How Low Can You Go?: Quantitative Risk Measures in Commodity Markets

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Johan Forsgren Uppsala University Department of Statistics Fall 2016

Supervisor: Lars Forsberg

How Low Can You Go?

Quantitative Risk Measures in Commodity Markets

Abstract

The volatility model approach to forecasting Value at Risk is complemented with modelling of Expected Shortfalls using an extreme value approach. Using three models from the GARCH family (GARCH, EGARCH and GJR- GARCH) and assuming two conditional distributions, normal Gaussian and Student t’s distribution, to make predictions of VaR, the forecasts are used as a threshold for assigning losses to the distribution tail. The Expected Shortfalls are estimated assuming that the violations of VaR follow the Generalized Pareto distribution, and the estimates are evaluated. The results indicate that the most efficient model for making predictions of VaR is the asymmetric GJR-GARCH, and that assuming the t distribution generates conservative forecasts. In conclusion there is evidence that the commodities are characterized by asymmetry and conditional normality. Since no comparison is made, the EVT approach can not be deemed to be either superior or inferior to standard approaches to Expected Shortfall modeling, although the data intensity of the method suggest that a standard approach may be preferable.

Key words: Value at Risk, Expected Shortfall, GARCH, EGARCH, Extreme Value Theory, GPD

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Acknowledgements

I am grateful for the invaluable advice and support from my supervisor, Dr. Lars Forsberg, without whom this thesis probably would be in accounting. Mr. Adam Gustafsson’s comments have also been of great importance, as well as the correctional read-throughs of Dr. Magnus Forsgren, Ms. Linnea Lantz and Ms. Marta Larsson.

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Table of Contents

1. Introduction ... 4

1.1 Purpose of the thesis ... 5

1.2 Earlier Research ... 5

2. Theoretical framework ... 6

2.1 Volatility models ... 6

2.1.1 ARCH ... 7

2.1.2 GARCH ... 7

2.1.3 Exponential GARCH ... 8

2.1.4 GJR-GARCH ... 9

2.2 Distributions ... 9

2.2.1 Gaussian Normal Distribution ... 9

2.2.2 Student t’s Distribution ... 10

2.3 Value at Risk ... 10

2.4 Evaluating Model Fit ... 11

2.4.1 Kupiec’s Test for Unconditional Coverage ... 12

2.4.2 Christoffersen’s Test for Independence ... 12

2.5 Extreme Value Theory ... 13

2.6 Expected Shortfall ... 14

2.7 Examining the Precision of Expected Shortfall Predictions ... 14

3. Data ... 16

4. Methodology ... 19

5. Results ... 20

5.1. Salmon Results ... 20

5.2 Lumber Results ... 21

5.3 Cocoa Results ... 21

6. Conclusions ... 22

6.1 Recommendations for further studies ... 23

6.2 Recommendations for Decision Makers ... 24

References ... 25

Appendix ... 28

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1. Introduction

The risky behavior that led to the 2008 financial crisis caused an increase in interest of risk management methods among both institutions as well as regulators. One of the most common measures of financial risk is Value at Risk (VaR), defined as the largest possible loss over some time horizon given a certain confidence level (Jorion 2007). Since financial returns are characterized by heteroscedasticity and volatility clustering, the modeling of VaR is efficiently performed using methods that allow the practitioner to model changes in conditional variance over time.

One of the first attempts to implement modelling of the conditional variance in a risk management context was made by investment bank JP Morgan in the late 1980’s. The banks risk man- agers began to model conditional variance using the RiskMetrics model (JP Morgan & Reuters 1996), with prominent results. The success of RiskMetrics caused an increased interest in the ARCH and GARCH family of models, introduced by Engle (1982) and Bollerslev (1986), as methods to forecasts conditional variance, making them popular in a risk management context.

As research of financial time series developed, the family increased, adding models like APARCH (Ding et al. 1993), EGARCH (Nelson 1991) and GJR-GARCH (Glosten et al. 1993).

The focus of research in financial time series also shifted to the effect of the assumed conditional distribution, as different results were obtained when assuming different distributions.

Some research has found support for the student t’s distribution being superior to the normal distribution (see for example Berggren and Folkelid 2014), although these results do not hold in all cases.

VaR does however not come without drawbacks. From a practitioner’s point of view, it is rea- sonable to study on losses that violate VaR and their expectation, known as the Expected Short- fall (ES). Expected Shortfall can therefore be used to answer the crucial question when large losses occur, how large are they? Since these large losses are located in the tail of the condi- tional distribution one can turn to Extreme Value Theory (EVT) in order to model their expected value. The usage of EVT in financial risk management is an increasing subject of interest since McNeil and Frey (2000) showed the advantages of applying two EVT methods, block maxima and peak over threshold, to risk management issues.

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1.1 Purpose of the thesis

The aim of this study is to compare predictions of VaR with 99% and 95% confidence, using three different volatility models: GARCH, EGARCH and GJR-GARCH. When forecasting the conditional volatility, the errors will be assumed to follow the normal and student t’s distribution conditionally. Using VaR as a threshold, the Expected Shortfalls will be modeled by fitting the Generalized Pareto distribution to the violations. The expectation is thereafter compared to the actual losses and evaluated using MAPE. The data used are the weekly spot prices and indexes of salmon, lumber and cocoa, ranging from 1^st of January 2004 to 31^st of December 2015.

1.2 Earlier Research

Previous research on equity and currency markets suggest that asymmetric models, like GJR- GARCH and EGARCH, tend to generate the most accurate forecasts of VaR (see for example So et al. 2005; Patton 2006). These results have been persistent in the case of commodities as well; Giot and Laurent (2003) showed that there is evidence for asymmetry and skewness in several commodity markets, including cocoa, oil and various industrial metals. Earlier research has suggested that the assumption of fat tailed conditional distributions tend to be more efficient when forecasting VaR for financial assets than the normal distribution (Bollerslev 1987; Hung et al. 2008), indicating that student t’s distribution is superior to the normal Gaussian distribution.

Extreme Value Theory’s applicability to financial risk management has risen in popularity over the past years. The seminal work by McNeil and Frey (2000) popularized the approach as they showed the advantages of EVT applications to VaR forecasting. Later research has shown mixed results due to the approach’s data intensity (Lindholm 2015). Using EVT only to model the expectation of the violations of VaR is however a somewhat unexplored area.

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2. Theoretical framework

A common definition of returns in financial time series is the log-returns, defined as in equation 1,

𝑟_" = 𝑙𝑛 𝑝_"

𝑝_"'( = 𝑙𝑛 𝑝_" − 𝑙𝑛 𝑝_"'( (1)

where 𝑟_" denotes the return between time t and t-‐‑1 and 𝑝_" is the price of some asset, in this study the spot price of some commodity, at time t. Following this transformation, the return can be further decomposed into the following process

𝑟_" = 𝜇 + 𝜀_" (2)

where 𝜇 is the unconditional mean and 𝜀_" is the error term. Assuming a zero unconditional mean (𝜇 = 0), leave the financial returns equal to the error term (𝑟_"= 𝜀_"). This assumption is examined in section 3 through descriptive statistics of the weekly returns. The error term is then assumed to be given as

𝜀_" = 𝜎_"𝑍_" (3)

with 𝜎_" being the conditional standard deviation of 𝜀_" given time t-‐‑1 and 𝑍_" is an independent identically distributed sequence with mean zero and variance one, 𝑍_"~𝑖𝑖𝑑 0, 1 . Following this, the error term 𝜀_" is independent identically distributed, with mean zero and variance 𝜎_"⁷ (𝜀_"~𝑖𝑖𝑑(0, 𝜎_"⁷)).

2.1 Volatility models

As a method to forecast the conditional volatility, the GARCH family of models is used. This study investigates the predictive power of three models: GARCH, GJR-GARCH and EGARCH. Each model is presented below including the ARCH model to give an introductory background of the theoretical framework of the family of models.

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2.1.1 ARCH

In order to model heteroscedasticity in British inflation, Engle (1982) derived the Auto Regres- sive Conditional Heteroscedasticity (ARCH) model. An ARCH (q) model forecasts the conditional variance as a function of squared previous error terms, with a general form as defined in equation 4:

𝜎_"⁷ = 𝛼_;+ 𝛼_<𝜀_"'<⁷

=

<>(

(4)

where 𝜀_"⁷ is the squared error term at time t and q represents the number of lags of the squared error term taken into account in the model. Letting 𝑞 = 1 generates an ARCH (1), as given by equation 5

𝜎_"⁷ = 𝛼_;+ 𝛼₍𝜀_"'(⁷ (5)

which is stationary and non-explosive under the assumptions that 𝛼₍ < 1 and 𝛼_; is a strictly positive number, 𝛼_; > 0, hold.

2.1.2 GARCH

A generalized approach to the ARCH model, known as the Generalized Auto Regressive Con- ditional Heteroscedasticity (GARCH) model, was introduced by Bollerslev (1986). Modelling the conditional variance using GARCH is similar to the ARMA approach of modelling the conditional mean, using lags of the squared error term and conditional volatility. On its general form of the GARCH (p, q) model is written as

𝜎_"⁷ = 𝛼_;+ 𝛼_<𝜀_"'<⁷

=

<>(

+ 𝛽_E𝜎_"'E⁷

F

E>(

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𝜎_"⁷ = 𝛼_;+ 𝛼₍𝜀_"'(⁷ + 𝛽₍𝜎_"'(⁷ (7)

The GARCH(1,1) is restricted by 𝛼_; > 0, 𝛼₍ > 0, 𝛽₍ > 0 and 𝛼₍+ 𝛽₍ < 1. The first three assumptions are necessary for the process to generate a positive conditional variance. The latter condition is obvious when observing the unconditional variance in equation 8

𝜎⁷ = 𝛼_;

1 − 𝛼₍+ 𝛽₍ (8)

making it clear that if 𝛼₍+ 𝛽₍ ≥ 1 the denominator will either be 0 or negative, making the unconditional variance non-stationary or negative.

2.1.3 Exponential GARCH

Research in financial time series has shown a correlation between negative returns and increased volatility and vice versa (see for example Tse et al. 2002). Using ARCH or GARCH to model the conditional variance does not allow this asymmetry to be taken into account. This drawback has led to some criticism and the development of models that allow for modelling including the effect of negative returns. One of these models is the Exponential GARCH (EGARCH) model (Nelson 1991), which models the natural logarithm of the conditional variance and includes a parameter for the asymmetric effect. On its general form the EGARCH is written as in equation 9

𝑙𝑛 𝜎_"⁷ = 𝛼_;+ 𝛼_< 𝜀_"'<

𝜎_"'< − 𝐸 𝜀_"'<

𝜎_"'< + 𝛾_K𝜀_"'K

𝜎_"'K𝐼_"'K+ 𝛽_E𝑙𝑛 𝜎_"'E⁷

F

E>(

M

K>(

=

<>(

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where 𝐼_"'K is a dummy variable taking the value 1 if the return in t-‐‑k is negative and 0 otherwise with the parameter 𝛾_K representing the impact of the asymmetric effect. Recalling equation 3 it is clear that _S^O^PQR

PQR = 𝑍_"'< . If 𝑍_"'< follow the normal distribution, its expectation can be written

as 𝐸 𝑍_"'< = ⁷_T and if 𝑍_"'< follow student t’s distribution its expected value will be 𝐸 𝑍_" =

7 U'7V ^WXY_Z

U'( V ^W_Z T.

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2.1.4 GJR-GARCH

Another model that allows for asymmetric modelling of the conditional variance is the GJR- GARCH model (Glosten et. al. 1993). The model differs from the GARCH model due to the implementation of an additional parameter to capture the asymmetric effect. On its general form the GJR-GARCH is written

𝜎_"⁷ = 𝛼_;+ 𝛼_<𝜀_"'<⁷

=

<>(

+ 𝛾_K𝜀_"'K⁷ 𝐼_"'K

M

K>(

+ 𝛽_E𝜎_"'E⁷

F

E>(

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where, as for the EGARCH model, 𝐼_"'K is a dummy variable, taking the value 1 if the return of lag k is negative and 0 otherwise, making 𝛾_K the impact of a negative return. If the parameter 𝛾_K = 0 the GJR-GARCH (p, q) is equal to a GARCH model (p, q).

2.2 Distributions

A necessary step when modeling both the conditional variance and Value at Risk is the assumption of the conditional distribution. In this study the conditional distribution is assumed to be either the normal Gaussian distribution or student t’s distribution.

2.2.1 Gaussian Normal Distribution

Assuming that the error terms conditionally follow the normal distribution, as seen in for example Engle (1982), is assuming that the conditional distribution have a probability density function as defined in equation 11

𝑓 𝜀_" = 1

𝜎_" 2𝜋𝑒𝑥𝑝 − 𝜀 − 𝜇 ⁷

2𝜎_"⁷

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where 𝜎_"⁷ is the conditional variance and 𝜇 is the mean of the error term at time t.

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2.2.2 Student t’s Distribution

While Engle (1982) assumed conditional normality among the error terms there are reasons to question the assumptions applicability to financial time series. Mandelbrot (1963) and Fama (1965) showed that financial returns follow a low peaked and fat tailed conditional distribution.

Bollerslev (1987) suggested assuming the student t’s distribution as the conditional distribution, finding support in later research (see section 1.2). The student t’s distribution, as suggested by Bollerslev, have the following density function:

𝑓 𝜀_" = Γ 𝜈 + 1

2 Γ 𝜈

2

'( 𝜈 − 2 𝜎_"^{7 '}⁽⁷ 1 + 𝜀_"⁷𝜎_"^'7 𝜈 − 2 ^' ^Ub(⁷ (12)

where 𝜈 is the degrees of freedom and a strict positive number, 𝜎_"⁷ is the conditional variance and Γ ∙ is the gamma function. For low values of 𝜈 the distribution has a low kurtosis and fat tails but as the degrees of freedom increase, the distributions tails become thinner and the kurtosis higher. If the degrees of freedom approaches infinity, 𝜈 → ∞, the distribution converges to the normal Gaussian distribution.

2.3 Value at Risk

Value at Risk (VaR) is defined as a threshold for the maximum loss over some time horizon given a certain confidence level, often 0.99 or 0.95 (Jorion 2007). Intuitively the idea of VaR is for it to be set so that a loss, or negative return, will not exceed the gauge with some certainty, formally expressed as

𝑃𝑟 𝑟_" > −𝑉𝑎𝑅 = 1 − 𝛼 (13)

where 𝑟_" is the return at time t. This implies that the probability of a loss exceeding VaR is 𝛼, which corresponds directly to the chosen confidence level so that

Pr 𝑟_" < −𝑉𝑎𝑅 = 𝑓 𝑟_"

'lmn

'o

𝑑𝑟_" = 𝛼

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where 𝑓(𝑟_") is the density function of the conditional distribution. As losses are defined as negative returns, VaR is located on the left side of the conditional distribution

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The main concepts of Value at Risk is illustrated in figure 1.

Figure 1, visual illustration of VaR

2.4 Evaluating Model Fit

After fitting a model to the data set and making VaR predictions it is crucial to evaluate whether the model has been able to fulfill the objective. In order to identify when VaR is violated an indicator variable defined as in equation 15

𝐼_" = 1 𝑖𝑓 𝑟^" < −𝑉𝑎𝑅

0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 (15)

Thus making the total number of violations ^u_">(𝐼_" = 𝑉. The total number of violations can thereafter be used to estimate the empirical size of violations 𝛼 =^l_u, where 𝑛 is the number of out-of-sample forecasts.

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2.4.1 Kupiec’s Test for Unconditional Coverage

One of the central tasks of a forecast of VaR is for it to generate an empirical size, 𝛼, that is not significantly different from the nominal size 𝛼. Kupiec (1995) developed a method for comparing the two sizes, testing the hypotheses that the empirical size equals the nominal against the alternative that they are not equal, formally

𝐻_;: 𝛼 = 𝛼 𝐻_m: 𝛼 ≠ 𝛼

where 𝛼 is the nominal size and 𝛼 is the empirical size. The unconditional test is performed via the likelihood ratio test

𝐿𝑅_z{ = 2ln 1 − 𝛼 1 − 𝛼

u'l 𝛼

𝛼

l

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where n is the number of out-of-sample forecasts and V is defined as in section 2.4. The likelihood ratio follows the Chi squared distribution with one degree of freedom (𝐿𝑅_z{~𝜒₍⁷) under the null hypothesis.

2.4.2 Christoffersen’s Test for Independence

For a volatility model to be useful it is crucial that it is able to quickly adapt to changes in the conditional variance. A limitation of Kupiec’s test is its inability to test for is the dependence among violations of VaR, making the test is unable to determine if a model is able to capture changes in conditional volatility. Christoffersen (1995) suggested a method for testing the independence of the violations, using the following test

𝐿𝑅_<u€ = −2𝑙𝑛 1 − 𝛼 ^•^‚‚^b•^Y‚ 𝛼 ^•^‚Y^b•^Y‚ + 2𝑙𝑛 1 − 𝜋_;( ^•^‚‚𝜋_;(^•^‚Y 1 − 𝜋₍₍ ^•^Y‚𝜋₍₍^•^YY (17)

where 𝐼_<E is the number of observations with value 𝐼 = 𝑖 followed by 𝐼 = 𝑗 and 𝜋_<E = ^•^R„

•R„

„ with, in this case 𝑖, 𝑗 = 0,1. The likelihood ratio is Chi squared distributed with one degree of freedom under the null hypothesis, 𝐿𝑅_<u€~𝜒₍⁷. It is worth mentioning that Christoffersen’s test using this design does not take into account dependence between violations that are more than one day

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apart and it is therefore not possible to identify dependence among violations two, three or four days apart.

2.5 Extreme Value Theory

The focus of Extreme Value Theory (EVT) is extreme events, rare in nature and located in the tail of the distributions as they deviate heavily from the mean or median of the process. One method included in EVT toolbox is the Peak Over Threshold (POT) approach, which is setting a threshold in a distribution and assigning the exceeding observations to the tail. Balkema and de Haan (1974) and Pickands (1975) showed that observations assigned to the tail through this methodology approximately follow the Generalized Pareto Distribution (GPD), defined as in equation 17

𝐺_†,z,‡ 𝑥 =

1 − 1 +𝜉

𝜎 𝑥 − 𝑢

'(†

, 𝑖𝑓 𝜉 ≠ 0 1 − 𝑒𝑥𝑝 − 𝑥 − 𝑢

𝜎 , 𝑖𝑓 𝜉 = 0

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where 𝜉 is the shape parameter, 𝑢 is the threshold and 𝜎 is a positive scaling function of 𝑢. The expected value of the function is

𝐸 𝑥 = 𝑢 + 𝜎

1 − 𝜉 𝑖𝑓 𝜉 < 1 (18)

The Generalized Pareto distribution does not have a finite expected value when 𝜉 ≥ 1. If 𝜉 and 𝑢 are simultaneously zero, the GPD is equivalent to the exponential distribution and if 𝑢 =^S_† and 𝜉 > 0 the GPD will be equal to the Pareto distribution. If the original distribution is the normal Gaussian distribution, the shape parameter is zero, 𝜉 = 0, and for the student t’s distribution the shape parameter is positive (𝜉 > 0).

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Figure 2, Illustration of Peak Over Threshold (POT)

2.6 Expected Shortfall

Despite the advantages of VaR the measure does not give any insight to the expectation of large losses. From a risk management point of view, it is however crucial knowledge to know what to expect once a large loss occur. Expected Shortfall is simply the conditional value of a loss following a violation of VaR, and is intuitively answering the question “when a loss violates VaR, what can I expect it to be?”. The Expected Shortfall is defined as in equation 19

𝐸𝑆_" = 𝐸 𝐿_" 𝐿_" > 𝑉𝑎𝑅_" = 𝑉𝑎𝑅_"+ 𝐸 𝐿_"− 𝑉𝑎𝑅_" 𝐿_" > 𝑉𝑎𝑅_" (19)

where it is clear that the expected shortfall is dependent on the forecast of VaR and therefore dependent on the chosen confidence level. The decomposition on the right hand side also suggest that the Expected Shortfall is dependent on the expected value of the violations of VaR, in this case defined as the expected value of the General Pareto distribution with VaR as the threshold. An illustration of forecasted VaR and Expected Shortfall is shown in figure 3.

Figure 3. Illustration of VaR (orange) and Expected Shortfall (black).

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2.7 Examining the Precision of Expected Shortfall Predictions

In order to evaluate the predictions of the Expected Shortfall and the actual violations of VaR, the Mean Absolute Percentage Error (MAPE) is used. The measure reveals how many percent the predictions deviate from the observed values and is calculated through

𝑀𝐴𝑃𝐸 = 100 𝑛

𝐴_<− 𝐹_<

𝐴_<

u

<>(

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where, in this case, 𝐴_< is the observed loss and 𝐹_< is the estimated Expected Shortfall.

The Expected Shortfalls are also evaluated through the Mean Squared Errors (MSE) and Root Mean Squared Errors (RMSE) and the results are presented in the appendix (items 3 and 4).

MSE is calculated through the expression presented in equation 21

𝑀𝑆𝐸 = 1

𝑛 𝐴_< − 𝐹_< ⁷

u

<>(

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where, as for MAPE, 𝐴_< is the observed loss and 𝐹_< is the forecast. RMSE is simply the root of the MSE, and is calculated as in equation 22.

𝑅𝑀𝑆𝐸 = 1

𝑛 𝐴_< − 𝐹_< ⁷

u

<>(

(22)

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3. Data

The data are weekly spot prices and -index of salmon, cocoa and lumber, acquired through the Eikon terminal, spanning from week 1 2004 to week 52 2015, with a total of 625 observations.

The time frame is limited by the available amount of data of the salmon spot, which was introduced in week 1 2004. Cocoa and lumber are standardized spot prices traded on the Chicago Mercantile Exchange, whilst the salmon spot is an index developed by the Norwegian seafood exchange Fish Pool. See appendix, item 5, for an extended decomposition of the spot index.

The pricing unit for respective commodity is Norwegian kronor (NOK) per kilogram for salmon, US dollars (USD) per 1000 board feet for lumber and US dollars per metric ton for cocoa.

The spot price of a commodity is defined as the price for instant delivery of a physical commodity (Chicago Mercantile Exchange Group 2016). A clear limitation of the spot prices is the small applicability to reality as commodities are most commonly sold via future or forward contracts. Price data of these types of contracts are however difficult to use as every contract differ from others and have a fixed expiration date, making it difficult to obtain a long consecutive time series of implied contract prices. The pricing of these contracts are also time dependent as their pricing changes as the contracts approach their expiration date. These utilities make the spot price appropriate for studies. Figures 4-6 gives a visualization of the weekly prices and returns for each commodity respectively and table 1 show the descriptive statistics of each commodity return series.

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Figure 4, Spot prices and weekly returns of salmon

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Figure 6, Spot prices and weekly return for cocoa

Table 1, Descriptive statistics weekly returns for respective commodity

Commodity Salmon Lumber Cocoa

Mean 0.0015 0.0019 0.0011

Median 0.0000 -0.0014 0.0003 Std. Dev 0.0557 0.0956 0.0358 Skewness 0.0332 0.0958 0.1243 Kurtosis 3.1551 7.0513 5.3670 Jarque-Bera 0.7414 429.071 147.75

p 0.6903 0.0000 0.0000

Looking at the descriptive statistics in table 1 it is clear that the mean of the weekly returns of respective commodity is not significantly different from zero, supporting the assumption of an unconditional zero mean for weekly returns. The Jarque-Bera test suggests that lumber and cocoa are not unconditionally normal distributed but it is not possible to reject unconditional normality for weekly returns of the salmon spot.

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4. Methodology

The ARMAX-GARCHK Toolbox, developed by Alexandros Gabrielsen (2011), is used in Matlab 2014b to make forecasts of VaR. The estimates of the Expected Shortfalls are done using applications within Matlab 2014b that allow for modelling the General Pareto distribution and its expectation when assigning certain observations to the distribution tail.

When predicting VaR using GARCH-models, the forecasts are made using rolling window estimation. This approach mean that a fixed window of observations is used to make a one-step ahead forecast for the out of sample observations. In practice this means that when predicting VaR for t+1 observations 0 through t are used to make the forecast. When predicting VaR for time t+2, observations 1 through t+1 are used, and so on. This method ensures that all predictions are made with the same number of observations, making them comparable. In this study the estimation window length will be 250 observations, resulting in 375 out of sample forecasts.

Using a larger window size, for example 500 observations, could be justified, but would result in a small number of out of sample forecasts due to limited amount of data. Another problem that could arise when using a larger window size is some change in the data generating process for the weekly returns, making the estimates of the parameters inaccurate when estimating the volatility models.

The estimates of the Expected Shortfalls are made using a similar methodology. As the threshold (VaR) and the scale parameter are known through the volatility modelling and VaR forecast the parameter to estimate is the shape parameter when assuming the t distribution. The shape parameter is set to be zero for generating the first predictions. Using the first two violations of VaR as data for the shape parameter then generates the next prediction and so forth. This results in some loss of data which may have a skewing impact on the predictions, especially when there is a small number of violations.

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5. Results

The result for each commodity is presented below. The empirical sizes are compared to the nominal using Kupiec unconditional coverage test (for extensive results see appendix, item 1) and significantly different results are marked with an asterisk. No models show any dependence among the violations according to Christoffersen’s test of independence (see the appendix, item 2).

5.1. Salmon Results

Table 2 display the results when predicting VaR and ES for salmon. It is clear that for 99%

VaR the GARCH-model gave the most accurate predictions, with the t-distribution being the most precise. The normal- and student t distribution generated the same results for EGARCH and GJR-GARCH, with both models being less accurate than GARCH. EGARCH also generated empirical sizes significantly larger than the nominal size 0.01 for both distributions. Turn- ing to 95% VaR it is clear that the models all have generated the same empirical size of violations with the t-distributed GJR-GARCH model being a slightly more conservative exception.

It is also worth noticing that all models generated conservative predictions, though no model generated an empirical size significantly smaller than 0.05.

Comparing the outcome of the violations to the Expected Shortfall estimates it is clear that for the 95% VaR forecasts the violations deviate more from the expected exceedance than for those at 99% VaR. Assuming student t’s distribution also generated the smallest deviations from the expectation. For the 95% confidence level the normal distributed GARCH (1,1) provides the most accurate predictions of the expected shortfall with the actual losses deviating on average 11.73% from the expectation.

Table 2, empirical size of violations for respective model and confidence level. *10% **5% ***1%

Model 99% VaR 95% VaR MAPE ES99% MAPE ES95%

GARCH(1,1)-N 0.0133 0.0453 8.25% 11.73%

GARCH(1,1)-t 0.0107 0.0453 5.85% 12.92%

EGARCH(1,1)-N 0.0240** 0.0453 7.27% 14.77%

EGARCH(1,1)-t 0.0240** 0.0453 6.84% 14.75 GJR-GARCH(1,1)-N 0.0160 0.0453 8.62% 13.11%

GJR-GARCH(1,1)-t 0.0160 0.0427 6.33% 14.31%

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5.2 Lumber Results

Table 3 display the results when predicting VaR and ES for lumber. For 99% VaR the results suggest that assuming the t-distribution results in conservative forecasts, as all t distributed models generated zero exceedances. The normal distributed models also generate the same empirical size, which is significantly higher than the empirical size. A similar trend is seen for the 95% VaR with the t distributed models generating forecasts of VaR that is significantly below the nominal size of 0.05. Assuming the normal distribution does however generate rather accurate forecasts, with the GJR-GARCH being the most precise.

When examining the Expected Shortfall predictions for violations of the 99% VaR, it is hard to draw conclusions regarding any differences between the distributions due to the lack of violations when assuming the t-distribution. Of the normal distributed models, the GJR-GARCH generates the smallest average deviation from the expectation. For 95% VaR the average deviations are significantly smaller when assuming the t-distribution rather than the normal distribution. The model that generates the smallest average deviations is the t-distributed GJR- GARCH followed by the t distributed EGARCH. Of the normal distributed models, the EGARCH model generates the smallest deviations followed by the GARCH model.

Model 99% VaR 95% VaR MAPE 99% ES MAPE 95% ES GARCH(1,1)-N 0.0293*** 0.0533 13.10% 26.35%

GARCH(1,1)-t 0.0000 0.0267** - 12.48%

EGARCH(1,1)-N 0.0293*** 0.0533 12.37% 23.69%

EGARCH(1,1)-t 0.0000 0.0240** - 12.95%%

GJR-GARCH(1,1)-N 0.0293*** 0.0507 10.70% 26.98%

GJR-GARCH(1,1)-t 0.0000 0.0267** - 11.98%

5.3 Cocoa Results

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asymmetric models are also generating empirical sizes significantly lower on a 10% signifi- cance level. Assuming conditional normality do not generate an empirical size significantly different from 0.01 for any of the models, with the GJR-GARCH being the most accurate. For 95% VaR the t distribution generates significantly lower sizes than the empirical size again.

Assuming conditional normality generates identical results between all models and the forecasts tend to be on the conservative side.

Turning to the expected shortfall for violations of the 99% VaR, the normal distributed EGARCH generates the smallest average absolute percentage deviation from the expected shortfall, with a 1.53% deviation, although generating a small number of violations. For violations of 95% VaR it appears that assuming the t-distribution generates the smallest deviations from the expected shortfall. The model that gives the smallest deviation is in this case the GARCH model, followed by EGARCH both assuming the t distribution and both generating significantly fewer violations than the nominal size.

Model 99% VaR 95% VaR MAPE ES99% MAPE ES95%

GARCH(1,1)-N 0.0053 0.0373 4.47% 8.75%

GARCH(1,1)-t 0.0000 0.0053*** - 6.18%

EGARCH(1,1)-N 0.0053 0.0373 1.53% 9.38%

EGARCH(1,1)-t 0.0027* 0.0267** 16.61% 8.98%

GJR-GARCH(1,1)-N 0.0080 0.0373 16.09% 9.25%

GJR-GARCH(1,1)-t 0.0027* 0.0267** 20.41% 12.79%

6. Conclusions

With all models showing independence among the violations of VaR the results suggest that all models are able to adapt to changes in conditional volatility for consecutive weeks. The results also indicate that assuming the t distribution generates more conservative forecasts of VaR for both the 99% and 95% confidence level. These findings contradict some of the earlier research (see 1.2) to some extent, as findings tend to suggest the t distribution as the superior distribution.

Assuming the normal distribution does instead generate more accurate predictions of VaR for several cases, for example as seen in the case of cocoa. The results suggest that the GJR-

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GARCH model, assuming normality, is the most accurate in most cases, being the superior model for every prediction excluding 99% VaR for salmon, where it is still sufficient to use.

The results of the Expected Shortfall reveal mixed outcomes. ES of violations of 99% VaR is best predicted when assuming a t distribution for salmon but for cocoa the normal distribution generates more accurate forecasts. These results may also be due to the small number of violations as a consequence of the highly conservative forecasts of VaR obtained when assuming the t distribution. The loss of data when applying this methodology is also a concern, as few violations will not make the GPD fit the actual tail distribution for t distributed data. There is however no clear evidence that this is an issue that affect the predictions of the Expected Short- falls. Since no comparison is made between this EVT approach and some standard approach to modeling the ES it is difficult to say whether this methodology have been more successful. The data intensity of the methodology is however a clear disadvantage.

An issue that this thesis does not touch upon is the problem occurring when there is a significant difference between the nominal and empirical sizes, but the predictions of the expected shortfalls are accurate. This problem does not appear to any larger extent, but is worth mentioning.

6.1 Recommendations for further studies

An expansion of this study may include a larger number of volatility models, such as the ARCH, APARCH or RiskMetrics models to further examine the model fit. Another expansion of the study may be to use alternative methods to forecast the conditional volatility, such as using Mixed Data Sampling (MIDAS) techniques, or the EVT approach as suggested by McNeil and Frey (2000). The number of conditional error distributions could also be expanded to include, for example, the General Error distribution (GED) or other heavy tail distributions.

Increasing the sample size will undoubtedly generate more violations of VaR, which would make it easier to conduct inference, though this would require excluding salmon, due to the limitation of data. For lumber or cocoa this expansion is possible, as there are daily data avail-

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Including similar commodities could reveal if there are similarities in model fit among similar commodities. One example could be examining if the same model is the best fit for sugar and cocoa, since the farming conditions of these products are similar, making macroeconomic events having a similar impact on the prices of the two commodities. Another extension of this study could also be the development of some method that evaluate both the precision of both VaR and ES forecasts. A common examination of the two measures would eliminate the issue where a small number of violations tend generate accurate ES predictions due to the lack of data.

6.2 Recommendations for Decision Makers

When examining the outcome of the forecasts it is not clear that any model stands out. However, the results indicate that the asymmetric GJR-GARCH tend to generate rather accurate results and would therefore be sufficient to use in risk management practices. The choice of conditional distribution tends to have an impact on the result, with the t distribution generating very conservative forecasts in many cases. This may imply that the normal distribution is advantageous when predicting 99% VaR, as the t distribution in some cases generated forecasts without any violations.

Since no comparison to alternative methods of forecasting Expected Shortfall is made it is difficult to say whether the presented EVT approach is superior to those or not. The slightly more extensive data requirements of the approach do however suggest that, if not more accurate, the method could be disregarded in favor of other approaches.

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References

Balkema, A. A., & De Haan, L. (1974), “Residual life time at great age”, The Annals of probability, 792-804.

Berggren, E. & Folkelid, F. (2014), “Which GARCH Model is Best for Value at Risk?”, Upp- sala University, Department of Statistics

Bollerslev, T. (1986), “Generalized Autoregressive Conditional Heteroscedasticity”, Journal of Econometrics 31, 307-327.

Bollerslev, T. (1987), "A conditionally heteroskedastic time series model for speculative prices and rates of return", The review of economics and statistics 69, 542-547.

Chicago Mercantile Exchange Group, “Glossary” (2017-01-06), Education http://www.cme- group.com/education/glossary.html (2017-01-06)

Christoffersen, P. (1998), “Evaluating Interval Forecasts”, International Economic Review 39, 841-862.

Ding, Z., Granger, C. & Engle, R. (1993) “A long memory property of stock market returns and a new model”, Journal of Empirical Finance 1, 83-106.

Engle, R. (1982), “Autoregressive Conditional Heteroscedasticity with Estimates of the Vari- ance of United Kingdom Inflation”, Econometrica 50, 987-1008.

Fama, Eugene F. (1965), “The Behavior of Stock-Market Prices”, The Journal of Business 38, 34-105.

Giot, P. & Laurent, S. (2003). “Market Risk in Commodity Markets: A VaR Approach”, En-

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Glosten, L., Jagannathan, R. och Runkle, D. (1993), “On the Relation between the Expected Value and the Volatility of the Nominal Excess Return on Stocks”, The Journal of Finance 48, 1779-1801.

Hung, J., Lee, M., Liu, H. (2008) “Estimation of Value at Risk for Energy Commodities via Fat Tailed GARCH Models”, Energy Economics, 30(3) 1173-1191.

Jorion, P. (2007), Value at risk: the new benchmark for managing financial risk. 3. ed. New York, McGraw-Hill.

J.P Morgan and Reuters (1996), “RiskMetrics-A technical document”, New York.

Kupiec P. (1995), “Techniques for Verifying the Accuracy of Risk Management Models", Journal of Derivatives 3, 73-84.

Lindholm, Dennis (2015) “Beyond the Value at Risk – A study on quantitative risk measure- ments”, Uppsala University, Department of Statistics.

Mandelbrot, B. (1963), “The Variation of Certain Speculative Prices”, The Journal of Busi- ness 36, 394-419.

McNeil, A. J., & Frey, R. (2000), “Estimation of tail-related risk measures for heteroscedastic financial time series: an extreme value approach”, Journal of Empirical Finance, 7(3), 271- 300.

Nelson, Daniel B. (1991), “Conditional Heteroskedasticity in Asset Returns: A New Ap- proach”, Econometrica 59, 347-370.

Patton, A. (2006), “Modelling Asymmetric Exchange Rate Dependence”, International Eco- nomic Review 47 (2), 527-556.

Pickands III, J. (1975), “Statistical inference using extreme order statistics”, The Annals of Statistics3 (1), 119-131.

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So, M., & Yu, P. (2005), “Empirical Analysis of GARCH Models in Value at Risk Estima- tion”, Journal of International Financial Markets Institutions and Money 16, 180-197.

Tse, Y. & Tsui, K. (2002), “A Multivariate Generalized Autoregressive Conditional Hetero- scedasticity Model With Time-Varying Correlations”, Journal of Business and Economic Sta- tistics 20 (3), 352-362.

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Appendix

1. Kupiec unconditional coverage test values for respective commodity Salmon

Model 99% VaR 95% VaR

GARCH(1,1)-N 0.3810 0.1772 GARCH(1,1)-t 0.0165 0.1772 EGARCH(1,1)-N 5.3330 0.1772 EGARCH(1,1)-t 5.3330 0.1772 GJR-GARCH(1,1)-N 1.1537 0.1772 GJR-GARCH(1,1)-t 1.1537 0.4458 Lumber

Model 99% VaR 95% VaR

GARCH(1,1)-N 9.3176 0.0859 GARCH(1,1)-t 0.0000 5.1410 EGARCH(1,1)-N 9.3176 0.0859 EGARCH(1,1)-t 0.0000 0.0859 GJR-GARCH(1,1)-N 9.3176 5.1410 GJR-GARCH(1,1)-t 0.0000 0.4458 Cocoa

Model 99% VaR 95% VaR

GARCH(1,1)-N 0.9938 1.3832 GARCH(1,1)-t 0.0000 21.1908 EGARCH(1,1)-N 0.9938 1.3832 EGARCH(1,1)-t 2.8768 5.1410 GJR-GARCH(1,1)-N 0.1627 1.3832 GJR-GARCH(1,1)-t 2.8768 5.1410

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2. Christoffersen’s test for independence results for respective commodity Salmon

Model 99% VaR 95% VaR

GARCH(1,1)-N 23.12 60.74

GARCH(1,1)-t 19.27 60.74

EGARCH(1,1)-N 37.05 60.74

EGARCH(1,1)-t 37.05 60.74

GJR-GARCH(1,1)-N 26.80 60.74

GJR-GARCH(1,1)-t 26.80 57.99

Lumber

Model 99% VaR 95% VaR

GARCH(1,1)-N 43.41 68.66

GARCH(1,1)-t - 40.28

EGARCH(1,1)-N 43.41 68.66

EGARCH(1,1)-t - 37.07

GJR-GARCH(1,1)-N 43.41 66.09

GJR-GARCH(1,1)-t - 40.28

Cocoa

Model 99% VaR 95% VaR

GARCH(1,1)-N 10.83 52.01

GARCH(1,1)-t - 15.20

EGARCH(1,1)-N 10.83 52.37

EGARCH(1,1)-t 6.02 40.28

GJR-GARCH(1,1)-N 15.20 52.37

GJR-GARCH(1,1)-t 6.02 40.28

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3. Mean Square Errors of Expected Shortfall forecasts for respective commodity Salmon

Model MSE ES 99% VaR MSE ES 95% VaR

GARCH(1,1)-N 0.0002 0.0002

GARCH(1,1)-t 0.0001 0.0003

EGARCH(1,1)-N 0.0002 0.0005

EGARCH(1,1)-t 0.0002 0.0005

GJR-GARCH(1,1)-N 0.0003 0.0005

GJR-GARCH(1,1)-t 0.0025 0.0006

Lumber

GARCH(1,1)-N 0.0009 0.0043

GARCH(1,1)-t - 0.0011

EGARCH(1,1)-N 0.0013 0.0026

EGARCH(1,1)-t - 0.0009

GJR-GARCH(1,1)-N 0.0008 0.0042

GJR-GARCH(1,1)-t - 0.0017

Cocoa

GARCH(1,1)-N 0.0000 0.0001

GARCH(1,1)-t - 0.0000

EGARCH(1,1)-N 0.0000 0.0001

EGARCH(1,1)-t 0.0002 0.0002

GJR-GARCH(1,1)-N 0.0007 0.0003

GJR-GARCH(1,1)-t 0.0007 0.0002

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4. Root Mean Squared Errors of Expected Shortfall forecasts for respective commodity Salmon

GARCH(1,1)-N 0.0144 0.0149

GARCH(1,1)-t 0.0100 0.0185

EGARCH(1,1)-N 0.0129 0.0216

EGARCH(1,1)-t 0.0127 0.0217

GJR-GARCH(1,1)-N 0.0179 0.0221

GJR-GARCH(1,1)-t 0.0502 0.0247

Lumber

GARCH(1,1)-N 0.0299 0.0656

GARCH(1,1)-t - 0.0336

EGARCH(1,1)-N 0.0362 0.0508

EGARCH(1,1)-t - 0.0303

GJR-GARCH(1,1)-N 0.0280 0.0650

GJR-GARCH(1,1)-t - 0.0410

Cocoa

GARCH(1,1)-N 0.0041 0.0078

GARCH(1,1)-t - 0.0044

EGARCH(1,1)-N 0.0018 0.0083

EGARCH(1,1)-t 0.0125 0.0123

GJR-GARCH(1,1)-N 0.0263 0.0184

GJR-GARCH(1,1)-t 0.0270 0.0150

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5. Composition of the Salmon Spot Index

The Fish Pool Spot index is a weighted average of distributors prices from Nasdaq Salmon Index of exporters demanded prices, Fish Pool European Buyers index which comprises de- mand of large European buyers and exporting statistics from Statistics Norway. The prices are thereafter weighted depended on the price of 3-6 kilograms’ salmons, which are further divided into three price categories with the following weights

Kg Weights 3-4 30%

4-5 40%

5-6 30%