHB a ABB a ABB a Skandia NB Holding Investor b SEB a NB Holding ABB a SHB a SEB a Investor b Volvo b Nokia Electrolux b Electrolux b Sandvik b Skandia Volvo b FSPB a

Portfolio 2 – Swedish equity small-cap portfolio Re-allocation period / Random numbers

95:1 95:2 96:1 96:2 97:1 97:2 98:1 98:2 99:1 99:2

71 30 15 62 66 37 11 64 52 34 34 75 64 58 74 18 39 15 79 81 12 80 36 24 23 34 13 56 56 2 49 46 6 12 70 19 4 74 49 52 52 34 31 23 77 39 81 6 53 30 31 58 28 46 4 26 24 47 25 65 20 45 54 74 25 52 38 26 36 74 23 51 14 81 47 67 75 69 40 72 77 11 41 14 71 15 20 21 20 23 37 77 45 71 59 29 68 47 33 28 Re-allocation period / Stocks

95:1 95:2 96:1 96:2 97:1

Elanders b Peab b Rottneros Elekta b KM b Wallenstam b Tryckindustri b Nobel Biocare Spendrups b Strålfors b Nolato Matteus B&N b Diös OEM b Skanditek Active b HL Display b Nolato Bong b Midway b Wallenstam b Piren b OEM b Svolder b Piren b Spendrups b Ljungberggruppenb Active b Finnveden b Elgruppen b Nea b Westergyllen b Strålfors b Fastpartner OEM b Hexagon b Munksjö Nordiska Sintercast a Svolder b Kalmar ind. IBS b Munksjö Elanders b Concordia b Svolder b Nea b Elanders b TV4 a

97:2 98:1 98:2 99:1 99:2 Concordia b Kalmar ind. Nobel Biocare Midway b Wallenstam b Bilia a IMS Rottneros H&Q NH Nordiska Wallenstam b Klippan Brio b Brio b Celsius b Doro Finnveden b Spendrups b Skanditek Midway b IMS NH Nordiska HL Display b Tivox b Peab b Heba b Diös Active b Fastpartner J&W Midway b Stena-Line b Heba b B&N b Strålfors b Scandiaconsult Tryckindustri b Senea Måldata b Esselte b Rottneros Elgruppen b Fjällräven b Elgruppen b OEM b Norrporten Ångpanneför. b Sintercast a Realia b Ljunberggruppen b

Portfolio 3 – Swedish equity mixed portfolio Re-allocation period / Random numbers

95:1 95:2 96:1 96:2 97:1 97:2 98:1 98:2 99:1 99:2

71* 14* 30* 19* 34* 38* 20* 39* 56* 77* 79* 47* 79* 32* 60* 6* 49* 14* 37* 56* 14 15 9 4 19 21 14 11 13 10 9 9 2 3 15 9 2 20 15 15 13 6 3 6 20 19 21 17 18 17 Re-allocation period / Stocks

95:1 95:2 96:1 96:2 97:1

Elanders b Munksjö Peab b Doro Wallenstam b H&Q Sintercast a H&Q Platzer b VLT b Perstorp b Trelleborg Lundbergs b SSAB b Bure Lundbergs b Lundbergs b Avesta Sheffield Höganäs b Trelleborg Kinnevik b Svedala Höganäs b Svedala Custos a Astra b Ericsson b Astra b Astra b Ericsson b Ericsson b Astra b Ericsson b Ericsson b Astra b Volvo b Volvo b Volvo b Volvo b ABB a Asea a Asea a Asea a ABB a Volvo b Electrolux b Sandvik b SHB a Sandvik b SHB a

97:2 98:1 98:2 99:1 99:2 Stena-Line b Elgruppen b IMS Brio b Svolder b HL-Display b Skanditek Munksjö Concordia b Brio b Latour Perstorp b Enator Kinnevik b NCC b JM b Avesta Sheffield Custos a Trelleborg Trelleborg Bure Latour SAS Atle SAS Ericsson b Ericsson b Ericsson b Ericsson b Ericsson b Astra b Astra b Astra b AstraZeneca AstraZeneca Volvo b Volvo b H&M b H&M b H&M b Sparbanken a FSPB. a FSPB. a SHB a ABB Ltd. a H&M b H&M SHB a FSPB. a Skandia

Appendix 4 – Results at the 95% confidence level

Performance criteria /Method

MRB RMSRB APV FoOC MNtADC AmoTE-tRM

MmoTE-tRM

CBRM-aAVoO

MRBfRM-StDLoC

OMX-HS 250d -0,033 0,146 0,173 0,943 1,039 1,282 3,038 0,232 0,013

Small-HS 250d -0,031 0,160 0,195 0,943 1,056 1,463 4,031 0,166 0,016

Mix-HS 250d -0,012 0,161 0,155 0,953 0,980 1,322 3,980 0,221 -0,013

OMX-HS 125d -0,083 0,131 0,401 0,939 1,076 1,303 2,843 0,303 -0,006

Small-HS 125d -0,049 0,164 0,378 0,941 1,069 1,532 5,149 0,150 0,010

Mix-HS 125d -0,077 0,151 0,388 0,939 1,063 1,453 3,857 0,247 0,000

OMX-EqWMA 50d 0,006 0,139 0,474 0,952 0,983 1,185 2,762 0,352 -0,004

Small-EqWMA 50d -0,001 0,148 0,498 0,947 1,025 1,371 2,986 0,244 0,017

Mix-EqWMA 50d -0,004 0,140 0,467 0,953 0,987 1,326 3,241 0,279 0,003

OMX-EqWMA 125d 0,050 0,107 0,230 0,953 0,953 1,146 2,585 0,255 0,008

Small-EqWMA 125d 0,040 0,129 0,264 0,951 0,996 1,334 4,149 0,151 0,028

Mix-EqWMA 125d 0,043 0,122 0,208 0,957 0,926 1,280 3,698 0,191 -0,015

OMX-EqWMA 250d 0,065 0,187 0,102 0,957 0,965 1,184 2,778 0,204 0,036

Small-EqWMA 250d 0,058 0,172 0,110 0,957 0,946 1,274 3,652 0,184 0,001

Mix-EqWMA 250d 0,076 0,185 0,118 0,966 0,899 1,217 3,500 0,183 -0,013

OMX-ExpWMA 0.94 -0,020 0,161 0,745 0,951 0,990 1,200 2,499 0,397 -0,023

Small-ExpWMA 0.94 -0,025 0,180 0,919 0,950 0,997 1,397 2,698 0,306 -0,034

Mix-ExpWMA 0.94 -0,032 0,165 0,808 0,943 1,069 1,300 3,012 0,335 0,055

OMX-ExpWMA 0.97 0,017 0,086 0,398 0,956 0,953 1,159 2,519 0,367 -0,024

Small-ExpWMA 0.97 0,009 0,094 0,472 0,955 0,966 1,327 2,693 0,266 -0,032

Mix-ExpWMA 0.97 0,006 0,089 0,427 0,955 0,959 1,264 2,987 0,300 -0,016

Appendix 5 – Results at the 99% confidence level

Performance criteria /Method

MRB RMSRB APV FoOC MNtADC AmoTE-tRM

MmoTE-tRM

CBRM-aAVoO

MRBfRM-StDLoC

OMX-HS 250d 0,001 0,167 0,228 0,984 1,052 1,173 2,221 0,252 0,024

Small-HS 250d 0,161 0,160 0,316 0,986 1,164 1,268 1,911 0,186 0,096

Mix-HS 250d 0,093 0,198 0,287 0,988 1,050 1,159 2,575 0,200 0,003

OMX-HS 125d -0,088 0,154 0,416 0,979 1,151 1,317 2,300 0,287 0,022

Small-HS 125d 0,027 0,196 0,640 0,974 1,432 1,566 2,230 0,177 0,192

Mix-HS 125d 0,003 0,183 0,571 0,978 1,280 1,489 2,621 0,204 0,123

OMX-EqWMA 50d 0,000 0,142 0,474 0,989 1,014 1,154 1,952 0,352 -0,013

Small-EqWMA 50d -0,053 0,159 0,498 0,977 1,221 1,418 2,111 0,244 -0,063

Mix-EqWMA 50d -0,040 0,144 0,467 0,982 1,181 1,298 2,291 0,279 -0,009

OMX-EqWMA 125d 0,043 0,105 0,230 0,989 1,009 1,161 1,827 0,255 0,024

Small-EqWMA 125d -0,015 0,106 0,264 0,977 1,232 1,447 2,933 0,151 -0,016

Mix-EqWMA 125d 0,005 0,103 0,208 0,980 1,177 1,361 2,614 0,191 0,034

OMX-EqWMA 250d 0,059 0,186 0,102 0,991 0,964 1,211 1,964 0,204 -0,008

Small-EqWMA 250d 0,002 0,148 0,110 0,982 1,159 1,534 2,582 0,184 -0,058

Mix-EqWMA 250d 0,037 0,174 0,118 0,987 1,032 1,244 2,474 0,183 -0,065

OMX-ExpWMA 0.94 -0,026 0,163 0,745 0,986 1,029 1,093 1,767 0,397 -0,024

Small-ExpWMA 0.94 -0,077 0,192 0,919 0,976 1,257 1,409 1,907 0,306 -0,059

Mix-ExpWMA 0.94 -0,068 0,173 0,808 0,980 1,169 1,319 2,129 0,335 -0,047

OMX-ExpWMA 0.97 0,011 0,090 0,398 0,990 0,992 1,096 1,781 0,367 -0,025

Small-ExpWMA 0.97 -0,044 0,108 0,472 0,978 1,169 1,409 1,903 0,266 0,000

Mix-ExpWMA 0.97 -0,031 0,097 0,427 0,982 1,134 1,271 2,111 0,300 -0,039

Appendix 6 – Mean Relative Bias

Mean Relative Bias, 95%

-0,105-0,090-0,075-0,060-0,045-0,030-0,0150,0000,0150,0300,0450,0600,0750,090

Portfolio-Method

MRB

Mean Relative Bias, 99%

-0,100-0,075-0,050-0,0250,0000,0250,0500,0750,1000,1250,1500,1750,200

Portfolio-Method

MRB

Appendix 7 – Root Mean Squared Relative Bias

Root Mean Squared Relative Bias, 95%

0,0650,0780,0910,1040,1170,1300,1430,1560,1690,1820,195

Portfolio-Method

RMSRB

Root Mean Squared Relative Bias, 99%

0,0650,0780,0910,1040,1170,1300,1430,1560,1690,1820,195

Portfolio-Method

RMSRB

Appendix 8 – Annualized Percentage Volatility

Annualized Percentage Volatility, 95%

0,000,080,160,240,320,400,480,560,640,720,800,880,96

Portfolio-Method

APV

Annualized Percentage Volatility, 99%

0,000,080,160,240,320,400,480,560,640,720,800,880,96

Portfolio-Method

APV

Appendix 9 – Fraction of Outcomes Covered

Fraction of Outcomes Covered, 95%

0,9300,9330,9360,9390,9420,9450,9480,9510,9540,9570,9600,9630,9660,969

Portfolio-Method

FoOC

Fraction of Outcomes Covered, 99%

0,9700,9720,9740,9760,9780,9800,9820,9840,9860,9880,9900,9920,994

Portfolio-Method

FoOC

Appendix 10 Fractions of Outcomes Covered - 95% level Z-value Probability Significance OMX-HS 250d 1,075 0,859 OMX-HS 125d 1,853 0,968 OMX-EqWMA 50d -0,350 0,363 OMX-EqWMA 125d -0,479 0,316 OMX-EqWMA 250d -1,127 0,130 OMX-ExpWMA 0.94 -0,220 0,413 OMX-ExpWMA 0.97 -0,998 0,159 SMALL-HS 250d 1,075 0,859 SMALL-HS 125d 1,464 0,928 SMALL-EqWMA 50d 0,428 0,666 SMALL-EqWMA 125d -0,091 0,464 SMALL-EqWMA 250d -1,127 0,130 SMALL-ExpWMA 0.94 0,039 0,516 SMALL-ExpWMA 0.97 -0,868 0,193 MIX-HS 250d -0,473 0,318 MIX-HS 125d 1,861 0,969 MIX-EqWMA 50d -0,473 0,318 MIX-EqWMA 125d -1,122 0,131 MIX-EqWMA 250d -2,418 0,008 * MIX-ExpWMA 0.94 1,212 0,887 MIX-ExpWMA 0.97 -0,862 0,194 Fractions of Outcomes Covered - 99% level Z-value Probability Significance OMX-HS 250d 2,117 0,983 * OMX-HS 125d 3,820 1,000 *** OMX-EqWMA 50d 0,414 0,661 OMX-EqWMA 125d 0,418 0,662 OMX-EqWMA 250d -0,437 0,331 OMX-ExpWMA 0.94 1,550 0,939 OMX-ExpWMA 0.97 -0,153 0,439 SMALL-HS 250d 1,550 0,939 SMALL-HS 125d 5,523 1,000 *** SMALL-EqWMA 50d 4,672 1,000 *** SMALL-EqWMA 125d 4,678 1,000 *** SMALL-EqWMA 250d 2,969 0,999 ** SMALL-ExpWMA 0.94 4,955 1,000 *** SMALL-ExpWMA 0.97 4,104 1,000 *** MIX-HS 250d 0,702 0,759 MIX-HS 125d 4,394 1,000 *** MIX-EqWMA 50d 2,974 0,999 ** MIX-EqWMA 125d 3,542 1,000 *** MIX-EqWMA 250d 0,986 0,838 MIX-ExpWMA 0.94 3,542 1,000 *** MIX-ExpWMA 0.97 2,690 0,996 **

Appendix 11 Fractions of Outcomes Covered – Difference between small-cap and OMX portfolios Z-value Probability Significance OMX vs SMALL HS 250d – 95% 0,000 0,500 OMX vs SMALL HS 125d – 95% -0,252 0,401 OMX vs SMALL EqWMA 50d – 95% 0,548 0,708 OMX vs SMALL EqWMA 125d – 95% 0,280 0,610 OMX vs SMALL EqWMA 250d – 95% 0,000 0,500 OMX vs SMALL ExpWMA 0.97 – 95% 0,184 0,573 OMX vs SMALL ExpWMA 0.97 – 95% 0,097 0,539 OMX vs SMALL HS 250d – 99% -0,327 0,372 OMX vs SMALL HS 125d – 99% 0,797 0,787 OMX vs SMALL EqWMA 50d – 99% 2,307 0,989 * OMX vs SMALL EqWMA 125d – 99% 2,308 0,990 * OMX vs SMALL EqWMA 250d – 99% 2,072 0,981 * OMX vs SMALL ExpWMA 0.94 – 99% 1,749 0,960 * OMX vs SMALL ExpWMA 0.97 – 99% 2,421 0,992 **

Appendix 12 - Multiple Needed to Attain Desired Coverage

Multiple Needed to Attain Desired Coverage, 95%

0,850,870,890,910,930,950,970,991,011,031,051,071,09

Portfolio-Method

MNtADC

Multiple Needed to Attain Desired Coverage, 99%

0,800,850,900,951,001,051,101,151,201,251,301,351,401,451,50

Portfolio-Method

MNtADC

Appendix 13 – Average Multiple of Tail Event to Risk Measure

Average Multiple of Tail Event to Risk Measure, 95%

0,951,001,051,101,151,201,251,301,351,401,451,501,551,60

Portfolio-Method

AMoTEtRM

Average Multiple of Tail Event to Risk Measure, 99%

0,900,951,001,051,101,151,201,251,301,351,401,451,501,551,60

Portfolio-Method

AMoTEtRM

Appendix 14 – Maximum Multiple of Tail Event to Risk Measure

Maximum Multiple of Tail Event to Risk Measure, 95%

1,752,002,252,502,753,003,253,503,754,004,254,504,755,005,255,50

Portfolio-Method

MMoTEtRM

Maximum Multiple of Tail Event to Risk Measure, 99%

1,501,651,801,952,102,252,402,552,702,853,00

Portfolio-Method

MMoTEtRM

Appendix 15 – Correlation between Risk Measure and Absolute Value of Outcome

Correlation between Risk Measure and Absolute Value of Outcome, 95%

00,040,080,120,160,2

0,240,280,320,360,4

0,44

Portfolio-Method

CbRMa-AVoO

Correlation between Risk Measure and Absolute Value of Outcome, 99%

0,000,040,080,120,160,200,240,280,320,360,400,44

Portfolio-Method

CbRMa-AVoO

Appendix 16 Correlation coefficient - 95% level t-value Probability Significance OMX-HS 250d 8,45 0,00 *** OMX-HS 125d 11,26 0,00 *** OMX-EqWMA 50d 13,29 0,00 *** OMX-EqWMA 125d 9,32 0,00 *** OMX-EqWMA 250d 7,36 0,00 *** OMX-ExpWMA 0.94 15,32 0,00 *** OMX-ExpWMA 0.97 13,98 0,00 *** SMALL-HS 250d 5,95 0,00 *** SMALL-HS 125d 5,37 0,00 *** SMALL-EqWMA 50d 8,92 0,00 *** SMALL-EqWMA 125d 5,40 0,00 *** SMALL-EqWMA 250d 6,64 0,00 *** SMALL-ExpWMA 0.94 11,36 0,00 *** SMALL-ExpWMA 0.97 9,78 0,00 *** MIX-HS 250d 8,03 0,00 *** MIX-HS 125d 9,00 0,00 *** MIX-EqWMA 50d 10,28 0,00 *** MIX-EqWMA 125d 6,90 0,00 *** MIX-EqWMA 250d 6,59 0,00 *** MIX-ExpWMA 0.94 12,58 0,00 *** MIX-ExpWMA 0.97 11,14 0,00 *** Correlation coefficient - 99% level t-value Probability Significance OMX-HS 250d 9,21 0,00 *** OMX-HS 125d 10,58 0,00 *** OMX-EqWMA 50d 13,29 0,00 *** OMX-EqWMA 125d 9,32 0,00 *** OMX-EqWMA 250d 7,36 0,00 *** OMX-ExpWMA 0.94 15,32 0,00 *** OMX-ExpWMA 0.97 13,98 0,00 *** SMALL-HS 250d 6,72 0,00 *** SMALL-HS 125d 6,38 0,00 *** SMALL-EqWMA 50d 8,92 0,00 *** SMALL-EqWMA 125d 5,40 0,00 *** SMALL-EqWMA 250d 6,64 0,00 *** SMALL-ExpWMA 0.94 11,36 0,00 *** SMALL-ExpWMA 0.97 9,78 0,00 *** MIX-HS 250d 7,23 0,00 *** MIX-HS 125d 7,36 0,00 *** MIX-EqWMA 50d 10,28 0,00 *** MIX-EqWMA 125d 6,90 0,00 *** MIX-EqWMA 250d 6,59 0,00 *** MIX-ExpWMA 0.94 12,58 0,00 *** MIX-ExpWMA 0.97 11,14 0,00 ***

Appendix 17 – Mean Relative Bias for Risk Measure Scaled to Desired Level of Coverage

Mean Relative Bias for Risk Measure Scaled to Desired Level of Coverage, 95%

-0,040-0,032-0,024-0,016-0,0080,0000,0080,0160,0240,0320,0400,0480,056

Portfolio-Method

MRBfRM-StDLoC

Mean Relative Bias for Risk Measure Scaled to Desired Level of Coverage, 99%

-0,080-0,058-0,036-0,0140,0080,0300,0520,0740,0960,1180,1400,1620,184

Portfolio-Method

MRBfRM-StDLoC

Appendix 18 – Frequenciesσ of Portfolio vs Normal Distribution

Left-tail

σ Frequency, OMX

Frequency, Normal

σ Frequency, SMALL

Frequency, Normal

σ Frequency, MIXED

Frequency, Normal

-4,9. 1 1 -4,9. 1 0 -4,9. 1 0 -4,8. 0 0 -4,8. 0 0 -4,8. 0 0 -4,7. 1 0 -4,7. 0 0 -4,7. 1 0 -4,6. 0 0 -4,6. 0 0 -4,6. 0 0 -4,5. 1 0 -4,5. 0 0 -4,5. 0 0 -4,4. 0 0 -4,4. 2 0 -4,4. 1 0 -4,3. 1 0 -4,3. 0 0 -4,3. 0 0 -4,2. 1 0 -4,2. 2 0 -4,2. 2 1 -4,1. 2 0 -4,1. 0 0 -4,1. 0 0 -4,0 1 0 -4,0 0 0 -4,0 0 0 -3,9. 2 0 -3,9. 0 0 -3,9. 0 0 -3,8. 0 2 -3,8. 1 0 -3,8. 1 0 -3,7. 0 0 -3,7. 0 0 -3,7. 1 0 -3,6. 1 1 -3,6. 1 0 -3,6. 1 0 -3,5. 1 0 -3,5. 1 0 -3,5. 1 1

Right-tail

σ Frequency, OMX

Frequency, Normal

σ Frequency, SMALL

Frequency, Normal

σ Frequency, MIXED

Frequency, Normal

3,6. 0 0 3,6. 0 0 3,6. 0 2 3,7. 1 1 3,7. 0 0 3,7. 0 0 3,8. 0 1 3,8. 0 0 3,8. 0 0 3,9. 1 1 3,9. 0 0 3,9. 1 0 4,0 1 0 4,0 0 0 4,0 0 0 4,1. 0 0 4,1. 0 0 4,1. 0 0 4,2. 1 0 4,2. 1 0 4,2. 0 0 4,3. 0 0 4,3. 0 0 4,3. 0 0 4,4. 1 0 4,4. 0 0 4,4. 0 0 4,5. 0 1 4,5. 0 0 4,5. 0 0 4,6. 0 1 4,6. 1 0 4,6. 0 0 4,7. 1 0 4,7. 0 0 4,7. 1 0 4,8. 0 0 4,8. 0 0 4,8. 0 0 4,9. 1 0 4,9. 0 0 4,9. 0 0 5,0 1 0 5,0 1 0 5,0 1 1