Values at Risk

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Values at Risk

Uppsala University Hampus Eliasson Bachelor Thesis in Statistics

Fall 2017

Supervisor: Lars Forsberg

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Abstract

This paper presents a small study on Value at Risk calculation for four different companies. Value at Risk, VaR, is a risk-metric, which measures the least amount that an asset can be expected to deviate from its value at a chosen probability level. A five and one percent VaR calculation is performed for the four American conglomerates chosen for the study, these are: Apple, Alphabet, Exxon Mobil and General Electric. The VaR is calculated in this paper via three different GARCH estimations for the conditional heteroscedasticity coupled with three different probability distributions for the error terms.

The GARCH models used are the Standard GARCH, the Exponential GARCH and the Threshold GARCH. Every GARCH model is tested with Normal, Students-T and Normal Inverse Gaussian errors. In order to evaluate the models, Kupiec’s test and Christoffersen’s test are conducted. Based on passing both tests for the one and five percent VaR simultaneously, the study yields several suitable model combinations for each company except for Apple. The suitable models seem to cluster around what probability distribution is used rather than what GARCH-model. The study lacks in capacity of comparing the efficiency of the suitable models to one another.

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Introduction

In financial economics there exists a need for measuring and managing different sorts of risk. For instance, asset specific risk can be seen as an assets probability to decrease or increase in value and to what magnitude, clearly a study of interest for financial analysts. These changes in price are called volatility, and by studying an assets historical volatility, projections about future volatility can be made. These projections can be used to calculate the risk-metric Value at Risk or VaR which measures the minimum change in asset value which can be expected with a certain degree of confidence. For example, for a five percent VaR estimate, the Value at Risk estimation presents the lowest amount that the value of an asset will shift in the most extreme five percent of outcomes. This risk-metric gained popularity in the 1990s following changes in the Basel Accords prompting banks to perform their own risk calculations where the VaR metric was one of the options (Chang et al. 2011). This paper will evaluate different GARCH approaches for calculating the Value at Risk of the daily returns of stocks of four large American companies: Apple, Alphabet, General Electric and Exxon Mobil. These companies were chosen since they are large well-known companies with easily accessible financial data. In order to calculate the Value at Risk, the assets historical volatility is analyzed. Since volatility in financial assets has been shown to cluster and be serially dependent, the most recent information may be used in order to predict a portion of future volatility. By using an “Autoregressive Conditional Heteroscedasticity” or ARCH model (Engle, 1982), the volatility conditional on past values can be used to predict changes in future volatility. The ARCH models which will be evaluated in this paper are the Generalized ARCH or GARCH (Bollerslev 1986), the Exponential GARCH or EGARCH (Nelson 1991) and the Threshold GARCH or TGARCH (Zakoïan, 1994). Although the conditional part of the volatility can be predicted with the use of GARCH modeling, an unpredictable stochastic part still exists. This random part of the volatility has to be mapped with a probability distribution function, this paper makes use of the Normal, the Students T, and the Normal Inverse Gaussian distributions. In order to evaluate the models, Kupiecs Exceedance test and Christoffersen’s Exceedance Independence Test are performed on the models.

Earlier research

To provide a framework on how statistical time series behave, the first article presented is a survey of previous research. Rama Cont’s article Empirical Properties of Asset Resurns: Stylized Facts and Statistical Issues (2001) provides a good survey of reports on what has become known as stylized facts in financial analysis. These stylized facts can be seen as empirically proven general behaviors of financial assets. The most relevant of these properties for our VaR analysis are presented in short: The first one, ‘Fat tails’, the propensity for financial returns to have non-normally distributed returns with fatter tail risk than the normal distribution would suggest. This can lead to an underestimation of financial risk if not taken into account. Since Value at Risk deals with these tail end probabilities, it is only reasonable to include different probability distributions with higher tail risk in the study. The second “stylized fact” of importance for this paper is that of ‘asymmetric responses’ to positive or

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negative returns. Simply put, the market seems react stronger towards negative than positive returns and thus higher volatility ought to be expected days following negative shocks than positive ones.

Hence it is reasonable to include a model which takes these asymmetric responses into account. The third one is ‘absence of serial correlation’, this characteristic tells us high returns today do not in any way indicate high returns tomorrow. Which brings us to the final stylized fact of importance for this paper ‘Volatility Clustering’. Contrary to returns, the volatility of those returns have been seen to serially correlated and prone to cluster. Thus a strong positive shock today does not alter the

probability of a positive shock tomorrow, it can however be shown that the probable level of intensity of tomorrows shock is affected. Robert Engle (1982) used this last characteristic with his ARCH model to forecast future volatility by using current data. For example, a sharp spike in volatility today increases the probability of having high volatility tomorrow. But as the third of these financial characteristics state, the direction of the shocks normally cannot be predicted by statistical methods, hence only a part of the intensity of tomorrows shock can be predicted. However by using the characteristic of asymmetric responses to positive and negative shocks this information may also be harnessed in order to predict the intensity of future volatility. The paper ”The use of GARCH models in VaR estimation” released in (2004) by Timotheos Angelidis, Alexandros Benos and Stavros

Degiannakis calculated and tested the VaR for 5 different portfolios constructed out of major stock indices. Highly similar to this paper they used the SGARCH, EGARCH and TGARCH coupled with the Normal, Students-t and GED distributions. The EGARCH and TGARCH models make use of directional information of previous shocks. However after conducting Kupiec´s and Christofferson´s tests no model stood out as significantly fitting more of the portfolios. However after comparing the significantly fitting models the asymmetrical EGARCH and TGARCH produced more accurate forecasts than the SGARCH, with EGARCH as the best performer. For the error distributions, the thicker tailed Students-T and the Generalized Error Distribution performed better than the Normal distribution. These results are well in line with the expectations garnered from studying the characteristics of financial time-series.

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Data and Descriptive Statistics

The companies chosen for this analysis are the well-known American industrial giants General Electric and Exxon Mobil as well as the tech giants Apple and Alphabet, the parent company of Google. The time series used in this paper have been downloaded from Google Finance via the

“Quantmod” package of the R Studio statistical software. The data has been gathered from 2007-01-03 up to 2017-12-13. The returns which will be analyzed are the log-returns (not dividend adjusted) constructed by taking the logged difference between the closing prices at time t and t-1. The companies are different from one another in what they produce, however all of them can be seen as relatively large.

Apple Log Returns: 2007-01-03 to 2017-12-13 Alphabet Log Returns: 2007-01-03 to 2017-12-13

Exxon Mobil Log Returns: 2007-01-03 to 2017-12-13 General Electric Log Returns: 2007-01-03 to 2017-12-13 Figure 1: Time series on logged daily returns scaled by 100

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Theoretical Background

Value at Risk

Value at Risk or VaR is a risk-metric, which means that it measures a certain type of risk, which is the minimum amount an asset is expected to change in value given previous market volatility. It is

calculated upon a chosen percentage level which shows the tail end risk. This percentage level is chosen by the investor but is usually 5 and 1 percent. Basically the VaR can be seen as a one sided confidence interval indicating what level the investor can expect the asset not to drop 1 – 𝛼 amount of the time. In other words, given an 𝛼 of 5 percent, the investor can expect the stock to stay above the 5 percent loss quantile 95 percent of the time. A VaR forecast based on GARCH estimates results in a dynamic estimation due to the GARCH models ability to constantly change the volatility forecast (Jondeau, Poon and Rockinger 2007). By inspecting the graphs in Figure 2 one might quickly spot the difference in the red dots indicating VaR forecast exceedances. The plot with an 𝛼 = 1% has

significantly less exceedances than its 𝛼 = 5% counterpart. This is fully reasonable and expected as the

‘safety bar’ has been lowered.

𝛼 = 1% VaR Exceedance plot of Apple Returns 𝛼 = 5% VaR Exceedance plot of Apple Returns Figure 2

Modelling the Conditional Heteroscedasticity

Robert F. Engle introduced the “Autoregressive Conditional Heteroscedasticity”, ARCH model, for time series analysis in (1982) and was awarded the Sveriges Riksbanks Prize in Economic Sciences in Memory of Alfred Nobel for this in 2003 (Nobelprize.org 2003). The ARCH model comes into use when there is a timewise correlation in an assets volatility. Where just like a regular MA(p) time series model the previous shocks are used in order to predict the future value of a series. The difference with the ARCH model being that it attempts to predict part of the volatility of the series instead of the real value. Engle showed that since the volatility 𝜖 is serially correlated it can be written as a function, part conditional on past values, the 𝜎 and a strictly random element z such as 𝜖_𝑡= 𝜎_𝑡𝑧_𝑡.

Equation 1.

𝜎_𝑡²= 𝜔 + ∑ 𝛼_𝑖

𝑞

𝑖=1

𝜖_𝑡−𝑖²

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Where 𝜎 is the conditional volatility at time t dependent on the previous squared returns 𝜖. The constants, 𝛼 explains how much of the conditional volatility at time t depends on each 𝜖 term.

(Jondeau, Poon and Rockinger 2007) By examining the time series presented in Figure 1 one can clearly see the clustering tendencies of the returns. The most obvious cluster is the one at the financial crisis of 08, this forceful increase in volatility can be seen in each time serie.

This paper uses variations of the GARCH model developed from the ARCH model by Tim Bollerslev (1986), which possesses similar qualities to the ARCH model. Three variation are used, the Standard GARCH, the Exponential GARCH and the Threshold GARCH. The latter variants are included since they allow for asymmetric responses between positive contra negative shocks.

GARCH

Since there is a long persistence in volatility the ARCH model needs an inclusion of many of the previous shocks in order to be a good fit. The “Generalized Autoregressive Conditional

Heteroscedasticity” or GARCH model by Tim Bollerslev (1986) reduces the amount of previous shocks necessary to include in an analysis by introducing the previous day forecast into the equation.

Equation 2

𝜎_𝑡² = 𝜔 + ∑ 𝛼_𝑖

𝑞

𝑖=1

𝜖_𝑡−𝑖² + ∑ 𝛽_𝑗𝜎_𝑡−𝑗²

𝑝

𝑗=1

Hence the forecasted conditional volatility 𝜎 is a function of its previous value and the shock of the previous day. Since the shock, defined as 𝜖_𝑡 = 𝜎_𝑡𝑧_𝑡 is a function of the forecast conditional variance and the random element 𝑧𝑡 it becomes clear that the forecasted conditional variance is a product of its last projection and the deviation from it. The GARCH process is covariance stationary iff ∑^𝑝_𝑖=1𝛼_𝑖+

∑^𝑞_𝑗=1𝛽_𝑗< 1, a sufficient condition for the error term 𝜖 to be strictly stationary. (Jondeau, Poon and Rockinger 2007)

EGARCH

The Exponential GARCH (Nelson, 1991) or EGARCH model is an asymmetric model which takes consideration of both the sign and the size of the shocks and can thus predict different effects of negative returns than positive returns. Again let us define 𝜖_𝑡 = 𝜎_𝑡𝑧_𝑡, that is the shock is equal to the conditional volatility times a random error.

Equation 3

𝑙𝑜𝑔 𝜎_𝑡²= 𝜔 + ∑ 𝛼𝑖 𝑞

𝑖=1

𝑔(𝑧𝑡−𝑖) + ∑ 𝛽𝑗log 𝜎_𝑡−𝑗²

𝑝

𝑗=1

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8 Equation 4

𝑔(𝑧_𝑡) = {(𝜃 + 𝛾)𝑧_𝑡− 𝛾𝐸(|𝑧_𝑡|) 𝑖𝑓 𝑧_𝑡 ≥ 0 (𝜃 − 𝛾)𝑧_𝑡− 𝛾𝐸(|𝑧|) 𝑖𝑓 𝑧𝑡 < 0

The function 𝑔(𝑧𝑡−𝑖) = 𝜃𝑧𝑡+ 𝛾[|𝑧𝑡| − 𝐸|𝑧_𝑡|] is easier understood rewritten in Equation 4 where the asymmetric response to a negative or positive shock becomes clear. 𝜃 and 𝛾 are constants determining the asymmetric responses and linearity. The term E(|𝑧_𝑡|) is the expectation of the absolute value of 𝑧_𝑡. Contrary to a probability distribution with mean 0 the expectation of an absolute value follows a folded probability distribution from the one which 𝑧 is assumed to follow since every value on the positive side becomes twice as likely when the negatives are transformed. For example the expectation of a folded Normal(0,1) distribution would be √²

𝜋 instead of 0 (Rachev et al. 2007).

TGARCH

The Threshold GARCH or TGARCH (Zakoïan, 1994) is an interesting model which models the volatility rather than the squared volatility. Like the EGARCH it is also allows for asymmetric responses between positive and negative shocks.

Equation 5

𝜎_𝑡 = 𝜔 + ∑[𝛼_𝑖|𝜖_𝑡−𝑖| + 𝛾_𝑖

𝑝

𝑖=1

Π_t−i⁻ |𝜖_𝑡−𝑖|] + ∑ 𝛽_𝑗

𝑞

𝑗=𝑖

𝜎_𝑡−𝑗

The alpha term indicates how much the past absolute errors contribute to volatility and the gamma term indicates how much difference it makes if the shock is negative. Π_𝑡⁻ is an indicator which takes the value 1 if 𝜖_𝑡 < 0 but takes the value 0 otherwise.

Probability Functions

The probability functions chosen for this paper are the Normal distribution, the Students T distribution and the Normal Inverse Gaussian distribution. The normal distribution was included to since it is the go-to distribution for new statistics students and the distribution that Robert F.Engle assumed in his groundbreaking work on ARCH models (1982). The Students t-distribution was chosen on the same premise as the normal-distribution, it also being well known and adopted. Moreover in (1987) Tim Bollerslev released a paper where he showed that the t-distribution better handles the leptokurtic qualities of financial returns than the normal distribution due to its fatter tails. The last distribution assumed, the Normal Inverse-Gaussian Distribution or NIG, also allows for fatter tail probabilities and has been shown as very successful at modeling financial data. For example a Paper released (2000) by Erik Bølviken and Fred Espen Benth showed that the NIG-distribution fit the log-return data of several Norwegian stocks ‘nearly perfectly’. Thus including the NIG-distribution will examine how the workhorses of basic econometrics, the Normal and Students t-distributions fare against a more leptokurtic distribution, the NIG-distribution.

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9 Kupiec’s Test

One of the earliest tests for Value at Risk modeling performance efficiency was Kupiec’s PF Coverage Test (Kupiec, 1995). This test is a likelihood ratio test which examines whether or not the total amount of exceedances by a back tested VaR model is feasible with regards to the chosen 𝛼. Simply put, it tests how likely it is that the VaR model is correct given the actual result.

Christoffersen’s Test

Christoffersen’s (1998) Exceedance Independence Test examines whether or not the VaR exceedances are independent from one another. Just like Kupiec’s test it is a likelihood ratio test. The test is

relevant since the application of GARCH models to the VaR estimates attempt to account for volatility clustering. Therefor the Exceedance Independence test gives a hint of whether or not the model raises the Value at Risk estimate quickly enough in times of turbulence.

Methodology

The VaR estimation and subsequent Kupiec and Christoffersen tests were all conducted using the Rugarch package in R. The GARCH models were all of the order (1, 1) that is they all used a single lag for the ARCH term and a single lag for the GARCH term, this was chosen for simplicity and in order to reduce the number of models to be estimated. The estimation window was set to 500 observations for every test, by the reasoning that in order to properly forecast the rare 1% events, a few of these events have to be included within the estimation window. The forecast length was set to +1 since any forecasts beyond that have such large variances making them difficult to use

productively. The estimation window was set to refit every 50 days in order to refresh the parameter estimations as market conditions changed. The Value at Risk alphas were set to 1 percent and 5 as is common practice (Jondeau, Poon and Rockinger 2007).

Comparative analysis

In order to assess the efficiency of the various Value at Risk prediction models for the four companies, the Kupiec’s Test and Christoffersen’s Exceedance Independace Test are performed. The first test,

“Kupiec’s test” tests whether or not an estimation model might be correct based upon the number of exceedances of the Value at Risk boundary the model exhibits (Kupiec, 1995). The second test,

“Christoffersen’s Exceedance Independence test”, tests whether or not the exceedances are

independent from one another (Christoffersen 1998), which indicates if the model is suitably flexible to swings in volatility. The probabilities of these test results are presented in the results section and will serve as a basis for approving or disapproving the models.

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Results

Examining the time series plots in Figure 3 gives us a picture of the behavioral differences of the different stocks. Apple and especially Alphabet seem prone to sharp turns with many spikes in volatility both positive and negative. General Electric looks more stable and Exxon Mobil hardly exhibits any spikes in volatility at all apart from the crash in 2008.

Apple Log Returns: 2007-01-03 to 2017-12-13 Alphabet Log Returns: 2007-01-03 to 2017-12-13

Exxon Mobil Log Returns: 2007-01-03 to 2017-12-13 General Electric Log Returns: 2007-01-03 to 2017-12-13 Figure 3: Time series on logged daily returns scaled by 100

In the following tables the Standard GARCH is abbreviated SG, the Exponential GARCH, EG and the Threshold GARCH, TG. The probability distributions are abbreviated Norm for Normal, Std for Students T and Nig for the Normal Inverse Gaussian Distribution. Combinations of the models and distributions are written for example SG-Norm for the Standard Gaussian model paired with assumed Normally-distributed errors.

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Significant difference from 𝐻₀ at 1, 5 and 10 percent level respectively = *, **, ***.

Apple,

sGARCH(1,1)

Std 𝛼 = 1%

Std 𝛼 = 5%

Norm 𝛼 = 1%

Norm 𝛼 = 5%

Nig 𝛼 = 1%

Nig 𝛼 = 5%

Exceedances 0,88 3,72 1,15 3,15 0,80 3,06

Kupiec P 58, 21 0,37* 47,59 0,00* 31,82 0,00*

Christofferson P

26,84 66,09 30,37 39,01 80,19 67,20

Apple,

eGARCH(1,1)

Std 𝛼 = 1%

Std 𝛼 = 5%

Norm 𝛼 = 1%

Norm 𝛼 = 5%

Nig 𝛼 = 1%

Nig 𝛼 = 5%

Exceedances 0,88 3,23 0,93 2,97 0,75 3,05

Kupiec P 58,21 0,00* 73,99 0,00* 21,94 0,00*

Christofferson P

75,45 12,79 59,25 4,76** 72,23 13,50

Apple, tGARCH(1,1)

Std 𝛼 = 1%

Std 𝛼 = 5%

Norm 𝛼 = 1%

Norm 𝛼 = 5%

Nig 𝛼 = 1%

Nig 𝛼 = 5%

Exceedances 0,79 3,28 0,93 2,88 0,75 2,83

Kupiec P 31,82 0,00* 73,99 5,87*** 21,94 3,21**

Christofferson P

44,20 4,81** 59,25 2,77** 42,08 2,34**

Table 1 Apple test results, showing exceedances in percent of the VaR models and the p values of Kupiec’s and Christoffersen’s tests.

Alphabet, sGARCH(1,1)

Std 𝛼 = 1%

Std 𝛼 = 5%

Norm 𝛼 = 1%

Norm 𝛼 = 5%

Nig 𝛼 = 1%

Nig 𝛼 = 5%

Exceedances 0,57 4,21 1,01 2,83 0,48 3,90

Kupiec P 2,81** 7,82** 92,43 3,21** 0,66* 1,30**

Christofferson P

63,12 90,07 79,20 48,10 22,76 97,75

Alphabet, eGARCH(1,1)

Std 𝛼 = 1%

Std 𝛼 = 5%

Norm 𝛼 = 1%

Norm 𝛼 = 5%

Nig 𝛼 = 1%

Nig 𝛼 = 5%

Exceedances 0,93 4,39 0,97 3,01 0,84 3,76

Kupiec P 73,99 17,52 90,69 0,00* 43,99 0,51*

Christofferson P

96,57 77,24 98,16 98,89 87,15 89,66

Alphabet, tGARCH(1,1)

Std 𝛼 = 1%

Std 𝛼 = 5%

Norm 𝛼 = 1%

Norm 𝛼 = 5%

Nig 𝛼 = 1%

Nig 𝛼 = 5%

Exceedances 0,88 4,21 0,93 3,01 0,75 3,68

Kupiec P 58,21 7,82*** 73,99 0,00* 21,94 0,26*

Christofferson P

94,61 84,75 88,95 94,60 63,87 91,72

Table 2 Alphabet test results, showing exceedances in percent of the VaR models and the p values of Kupiec’s and Christoffersen’s tests.

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sGARCH(1,1)

Std 𝛼 = 1%

Std 𝛼 = 5%

Norm 𝛼 = 1%

Norm 𝛼 = 5%

Nig 𝛼 = 1%

Nig 𝛼 = 5%

Exceedances 0,88 4,92 1,11 4,43 0,79 4,12

Kupiec P 58,07 86,16 61,18 20,77 31,72 4,88

Christofferson P

80,35 64,53 42,91 55,94 91,72 49,56

Exxon Mobil, eGARCH(1,1)

Std 𝛼 = 1%

Std 𝛼 = 5%

Norm 𝛼 = 1%

Norm 𝛼 = 5%

Nig 𝛼 = 1%

Nig 𝛼 = 5%

Exceedances 1,01 4,87 1,19 4,43 0,66 4,07

Kupiec P 92,60 78,59 36,21 20,77 8,84*** 3,81**

Christofferson P

29,88 6,20 17,38 15,77 14,51 6,72***

Exxon Mobil, tGARCH(1,1)

Std 𝛼 = 1%

Std 𝛼 = 5%

Norm 𝛼 = 1%

Norm 𝛼 = 5%

Nig 𝛼 = 1%

Nig 𝛼 = 5%

Exceedances 0,79 4,65 1,15 4,25 0,62 3,81

Kupiec P 31,72 44,60 47,72 9,61*** 5,14*** 0,69*

Christofferson P

76,33 8,60*** 34,84 19,62 4,66** 5,99***

Table 3 Exxon test results, showing exceedances in percent of the VaR models and the p values of Kupiec’s and Christoffersen’s tests.

General Electric, sGARCH(1,1)

Std 𝛼 = 1%

Std 𝛼 = 5%

Norm 𝛼 = 1%

Norm 𝛼 = 5%

Nig 𝛼 = 1%

Nig 𝛼 = 5%

Exceedances 0,84 5,36 1,24 4,74 0,75 4,96

Kupiec P 43,87 43,34 26,72 57,21 21,86 93,83

Christofferson P

67,47 69,96 3,03** 66,92 58,10 86,06

General Electric, eGARCH(1,1)

Std 𝛼 = 1%

Std 𝛼 = 5%

Norm 𝛼 = 1%

Norm 𝛼 = 5%

Nig 𝛼 = 1%

Nig 𝛼 = 5%

Exceedances 0,93 4,74 1,41 4,34 0,84 4,74

Kupiec P 73,84 57,21 6,02*** 14,39 43,87 57,21

Christofferson P

4,51** 72,16 8,23*** 90,79 33,66 99,10

General Electric, tGARCH(1,1)

Std 𝛼 = 1%

Std 𝛼 = 5%

Norm 𝛼 = 1%

Norm 𝛼 = 5%

Nig 𝛼 = 1%

Nig 𝛼 = 5%

Exceedances 0,97 5,00 1,37 4,43 0,84 4,60

Kupiec P 90,52 98,45 9,10*** 20,77 43,87 38,92

Christofferson P

2,68** 99,13 4,35** 84,42 39,77 87,21

Table 4 General Electric test results, showing exceedances in percent of the VaR models and the p values of Kupiec’s and Christoffersen’s tests.

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Company VaR 𝛼 = 1% VaR 𝛼 = 5%

Apple All models pass tests. TG-Norm

Alphabet SG-Norm, EG-Std, EG-Norm, EG- Nig, TG-Std, TG-Norm, TG-Nig

SG-Std, EG-Std, TG-Std

Exxon Mobil All pass SG-Std, SG-Norm, EG-Std, EG-Norm,

TG-Std, TG-Norm

General Electric All pass All pass

Table 5 Models that pass Kupiec’s test at 5% level.

The first test performed, Kupiecs test of whether the model exhibits the correct amount of exceedances or not yielded varying results for the different companies. Interestingly enough all models pass the test for Apple with a 1% 𝛼. However for the 5% 𝛼 only the TG-Norm combination proved successful. For Alphabet most models passed the 1% VaR but only the models with assumed t-distributed errors passed the 5% VaR test. For Exxon Mobil all models passed the tests save for the ones with assumed Normally Inverse Gaussian distributed errors at 5% VaR. For General Electric any model seems to do.

Apple All pass SG-Std, SG-Norm, SG- Nig, EG-Std,

EG-Nig

Alphabet All pass All pass

Exxon Mobil All pass save for TG-Nig All pass General Electric SG-Std, SG-Nig, EG-Norm, EG-

Nig, TG-Nig

All pass Table 6 Models that pass Christoffersen’s test at 5% level.

Christoffersen’s Exceedance Independence Test show that the exceedances of Apple are all

independent from one another at the 1% VaR. For the 5% VaR however only the Standard GARCH models and two Exponential GARCH models, the EG-Std and the EG-Nig pass the independence test.

For Alphabet all of the models pass the test. All models pass the test save for the Threshold GARCH with NIG distributed standard errors at the 1% VaR. For the 5% VaR all models pass. The models which fit for General Electric with the 1% VaR are all of the models paired with the Nig errors as well as the Standard GARCH with t-distributed errors and the Exponential GARCH with normal-

distributed errors. For the 5% VaR all of the models fit.

Apple All pass None

Alphabet SG-Norm, EG-Std, EG-Norm, EG- Nig, TG-Std, TG-Norm, TG-Nig

SG-Std, EG-Std, TG-Std

Exxon Mobil All pass save for TG-Nig SG-Std, SG-Norm, EG-Std, EG-Norm, TG-Std, TG-Norm

General Electric SG-Std, SG-Nig, EG-Norm, EG- Nig, TG-Nig

All pass Table 7 Models that pass both Kupiec’s and Christoffersen’s tests at the 5% level.

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The models which passed both the ‘Kupiec’s Test’ of the proportion of exceedances and the

‘Christoffersen’s Exceedance Independence Test’ were: For Apple at the 1% VaR, all of the models passed but none at the 5% VaR level. For Alphabet all but the Standard GARCH paired with the Nig and T-distributions passed. At the 5% level however all of the GARCH models proved succesfull but only if performed with assumed T-distributed errors. Exxon Mobil was properly modeled according to both tests at the 1% VaR level with all models and distributions except for the Threshold GARCH with assumed NIG-errors. For the 5% VaR all of the models passed both tests except for those with assumed NIG-errors. For General Electric on the other hand all of the models with NIG-errors passed the tests for a 1% VaR as well as the SGARCH with T-distributed errors and the EGARCH with Normally-distributed errors. At the 5% VaR all of the models passed both tests.

Company

Apple None

Alphabet EG-Std, TG-Std

Exxon Mobil SG-Std, SG-Norm, EG-Std, EG-Norm, TG-Std, TG-Norm General Electric SG-Std, SG-Nig, EG-Norm, EG-Nig, TG-Nig

Table 8 Models which passed both tests for both 1% and 5% VaR estimations.

No GARCH-Probability function combination properly managed to capture both the 1% and 5% VaR for Apples returns. For Alphabet the Exponential GARCH and the Threshold GARCH paired with student T-distributions successfully modelled the VaR at both 1 and 5% with a suitable number of exceedances and exceedance independence. Exxon Mobil was properly modeled by all of the models except those using NIG-distributed errors. General Electric on the other hand was properly modeled for both 1 and 5% VaR estimates by any model paired with the NIG-distribution as well as the combination SGARCH with T-distributed errors and the EGARCH with Normally-distributed errors.

Analysis and Conclusion

The Value at Risk for the two tech giants proved to be more difficult to model than the petroleum producer Exxon Mobil or the conglomerate General Electric. A quick examination of the time series plots for the companies gives a hint of the tech stocks being more unpredictable and volatile with more sharp spikes in volatility. The tests did not yield a single valid candidate model which could accurately predict Apples Value at Risk for alphas of 1 and 5 percent simultaneously. For Alphabet the Students-t distribution combined with either of the direction sensitive EGARCH or TGARCH models proved to be effective predictive models. The Value at Risk of Exxon Mobil was accurately predicted by any one of the SGARCH, EGARCH or TGARCH models given an assumption of either Students-t or Normally-distributed errors however no model with assumed Normal Inversed Gaussian errors was successful. General Electric on the other hand gave clear results favoring the Normal Inverse Gaussian distribution where all of the models with assumed NIG errors were significant and only one model each for the Students-t and Normal distributions.

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Out of the probability distributions tested, the Students-T distribution yielded the most successfull models with 6, wheras only 4 combinations with normally distributed and 3 combinations with NIG distributed errors were successful. That a fat-tailed distribution won the bout is as should be expected since the error distributions seen in financial timeseries are usually fat-tailed as per the stylized facts presented by Rama Cont (2001). However as pertains to the GARCH models, the EGARCH model scored 5 successful models wheras the other scored 4 each. These small differences can hardly be taken as evidence of any models superiority over another for a broader population of companies.

Although one interesting finding is a clear clustering of results, which seem to indicate that the probability distribution assumed is more important for successful modeling than what GARCH model is used. The asymmetric EGARCH and TGARCH models did not stick out as being particularly more efficient than the Standard GARCH in this small survey. This result is similar to the one that

Timotheos Angelidis, Alexandros Benos and Stavros Degiannakis (2004) presented when they only used Kupiec´s and Christofferson´s tests for evaluating fit. The very differing results for each company suggests that it is highly unlikely that a single model would fit every stock but rather that similar surveys would have to be performed on a case by case basis in order to properly calculate their VaR estimates.

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References

Bollerslev, Tim (1987) A Conditionally Heteroscedastic Time Series Model for Speculative Prices and Rates of Return. The Review of Economics and Statistics Vol 69 (No 3) pp 542-547.

Bølviken, Erik and Benth, Fred Espen (2000) Quantification of Risk in Norwegian Stocks via the Normal Inverse Gaussian Distribution. Proceedings AFIR 2000. Tromsø, Norway

Chia‐lin Chang, Juan‐Ángel Jiménez‐Martín, Michael McAleer, Teodosio Pérez‐Amaral, (2011) "Risk management of risk under the Basel Accord: forecasting value‐at‐risk of VIX futures", Managerial Finance, Vol. 37 Issue: 11, pp.1088-110

Christoffersen, Peter (1998). Evaluating interval forecasts. International Economic Review. Vol. 39 (No. 4) pp 841-862.

Cont, Rama (2001) Empirical properties of asset returns: stylized facts and statistical issues.

Quantitative Finance Vol. 1 pp 223-236

Engle, Robert F. and Bollerslev, Tim (1986) Modelling the Persistence of Conditional Variances.

Econometric Reviews Vol 5 1986 (No. 1) pp 1-50.

Engle, Robert F. (1982) Autoregresive Conditional Heteroscedasticity with estimates of the Variance of United Kingdom Inflation. Econometrica. Vol. 50 (No. 4)

Joneau, Eric, Poon, Ser-Huang, Rockinger, Michael (2007) Financial Modeling Under Non-Gaussian Distributions. London: Springer

Kupiec, Paul H. (1995). Techniques for Verifying the Accuracy of Risk Measurement Models. The J. of Derivatives. Vol. 3 (No. 2)

Nelson, Daniel B. (1991) Conditional Heteroscedasticity in Asset Returns: A New Approach.

Econometric Society. Vol. 59 (No. 2) pp 347-370.

Nobelprize.org (2003) The Prize in Economic Sciences 2003 - Press Release

http://www.nobelprize.org/nobel_prizes/economic-sciences/laureates/2003/press.html [2018-01- 02]

Timotheos Angelidis, Alexandros Benos, Stavros Degiannakis, (2004) ”The use of GARCH models in VaR estimation”, Statistical Methodology, Vol. 1, pp.105-128.

Rachev, Svetlozar T., Mittnik, Stefan, Fabozzi Frank J., Focardi, Jai, Teo (2007) Financial Econometrics From Basic to advanced modeling techniques, Hoboken: Wiley

Zakoïan, Jean-Michel (1994) Threshold heteroskedastic models Journal of Economic Dynamics and Control. Vol. 18 (No. 5) pp 931-955

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Appendix 1

Parameter estimations for VaR models. Coded as follows, AP: Apple, Alp: Alphabet, Ge: General Electric, Ex: Exxon Mobil. SG: Standard Garch, EG: Exponential Garch, TG: Threshold Garch. Norm:

Normal distribution, STD: Students-T, NIG: Normal Inverse Gaussian.

AP SG Norm omega alpha1 beta1

1 0,306295 0,117368 0,852631

2 0,289923 0,111899 0,860874

3 0,283519 0,104876 0,864882

4 0,232039 0,101662 0,87201

5 0,144027 0,094668 0,888518

6 0,155467 0,088604 0,891508

7 0,161511 0,089527 0,888584

8 0,18896 0,09512 0,879193

9 0,176565 0,091195 0,882734

10 0,142276 0,084339 0,893445

11 0,103314 0,081919 0,901413

12 0,085867 0,079279 0,906624

13 0,07682 0,079435 0,907511

14 0,070917 0,075933 0,911583

15 0,088311 0,082912 0,902009

16 0,087937 0,081006 0,903246

17 0,094326 0,081068 0,901195

18 0,1091 0,086735 0,893265

19 0,108737 0,084589 0,894632

20 0,098549 0,081638 0,89886

21 0,09449 0,078428 0,903627

22 0,092832 0,076696 0,906623

23 0,096978 0,076547 0,905554

24 0,099123 0,076948 0,904054

25 0,099149 0,074594 0,906015

26 0,093776 0,076166 0,904987

27 0,105785 0,074991 0,903243

28 0,131753 0,080346 0,891953

29 0,12363 0,08368 0,89002

30 0,114952 0,078589 0,896195

31 0,10898 0,077542 0,898034

32 0,111094 0,077534 0,897486

33 0,110002 0,077923 0,896885

34 0,108792 0,080962 0,89411

35 0,112934 0,082448 0,891877

36 0,114504 0,082183 0,891371

37 0,117041 0,082789 0,890319

38 0,119638 0,080305 0,891641

39 0,116017 0,080681 0,891484

40 0,128347 0,082191 0,88658

41 0,124604 0,084127 0,885156

42 0,124536 0,082293 0,886202

43 0,108596 0,082884 0,889524

44 0,106879 0,080629 0,891624

45 0,10799 0,080662 0,890925

46 0,103081 0,079111 0,893261

Ap SG STD omega alpha1 beta1 shape

1 0,280452 0,106569 0,865185 7,552712

2 0,250252 0,099732 0,876185 8,628924

3 0,241469 0,091852 0,881578 8,940503

4 0,191784 0,086847 0,890524 8,612583

5 0,099771 0,078179 0,910045 7,932404

6 0,106238 0,071994 0,914422 7,786365

7 0,109212 0,073199 0,912212 7,671384

8 0,116409 0,084319 0,902032 7,148482

9 0,105678 0,080005 0,905979 7,165117

10 0,081673 0,073778 0,915034 6,948246

11 0,045744 0,073736 0,921408 6,558602

12 0,050955 0,074028 0,919727 6,703125

13 0,047392 0,073932 0,919827 6,785395

14 0,052095 0,071887 0,920115 7,192561

15 0,063707 0,077296 0,9133 7,179022

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16 0,062672 0,07448 0,915561 7,193273

17 0,058988 0,071878 0,918469 6,825677

18 0,065508 0,072168 0,917326 6,584304

19 0,066429 0,072347 0,91654 6,405191

20 0,064916 0,070642 0,917642 6,626739

21 0,067776 0,070185 0,918271 6,510747

22 0,067828 0,065532 0,922754 6,084107

23 0,070254 0,065041 0,922323 6,173944

24 0,069931 0,064951 0,922061 6,055032

25 0,068577 0,062237 0,924872 5,951644

26 0,065729 0,063349 0,923726 6,009311

27 0,063125 0,057996 0,929333 5,746969

28 0,061039 0,060029 0,928134 5,438589

29 0,057945 0,062331 0,926141 5,466586

30 0,047567 0,057502 0,933312 5,277323

31 0,046654 0,05625 0,934185 5,379759

32 0,051165 0,05753 0,931924 5,448957

33 0,05259 0,057546 0,931035 5,56857

34 0,05503 0,061107 0,927011 5,565364

35 0,058077 0,062753 0,925067 5,493149

36 0,058236 0,061514 0,925926 5,496248

37 0,060546 0,061978 0,925014 5,532824

38 0,062639 0,062314 0,924129 5,413505

39 0,059899 0,062465 0,924232 5,38935

40 0,061753 0,063307 0,923223 5,226812

41 0,059137 0,064321 0,922661 5,206073

42 0,037897 0,058977 0,934009 4,912027

43 0,022132 0,055736 0,941592 4,790828

44 0,022877 0,054237 0,942826 4,723069

45 0,022476 0,05286 0,944099 4,713116

46 0,02096 0,052304 0,944997 4,689189

Ap SG Nig omega alpha1 beta1 skew shape

1 0,271749 0,1024 0,869902 -0,23344 2,801577

2 0,235761 0,096505 0,881012 -0,21243 3,405778

3 0,223342 0,089295 0,886888 -0,21718 3,546147

4 0,185507 0,084791 0,89439 -0,20256 3,182571

5 0,102629 0,077378 0,911476 -0,20249 2,782664

6 0,112232 0,071524 0,914814 -0,16633 2,605645

7 0,110577 0,070257 0,91569 -0,17222 2,478097

8 0,126232 0,081177 0,904094 -0,1436 2,170659

9 0,116294 0,077628 0,907082 -0,13868 2,210168

10 0,09151 0,07222 0,915448 -0,13589 2,088476

11 0,055146 0,07199 0,921703 -0,12335 1,905525

12 0,054335 0,071394 0,922412 -0,14206 1,904541

13 0,04913 0,071241 0,922729 -0,13959 1,974012

14 0,054265 0,069744 0,922501 -0,13367 2,147448

15 0,065635 0,075072 0,915727 -0,1279 2,141287

16 0,064608 0,072618 0,917535 -0,12147 2,160172

17 0,062339 0,070186 0,92031 -0,13222 1,946823

18 0,069364 0,071642 0,917645 -0,10605 1,863893

19 0,070262 0,072006 0,916588 -0,10537 1,800968

20 0,068414 0,0705 0,917776 -0,11157 1,884465

21 0,069994 0,069543 0,918866 -0,10519 1,849406

22 0,069401 0,065051 0,923139 -0,10602 1,709338

23 0,072499 0,064955 0,922259 -0,11422 1,753776

24 0,072475 0,065185 0,921454 -0,10054 1,703756

25 0,071313 0,062684 0,924016 -0,10433 1,661657

26 0,06905 0,063878 0,922719 -0,10546 1,68626

27 0,067426 0,058609 0,927841 -0,10939 1,598871

28 0,069566 0,060893 0,92514 -0,09432 1,457398

29 0,066554 0,063375 0,922992 -0,09506 1,465808

30 0,054728 0,058158 0,930759 -0,10329 1,390271

31 0,053228 0,056827 0,931931 -0,10586 1,436587

32 0,057786 0,058166 0,929681 -0,10482 1,454724

33 0,059223 0,058415 0,92856 -0,10094 1,509052

34 0,061296 0,062117 0,924464 -0,10178 1,509615

35 0,064189 0,0635 0,922654 -0,10246 1,482242

36 0,06415 0,062269 0,923476 -0,09649 1,481956

37 0,066292 0,062586 0,922697 -0,09748 1,50153

38 0,068239 0,06255 0,922054 -0,09889 1,458007

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39 0,065608 0,063071 0,921839 -0,10382 1,440473

40 0,069222 0,064289 0,919798 -0,09428 1,367699

41 0,066179 0,06561 0,919051 -0,09633 1,353779

42 0,047274 0,059721 0,929979 -0,09459 1,226565

43 0,030333 0,05598 0,938126 -0,08611 1,18111

44 0,02961 0,053706 0,940426 -0,09068 1,154434

45 0,029047 0,052317 0,94173 -0,0895 1,152599

46 0,02724 0,05168 0,942733 -0,08699 1,143515

Ap TG Norm omega alpha1 beta1 eta11

1 0,180715 0,086261 0,871457 0,847349

2 0,175156 0,082026 0,876823 0,870442

3 0,166155 0,072457 0,886957 0,963364

4 0,156576 0,075406 0,887102 0,968461

5 0,129607 0,079976 0,892184 0,872457

6 0,1263 0,079393 0,893149 0,854652

7 0,130884 0,084667 0,88641 0,817514

8 0,151275 0,094154 0,871789 0,828805

9 0,15111 0,096318 0,868536 0,792561

10 0,140502 0,097954 0,870477 0,760016

11 0,124498 0,102256 0,872415 0,669938

12 0,112735 0,104168 0,875098 0,636062

13 0,105775 0,109848 0,872433 0,571643

14 0,097373 0,106896 0,877497 0,543323

15 0,107838 0,111671 0,869386 0,561745

16 0,105144 0,109804 0,871254 0,537431

17 0,114269 0,115595 0,863032 0,552825

18 0,124568 0,120121 0,855347 0,569479

19 0,124656 0,118909 0,855817 0,585547

20 0,12004 0,117755 0,858137 0,58004

21 0,116004 0,111265 0,864783 0,598125

22 0,107732 0,105947 0,873035 0,592577

23 0,108923 0,10561 0,872349 0,592899

24 0,108632 0,108212 0,869796 0,576489

25 0,113794 0,1091 0,866756 0,592427

26 0,109215 0,110366 0,867288 0,575574

27 0,108714 0,10946 0,868032 0,568855

28 0,111893 0,113144 0,863924 0,573524

29 0,102916 0,113487 0,867504 0,559375

30 0,102875 0,110158 0,869581 0,558516

31 0,100948 0,110156 0,870123 0,554997

32 0,101637 0,108883 0,87069 0,567321

33 0,099355 0,108707 0,871327 0,556001

34 0,095779 0,11126 0,870743 0,52616

35 0,096493 0,109845 0,871372 0,538218

36 0,097098 0,109465 0,871075 0,535557

37 0,094443 0,106728 0,87412 0,529881

38 0,095277 0,1049 0,875042 0,550762

39 0,094336 0,107036 0,873379 0,536101

40 0,100658 0,110601 0,867779 0,55965

41 0,097276 0,112749 0,867411 0,540246

42 0,105273 0,110515 0,864965 0,585699

43 0,0942 0,10977 0,870895 0,570139

44 0,095308 0,108662 0,870786 0,563666

45 0,095672 0,109144 0,870042 0,559682

46 0,094378 0,108614 0,87077 0,555273

AP TG STD omega alpha1 beta1 eta11 shape

1 0,148243 0,08717 0,882366 0,714029 8,458949

2 0,14439 0,081568 0,888026 0,765527 10,03576

3 0,137768 0,071204 0,897879 0,858634 10,58898

4 0,128879 0,072599 0,899265 0,897218 10,33871

5 0,100944 0,07575 0,906358 0,824481 9,243207

6 0,098152 0,07359 0,908611 0,824092 9,030634

7 0,103731 0,080584 0,900459 0,780139 8,881434

8 0,124752 0,098349 0,879775 0,747539 8,594958

9 0,122683 0,099481 0,87818 0,717381 8,484088

10 0,111662 0,099428 0,881853 0,693798 8,197386

11 0,092312 0,105491 0,884502 0,597164 7,485707

12 0,087579 0,107667 0,884364 0,584208 7,510735

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13 0,082012 0,113848 0,881036 0,531041 7,382313

14 0,07552 0,108117 0,887192 0,507344 7,670606

15 0,086211 0,113782 0,87837 0,527845 7,913729

16 0,084735 0,111536 0,880105 0,516158 7,766613

17 0,090149 0,115581 0,874903 0,522796 7,578769

18 0,102137 0,121021 0,865822 0,538623 7,563055

19 0,100363 0,118847 0,867933 0,550177 7,398828

20 0,099236 0,11726 0,869008 0,553417 7,589723

21 0,096544 0,111616 0,874355 0,565739 7,436208

22 0,095779 0,108706 0,876862 0,557199 6,896772

23 0,09558 0,106815 0,877893 0,557456 7,011511

24 0,094357 0,109074 0,87617 0,543267 6,842223

25 0,098462 0,109139 0,874298 0,560737 6,765411

26 0,094535 0,109402 0,87525 0,546053 6,805992

27 0,089332 0,106217 0,87984 0,528333 6,502667

28 0,082804 0,109131 0,880978 0,506973 6,118552

29 0,076581 0,107606 0,884646 0,500364 6,163383

30 0,070388 0,103634 0,890619 0,491342 5,891979

31 0,070819 0,10325 0,890318 0,495259 5,992366

32 0,072351 0,102288 0,890278 0,512287 6,081848

33 0,071918 0,102046 0,890119 0,50855 6,186217

34 0,06951 0,104549 0,889157 0,484965 6,157731

35 0,0699 0,102872 0,890205 0,497387 6,104658

36 0,070403 0,10177 0,890578 0,499372 6,102183

37 0,068548 0,098905 0,893333 0,496304 6,117314

38 0,068789 0,097868 0,893966 0,500009 6,000402

39 0,067423 0,099312 0,893143 0,486616 5,965581

40 0,071574 0,104483 0,887626 0,501944 5,831041

41 0,06899 0,106479 0,88716 0,48925 5,794049

42 0,060414 0,107455 0,891563 0,493113 5,457021

43 0,048605 0,103719 0,900483 0,480456 5,328033

44 0,048549 0,102217 0,901627 0,475196 5,208435

45 0,048789 0,102238 0,901461 0,476339 5,179784

46 0,047073 0,100994 0,903102 0,466645 5,137368

Ap TG Nig omega alpha1 beta1 eta11 skew shape

1 0,143467 0,081675 0,888092 0,719207 -0,19879 3,125936

2 0,139836 0,076184 0,893688 0,793629 -0,19461 4,105617

3 0,130344 0,066501 0,904244 0,879368 -0,19414 4,279005

4 0,123883 0,067534 0,905275 0,927169 -0,18568 4,002017

5 0,096566 0,070127 0,912628 0,848887 -0,18457 3,363061

6 0,094591 0,068001 0,914498 0,857418 -0,15213 3,177279

7 0,097141 0,072728 0,909274 0,812376 -0,15203 3,005501

8 0,123612 0,091655 0,885551 0,775263 -0,1214 2,823638

9 0,122439 0,093412 0,883159 0,741458 -0,11932 2,798109

10 0,111743 0,094028 0,886201 0,711562 -0,11523 2,630058

11 0,092814 0,099951 0,88869 0,61164 -0,10497 2,288208

12 0,086098 0,101276 0,890205 0,5951 -0,12078 2,240762

13 0,080399 0,107459 0,886779 0,535103 -0,11191 2,218307

14 0,074292 0,102152 0,892552 0,514258 -0,11194 2,355579

15 0,085105 0,108089 0,883476 0,53614 -0,10865 2,458972

16 0,084103 0,106421 0,884546 0,525217 -0,10736 2,411954

17 0,091101 0,111173 0,878424 0,534385 -0,11844 2,263739

18 0,103503 0,117633 0,86818 0,547347 -0,0916 2,275186

19 0,1017 0,115865 0,869962 0,557618 -0,09235 2,209395

20 0,100488 0,114418 0,871071 0,560516 -0,098 2,284479

21 0,097308 0,108787 0,876415 0,571934 -0,09054 2,247435

22 0,095909 0,105534 0,879382 0,564362 -0,09427 2,060143

23 0,095448 0,103642 0,880557 0,56252 -0,10089 2,113002

24 0,094469 0,106291 0,87833 0,54811 -0,08771 2,036835

25 0,098414 0,106387 0,876535 0,564793 -0,09206 2,001678

26 0,094862 0,106812 0,877256 0,550203 -0,09394 2,018506

27 0,090375 0,103783 0,881297 0,534296 -0,09694 1,911987

28 0,086162 0,107236 0,880801 0,519569 -0,08034 1,742486

29 0,079891 0,105927 0,884365 0,512819 -0,08004 1,759739

30 0,074048 0,102025 0,890064 0,504208 -0,09154 1,646606

31 0,074404 0,101612 0,889863 0,508092 -0,0945 1,698156

32 0,076125 0,100922 0,889608 0,525892 -0,096 1,72795

33 0,075486 0,100714 0,889498 0,521512 -0,09272 1,778862

34 0,072905 0,103426 0,888433 0,495479 -0,09426 1,767482

35 0,073225 0,101757 0,889421 0,506341 -0,09479 1,745234

Values at Risk

Values at Risk

Uppsala University Hampus Eliasson Bachelor Thesis in Statistics

Fall 2017

Supervisor: Lars Forsberg

Abstract

Contents

Introduction

Data and Descriptive Statistics

Theoretical Background

Methodology

Results

Analysis and Conclusion

References

Appendix 1