Incorporating Discontinuities in Value-at- Risk via the Poisson Jump Diffusion Model and Variance Gamma Model

Name: Brendan Chee-Seng Lee (2231096) Degree: FINSCR2574 Masters of Commerce (Honours) Year of Submission: 2007 The University of New South Wales Thesis/Dissertation Sheet

Surname of Family name: Lee

First name: Brendan Other name/s: Chee-Seng

Abbreviation for degree as given in the University calendar: FINSCR2574 MComm (Honours)

School: Banking and Finance Faculty: Commerce

Title: Incorporating Discontinuities in Value-at-Risk via the Poisson Jump Diffusion Model and Variance Gamma Model

Abstract

We utilise several asset pricing models that allow for discontinuities in the returns and volatility time series in order to obtain estimates of Value-at-Risk (VaR). The first class of model that we use mixes a continuous diffusion process with discrete jumps at random points in time (Poisson Jump Diffusion Model). We also apply a purely discontinuous model that does not contain any continuous component at all in the underlying distribution (Variance Gamma Model). These models have been shown to have some success in capturing certain characteristics of return distributions, a few being leptokurtosis and skewness. Calibrating these models onto the returns of an index of Australian stocks (All Ordinaries Index), we then use the resulting parameters to obtain daily estimates of VaR. In order to obtain the VaR estimates for the Poisson Jump Diffusion Model and the Variance Gamma Model, we introduce the use of an innovation from option pricing techniques, which concentrates on the more tractable characteristic functions of the models. Having then obtained a series of VaR estimates, we then apply a variety of criteria to assess how each model performs and also evaluate these models against the traditional approaches to calculating VaR, such as that suggested by J.P. Morgan’s RiskMetrics. Our results show that whilst the Poisson Jump Diffusion model proved the most accurate at the 95% VaR level, neither the Poisson Jump Diffusion or Variance Gamma models were dominant in the other performance criteria examined. Overall, no model was clearly superior according to all the performance criteria analysed, and it seems that the extra computational time required to calibrate the Poisson Jump Diffusion and Variance Gamma models for the purposes of VaR estimation do not provide sufficient reward for the additional effort than that currently employed by Riskmetrics.

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Incorporating Jumps in Value-at-Risk

via the Poisson Jump Diffusion Model

and Variance Gamma Model

Name: Brendan Chee-Seng Lee (2231096)

Degree: FINSCR2574 Masters of Commerce (Honours)

Year of Submission: 2007

Supervisor: Dr David Colwell

ABSTRACT

We utilise several asset pricing models that allow for jumps in the return time series in order to obtain estimates of Value-at-Risk (VaR). The first class of model that we use mixes a normal diffusion process with discrete jumps at random points in time

(Poisson Jump Diffusion Model). We also apply a pure jump model that does not contain any continuous component at all in the underlying distribution (Variance

Gamma Model). These models have been shown to have some success in capturing certain characteristics of return distributions, a few being leptokurtosis and skewness.

Calibrating these models onto the returns of an index of Australian stocks (All

Page 1 of 169 Ordinaries Index), we then use the resulting parameters to obtain daily estimates of

VaR. In order to obtain the VaR estimates for the Poisson Jump Diffusion Model and the Variance Gamma Model, we introduce the use of an innovation from option pricing techniques, which concentrates on the more tractable characteristic functions of the models. Having then obtained a series of VaR estimates, we then apply a variety of criteria to assess how each model performs and also evaluate these models against the traditional approaches to calculating VaR, such as that suggested by J.P.

Morgan’s RiskMetrics. Our results show that whilst the Poisson Jump Diffusion model proved the most accurate at the 95% VaR level, neither the Poisson Jump

Diffusion or Variance Gamma models were dominant in the other performance criteria examined. Overall, no model was clearly superior according to all the performance criteria analysed, and it seems that the extra computational time required to calibrate the Poisson Jump Diffusion and Variance Gamma models for the purposes of VaR estimation do not provide sufficient reward for the additional effort than that currently employed by Riskmetrics.

Page 2 of 169 TABLE OF CONTENTS

Section 1: Introduction 5

Section 2: Value at Risk 9

Section 3: Stochastic Volatility & Jump-Diffusion Models 24

Section 4: Pure Jump Models 45

Section 5: Data and Descriptive Statistics 52

Section 6: Methodology 59

Section 7: Results 78

Section 8: Conclusion 106

Bibliography 110

Appendix 1: Deriving the Characteristic Function of the

Poisson Jump Diffusion Model 117

Appendix 2: Estimation of Variance Gamma Model using

Method of Moments Procedure 120

Page 3 of 169 Appendix 3: Maximum Likelihood Estimators of the Normal

Distribution 125

Appendix 4: Density Function of the Poisson Jump Diffusion

Model 127

Appendix 5: Matlab Code For Iterating the VaR for the

Variance Gamma Model 129

Appendix 6: Matlab Code For Iterating the VaR for the

Poisson Jump Diffusion Model 134

Appendix 7: VaR Estimates vs Actual Returns 140

Appendix 8: Portfolio Construction Application 150

Page 4 of 169 1. INTRODUCTION

When one is examining tools such as Value-at-Risk (VaR), they essentially have one foot in the area of risk management/stochastic optimization and the other in the realm of asset pricing. Researchers and practitioners alike have become increasingly aware of the importance of risk management. Financial disasters still occur – such as the

1997 Asia economic collapse and the near-collapse of Long Term Capital

Management in 1998. Domestically, recent newspaper headlines such as the National

Australia Bank’s $360mn in FOREX trading losses underscore the need for market participants to develop robust and accurate risk management techniques.

One technique advanced in literature as a method to measure financial risk is Value- at-Risk (VaR), which was developed in response to the financial disasters of the early

1990s that engulfed Barings, Orange County, Metallgesellschaft, Daiwa and many others.1 Intuitively, VaR summarises the worst loss over a target horizon with a given level of confidence. That is, VaR measures the risk that the market value of a portfolio will decline as a result of changes in such variables as interest rates, foreign exchange rates, equity prices, or commodity prices. VaR has become recognized as the new benchmark, with the Basle Committee on Banking Supervision (1994) and the Derivatives Policy Group (1994) endorsing the use of such models to measure

1 These financial disasters included the infamous Nicholas Leeson, who lost $1.3bn for Barings from derivatives trading; and the $1.81bn loss of Orange County from highly leveraged agency note strategies. Further details can be found in Jorion (2001).

Page 5 of 169 market risk. The unveiling of RiskMetrics by J.P. Morgan in October 1994, has also increased the popularity of VaR models amongst financial as well as non financial firms.2 But an important question that our thesis is based around is how well do these

VaR models perform in practice? How do actual losses compare to the theoretical losses predicted by the VaR models? How conservative are the models actually employed? How well do VaR models track changes in underlying risk over time?

Pivotal to the success of a VaR model is the asset pricing model underlying the statistical techniques used to calculate the VaR measure. Standard methods for calculating VaR have historically not done a good job capturing things like event risk and infrequent large price movements caused by information shocks. The more common methodologies for calculating VaR, including the approach suggested by the

RiskMetrics model, assumes away jumps and temporal correlation in a time series whilst make the underlying assumption of a normal or Gaussian return-loss distribution. Evidence abounds that the distribution of returns (whether we are talking about equity prices, exchanges rates or commodity prices) is far from normal, exhibiting characteristics such as leptokurtosis and skewness. Leptokurtosis or “fat tails” are especially problematic for VaR measurements, which focus on the tails of the distribution. Non parametric methodologies such as historical simulation allow for the adoption of modeling approaches that make no assumptions regarding the

2 J.P. Morgan provided a database system, RiskMetrics, which initially measured risk of 300 financial instruments across 14 countries. Companies are able to integrate RiskMetrics with their own positions to produce their own Value-at-Risk measure.

Page 6 of 169 statistical distribution. However, these methodologies are sensitive to the amount of data used and do not handle jumps well.

We apply two well known classes of models to this particular area of risk management in an attempt to more accurately represent the possible loss of a portfolio of financial assets. The first class of model that we examine mixes a normal diffusion process with discrete jumps. In particular, we analyse the well known

Poisson Jump Diffusion model, first put forward by Merton (1976). The argument for using this model, is that the normal diffusion part is responsible for the usual fluctuations in the return series and the jump part accounts for the extreme events.

The second class of model we examine (pure jump model) is a pure jump model that does not contain any continuous component at all. In particular, we are interested in the Variance Gamma model advocated by Madan (1999), which arises by evaluating a Brownian motion with drift at independent random times given by a Gamma process. Both of these models have been found by numerous authors to better explain the characteristics of an asset stochastic process. However, these models have generally been used as alternative models for the dynamics of asset prices in the option pricing context and never in the context of VaR (to our knowledge).

Using these models, we arrive at estimates of 95th and 99th percentile VaR for an index of Australian stocks (All Ordinaries Index). The actual benefits derived from the VaR estimates depend crucially on the quality and accuracy of the models used to obtain the estimates. Consequently, we then apply a variety of criteria to assess how each model performs. Following Engle and Gizycki (1999), we focus on three

Page 7 of 169 aspects of model performance – model accuracy, model conservatism and model efficiency.

The outline of the remainder of this thesis is as follows. In Section 2 we review the concept of Value-at-Risk. We look at the traditional methods of estimating these risk measures, both parametric and non-parametric, as well as their shortcomings. We also introduce the criteria used in assessing how each model performs under the VaR approach. In Section 3 we introduce and provide the basic setup for the dynamics of an underlying asset according to the jump-diffusion class of models. We examine various models that have been employed over time in an attempt to capture the pervading distributional characteristics of asset return processes. The Poisson Jump

Diffusion model is defined and we also present a generalized bi-dimensional jump diffusion model which may be used for future research. In Section 4, we define the

Variance Gamma process and present its properties. In Section 5 we outline the data used and provide descriptive statistics. In Section 6 we provide details on the steps required for our thesis. We outline the methodologies used to estimate our models’ parameters as well as obtaining our VaR forecasts. Obtaining the VaR estimates for the Poisson Jump Diffusion and Variance Gamma models have not been done before in literature (to our knowledge), and we use an innovation based on option pricing to obtain these. We also outline the methodologies used to assess the performance of each model underlying the VaR estimation. In Section 7 we present and discuss our empirical findings. Section 8 concludes our thesis.

Page 8 of 169 2. VALUE-AT-RISK

We will firstly turn to a review of the popular risk management tool Value at Risk

(VaR) and introduce various criteria used to evaluate the effectiveness of VaR as a risk measurement technique.

2.1 Introduction to Value-at-Risk

Intuitively, VaR summarises the worst loss over a target horizon with a given level of confidence. As one of our references states “VaR answers the question how much can I lose with x% probability over a pre-set horizon”.3 Till Guldimann, while head of global research at J.P. Morgan, created the term value at risk and VaR became widely publicised through the Group of Thirty report published in July 1993. Today, many banks, brokerage firms, investment funds and nonfinancial corporations use

VaR to gauge their financial risk. This was helped by the effort of J.P. Morgan, which unveiled its RiskMetrics system in October 1994. RiskMetrics provides a database, as well as a technical manual, which firms can use to produce their own

VaR measures. This RiskMetrics system has engaged the interest of financial as well as nonfinancial firms. However, standard methods for calculating VaR have historically not done a good job capturing things like event risk, for example, the

RiskMetrics methodology of calculating VaR makes the underlying assumption of a normal or Gaussian distribution of returns (which equivalent to assuming that prices

3 J.P. Morgan (1996) RiskMetrics Technical Document, 4th Edition, J.P. Morgan New York.

Page 9 of 169 follow a Geometric Brownian motion-a continuous process).

Page 10 of 169 2.2 Computing Value-at-Risk

Perhaps the greatest advantage of VaR is that it summarises in a single number the downside risk of an institution due to financial market variables. We now turn to a more formal definition of VaR. Essentially, VaR describes the quantile of the projected distribution of gains and losses over the target horizon. If α is the selected confidence interval (ie. 95th or 99th confidence level), VaR corresponds to the 1-α lower-tail level. For example, with a 95% confidence level, VaR should be such that the return distribution f(x) exceeds it 95% of the total number of observations in the distribution. In its general form, the VaR at the α confidence level can be inferred from:

∞ )( dxxf = α ∫ xVaR )( α .

Figure 1 below shows this graphically and illustrates the VaR for a distribution of returns calculated over the holding period using a confidence interval of 99%.

Throughout our analysis, the holding period is one day (ie we calculate daily VaR) and we set the estimation windows as one of three horizons 1250, 1,500 and 1,875 days (or about 5 years, 6 or 7.5 years). These are longer horizons than those used in existing literature by Hendricks (1996) and Engel and Gizycki (1999) for several reasons. Firstly, as highlighted in the literature that utilizes jump models, a sufficiently long time period must be used to capture jumps that may occur infrequently. Secondly, in calculating VaR using historical data, a long time period must be used in order to get a more accurate picture of the security’s behavior.

Page 11 of 169 Thirdly, we wanted to include the crash of 1987 as this would test the models’ ability to pick up extreme events and hence we do not remove the outliers as many other empirical examinations do.

99%

Value at Risk

Figure 1: Value at Risk

VaR can be calculated in a number of ways. The first way is to approximate the distribution by a parametric approximation, such as a normal distribution, in which case VaR is determined from the estimated standard deviation. The second way is to consider the actual empirical distribution by means of historical simulation.

2.2.1 Parametric Approximation

The VaR computation can often be simplified if the distribution can be assumed to belong to a parametric family, such as the normal distribution. This approach is called parametric because it involves the estimation of parameters, such as the standard deviation, instead of just reading the quantile off the empirical distribution.

Page 12 of 169 The most basic way of finding a particular VaR for a single asset is to assume the return of the asset is normally distributed as documented by Linsmeier and Pearson

(1996) and Pan (1997). If we assume that the distribution f(x) from above follows a

Gaussian process, then the VaR at α confidence level for one unit of this security is calculated by:

Φ xVaR 1))(( −= α α , −1 ∴ α xVaR Φ−= )()( σαμ where Φ-1 is the inverse cumulative standard normal density function and the mean, μ

(mean profit/loss, often assumed to be zero), and standard deviation σ (standard deviation of profit/loss), can be estimated using historical data.

For a portfolio containing n stocks, the VaR at α confidence level for one unit of this portfolio is given by:

T −1 T α )( wPVaR Φ−= αμ )( ∑ ww , where ∑ is the covariance matrix for the securities, w a column of vector whose elements are the weights for each security, μ a column vector of mean returns and Φ-1 is the inverse cumulative normal density function.

Generally the means (μ) and variances (σ or∑) are obtained by using historical data.

Whilst using historical data inherently assumes that the future will be like the past, different parametric approaches often define the past quite differently and make different assumptions about how markets will behave in the future. For example, two

Page 13 of 169 well known approaches to parametric approximation are the “equally weighted moving average” approach and the “exponentially weighted moving average method”. The equally weighted moving average approach is the more straightforward, and calculates a given portfolio’s variance using a fixed amount of historical data. The return covariances and variances are assumed to be constant over the period of estimation. The exponentially weighted moving average approach, which was devised by J.P. Morgan and Reuters, emphasise the recent observations by using exponentially weighting moving averages of squared deviations. Because the weights decline exponentially, the most recent observations receive much more weight than earlier observations. This approach clearly aims to capture short-term movements in volatility, which is the focus of conditional volatility forecasting. In fact, this method is equivalent to the IGARCH(1,1) family of popular conditional volatility models4. However, the reliance on recent data effectively reduces the overall sample size, increasing the possibility of measurement error. In the limiting case, relying on only yesterday’s observations would produce highly variable and error-prone risk measures.

Whilst the assumption of normality simplifies VaR calculations, the advantages of this simplicity must be weighed against whether the normal approximation is realistic. A large body of evidence that suggests that the tails of the distributions will be fatter than that predicted by the normal distribution. This is especially relevant for

VaR measurements, which focus on the tails of the distribution, and suggests that

4 Engle and Bollerslev (1986)

Page 14 of 169 another distribution may fit the data better. As the forms of our models are known, we will be taking the parametric approach to estimating VaR. However, evaluating

∞ the VaR from )( dxxf = α is not as straight forward for the Poisson Jump ∫ xVaR )( α

Diffusion and Variance Gamma models as if we had assumed a simple distribution such as the Gaussian process, and we were required to use an innovation from option pricing techniques5. This is discussed in more detail in Section 6.

2.2.2 Non parametric approaches

Historical simulation is a nonparametric technique in that it has no parameters to estimate as volatilities and correlations are already manifested within the historical data. Rather than employing actual observations to calculate the mean and standard deviation of the underlying securities distribution, historical simulation approaches use the actual percentiles of the observation period as VaR measures. As this method does not rely on valuation models, it is not prone to model risk. By relying on actual asset prices, historical simulation approaches make no assumptions of normality or serial independence. However, relaxing these assumptions also implies that historical simulation approaches do not easily accommodate translations between multiple percentiles and holding periods. The only decision to make for historical simulation is how many days of historical data to use. This however, is also the main problem with historical simulation as documented by authors such as Gibson (2001), Hendriks

(1996) and Dowd (2000). Gibson finds that typically historical simulations are done

5 Bakshi and Madan (2000)

Page 15 of 169 with too short a sample of historical data to accurately estimate the probability of jumps. This is because if the jumps are not present in the data used for simulation, they will not be present in the modeled distribution either. Hendricks finds that the historical VaR is extremely sensitive to the length of the measurement horizon and states that extreme percentiles such as the 95th and 99th are very difficult to estimate accurately with small samples. Another problem as cautioned by Mahoney (1995) is that past extreme returns could be a poor predictor of future extreme returns. The reason is that by its very nature, historical simulation has nothing to say about the probability outcomes which are worse than the sample minimum return.

Whilst Monte Carlo simulation methods are not strictly nonparametric, it is worth mentioning under this Section due to its similarity to the historical simulation approach. Monte Carlo approaches approximate the behaviour of asset prices by drawing from a prespecified stochastic process to generate random price paths. The

VaR can then be read off directly from the distribution of simulated asset values. Like the historical simulation approach, Monte Carlo analysis expresses the returns as a histogram. This approach can provide a much greater range of outcomes than historical simulation, and it is flexible enough to account for time variation in volatility, fat tails and extreme scenarios. Any distribution may be simulated, as long as the necessary parameters of the assumed distribution can be estimated. However, all this makes Monte Carlo analysis the most expensive and time-consuming method, and tends to make it unsuitable for large, complex portfolios. Another potential weakness of the method is model risk, as Monte Carlo relies on the specific stochastic processes used to represent underlying risk factors.

Page 16 of 169 2.2.3 Semi parametric approaches

Danielsson and de Vries (1997) propose a semi-parametric method for VaR evaluation that seeks to improve on the VaR estimate derived from historical simulation, without the constraints of using a parametric approach. The authors recognised that the shortcoming of the parametric approach is that the estimation of model parameters are biased to the centre of the distribution and hence perversely a method designed to predict common events well is used to predict extreme events

(which is what VaR is about). The authors’ approach only modeled the tail events parametrically, whilst historical simulation is used for common observations. The methodology involves ranking all hypothetical profits and losses on a relative basis and then calibrating them with the likelihood function to calculate the tail index. The semi-parametric method is compared with VaR’s calculating using historical simulation approaches and the RiskMetrics approach (which is the exponentially weighted moving average method described earlier). The authors find that the

RiskMetrics analysis underpredicts the VaR while historical simulation overpredicts the VaR. However, the estimates obtained from applying the semi-parametric method are more accurate in the VaR prediction as it provided a VaR measure in between the other two.

Page 17 of 169 2.3 Evaluating Value-at-Risk

We now turn to a discussion on how to evaluate the effectiveness of VaR as a risk measurement technique. The actual benefits derived from the VaR estimates depend crucially on the quality and accuracy of the models used to obtain the estimates.

From a practitioner’s point of view, inaccurate models which misstate a portfolio’s or a bank’s true risk exposure are not useful. Thus VaR estimates should be accompanied by some form of validation. Authors such as Engle and Gizycki (1999) and Hendrick (1996) look at various ways to assess VaR model performance. Engle and Gizycki (1999) focus on three aspects of model performance – Model Accuracy,

Model Conservatism and Model Efficiency. Accuracy looks at the proportion of losses that exceed the VaR estimate compared to the model’s stated level of confidence. Supervisory bodies are not only concerned with the accuracy of potential models, but also the level of conservatism regarding the estimated measure of risk. A conservative model can be defined as one that tends to yield high estimates of risk relative to other models. In addition, firms are concerned about model efficiency. An efficient model is one that is strongly correlated with the portfolio’s true risk exposure. A conservative, but inefficient model would tend to overestimate risk in periods of low risk. An efficient model on the other hand, provides sufficient conservatism to satisfy supervisory body’s requirements whilst at the same time minimizing the capital that must be held. We present here a variety of different metrics which provide an indication of model performance.

Page 18 of 169 2.3.1 Model Accuracy

Essentially, the models used in VaR estimation are only useful insofar as they predict risk reasonably well. Backtesting is a formal statistical framework that consists of verifying that actual losses are in line with projected losses. This involves systematically comparing the history of VaR forecasts with their associated security/portfolio returns.

Lopez (1999) identified a general approach to assessing the accuracy of VaR models.

It involves specifying a general loss function and then obtaining a numerical score that reflects the practitioner’s concern. Simulation results have indicated that this technique can distinguish between VaR models and is less prone to model misspecification. Lopez proposes a general loss function for security i at time t:

⎧ Δ ti + VARPf ,1, ti if),( ti +1, <Δ VARP ,ti LR ti +1, = ⎨ , ⎩ Δ ti + VARPg ,1, ti if),( ti +1, ≥Δ VARP ,ti where f() and g() are functions that satisfy f() ≥ g() and ΔP represents the realized profit or loss. We consider two forms of loss functions – a binary loss function, based only on whether a given day’s loss is larger or smaller than a VaR estimate, and a quadratic loss function, which take account of the magnitude of those losses that are larger than the VaR estimate.

The binary loss function treats any loss larger than the VaR estimate as an

‘exception’, however ignores the magnitude of the exception. Each loss that exceeds

Page 19 of 169 the VaR is assigned an equal weight of unity, while all other profits and losses have zero weight:

⎧ if1 ti +1, <Δ VARP ,ti L ti +1, = ⎨ . ⎩ if0 ti +1, ≥Δ VARP ,ti

If a VaR model is truly providing the level of coverage defined by its confidence level, then the average binary loss function over the full sample (ie the average over all t in the study) will be equal to 0.05 for the 95th percentile VaR and 0.01 for the

99th percentile VaR.

This binary loss function provides the simplest method to verify the accuracy of a

VaR as it is essentially examining its failure rate (ie. the proportion of times VaR is exceeded in a given sample). This allows us to utilize a likelihood-ratio test specified by Kupiec (1995) in order to test whether the sample point estimate (provided by the binary loss function) is statistically consistent with the VaR model’s prescribed confidence level. Suppose we have a α-VaR estimate for a total of T days. Define N as the number of failures and N/T as the failure rate. If the model is adequate, N/T should be an unbiased measure of p=1-α. This approach is fully non-parametric and the formal test statistic, obtained by modeling the number of failures as a binomial probability distribution is the Likelihood Ratio (LR) statistic:

N −NT ⎛ − pp )1( ⎞ 2 LR −= ln2 ⎜ ⎟ ~ χ 1 , ⎜ N −NT ⎟ ⎝ − TNTN )1()( ⎠

Page 20 of 169 where p is the probability of a failure under the null hypothesis that the actual proportion of failures is equal to the theoretical value, that is, H0: p=1-α. We would reject the null hypothesis (that the VaR model is accurate) at the 5% level if

LR>3.841. Statistical decision theory has shown that this test is the most powerful among its class (Jorion 2001).

The quadratic loss function takes into account the magnitude of the exception and also penalizes large exceptions more severely than a linear or binary measure:

2 ⎧ (1 ti +1, −Δ+ VaRP ,ti if) ti +1, <Δ VaRP ,ti L ti +1, = ⎨ . ⎩0 if ti +1, ≥Δ VaRP ,ti

Lopez (1999) found that the quadratic loss function provides a more powerful measure of model accuracy than the binary loss function, and attributed it to the use of the additional information embedded within the magnitude of the exception. In addition to taking account of the magnitude of the exception, the application of the quadratic functional form penalizes large exceptions more severely than a linear or binary measure. Sarma et al (2000) suggest that this form of loss function captures the goals of financial regulators, referring to it as a regulatory loss function.

2.3.2 Model Conservatism

These tests consider the relative size and variability in the VaR estimates produced by different models. This enables researchers to assess whether a particular model produces higher risk estimates relative to other models. Two related performance criterion were suggested by Hendricks (1996). The first criterion called the “mean

Page 21 of 169 relative bias statistic” (MRB) captures the extent to which different VaR models produce risk measures of similar average size. Given T time period and N VaR models, the MRB of model i is computed as:

T N 1 it −VaRVaR t 1 MRBi = ∑ , where VaRt = ∑VaRit . T t=1 VaRt N i=1

The second criterion called the “root mean squared relative bias” is a related measure and captures the degree to which the risk measures tend to vary around the all-model average risk measure. The statistic below captures two effects: the extent to which the average risk estimate provided by a given model systematically differs from the all-model average and the variability of each model’s risk estimate.

2 1 T ⎛ −VaRVaR ⎞ 1 N RMSRB = ⎜ it t ⎟ where VaR = VaR . i ∑⎜ ⎟ t ∑ it T t=1 ⎝ VaRt ⎠ N i=1

2.3.3 Model Efficiency

An efficient model is one that produces a risk measure strongly correlated with the portfolio’s true risk exposure. Intuitively, an efficient model is one that accurately tracks the evolution of risk exposures through time. A more efficient VaR model provides more precise resource allocation signals to traders and the financial firm as a whole.

A simple efficiency test is to measure the correlation between the VaR estimate and the absolute value of each day’s profit or loss. To the extent that the VaR is tracking

Page 22 of 169 the increased variance of the underlying return distribution, there should be some correlation between the size of the returns and the VaR. Given our discussion on the non-normality of return distributions outlined in Section 3, it is clearly appropriate to consider a correlation measure that does not rely on the assumption that the two series are normally distributed. Engel and Gizycki (1999) note that a rank correlation coefficient provides such a measure. We let X denote the series of return magnitudes and Y the VaR series. Ranks 1,2,…,T are assigned to the X’s and Y’s according to their relative size. Let Ui be the rank of Xi and Vi be the rank of Yi then the rank correlation coefficient, R, is defined as the correlation between U and V, that is,

= (),VUCorrR .

⎛ 1 ⎞ For sufficiently large samples, NR ⎜ ,0~ ⎟ . ⎝ n −1⎠

Page 23 of 169 3. STOCHASTIC VOLATILITY & JUMP-DIFFUSION

MODELS

In this Section we will examine some of the models that have been employed over time in an attempt to capture the pervading distributional characteristics of asset return processes. We will follow the evolution of the jump-diffusion class of models and introduce its dynamics as it relates to an asset return series. For this thesis, we are interesting in utilizing the Poisson Jump Diffusion model first put forward by

Merton (1976). We present a definition of this model and explain its components.

We finish with a brief look at a generalized form bi-dimensional affine jump model, which nests many of the jump diffusion models discussed in this Section. This more generalized model is a suggested area for future research.

3.1 The Gaussian Model

Research done by several prominent academics over the years have made the general assumption that stock returns follow a normal distribution. Indeed, this was the basic assumption for the pioneering works of Markowitz (1958), Sharpe (1964), Linter

(1965) and Mossin (1966). Markowitz really laid the foundations in 1958 with his

Modern Portfolio Theory and its assumption of normally distributed returns. For comparison purposes of our two models, we will also model the Gaussian distribution, which is still being used commonly by academics and practitioners. This model is normal and does not incorporate jumps, so it serves as a good sounding

Page 24 of 169 block as to whether the jumps we have incorporated add any value to the empirical strength of our models. The Gaussian distribution is simply given by:

2 1 ⎛ x − μ ⎞ xf )( = exp⎜− ⎟ , 2πσ ⎝ σ ⎠ where μ is the mean andσ is the standard deviation.

The Gaussian model is also the underlying distribution for the popular RiskMetrics methodologies of calculating VaR. We will utilize both the equally weighted moving average approach as well as the exponentially weighted moving average method

(introduced in Section 2.2.1) to obtain estimates of the volatility parameters. Both these approaches are discussed further in Section 6.

Much work has been done surrounding the statistical distribution of assets returns in the stock market and the foreign exchange market. Empirically, the evidence that has been collected from time series analysis regarding the stochastic process of securities such as stocks has been sizeable. As mentioned above, many practitioners in the past have adhered to the basic assumption of normality in stock returns as formalised by

Markowitz in his Modern Portfolio Theory. However, the finance and financial econometrics literature have raised serious doubts that this assumption holds: hardly any time series of returns can be described reliably with mean and variance only, and the existence of skewness, excess kurtosis, and autocorrelation seems to be the rule rather than the exception.

Page 25 of 169 That stock return processes exhibit skewness and leptokurtosis is undisputed. That is, financial returns have “fatter tails” than one would expect from a normal distribution.

This is supported by an overwhelming amount of research. Figure 2 below highlights the difference in the distribution of returns between a normal distribution and a distribution which is characterised by excess kurtosis, i.e. fat tails.

Figure 2

Normal Dist. Fat Tails

These facts have prompted both researchers and practitioners alike to look to alternatives besides the traditional continuous diffusion processes such as the

Brownian or Gaussian motion framework. One of the key features of Gaussian distributions is the thin exponentially decaying tails. This makes large changes in the asset prices much less probable than actually seen in the market. Thus models based on this distribution do badly at predicting extreme events, which is what we are interested in when we are researching VaR.

Page 26 of 169 It is also important for models pricing assets to take into account non-constant volatility. The Black Scholes model, the most celebrated approach to pricing options, assumes a constant mean and volatility. However, if the implied volatilities are calculated by inverting the Black Scholes formula using actual prices of options traded in the market and plotted against strike price, we observe a highly non- constant plot known as the “volatility smile” or skew. This gives us the motivation for examining models with non-constant volatility.

Some authors have attempted to explain the empirical distributional observations by using stationary processes with fatter tails than the normal distribution. Authors such as Fama (1965) and Mandelbrot and Taylor (1967) proposed to use a stable Paretian distribution in favour of the normal distribution. Rogalski and Vinso (1978) and others looked at using the Student’s t distribution, which also has fatter tails than a normal distribution. Evidence have shown that using these distributions provide a better fit than the benchmark Gaussian distribution, however they still fail to capture behaviors such as skewness that are exhibited in stock returns. Various time series models have been employed in an attempt to capture these pervading distributional features outlined above, which leads us onto the next sub-Section.

Page 27 of 169 3.2 Introduction of Time Varying Parameters

The next class of models is the utilization of a distribution with time varying parameters. This evolvement is important to understand the diffusion part of the jump-diffusion model. That volatility is non-constant and time-varying is substantially documented by authors such as Bollerslev, Chou and Kroner (1992) who provide an extensive survey of ARCH/GARCH literature. Stochastic volatility processes, such as the ARCH processes were introduced by Engle in 1982 to account for the lack of stationary volatility. In an ARCH(q) process, the volatility at time t is a function of the observed data at t-1, t-2, ….., t-q. In 1986, Bollersev introduced the

GARCH(q,p), or generalized ARCH process, where volatility at time t depends on the observed data at t-1, t-2, ….., t-q as well as volatilities at t-1, t-2, ….., t-p. This class of models differed greatly from the previous classes in that it allowed the time- varying characteristics prevalent in observed asset prices.

The traditional Black Scholes model assumes that the proportional price change of an

asset follows a process given by: = tt (μ + σdWdtSdS t ), where Wt is a Brownian motion and μ and σ are the constant mean and volatility of the model. Heston (1993) uses this same return process, but assumes that the variance is now driven by a mean reverting stochastic process, which may be correlated with the stochastic process for returns. Heston used the Stochastic Volatility model (which is shown below) to demonstrate that a model allowing for arbitrary correlation between volatility and spot asset returns is important for explaining return skewness and strike-price biases in the Black Scholes model. Some of the fat tails in actual data comes from this time-

Page 28 of 169 varying volatility itself as acknowledged by Duffie and Pan (1997) and Gibson

(2001). However actual market data are typically found to have fatter tails than would come from time-varying volatility alone. To match actual market data, an additional source of fat tails is needed such as jumps combined with stochastic volatility. Heston’s model assumes that the spot asset at time t and the volatility can be shown by the following:

)( μ += tSdztvSdttdS )()( 1 θω +−= σ 2 tdztvdttvtdv )()())(()(

Where z1(t) and z2(t) are Wiener processes, with correlation of ρ. The μ and

ω θ − tv ))(( are the respective drift factors and similarly, tv )( andσ tv )( are the respective standard deviation factors.

Page 29 of 169 3.3 Combining Stochastic Volatility and Jump-Diffusion models

3.3.1 The First Attempts at Using these Models - Jump Diffusions with Jumps

Only in the Return Series

The combining of stochastic volatility with discrete jumps has attractive properties.

The continuous stochastic volatility can accommodate clustered changes in volatility however its ability to generate enough negative skewness or excess kurtosis is limited. The jump process can internalize skewness and kurtosis and accounts for infrequent large price movements caused by information shocks. Additionally as noted by authors such as Bates (2000) and Pan (2001) in the context of pricing options, jumps primarily affect short-maturity options, whereas stochastic volatility primarily affects longer-maturity options. The papers mentioned below all have found that incorporating jumps improved the empirical performance of the models:

Merton (1976) suggested a jump-diffusion model in which stock prices are generated by a geometric Brownian motion with Poisson jumps superimposed on them. In this model, he derived implied option prices, assuming that the jump component represents idiosyncratic risk only. That is, any deviations from the log-normality of observed stock returns at any frequency can be attributed to jumps.

Jorion (1989) was one of the first authors to consider combining fat-tailed independent shocks with time-varying volatility and he did so for both the stock market and the foreign exchange market. Jorion considered two models in his 1989 study - a mixed jump-diffusion/ARCH model and a simple diffusion process with

Page 30 of 169 time-varying parameters (the first-order ARCH process). The purpose was to compare the empirical fit of these two distributions as alternatives to the usual continuous diffusion process that is the traditional multivariate Gaussian process.

The author then tested and compared the two models against each other. This was to answer the question whether allowing for stochastic time-varying second moments can fully account for the observed fat tails, or whether including jumps adds any value.

Jorion found that for exchange rates, the combined jump-diffusion and ARCH model was statistically superior to the pure diffusion process (ie. including jumps improved the empirical fit of the models). This was found for both monthly and weekly data, although the evidence was stronger for weekly data. More importantly for our purposes, the author also found evidence that jumps are present in the distribution of weekly exchange rates even after explicitly accounting for heteroscedasticity. This, in conjunction with evidence found by Hsieh (1988) and Engle and Bollerslev (1986) that the residuals of more complex ARCH and GARCH models exhibited substantial leptokurtosis, suggests that instead of fitting more complicated stochastic volatility models, one should add a jump component to account for the fat tails in financial return data.

Bakshi, Cao and Chen (1997) empirically examined different underlying price processes and compared how each generalization can improve option pricing and hedging. To do this, they look at the in- and out-of-sample pricing errors to reflect each model’s static performance and hedging errors to reflect each model’s dynamic

Page 31 of 169 performance. At the conclusion of their paper, they found that combining stochastic jumps and volatility yielded the most accurate pricing and internal consistency.

Having negative “risk neutral” jumps were important to capture the “smirkiness” exhibited in the cross-sectional options data.

3.3.2 Evolution of the Jump Diffusion model – affine models

The earlier authors of the jump diffusion model assumed that the stochastic volatility component and the jump components were independent. This implies that the day after a jump, such as a crash, another jump is equally likely. The class of “affine” models relaxed this assumption of mutual orthogonality, by allowing the jump intensity to be related to volatility.

The affine models provide tractable option pricing and estimation processes, and are based on the assumption that the state vector X follows an affine jump-diffusion.

This means a jump-diffusion where the drift vector, the “instantaneous” covariance matrix and the jump intensities all have affine dependence on the state vector. This class of models has produced papers that allow the jump intensity to vary with time including Bates (2000), Pan (2001), Duffie, Pan and Singleton (2000), and Anderson,

Benzoni and Lund (2001).

Pan attempts to identify and quantify the extent to which jump risk is priced in the market compared to the diffusive price shocks and the diffusive volatility shocks.

Pan (1999) assumes the stock return dynamics is characterized by the model adopted by Bates (2000). In this model the underlying return process has three sources of

Page 32 of 169 uncertainty namely, diffusive return shocks, volatility shocks and jump risks. Pan

(1999) allows the jump intensity defined as λ(t) to be a linear specification of the instantaneous volatility level in order to allow for the possibility that when the market is more volatile, the jump-risk premium implicit in option prices becomes higher.

That is, they specify the jump arrival intensity as λ(t)=λVt. The authors find that when estimating jump risk premia in isolation, the jump risk premia is highly positive and highly significant, indicating that fear of large adverse price moments is reflected in option prices through a large jump-risk premium. However, once estimating jump- and volatility-risk simultaneously, the jump-risk premium more than halves and also becomes barely significant.

In a related study, Anderson, Benzoni and Lund (2001) find that a continuous time model should include discrete jumps as well as stochastic volatility with pronounced negative relationship between return and volatility innovations. The authors also apply their model to the derivatives market to examine the relationship between their estimated models and option prices. They find that there is a general correspondence between the implied return dynamics from option prices to the dominant characteristics of the equity return process. The authors find that allowing the jump intensity to depend on volatility does not improve the fit. However, models that incorporate discrete jumps and stochastic volatility, with return innovations and diffusion volatility strongly and negatively correlated accommodate the main features of the daily S&P500 returns.

Page 33 of 169 Bates (2000) looks at a potential explanation behind the substantially negatively skewed distribution implicit in options prices subsequent to the 1987 crash. The two competing hypotheses were a stochastic volatility model with negative correlation between underlying asset return and volatility shocks; and a stochastic volatility jump-diffusion model with time-varying jump risk. Bates (2000) assumes a two- factor geometric jump-diffusion model as there is evidence from currency options6 which indicate that one-factor models can do a poor job in capturing the term structures of implicit volatilities over time and the two-factor model would do better.

This model nests the two competing hypotheses and the instantaneous conditional jump frequency is thus defined as λ(t)=λ0+λ1V1t+λ2V2t for the stochastic volatility jump-diffusion alternative. The models had 4 parameters in most instances and 5 for what the authors termed “constrained” estimates – which involved “smoothing” the state variable sample paths. Bates found that the stochastic volatility jump-diffusion two factor model achieved the lowest overall root mean squared error, which was used as a broad summary measure of model performance. Bates also found that the elementary premise behind the stochastic volatility model was justified and there is significant negative correlation between index and implicit volatility shocks.

However, the stochastic volatility model cannot sufficiently explain the empirical negative implicit skewness nor can the jump model with finite variance match longer- maturity option prices.

6 Evidence can be found in Taylor and Xu (1994) and Bates (1996)

Page 34 of 169 3.3.3 Jump Diffusions with Jumps in both the Return and Volatility Series

A limitation of the volatility specifications by these authors mentioned up to this point, and admitted by Pan (2001), Bates (2000) and others was that it does not allow volatility to jump. Empirical evidence in the existing literature indicates that the conditional volatility of returns increases rapidly, which cannot be generated simply by using a diffusive specification for volatility and jumps in returns. Anderson,

Benzoni and Lund (2000) found that an implausibly high parameter value in the volatility specification would be needed to accommodate the volatility smile/smirk observed in actual options data. That is, a possible misspecification of the volatility factor resulted in a high “volatility of volatility” parameter. This empirical problem could possibly be alleviated by a model incorporating two volatility factors, or incorporating jumps into the volatility process. Similar conjectures were also presented by Bates (2000) and Bakshi, Cao and Chen (1997) who conclude that their affine stochastic volatility models with jumps in returns only do not allow for a degree of volatility of volatility suffice to explain the substantial smirk in the implied volatilities of index option prices (Duffie, Pan and Singleton (2000)). That is, the diffusive specification for volatility and jumps in returns does not allow implied volatility to increase fast enough to generate sufficient skewness and leptokurtosis.

One shortcoming of the model used by Bates (2000) and Pan (2001) is that on days of low implied volatility they observe a large volume of quotes for deep in-the-money puts, whose prices their models cannot price. A possible solution is to allow jumps in volatility, with jump arrival rates more frequent or with larger jump amplitudes, when implied volatility is low.

Page 35 of 169 The advantage of jumps in volatility stems from the persistent nature of the diffusive volatility and the transient nature of the return jumps within the process. Whilst jumps in returns can generate large movements to explain events such as the 1987 crash, the impact of the jump is only transient and does not impact future distribution of returns. However, the diffusive volatility component is persistent, albeit can only increase gradually via a sequence of small normally distributed increments due to it being driven by a Brownian motion. Modeling jumps in volatility captures the gap between jumps in returns and diffusive volatility by providing a rapidly moving but persistent factor that drives the conditional volatility of returns.

Papers that consider models with jumps to the volatility include Eraker, Johannes and

Polson (2003), Pezzo and Uberti (2002), Duffie, Pan and Singleton (2000).

Duffie, Pan and Singleton (2000) use an affine jump-diffusion setting that allows correlated jumps in both the return and volatility process. Jumps may be correlated if their amplitudes are drawn from correlated distributions or if there is correlation between the stochastic jump arrival intensities. These authors examine an example with three types of jumps incorporated. Jumps in the return process with jump sizes normally distributed; jumps in the volatility process with jump sizes exponentially distributed; and simultaneous jumps in both the return process and volatility process.

The authors find that adding a jump to the volatility process causes a substantial decline in the volatility of the diffusion component of volatility. It also lowers the implied volatilities and attenuates the systematic overpricing of out of the money calls, at least for options that are not too far out of the money.

Page 36 of 169 Eraker, Johannes and Polson (2003) consider two models with jumps in volatility and returns. One with contemporaneous jump arrival times and correlated jump sizes and another with independent arrivals and sizes, both introduced by Duffie, Pan and

Singleton (2000). The authors use a likelihood-based estimation procedure and provide estimates of the unobserved jump times, jump sizes and spot volatilities.

They find that models without jumps in volatility are misspecified as they do not have a component driving the conditional volatility of returns, which is rapidly moving.

The model that allows volatility to jump independently of returns provides additional flexibility, although the authors warn that the model is harder to estimate with their methodology as jumps in volatility are not signaled by a jump in returns.

Pezzo and Uberti (2002) also examine how this model proposed by Duffie, Pan and

Singleton (2000) match volatility patterns extracted from real data. They find that the affine Jump Diffusion model with jumps in both returns and volatility outperform the

Stochastic Volatility model in forecasting variance. In Pezzo and Uberti (2002), the authors propose an alternate double jump model using a different positive distribution for the jumps in volatility. As opposed to Duffie, Pan and Singleton (2000) and

Eraker, Johannes and Polson (2003), who assume jumps in the volatility process have an exponentially distributed jump size, Pezzo and Uberti presuppose the same jumps to follow an inverted Gamma process. The authors find that their model has better predictive power in forecasting returns and volatility. This would be of interest for further study into simulating VaR bounds, as VaR bounds depend heavily on predicting volatility.

Page 37 of 169 3.4 Definition of the Poisson Jump Diffusion Model

For this thesis, we are interested in utilizing the Poisson Jump Diffusion model, first put forward by Merton (1976). We assume that the stock price series St can be described by a normal diffusion part and a discontinuous jump part, where the normal part is responsible for the usual fluctuation in xt and the jump part accounts for the extreme events. The Poisson Jump Diffusion model can be represented by the stochastic equation below. Let St be the price process of the index and let Y=ln(S):

⎛ 1 2 ⎞ dYt ⎜ −−= 1 ⎟ σσμλμ 1 t ++ JdNdWdt t , ⎝ 2 ⎠

⎛ 1 2 ⎞ where ⎜ −− σμλμ 1 ⎟ is the drift term, σ1 is the volatility of the diffusion part and ⎝ 2 ⎠

Wt is a standard Brownian motion. The jump component is represented by the JdNt term.

In order to estimate the Jump Diffusion model, it is necessary to make restrictions on the jump size J. We impose a distributional assumption on J such that likelihood

2 estimation is attainable. The size of the random jump is given by J ~ N(µ2, σ2 ) and the number of jumps is given by the Poisson process Nt, which is assumed to be independent of the Brownian motion Wt. We assume that the intensity of the Poisson process is given by λ per unit time. That is, λ is the mean number of jumps per unit time (which in our case is a trading year). Thus dNt ~ Poisson(λdt).

Page 38 of 169 3.4.1 Characteristic function of the Poisson Jump Diffusion Model

We show in Appendix 1 that the characteristic function of the general form of the

⎛ 1 2 ⎞ process dYt ⎜ −−= 1 ⎟ σσμλμ 1 t ++ JdNdWdt t can be written as: ⎝ 2 ⎠

⎧ ⎛ 1 2 ⎞ 1 22 ⎫ uJ φ tx )( = exp)( ⎨uu ⎜ −− σμλμ ⎟ ()z −++ 1)( λθσ tutut ⎬ where θJ(u) = eE ][ , and ⎩ ⎝ 2 ⎠ 2 ⎭

μ = θ J −1)1( . Therefore, we can write:

⎧ 1 2 2 ⎫ φ tx )( exp)( ⎨ μ ()()J uutuutuu θθσ J −−−+−+= 1)1)1(()( λt⎬. ⎩ 2 ⎭

The approach we use is to assume that jumps are normally distributed, with mean µ2, and standard deviation σ2. This model makes VaR calculations more tractable and straightforward to implement, as returns become conditionally normal, after

1 2 2 2 + 2 uu σμ 2 conditioning on the number of jumps. In this case, θJ(u) = e and the characteristic function can be written as:

1 2 2 1 2 ⎧ 1 2 2 2 + 2 uu σμ 2 2 + 2σμ 2 ⎫ φ tx )( exp)( ⎨ μ )( σ +−+= (etuutuu (eu )−−− 11 )λt⎬ . ⎩ 2 ⎭

The characteristic function allows us a way of estimating the parameters of the

Poisson Jump Diffusion model using a Maximum Likelihood Estimation procedure.

This is outlined in more detail in Section 6.

Page 39 of 169 3.5 For Future Research: The Generalized Bi-dimensional Jump

Model in an Affine Setting

As mentioned above in Section 3.3, research suggests that jump diffusion models with jumps in both the returns and volatility series appear to be superior for predicting volatility. For the purpose of future study, this would be an area of interest for the calculation of VaR, which depends heavily on predicting volatility. We present here a generalized form of the bi-dimensional jump diffusion model, and describe its characteristics. Section 6 of our thesis introduces a framework for parameter estimation as well as VaR estimation using the characteristic function of distributions, which may be applied for this generalized jump diffusion model in future research. We have included a summary of the derivation of the bi-dimensional jump diffusion model characteristic function here as it represented substantial work done by us, however we were unable to use this characteristic function in our VaR estimation framework due to its complexity and programming limitations.

Below we present a bi-dimensional affine jump-diffusion model, involving the stock return Y as well as the volatility V in some state space D⊂R2 solving the following stochastic equations:

Y ⎛ 1 ⎞ 01 dW 1 ⎛ y y ⎞ ⎛ t ⎞ ⎜ μλςμ −−− V ⎟ ⎛ ⎞⎛ t ⎞ dNZ t d⎜ ⎟ = t + Vdt ⎜ ⎟⎜ ⎟ + ⎜ ⎟ , 2 t − 2 2 v v ⎜V ⎟ ⎜ ⎟ ⎜ 1− σρρσ ⎟⎜dW ⎟ ⎜ dNZ ⎟ ⎝ t ⎠ ⎝ ⎠⎝ t ⎠ t ⎝ t ⎠ ⎝ v −Vvk t − )( ⎠ where the stochastic volatility process is modeled by the autonomous mean reverting

“square root process”, with constant long-run mean v if there is no jump in the

Page 40 of 169 2 volatility, mean reversion rate kv and volatility coefficient σv . If there is no jump in the volatility, then the long run mean of V is given by E(Vt)= v . However, with

μ λ jumps in the volatility, )( vVE += vv .7 t k

y v Nt and Nt are Poisson processes with constant intensities λy and λv. Jump sizes in returns and volatility are given by Zy and Zv respectively. The jump amplitude of the

2 y 2 returns follows a normal distribution with mean μy and variance σy (ie Z ~N(μy, σy )).

The jump amplitude of the volatility follows an exponential distribution with mean μv

v y v (ie Z ~exp(μv)). W=[W , W ] is a 2-dimensional adapted-standard Brownian motion.

The return drift is given by μ..

As can be seen, this model specification nests many of the popular models used for option pricing which we discussed earlier under Section 3:

7 To calculate this long-run mean, integrate the SDE to get

v t t N t ⎛ ⎞ ⎛ ⎞ ⎛ v ⎞ λμ ⎜ −+= )()()( ⎟ + ⎜ σ ⎟ + ⎜ ⎟vZEdWVEdsVsvkEVEVE + vv t 0 ⎜ ∫ ⎟ ⎜ ∫ ssv ⎟ ⎜∑ j ⎟ k ⎝ 0 ⎠ ⎝ 0 ⎠ ⎝ j =1 ⎠

v Since the stochastic integral against the Brownian motion is mean zero, ZE j )( = μv and the jump arrivals are Poisson, we have:

μ λ += + )(()()( + μ λ ttVtEvkVEVE )( vVE +=⇒ vv t t vv t k

Page 41 of 169 • (SV) Without jumps (equivalent to Heston’s (1993) square root stochastic

volatility model). λy = λ v = λ c = 0

• (SVJ) Jumps in return series, but no jumps in volatility (equivalent to Bate’s

(1996) model with normally distributed jumps in returns).

2 y ,0 v λλλ c ==> 0 and y NZ σμ yy ),(~ .

• (SVIJ) Jumps in return series and jumps in the volatility (first suggested by

Duffie, Pan and Singleton (2000)). The jumps are independently arriving,

with the jumps in the returns following a normal distribution and the jumps in

the volatility following an exponential distribution.

y 2 v )exp(~ and ),(~ ,0 ,0, vy ,0, λλλ c => ,0 NZ σμ yy ),(~ and Z μv )exp(~ .

• (SVCJ) Jumps in the return series and volatility, however there can be

simultaneous correlated jumps in the return and volatility series.

y 2 v vy 2 ),(~| and )exp(~ , ),(~ ,0, ,0 c ,0 λλλ vy => ,0, NZ σμ yy ),(~ , Z μv )exp(~ and NZZ , + z σρμ , ycvJyc ),(~|

If we let Xt = [Yt, Vt]’, then our proposed model can be written as a bi-dimensional

(n=2) affine jump-diffusion model in the following way:

t μ t += σ )()( + dZdWXdtXdX ttt

In the affine setting, the functions μ, σσ’ and λ, are assumed to be affine on D and they can be represented as follows:

2 2x2 μ(x) = Ko + K1x, with K=(Ko, K1) ∈ R xR

2x2 2x2x2 σ(x)σ’(x) = Ho + H1x, with H=(H0, H1) ∈ R xR

Page 42 of 169 2 λ(x)=lo+l1x, where l=(lo, l1) ∈ RxR

In this setting, Heston (1993) and Scott (1997) showed that the probability distribution function of Xt can be achieved by using the Inverse Fourier Transform of the characteristic function φ, which admits the following closed form representation as an exponential of an affine function:

1 t ++ ββα 2 )()()( VtYtt t φ 21 tt ),,,,( = etVYuu

The coefficients of the characteristic function above turn out to be solutions of the following complex-valued ODE, that are at the basis of the tractability of affine processes:

β1 = 0' 1 1 1 ' k 2 −−+= 2 2 − ρσββσβββββ 2 2 21 2 1 2 2 21 y y v v c c (' ) 1 −−−−−= lvk 02 βθββμλςμα 1 −− l0 βθ 2 −− l0 ββθ 21 − )1),(()1)(()1)((

with boundary conditions β(T)=u and α(T)=0.

The solution to the above ODE and the coefficients of the characteristic function of the bi-dimensional affine jump model can be given by:

β = u11

()1−− ea −− tTm )( β t)( = , 2 ()1)(2 −+− enmm −− tTm )(

Page 43 of 169 22 where, 1 ρσ σ 1 −+−= uukum 1 )1()( , = 1 ρσ − kun )( , = 1 (1− uua 1 )

⎡ + nm )( 2 ⎡ −+− enmm −− tTm )( )1)((2 ⎤⎤ t ςμα μλ )()()()( −−−−−= vktTutTu tT )( +− ln 1 1 ⎢ 2 2 ⎢ ⎥⎥ ⎣ σ σ ⎣ 2m ⎦⎦

y ⎡ ⎛ 1 2 2 ⎞ ⎤ + exp + σμ − − tTuul )(1 0 ⎢ ⎜ y 1 y 1 ⎟ ⎥ ⎣ ⎝ 2 ⎠ ⎦

v ⎡ 2μ a ⎡ ( −+ μ anm ) ⎤ μ a ⎤ l −+ v 1ln − v ()1 − e −− tTm )( − v − tT )( 0 ⎢ 2 2 ⎢ ⎥ ⎥ ⎣ −− μ v anm )( ⎣ 2m ⎦ ()+− μ v anm ⎦

−− tTm )( c ⎡ − BCT 1 ⎛ + CAe ⎞⎤ + l + ln 0 ⎢ ⎜ ⎟⎥ ⎣ ACC ⎝ + CA ⎠⎦

( μ ,vc ++−−= ρ μ vcJ 1, + ρ μ vcJ 1, numuanmA ) = ()− μρμρ b b μ b ++−−−−+ b nTentntetaTmemttmenutmutB vcJ 1, vcJ 1, ,vc ))(( b −−−= tTnmeCt ))((

(),vc +−+−= vcJ 1, − μρμρμ vcJ 1, numuanmC

This is the “state of the art” in option pricing, however it does not seem to be possible to estimate all of the parameters for this model at this time – particularly the parameters for the jumps in volatility. We mention this model as a possibility for future research.

Page 44 of 169 4. PURE JUMP MODELS

A drawback with the stochastic volatility and jump-diffusion models described above, is the parameter instability that is exhibited as documented by authors such as

Bates (1996) and Madan (1999). This lack of robustness is due to the property of infinite variation inherent in these sorts of models. Infinite variation means that the sum of absolute price moves is infinity in any interval8 and for this reason, Madan strongly advocates against the use of stochastic volatility and jump-diffusion models to model price dynamics. In contrast, pure jump models, as advocated by Madan, have finite variation, which enhances their robustness and thereby increases relevance for economic modeling. Pure jump or purely discontinuous models, as the names suggest, have no continuous component, with discontinuities infinite in number.

4.1 The argument for using pure jump processes

Madan uses economic analysis, combined with deep structural mathematical results, to point to the use of pure jump price processes over continuous price processes. In summary, under the no-arbitrage hypothesis, a price process is a semi martingale

(which can be decomposed into a martingale plus a very general model for the drift).

Every semi martingale can be written as a time-changed Brownian motion. Hence,

ε >0 8 A process of finite variation requires that the integral ∫ )( dxxxk be finite. In such processes the 0 sum of their moves in absolute magnitude is finite.

Page 45 of 169 the study of price processes is reduced to the study of time changes for Brownian motion whereby the time change may be determined either independently or dependent on the Brownian motion. Now, the crux of the author’s argument is that a price process can only be continuous if the time change is continuous, locally deterministic and non-random. Intuitively, one may view these time changes as measures of economic activity such as the arrival of new information, buy and sell orders or trades. One would expect some elements of randomness and local uncertainty from these activities, pointing to a class of discontinuous price processes.

Additionally, if the time change was continuous, as must be the case for a continuous price process, then the process must be locally Gaussian. There is considerable evidence cited against the likelihood of his possibility, pointing again to the use of pure jump models. Madan concludes by stating “The need to add on an additional continuous process onto a functioning purely discontinuous process must in our view be argued for on theoretical and empirical grounds.” (Madan 1999).

Page 46 of 169 4.2 The Variance Gamma Model

The Variance Gamma model, first introduced by Madan and Seneta (1990) is a particularly tractable and parsimonious subclass of the purely discontinuous processes with finite variation and infinite arrival rates. It is constructed out of two very well know processes, the Brownian motion and the Gamma process. The model arises by evaluating a Brownian motion with drift at independent random times given by a Gamma process. The process is parametrically parsimonious when compared to the well known Black and Scholes model, with two additional parameters that provide control over skewness and kurtosis. The volatility of the Gamma process provides control over kurtosis, while the drift in the Brownian motion before the time change, controls skewness.

A structural property of the model examined is the complete monotonicity of the arrival rates in jump size, whereby large jumps occur at a smaller rate than small jumps. This is a reasonable property as market participants placing buy orders face price increases and those placing sell orders face price decreases. Both types of market participants have an incentive to minimize these impacts, which is seen in practice by large institutional investors doing block trades and spreading their trades out over time. Another structural property of the model is the infinite arrival rates of jumps, which respects the intuition underlying the path continuity of Brownian motion as a model.

The evidence supporting pure jump models has been encouraging thus far. Madan

(1990) argues that the Variance Gamma model has the advantages of possessing

Page 47 of 169 qualities such as long tailedness relative to the normal for daily returns, with returns over longer periods approaching normality (Fama 1965); finite moments for at least the lower powers of returns; consistency with an underlying, continuous time stochastic process; and elliptical multivariate unit period distributions, which thereby maintain validity of the capital asset pricing model

Madan, Carr and Chang (1998) show that while option pricing errors from the Black

Scholes model are observed to be correlated with the degree of moneyness and maturity of the option, the Variance Gamma model is relatively free of these biases.

Madan (1999) summarises his case for using pure jump models. In this model, variance is constant, but the price process is purely discontinuous. The authors match the success of Brownian motion in option pricing and portfolio management with the success of the purely discontinuous Variance Gamma process obtained on time changing Brownian motion by a Gamma process. The improvement in option pricing is apparent in the elimination of the implied volatility smile in the strike direction.

The authors were also able to study the optimal management of portfolios of derivative securities, a question relatively untouched by the jump-diffusion literature.

4.2.1 The Variance Gamma Process

Madan, Carr and Chang (1998) explain the general form of the Variance Gamma model. Their paper generalizes the previous formulations by Madan and Seneta

(1990) and Madan and Milne (1991). The Madan and Seneta symmetric Variance

Gamma model, considers symmetric returns whereas the Madan and Milne

Page 48 of 169 asymmetric Variance Gamma model was intended for option pricing by skewing the former model in a general equilibrium setting.

The Variance Gamma model has three parameters σ (volatility of the Brownian motion), θ (drift in the Brownian motion with drift) and v (variance rate of the

Gamma time-change), which take into account the variance, skewness and kurtosis of the price process. Specifically, the Variance Gamma model is obtained as a

Brownian motion with drift, evaluated at a random time γ(t):

t θγ )( += σWtx γ t)( ,

where Wt is a standard Brownian motion with drift θ and volatilityσ, and γ(t) is a

Gamma process evaluated at t. The Gamma process is an infinitely divisible one, obtained by adding independent increments which follow a Gamma random variable.

That is, the increments, x, are independent and identically distributed over nonoverlapping intervals of equal length t, with the increments having the Gamma distribution:

t −1 x x v − )exp( v γ t)( xf )( = , t t v v Γ )( v where Γ(x) is the Gamma function.

Page 49 of 169 The mean of the Gamma density is t and the variance is vt. Thus, the average random time-change in t units of calendar time is t and its variance is proportional to the length of the interval.

4.2.2 Characteristic Function of the Variance Gamma Process

The Gamma density may also be described by its characteristic function, which is obtained by the inverse Fourier transform of the density function given above:

φγ t)( = γ tiuEu ))((exp()( t . ⎛ 1 ⎞ v = ⎜ ⎟ ⎝1− iuv ⎠

As the process is an infinitely divisible one, of independent and identically distributed increments over non-overlapping intervals of equal length, the Gamma process may be described in terms of the Levy measure (Revuz and Yor (1994)):

⎧ − x )exp( ⎪ v γ x)( dxk = ⎨ , x > 0for . ⎪ vx ⎩⎪ ,0 otherwise

Since the Levy measure has an infinite integral, we see that the Gamma process has an infinite arrival of jumps, most of which are small as is indicated by the concentration of the Levy measure at the origin. The process is pure jump and may be approximated as a compound Poisson process.

Page 50 of 169 The characteristic function of the Variance Gamma process is then evaluated by conditioning on the Gamma process first and then employing the characteristic of the

Gamma process itself. Specifically:

t ⎛ ⎞ v ⎜ ⎟ 1 φ = tiuxEu ))]([exp()( = ⎜ ⎟ . tx )( ⎜ σ 2v ⎟ ⎜1 θviu +− u 2 ⎟ ⎝ 2 ⎠

The moments and central moments of the Variance Gamma model are easily obtained by differentiating the above characteristic function. We show in Appendix 2, that the central moments are given by:

[])( = θttxE []()− [])()( 2 ()+= σθ 22 tvtxEtxE

[]()− []txEtxE 3 ()+= 32)()( 223 θσθ tvv []()− []txEtxE 4 ()4 += 123)()( 34222 tvvv ()4 22 ++++ 3636 θθσσθθσσ tvv 224

Which allows us a way of estimating the parameters of the Variance Gamma model easily enough using a Method of Moments estimation procedure. This is outlined in more detail in Section 6.

Page 51 of 169 5. DATA AND DESCRIPTIVE STATISTICS

In this thesis, we will be looking at the Australian stock market as the majority of studies done in this field thus far have concentrated on American stocks. The characteristics that we are looking to capture, such as leptokurtosis, skewness and time clustering are especially prevalent in volatile markets and stocks. For this reason, we would like to examine stocks that have data going back a substantial period to at least 1987, to incorporate the well known October 1987 market crash.

We obtain the daily closing prices of the All Ordinaries Index from IRESS market technology. Our data is for the period from the 4th January 1985 to 5th September

⎛ Pt ⎞ 2003. We take the natural logarithm of prices ln⎜ ⎟ to get the return series for the ⎝ Pt−1 ⎠ index. This gives us a total of 4869 observations.

It is important that a long period is chosen for several reasons. Firstly, as highlighted in the literature that utilizes jump models, a sufficiently long time period must be used to capture jumps that may occur infrequently. Secondly, in calculating VaR using historical data, a long time period must be used in order to get a more accurate picture of the security’s behavior. Thirdly, we wanted to include the crash of 1987 as this would test the models’ ability to pick up extreme events and hence we do not remove the outliers as many other empirical examinations do.

Page 52 of 169 5.1 Descriptive Statistics

Table 1 reports the descriptive statistics of the daily and fortnightly returns on the All

Ordinaries Index. The total number of observations is 4869 for the daily returns and

973 for the fortnightly returns and ranges from the 4th January 1985 to 5th September

2003. We also perform the Jarque-Bera Normality test of normality on the data to get a very crude idea of whether the data follows a normal distribution. The Jarque-Bera

Normality test is an asymptotic or large sample test that is based on skewness and kurtosis. The test statistic used for the test is given by:

⎡ 2 KS − )3( 2 ⎤ nJB ⎢ += ⎥ , ⎣ 6 24 ⎦ where n is the sample size, S is the skewness coefficient and K is the kurtosis coefficient. For a normally distributed variable, skewness should be equal to 0 and kurtosis should be equal to 3. Under the null hypothesis of normality, the JB statistic given above asymptotically follows a chi-square distribution with 2 degrees of

2 freedom (ie. χ 2). We reject the null hypothesis if the p-value is sufficiently low enough. From Table 1 below, we clearly reject the hypothesis of normality in the return distribution of the All Ordinaries index. This suggests that the methods using the normality assumption to calculate VaR are flawed and this assumption needs to be relaxed.

Page 53 of 169

Table 1 Descriptive Statistics

This Table reports the descriptive statistics of the daily and fortnightly returns on the All Ordinaries Index. The total number of observations is 4869 for the daily returns and 973 for the fortnightly returns and ranges from the 4th January 1985 to 5th September 2003. The Jarque-Bera test of normality is performed. The p-value of the test is given below in the parenthesis.

All Ordinaries Index Daily Returns Fortnightly Returns Number of Observations 4869 973 Mean 0.03010% 0.29941% Median 0.02075% 0.42493% Maximum 5.73870% 9.29057% Minimum -28.75850% -50.48325% Standard Deviation 0.96593% 3.43773% Skewness -6.07122 -4.93477 Kurtosis 167.55901 65.21944 Jarque-Bera Normality Test 5523699.238 160896.381 p-value (0.00000) (0.00000)

Figures 3 and 4 below show histograms of the daily and fortnightly returns of the data set. From the descriptive statistics table, we can see that skewness of both the daily and fortnightly data is negative. Negative skewness indicates a distribution with an

Page 54 of 169 asymmetric tail extending toward more negative values. The figures below clearly show this asymmetric tail, however the tails are not as fat as we would have thought.

The presence of skewness further suggests that symmetric distributions such as the normal and t distribution are inappropriate for modeling returns. Kurtosis describes how flat or peaked the distribution is compared to a normal distribution, with positive kurtosis indicating a relatively peaked distribution as seen in the figures below.

Figure 3: Histogram of All Ordinaries Index Daily Returns

1600

1400

1200

1000

800 Frequency 600

400

200

0

6 1 0 6 5 1 0 5 0 1 5 .29 .28 .27 .2 .25 .24 .23 .22 .2 .2 .19 .18 .17 .1 .1 .14 .13 .12 .1 .1 .09 .08 .07 .0 .04 .03 .02 0 0 .0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0.06-0 -0 -0 -0 -0.010. 0. 0.020.030.040

Page 55 of 169 Figure 4: Histogram of All Ordinaries Index Fortnightly Returns

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150 Frequency

100

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0 7 3 9 3 0 6 2 8 4 .5 .4 .4 .3 .35 .31 .27 .2 .2 .1 .1 .0 .0 .04 07 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 0.00 0 0.

Figures 5 and 6 below further examine the characteristics of the returns. The figures below show extremely large positive and negative values, particularly during the well known October 1987 crash and also in various other periods. These indicate possible presence of jumps during the observed period and again suggest that models that can account for jumps in the underlying asset return process should be more appropriate and hence produce better VaR estimates than the traditional models.

Page 56 of 169 Figure 5: Daily Returns All Ords 4 Jan 1985 to 5 Sep 2003

10.00%

5.00%

0.00%

-5.00%

-10.00%

-15.00% Daily Returns Daily -20.00%

-25.00%

-30.00%

-35.00% Jan-85 Jan-86 Jan-87 Jan-88 Jan-89 Jan-90 Jan-91 Jan-92 Jan-93 Jan-94 Jan-95 Jan-96 Jan-97 Jan-98 Jan-99 Jan-00 Jan-01 Jan-02 Jan-03

Date

Figure 6: Fortnightly Returns All Ords 4 Jan 1985 to 5 Sep 2003

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-30.00% Fortnightly Returns Fortnightly -40.00%

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-60.00% Jan-85 Jan-86 Jan-87 Jan-88 Jan-89 Jan-90 Jan-91 Jan-92 Jan-93 Jan-94 Jan-95 Jan-96 Jan-97 Jan-98 Jan-99 Jan-00 Jan-01 Jan-02 Jan-03

Date

Page 57 of 169 In the next Section, we will revisit the models that we will be using in this thesis, as well as outline the estimation methodologies we will use.

Page 58 of 169 6. METHODOLOGY

Our methodology in this thesis has 3 main steps which are required to obtain our results:

1) Firstly, we take a point in time and, utilizing the previous 1250, 1500 and 1875

observations as an estimation window, we obtain parameter estimates for each of

the various models that we are analyzing. During this step, we utilise various

parameter estimation methodologies such as those based on Maximum Likelihood

Estimation and Method of Moments.

2) Secondly, using these parameter estimates, we then obtain an estimate for the

VaR measure at that point time from each of the various models. For the Gaussian

model, the obtaining of an estimate of VaR is relatively straight forward and is

based on solving the integral of a known distribution. For the Poisson Jump

Diffusion and Variance Gamma model, however, we introduce the use of an

innovation from option pricing techniques, which concentrates on the more

tractable characteristic functions of the models, to obtain our estimates of VaR.

This is described in more detail in Section 6.2 below,

We roll our estimation window forward over our entire data sample, repeating

steps 1) and 2) at each point in time. This gives us a (very large) series of

parameter and VaR estimates for each of the various models analysed.

3) Thirdly, we then conduct performance tests on our VaR estimates which are

implicitly testing the suitability of the underlying VaR models used. As

Page 59 of 169 mentioned in the earlier Section of this thesis, we will examine the following

properties – model accuracy, model conservatism and model efficiency.

This methodology bears resemblance to that followed by other authors such as

Hendricks (1996) and Khanthavit and Srisopitsawat (2002). Key difference are in the models utilized and hence in the parameter and VaR estimation methodologies employed (these estimation methodologies are outlined in more detail in the rest of this Section), and also the size of the estimation windows. Hendricks did the simulation using 50, 100, 250, 500 and 1250 day observation windows. We use longer periods for estimating the parameters, since we are concerned with jumps and if the window range is too short, then the stochastic jumps may not be present because the abnormal circumstances that lead to jumps do not occur frequently.

6.1 Parameter Estimation Methodology

6.1.1 Gaussian Model

For comparison purposes, we model the commonly used Gaussian distribution. This model is continuous and does not incorporate jumps, so it serves as a good sounding block as to whether the jumps we have incorporated add any value to the empirical strength of our models. The Gaussian distribution is simply given by:

1 ⎛ ()x − μ 2 ⎞ xf )( = exp⎜− ⎟ , ⎜ 2 ⎟ 2πσ ⎝ 2σ ⎠ where μ is the mean and σ is the standard deviation.

Page 60 of 169 The Gaussian model is also the underlying distribution for the popular RiskMetrics methodologies of calculating VaR. We will utilize both the equally weighted moving average approach as well as the exponentially weighted moving average method to obtain estimates of the volatility parameters. The mean of the distribution can be simply estimated by the sample mean:

1 n ˆ μ x == ∑ xi . n i=1

The above estimator is the maximum likelihood estimator (MLE) for the normal distribution as shown in Appendix 3. This estimator of the mean is unbiased. We utilise two approaches – the equally weighted moving average approach as well as the exponentially weighted moving average method, to obtain estimates of the volatility parameter.

The calculation of the standard deviation under the equally weighed moving average approach is:

1 t−1 ˆ 2 σ t = ∑ s − xx )( , k −1 −= kts

where σt denotes the estimated standard deviation at the beginning of day t, the parameter k defines the number of days included in the moving average (“observation period”), xs is the security return on day s and μ is the mean return. This is an unbiased estimator of the standard error of the normal distribution.

Page 61 of 169 The calculation of the standard deviation under the exponentially weighted moving average approach is:

t−1 ~ st −− 1 2 t −= ∑λλσ s − xx )()1( , −= kts where λ denotes the ‘decay’ function whereby the model gives stronger weighting to the more recent data in determining the present or future volatility. The lower the decay factor, the faster the decay in the influence of a given observation, i.e. the less weight is put on past observations. We will use decay factors of 0.94, 0.96 and 0.99.

A decay factor of 0.94 implies that the VaR measure relies more on recent observations than a decay factor of 0.99, resulting in a higher level of variability.

Whilst relying heavily on recent data is important when trying to capture short-term movements in volatility, it also effectively reduces the sample size and increases the probability of measurement error. In the extreme case, where the decay factor limits to 0, the volatility measure only depends on yesterday’s observation, which would produce highly variable and error-probe risk measures.

6.1.2 Poisson Jump Diffusion Model

We introduced the Poisson Jump Diffusion Model in Section 3.4 as:

⎛ 1 2 ⎞ dYt ⎜ −−= 1 ⎟ σσμλμ 1 t ++ JdNdWdt t , ⎝ 2 ⎠

Page 62 of 169 ⎛ 1 2 ⎞ where ⎜ −− σμλμ 1 ⎟ is the drift term, σ1 is the volatility of the diffusion part and ⎝ 2 ⎠

Wt is a standard Brownian motion. The jump component is represented by the JdNt

2 term. The size of the random jump is given by J ~ N(µ2, σ2 ) and the number of jumps is given by the Poisson process Nt, which is assumed to be independent of the

Brownian motion Wt. We assume that the intensity of the Poisson process is given by

λ per unit time. That is, λ is the mean number of jumps per unit time (which in our case is a trading year). Thus dNt ~ Poisson(λdt).

uJ We write θJ(u)=E(e ) for the characteristic function of J, and let μ = θJ(1)-1 as the expected jump size. With this assumption, the stock has expected return, µ, i.e.

µT E(ST)=S0e . Our approach then assumes that jumps are normally distributed

2 J~N(µ2,σ2 ), which makes VAR calculations more straightforward, as returns become conditionally normal, after conditioning on the number of jumps. In this case,

1 1 + uu 2σμ 2 + σμ 2 2 2 2 2 22 θ J )( = eu and θμ J 1)1( e −=−= 1. For ease of parameter estimation,

⎛ 1 2 ⎞ we make a reparameterization in the following manner. Let 1 ⎜ −−= σμλμμ 1 ⎟ ⎝ 2 ⎠ and our stochastic equation becomes:

t μ1 σ 1 t ++= JdNdWdtdY t .

The discrete form of the above model can be written as:

Yx tt 1 1 τεστμ Δ++=Δ= NJ t ,

Page 63 of 169 where ε ~ N(0,1), and Δt=τ. We show in Appendix 4 that the density function of xt is given by:

∞ n −λτ ()λτ 22 t )( = ∑exf ()t ; ()+Φ nx , ()121 + nσστμμτ 2 . n=0 n!

In order to practically perform our estimation, we follow the approach of Ball and

Torous (1983, 1985) to discretize and truncate the above density function such that the density function contains a finite number of terms N. We use N=10 in our estimation process and our approximation of xt becomes a mixture of (N+1) normals:

10 n λΔ− ()λτ 2 2 t )( = ∑exf ()t ; ()+Φ nx , ()121 + nσστμμτ 2 . n=0 n!

We can then derive the log-likelihood function which can be used to obtain the

Maximum Likelihood Estimators of the parameters. However, Honoré (1998) shows that simply using MLE at this point, can result in inconsistent estimates, as the jump diffusion model is equivalent to a discrete mixture of N normally distributed variables, where N goes to infinity. Thus from mixture-of-distributions literature

(Kiefer (1978) and Hamilton (1994)) the likelihood function for some parametric specifications is unbounded which causes inconsistency of standard MLE.

Honoré (1998) suggests using the following procedure to obtain consistent and asymptotically normally distributed estimates. We fix the volatility parameters σ1 and σ2 to be in a compact set. We make a reparametrisation, for a fixed positive m we

2 2 let σ2 =mσ1 . Our log likelihood function becomes:

Page 64 of 169 n ⎡ 10 ()λτ k 1 ⎛ ()()+− kx μμτ 2 ⎞⎤ LLF = log e λΔ− exp⎜− i 21 ⎟ . ∑∑⎢ ⎜ 2 ⎟⎥ i==1 ⎣⎢k 0 k! πτσ ()12 + km ⎝ ()12 + km στ 1 ⎠⎦⎥

In our parameter estimation step for the Poisson Jump Diffusion model, we run our

Maximum Likelihood Estimation for m ranging from 0 to 0.1. The final parameter estimates are then taken at the value of m which gives the maximum log likelihood.

Figure 7: Maximising the Log Likelihood 2,786

2,784

2,782

2,780

2,778

2,776

2,774

Log Likelihood 2,772

2,770 Value of m where log likelihood is maximised 2,768 for parameter estimation 2,766

2,764 - 0.004 0.004 0.008 0.012 0.016 0.020 0.024 0.028 0.032 0.036 0.040 0.044 0.048 0.052 0.056 0.060 0.064 0.068 0.072 0.076 0.080 0.084 0.088 0.092 0.096 0.100 m

Our final parameter estimates, after transforming the reparameterization back to its original form for our original stochastic equation:

⎛ 1 2 ⎞ dYt ⎜ −−= 1 ⎟ σσμλμ 1 t ++ JdNdWdt t , can be calculated by: ⎝ 2 ⎠

Page 65 of 169 ~ ~ ~ ~ ~ , , 121 ,λσμμ and m are all by thegiven estimation procedure above, :obtain toable are then we are toable :obtain ~~~ 2 = mσσ 1

~ 1 ~ 2 ~ + σμ 22 μ e 2 −= 1

~~ 1 2 ~~ ++= σμλμμ ~ 1 2 1

6.1.3 Variance Gamma Model

We introduced the Variance Gamma Model in Section 4.2 as:

t θγ )( += σWtx γ t)( , which is a Brownian motion with drift θ and volatility σ evaluated at the gamma time

γ(t). The characteristic function of the Variance Gamma model is given by the following:

t ⎛ ⎞ v ⎜ ⎟ 1 φ = tiuxEu ))]([exp()( = ⎜ ⎟ . tx )( ⎜ σ 2v ⎟ ⎜1 θviu +− u 2 ⎟ ⎝ 2 ⎠

We use a method of moments estimation procedure to obtain the parameters θ, σ and v. The method of moments is a very simple procedure for finding an estimator for one or more population parameters. The kth moment of a random variable, taken about the origin is:

k μ’k = E(x )

Page 66 of 169 Where the corresponding kth sample moment is the average:

n 1 k m'k = ∑ xi . n i=1

The method of moments solves for the required parameters by setting μ = m'' kk .

Since the central moments are more manageable than the raw moments, we will actually use the central moments in our method of moments estimation process.

Using the central moments, the kth central moment of a random variable is:

~ k μ k xE −= μ)(' where μ represents the mean or E(x).

The corresponding kth sample central moment is given by:

n n ~ 1 k 1 m'k ∑()i −= xx where x is the sample mean given by x = ∑ xi n i=1 n i=1

Our method of moments estimation process utilising the central moments solves for ~ ~ the required parameters by setting μ' = m'kk . The central moments are given by:

[])( = θttxE []()− [])()( 2 ()+= σθ 22 tvtxEtxE , []()− []txEtxE 3 ()+= 32)()( 223 θσθ tvv []()− []txEtxE 4 ()4 += 123)()( 34222 tvvv ()4 22 ++++ 3636 θθσσθθσσ tvv 224 where since we are dealing with daily returns, we let t=1/250 for the number of trading days in a year. From the above, we can observe as per Madan Carr and

Chang (1998) that skewness is zero if θ =0. Furthermore, in the case of θ=0, we have

Page 67 of 169 that the fourth central moment divided by the square of the second central moment or the kurtosis is 3(1+v). This leads to the interpretation that the parameter v controls kurtosis and is in fact (for θ=0) the percentage excess kurtosis over the kurtosis of the normal distribution, which is 3.

Page 68 of 169 6.2 Value-at-Risk Estimation Methodology

In order to examine how the length of data used affects the accuracy of the VaR measures, we will use observation or estimation periods of 1250, 1500 and 1875 days

(or about 5 years, 6 or 7.5 years), rolling forward for each estimate of the model parameters and the VaR measure. That is, we estimate then re-estimate the models every day and roll the 1250- 1500- or 1875-day window to the end of the sample.

This approach was also followed by other authors such as Hendricks (1996) and

Khanthavit and Srisopitsawat (2002), however Hendricks did the simulation using 50,

100, 250, 500 and 1250 day observation windows. We use longer periods for estimating the parameters, since we are concerned with jumps and if the window range is too short, then the stochastic jumps may not be present because the abnormal circumstances that lead to jumps do not occur frequently. We calculate 95% and 99%

VaR measures from 16th March 1992 to 5th September 2003 for each of the models described above, using the relevant observation period. In each case, we draw from the 1200, 1500 or 1875 days preceding the date for which the VaR calculation is made. For example, the 1250-day equally weighted moving average estimate for a given date would be based on the 1250 days of historical data preceding the given date. Next, we calculate the actual stock returns from 16th Mach 1992 to 5th

September 2003 for each of the models described above. We can then assess the performance of each VaR approach for the stocks by backtesting, where we compare the VaR estimates generated with the realized returns for each given day.

Page 69 of 169 6.2.1 Gaussian Model

In its general form, the VaR at the α confidence level can be inferred from:

∞ )( dxxf = α , where f is the probability density function. ∫ xVaR )( α

The main advantage of the normal distribution is its known form and tractability.

Many statistical packages can integrate the normal distribution analytically. In order to derive the 95% VaR using the Gaussian model we use Microsoft Excel’s

NORMINV − ,1( ˆ,σμα ˆ 2 ) , function to calculate the following integral and solve for the VaR:

2 α xVaR )( 1 ⎛ ()x − μˆ ⎞ exp⎜− ⎟dx 1−= α . ∫ ∞− ⎜ 2 ⎟ 2 σπ ˆ ⎝ 2σˆ ⎠

6.2.2 Poisson Jump Diffusion Model and Variance Gamma Model

In contrast to using the Normal distribution, evaluating the VaR from

∞ )( dxxf = α is not as straight forward for our Poisson Jump Diffusion Model ∫ xVaR )( α and Variance Gamma models, and we introduce the use of an innovation from option pricing techniques, which concentrates on the more tractable characteristic functions of the models. Bakshi and Madan (2000) show that very generally one may write a call option price in the form of:

())0(,);0( 1 −∏= −rtKStKSc )exp( ∏ 2 ,

Page 70 of 169 where Π1 and Π2 are complementary distribution functions obtained on computing the integrals

− ln Kiu 1 1 ∞ ⎡e φ tS )(ln − iu )( ⎤ 1 +=∏ Re⎢ ⎥du ∫o 2 π ⎣⎢ φ tS )(ln −iiu )( ⎦⎥ . − ln Kiu 1 1 ∞ ⎡e φ tS )(ln u)( ⎤ 2 +=∏ Re⎢ ⎥du ∫o 2 π ⎣⎢ iu ⎦⎥

K refers to the exercise price, and φlnS(t)(u) refers to the characteristic function of the log of the stock price. Examining the equations above, the similarity to the much used Black Scholes formulas are obvious and the Π2 refers to the probability that the option will be exercised. That is, Π2 calculates the probability P(ST>K) where ST is the stock price at time T. Hence our calculation for calculating the VaR

∞ )( dxxf = α is essentially the same as calculating the distribution function Π2. ∫ xVaR )( α

In order to calculate the VaR, we need to solve the following integral for VaR:

∞ )( dxxf = α , ∫ xVaR )( α where α=0.95 and 0.99 (representing the 95% and 99% VaR respectively).

From above, this is equivalent to solving the following for VaR:

−iuVaR 1 1 ∞ ⎡e α ˆ ⎤ + Re φ tX )( )( duu = α , ∫0 ⎢ ⎥ 2 π ⎣ iu ⎦

Page 71 of 169 where φˆ is the characteristic functions of our models with the estimated parameters obtained as outlined in Section 6.1. We perform the calculations required above in

Matlab using an iterative procedure and the codes are included in Appendix 5 and 6.

Our optimisation procedure uses an initial step size α and at each step of the algorithm, it finds the next iterate xk+1 of the form:

+1 += daxx kkkk ,

−iuVaR 1 1 ∞ ⎡e α ˆ ⎤ where xk denotes the current iterate of VaR solving + Re φ tX )( )( duu , ∫0 ⎢ ⎥ 2 π ⎣ iu ⎦ dk is the current search direction which is determined by whether the current iterate of

VaR results in a probability that is higher or lower than the desired confidence level and ak is the scalar step size, which halves at each iteration so that our solution converges to our VaR estimate.

The reason for introducing this methodology is due to the relatively easier tractability of the characteristic function as opposed to the distributions of the Poisson Jump

Diffusion Model and the Variance Gamma model.

As introduced in Section 4, the characteristic function of the Variance Gamma Model is given by:

t ⎛ ⎞ v ⎜ ⎟ 1 φ = tiuxEu ))]([exp()( = ⎜ ⎟ . tx )( ⎜ σ 2v ⎟ ⎜1 θviu +− u 2 ⎟ ⎝ 2 ⎠

Page 72 of 169 As introduced in Section 3, the characteristic function of the Poisson Jump Diffusion

Model is given by:

1 2 2 1 2 ⎧ 1 2 2 2 + 2 uu σμ 2 2 + 2σμ 2 ⎫ φ tx )( exp)( ⎨ μ )( σ +−+= (etuutuu (eu )−−− 11 )λt⎬. ⎩ 2 ⎭

The parameters in the characteristic functions above are obtained as outlined in

Section 6.1 to give the estimated characteristic function. As mentioned previously, since we are dealing with daily returns, we let t=1/250 for the number of trading days in a year.

Page 73 of 169 6.3 Methodology for Conducting Performance Tests

To test the relative performance of our model, we compare the VaR estimates obtained from the Poisson Jump Diffusion model and the Variance Gamma model with the Gaussian model. As outlined in Section 3.1, we utilise a Gaussian model under two approaches – the equally weighted moving average approach as well as the exponentially weighted moving average method to obtain estimates of the volatility parameters.

6.3.1 Accuracy Testing

As outlined in Section 2.3, we utilize both the binary and quadratic loss function to analyse the relative accuracy of our VaR models. Utilising Excel we compare our series of VaR estimates with the series of actual returns/losses experienced from the

All Ordinaries Index and calculate the:

Binary Loss Function which is given by,

⎧ if1 ti +1, <Δ VARP ,ti L ti +1, = ⎨ , ⎩ if0 ti +1, ≥Δ VARP ,ti whereby, we can then conduct a likelihood-ratio test as specified by Kupiec (1995) to formally examine whether the sample point estimate (provided by the binary loss function) is statistically consistent with the VaR model’s prescribed confidence level.

Then, in order to confirm our results, we calculate the:

Page 74 of 169 Quadratic Loss Function which is given by,

2 ⎧ (1 ti +1, −Δ+ VaRP ,ti if) ti +1, <Δ VaRP ,ti L ti +1, = ⎨ , ⎩0 if ti +1, ≥Δ VaRP ,ti which takes into account the magnitude of the exception and also penalizes large exceptions more severely than a linear or binary measure. However, we cannot use the Kupiec likelihood-ratio statistic as this loss function does not give rise to a binomial experiment (which looks at independent trials resulting in a “success” with a certain probability p and “failure” with probability 1-p). What the quadratic function allows us to do however, is to analyse the relative accuracy of each of the VaR models and see whether the rankings are the same as that produced by the binary loss function.

6.3.2 Conservatism Testing

As outlined in Section 2.3, we utilize both the Mean Relative Bias (MRB) statistic as well as the Root Mean Squared Relative Bias (RMSRB) statistic to examine the relative size and variability of the VaR estimates produced by our models.

To calculate the MRB statistic, we follow a 4-stage process: First, we calculate the

VaR measures for each our models used for each observation date. Second, we average the VaR measures for each observation to obtain the average risk measure for that observation date. Third, we calculate the percentage difference between each model’s VaR measure and the average risk measure for each observation date (this gives us the daily relative bias figures). Fourth, we average the daily relative bias

Page 75 of 169 figures for a given VaR approach across the entire data sample to obtain the MRB statistic.

Intuitively, this procedure results in a measure of size for each VaR approach that is relative to the average of all the approaches. The MRB is measured as a percentage, so for example, a MRB of 10% implies that a given VaR approach is 10% larger, on average, than the average of all the approaches looked at.

We perform this analysis through Excel by lining up the series of VaR estimates produced by our models and for each time period T, and given our N VaR models, we calculate:

T N 1 it −VaRVaR t 1 MRBi = ∑ where VaRt = ∑VaRit . T t=1 VaRt N i=1

The RMSRB statistic looks at the degree to which the different VaR measures tend to vary around the average risk measure for a given observation date. This can be compared to a standard deviation calculation.

Again, we perform this analysis through Excel by lining up the series of VaR estimates produced by our models and for each time period T, and given our N VaR models, we calculate:

2 1 T ⎛ −VaRVaR ⎞ 1 N RMSRB = ⎜ it t ⎟ where VaR = VaR . i T ∑⎜ ⎟ t N ∑ it t=1 ⎝ VaRt ⎠ i=1

Page 76 of 169 After we have calculated the above statistics for all our VaR approaches, we can rank the VaR models accordingly according to how conservative they are.

6.3.3 Efficiency Testing

As outlined in Section 2.3, we utilize a rank correlation coefficient to examine the correlation between the size of the returns and our VaR estimates. The purpose of these performance criteria is to assess how well the VaR measures adjust over time to underlying changes in risk. That is, how closely do changes in the VaR measures correspond to actual changes in the risk of a portfolio. To the extent that the VaR is tracking the increased variance of the underlying return distribution, there should be some correlation between the magnitude of the returns and the VaR.

To obtain the rank correlation coefficient, R, we line up the series of returns and VaR estimates in Excel. We calculate the magnitude of returns by simply using the Excel function ABS(number) to get the absolute values. Then letting X denote the series of return magnitudes and Y the VaR series. Ranks 1,2,…,T are assigned to the X’s and

Y’s according to their relative size by using the Excel function RANK(number,range).

Let Ui be the rank of Xi and Vi be the rank of Yi then the rank correlation coefficient,

R, is defined as the correlation between U and V. This is obtained simply by utilizing the Excel function CORREL(range1, range2).

Page 77 of 169 7. RESULTS

7.1 Parameters Estimation Results

7.1.1 Gaussian Model

The first approach we use assumes normality, which simplifies the value-at-risk calculations. The Gaussian distribution is simply given by:

2 1 ⎛ x − μ ⎞ xf )( = exp⎜− ⎟ , 2πσ ⎝ σ ⎠ where μ is the mean andσ is the standard deviation. As previously noted, we use the equally weighted moving average and the exponentially weighted moving average approaches to estimating the standard deviation. With regards to the exponentially weighted moving average approach, a decay factor of 0.94 relies more on recent observations than a decay factor of 0.99.

The estimation of the mean of the returns is largely indifferent to the size of the estimation window when looking at the entire sample. The mean parameter is relatively lower during the subset period of 1992-1995 compared to the other periods.

This is because of the significantly negative returns recorded in the October 1987 crash (note that the subset dates refer to the dates at which the VaR is calculated, however they are utilising historical data going back 1250, 1500 or 1875 days). The standard deviation based on the equally weighted approach appears to be slightly higher in the subset period of 1992-1995. However, using the exponentially weighted

Page 78 of 169 approach it is higher in the subset period of 1996-1999. Again this is because of the significantly negative returns recorded in October 1987 have such a large influence on the subset period of 1992-1995. This influence is reduced through the use of a decay factor, which results in its influence decreasing as more recent observations are given a heavier weighting. As expected, the estimation window (1250, 1500 or 1875) has little influence on the standard deviations estimated using the exponentially weighted approach. This is because, the most recent observations are given the most weighting, hence the additional observations included by using a longer estimation window are given very small weighting.

The Table below records the parameter estimation results for the Gaussian Model.

Page 79 of 169 Table 2 Parameter Estimates Gaussian Models

This table presents the paramater estimates for our Gaussian Model using estimation periods of 1250, 1500 and 1875 days. Since we re-estimate the model every day for the backtesting period by rolling the estimation window forward to the end of the sample, we present here the average of the parameter estimates along with the standard deviation to indicate the variability of the estimate over that period as we roll the estimation window forward. To indicate the variability of results over time, we report the average over the entire sample period as well as 3 subsets of the sample period. Note that the subset dates refer to the dates at which the VaR is calculated, however they are utilising historical data going back 1250, 1500 or 1875 days

Panel A: Estimation period of 1250 days Entire Sample 1992-1995 1996-1999 2000-2003

Average Std Dev Average Std Dev Average Std Dev Average Std Dev

Gaussian Distribution µ 0.00022 0.00013 0.00014 0.00014 0.00029 0.00008 0.00023 0.00010 σ (Equally Weighted) 0.00849 0.00098 0.00883 0.00153 0.00806 0.00036 0.00861 0.00031 σ (Exponentially Weighted Moving Average: λ=0.94) 0.00775 0.00233 0.00761 0.00161 0.00830 0.00269 0.00729 0.00242 σ (Exponentially Weighted Moving Average: λ=0.97) 0.00787 0.00191 0.00768 0.00131 0.00840 0.00227 0.00749 0.00189 σ (Exponentially Weighted Moving Average: λ=0.99) 0.00802 0.00128 0.00783 0.00081 0.00845 0.00165 0.00777 0.00107

Panel B: Estimation period of 1500 days Entire Sample 1992-1995 1996-1999 2000-2003 Average Std Dev Average Std Dev Average Std Dev Average Std Dev

Gaussian Distribution µ 0.00022 0.00012 0.00011 0.00013 0.00029 0.00005 0.00024 0.00009 σ (Equally Weighted) 0.00882 0.00140 0.00986 0.00202 0.00809 0.00035 0.00855 0.00011 σ (Exponentially Weighted Moving Average: λ=0.94) 0.00775 0.00233 0.00761 0.00161 0.00830 0.00269 0.00730 0.00241 σ (Exponentially Weighted Moving Average: λ=0.97) 0.00787 0.00191 0.00768 0.00131 0.00840 0.00227 0.00750 0.00189 σ (Exponentially Weighted Moving Average: λ=0.99) 0.00803 0.00128 0.00783 0.00081 0.00845 0.00165 0.00777 0.00107

Panel C Estimation period of 1875 days Entire Sample 1992-1995 1996-1999 2000-2003 Average Std Dev Average Std Dev Average Std Dev Average Std Dev

Gaussian Distribution µ 0.00023 0.00009 0.00019 0.00011 0.00025 0.00006 0.00025 0.00007 σ (Equally Weighted) 0.00913 0.00152 0.01089 0.00152 0.00814 0.00017 0.00839 0.00010 σ (Exponentially Weighted Moving Average: λ=0.94) 0.00775 0.00233 0.00761 0.00161 0.00830 0.00268 0.00730 0.00242 σ (Exponentially Weighted Moving Average: λ=0.97) 0.00787 0.00191 0.00768 0.00131 0.00840 0.00227 0.00750 0.00189 σ (Exponentially Weighted Moving Average: λ=0.99) 0.00803 0.00127 0.00783 0.00081 0.00845 0.00165 0.00777 0.00107

Page 80 of 169 Note that the subset dates refer to the dates at which the VaR is calculated, however they are utilising historical data going back 1250, 1500 or 1875 days. Ie for the estimation period of: 1250 days - Entire Sample utilises returns recorded by the ASX All Ordinaries Index over 1 June 1987 to 5 September 2003; - 1992-1995 utilises returns recorded by the ASX All Ordinaries Index over 1 June 1987 to 31 December 1995; - 1996-1999 utilises returns recorded by the ASX All Ordinaries Index over 18 March 1991 to 31 December 1999; and - 2000-2003 utilises returns recorded by the ASX All Ordinaries Index over 20 March 1995 to 5 September 2003 1500 days - Entire Sample utilises returns recorded by the ASX All Ordinaries Index over 16 June 1986 to 5 September 2003; - 1992-1995 utilises returns recorded by the ASX All Ordinaries Index over 16 June 1986 to 31 December 1995; - 1996-1999 utilises returns recorded by the ASX All Ordinaries Index over 2 April 1990 to 31 December 1999; and - 2000-2003 utilises returns recorded by the ASX All Ordinaries Index over 4 April 1994 to 5 September 2003 1875 days - Entire Sample utilises returns recorded by the ASX All Ordinaries Index over 4 January 1985 to 5 September 2003; - 1992-1995 utilises returns recorded by the ASX All Ordinaries Index over 4 January 1985 to 31 December 1995; - 1996-1999 utilises returns recorded by the ASX All Ordinaries Index over 24 October 1988 to 31 December 1999; and - 2000-2003 utilises returns recorded by the ASX All Ordinaries Index over 26 October 1992 to 5 September 2003

Page 81 of 169 7.1.2 Poisson Jump Diffusion Model

The parameters estimated below are for the Poisson Jump Diffusion model

⎛ 1 2 ⎞ introduced in Section 3.4 as dYt ⎜ −−= 1 ⎟ σσμλμ 1 t ++ JdNdWdt t , ⎝ 2 ⎠

⎛ 1 2 ⎞ where ⎜ −− σμλμ 1 ⎟ is the drift term, σ1 is the volatility of the diffusion part and ⎝ 2 ⎠

Wt is a standard Brownian motion. The jump component is represented by the JdNt

2 term. The size of the random jump is given by J ~ N(µ2, σ2 ) and the number of jumps is given by the Poisson process Nt, which is assumed to be independent of the

Brownian motion Wt. We assume that the intensity of the Poisson process is given by

λ per unit time. That is, λ is the mean number of jumps per unit time (which in our case is a trading year). Thus dNt ~ Poisson(λdt).

⎛ 1 2 ⎞ The drift and volatility of the diffusion part (⎜ −− σμλμ 1 ⎟ and σ1 respectively) is ⎝ 2 ⎠ similar to the mean and standard deviation estimated for the Gaussian model in

Section 7.1.1. The main difference is that the volatility of the diffusion part is slightly lower for the Poisson Jump Diffusion Model. This is expected, because adding jumps to the returns has the effect of reducing the demands on the diffusion process (the large movements in the return series can be generated by the jump process). Lower average diffusion volatility indicates a less volatile and more persistent diffusion part. With the drift term, a similar pattern is seen as with the

Gaussian model, with the drift component uniformly lower in the subset period of

1992-1995 due to the large negative returns in October 1987 when the estimation

Page 82 of 169 windows of 1250 and 1500 days are used (note that the subset dates refer to the dates at which the VaR is calculated, however they are utilising historical data going back

1250, 1500 or 1875 days). However, for the estimation window of 1875 days, this is less prevalent, possibly demonstrating that the longer estimation windows are better for jump diffusion models.

Moving on to the jump component, the mean jump size, µ2, is negative, indicating that if a large return occurs on the ASX All Ordinaries Index, on average it is in the negative direction. The number of jumps varies significantly within a period as the estimation window is rolled forward. This is consistent with findings that in time periods of multiple large movements, jump diffusion models generate these movements by clusters of jumps. Consequently, as our estimation window rolls forward and includes large negative or positive movements, our jump diffusion model generates clusters of jumps and the number of jumps is estimated to increase for a time.

The Table below records the parameter estimation results for the Poisson Jump

Diffusion Model.

Page 83 of 169 Table 3 Parameter Estimates Poisson Jump Diffusion Model

This table presents the paramater estimates for the Poisson Jump Diffusion Model using estimation periods of 1250, 1500 and 1875 days. Since we re-estimate the model every day for the backtesting period by rolling the estimation window forward to the end of the sample, we present here the average of the parameter estimates along with the standard deviation to indicate the variability of the estimate over that period as we roll the estimation window forward. To indicate the variability of results over time, we report the average over the entire sample period as well as 3 subsets of the sample period. Note that the subset dates refer to the dates at which the VaR is calculated, however they are utilising historical data going back 1250, 1500 or 1875 days.

Panel A: Estimation period of 1250 days Entire Sample 1992-1995 1996-1999 2000-2003

Average Std Dev Average Std Dev Average Std Dev Average Std Dev

Poisson Jump Diffusion Model µ 0.00025 0.00011 0.00018 0.00011 0.00032 0.00008 0.00026 0.00010

µ 2 -0.00603 0.00907 -0.00439 0.00336 -0.00324 0.00277 -0.01076 0.01426 σ 1 0.00728 0.00101 0.00762 0.00099 0.00676 0.00123 0.00751 0.00030 σ 2 0.00139 0.00054 0.00201 0.00034 0.00113 0.00039 0.00103 0.00016 λ 0.11161 0.37998 0.07523 0.40568 0.19797 0.49456 0.05505 0.04411

Panel B: Estimation period of 1500 days Entire Sample 1992-1995 1996-1999 2000-2003 Average Std Dev Average Std Dev Average Std Dev Average Std Dev

Poisson Jump Diffusion Model µ 0.00025 0.00010 0.00018 0.00010 0.00031 0.00005 0.00027 0.00009

µ 2 -0.00320 0.00334 -0.00587 0.00419 -0.00058 0.00090 -0.00328 0.00098 σ 1 0.00684 0.00140 0.00777 0.00007 0.00561 0.00176 0.00721 0.00023 σ 2 0.00117 0.00060 0.00195 0.00021 0.00065 0.00030 0.00095 0.00008 λ 0.27031 0.58255 0.02293 0.01185 0.67529 0.84718 0.08440 0.03511

Panel C Estimation period of 1875 days Entire Sample 1992-1995 1996-1999 2000-2003 Average Std Dev Average Std Dev Average Std Dev Average Std Dev

Poisson Jump Diffusion Model µ 0.00028 0.00009 0.00028 0.00012 0.00027 0.00006 0.00028 0.00007

µ 2 -0.00449 0.00432 -0.00880 0.00406 -0.00024 0.00043 -0.00467 0.00145 σ 1 0.00653 0.00194 0.00737 0.00142 0.00503 0.00232 0.00728 0.00031 σ 2 0.00122 0.00061 0.00187 0.00051 0.00067 0.00031 0.00114 0.00018 λ 0.40218 0.79748 0.18445 0.57687 0.91799 1.04278 0.06488 0.07248

Page 84 of 169 7.1.3 Variance Gamma Model

We introduced the Variance Gamma Model in Section 4.2 as:

t θγ )( += σWtx γ t)( , which is a Brownian motion with drift θ and volatility σ evaluated at the gamma time

γ(t). Recall that γ(t) is a gamma process with variance vt. As noted previously, the drift component θ gives an indication of skewness. In our results, the value of θ is positive, indicating positive skewness (which is in contrast to that predicted in our descriptive statistics Section 5.1. In addition, the parameter v gives an indication of excess kurtosis, with a positive value implying excess kurtosis or fatter tails, which is consistent with what is expected.

The Table below records the parameter estimation results for the Variance Gamma

Model.

Page 85 of 169 Table 4 Parameter Estimates Variance Gamma Model

This table presents the paramater estimates for our Variance Gamma Model using estimation periods of 1250, 1500 and 1875 days. Since we re-estimate the model every day for the backtesting period by rolling the estimation window forward to the end of the sample, we present here the average of the parameter estimates along with the standard deviation to indicate the variability of the estimate over that period as we roll the estimation window forward. To indicate the variability of results over time, we report the average over the entire sample period as well as 3 subsets of the sample period. Note that the subset dates refer to the dates at which the VaR is calculated, however they are utilising historical data going back 1250, 1500 or 1875 days.

Panel A: Estimation period of 1250 days Entire Sample 1992-1995 1996-1999 2000-2003

Average Std Dev Average Std Dev Average Std Dev Average Std Dev

Variance Gamma Distribution θ 0.05572 0.03222 0.03414 0.03579 0.07363 0.02070 0.05849 0.02480 v 0.01429 0.04091 0.03066 0.06818 0.00507 0.00328 0.00744 0.00166 σ 0.13460 0.01535 0.13994 0.02396 0.12784 0.00573 0.13646 0.00496

Panel B: Estimation period of 1500 days Entire Sample 1992-1995 1996-1999 2000-2003

Average Std Dev Average Std Dev Average Std Dev Average Std Dev

Variance Gamma Distribution θ 0.05448 0.03035 0.02852 0.03198 0.07316 0.01256 0.06092 0.02307 v 0.03469 0.07643 0.09219 0.11278 0.00454 0.00259 0.00817 0.00084 σ 0.13978 0.02199 0.15609 0.03167 0.12824 0.00551 0.13552 0.00181

Panel C Estimation period of 1875 days Entire Sample 1992-1995 1996-1999 2000-2003 Average Std Dev Average Std Dev Average Std Dev Average Std Dev

Variance Gamma Distribution θ 0.05863 0.02246 0.04806 0.02763 0.06412 0.01632 0.06356 0.01801 v 0.06450 0.10166 0.18141 0.10406 0.00605 0.00258 0.00749 0.00099 σ 0.14423 0.02327 0.17112 0.02321 0.12907 0.00271 0.13299 0.00159

Page 86 of 169 7.2 VaR Simulation Results

In Table 5 below, we display the statistics for our VaR simulation procedure.

Page 87 of 169 Table 5 Value-at-Risk Simulation Results

This table presents the simulation results for our VaR using estimation periods of 1250, 1500 and 1875 days. Since we re- estimate the model every day for the backtesting period by rolling the estimation window forward to the end of the sample, we present here the average of the parameter estimates along with the standard deviation. To indicate the variability of results over time, we report the average over the entire sample period as well as 3 subsets of the sample period. Note that the subset dates refer to the dates at which the VaR is calculated, however they are utilising historical data going back 1250, 1500 or 1875 days.

Panel A: Estimation period of 1250 days Entire Sample 1992-1995 1996-1999 2000-2003 Average Std Dev Average Std Dev Average Std Dev Average Std Dev

95% VaR Gaussian (Equally Weighted) -0.01374 0.00169 -0.01439 0.00263 -0.01297 0.00063 -0.01392 0.00047 Gaussian (Exp Weighted:Lambda=0.94) -0.01253 0.00383 -0.01238 0.00263 -0.01336 0.00443 -0.01177 0.00398 Gaussian (Exp Weighted:Lambda=0.97) -0.01273 0.00314 -0.01250 0.00214 -0.01352 0.00375 -0.01210 0.00310 Gaussian (Exp Weighted:Lambda=0.99) -0.01298 0.00209 -0.01274 0.00133 -0.01361 0.00273 -0.01254 0.00173 Variance Gamma -0.01273 0.00093 -0.01273 0.00141 -0.01244 0.00048 -0.01306 0.00047 Poisson Jump Diffusion -0.01273 0.00056 -0.01300 0.00064 -0.01254 0.00052 -0.01265 0.00036

99% VaR Gaussian (Equally Weighted) -0.01953 0.00236 -0.02041 0.00367 -0.01847 0.00088 -0.01979 0.00069 Gaussian (Exp Weighted:Lambda=0.94) -0.01781 0.00541 -0.01756 0.00373 -0.01902 0.00627 -0.01674 0.00562 Gaussian (Exp Weighted:Lambda=0.97) -0.01809 0.00444 -0.01774 0.00303 -0.01924 0.00530 -0.01720 0.00439 Gaussian (Exp Weighted:Lambda=0.99) -0.01845 0.00296 -0.01807 0.00188 -0.01937 0.00386 -0.01783 0.00245 Variance Gamma -0.01795 0.00131 -0.01795 0.00199 -0.01754 0.00067 -0.01841 0.00066 Poisson Jump Diffusion -0.01809 0.00079 -0.01848 0.00091 -0.01782 0.00073 -0.01798 0.00051

Panel B: Estimation period of 1500 days Entire Sample 1992-1995 1996-1999 2000-2003 Average Std Dev Average Std Dev Average Std Dev Average Std Dev

95% VaR Gaussian (Equally Weighted) -0.01429 0.00238 -0.01610 0.00340 -0.01301 0.00059 -0.01382 0.00024 Gaussian (Exp Weighted:Lambda=0.94) -0.01253 0.00383 -0.01241 0.00266 -0.01337 0.00442 -0.01176 0.00398 Gaussian (Exp Weighted:Lambda=0.97) -0.01273 0.00315 -0.01253 0.00217 -0.01352 0.00373 -0.01209 0.00311 Gaussian (Exp Weighted:Lambda=0.99) -0.01298 0.00208 -0.01276 0.00134 -0.01361 0.00270 -0.01253 0.00174 Variance Gamma -0.01230 0.00131 -0.01144 0.00193 -0.01249 0.00045 -0.01296 0.00028 Poisson Jump Diffusion -0.01246 0.00058 -0.01283 0.00047 -0.01211 0.00062 -0.01246 0.00036

99% VaR Gaussian (Equally Weighted) -0.02030 0.00334 -0.02281 0.00477 -0.01852 0.00082 -0.01964 0.00031 Gaussian (Exp Weighted:Lambda=0.94) -0.01782 0.00542 -0.01759 0.00376 -0.01902 0.00625 -0.01673 0.00563 Gaussian (Exp Weighted:Lambda=0.97) -0.01810 0.00445 -0.01776 0.00307 -0.01925 0.00528 -0.01720 0.00440 Gaussian (Exp Weighted:Lambda=0.99) -0.01845 0.00295 -0.01809 0.00190 -0.01937 0.00383 -0.01783 0.00247 Variance Gamma -0.01734 0.00185 -0.01613 0.00273 -0.01761 0.00064 -0.01828 0.00039 Poisson Jump Diffusion -0.01771 0.00082 -0.01824 0.00066 -0.01721 0.00088 -0.01771 0.00051

Page 88 of 169 Panel C Estimation period of 1875 days Entire Sample 1992-1995 1996-1999 2000-2003 Average Std Dev Average Std Dev Average Std Dev Average Std Dev

95% VaR Gaussian (Equally Weighted) -0.01478 0.00253 -0.01772 0.00249 -0.01313 0.00028 -0.01354 0.00016 Gaussian (Exp Weighted:Lambda=0.94) -0.01252 0.00383 -0.01232 0.00267 -0.01340 0.00441 -0.01175 0.00397 Gaussian (Exp Weighted:Lambda=0.97) -0.01272 0.00314 -0.01244 0.00219 -0.01356 0.00372 -0.01208 0.00309 Gaussian (Exp Weighted:Lambda=0.99) -0.01297 0.00208 -0.01268 0.00139 -0.01365 0.00268 -0.01252 0.00173 Variance Gamma -0.01181 0.00170 -0.00993 0.00180 -0.01264 0.00033 -0.01286 0.00001 Poisson Jump Diffusion -0.01222 0.00072 -0.01268 0.00054 -0.01172 0.00080 -0.01229 0.00036

99% VaR Gaussian (Equally Weighted) -0.02100 0.00356 -0.02514 0.00352 -0.01868 0.00039 -0.01926 0.00022 Gaussian (Exp Weighted:Lambda=0.94) -0.01780 0.00541 -0.01751 0.00377 -0.01906 0.00624 -0.01672 0.00562 Gaussian (Exp Weighted:Lambda=0.97) -0.01808 0.00444 -0.01768 0.00308 -0.01929 0.00526 -0.01719 0.00438 Gaussian (Exp Weighted:Lambda=0.99) -0.01844 0.00295 -0.01801 0.00194 -0.01941 0.00380 -0.01782 0.00245 Variance Gamma -0.01679 0.00241 -0.01411 0.00256 -0.01796 0.00046 -0.01828 0.00002 Poisson Jump Diffusion -0.01737 0.00102 -0.01802 0.00076 -0.01666 0.00113 -0.01746 0.00052

Figures 8 to 13 show the estimated VaR using the Gaussian (Equally Weighted)

approach, the Gaussian (Exponentially Weighted λ=0.94, 0.97 and 0.99), Variance

Gamma and Poisson Jump Diffusion Model against the actual returns using a 1250

estimation window.

Looking at Figure 8, the Gaussian Equally Weighted approach, we notice that the

VaR estimates for the period 16th March 1992 to 4th August 1992, are noticeably larger than the later periods. This is due to these periods including the October 1987 crash in the estimation of parameters and subsequent VaR estimate and assigning an equal weight to this period as with the more recent periods (note that the subset dates refer to the dates at which the VaR is calculated, however they are utilising historical data going back 1250, 1500 or 1875 days). In contrast, looking at the Gaussian

Exponentially Weighted results in Figures 9 to 11, as we noted in Section 6.1.1, a decay factor of 0.94 implies that the VaR measure relies more on recent observations than a decay factor of 0.99, resulting in a higher level of variability of the VaR estimates. Additionally, the October 1987 crash doesn’t have as big an impact on the

Page 89 of 169 earlier estimates of VaR, as it is assigned a lower weighting. Instead, the severe stock market declines experienced in later years such as 1997 with the Asian crisis resulted in the subsequent VaR measures increasing for the next period. The problem with this approach, as with all our approaches that rely on historical data, the VaR only increases severely after the event has actually happened.

In Appendix 7, we present the graphs using the 1250, 1500 and 1875 day windows.

It can be clearly seen that the shorter estimation windows are more prone to swings in the data as the results rely more on recent events, allowing estimates to capture more of the volatility in the market. The VaR estimates using a longer window length tend to be more stable, as expected.

Figure 8 Actual Return vs VaR: Gaussian (Equally Weighted) 1250-day estimation window

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2 2 3 4 7 8 1 2 3 9 93 9 95 97 9 98 00 01 02 02 0 r 9 l r 9 r 9 l r 0 l r 0 l v a Ju a Jul 93 ov ar Jul 94 Jul ar 96Jul 96 a Ju ar Jul 98 Jul 99 ar Jul 00 a Ju a Ju ar Jul 03 M Nov 92M N M Nov 94Mar 95 Nov 95M Nov 96M Nov 97M Nov Mar 99 Nov 99M Nov 00M Nov 01M No M 95% VaR Actual Returns 99% VaR

Page 90 of 169 Figure 9 Actual Return vs VaR: Gaussian (Exponentially Weighted Lambda=0.94) 1250-day estimation window

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2 6 9 0 9 92 93 94 95 97 97 98 98 01 02 02 03 r 92 l r 93 r 97 l r 01 l r 02 l v a Ju a Jul 93 ov ar Jul 94 Jul ar 9 Jul 96 a Ju ar Jul 98 Jul 99 ar 0 Jul 00 a Ju a Ju ar Jul 03 M Nov M N M Nov 94Mar 95 Nov 95M Nov 96M Nov M Nov Mar 9 Nov 99M Nov 00M Nov 01M No M 95% VaR Actual Returns 99% VaR

Figure 10 Actual Return vs VaR: Gaussian (Exponentially Weighted Lambda=0.97) 1250-day estimation window

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2 2 3 4 7 8 1 2 3 9 93 9 95 97 9 98 00 01 02 02 0 r 9 l r 9 r 9 l r 0 l r 0 l v a Ju a Jul 93 ov ar Jul 94 Jul ar 96Jul 96 a Ju ar Jul 98 Jul 99 ar Jul 00 a Ju a Ju ar Jul 03 M Nov 92M N M Nov 94Mar 95 Nov 95M Nov 96M Nov 97M Nov Mar 99 Nov 99M Nov 00M Nov 01M No M 95% VaR Actual Returns 99% VaR

Page 91 of 169 Figure 11 Actual Return vs VaR: Gaussian (Exponentially Weighted Lambda=0.99) 1250-day estimation window

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2 6 9 0 9 92 93 94 95 97 97 98 98 01 02 02 03 r 92 l r 93 r 97 l r 01 l r 02 l v a Ju a Jul 93 ov ar Jul 94 Jul ar 9 Jul 96 a Ju ar Jul 98 Jul 99 ar 0 Jul 00 a Ju a Ju ar Jul 03 M Nov M N M Nov 94Mar 95 Nov 95M Nov 96M Nov M Nov Mar 9 Nov 99M Nov 00M Nov 01M No M 95% VaR Actual Returns 99% VaR

Figure 12 Actual Return vs VaR: Variance Gamma 1250-day estimation window 8.00%

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2 2 3 4 7 8 1 2 3 9 93 9 95 97 9 98 00 01 02 02 0 r 9 l r 9 r 9 l r 0 l r 0 l v a Ju a Jul 93 ov ar Jul 94 Jul ar 96Jul 96 a Ju ar Jul 98 Jul 99 ar Jul 00 a Ju a Ju ar Jul 03 M Nov 92M N M Nov 94Mar 95 Nov 95M Nov 96M Nov 97M Nov Mar 99 Nov 99M Nov 00M Nov 01M No M 95% VaR Actual Returns 99% VaR

Page 92 of 169 Figure 13 Actual Return vs VaR: Poisson Jump Diffusion 1250-day estimation window 8.00%

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2 6 9 0 9 92 93 94 95 97 97 98 98 01 02 02 03 r 92 l r 93 r 97 l r 01 l r 02 l v a Ju a Jul 93 ov ar Jul 94 Jul ar 9 Jul 96 a Ju ar Jul 98 Jul 99 ar 0 Jul 00 a Ju a Ju ar Jul 03 M Nov M N M Nov 94Mar 95 Nov 95M Nov 96M Nov M Nov Mar 9 Nov 99M Nov 00M Nov 01M No M 95% VaR Actual Returns 99% VaR

Page 93 of 169 7.3 Performance Testing Results

7.3.1 Accuracy

Table 6 Accuracy of VaR Measures - Fraction of Outcomes Exceeding VaR

This table tabulates the fraction of outcomes where the realised losses exceed the VaR risk measure. For the 95th percentile VaR, realised losses should exceed the VaR measure 5% of the time. For the 99th percentile VaR, realised losses should exceed VaR 1% of the time.

Panel A: Fraction of Outcomes Exceeding VaR (Binary Loss Function) 1250 days 1500 days 1875 days Model Average Std Dev Average Std Dev Average Std Dev 95th Percentile VaR Measures Gaussian (Equally Weighted) 4.18% 0.2001 3.97% 0.1954 3.47% 0.1831 Gaussian (Exp Weighted:Lambda=0.94) 5.58% 0.2295 5.68% 0.2315 5.68% 0.2315 Gaussian (Exp Weighted:Lambda=0.97) 5.28% 0.2236 5.38% 0.2256 5.31% 0.2243 Gaussian (Exp Weighted:Lambda=0.99) 4.71% 0.2119 4.71% 0.2119 4.61% 0.2097 Variance Gamma 4.71% 0.2119 5.51% 0.2282 6.48% 0.2462 Jump Diffusion Model 4.91% 0.2161 5.08% 0.2196 5.44% 0.2269

99th Percentile VaR Measures Gaussian (Equally Weighted) 1.27% 0.1120 1.17% 0.1075 0.94% 0.0963 Gaussian (Exp Weighted:Lambda=0.94) 1.80% 0.1331 1.80% 0.1331 1.80% 0.1331 Gaussian (Exp Weighted:Lambda=0.97) 1.74% 0.1307 1.77% 0.1319 1.77% 0.1319 Gaussian (Exp Weighted:Lambda=0.99) 1.57% 0.1243 1.54% 0.1230 1.54% 0.1230 Variance Gamma 1.47% 0.1204 1.94% 0.1379 2.57% 0.1583 Jump Diffusion Model 1.37% 0.1162 1.57% 0.1243 1.74% 0.1307

Panel B: Fraction of Outcomes Exceeding VaR (Quadratic Loss Function) 1250 days 1500 days 1875 days Model Average Std Dev Average Std Dev Average Std Dev 95th Percentile VaR Measures Gaussian (Equally Weighted) 4.21% 0.2021 4.01% 0.1975 3.51% 0.1853 Gaussian (Exp Weighted:Lambda=0.94) 5.62% 0.2312 5.72% 0.2331 5.72% 0.2332 Gaussian (Exp Weighted:Lambda=0.97) 5.32% 0.2254 5.42% 0.2274 5.35% 0.2261 Gaussian (Exp Weighted:Lambda=0.99) 4.75% 0.2139 4.75% 0.2139 4.65% 0.2117 Variance Gamma 4.75% 0.2139 5.56% 0.2303 6.53% 0.2483 Jump Diffusion Model 4.95% 0.2180 5.12% 0.2215 5.49% 0.2289

99th Percentile VaR Measures Gaussian (Equally Weighted) 1.29% 0.1141 1.19% 0.1098 0.96% 0.0988 Gaussian (Exp Weighted:Lambda=0.94) 1.82% 0.1347 1.82% 0.1347 1.82% 0.1347 Gaussian (Exp Weighted:Lambda=0.97) 1.76% 0.1324 1.79% 0.1336 1.79% 0.1336 Gaussian (Exp Weighted:Lambda=0.99) 1.59% 0.1264 1.56% 0.1251 1.56% 0.1251 Variance Gamma 1.49% 0.1226 1.96% 0.1399 2.60% 0.1602 Jump Diffusion Model 1.39% 0.1183 1.59% 0.1263 1.76% 0.1326

Page 94 of 169 Our first performance criteria addresses the fundamental goal of the VaR measures – whether they cover losses actually realized on the ASX All Ordinaries as they are intended to. As can be seen from Table 6 above and Figure 14 below, the actual fraction of times that the 95% VaR is exceeded is generally pretty accurate from all the models. The Gaussian Exponentially Weighted models with the decay factors of

0.94 and 0.97 systematically tend to understate the rise (ie the actual fraction of times that the 95% VaR is exceeded greater than that expected). The accuracy is formally demonstrated in Table 7.

Figure 14 Fraction of Outcomes Exceeding VaR (1250 day estimation window)

6.0%

5.0%

4.0%

3.0%

2.0%

1.0%

0.0% Equally Weighted Exponentially Weighted Exponentially Weighted Exponentially Weighted Variance Gamma Jump Diffusion Model (lambda=0.94) (lambda=0.97) (lambda=0.99)

Page 95 of 169 Table 7 Accuracy of VaR Measures - Likelihood Ratio Tests

This table displays the results from the likelihood ratio tests based on the number of failures from our backtesting. The null hypothesis we are testing is that the actual proportion of failures is equal to the theoretical value. We have displayed the likelihood ratio statistics below for each model, as well as the associated p-value. The null hypothesis is accepted at the 5% level if the p-value is greater than 0.05

Likelihood Ratio Tests(Binary Loss Function) 1250 days 1500 days 1875 days Model LR Stat p-value LR Stat p-value LR Stat p-value 95th Percentile VaR Measures Gaussian (Equally Weighted) 2.5204 0.1124 4.1499 0.0416 10.9362 0.0009 Gaussian (Exp Weighted:Lambda=0.94) 0.8983 0.3432 1.2790 0.2581 1.2079 0.2717 Gaussian (Exp Weighted:Lambda=0.97) 0.2225 0.6371 0.4259 0.5140 0.2933 0.5881 Gaussian (Exp Weighted:Lambda=0.99) 0.2839 0.5941 0.2890 0.5909 0.5282 0.4674 Variance Gamma 0.2923 0.5887 0.7349 0.3913 5.3335 0.0209 Jump Diffusion Model 0.0270 0.8696 0.0187 0.8912 0.5708 0.4499

99th Percentile VaR Measures Gaussian (Equally Weighted) 3.0237 0.0821 1.2708 0.2596 0.2456 0.6202 Gaussian (Exp Weighted:Lambda=0.94) 15.2225 0.0001 15.2225 0.0001 16.4911 0.0000 Gaussian (Exp Weighted:Lambda=0.97) 14.6114 0.0001 15.5485 0.0001 15.5485 0.0001 Gaussian (Exp Weighted:Lambda=0.99) 10.0518 0.0015 8.4701 0.0036 9.1760 0.0025 Variance Gamma 2.8754 0.0899 7.8065 0.0052 14.6930 0.0001 Jump Diffusion Model 3.9195 0.0477 7.7322 0.0054 11.2396 0.0008

Table 7 above shows a likelihood ratio test for assessing VaR accuracy. Kupiec

(1995) presents a likelihood-ratio test that can be applied to test whether the sample

point estimate is statistically consistent with the VaR model’s prescribed confidence

level. Table 7 tabulates the results of the likelihood ratio tests of the accuracy of each

of the models in achieving the 95% VaR. For all the models except for the Gaussian

Equally Weighted model (1500 and 1875 day estimation windows) and the Variance

Gamma model (1875 day estimation window) the null hypothesis is accepted. That

is, that the actual proportion of failures is statistically equal to the theoretical value.

Based on these results, the most accurate models for the 95% VaR is the Poisson

Jump Diffusion model (1250 and 1500 day estimation windows).

Compared with the 95% VaR results, the 99% VaR results exhibit a more widespread tendency to fall short of the desired level of coverage (ie underestimate the risk).

Page 96 of 169 From Table 6 and Figure 14, it can be seen that the actual fraction of times that 99%

VaR is exceeded appears to be above the expected 1% for pretty much all the models used. In Table 7, only the Gaussian Equally Weighted model and the Variance

Gamma model (1250 day estimation window) are shown to be statistically accurate.

Thus, it seems that in general, the models are found to be more accurate when forecasting the 95% VaR compared to the 99% VaR level. Moreover, the model that was the least accurate at the 95% VaR level, being the Gaussian Equally Weighted model turned out to be the most accurate at the 99% VaR level.

Lastly, we can see an effect of changing the estimation window or length of data used to calibrate the VaR models. For the 95% VaR estimates, it seems that increasing the estimation window length reduces the significance of the likelihood ratio tests (ie that p-values decrease). In other words, the models become less accurate for VaR estimation as the window length is extended. This pattern is not as clear for the 99%

VaR estimations, with the exception of the Variance Gamma model and the Poisson

Jump Diffusion model, which continue to show a worsening in accuracy as the estimation window lengthens. In contrast, the Gaussian Equally Weighted model gets more accurate as the estimation window lengthens, suggesting that accurate estimates of extreme percentiles require the use of long estimation periods for this model.

7.3.2 Conservatism

The next performance measure, the mean relative bias (MRB) statistic, as outlined in

Section 2.3 looks at the relative sizes and hence the average conservatism of the various models employed. The MRB statistic captures the extent to which the

Page 97 of 169 different models produce risk estimates of similar size. If all the models produced exactly the same VaR estimate, then the MRB would be 0 across the board. Recall that given T time period and N VaR models, the MRB of model i is computed as:

T N 1 it −VaRVaR t 1 MRBi = ∑ , where VaRt = ∑VaRit . T t=1 VaRt N i=1

Figure 15 and Table 8 suggest that the Gaussian Equally Weighted model is the most conservative VaR measure, providing a risk measure that is on average approximately

8% higher than the average across the models. In contrast, the Gaussian

Exponentially Weighted models produce the least conservative VaR estimates. From

Table 8, as the length of the estimation window lengthens, the trend for the Gaussian models at both 95% and 99% level, is for the MRB statistic to increase. That is, longer lengths of data produces larger and more conservative average VaR estimates.

This result is not observed for the Variance Gamma and Poisson Jump Diffusion models however. The Figures suggests these two models, along with the

Exponentially Weighted model (with λ=0.99) produce VaR measures close to the average across the models.

The Root Mean Squared Relative Bias (RMSRB) statistic, as outlined in Section 2.3, measures the degree to which the risk measures tend to vary around the all-model average risk measure for a point in time. Recall that the RMSRB statistic is given by:

2 1 T ⎛ −VaRVaR ⎞ 1 N RMSRB = ⎜ it t ⎟ where VaR = VaR . i T ∑⎜ ⎟ t N ∑ it t=1 ⎝ VaRt ⎠ i=1

Page 98 of 169 Figure 16 and Table 9 suggest that the variability of the models around the all-model average risk measure is quite low, below 3.5%. The Gaussian Equally Weighted model exhibits the most variability about the all-model average, which can largely be attributed to the relative conservatism of its average risk estimate. Similarly, the high

RMSRB of the Gaussian Exponentially Weighted models are attributable to their relatively more aggressive average risk estimates.

Page 99 of 169 Figure 15: Mean Relative Bias (1250 day estimation window, entire sample period)

10.0%

8.0%

6.0%

4.0%

2.0%

0.0%

-2.0%

-4.0%

-6.0% Equally Weighted Exponentially Exponentially Exponentially Variance Gamma Poisson Jump Weighted (λ=0.94) Weighted (λ=0.97) Weighted (λ=0.99) Diffusion

95% VaR 99% VaR

Figure 16: Root Mean Squared Relative Bias (1250 day estimation window, entire sample period)

4.0%

3.5%

3.0%

2.5%

2.0%

1.5%

1.0%

0.5%

0.0% Equally Weighted Exponentially Exponentially Exponentially Variance Gamma Poisson Jump Weighted (λ=0.94) Weighted (λ=0.97) Weighted (λ=0.99) Diffusion

95% VaR 99% VaR

Page 100 of 169 Table 8 Conservatism of VaR Measures - Mean Relative Bias (MRB)

This table presents the MRB statistics based on our VaR estimates using estimation periods of 1250, 1500 and 1875 days. Intuitively, this procedure results in a measure of size for each VaR approach that is relative to the average of all the approaches. The MRB is measured as a percentage, so for example, a MRB of 10% implies that a given VaR approach is 10% larger, on average, than the average of all the approaches looked at. To indicate the variability of results over time, we report the average over the entire sample period as well as 3 subsets of the sample period. Note that the subset dates refer to the dates at which the VaR is calculated, however they are utilising historical data going back 1250, 1500 or 1875 days

Panel A: Estimation period of 1250 days Entire Sample Period 1992-1995 1996-1999 2000-2003 Model Average Std Dev Average Std Dev Average Std Dev Average Std Dev 95th Percentile VaR Measures Gaussian (Equally Weighted) 7.6% 16.6% 11.8% 23.0% 0.5% 9.8% 11.1% 11.2% Gaussian (Exp Weighted:Lambda=0.94) -4.6% 17.1% -5.3% 13.6% 0.2% 16.9% -9.2% 19.2% Gaussian (Exp Weighted:Lambda=0.97) -2.7% 12.0% -4.2% 9.4% 1.8% 12.2% -5.9% 12.8% Gaussian (Exp Weighted:Lambda=0.99) 0.3% 7.8% -1.8% 4.8% 3.5% 9.7% -1.1% 6.8% Variance Gamma -0.4% 11.0% -1.4% 11.3% -3.6% 9.6% 4.2% 10.6% Poisson Jump Diffusion -0.2% 10.7% 0.9% 8.2% -2.4% 13.1% 0.9% 9.9%

99th Percentile VaR Measures Equally Weighted 7.8% 16.6% 11.9% 23.0% 0.7% 9.7% 11.2% 11.2% Exponentially Weighted (λ =0.94) -4.5% 17.1% -5.2% 13.6% 0.4% 16.8% -9.0% 19.1% Exponentially Weighted (λ =0.97) -2.5% 12.0% -4.1% 9.4% 2.0% 12.1% -5.7% 12.7% Exponentially Weighted (λ =0.99) 0.4% 7.8% -1.7% 4.8% 3.6% 9.6% -0.9% 6.8% Variance Gamma -1.0% 10.9% -2.0% 11.1% -4.3% 9.5% 3.5% 10.5% Poisson Jump Diffusion -0.1% 10.7% 1.1% 8.1% -2.4% 13.0% 1.0% 9.9%

Panel B: Estimation period of 1500 days Entire Sample Period 1992-1995 1996-1999 2000-2003 Model Average Std Dev Average Std Dev Average Std Dev Average Std Dev 95th Percentile VaR Measures Gaussian (Equally Weighted) 12.0% 20.2% 24.3% 27.0% 1.3% 10.4% 10.9% 11.0% Gaussian (Exp Weighted:Lambda=0.94) -4.4% 17.3% -5.6% 13.7% 0.6% 17.1% -8.7% 19.3% Gaussian (Exp Weighted:Lambda=0.97) -2.4% 12.2% -4.4% 9.6% 2.2% 12.5% -5.5% 12.8% Gaussian (Exp Weighted:Lambda=0.99) 0.5% 8.2% -2.0% 5.3% 4.0% 10.1% -0.6% 6.9% Variance Gamma -3.4% 14.3% -11.4% 17.0% -2.8% 9.7% 4.0% 10.7% Poisson Jump Diffusion -2.2% 10.8% -0.9% 7.1% -5.3% 13.5% -0.1% 9.8%

99th Percentile VaR Measures Gaussian (Equally Weighted) 12.1% 20.1% 24.3% 26.9% 1.5% 10.3% 11.0% 11.0% Gaussian (Exp Weighted:Lambda=0.94) -4.3% 17.2% -5.5% 13.7% 0.8% 17.0% -8.5% 19.2% Gaussian (Exp Weighted:Lambda=0.97) -2.3% 12.2% -4.3% 9.6% 2.4% 12.4% -5.3% 12.7% Gaussian (Exp Weighted:Lambda=0.99) 0.6% 8.2% -2.0% 5.4% 4.1% 10.0% -0.5% 6.9% Variance Gamma -4.1% 14.1% -11.9% 16.8% -3.5% 9.6% 3.3% 10.6% Poisson Jump Diffusion -2.1% 10.7% -0.6% 7.0% -5.3% 13.4% 0.0% 9.7%

Panel C: Estimation period of 1875 days Entire Sample Period 1992-1995 1996-1999 2000-2003 Model Average Std Dev Average Std Dev Average Std Dev Average Std Dev 95th Percentile VaR Measures Gaussian (Equally Weighted) 16.0% 20.2% 36.9% 18.0% 2.3% 10.8% 9.5% 10.9% Gaussian (Exp Weighted:Lambda=0.94) -4.2% 17.2% -5.9% 13.2% 0.9% 17.2% -8.1% 19.4% Gaussian (Exp Weighted:Lambda=0.97) -2.2% 12.3% -4.7% 9.0% 2.6% 13.0% -4.8% 12.8% Gaussian (Exp Weighted:Lambda=0.99) 0.8% 8.6% -2.3% 4.9% 4.3% 11.1% 0.1% 6.8% Variance Gamma -6.6% 17.7% -22.3% 18.3% -1.6% 10.7% 4.1% 10.8% Poisson Jump Diffusion -3.7% 11.0% -1.7% 7.3% -8.5% 13.4% -0.7% 9.6%

99th Percentile VaR Measures Gaussian (Equally Weighted) 16.0% 20.1% 36.7% 17.9% 2.3% 10.8% 9.5% 10.8% Gaussian (Exp Weighted:Lambda=0.94) -4.2% 17.1% -5.9% 13.1% 0.9% 17.2% -8.0% 19.3% Gaussian (Exp Weighted:Lambda=0.97) -2.2% 12.2% -4.7% 8.9% 2.6% 12.9% -4.8% 12.8% Gaussian (Exp Weighted:Lambda=0.99) 0.8% 8.6% -2.2% 4.9% 4.3% 11.1% 0.1% 6.8% Variance Gamma -6.7% 17.7% -22.3% 18.3% -1.6% 10.7% 4.0% 10.7% Poisson Jump Diffusion -3.8% 11.0% -1.6% 7.2% -8.5% 13.4% -0.8% 9.6%

Page 101 of 169 Table 9 Conservatism of VaR Measures - Root Mean Squared Relative Bias (RMSRB)

This table presents the RMSRB statistics based on our VaR estimates using estimation periods of 1250, 1500 and 1875 days. The RMSRB statistic looks at the degree to which the different VaR measures tend to vary around the average risk measure for a given observation date. This can be compared to a standard deviation calculation. To indicate the variability of results over time, we report the average over the entire sample period as well as 3 subsets of the sample period. Note that the subset dates refer to the dates at which the VaR is calculated, however they are utilising historical data going back 1250, 1500 or 1875 days

Panel A: Estimation period of 1250 days Entire Sample Period 1992-1995 1996-1999 2000-2003 Model 95th Percentile VaR Measures Gaussian (Equally Weighted) 3.3% 6.7% 1.0% 2.5% Gaussian (Exp Weighted:Lambda=0.94) 3.1% 2.1% 2.9% 4.5% Gaussian (Exp Weighted:Lambda=0.97) 1.5% 1.1% 1.5% 2.0% Gaussian (Exp Weighted:Lambda=0.99) 0.6% 0.3% 1.1% 0.5% Variance Gamma 1.2% 1.3% 1.1% 1.3% Poisson Jump Diffusion 1.2% 0.7% 1.8% 1.0%

99th Percentile VaR Measures Gaussian (Equally Weighted) 3.3% 6.7% 1.0% 2.5% Gaussian (Exp Weighted:Lambda=0.94) 3.1% 2.1% 2.8% 4.4% Gaussian (Exp Weighted:Lambda=0.97) 1.5% 1.1% 1.5% 1.9% Gaussian (Exp Weighted:Lambda=0.99) 0.6% 0.3% 1.1% 0.5% Variance Gamma 1.2% 1.3% 1.1% 1.2% Poisson Jump Diffusion 1.1% 0.7% 1.7% 1.0%

Panel B: Estimation period of 1500 days Entire Sample Period 1992-1995 1996-1999 2000-2003 Model 95th Percentile VaR Measures Gaussian (Equally Weighted) 5.5% 13.2% 1.1% 2.4% Gaussian (Exp Weighted:Lambda=0.94) 3.2% 2.2% 2.9% 4.5% Gaussian (Exp Weighted:Lambda=0.97) 1.6% 1.1% 1.6% 1.9% Gaussian (Exp Weighted:Lambda=0.99) 0.7% 0.3% 1.2% 0.5% Variance Gamma 2.2% 4.2% 1.0% 1.3% Poisson Jump Diffusion 1.2% 0.5% 2.1% 1.0%

99th Percentile VaR Measures Gaussian (Equally Weighted) 5.5% 13.2% 1.1% 2.4% Gaussian (Exp Weighted:Lambda=0.94) 3.1% 2.2% 2.9% 4.4% Gaussian (Exp Weighted:Lambda=0.97) 1.5% 1.1% 1.6% 1.9% Gaussian (Exp Weighted:Lambda=0.99) 0.7% 0.3% 1.2% 0.5% Variance Gamma 2.2% 4.2% 1.0% 1.2% Poisson Jump Diffusion 1.2% 0.5% 2.1% 0.9%

Panel C: Estimation period of 1875 days Entire Sample Period 1992-1995 1996-1999 2000-2003 Model 95th Percentile VaR Measures Gaussian (Equally Weighted) 6.7% 16.8% 1.2% 2.1% Gaussian (Exp Weighted:Lambda=0.94) 3.1% 2.1% 3.0% 4.4% Gaussian (Exp Weighted:Lambda=0.97) 1.5% 1.0% 1.7% 1.9% Gaussian (Exp Weighted:Lambda=0.99) 0.7% 0.3% 1.4% 0.5% Variance Gamma 3.6% 8.3% 1.2% 1.3% Poisson Jump Diffusion 1.4% 0.6% 2.5% 0.9%

99th Percentile VaR Measures Gaussian (Equally Weighted) 6.6% 16.7% 1.2% 2.1% Gaussian (Exp Weighted:Lambda=0.94) 3.1% 2.1% 2.9% 4.4% Gaussian (Exp Weighted:Lambda=0.97) 1.5% 1.0% 1.7% 1.9% Gaussian (Exp Weighted:Lambda=0.99) 0.7% 0.3% 1.4% 0.5% Variance Gamma 3.6% 8.3% 1.2% 1.3% Poisson Jump Diffusion 1.3% 0.5% 2.5% 0.9%

Page 102 of 169 7.3.3 Efficiency

The last performance criteria we look at assesses how well the VaR measures adjust over time to underlying changes in risk. In other words, how closely do changes in the VaR measures track to actual return outcomes achieved in our data set.

The most obvious observation is that the VaR estimates produced by the Gaussian

Exponentially Weighted models record the highest correlation to the actual return outcomes in the All Ordinaries Index. This implies that these approaches tend to track changes in risk over time more accurately than the other models. The Gaussian

Exponentially Weighted models with a decay factor of 0.94 records a higher correlation than those using a decay factor of 0.99. This makes sense because as market conditions change over time, more emphasis on recent information is more helpful in tracking changes in risk.

The worse performing models under this criteria are the Gaussian Equally Weighted model and the Poisson Jump Diffusion model. Whilst these models may perform better under accuracy and conservatism, the results here indicate that these models reveal little about actual changes in market risk over time.

Page 103 of 169

Figure 17: Correlation between VaR and absolute value of outcome (1250 day estimation window, entire sample period) 30.0%

25.0%

20.0%

15.0%

10.0%

5.0%

0.0% Equally Weighted Exponentially Exponentially Exponentially Variance Gamma Poisson Jump Weighted (λ=0.94) Weighted (λ=0.97) Weighted (λ=0.99) Diffusion

95% VaR 99% VaR

Page 104 of 169 Table 10 Efficiency of VaR Measures - Correlation of VaR Estimates and Magnitude of Actual Returns

This table presents the correlation of our VaR estimates using estimation periods of 1250, 1500 and 1875 days with the magniture of the actual returns. The purpose of these performance criteria is to assess how well the VaR measures adjust over time to underlying changes in risk. To indicate the variability of results over time, we report the average over the entire sample period as well as 3 subsets of the sample period. Note that the subset dates refer to the dates at which the VaR is calculated, however they are utilising historical data going back 1250, 1500 or 1875 days

Panel A: Estimation period of 1250 days Model Entire Sample Period 1992-1995 1996-1999 2000-2003 95th Percentile VaR Measures Gaussian (Equally Weighted) 0.4% 0.9% 12.2% 2.8% Gaussian (Exp Weighted:Lambda=0.94) 16.8% 12.0% 13.4% 21.7% Gaussian (Exp Weighted:Lambda=0.97) 14.8% 11.6% 12.6% 17.3% Gaussian (Exp Weighted:Lambda=0.99) 11.3% 8.9% 12.6% 9.5% Variance Gamma 6.7% 18.5% 12.2% 4.8% Poisson Jump Diffusion 0.5% 9.0% -5.0% 9.8%

99th Percentile VaR Measures Gaussian (Equally Weighted) 0.5% 1.0% 12.0% 2.3% Gaussian (Exp Weighted:Lambda=0.94) 16.8% 12.1% 13.4% 21.7% Gaussian (Exp Weighted:Lambda=0.97) 14.8% 11.7% 12.6% 17.2% Gaussian (Exp Weighted:Lambda=0.99) 11.4% 9.2% 12.6% 9.4% Variance Gamma 6.7% 18.5% 12.2% 4.8% Poisson Jump Diffusion 0.5% 9.0% -5.0% 9.8%

Panel B: Estimation period of 1500 days Model Entire Sample Period 1992-19951996-1999 2000-2003 95th Percentile VaR Measures Gaussian (Equally Weighted) -1.5% -1.5% 8.5% 1.8% Gaussian (Exp Weighted:Lambda=0.94) 16.8% 12.1% 13.5% 21.7% Gaussian (Exp Weighted:Lambda=0.97) 14.9% 11.7% 12.6% 17.2% Gaussian (Exp Weighted:Lambda=0.99) 11.5% 9.5% 12.5% 9.5% Variance Gamma 3.8% 8.8% 8.8% 3.7% Poisson Jump Diffusion 0.9% 10.3% -6.9% 9.3%

99th Percentile VaR Measures Gaussian (Equally Weighted) -1.8% -2.0% 8.5% 1.5% Gaussian (Exp Weighted:Lambda=0.94) 16.8% 12.2% 13.4% 21.7% Gaussian (Exp Weighted:Lambda=0.97) 14.9% 11.7% 12.6% 17.1% Gaussian (Exp Weighted:Lambda=0.99) 11.6% 9.6% 12.5% 9.4% Variance Gamma 3.8% 8.8% 8.8% 3.7% Poisson Jump Diffusion 0.9% 10.3% -6.9% 9.3%

Panel C: Estimation period of 1875 days Model Entire Sample Period 1992-1995 1996-1999 2000-2003 95th Percentile VaR Measures Gaussian (Equally Weighted) 1.0% 13.2% 6.3% -0.2% Gaussian (Exp Weighted:Lambda=0.94) 16.9% 12.4% 13.4% 21.8% Gaussian (Exp Weighted:Lambda=0.97) 14.9% 12.0% 12.5% 17.2% Gaussian (Exp Weighted:Lambda=0.99) 11.7% 10.0% 12.6% 9.5% Variance Gamma -5.1% -5.8% 0.4% -2.3% Poisson Jump Diffusion 0.4% 11.3% -6.6% 6.1%

99th Percentile VaR Measures Gaussian (Equally Weighted) 1.1% 13.0% 6.1% -0.2% Gaussian (Exp Weighted:Lambda=0.94) 16.9% 12.3% 13.3% 21.7% Gaussian (Exp Weighted:Lambda=0.97) 14.9% 11.9% 12.5% 17.2% Gaussian (Exp Weighted:Lambda=0.99) 11.7% 10.0% 12.6% 9.5% Variance Gamma -5.1% -5.8% 0.4% -2.3% Poisson Jump Diffusion 0.4% 11.3% -6.6% 6.1%

Page 105 of 169 8. CONCLUSION

We have employed the use of jump models to obtain estimates of Value-at-Risk. The motive of using models such as the Poisson Jump Diffusion model and the Variance

Gamma model is that return distributions typically exhibit leptokurtosis and skewness. It was thought that these models could explain such characteristics better than the standard Gaussian models that typically underlie commonly used risk optimization methods such as Riskmetrics.

We calibrated a number of models onto the returns of the Australian All Ordinaries

Index. Our parameter estimation utilized both the Maximum Likelihood and Method of Moments estimation methodologies. In estimating the VaR for the models, we introduced a methodology based on option pricing techniques, where the more tractable characteristic functions of the models could be utilized for the estimation of

VaR. This framework can be used for future research for testing more complicated models which may be able to better capture the pervading characteristics in the market and improve the accuracy of extreme event forecasting. Future study may seek to look at models such as a more general form of the jump diffusion model, which can incorporate jumps in the volatility as well as jumps in the returns.

Research has found that models with jumps in volatility allow the volatility component to move fast enough to generate the period of large market movements found in practice.

Page 106 of 169 In terms of the overall performance of the various models employed, according to the criteria we used to assess the effectiveness of the VaR estimates none of the models really dominated in every category.

In general, we found that all the models are more accurate at predicting the 95% VaR rather than the 99%. The Gaussian Exponentially Weighted models with decay factors of 0.94 and 0.97 systematically underestimated the risk, however in general, all the models employed were statistically accurate at predicting the 95% VaR, with the exception of the Gaussian Equally Weighted model and the Variance Gamma model. The Poisson Jump Diffusion model proved the most accurate at estimating the 95% VaR. In general, all the models underestimated the 99% VaR, with the fraction of outcomes exceeding the estimated VaR greater than 1% of the time in most cases. Moreover, the model that was the least accurate at the 95% VaR level, being the Gaussian Equally Weighted model turned out to be the most accurate at the

99% VaR level. We also observed at the 95% VaR level, that as the length of the estimation window increases, in general the accuracy decreased. This could suggest a tradeoff between parameter estimation accuracy and VaR accuracy, as a sufficiently long enough estimation window is required to capture the impact of extreme events.

This same pattern is not observed at the 99% VaR level, with the length of estimation window not having a direct relationship with the accuracy of the VaR estimate.

The results for the conservatism and efficiency performance criteria suggest that the

Gaussian Equally Weighted model and Exponentially Weighted models perform relatively better according to these measures. The Gaussian Equally Weighted model

Page 107 of 169 was the most conservative of all the models tested and the Gaussian Exponentially

Weighted models were the most efficient of all the models tested. Whilst accuracy is probably the most important performance criteria, both of these other performance criteria are still important to practitioners. The level of conservatism may have implications on the amount of capital that would need to be set aside in practice, whilst a conservative but inefficient model would tend to overestimate risk in periods of low risk.

Our results show that whilst the Poisson Jump Diffusion model proved the most accurate at the 95% VaR level, neither the Poisson Jump Diffusion or Variance

Gamma models were dominant in the other performance criteria examined.

Additionally, neither of these models worked particularly well at the 99% (extreme percentiles), although admittedly neither did any of the other models currently employed by practitioners. The model that seemed to perform reasonably well in each of the performance criteria was the Gaussian Exponentially Weighted model with decay factor of 0.94. This model achieved good accuracy at the 95% VaR level, as well as good conservatism and efficiency. However, no model was clearly superior according to all the performance criteria analysed, and it seems that the extra computational time required to calibrate the Poisson Jump Diffusion and Variance

Gamma models for the purposes of VaR estimation do not provide sufficient reward for the additional effort than that currently employed by Riskmetrics.

For future research, it would be of interest to try to apply the more sophisticated bi- dimensional affine models. Option pricing theory suggests that these models are

Page 108 of 169 needed to capture some of the more subtle features of prices, and so it is natural to ask whether or not these models would improve VaR calculations. Econometrically, however, they are difficult, if not impossible, models to estimate, at this time.

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Page 116 of 169 APPENDIX 1: DERIVING THE CHARACTERISTIC

FUNCTION OF THE POISSON JUMP DIFFUSION

MODEL

We consider the following model. Assume that S is the price process of a security that pays dividends at a constant proportional rate, ζ , and let Y = ln(S). Then,

1 2 dYt = ( 2 1 ) σσμλζμ 1 ++−−− dNJdWdt ttt .

Here μ is the cum-dividend expected return, and J is the random jump size. In

uJ general, if we write θJ(u) := [eE ], then setting μ = θ z −1)1( gives the usual result

ζμ −− tT ))(( that E[ST|Ft] = t eS .

Taking the exponential of Yt, we see that we need to find

T [exp{ dNJE }] = [exp{ JE }] ∫ ss ∑ Tn t n ≤≤ TTt

= [exp{[ − NNJEE )]](|} ∑ Tn tT n ≤≤ TTt

−NN tT )( = EE θ J − NN tT ]]|)1([[

∞ n λθ − tT )]([)1( n = ∑ J e λ −− tT )( n=0 n!

= λ −− tT )( ee J λθ −tT )()1(

Page 117 of 169 T = − λθ ds})1)1((exp{ . ∫ J t

th Here, Tn represents the random time at which the n jump occurs.

Generalizing this, we find that, for arbitrary u,

T [exp{ dNJuE }] = [exp{ JuE }] ∫ ss ∑ Tn t n ≤≤ TTt

= [exp{[ − NNJuEE )]](|} ∑ Tn tT n ≤≤ TTt

−NN tT )( = θ J |)([[ − NNuEE tT ]]

∞ n λθ − tTu )]([)( n = ∑ z e λ −− tT )( n=0 n!

= λ −− tT )( ee J λθ −tTu )()(

T = − λθ dsu })1)((exp{ . ∫ J t

So, the characteristic function is (generally, we’ll assume ζ = 0)

SSu tT )/ln( 1 2 1 22 t exp]|[ {uFeE ( 2 σμλζμ ) +−−−−= 2 σ ()J λθ −−+− tTutTutT )(1)()()( }

1 2 2 = { μ 2 σ ( J uutTuutTu (θθ z ) )λ −−−−+−−+− tT )(11)1()()()()(exp }.

Page 118 of 169 Our approach is to assume that jumps are normally distributed, with mean μ2 and standard deviation σ2. This model makes VaR calculations straightforward, as returns become conditionally normal, after conditioning on the number of jumps. In

1 2 2 2 + 2uu σμ 2 this case, θJ(u) = e . Writing t in place of (T-t), we obtain the characteristic function

1 2 2 1 2 ⎧ 1 2 2 2 + 2 uu σμ 2 2 + 2σμ 2 ⎫ φ tx )( exp)( ⎨ μ )( σ +−+= (etuutuu (eu )−−− 11 )λt⎬ ⎩ 2 ⎭

Page 119 of 169 APPENDIX 2 – ESTIMATION OF VARIANCE GAMMA

MODEL USING METHOD OF MOMENTS PROCEDURE

The characteristic function of the Variance Gamma model is given by the following:

t ⎛ ⎞ v ⎜ ⎟ 1 φ = tiuXEu ))]([exp()( = ⎜ ⎟ tX )( ⎜ σ 2v ⎟ ⎜1 θviu +− u 2 ⎟ ⎝ 2 ⎠

Now, we want to use method of moments to estimate the parameters θ, σ and v. The method of moments is a very simple procedure for finding an estimator for one or more population parameters. The kth moment of a random variable, taken about the origin is:

μ’k = E(Xk)

Where the corresponding kth sample moment is the average:

n 1 k m'k = ∑ X i n i=1

The method of moments solves for the required parameters by setting μ = m'' kk .

Given a characteristic function, we can generate raw moments:

k k )( ⎡d φ ⎤ k φ )0( = ⎢ k ⎥ = i μ'k ⎣ du ⎦ u=0

Page 120 of 169 Now, deriving the characteristic function above, we can obtain the moments of the variance gamma model and compare then to the sample moments to estimate the parametersθ,σ and v. Since the central moments are more manageable than the moments, we will actually use the central moments in our method of moments estimation process. Using the central moments, the kth central moment of a random variable is:

~ k μ k XE −= μ)(' where μ represents the mean or E(X).

The corresponding kth sample central moment is the average:

n n ~ 1 k 1 m'k ∑()i −= xX where x is the sample mean given by x = ∑ X i n i=1 n i=1

Our method of moments estimation process utilising the central moments solves for ~ ~ the required parameters by setting μ' = m'kk .

Now, as already mentioned, the characteristic function of the Variance Gamma model is given by the following:

t ⎛ ⎞ v ⎜ ⎟ 1 φ = tiuXEu ))]([exp()( = ⎜ ⎟ tX )( ⎜ σ 2v ⎟ ⎜1 θviu +− u 2 ⎟ ⎝ 2 ⎠

In order to calculate the moments, we need to derive the characteristic function given above with respect to the u.

Page 121 of 169 t ⎛ ⎞ v ⎜ ⎟ 1 ⎜ ⎟ ( +− σθ 2vuit ) ⎜ σ 2v ⎟ 1 θviu +− u 2 d ⎜ ⎟ φ u)( = ⎝ 2 ⎠ du tX )( σ 2v 1( θviuv +− u 2 ) 2

Evaluated at u=0, this is

d φ tX )( )( = itu θ du u=0

Thus we have the first moment given as (cancelling the i which is on both sides of the equation):

)( = itXiE θ

)( = tXE θ

Similarly, taking the 2nd, 3rd and 4th derivative, evaluated at u=0 and cancelling the i’s given on both sides of the equations gives us the following moments:

2 d 2222222 2 φ tX )( )( −+= titvitiu σθθ du u=0 2 ()( vttXE ++= σθθ 222 )

3 d 22332333 332 2 3 φ tX )( )( θσθθ −+−+= 3233 vtiitvitvititu σθθ du u=0 3 )( ()2 vttXE 32 σθθθ 22 ()+++= vt

Page 122 of 169 4 d 42222244222223443444 4 φ tX )( )( −+= 66 σθθθ +11 θ −18 + 3tvitivtitvititu σσθ du u=0 6 443 −+ 12 2222 + 3 σσθθ 4vtvtiitv 4 )( ()()24 22 4 ++++= 1256 vtvttvttXE ++ 63 v θσθσθσθθ 42422

Now the central moments can be given by:

()μ 2 2 XEXEXE )(2)( +−=− μμ 2 ()μ 3 3 −=− 2 + XEXEXEXE )(3)(3)( − μμμ 32 ()μ 4 4 −=− 3 μ + μ 22 − XEXEXEXEXE )(4)(6)(4)( + μμ 43 where μ represents the mean or E(X).

The second central moment is given by:

()μ 2 2 XEXEXE )(2)( +−=− μμ 2 ()vtt 222 −++= ()tθσθθ 2 ()+= σθ 22 tv

The third central moment is given by:

()μ 3 3 −=− 2 + XEXEXEXE )(3)(3)( − μμμ 32 ()2 vtt 32 σθθθ 22 ()3 ()222 +++−+++= 2()ttvttvt θθσθθ 2 ()+= 32 223 θσθ tvv

The fourth central moment is given by:

Page 123 of 169 ()μ 4 4 −=− 3 μ + μ 22 − XEXEXEXEXE )(4)(6)(4)( + μμ 43 ()()24 22 4 ++++= 1256 vtvttvtt 42422 (2 vttv 32463 σθθθθσθσθσθθ 22 )()+++−++ vt 6 ()222 ()2 −+++ 3 ()ttvtt θθσθθ 4 ()4 += 123 34222 tvvv ()4 22 ++++ 3636 θθσσθθσσ tvv 224

Page 124 of 169 Appendix 3 – Maximum Likelihood Estimators of the

Normal Distribution

The method of Maximum Likelihood is based on the logic of maximising the joint probability of observing a set of given data.

Now, the Normal distribution is given by:

1 ⎛ ()x − μ 2 ⎞ xf )( = exp⎜− ⎟ ⎜ 2 ⎟ 2πσ ⎝ 2σ ⎠

Let X1, X2…,Xn be a random sample from the normal distribution with mean of μ and variance of σ2. Letting L be the joint density of the sample (or the likelihood function), we have:

2 2 L σμ = 21 xxxf n σμ ),|,...,,(),( n n ⎛ 1 ⎞ 2 ⎛ 1 2 ⎞ = exp − x − μ ⎜ 2 ⎟ ⎜ 2 ∑()i ⎟ ⎝ 2πσ ⎠ ⎝ 2σ i=1 ⎠

Now, maximising the likelihood function L is the same as maximising the log of the likelihood function lnL as it is a monotonically increasing function of L.

nn 1 n L σμ 2 ),(ln ln 2 2ln πσ −−−= x − μ 2 2 ∑()i 2 2 2σ i=1

The Maximum Likelihood Estimators (MLE’s) of μ and σ2 are the values that make lnL(μ, σ2) a maximum. Taking the derivatives with respect to μ and σ2, we obtain:

Page 125 of 169 ∂ 1 n L σμ 2 ),(ln = x − μ 2 ∑()i ∂μ σ i=1 and

∂ n 11 n L σμ 2 ),(ln +−= x − μ 2 2 2 4 ∑()i ∂σ 2 2σσ i=1

Setting these derivatives equal to zero and solving simultaneously, we obtain the

MLE of the normal distribution μˆ and σˆ 2

1 n x μˆ =− 0 2 ∑()i σˆ i=1 n ˆ ∑ i nx μ =− 0 i=1 1 n ˆ μ ∑ i == xx n i=1

Substituting for μˆ in the second equation and solving for σˆ 2 we obtain

n 1 1 n +− ()xx 2 =− 0 2 ˆ 2 2σσ ˆ 4 ∑ i i=1 1 n ˆ 2 2 σ ∑()i −= xx n i=1

Page 126 of 169 APPENDIX 4 – DENSITY FUNCTION OF THE POISSON

JUMP DIFFUSION MODEL

Poisson Jump Diffusion Model is defined as:

⎛ 1 2 ⎞ dYt ⎜ −−= 1 ⎟ σσμλμ 1 t ++ JdNdWdt t ⎝ 2 ⎠

The jump component is represented by the JdNt term. The size of the random jump is

2 given by J ~ N(τµ2, τσ2 ) and the number of jumps is given by the Poisson process Nt, which is assumed to be independent of the Brownian motion Wt. We assume that the intensity of the Poisson process is given by λ per unit time. That is, λ is the mean number of jumps per unit time (which in our case is a trading year). Thus dNt ~

Poisson(λdt).

⎛ 1 2 ⎞ Let 1 ⎜ −−= σμλμμ 1 ⎟ and our stochastic equation becomes: ⎝ 2 ⎠

t μ1 σ 1 t ++= JdNdWdtdY t

The discrete form of the above model can be written as:

Yx tt 1 1 τεστμ Δ++=Δ= NJ t where ε ~ N(0,1), and Δt=τ.

Page 127 of 169 2 If ΔN t = 0 (ie we have no jumps), then xt ~ N(µ1τ, σ1 τ)

2 If ΔN t = n where n > 0 (ie we have n jumps), then JΔNt ~ N(nµ2 τ, nσ1 τ)

2 2 The sum of two independent normal distributions N(µ1τ, σ1 τ) and N(nµ2 τ, nσ1 τ)

2 will give the normal distribution N(τ(µ1+ nµ2), τ(1+nm) σ1 ).

~ 2 Thus, for each n ≥ 0 , we have that xt N(τ(µ1+ nµ2), τ(1+nm) σ1 )

Thus, the density function of xt is:

∞ n −λτ ()λτ 2 2 t )( = ∑exf ()t ; ()+Φ nx , ()121 + nσστμμτ 2 n=0 n!

And xt is an infinite mixture of normals

Page 128 of 169 APPENDIX 5 – MATLAB CODE FOR ITERATING THE

VAR FOR THE VARIANCE GAMMA MODEL

Appendix 5.1 Defining the Characteristic Function

function Value=phiVG(u,X,ui,t,theta,v,sigma)

im=i*ui;

A=i*(u+im).*theta.*v;

B=(sigma^2.*v.*(u+im).^2)./2;

phi=(1./(1-A+B)).^(t/v);

Value=real((exp(-i*(u+im).*X)./(i*(u+im))).*phi);

Appendix 5.2 Iterating a solution to find 95% and 99% VaR

ParamsVG = xlsread('C:\Bren Data\Thesis\Matlab\VarianceGamma\ParamVG.xls');

VARVG95=[]

VARVG99=[] q95=[];

Page 129 of 169 q99=[];

t=1/250; ui=0;

warning off format long

%Calculating the 95% VARs for j=1:3

for i=1:2994

theta=ParamsVG(i,j+j+j-2);

v=ParamsVG(i,j+j+j-1);

sigma=ParamsVG(i,j+j+j);

alpha=0.5;

X=0-alpha;

kmax=5000;

q=(1/2)+(1/pi)*quad(@phiVG,0,kmax,[],[],X,ui,t,theta,v,sigma);

while abs(0.95-q)>eps & alpha>1e-20

Page 130 of 169 alpha=alpha/2;

if q>0.95

X=X+alpha;

else

X=X-alpha;

end

q=(1/2)+(1/pi)*quad(@phiVG,0,kmax,[],[],X,ui,t,theta,v,sigma);

end

VARVG95(i,j)=X;

q95(i,j)=q;

i=i+1;

j

i end

j=j+1; end

save VARVG95.out VARVG95 -ASCII save q95.out q95 -ASCII

%Calculating the 99% VARs

Page 131 of 169 for j=1:3

for i=1:2994

theta=ParamsVG(i,j+j+j-2);

v=ParamsVG(i,j+j+j-1);

sigma=ParamsVG(i,j+j+j);

alpha=0.5;

X=0-alpha;

kmax=5000;

q=(1/2)+(1/pi)*quad(@phiVG,0,kmax,[],[],X,ui,t,theta,v,sigma);

while abs(0.99-q)>eps & alpha>1e-20

alpha=alpha/2;

if q>0.99

X=X+alpha;

else

X=X-alpha;

end

q=(1/2)+(1/pi)*quad(@phiVG,0,kmax,[],[],X,ui,t,theta,v,sigma);

end

VARVG99(i,j)=X;

Page 132 of 169 q99(i,j)=q;

i=i+1;

j

i end

j=j+1; end

save VARVG99.out VARVG99 -ASCII save q99.out q99 -ASCII

VARVG95

VARVG99

Page 133 of 169 APPENDIX 6 – MATLAB CODE FOR ITERATING THE

VAR FOR THE POISSON JUMP DIFFUSION MODEL

Appendix 6.1 Defining the Characteristic Function

function Value=phiJD(u,X,ui,t,mu,mu2,sigma1,sigma2,lambda)

im=i*ui;

A=(u+im).*mu.*t;

B=((u+im).^2-(u+im)).*sigma1.^2.*t./2;

C=exp((u+im).*mu2+(u+im).^2.*sigma2.^2./2);

D=(u+im).*(exp(mu2+sigma2.^2./2)-1);

phi=exp(A+B+(C-D-1).*lambda.*t);

Value=real((exp(-i*(u+im).*X)./(i*(u+im))).*phi);

Appendix 6.2 Iterating a solution to find 95% and 99% VaR

ParamsJD = xlsread('C:\Bren Data\Final Thesis\JDParams1875.xls');

Page 134 of 169 VARJD951875=[]; qJD951875=[];

VARJD991875=[]; qJD991875=[];

t=1/250; ui=0;

warning off format long

%Calculating the 95% VARs for j=1:1

for i=1:2994

mu=ParamsJD(i,j+j+j+j+j-4);

mu2=ParamsJD(i,j+j+j+j+j-3);

sigma1=ParamsJD(i,j+j+j+j+j-2);

sigma2=ParamsJD(i,j+j+j+j+j-1);

lambda=ParamsJD(i,j+j+j+j+j);

alpha=0.5;

Page 135 of 169 X=0-alpha;

kmax=5000;

q=(1/2)+(1/pi)*quad(@phiJD,0,kmax,[],[],X,ui,t,mu,mu2,sigma1,sigma2,lambda);

while abs(0.95-q)>eps & alpha>1e-20

alpha=alpha/2;

if q>0.95

X=X+alpha;

else

X=X-alpha;

end

q=(1/2)+(1/pi)*quad(@phiJD,0,kmax,[],[],X,ui,t,mu,mu2,sigma1,sigma2,lambda);

end

VARJD951875(i,j)=X;

qJD951875(i,j)=q;

i=i+1;

j

i end

Page 136 of 169 j=j+1; end

save VARJD951875.out VARJD951875 -ASCII save qJD951875.out qJD951875 -ASCII

%Calculating the 99% VARs for j=1:1

for i=1:2994

mu=ParamsJD(i,j+j+j+j+j-4);

mu2=ParamsJD(i,j+j+j+j+j-3);

sigma1=ParamsJD(i,j+j+j+j+j-2);

sigma2=ParamsJD(i,j+j+j+j+j-1);

lambda=ParamsJD(i,j+j+j+j+j);

alpha=0.5;

X=0-alpha;

kmax=5000;

q=(1/2)+(1/pi)*quad(@phiJD,0,kmax,[],[],X,ui,t,mu,mu2,sigma1,sigma2,lambda);

while abs(0.99-q)>eps & alpha>1e-20

Page 137 of 169 alpha=alpha/2;

if q>0.99

X=X+alpha;

else

X=X-alpha;

end

q=(1/2)+(1/pi)*quad(@phiJD,0,kmax,[],[],X,ui,t,mu,mu2,sigma1,sigma2,lambda);

end

VARJD991875(i,j)=X;

qJD991875(i,j)=q;

i=i+1;

j

i end

j=j+1; end

save VARJD991875.out VARJD991875 -ASCII save qJD991875.out qJD991875 -ASCII

VARJD951875

Page 138 of 169 qJD951875

VARJD991875 qJD991875

Page 139 of 169 APPENDIX 7 – VALUE-AT-RISK ESTIMATES VS

ACTUAL RETURNS

Figure 8 Actual Return vs VaR: Gaussian (Equally Weighted) 1250-day estimation window

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2 6 9 0 9 92 93 94 95 97 97 98 98 01 02 02 03 r 92 l r 93 r 97 l r 01 l r 02 l v a Ju a Jul 93 ov ar Jul 94 Jul ar 9 Jul 96 a Ju ar Jul 98 Jul 99 ar 0 Jul 00 a Ju a Ju ar Jul 03 M Nov M N M Nov 94Mar 95 Nov 95M Nov 96M Nov M Nov Mar 9 Nov 99M Nov 00M Nov 01M No M 95% VaR Actual Returns 99% VaR

Page 140 of 169 Figure 9 Actual Return vs VaR: Gaussian (Exponentially Weighted Lambda=0.94) 1250-day estimation window

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2 6 9 0 9 92 93 94 95 97 97 98 98 01 02 02 03 r 92 l r 93 r 97 l r 01 l r 02 l v a Ju a Jul 93 ov ar Jul 94 Jul ar 9 Jul 96 a Ju ar Jul 98 Jul 99 ar 0 Jul 00 a Ju a Ju ar Jul 03 M Nov M N M Nov 94Mar 95 Nov 95M Nov 96M Nov M Nov Mar 9 Nov 99M Nov 00M Nov 01M No M 95% VaR Actual Returns 99% VaR

Figure 10 Actual Return vs VaR: Gaussian (Exponentially Weighted Lambda=0.97) 1250-day estimation window

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2 2 3 4 7 8 1 2 3 9 93 9 95 97 9 98 00 01 02 02 0 r 9 l r 9 r 9 l r 0 l r 0 l v a Ju a Jul 93 ov ar Jul 94 Jul ar 96Jul 96 a Ju ar Jul 98 Jul 99 ar Jul 00 a Ju a Ju ar Jul 03 M Nov 92M N M Nov 94Mar 95 Nov 95M Nov 96M Nov 97M Nov Mar 99 Nov 99M Nov 00M Nov 01M No M 95% VaR Actual Returns 99% VaR

Page 141 of 169 Figure 11 Actual Return vs VaR: Gaussian (Exponentially Weighted Lambda=0.99) 1250-day estimation window

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2 6 9 0 9 92 93 94 95 97 97 98 98 01 02 02 03 r 92 l r 93 r 97 l r 01 l r 02 l v a Ju a Jul 93 ov ar Jul 94 Jul ar 9 Jul 96 a Ju ar Jul 98 Jul 99 ar 0 Jul 00 a Ju a Ju ar Jul 03 M Nov M N M Nov 94Mar 95 Nov 95M Nov 96M Nov M Nov Mar 9 Nov 99M Nov 00M Nov 01M No M 95% VaR Actual Returns 99% VaR

Figure 12 Actual Return vs VaR: Variance Gamma 1250-day estimation window 8.00%

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2.00%

0.00%

-2.00%

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2 2 3 3 4 4 5 5 6 7 7 8 8 9 9 0 1 1 2 2 3 3 9 9 93 9 9 9 96 9 9 98 9 0 00 0 0 01 0 0 v r v 9 r v v v 9 l v v Jul 92 Jul Jul 94 Jul 95 a Jul 96 Jul 97 Jul Jul 99 Ju o Jul Jul 02 Jul Mar Nov 9 Mar 9 No Ma No Mar Nov 9 M No Mar Nov 9 Mar 9 No Mar No Mar 00 N Mar No Mar Nov 0 Mar 0 95% VaR Actual Returns 99% VaR

Page 142 of 169 Figure 13 Actual Return vs VaR: Poisson Jump Diffusion 1250-day estimation window 8.00%

6.00%

4.00%

2.00%

0.00%

-2.00%

-4.00%

-6.00%

-8.00%

2 2 3 3 4 4 5 5 6 7 7 8 8 9 9 0 1 1 2 2 3 3 9 9 93 9 9 9 96 9 9 98 9 0 00 0 0 01 0 0 v r v 9 r v v v 9 l v v Jul 92 Jul Jul 94 Jul 95 a Jul 96 Jul 97 Jul Jul 99 Ju o Jul Jul 02 Jul Mar Nov 9 Mar 9 No Ma No Mar Nov 9 M No Mar Nov 9 Mar 9 No Mar No Mar 00 N Mar No Mar Nov 0 Mar 0 95% VaR Actual Returns 99% VaR

Figure 14 Actual Return vs VaR: Gaussian (Equally Weighted) 1500-day estimation window

8.00%

6.00%

4.00%

2.00%

0.00%

-2.00%

-4.00%

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2 2 3 3 4 4 5 5 6 7 7 8 8 9 9 0 1 1 2 2 3 3 9 9 93 9 9 9 96 9 9 98 9 0 00 0 0 01 0 0 v r v 9 r v v v 9 l v v Jul 92 Jul Jul 94 Jul 95 a Jul 96 Jul 97 Jul Jul 99 Ju o Jul Jul 02 Jul Mar Nov 9 Mar 9 No Ma No Mar Nov 9 M No Mar Nov 9 Mar 9 No Mar No Mar 00 N Mar No Mar Nov 0 Mar 0 95% VaR Actual Returns 99% VaR

Page 143 of 169 Figure 15 Actual Return vs VaR: Gaussian (Exponentially Weighted Lambda=0.94) 1500-day estimation window

8.00%

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-4.00%

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2 2 3 3 4 4 5 5 6 7 7 8 8 9 9 0 1 1 2 2 3 3 9 9 93 9 9 9 96 9 9 98 9 0 00 0 0 01 0 0 v r v 9 r v v v 9 l v v Jul 92 Jul Jul 94 Jul 95 a Jul 96 Jul 97 Jul Jul 99 Ju o Jul Jul 02 Jul Mar Nov 9 Mar 9 No Ma No Mar Nov 9 M No Mar Nov 9 Mar 9 No Mar No Mar 00 N Mar No Mar Nov 0 Mar 0 95% VaR Actual Returns 99% VaR

Figure 16 Actual Return vs VaR: Gaussian (Exponentially Weighted Lambda=0.97) 1500-day estimation window

8.00%

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0.00%

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2 2 3 3 4 4 5 5 6 7 7 8 8 9 9 0 1 1 2 2 3 3 9 9 93 9 9 9 96 9 9 98 9 0 00 0 0 01 0 0 v r v 9 r v v v 9 l v v Jul 92 Jul Jul 94 Jul 95 a Jul 96 Jul 97 Jul Jul 99 Ju o Jul Jul 02 Jul Mar Nov 9 Mar 9 No Ma No Mar Nov 9 M No Mar Nov 9 Mar 9 No Mar No Mar 00 N Mar No Mar Nov 0 Mar 0 95% VaR Actual Returns 99% VaR

Page 144 of 169 Figure 17 Actual Return vs VaR: Gaussian (Exponentially Weighted Lambda=0.99) 1500-day estimation window

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2 2 3 3 4 4 5 5 6 7 7 8 8 9 9 0 1 1 2 2 3 3 9 9 93 9 9 9 96 9 9 98 9 0 00 0 0 01 0 0 v r v 9 r v v v 9 l v v Jul 92 Jul Jul 94 Jul 95 a Jul 96 Jul 97 Jul Jul 99 Ju o Jul Jul 02 Jul Mar Nov 9 Mar 9 No Ma No Mar Nov 9 M No Mar Nov 9 Mar 9 No Mar No Mar 00 N Mar No Mar Nov 0 Mar 0 95% VaR Actual Returns 99% VaR

Figure 18 Actual Return vs VaR: Variance Gamma 1500-day estimation window 8.00%

6.00%

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0.00%

-2.00%

-4.00%

-6.00%

-8.00%

2 2 3 3 4 4 5 5 6 7 7 8 8 9 9 0 1 1 2 2 3 3 9 9 93 9 9 9 96 9 9 98 9 0 00 0 0 01 0 0 v r v 9 r v v v 9 l v v Jul 92 Jul Jul 94 Jul 95 a Jul 96 Jul 97 Jul Jul 99 Ju o Jul Jul 02 Jul Mar Nov 9 Mar 9 No Ma No Mar Nov 9 M No Mar Nov 9 Mar 9 No Mar No Mar 00 N Mar No Mar Nov 0 Mar 0 95% VaR Actual Returns 99% VaR

Page 145 of 169 Figure 19 Actual Return vs VaR: Poisson Jump Diffusion 1500-day estimation window 8.00%

6.00%

4.00%

2.00%

0.00%

-2.00%

-4.00%

-6.00%

-8.00%

2 2 3 3 4 4 5 5 6 7 7 8 8 9 9 0 1 1 2 2 3 3 9 9 93 9 9 9 96 9 9 98 9 0 00 0 0 01 0 0 v r v 9 r v v v 9 l v v Jul 92 Jul Jul 94 Jul 95 a Jul 96 Jul 97 Jul Jul 99 Ju o Jul Jul 02 Jul Mar Nov 9 Mar 9 No Ma No Mar Nov 9 M No Mar Nov 9 Mar 9 No Mar No Mar 00 N Mar No Mar Nov 0 Mar 0 95% VaR Actual Returns 99% VaR

Figure 20 Actual Return vs VaR: Gaussian (Equally Weighted) 1875-day estimation window

8.00%

6.00%

4.00%

2.00%

0.00%

-2.00%

-4.00%

-6.00%

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2 2 3 3 4 4 5 5 6 7 7 8 8 9 9 0 1 1 2 2 3 3 9 9 93 9 9 9 96 9 9 98 9 0 00 0 0 01 0 0 v r v 9 r v v v 9 l v v Jul 92 Jul Jul 94 Jul 95 a Jul 96 Jul 97 Jul Jul 99 Ju o Jul Jul 02 Jul Mar Nov 9 Mar 9 No Ma No Mar Nov 9 M No Mar Nov 9 Mar 9 No Mar No Mar 00 N Mar No Mar Nov 0 Mar 0 95% VaR Actual Returns 99% VaR

Page 146 of 169 Figure 21 Actual Return vs VaR: Gaussian (Exponentially Weighted Lambda=0.94) 1875-day estimation window

8.00%

6.00%

4.00%

2.00%

0.00%

-2.00%

-4.00%

-6.00%

-8.00%

2 2 3 3 4 4 5 5 6 7 7 8 8 9 9 0 1 1 2 2 3 3 9 9 93 9 9 9 96 9 9 98 9 0 00 0 0 01 0 0 v r v 9 r v v v 9 l v v Jul 92 Jul Jul 94 Jul 95 a Jul 96 Jul 97 Jul Jul 99 Ju o Jul Jul 02 Jul Mar Nov 9 Mar 9 No Ma No Mar Nov 9 M No Mar Nov 9 Mar 9 No Mar No Mar 00 N Mar No Mar Nov 0 Mar 0 95% VaR Actual Returns 99% VaR

Figure 22 Actual Return vs VaR: Gaussian (Exponentially Weighted Lambda=0.97) 1875-day estimation window

8.00%

6.00%

4.00%

2.00%

0.00%

-2.00%

-4.00%

-6.00%

-8.00%

2 2 3 3 4 4 5 5 6 7 7 8 8 9 9 0 1 1 2 2 3 3 9 9 93 9 9 9 96 9 9 98 9 0 00 0 0 01 0 0 v r v 9 r v v v 9 l v v Jul 92 Jul Jul 94 Jul 95 a Jul 96 Jul 97 Jul Jul 99 Ju o Jul Jul 02 Jul Mar Nov 9 Mar 9 No Ma No Mar Nov 9 M No Mar Nov 9 Mar 9 No Mar No Mar 00 N Mar No Mar Nov 0 Mar 0 95% VaR Actual Returns 99% VaR

Page 147 of 169 Figure 23 Actual Return vs VaR: Gaussian (Exponentially Weighted Lambda=0.99) 1875-day estimation window

8.00%

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4.00%

2.00%

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-8.00%

2 2 3 3 4 4 5 5 6 7 7 8 8 9 9 0 1 1 2 2 3 3 9 9 93 9 9 9 96 9 9 98 9 0 00 0 0 01 0 0 v r v 9 r v v v 9 l v v Jul 92 Jul Jul 94 Jul 95 a Jul 96 Jul 97 Jul Jul 99 Ju o Jul Jul 02 Jul Mar Nov 9 Mar 9 No Ma No Mar Nov 9 M No Mar Nov 9 Mar 9 No Mar No Mar 00 N Mar No Mar Nov 0 Mar 0 95% VaR Actual Returns 99% VaR

Figure 24 Actual Return vs VaR: Variance Gamma 1875-day estimation window 8.00%

6.00%

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2.00%

0.00%

-2.00%

-4.00%

-6.00%

-8.00%

2 2 3 3 4 4 5 5 6 7 7 8 8 9 9 0 1 1 2 2 3 3 9 9 93 9 9 9 96 9 9 98 9 0 00 0 0 01 0 0 v r v 9 r v v v 9 l v v Jul 92 Jul Jul 94 Jul 95 a Jul 96 Jul 97 Jul Jul 99 Ju o Jul Jul 02 Jul Mar Nov 9 Mar 9 No Ma No Mar Nov 9 M No Mar Nov 9 Mar 9 No Mar No Mar 00 N Mar No Mar Nov 0 Mar 0 95% VaR Actual Returns 99% VaR

Page 148 of 169 Figure 25 Actual Return vs VaR: Poisson Jump Diffusion 1875-day estimation window 8.00%

6.00%

4.00%

2.00%

0.00%

-2.00%

-4.00%

-6.00%

-8.00%

2 2 3 3 4 4 5 5 6 7 7 8 8 9 9 0 1 1 2 2 3 3 9 9 93 9 9 9 96 9 9 98 9 0 00 0 0 01 0 0 v r v 9 r v v v 9 l v v Jul 92 Jul Jul 94 Jul 95 a Jul 96 Jul 97 Jul Jul 99 Ju o Jul Jul 02 Jul Mar Nov 9 Mar 9 No Ma No Mar Nov 9 M No Mar Nov 9 Mar 9 No Mar No Mar 00 N Mar No Mar Nov 0 Mar 0 95% VaR Actual Returns 99% VaR

Page 149 of 169 APPENDIX 8 – PORTFOLIO CONSTRUCTION

APPLICATION

Introduction

VaR forecasts are one of the leading approaches currently employed by banks to measure portfolio risk. Portfolio managers in practice may be required to compute

VaR for their portfolios in order to evaluate their performance. In this Appendix 8, we conduct a further empirical analysis of VaR on an equally-weighted portfolio (EW

Portfolio) and a value-weighted portfolio (VW Portfolio), constructed out of stocks contained in the ASX 20 Index (the Portfolios). In order to evaluate how the

Portfolios perform in regards to VaR, we obtain estimates of the 95th and 99th percentile VaR using both the historical returns of the ASX 20 Index and the historical returns of the Portfolios. We lastly apply the criteria of model accuracy to assess how each of the Portfolios performs against its VaR expectations.

The Methodology described below largely mirrors the methodology used in Section 6 of the thesis, with the additional step of constructing the Portfolio. This allows us to consider whether working with the constructed Portfolios is similar to working with an index (as what was studied in the thesis).

Methodology

We follow a methodology that is a simplified version of that outlined in Section 6 of the thesis, with the additional step of constructing the Portfolio.

Page 150 of 169 1) Firstly, we construct the returns of the Portfolios. To do this, we obtain the

member listing of the ASX 20 Index as at the first trading day of each month

over the analysis period (from 1st January 2003 to 31st December 2007). If

there is a removal or addition to the ASX 20 Index as reflected in a change of

the member constituents at the start of the month, it is reflected from that

month in the Portfolios by either removing or adding it to the Portfolio return

calculations. This monthly review and rebalancing of the Portfolio is

considered a reasonable assumption as portfolio managers would have to take

into consideration transaction costs involved with trades. For the EW

Portfolio, it is assumed that the stocks are equally weighted within the

Portfolio and the return of the EW Portfolio is simply the average of the

constituent stock returns. The VW Portfolio assumes that the stocks are held

according to their ASX 20 Index weighting as at the start of each month. This

weighting is updated on the first trading day of each month. The return of the

VW Portfolio is then the weighted average of the constituent stock returns.

2) Secondly, we take a point in time and utilize the previous 52 observations (1

year) as the estimation window to obtain parameter estimates for the model

used. For the EW and VW Portfolio, we utilize the previous 52 observations

and use them to find the variance-covariance matrix. Then, we use the

portfolio weights as at that point in time as determined in step (1) above to

find the variance of the portfolio . In matrix notation, and for a portfolio of N

2 securities, the variance of the portfolio can be compactly denoted by σ p =

Page 151 of 169 xTΩx, where Ω is the variance-covariance matrix for the N securities and x is a

T column vector of weights, x = [x1, x2,...,xN]; that is,

2 ⎡⎤σσ112L σ 1N ⎡ x 1⎤ ⎢⎥NN σ 2 = xTΩx = [x , x ,...,x ] ⎢ ⎥ = xσ x p 1 2 N ⎢⎥MM OM⎢ M⎥ ∑∑ iijj ⎢⎥2 ⎢ ⎥ ij==11 ⎣⎦σσNN12L σ N⎣x N⎦

2 (where σii = σ i ).

The details of how the parameter estimation is conducted for the ASX 200 Index is outlined in Section 6.1.1. We focus only on the Gaussian model and use the equally weighted moving average approach to obtain estimates of the volatility parameters. We obtain parameter estimates using the return series of the ASX 20 Index and the Portfolios.

3) Thirdly, using these parameter estimates, we can then obtain an estimate for

the 95% and 99% VaR measures at that point in time. The details of how this

VaR estimation is conducted is outlined in Section 6.2.1. The first estimation

window for parameter estimation is from the period 1st January 2003 to 31st

December 2003. Thus the first VaR estimate is as at the first week of January

2004.

We roll the estimation window forward over our entire data sample, repeating

steps 2) and 3) at each point in time. For every VaR calculation point, we

draw from the 52 observations preceding the date for which the VaR

calculation is made. This gives us a series of parameter and VaR estimates

Page 152 of 169 that we can use to test the performance of the Portfolios against by

backtesting.

4) Fourthly, we then conduct an accuracy performance test by comparing the

VaR estimates obtained against the actual returns of the Portfolios and the

ASX 20 Index. The details of how this accuracy performance test is

conducted is outlined in Section 6.3.1. In contrast to the objective set out in

Section 6, where the VaR performance testing was a way of implicitly testing

the suitability of the underlying VaR models used, here we are interested in

testing the performance of the Portfolios against the VaR estimates obtained.

We test the Portfolios against VaR estimates calculated using both the ASX

20 Index and the Portfolio historical returns. For comparison purposes, we

also test the performance of the ASX 20 Index against VaR estimates

calculated using the ASX 20 Index historical returns.

Page 153 of 169 Data and Descriptive Statistics

We have utilized historical data over the period from 1st January 2003 to 31st

December 2007. We have chosen to use a smaller index such as the ASX 20 Index so

that the construction of the Portfolios is not overly onerous and we can focus on the

back-testing of VaR against actual returns achieved. For the purpose of this exercise,

we use weekly returns. We obtain weekly closing prices of the ASX 20 Index from

Bloombergs for the period from 1st January 2003 to 31st December 2007. We construct the Portfolio over this same period. We take the natural logarithm of prices

⎛ Pt ⎞ ln⎜ ⎟ to get the weekly return series for the ASX 20 Index and the Portfolios. ⎝ Pt−1 ⎠

This gives us a total of 260 observations for each of the ASX 20 Index and the

Portfolios.

Table 11 Descriptive Statistics

This table reports the descriptive statistics of the weekly returns of the ASX 20 Index and the Portfolios. The total number of observations is 260 and ranges from the 1st January 2003 to the 31st December 2007. The Jarque-Bera test of normality is performed. The p-value of the test is given below in the parenthesis.

ASX 20 Index EW Portfolio VW Portfolio Number of Observations 260 260 260 Mean 0.257% 0.239% 0.238% Median 0.356% 0.302% 0.311% Maximum 7.505% 7.029% 7.559% Minimum -5.517% -5.692% -5.470% Standard Deviation 1.590% 1.542% 1.583% Skewness -0.30067 -0.31658 -0.26551 Kurtosis 2.53415 2.58316 2.51266 Jarque-Bera Test 6.268 6.225 5.628 p-value (0.0435) (0.0445) (0.0600) Correlation with ASX 20 Index 1.0000 0.9646 0.9989

Page 154 of 169

From the descriptive statistics table, we can again see that skewness of the returns is negative. Negative skewness indicates a distribution with an asymmetric tail extending towards more negative values. The Jarque-Bera test of normality suggests that the returns data is not normal for the ASX 20 index and EW Portfolio, and normal for the VW Portfolio at the 5% level. As expected, the VW Portfolio returns are highly correlated with the ASX 20 Index returns, since the VW portfolio effectively rebalances to the ASX 20 Index weightings every month.

Page 155 of 169 Results

Portfolio Construction

Figure 26: EW Portfolio vs ASX 20 Index

250

200

150 ASX 20 Index

Portfolio 100 Rebased to 100

50

- Jan 03 Jul 03 Jan 04 Jul 04 Jan 05 Jul 05 Jan 06 Jul 06 Jan 07 Jul 07

Date

Figure 27: VW Portfolio vs ASX 20 Index

250

200

150 ASX 20 Index

Portfolio 100 Rebased to 100

50

- Jan 03 Jul 03 Jan 04 Jul 04 Jan 05 Jul 05 Jan 06 Jul 06 Jan 07 Jul 07

Date

Page 156 of 169 The figures above shows the cumulative returns of the ASX 20 Index (rebased to

100) against the cumulative returns of the Portfolios (rebased to 100). The Portfolios track the ASX 20 Index reasonably well, with differences accounted for by different weightings in the stocks and also a timing lag in the adding or removing of stocks to the Portfolio.

Page 157 of 169 Parameters Estimation Results

Table 12 Parameter Estimates

This table presents the paramater estimates for the Gaussian Model using an estimation period of 52 observations (1 year). Since we re-estimate the model every week for the backtesting period by rolling the estimation window forward to the end of the sample, we present here the average of the parameter estimates along with the standard deviation to indicate the variability of the estimate over that period as we roll the estimation window forward.

Calculated from Calculated from Calculated from ASX 20 Index EW Portfolio VW Portfolio Returns Returns Returns

Average Std Dev Average Std Dev Average Std Dev

Gaussian Distribution µ 0.00301 0.00100 0.00297 0.00081 0.00286 0.00102 σ 0.01428 0.00286 0.01384 0.00289 0.01384 0.00289

The Gaussian distribution is simply given by:

1 ⎛ ()x − μ 2 ⎞ xf )( = exp⎜− ⎟ , ⎜ 2 ⎟ 2 σπ ⎝ σ ⎠

where μ is the mean andσ is the standard deviation. The Table above records the parameter estimates for the Gaussian model applied to the return series of the ASX 20

Index and the Portfolio.

Page 158 of 169 VaR Simulation Results

Table 13 Value-at-Risk Simulation Results

This table presents the simulation results for our VaR using estimation periods of 52 observations. Since we re-estimate the model every day for the backtesting period by rolling the estimation window forward to the end of the sample, we present here the average of the parameter estimates along with the standard deviation.

Calculated from Calculated from Calculated from ASX 20 Index EW Portfolio VW Portfolio Returns Returns Returns Average Std Dev Average Std Dev Average Std Dev

95% VaR Gaussian -2.048% 0.0043 -1.980% 0.0047 -1.990% 0.0044

99% VaR Gaussian -3.021% 0.0063 -2.923% 0.0066 -2.933% 0.0064

The Table above displays our statistics for the VaR simulation procedure. We have estimated a series of VaR estimates using the ASX 20 Index historical returns as well as the Portfolios historical returns. Using the Portfolios historical returns to estimate

VaR results in a slightly less conservative VaR estimate on average.

The Figures below illustrate the actual returns of the Portfolio and the ASX 20 Index against the VaR estimates.

Page 159 of 169 Figure 28: EW Portfolio Actual Returns vs VaR estimates (calculated from ASX 20 Index)

8.00%

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4.00%

2.00%

0.00%

-2.00%

-4.00%

-6.00%

-8.00% Jul 04 Jul Jul 05 Jul Jul 06 Jul Jul 07 Jul Jan 04 Jan 05 Jan 06 Jan 07 Mar 04 Mar 05 Mar 06 Mar 07 Sep 04 Nov 04 Sep 05 Nov 05 Sep 06 Nov 06 Sep 07 Nov 07 May 04 May 05 May 06 May 07 95% VaR Actual Returns 99% VaR

Figure 29: VW Portfolio Actual Returns vs VaR estimates (calculated from ASX 20 Index)

10.00%

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6.00%

4.00%

2.00%

0.00%

-2.00%

-4.00%

-6.00%

-8.00% Jul 04 Jul Jul 05 Jul Jul 06 Jul Jul 07 Jul Jan 04 Jan 05 Jan 06 Jan 07 Mar 04 Mar 05 Mar 06 Mar 07 Sep 04 Nov 04 Sep 05 Nov 05 Sep 06 Nov 06 Sep 07 Nov 07 May 04 May 05 May 06 May 07 95% VaR Actual Returns 99% VaR

Page 160 of 169 Figure 30: EW Portfolio Actual Returns vs VaR estimates (calculated from EW Portfolio)

8.00%

6.00%

4.00%

2.00%

0.00%

-2.00%

-4.00%

-6.00%

-8.00% Jul 04 Jul Jul 05 Jul Jul 06 Jul Jul 07 Jul Jan 04 Jan 05 Jan 06 Jan 07 Mar 04 Mar 05 Mar 06 Mar 07 Sep 04 Nov 04 Sep 05 Nov 05 Sep 06 Nov 06 Sep 07 Nov 07 May 04 May 05 May 06 May 07 95% VaR Actual Returns 99% VaR

Figure 31: VW Portfolio Actual Returns vs VaR estimates (calculated from VW Portfolio)

10.00%

8.00%

6.00%

4.00%

2.00%

0.00%

-2.00%

-4.00%

-6.00%

-8.00% Jul 04 Jul Jul 05 Jul Jul 06 Jul Jul 07 Jul Jan 04 Jan 05 Jan 06 Jan 07 Mar 04 Mar 05 Mar 06 Mar 07 Sep 04 Nov 04 Sep 05 Nov 05 Sep 06 Nov 06 Sep 07 Nov 07 May 04 May 05 May 06 May 07 95% VaR Actual Returns 99% VaR

Page 161 of 169 Figure 32: ASX 20 Index Actual Returns vs VaR estimates (calculated from ASX 20 Index)

8.00%

6.00%

4.00%

2.00%

0.00%

-2.00%

-4.00%

-6.00%

-8.00% Jul 04 Jul Jul 05 Jul Jul 06 Jul Jul 07 Jul Jan 04 Jan 05 Jan 06 Jan 07 Mar 04 Mar 05 Mar 06 Mar 07 Sep 04 Nov 04 Sep 05 Nov 05 Sep 06 Nov 06 Sep 07 Nov 07 May 04 May 05 May 06 May 07 95% VaR Actual Returns 99% VaR

Page 162 of 169 Performance Testing Results - Accuracy

Table 14 Accuracy of VaR Measures - Fraction of Outcomes Exceeding VaR EW Portfolio

This table tabulates the fraction of outcomes where the realised losses exceed the VaR risk measure. For the 95th percentile VaR, realised losses should exceed the VaR measure 5% of the time. For the 99th percentile VaR, realised losses should exceed VaR 1% of the time.

Panel A: Fraction of Outcomes Exceeding VaR (Binary Loss Function) Portfolio Actual Portfolio Actual ASX 20 Index Returns vs VaR Returns vs VaR Actual Returns vs estimates estimates VaR estimates (calculated from (calculated from (calculated from ASX 20 Index) Portfolio) ASX 20 Index) Model Average Std Dev Average Std Dev Average Std Dev 95th Percentile VaR Measures Gaussian 5.74% 0.2332 6.22% 0.2421 6.22% 0.2421

99th Percentile VaR Measures Gaussian 3.83% 0.1942 3.83% 0.1923 4.31% 0.2035

Panel B: Fraction of Outcomes Exceeding VaR (Quadratic Loss Function) Portfolio Actual Portfolio Actual ASX 20 Index Returns vs VaR Returns vs VaR Actual Returns vs estimates estimates VaR estimates (calculated from (calculated from (calculated from ASX 20 Index) Portfolio) ASX 20 Index) Model Average Std Dev Average Std Dev Average Std Dev 95th Percentile VaR Measures Gaussian 5.95% 0.2417 6.45% 0.2512 6.45% 0.2511

99th Percentile VaR Measures Gaussian 3.91% 0.1965 3.93% 0.1973 4.40% 0.2080

Page 163 of 169

Table 15 Accuracy of VaR Measures - Fraction of Outcomes Exceeding VaR VW Portfolio

This table tabulates the fraction of outcomes where the realised losses exceed the VaR risk measure. For the 95th percentile VaR, realised losses should exceed the VaR measure 5% of the time. For the 99th percentile VaR, realised losses should exceed VaR 1% of the time.

Panel A: Fraction of Outcomes Exceeding VaR (Binary Loss Function) Portfolio Actual Portfolio Actual ASX 20 Index Returns vs VaR Returns vs VaR Actual Returns vs estimates estimates VaR estimates (calculated from (calculated from (calculated from ASX 20 Index) Portfolio) ASX 20 Index) Model Average Std Dev Average Std Dev Average Std Dev 95th Percentile VaR Measures Gaussian 6.22% 0.2421 5.74% 0.2332 6.22% 0.2421

99th Percentile VaR Measures Gaussian 4.31% 0.2052 4.31% 0.2035 4.31% 0.2035

Panel B: Fraction of Outcomes Exceeding VaR (Quadratic Loss Function) Portfolio Actual Portfolio Actual ASX 20 Index Returns vs VaR Returns vs VaR Actual Returns vs estimates estimates VaR estimates (calculated from (calculated from (calculated from ASX 20 Index) Portfolio) ASX 20 Index) Model Average Std Dev Average Std Dev Average Std Dev 95th Percentile VaR Measures Gaussian 6.43% 0.2504 5.97% 0.2428 6.45% 0.2511

99th Percentile VaR Measures Gaussian 4.39% 0.2074 4.41% 0.2082 4.40% 0.2080

With the VaR estimates obtained, we now turn to examining how the Portfolio performs against these VaR estimates. We look at the fundamental goal of the VaR measures – whether they cover losses actually realized by the Portfolios as they are intended to. As can be seen from the Tables above and the Figures below, the actual fraction of times that the 95% VaR is exceeded is larger than 5%. Likewise the 99%

Page 164 of 169 VaR results exhibit the same tendency to fall short of the desired level of coverage (ie underestimate the risk). It can be seen that the actual fraction of times that the 99%

VaR is exceeded is consistently above the expected 1%. These observations are consistent with fat tailed characteristic of returns. Thus, as suggested in Section 1 of the thesis, utilizing models that can better capture the non-normal characteristics inherent in return distributions could potentially result in more accurate estimates of

VaR.

Figure 33: Fraction of Outcomes Exceeding VaR (EW Portfolio)

7.0%

6.0%

5.0%

95% VaR

4.0% 99% VaR

3.0%

2.0%

1.0%

0.0% Portfolio Actual Returns vs VaR estimates Portfolio Actual Returns vs VaR estimates ASX 20 Index Actual Returns vs VaR (calculated from ASX 20 Index) (calculated from Portfolio) estimates (calculated from ASX 20 Index)

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Figure 34: Fraction of Outcomes Exceeding VaR (VW Portfolio)

7.0%

6.0%

5.0%

95% VaR

4.0% 99% VaR

3.0%

2.0%

1.0%

0.0% Portfolio Actual Returns vs VaR estimates Portfolio Actual Returns vs VaR estimates ASX 20 Index Actual Returns vs VaR (calculated from ASX 20 Index) (calculated from Portfolio) estimates (calculated from ASX 20 Index)

In the Tables below we formally test accuracy. The Tables below shows the likelihood ratio test for assessing VaR accuracy. The likelihood ratio tests of the accuracy of the Portfolios and the ASX 20 Index in achieving the 95% VaR suggests that the null hypothesis is accepted at the 5% level in all cases. That is, that the actual proportion of failures is statistically equal to the theoretical value. On the other hand, the likelihood ratio test of the Portfolios and the ASX 20 Index in achieving the 99%

VaR suggests that the null hypothesis is rejected at the 5% level in all cases.

Page 166 of 169 Table 16 Accuracy of VaR Measures - Likelihood Ratio Tests EW Portfolio

This table displays the results from the likelihood ratio tests based on the number of failures from our backtesting. The null hypothesis we are testing is that the actual proportion of failures is equal to the theoretical value. We have displayed the likelihood ratio statistics below for each model, as well as the associated p-value

Likelihood Ratio Tests(Binary Loss Function) Portfolio Actual Portfolio Actual ASX 20 Index Returns vs VaR Returns vs VaR Actual Returns vs estimates estimates VaR estimates (calculated from (calculated from (calculated from ASX 20 Index) Portfolio) ASX 20 Index) Model LR Stat p-value LR Stat p-value LR Stat p-value 95th Percentile VaR Measures Gaussian (Equally Weighted) 0.1005 0.7512 0.2649 0.6068 0.2649 0.6068

99th Percentile VaR Measures Gaussian (Equally Weighted) 4.2678 0.0388 4.2678 0.0388 5.5131 0.0189

Table 17 Accuracy of VaR Measures - Likelihood Ratio Tests VW Portfolio

This table displays the results from the likelihood ratio tests based on the number of failures from our backtesting. The null hypothesis we are testing is that the actual proportion of failures is equal to the theoretical value. We have displayed the likelihood ratio statistics below for each model, as well as the associated p-value

Likelihood Ratio Tests(Binary Loss Function) Portfolio Actual Portfolio Actual ASX 20 Index Returns vs VaR Returns vs VaR Actual Returns vs estimates estimates VaR estimates (calculated from (calculated from (calculated from ASX 20 Index) Portfolio) ASX 20 Index) Model LR Stat p-value LR Stat p-value LR Stat p-value 95th Percentile VaR Measures Gaussian (Equally Weighted) 0.2649 0.6068 0.1005 0.7512 0.2649 0.6068

99th Percentile VaR Measures Gaussian (Equally Weighted) 5.5131 0.0189 5.5131 0.0189 5.5131 0.0189

Page 167 of 169 Conclusion

Whilst our thesis utilized the VaR back-testing mainly as a way to test the usefulness of certain models, in this Appendix 8 we have undertaken a further empirical analysis that looks at the actual returns of constructed Portfolios against VaR estimates in order to gauge how these Portfolios performs against expectations and to illustrate how this backtesting might be used in practice.

Using the Gaussian model, we obtained parameter estimates from historical returns over the period from 1st January 2003 to 31st December 2007, which were then used to simulate VaR expectations. We obtained VaR estimates using historical returns of the ASX 20 Index, as well as the historical returns of an equally weighted Portfolio and value weighted Portfolio, both constructed from stocks included in the ASX 20

Index over the same period. We compared the performance of the Portfolios against both sets of VaR estimates (ie the VaR estimates calculated from the ASX 20 Index historical returns and the VaR estimates calculated from the Portfolios historical returns). We also compared the performance of the ASX 20 Index against VaR estimates calculated from the ASX Index historical returns only.

We find that in general, the actual fraction of times that the 95% and 99% VaR is exceeded is larger than expectations (ie 5% of 1% respectively). That is the VaR estimates underestimate the level of risk and suggests that the return distributions have certain characteristics that could potentially be captured better with models that can account for fat tails. However, the formal likelihood ratio tests found that the

Page 168 of 169 model does a statistically adequate job at estimating the 95% VaR, but not the 99%

VaR.

The Methodology utilized in this Appendix largely mirrors the methodology used in

Section 6 of the thesis, with the additional step of constructing the Portfolio. We find that working with the constructed Portfolios is similar to working with an index (as what was studied in the thesis). The results illustrate that the constructed Portfolios behave and have similar characteristics to the index on which they are based. In addition, the constructed Portfolios perform much like the index when assessing the accuracy of using VaR to estimate risk.

Future research may continue to examine more sophisticated models that we have touched on in the thesis that could potentially capture the pervading characteristics exist in returns and produce more accurate VaR estimates. As previously concluded, it would be of interest to try to apply the more sophisticated bi-dimensional affine models. Option pricing theory suggests that these models are needed to capture some of the more subtle features of prices, and so it is natural to ask whether or not these models would improve VaR calculations. Econometrically, however, they are difficult, if not impossible, models to estimate, at this time.

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