Estimating value-at-risk using extreme value theory : new evidence from commodity markets.

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Amsterdam Business School

Master in International Finance

Estimating Value-at-Risk using Extreme Value Theory:

New Evidence from Commodity Markets

By

Tzu-Yun Katherine Lin

Supervised by

Dr. Chris Florackis (University of Liverpool)

September 2014

University of Amsterdam

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Estimating Value-at-Risk using Extreme Value Theory:

New Evidence from Commodity Markets

TZU-YUN KATHERINE LIN

ABSTRACT

The Value-at-Risk model using extreme value theory, expected to provide good risk measurement in extreme cases, is studied and examined. We use extreme value theory to estimate Value-at-Risk for the case of commodity markets. Our analysis is based on sample of four precious metals and two energy products over the period 2000-2010. The VaR model based on EVT are compared against traditional VaR methods such as historical simulation, Monte Carlo simulation, GARCH model etc. Our findings show that VaR model measured using EVT fits well among all VaR approaches assessed, with performance only behind GARCH approach.

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1. Introduction

The recent financial crisis highlighted the need for effective risk management- a crucial reason being the integration of the international markets, where any event happening in one corner in the world spreads massive consequences worldwide. Following the cross-border market, the nature of business models and trading became creative and thus complex, so does the risk factors involved. The complexity at certain point got beyond mankind’s naive expectation and caused several financial disasters. Some examples include the ’08 crisis and the burst of 90’s bubble; when these shocks occurred, public sectors and private businesses were crashed by the unexpected market downturn, many even within one day. As the action to prevent such incidences, effort to predict from economics side of view and to alert from risk control side of view has been put in. Studies and practical tools of risk management have been growing during the past few decades in coordination with the needs.

Among all markets, the one of commodities deserves attention for many reasons. First of all, it connects to manufacturing chains as materials; commodity price fluctuation directly influences the entities’ profit margin. Secondly, the primary business in commodity market is out-of-imagination large; and the business group usually owns many businesses in other industries. The butterfly effect is expected when extreme events or shocks occur. Furthermore, commodity trading had grown so well that its derivatives market is developed. Commodity and its derivatives are largely viewed as an investment tool, and when it comes to investment warzone, price fluctuation is many-times amplified. There is a period for commodity worth us to notice – 2000 to 2010. Some may refer the commodities bubble to 2006 to 2008, but if we look at bigger picture of price behavior the full decade should be studied. Most fundamental

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commodities show a rapid upward trend during this period; even after the 2008 shock many of them still count for three to four times their prices in the beginning of 2000. As price movements show, the growth path is much associated with global economy as expected when sectors get integrated. The phenomena also applies to shocks commonly appeared among commodities around the end of 2008. The volatile period lasts to roughly mid-2009. (Figure A. and B. in Appendix.)

This thesis aims to assess one of the well-structured approaches, the Extreme Value Theory approach, for estimating the important risk management indicator Value-at-Risk, in order to discover whether this VaR model can be used as a risk management tool for commodity. Value-at-Risk is a commonly monitored risk indicator in practice. The use of VaR is recent but the assess-ability has made it the “new science of risk management” (Dowd, 2008). Value-at-Risk is an attempt to provide a single number that summarizes the total risk in a portfolio. (Hull, 2012, pp.104) It measures the maximum loss due to market fluctuations over a certain period with a given probability. VaR is comprehensible to express extreme risk as only one value is output. It is a scientific language for translating management guidelines into actual numbers. (Embrechts, Resnick, and Samorodnitsky, 1999) To compute the value, many approaches and their extensions have been produced. They are roughly categorized as follows: i) historical simulation, ii) variance-covariance model, and iii) stochastic model simulation. For comparison purpose, some conventional approaches will be conducted on the same datasets for VaR estimation.

The extreme value theory is first developed by Gnedenko in 1943. It is a statistical theory of extreme events; the two major results are: first, a series of maxima (minima)

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can be modeled and under certain conditions the distribution is shown to converge to generalized extreme value (GEV) distribution.1 The second results that the behavior of the excess loss over a given high threshold converges to a limiting distribution- generalized Pareto distribution (GPD). (Bensalah, 2000) Convergence enables us to have observation specifying the tails. The second result will be our application to use on VaR, also called Peak-over-Threshold (POT) method.

EVT has been applied to extreme events measurements in many occasions, e.g. prediction of natural disasters, yet its history as a regular tool in financial risk management is short, especially in commodity market. Our work complements a series of studies on the subject such as VaR using EVT in stock market and VaR measurement for energy commodity in ‘90s. We extend these studies by focusing on the period 2000-2010 and in commodities market. The proposed period 2000-2010 is worth to look into because it is when the so called commodity bubble blown bigger and bigger. The commodity markets was boosted for several reasons during the time: commodity previously being ignored as an investment opportunity because the earlier economy boom and the dotcom bubble, rapid mega-constructions of emerging markets, fast-growing speculative actions, the depreciated dollar as most commodities are priced in dollar, and the distrust against traditional investing instruments after several crisis which making commodity a huge tool. Nevertheless, many factors can be foreseen to also cause the bubble popped, for instance the ghost-cities created in China would stop the excessive mega-projects at a point and thus the huge need of metal commodity. (Colombo, 2011) The common sharp drop of commodity price from late 2008 is an example of such instability. As many commodities are observed

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to have similar price movements during the ’08 crisis and as the price drop can possibly interpreted as a bubble pop, the study will focus much on the period. Extreme value theory is purposed to capture extreme events thus the thesis expects that VaR measured by EVT results a better quantification, efficiently, practically and flexibility.

2. Literature Review

The history of Value-at-Risk as one of the indicators for risk management is short. It was only introduced in late 1980s. (Linsmeier and Pearson, 1996) But it soon raised regulator’s interests that the Basel committee in 1995 proposed VaR to be in the monitoring scope. Further on many regulatory associations named it as one of the major risk management tools. Value-at-Risk is a good indicator when describing risk. The concept is that given a determined confidence level (95% or 99% commonly used) the actual return (gain/loss) will not exceed the value of VaR derived. The idea is to answer the what-if question when worst scenario happens. (Harper, 2009) Many approaches to calculate VaR have been developed, and this thesis will mainly focus on the extreme value theory approach.

Using extreme value theory to measure VaR is first proposed by Diebold, Schuerman, and Stroughair in 1998. The ground of EVT taking place is that the result of the classical extremes theory suggests once critical conditions are satisfied, the probabilities associated with properly normalized distribution will converge to certain distribution functions (Neftci, 2000). McNeil’s work (1999) demonstrated the application of EVT on extreme event’s valuation and to risk management, precisely

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on VaR’s percentile observation. Neftci (2000) further explained in depth the tail estimation and the clear description from the work values much on inserting EVT in VaR measurement. There are also extensions of extreme value theory not to measure VaR itself but to apply on existing VaR estimations, for instance Marimoutou, Raggad, and Trabelsi's work (2009), which improves GARCH-VaR model by extreme value theory.

The implements of VaR method using EVT are published such as Hsu, Huang, and Chiou’s empirical study on portfolio risk in Asian markets (2012) and Ozun, Cifter, and Yılmazer’s empirical work on emerging market stock returns (2010) and others on diversified markets/periods. However there is not much discussion to evident extreme value theory VaR measurement from commodity market. Žiković’s work on oil futures returns (2011) is one example but the number of published study is relatively few. Thus we will conduct the proposed VaR approach and seek for evidence from several commodities’ market data.

The objective of this thesis, extreme value theory approach, is rather young among existing recognized VaR measurements. There are existing VaR approaches commonly considered well developed and they mainly fall in three categories:

i) Historical Simulation; ii) Variance-Covariance Model; iii) Stochastic Model

Historical Simulation assumes that historical movements are representative for future movements. One popular approach of this category is simple historical simulation

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which predicts tomorrow’s movement from distribution built on movements of past sample directly. The approach is straight-forward to conduct. However it has flaws such as sensitive to large loss and responding to asymmetric movements (loss VaR is affected largely by downside shock but rarely by upside shock) and that its lack of flexibility accumulates the errors. (Pritsker, 2006) Historical simulation with ARMA forecast (HASF) is another approach under this category that receives much attention. Cabedo and Moya (2003) introduced an improved VaR method based on simple historical simulation method in response to the need of risk quantification of oil price. HSAF features practical feasibility and introduces a regressed forecasting model (ARMA) to create flexibility. As shown in their study, HSAF is proved to model VaR more better than HS when there are shocks in the sample and more efficient than variance-covariance model (GARCH(1,1) model), applying to oil market (oil spot price from 1992 – 1999).

The variance-covariance model assumes the potential movement (return) is related to volatility; therefore modeling volatility stands a big portion in methodologies under this category. Methodologies in this categories differs in assigning the level of importance for historical information, e.g. equally weighted or exponentially weighted to past information. Yet an autoregressive conditional heteroskedasticity (ARCH) model, introduced by Engle (1982) and the generalized ARCH model developed by Bollerslev (1986) may provide the best chance to capture reality as it assumes the combination of past return and volatility and the relations are not assigned but estimated. (Bera and Higgins, 1993) The model resolves issues of volatility clustering and asymmetric impact of shocks.

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The third category – Stochastic Models (Monte Carlo Simulation), comparing other two takes less information from past. This is to assume that the future follows a pre-determined distribution; simulation will be fulfilled when pseudo random variables are generated. Historical data are only extracted for distribution statistics or for reference to select variables. (Hull and White, 1987) Apart from extreme value theory, methodologies in every category will be chosen to perform VaR estimation on the same scope and dataset in this thesis: Historical Simulation, HSAF, Monte Carlo Simulation, Delta Normal, and GARCH model.

3. Value-at-Risk Models

The thesis aims to assess the quality to measure Value-at-Risk using extreme value theory. The evidence will be found in commodity market as mentioned in section 2 and we expect extreme value theory to provide a relatively effective measurement. The hypothesis of better fitness from EVT comes from the aim and characteristics of the theory. The definition of fitness will be demonstrated as well. To understand application of Extreme Value Theory in Value-at-Risk measuring, we have to look at the theory itself and power law provides explanation why extreme value theory is useful. The power law states that the possibility of a variable’s value, especially for extreme cases (in other word - tails), can be almost correctly estimated:

> = ∙ ; equivalent to > = − ∙

The law gives us a chance to improve VaR estimation as it suggests that the tails of a distribution, even at the very high percentile, can be estimated correctly at a satisfactory level (depending on the assumed confidence level of VaR). Knowing it is possible to improve VaR measurement, we’ll introduce Extreme Value Theory as a

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tool to fulfill what is indicated by power law because the theory “shows that the tails of a wide range of different probability distributions share common properties”. (Hull, 2012, pp. 167).

Figure 1. Extreme value theory’s focus in probability distribution (POT method) Extreme value

theory (Peak-over-Threshold method) focuses on the distribution over a threshold where > .

The extreme value theory provides insights of extreme cases – there exists a distribution that fits the conditional probability of extreme cases. There are several applications of EVT; we will use the method of Peak-over-Threshold, meaning we specially observe distribution over a pre-decided high threshold. In figure 1., if we would like to know the probability of v, instead of direct unconditional probability, we look into conditional probability and then manipulate it into unconditional probability. Firstly, the conditional probability of the shadow area in figure 1. F y can be written as

= _{1 −}+ − . 3.1

Under the condition of v > . As Gnedenko demonstrated in 1943, this conditional probability F y has a distribution which converges to a generalized Pareto distribution (GPD) where u is large. Generalized Pareto distribution is written as

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The coefficient ξ is the tail index describing how thick the tail is, commonly falling in the range 0.1-0.4 for financial data. To provide a comparison, ξ of normal distribution is zero. The second coefficient β is the scale parameter. These two inputs must be estimated unlike that the threshold can be selected. Then we compute the unconditional probability of variable x exceeding threshold u by combining equation 1 and 2.

> = = + −

= 1 − ∙ $%,' −

And thus, > = ∙ 41 + ( ∙ −_{) 5} * %⁄ . 3.3

where 6₆7 = 1 − is observed from empirical data, and coefficients ξ and β are to be estimated by maximum likelihood estimation (MLE). Equation 3.3 is the solution for extreme distribution by extreme value theory.

By definition, VaR is measured by solving equation 89: = ;∗ , where ;∗ is the assumed VaR confidence level 99%. For extreme value theory, this is equivalent to solving 89: = ;∗ = 1 − > . The result of x will be the VaR we are after. Thus

;∗_{= 1 −} _{∙ 41 + ( ∙}89: −

) 5

* %⁄

To write as an equation for VaR, this is 89: = +'_%∙ =>67

6 ∙ 1 − ;∗ ? %

− 1@ (eq. 3.4)

Straight forward as equation 3.4, extreme value theory suggests that once the threshold is decided and the generalized Pareto distribution of tails are estimated by MLE, VaR is ready to output.

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The choice of threshold u is built in for few statistics software along with the GPD estimation. However study shows that as long as the threshold is high to a point, it is sufficient to estimate coefficients of tail distribution without unacceptable variation from truth. The point around 95% of the full distribution is commonly believed a fine selection. This thesis decides the threshold by roughly 95% (the 117th in a 2348 data-point sample). Then the coefficients ξ and β are to be estimated upon given information. We build the MLE material by GPD’s probability density function (logarithm presentation). The coefficients are decided when the function is maximized: ∑ B_'*∙ C1 + ( ∙DE ' F * %⁄ G 67 HI* . 3.5

By this function, the _H below the threshold is excluded from GPD that we are building. The sample will be resized to K = 1,2, . . , and this is how extreme value theory approach focuses on tails. One thing to note is that the estimation is not applicable for both sides of tails. This is because for one, for each estimation we only take information of one tail into account and for the other, extreme value theory does not assume the full data scope distribution thus we cannot expect one tail is able to explain the other tail.

The thesis includes other classical approaches of VaR measurement for comparison. These are Historical Simulation, Historical Simulation with ARMA Forecast, Delta Normal, Monte Carlo Simulation, and GARCH. The results will be assessed by back testing in section 4.2; not only a simple test on numbers of exceptions but also a test for model features (conditional coverage and independence) on hit sequence will be conducted.

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4. Empirical analysis

4.1 Data and descriptive statistics

As discussed in section 1 the importance of observation on commodity variation and in section 3 the prospective improvement extreme value theory may bring to VaR measurement, the empirical assessment of VaR approaches will be conducted with datasets of spot price of six commodities, mainly in energy and in precious metal sectors. The price movements are shown in Figure A in appendix.

The energy sector includes oil and gas considering they are seen as main drivers in the energy market:

UK Brent Crude Oil. Brent oil is light crude oil extracted from the North Sea and nowadays commonly used as a price reference for crude oil transaction globally. The price is quoted as US dollar per barrel ($/BBL).

Henry Hub Natural Gas. The price of Henry Hub gas is considered a representative of North America market. Price is quoted as US dollar per millions of British thermal units ($/MMBTU).

The precious metal sector includes gold, silver, aluminum, and copper:

London Bullion Market Gold. The data is again representative as the market in which it is traded, LBM, is the largest OTC trading center of gold and silver in the world. Gold is selected not only for being an indicator among commodities but also by its unique role as a popular asset to hold and investment tool. Gold price is quoted as US dollar per troy ounce ($/Troy OZ).

Handy & Harman Silver. Silver is selected for its close relationship of price movement with gold as shown in Figure A in appendix. The Handy and Harman silver price was firstly served as quote from London to American demand in late

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19th century and now only as the lowest price the Handy & Harman cooperation could acquire; yet for the company’s position as leading supplier, its price is taken world widely as a reference. The price is US dollar per troy ounce quoted ($/Troy OZ).

London Metal Exchange Aluminum Alloy. Apart from gold and silver, the thesis also selects few major precious metal commodities of which Aluminum is included. The LME market Price is quoted as US dollar per metric ton ($/MT). London Metal Exchange Copper (Grade A). The Copper selected is copper cathode, also named as grade A, which is high purity. LME is the largest trade center for precious metal other than gold and silver and copper cathode is quoted as US dollar per metric ton ($/MT).

The scope of raw data input ranges from 2000 to 2010 in daily spot price. We choose spot price instead of derivatives price to simplify the analysis. The factors to derivatives price movements may be so complex that shift our focus on measurement. We obtained time series data through Datastream database in the form of price and it is processed as log return before measuring VaR. The source includes Thomson Reuter and N.Y. exchange.

The dataset is segregated into two periods: in-sample period from 2000 to 2008 for constructing VaR models (2348 days) and out-of-sample period 2009 to 2010 for model fitness assessment (522 days). This is decided based on the common price movement among selected commodities. Most commodities selected show upward trend throughout the decade and especially a boom in the period of 2006 – 2008. Specifically, the boom ends around the first half of 2008 and a sharp drop follows

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after price peak at mid-2008. At the end of 2008 price is generally at trough and it bounced back in 2009, recovering in rapid speed onwards. Figure B in Appendix demonstrates the returns movements, and it is obvious that market was rather volatile around the start of the year 2009, though gas seems to have one year lag. Thus sample period is designed in such way so that the crisis time is considered and it is made possible to put emphasis on these volatile times.

4.2 Empirical result

4.2.1 Implementing empirical research

Each Value-at-Risk approach processes in-sample data in different way cooperated with its characteristics, and coordinately derives a measurement on out-of-sample price deviation (return). Below describes the steps they follow; the results are analyzed in next section.

Stage 1. Model in-sample data and window-shuffle afterwards

For historical simulation, this is to construct scenarios; each return from the previous trading day is taken as a deviation scenario, and 2348 scenarios are constructed as a deviation model for each data point in out-of-sample period. This involves shuffling, where the current out-of-sample data point will be included in the scenario sample for the next out-of-sample data point.

For HSAF approach, in stage 1 we model the in-sample return series using an AR model. AR model takes into account the clustering and dependence of the time series itself, that is, autocorrelation. The estimation of AR Models by EViews can be found

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in Appendix D.1. The AR models used by each commodity are shown in table 4.1 and the Ljung-Box test (Q-Statistics) is performed accordingly to examine whether the residual autocorrelation is dismissed through AR modeling. Ideally, after modeling we expect insignificant p-values from Ljung-Box test, indicating the disappearance of autocorrelation. The AR return models eliminate autocorrelation rather successfully in oil, gold, and silver. The residual autocorrelations remain significant after AR modeling for the other three commodities. The relatively poor modeling may be because the data themselves were significantly autocorrelated throughout 36 lags2; thus selection of a slightly less significant lag (for instance, 0.2% significance) due to limitation would not make a good AR model accordingly.

Table 4.1 Ljung-Box test (Q-Statistics) for the ARMA model with selected lags. Autoregressive

moving average model consider autocorrelation in the data; with proper selections of lags, it captures the effect of clustering. The in-sample returns are modeled by ARMA which the lags are pre-selected by correlogram and Q-statistics on autocorrelation. Q-statistics reveals the level of autocorrelation within the assigned lags and in this table this is presented as p-value, meaning the possibility of autocorrelation dismissed. Ideally after AR modeling, the p-value should be larger than 5% significant level. However the AR model for Gas, Aluminum, and Copper seem not to fit. This may be due to its lack of valid ground to apply ARMA model in the first place. (The first autocorrelation inspection did not show a valid lag for these commodities)

For delta normal approach, since the in-sample distribution is assumed to be normal, we compute the missing numbers in stage one – mean and standard deviation, from 2348 sample points for every day in out-of-sample period. Shuffling is involved as well in delta normal approach.

2

The limitation of EViews correlogram is 36 lags. Therefore estimation of autocorrelation level for more lags is not accessible.

Oil Gas Gold Silver Aluminum Copper

AR Model AR(5) AR(2) AR(17) AR(11) AR(1) AR(1)

Q(12) 4.7% 1.0% 29.5% 33.4% 0.0% 0.0%

Q(24) 12.7% 0.4% 10.1% 11.3% 0.0% 0.0%

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For Monte Carlo simulation we do not make any assumptions about the underlying distribution. Instead, by the outputted sample mean and variance, we simulate 10,000 times (i) with a random factor (Zi) inserted. See equation 4.1. For each out-of-sample data point, the simulated 10,000 returns form a model. The code we build for such model using Matlab is shown in Appendix D.2.

LKM 9N O H = K L P M9 QOP ; 9 K KN RH ∗ LN. O K9NK − M 9 ,

Sℎ RH KL 9 9 O M M NS 0 N 1 (eq. 4.1)

For the case of the GARCH model, we model variance instead of return, assuming the relation of which is as equation 4.2. The table 4.2 shows the estimation of coefficients for Variance GARCH model by EViews. (Also shown in Appendix D.3) Out-of-sample variance is then forecasted.

89 K9 Q V = W*+ WX∗ : N V *X+ WY∗ 89 K9 Q V * (eq. 4.2)

Table 4.2 Coefficients estimated for GARCH model by EViews. In-sample returns are modeled by

GARCH. Variance GARCH model combine the effect of past return and past variance and the future is compute by Variance_` = C_*+ C_X∗ Return_{` *}X+ C_Y∗ Variance_{` *} which the coefficients can be estimated by statistics software.

For the approach using extreme value theory, we build a model only on distribution tail. Following the steps in section 3, the threshold u and the coefficients of Generalized Pareto distribution (tail parameterξand scale parameter β) are estimated by maximum likelihood estimation. The thesis build this model using Excel Solver, and the VBA code is shown in Appendix D.4.

Coefficients Oil Gas Gold Silver Aluminum Copper C(1) 0.000015300 0.000046800 0.000001010 0.000000252 0.000005640 0.000001510 C(2) 0.053132000 0.115017538 0.042648501 0.037084000 0.134553526 0.052392625 C(3) 0.922786000 0.874603753 0.949439002 0.965718000 0.827325102 0.944365909

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Stage 2. Observe in-sample model and make adjusting factor for deriving VaR

Stage 2 is required only by HSAF approach. After modeling in-sample return, the error from in-sample forecast is obtained and processed into a distribution. We observe the 99th percentile of both tails (2348 data points); the percentile is to be used as an adjusting factor in quantifying VaR in next stage. Again, shuffling involves in the stage.

Stage 3. Quantification of Value-at-Risk

In stage 3, the Value-at-Risk of the out-of-sample period is quantified. Given the models built in stage 1 and adjusting factor in stage 2, each approach computes in different ways in accordance to the assigned likelihood (that is, 99% in this thesis.)

Historical simulation derives VaR by searching for 99th percentile from the constructed return scenarios (and 1th percentile for negative VaR). HSAF approach makes use of the in-sample-built AR model to forecast the out-of-sample return, and it corrects the forecast into VaR using the 99th/1th percentile from the previous forecast errors which is the adjusting factor derived in stage 2. Delta Normal approach derives VaR using the formula

89: = c ∙ d * _{; − e (eq. 4.3)}

with the required statistical information extracted in stage 1. Monte Carlo takes similar action as HS approach except it searches for 99th/1th percentile on the 10,000 times of simulation. The GARCH approach also makes use of the formula 4.3 to derive VaR with volatility which is modeled in stage 1. The extreme value VaR approach constructs VaR using the estimated distribution and equation 3.4.

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After model building and VaR measurement, the actual and the estimated will be put together in order for investigation on fitness, which will be introduced in the next section. The visualized VaR measurement of each approach is presented in Figures C.1 to C.6 in Appendix. The actual return and VaR are shown in the same figure and thus we can study the appearance of exceptions. Each VaR approach also reveal its characteristics and emphasis of modeling, for instance GARCH shows dynamic VaR paths due to its assumption of variable volatility.

4.2.2 Back Testing

To assess the fitness of VaR models, we introduce back-testing here. Whenever the VaR is hit by the actual return, an exception is marked. According to the assumed likelihood level 99%, there are no more than 5 exceptions (522 days * 0.01 = 5.22 days). However the real world is not perfect, especially at the crisis times; models deliver more exceptions than expected from time to time. Therefore we use back testing to assess whether the number of exceptions is acceptable or it is an exceptional case (unacceptable).

Back-test addresses the probability of the model having the observed number or more of exceptions. It is based on binomial distribution, where the result of each out-of-sample data point is indeed shown exactly in two cases – non-exception and exception. Recalling the statistics of binomial distribution, the mean of such is ; and the standard deviation is f ; ∗ 1 − ; , where p is the expected probability of exceptions (in this case, 1 – assumed likelihood level 99%). The number of exceptions will be normalized as g = hijklmkn kDokpVHh6j q,r skt6

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distribution can be applied3. The cumulated probability of having fewer exceptions than observed is then d g , and thus the probability (p-value) of the model marking such number of exceptions will be 1 − d g . Statistically, when the outcome p-value is less than 5%, the model/hypothesis is rejected; this means that the outcome only happens by less than 5% chances based on hypothesis, which is viewed as a significantly exceptional case in statistics. The models will not be rejected if the number of exceptions is no more than nine (sampling 522 data points).

Table 4.3 reveals the results across commodities and VaR approaches. The VaR approaches are in order of performance among six commodities. All cases are rejected exception one; the result suggests that all models we test are not considered fit statistically. We can still interpret the performance depending on the numbers of exceptions. Yet if we take consider mentioned volatile period out of scope, the exception record is rather quiet throughout the sample period. The impact makes the possible reasons for rejections worth to discuss. From Figure B in Appendix, it is obvious that there are some periods that the returns of many commodities are quite fluctuate and many of the exceptions happen during this period. A common period is in early 2009. The fluctuation is believed due to the pop of 2006-2008 commodity bubble, or of the fast growing trend in commodity from 2000 if to trace back from the start. The upward trend from 2000 is much related to the nature change of worldwide trading. When the condition of global market changes, commodity market is largely affected as well. For Aluminum and Copper, similar situation also appears around May 2010, with lower visibility in Copper. This can be suspected a link to Gold’s

3_{During the normalization, the observed number of exceptions will be subtracted by 0.5 or 1 due to}

our definition of the final outcome probability as in having the observed number or more. Thus the case of exactly observed exceptions should be excluded from the cumulative probability.

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variation during 2009 year-end, possibly caused by gold purchases by Central Banks and further correction to the high price. The volatile period for Gas, different from other commodities, is in the third quarter of 2009.

Table 4.3 Back-testing on the numbers of exceptions. Now that the VaR for out-of-sample return are

estimated, exceptions will be marked whenever the actual return exceeds VaR measurement. Fitness assessment will be conducted via back test on the numbers of exceptions. By binomial distribution, the numbers of exceptions will be normalized z = L O Q ;NK L − 0,5 − e c⁄ and the probability of having fewer exceptions than observed is then N z , and thus the probability (p-value) of the model marking such number of exceptions will be 1 − N z . The back-test p-value is presented in third column. The model is accepted as fit when p-value is larger than 5%. In this case, only GARCH succeeds in estimating VaR for Oil. However it is also possible to assess base on the numbers of exceptions. In this sense, the general order from good VaR estimation to worse ones is GARCH, extreme value theory, HSAF, historical simulation, Monte Carlo simulation, and then Delta Normal.

Despite the common rejections, GARCH performs the best in terms of numbers of exceptions. In a difficult time scope to measure VaR as mentioned in the previous paragraph, GARCH still delivers one non-rejected result, insignificant 27% of p-value for oil. This may indicate that updating variance (Unique feature of GARCH) plays a

VaR

Approach Actl. Excp Actl. P Back test Reject/N Actl. Excp Actl. P Back test Reject/N

GARCH 7 1.3% 27.0% Not Rej. 12 2.3% 0.7% Reject

EVT 11 2.1% 1.8% Reject 12 2.3% 0.7% Reject

HSAF 12 2.3% 0.7% Reject 10 1.9% 4.0% Reject

HS 12 2.3% 0.7% Reject 12 2.3% 0.7% Reject

Monte Carlo 20 3.8% 0.0% Reject 15 2.9% 0.0% Reject

Delta Normal 19 3.6% 0.0% Reject 16 3.1% 0.0% Reject

VaR

GARCH 12 2.3% 0.7% Reject 13 2.5% 0.3% Reject

HS 10 1.9% 4.0% Reject 12 2.3% 0.7% Reject

VaR

GARCH 24 4.6% 0.0% Reject 11 2.1% 1.8% Reject

HS 31 5.9% 0.0% Reject 15 2.9% 0.0% Reject

Oil Gas

Gold

Silver

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role in improving VaR measurement. We would also expect this outcome as variable volatility is reality and volatility must differ much in a time scope as long as ten years. Behind GARCH, it follows the group of extreme value theory, historical simulation, and HSAF approach. It is hard to give a firm statement which is a better measure by only numbers of exceptions, but the result of VaR using extreme value theory does looks slightly fitter than other two. Monte Carlo simulation and Delta Normal approach are listed the last two; their relatively poor outcome may be a result from that they both request input of full distribution assumption. There is risk to do so as the assumption can be wrong and building model on an incorrect set of assumption would shift more from truth than building a rather inflexible model without simulation basing on certain assumption. Comparing to Historical Simulation and HSAF, Monte Carlo simulation and Delta Normal approach model worse; it is reasonable to suspect that a distribution of ten-year time scope leads to incorrect assumption as part of the information can be outdated and irrelevant. In the same sense, we would also have expected GARCH and extreme value theory capture variation better for that they do not only mitigate mentioned risk but also create flexibility (distributions are estimated in the process but not assumed).

4.2.3 Hit Sequence Assessment

Back-testing on the numbers of exceptions is the major exam in evaluating VaR estimations. However once all models are rejected as in our case, analysis of the already rejected numbers of exceptions seems to lose its solid ground. One can easily challenge the result with the fact that most exceptions occurs in certain period and the fitness of model may differ if we took those periods out of consideration. Thus to

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provide another view of model fitness and to take clustering during those difficult periods into examination, this thesis also conduct a back testing on Conditional Coverage and Independence using the Hit Sequence4 as test objective proposed by Christoffersen in 1998.

First, for each model on each commodity, a dummy variable is defined and the hit sequence is formed as time series:

wVx* = = 1,_0, _{KP 89: KL N ℎKN Nℎ 9QN 9 N}KP Nℎ 9QN 9 N ℎKNL 89: y

In other words, an exception marks I_`x* as one or else zero. The number of exceptions is as T_|}` = ∑ I~_`I* _` , and the percentage of exceptions P(actual return hits or over VaR) is equivalent to T_|}`⁄ . This percentage would be the same as p (1-VaR T confidence level 99%) under the assumption that model is accurately estimated.

V 9QN 9 N ℎKNL 89: = •€HV⁄• = ; (eq. 4.4)

The assumption, if correct, indicates both unconditional coverage and independence. Equation 4.4 demonstrates that the probability to mark an exception of any data point should be the same (p) given any information at any time. Thus we expect no relation between data points - independence. Whether it is an exception today does not affect tomorrow’s probability to be an exception. This can be assessed by a regression of wVx*− ; on wV. Under hypothesis of independence as presented below, the coefficient estimated would equals to zero:

•6 ‚‚: wVx*= 1|wV = 0 = wVx*= 1|wV = 1

The conditional coverage further indicates that the probability of a data point marking an exception is exactly p in accordance to the assumed VaR confidence level. This can

4

More details of back testing can refer to Christoffersen's book “Elements of Financial Risk Management”, chapter 13. (2012)

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be tested by a regression of w_Vx*− ; on a constant and conditional coverage expects the constant to be zero so that w_Vx*= ; would be valid assumption:

•6 ‚‚: wVx*= 1|wV = 0 = wVx*= 1|wV = 1 = ;

However, the above null hypothesis shows that the independence is to be considered before conditional coverage hypothesis can be valid. Otherwise the actual regressed probability level is only checked “on average” while the clustering (dependence) may involve. Thus the regression will be revised as

I`x*− p = bq+ b*∙ I`+ error`x*

to take both hypothesis into account. The hypothesis of Independence which can be assessed by T-test is then •_q*: _* = 0 , and that of Conditional Coverage which requires a joint F-test is •_qX: _q = _* = 0 . (Christoffersen, 2012, pp.308)

The result of such tests is shown in Table 4.3; VaR approaches are listed in order of general performance. Different from simple exception back-testing, the hit sequence back-testing enables us to further investigate the level of fitness in terms of outcome significance. T-statistic and F-statistic are presented and from Table 4.3 the result is clear and easy to analyze. Again, GARCH model delivers the best outcome among all and especially outstands in the conditional coverage. This is straight forward as GARCH itself features dynamic variance in VaR modeling while other approaches analyzed in this thesis assume a fixed volatility for entire sample period. GARCH’s real-life assumption, the updated volatility, empowers conditional coverage since the assumed VaR confidence level is to capture the real-life return variation. Extreme Value Theory follows as the second best fitting model; this may be mostly due to the fact that EVT does not model the whole sample but only tails. Direct observation on

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tails eliminates irrelevant noise thus derives a focused and fitter distribution for VaR. However, the issue of unconditional coverage is not resolved; the F-test passed (coefficients are not rejected to be zero, statistically describing) only for gold among six commodities. Yet regarding all models assuming fixed volatility, EVT makes the most conditional one. The order of VaR estimation quality in the hit sequence back test is generally aligned with that in the simple exception back test. After EVT approach come Historical Simulation, Historical Simulation with ARMA Forecast, Monte Carlo Simulation, and the last Delta Normal approach. As the order may suggest, it is unwise to assuming normal distribution on financial data, especially VaR which tails are examined. All in all, we can see the value added using extreme value theory to estimate VaR, as it does improve preciseness comparing most of the approaches we assessed.

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Table 4.3 Back-testing on Conditional Coverage and Independence based on the hit sequence.

Simple back test on numbers of exceptions did not work out for assessment (see table 4.2), and thus back-test on hit sequence is further conducted. On top of the numbers of exceptions, the time series, Hit Sequence, is formed for exception to be marked (w_V= 1 or not w_V= 0 . This test will examine coverage and independence of models. Conditional coverage is reached when probability to mark an exception at a data point is exactly associated with pre-determined confidence level (1-p); the hypothesis is H_{‰ ŠŠ}: P I_`x*= 1|I_`= 0 = P I_`x*= 1|I_`= 1 = p . Independence means that the conditional coverage is valid regardless the result of other data points; the hypothesis is H‰ ŠŠ: P I`x*= 1|I`= 0 = P I`x*= 1|I`= 1 . The regression goes I`x*− p = bq+ b*∙ I`+

error`x*. Conditional coverage is said to be featured if H_qX: b_q= b_*= 0 is not rejected by F-joint test (last two columns); independence is featured when H_q*: b_*= 0 is not rejected by t-test (first two columns). The hit sequence back test does provide a clearer insight. GARCH performs the best again, and extreme value theory follows. The order of performance basically is the same as simple back test: GARCH, extreme value theory, historical simulation, HSAF, Monte Carlo, and Delta Normal approach.

5. Conclusion

The aim of this thesis is to study and assess the Value-at-Risk approach using extreme value theory. Having sensed EVT’s potential to measure risks and the need of extreme events risk management in commodity markets, the thesis will use the data from commodity market during 2000-2010, the so called commodity bubble period, to

Hit Test t(b1=0) p-value F(b0,b1=0) p-value t(b1=0) p-value F(b0,b1=0) p-value

GARCH -0.31 75.7% 0.28 75.6% -0.54 59.1% 2.10 12.3% EVT 3.79 0.0% 8.78 0.0% 10.03 0.0% 52.63 0.0% HS 5.44 0.0% 16.88 0.0% 10.03 0.0% 52.63 0.0% HSAF 5.44 0.0% 16.88 0.0% 4.27 0.0% 10.33 0.0% Monte Carlo 3.88 0.0% 13.37 0.0% 11.51 0.0% 70.32 0.0% Delta Normal 4.18 0.0% 14.10 0.0% 10.53 0.0% 60.02 0.0%

GARCH -0.54 59.1% 2.10 12.3% 1.22 22.5% 3.13 4.5% EVT -0.45 65.6% 1.27 28.3% 3.79 0.0% 8.78 0.0% HS -0.45 65.6% 1.27 28.3% 3.39 0.1% 7.74 0.1% HSAF -0.54 59.1% 2.10 12.3% 3.79 0.0% 8.78 0.0% Monte Carlo 0.28 78.3% 5.70 0.4% 2.33 2.0% 12.04 0.0% Delta Normal 0.17 86.2% 6.18 0.2% 2.16 3.2% 12.17 0.0%

GARCH -1.10 27.1% 8.31 0.0% -0.49 62.3% 1.67 18.9% EVT 1.84 6.7% 12.58 0.0% 2.23 2.6% 6.25 0.2% HS 1.69 9.2% 12.85 0.0% 2.47 1.4% 6.36 0.2% HSAF 2.48 1.3% 14.57 0.0% 2.47 1.4% 6.36 0.2% Monte Carlo 4.03 0.0% 31.52 0.0% 2.90 0.4% 16.87 0.0% Delta Normal 3.16 0.2% 27.51 0.0% 2.73 0.7% 16.90 0.0% Oil Gas Gold Silver Aluminium Copper

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build our VaR model on and highlight the 2008-2009 volatile period. Other conventional VaR approaches with different characteristics are conducted in the thesis as well for comparison; these are historical simulation, HSAF, Delta Normal approach, Monte Carlo simulation, and GARCH model.

The outcome of assessment shows that extreme value theory does provide a relatively good risk measure, despite that all models are rejected because of certain volatile periods. Among six approaches we examine, the VaR estimation using EVT ranks the second.

With the help of back testing, we are given chances to state the fitness of VaR approaches whether the phenomena of exceptions is exceptional - in another word, whether the VaR model is rejected. However, the first back test conducted on the numbers of exceptions does not provide much information. The p-value (significance of inaccurate measurement; the lower, the more significant and more inaccurate) shows that every models on every commodities are rejected, except one occasion. Although from the numbers of exceptions we can tell the order of fitness level of VaR approaches, the statement would not be solid-grounded as they are all rejected. On top of that, frequent exceptions are spotted in certain period; this may loosen the validity to assess upon these exceptions happened in such period. For this part of back test, the performance of VaR approaches is in order of: GARCH, extreme value theory, HSAF, HS, Monte Carlo, and the last Normal Delta.

To provide additional proof for assessment, the thesis introduces another back test based on hit sequence. The exceptions now form a time series instead of a count of

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incidences. The back test examines two features: conditional coverage (confidence level is matched) and independence. The result of the second back test does give us more insights to fitness and allows us to state fitness with more ground. The order of performance from this part is similar to the first back test: GARCH, extreme value theory, HS, HSAF, Monte Carlo, and the last Normal Delta. From both tests, we can say extreme value theory, although is not the most fitted and is rejected, does provide a relatively fitter VaR measurement.

Regarding the difficulty faced in determining model fitness, common rejections are a matter of concern. This raises the question: Is VaR at all suitable to be a risk indicator for commodity? Are commodity markets so special that VaR is not applicable? The uncommon volatile period should be taken into account when trying to answer these questions. Therefore a less solid but positive answer is that Value-at-Risk is not an invalid indicator of risk management for commodity. VaR is not expected to perform better on other market (i.e. equity) during those volatile crisis times and downturn shocks. If to look at most of the time outside of those periods in our scope, VaR is seldom hit by the actual. Thus it can be argued that, Value-at-Risk can be a useful tool of risk monitoring for the case of commodity markets.

Although difficulties are encountered during measurement, for instance in-sample volatile period, issues can be fixed by considering adding weights to such period or extending the model to consider correlation among commodities and their co-movements.5 It is understandable that extreme value theory is one of the commonly used approaches in practice. The risk management departments from both

5

Aloui, Aïssa, Hammoudeh, and Nguyen’s work in 2014 made extension of co-movements into GARCH VaR measurement.

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public and private sector now appreciate risk measures of good quality, especially the ones with capability dealing with extreme losses; and extreme value theory is one of them.

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References

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Christoffersen P.F., 1998, "Evaluating interval forecasts," International economic review (1998): 841-862

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Danielsson J., 2000, "(Un)Conditionalityof Risk Forecasting," Mimeo, London School of Economics

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Energy Economics Vol. 30, 3156-3171

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value theory in estimating value-at-risk: empirical evidence from Asian emerging markets," Review of Quantitative Finance and Accounting, 2012, Vol.39 (4), 447-468

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Jorion P., 1997, "Value at Risk: The New Benchmark for Controlling Market Risk," Irwin Professional Publishing: Illinois

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using Extreme Value Theory," Department Mathematik, ETH Zürich

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APPENDIX

A. Figure: Price movement of selected commodities for the period 2000-2010

The figures show the spot price movement during the whole sample period 2000-2010; the commodities include UK Brent Crude Oil ($/BBL), Henry Hub Natural Gas ($/MMBTU), London Bullion Market Gold ($/Troy OZ), Handy & Harman Silver ($/Troy OZ), London Metal Exchange Aluminum Alloy ($/MT), and London Metal Exchange Copper - Grade A ($/MT). Source from the database Datastream.

0 20 40 60 80 100 120 140 160 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 Brent Crude Oil

0 4 8 12 16 20 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 Henry Hub Natural Gas

0 200 400 600 800 1000 1200 1400 1600 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 LBM Gold 0 500 1000 1500 2000 2500 3000 3500 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 Handy & Harman Silver

500 1000 1500 2000 2500 3000 3500 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 LME Aluminum Alloy

0 2000 4000 6000 8000 10000 12000 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 LME Copper (Grade A)

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B. Figure: Return behavior of selected commodities for the period 2000-2010

The figures show the daily logarithm return behavior of spot prices during the whole sample period 2000-2010; the commodities include UK Brent Crude Oil, Henry Hub Natural Gas, London Bullion Market Gold, Handy & Harman Silver, London Metal Exchange Aluminum Alloy, and London Metal Exchange Copper - Grade A. The logarithm return is derived from price using formula: Return = ln Price_`⁄Price_{` *}

-20% -15% -10% -5% 0% 5% 10% 15% 20% 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 Brent Crude Oil

-60% -40% -20% 0% 20% 40% 60% 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 Henry Hub Natural Gas

-8% -6% -4% -2% 0% 2% 4% 6% 8% 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 LBM Gold -15% -10% -5% 0% 5% 10% 15% 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 Handy & Harman Silver

-15% -10% -5% 0% 5% 10% 15% 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 LME Aluminum Alloy

-15% -10% -5% 0% 5% 10% 15% 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 LME Copper (Grade A)

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C. Visualized Value-at-Risk Measurement

The out-of-sample Value-at-Risk is visualized in below figures and is presented in terms of returns. Six commodities are modeled by six VaR approaches. The deviate data lines in the middle are actual daily log-returns; the (red) upper data lines are estimated 99% positive/gain VaR and the (green) lower data lines are estimated 99% negative/loss VaR. Whenever the actual return exceeds the estimated VaR, it is marked as a exception, and the fewer the exceptions exist the fitter the VaR approach models, to simply demonstrate the idea of back testing assessment.

C.1 Figure: VaR measurement by Historical Simulation

-15% -10% -5% 0% 5% 10% 15% 20%

Brent Crude Oil

-25% -20% -15% -10% -5% 0% 5% 10% 15% 20% 25%

Henry Hub Natural Gas

-6% -4% -2% 0% 2% 4% 6% LBM Gold

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-10% -8% -6% -4% -2% 0% 2% 4% 6% 8% 10%

Handy & Harman Silver

-10% -8% -6% -4% -2% 0% 2% 4% 6% 8% 10%

LME Aluminum Alloy

-8% -6% -4% -2% 0% 2% 4% 6% 8% 10%

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C.2 Figure: VaR measurement by Historical Simulation with ARMA Forecast

HSAF involves ARMA modeling on out-of-sample return and the forecast is shown in the figure as well. The (red) upper data lines are estimated 99% positive/gain VaR and the (green) lower data lines are estimated 99% negative/loss VaR; the (purple) relatively calm data line in between positive and negative is the ARMA forecast of out-of-sample returns.

-15% -10% -5% 0% 5% 10% 15% 20%

Brent Crude Oil

-30% -20% -10% 0% 10% 20% 30%

-6% -4% -2% 0% 2% 4% 6% 8% LBM Gold

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-10% -8% -6% -4% -2% 0% 2% 4% 6% 8% 10%

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C.3 Figure: VaR measurement by Delta Normal Approach -15% -10% -5% 0% 5% 10% 15% 20%

Brent Crude Oik

-30% -20% -10% 0% 10% 20% 30%

-6% -4% -2% 0% 2% 4% 6% 8% LBM Gold

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-10% -8% -6% -4% -2% 0% 2% 4% 6% 8% 10%

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C.4 Figure: VaR measurement by Monte Carlo Simulation -15% -10% -5% 0% 5% 10% 15% 20%

Brent Crude Oil

-30% -20% -10% 0% 10% 20% 30%

-6% -4% -2% 0% 2% 4% 6% 8% LBM Gold

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-10% -8% -6% -4% -2% 0% 2% 4% 6% 8% 10%

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C.5 Figure: VaR measurement by GARCH Model -15% -10% -5% 0% 5% 10% 15% 20%

Brent Crude Oil

-30% -20% -10% 0% 10% 20% 30%

-6% -4% -2% 0% 2% 4% 6% 8% LBM Gold

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-15% -10% -5% 0% 5% 10% 15%

-10% -8% -6% -4% -2% 0% 2% 4% 6% 8% 10%

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C.6 Figure: VaR measurement by Extreme Value Theory Approach -15% -10% -5% 0% 5% 10% 15% 20%

Brent Crude Oil

-30% -20% -10% 0% 10% 20% 30%

-6% -4% -2% 0% 2% 4% 6% 8% LBM Gold

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-10% -8% -6% -4% -2% 0% 2% 4% 6% 8% 10%

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D. Estimations during VaR measurement process

This section includes extra materials for reference that are involved during the process of VaR measurement. This could be estimations by Eviews or Matlab.

D.1 HSAF approach – Estimation of ARMA model for return

HSAF approach requires return modeling in the process. The ARMA model takes information from in-sample data and will be applied on out-of-sample. The estimation is done by EViews. The lags to use for AR modeling is preselected from the Q-statistics in correlogram.

D.1.e Table: LME Aluminum Alloy - AR(1) Model D.1.f Table: LME Copper (Grade A) - AR(1) Model D.1.a Table: Brent Crude Oil - AR(5) Model D.1.b Table: Henry Hub Natural Gas - AR(2) Model

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D.2 Monte Carlo Simulation – Simulation by Matlab

After obtaining the distribution statistics assumption, simulation is conducted in Matlab. Below is the Matlab script generating VaR for both tails.

%% montecarlo simulation - return

filename = 'xxxx.xlsx'; % variable: the input excel file

sheet = 7; % variable (number): the input worksheet

% input distribution assumption

mu = xlsread(filename, sheet, 'F2355:F2876'); % mean of 2348 samples

sigma = xlsread(filename, sheet, 'G2355:G2876'); % sigma of 2348 samples

posVaR = zeros(522,1); negVaR = zeros(522,1);

% simulate for each out-of-sample data point

for i = 1:522

z = rand(10000,1); % random number generator

r = norminv(z).*sigma(i,1) - mu(i,1); % simulate 10,000 paths

posVaR(i,1) = prctile(r,99); negVaR(i,1) = prctile(r,1);

end

xlswrite(filename, posVaR, sheet, 'D2355'); % output right tail VaR

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D.3 GARCH approach – Model estimation by EViews

GARCH approach conducts modeling on variance for in-sample data and applies it on out-of-sample period. The model is built as

Variance`= C* + CX∗ Return` *X+ CY∗ Variance` *

D.3.a Table: Brent Crude Oil D.3.b Table: Henry Hub Natural Gas

D.3.c Table: LBM Gold D.3.d Table: Handy& Harman Silver

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D.4 Extreme Value Theory – Coefficients estimated by MLE using VBA Solver

As described in section 3 (also see GPD tail distribution function below eq. 3.2), the tail parameter ξ and scale parameter β are to be estimated (using MLE function eq. 3.5) then VaR can be measured. This estimation is conducted in Excel VBA using Solver and it is written by parts. Below scripts only show estimation of one tail; the other tail is with the same process as long as the input data is inversely manipulated. The workbook is designed as , where dataset-x contains all the dataset, EVT estimate tab performs estimation using solver, and results tab marks the coefficients of each out-of-sample data point for further VaR derivation using equation 3.4.

Gξ,β y = 1 − 1 + ξ ∙ Ž β * ξ⁄_{eq. 3.2 ;} ∑ ln B*_β∙ C1 + ξ ∙•‘ β F * ξ⁄ G ‰’ }I* eq. 3.5 ; VaR = u +β ξ∙ => ‰’ ‰ ∙ 1 − p∗ ? ξ − 1@ eq. 3.4

D.4.a Import past information: past return

Firstly the past return is taken in and sorted descending according to the level of price deviation.

Sub InputNeg()

Application.ScreenUpdating = False

i = Sheets("EVT estimate").Range("I3").Value

'counter: finished out-of sample data points

Sheets("dataset-x").Activate

Range("E7:E2354").Offset(i, 0).Select 'Copy return of 2348 samples

Selection.Copy

Sheets("EVT estimate").Activate

Sheets("EVT estimate").Range("A9").Select

Selection.PasteSpecial Paste:=xlPasteValues 'paste samples and sort

Application.CutCopyMode = False

ActiveWorkbook.Worksheets("EVT estimate").Sort.SortFields.Clear ActiveWorkbook.Worksheets("EVT estimate").Sort.SortFields.Add

Key:=Range("A9:A2356"), SortOn:=xlSortOnValues, Order:=xlDescending With ActiveWorkbook.Worksheets("EVT estimate").Sort

.SetRange Range("A8:A2356") .Header = xlYes .Orientation = xlTopToBottom .SortMethod = xlPinYin .Apply End With Sheets("EVT estimate").Range("G3").Select End Sub

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D.4.b Coefficients estimation using Solver

The Solver conducts a maximum likelihood estimation, where sum of likelihood is maximized in below table. (The function of individual likelihood is “=IF(return>u, LN((1/beta)*((1+(xi*(return-u)/beta)))^(-1/xi-1)),0)” ). The threshold u in the table is put as “equals to 117th highest sorted price deviation” which includes around 5% of original data (counted in n_u cell) into tail distribution. The likelihood is subjected to xi and beta in the table; xi and beta is updated for each data point.

Sub Solve() Application.ScreenUpdating = False Application.Run "SOLVER.XLAM!Solver.Solver2.Auto_open" Sheets("EVT estimate").Activate SolverReset

SolverOK SetCell:=Range("G3"), MaxMinVal:=1, ByChange:=Range("B3:C3") SolverSolve UserFinish:=True

SolverFinish KeepFinal:=1 End Sub

D.4.c Record coefficients

After coefficients are estimated, below script export them to the sheet results. These data will be used in calculation later for VaR.

Sub OutputNeg()

Application.ScreenUpdating = False

i = Sheets("EVT estimate").Range("I3").Value

'Copy coefficients estimated

Sheets("EVT estimate").Range("B3:F3").Select Selection.Copy Sheets("results").Activate Range("C2:G2").Offset(i, 0).Select Selection.PasteSpecial Paste:=xlPasteValues Sheets("EVT estimate").Range("I3").Value = _

Sheets("EVT estimate").Range("I3").Value + 1 'move to next

Sheets("EVT estimate").Activate Range("G3").Select

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D.4.d Execution whole process

Above scripts are put in one macro and is called automatically. Per execution the estimation of full out-of-sample period will be conducted.

Sub Estimate() Application.ScreenUpdating = False kNeg = 0 Do While kNeg < 522 InputNeg Solve OutputNeg

kNeg = Sheets("EVT estimate").Range("I3").Value Loop

Estimating value-at-risk using extreme value theory : new evidence from commodity markets.

Amsterdam Business School

Master in International Finance

Estimating Value-at-Risk using Extreme Value Theory:

New Evidence from Commodity Markets

By

Tzu-Yun Katherine Lin

Supervised by

Dr. Chris Florackis (University of Liverpool)

September 2014

University of Amsterdam

Estimating Value-at-Risk using Extreme Value Theory:

New Evidence from Commodity Markets

TZU-YUN KATHERINE LIN

ABSTRACT

TABLE OF CONTENTS

1.

Introduction

2.

Literature Review

3.

Value-at-Risk Models

4.

Empirical analysis

5.

Conclusion

References

APPENDIX