Estimating risk measures
using an EVT-GARCH approach
Joran Houtman
Afstudeerscriptie voor de
Bachelor Actuari¨ele Wetenschappen Universiteit van Amsterdam
Faculteit Economie en Bedrijfskunde Amsterdam School of Economics
Auteur: Joran Houtman
Studentnr: 5833361
Email: johannes.houtman@student.uva.nl
Datum: June 19, 2014
Estimating risk measures using EVT-GARCH — Joran Houtman iii
Abstract
In this thesis, a method is proposed for quantifying and estimating the risk of investing in Mexico, Indonesia, Nigeria, and Turkey (the MINT countries). In this case, the risk of investing is defined as a combination of the probability and magnitude of large losses. Appropriate risk measures to quantify this type of risk are Value-at-risk (VaR) and Ex-pected shortfall (ES). In order to account for any leptokurtosis and asymmetry in the return series, Extreme value theory (EVT) is used to estimate a generalized Pareto dis-tribution (GPD). This avoided having to make assumptions on the specific disdis-tribution of the return series. However, EVT needs approximately independent and identically distributed data series as an input. Therefore, the return series were first approximated using a generalized autoregressive conditional heteroskedasticity (GARCH) model. This model filtered out any volatility clustering and gave the opportunity to make predictive statements.
The results showed that, conditional on current volatility, Indonesia, Nigeria, and Turkey all had a higher probability of large losses compared to the benchmark S&P500, while Mexico was similar in risk and return to the benchmark. However, the dynamic model needs further research and backtesting before it can be used in practice.
Keywords Extreme value theory, Generalized Pareto distribution, GARCH models, Risk
Preface v
1 Introduction 1
2 Theory 3
2.1 Basic probability theory . . . 3
2.2 Extreme value theory . . . 4
2.2.1 The Fisher-Tippett theorem. . . 4
2.2.2 The Pickands-Balkema-de Haan theorem. . . 5
2.2.3 Estimating the unconditional tail distribution . . . 6
2.3 The GARCH model . . . 6
2.4 Risk measures . . . 7
2.4.1 Choosing the right risk measures . . . 8
2.4.2 Estimation theory of risk measures . . . 9
3 Data and methods 11 3.1 The MINT countries . . . 11
3.2 Exploratory data analysis . . . 12
3.2.1 Mexico exploratory data analysis . . . 13
3.2.2 Indonesia exploratory data analysis. . . 14
3.2.3 Nigeria exploratory data analysis . . . 15
3.2.4 Turkey exploratory data analysis . . . 16
3.2.5 Benchmark exploratory data analysis. . . 16
3.3 Obtaining stationary residuals. . . 16
3.4 Selecting a threshold . . . 16
3.5 Estimating the risk measures . . . 17
4 Results and discussion 18 4.1 GARCH model results . . . 18
4.2 GPD estimation results . . . 21
4.3 Risk measure estimates . . . 22
5 Conclusion 25
Bibliography 26
Appendix A: Graphs 29
Appendix B: GARCH tests 40
Preface
It took some time before reaching this point, but I can finally say that with great pleasure I have been working on this thesis. I want to thank my supervisor, dr. Sami Umut Can, for giving close technical guidance and helping to define the boundaries of the subject. I want to thank drs. Rob van Hemert for his general guidance on structuring and writing style. I specifically want to thank my girlfriend, who has been the best support and most understanding for the past years, during which I have spent too many evenings and weekends studying. Lastly, I also want to thank my parents for their support and enabling me to grab this opportunity.
Chapter 1
Introduction
Most people have probably heard the acronym BRIC before. The term refers to a group of the four biggest and fastest growing emerging markets; namely Brazil, Russia, India and China (O’Neill, 2001). Fewer people have probably heard of the creator of this term, Jim O’Neill, the former chairman of Goldman Sachs Asset Management. With such a status, merely making up a new acronym for four countries is enough to spark the interest of researchers and investment managers across the globe. Recently he has introduced the MINT countries (O’Neill,2013): Mexico, Indonesia, Nigeria and Turkey. According to O’Neill, they share very favorable demographics and have interesting eco-nomic prospects, implying this results in potentially high returns.
When investing or trading in these countries, it is the risk manager’s job to limit the downside risk, while still profiting from positive returns. One way of achieving this is not to trade when the risk of large losses is too high. In other words, only play when the odds are in your favor. A necessary step is to define what ‘large losses’ are and when the risk is ‘too high’. Therefore, a method is needed to quantify large losses and estimate the probability of them occurring. This probability could be estimated over any time frame, but since extreme positive and negative outcomes could cancel each other out in the long run, it will make more sense to focus on daily probabilities. Therefore the emphasis of this paper will be to provide dynamic daily estimators of risk for the four MINT countries. Based on the differences between these daily estimators of risk, the MINT countries can be compared to conclude if it makes sense to group these countries at all.
There are four major obstacles which need to be overcome when selecting an appro-priate method to estimate the probability of large losses. First and foremost, the data generating process (or distribution) is unknown. Often this is ‘solved’ by assuming a normal or some other distribution and then choosing the parameters of this distribution to best fit the historical data. However, assuming a normal distribution is misleading when the goal is to accurately estimate extreme losses, since most of the mass of the distribution will usually be away from the tails. This results in a very good fit for most of the data, but at the cost of accuracy in the tails. A second problem is leptokurtosis, also known as ‘fat’ or ‘heavy’ tails. This means more mass than expected is in the tails of the distribution. In other words, large losses tend to realize more often then would normally be expected. AsMcNeil and Frey (2000) indicate, this is a very common characteristic of financial time-series. A third obstacle the method needs to address is autocorrelation, literally the ”correlation with itself”. Since this is practically what defines a financial time series, many methods exist to test and correct for autocorrelation. The last main obstacle for providing reliable estimates is volatility clustering. This phenomenon means
that periods of high volatility tend to stay relatively volatile and periods of low volatility tend to stay relatively calm. When not correcting or ‘conditioning’ for this phenomenon, the assumption is implicitly made that markets always have the same volatility (i.e. they exhibit homoscedasticity). This results in overestimating the risk in calm periods and, more dangerously, underestimating the risk in high-volatility periods.
A method to handle the first two problems has been set out by McNeil (1997) building on the foundations of extreme value theory (EVT) laid out by Fisher and Tippett (1928), Gnedenko (1943), and more recently De Haan and Ferreira (2007), among others. Later McNeil and Frey (2000) expand their framework to incorporate the conditioning for autocorrelation and volatility clustering in financial time series by using a generalized autoregressive conditional heteroscedasticity (GARCH) model. This EVT-GARCH approach gives a framework to estimate the distribution of the tails with minimal assumptions about the data generating process. Using this approach, tail risk measures like value-at-risk (VaR) and expected shortfall (ES) will be estimated.
The EVT-GARCH approach will be applied to the gross MSCI indices of the MINT countries in local currencies. The data consists of the llast 2000 daily returns over a time frame of approximately 8 years. As a benchmark, the S&P 500 daily gross returns will be used. This gives the possibility to compare the excess risk for the MINT countries in relation to a developed market. As is often done in practice, the data will be transformed to the negative log returns, so that large losses are positive and give an approximate percentage change.
For each MINT country, an AR-GARCH(1,1) model will be estimated, resulting in a stationary residual series. These residuals are independent and identically distributed (iid) and can be used for EVT. The GARCH model itself will yield the one-day ex-pected return and volatility forecast for estimating the risk measures. For each residual series a positive threshold will be selected which will mark the start of the tail of the distribution. The rest of the data is not relevant and will be discarded. A generalized Pareto distribution (GPD) will be fitted to the remaining data using maximum likeli-hood estimation (MLE). Using this fitted distribution, the current VaR and ES can be estimated. When combining these estimates with the one-day forecast expected return and volatility, the next day VaR and ES can be estimated. These estimators are dy-namic since they can be updated every day to incorporate the latest information and give tomorrow’s predictions. Management can then decide if they want to trade and take the risk, or withdraw from trading for that day.
The findings of applying the EVT-GARCH approach for the MINT countries are positive. It will be shown that this method results in a reasonably good description of the sample data of large losses for all MINT countries. Based on the fitted distributions, the risk measures are estimated and compared. The conditional forecast risk measures will show that Mexico currently has a different risk profile, compared to the other MINT countries.
The structure of this paper will be as follows. First the theory behind the EVT-GARCH model and the risk measures VaR and ES will be presented. In Data and Methods, the dataset will be analyzed and the aforementioned methods will be applied. After this the results will be compared and interpreted before reaching the conclusion.
Chapter 2
Theory
In this section the theory used for this research will be presented. For completeness, some basic probability theory will be discussed, after which the extreme value theory will be given in more detail. A short explanation of the GARCH model is given as well, where some familiarity with econometrics is assumed. Lastly, the characteristics of two risk measures will be discussed, combined with the theory for estimating them.
2.1
Basic probability theory
When making statements about the probability of certain events occurring in the future, the first thing to consider is what has happened in the past. The main assumption is that these past events, or more formally outcomes of a random variable, follow a certain pattern. It is this pattern, also know as a data generating process or joint probability distribution, that needs to be estimated. It is impossible to know the exact process, but a function can be fitted to historical data that assigns a probability to every possible outcome. We can define a random variable and a cumulative probability distribution function as follows:
Definition 2.1.1. The variable X is a continuous random variable if it is a function X : Ω → R, where Ω is the set of all possible outcomes of a random process.
Definition 2.1.2. A continuous random variable X has a cumulative distribution func-tion FX(x) = P(X ≤ x) , where P(X ≤ x) =
Rx
−∞fX(r)dr and fX(x) is the probability density function of X.
The parameters of the cdf should be chosen in a way that the cdf follows the sample data as close as possible. When using maximum likelihood estimation (MLE) the n parameters θ = {θ1, θ2, ..., θn} of the cdf are chosen such that the probability of the realized outcome of the random variable actually being generated by the estimated cdf is maximized. The set of distribution functions FX(x|θ) where θ ∈ Θ is called the family of distributions. Here Θ is the set of all possible parameter values.
It should be clear that in order to estimate any sort of risk measures, probabilities of a random variable’s outcomes need to be estimated. However, before these can be estimated, a family of distribution functions for the data generating process needs to be selected and its parameters need to be estimated. Since the aim of this paper is to minimize the exposure to large losses, risk measures are needed that focus on the tail of the distributions. Extreme value theory provides a family of limiting distributions that fit the tails of a large selection of distributions, with the distinct advantage that no assumptions are needed about the actual distribution of the data generating process.
2.2
Extreme value theory
Two main methods for applying EVT will be discussed, namely the block maxima method and the peaks-over-threshold (POT) method. Both of these methods require the assumption that the (adjusted) historical data are realizations of independent, iden-tically distributed (iid) random variables (McNeil, 1997). The methods differ in the way they select extreme values. For the block maxima method the sample is divided in non-overlapping blocks of size n and the maximum of each block is selected to create a new series, which can be used for estimating the extreme value distribution. This method has the disadvantage that when multiple high values occur in one period, only the largest is selected, and if no extreme values occur in a period, a relative ’normal’ value is selected as the maximum. Therefore, especially in homoscedastic time-series, valuable data is discarded (Ferreira and de Haan,2013). Since extreme values are rare by definition, discarding any extreme values could easily lead to wrong conclusions. The POT method uses all sample outcomes larger than a predefined threshold, therefore not discarding any extreme values. However, this gives rise to the problem of selecting an appropriate threshold, which will be discussed later.
2.2.1 The Fisher-Tippett theorem
The Fisher-Tippett theorem is the main reason why EVT is an attractive method for modeling the tail behavior of distribution functions. It needs minimal assumptions on the actual distribution of the extreme values of a random variable. Fisher and Tippett
(1928) show that in the case where a series of random variables are iid, and when adequately rescaled, the cdf of the extreme values converge to a ”generalized extreme value distribution”.
Definition 2.2.1. Let X1, X2, ... be a sequence of iid random variables from an unknown cdf FX(x). Let Yn= Mn− bn/an, where Mn= max X1, ..., Xn. Suppose there exist sequences an∈ R+ and bn∈ R such that
P Mn− bn an
≤ x
= Fn(anx + bn) → H(x), as n → ∞, (2.1)
then Yn converges in distribution to H(x) and FX(x) is in the maximum domain of attraction of H(x), that is FX(x) ∈ MDA H(x).
The theorem ofFisher and Tippett(1928) specifies the limiting distribution function for when FX(x) is in the maximum domain of attraction of H(x):
Theorem 2.2.1. When FX(x) ∈ MDA H(x)
then H(x) is of the type generalized extreme value (GEV) distribution Hξ,µ,σ(x) where Hξ,µ,σ(x) = Hξ (x − µ)/σ and
Hξ(x) = (
exp − (1 + ξx)−1/ξ if ξ 6= 0,
exp − e−x if ξ = 0, (2.2)
for 1 + ξx > 0.
The parameter ξ of the GEV distribution is the shape parameter and determines how heavy the tails are. For ξ > 0 the GEV is heavy tailed and is called the Fr´echet distribution, for ξ = 0 the GEV has normal to moderately fat tails and is called the Gumbel distribution and for ξ < 0 the GEV is short tailed (i.e. has a finite right endpoint) and is called the Weibull distribution. When fitting the GEV to sample block
Estimating risk measures using EVT-GARCH — Joran Houtman 5
maxima data, the method is referred to as the earlier introduced block maxima method (Embrechts et al., 1997). However, the alternative approach given by Balkema and De Haan(1974) andPickands (1975) will be used for this research.
2.2.2 The Pickands-Balkema-de Haan theorem
The application of the Pickands-Balkema-de Haan theorem is also referred to as the previously introduced peaks-over-threshold method. For this method it is important to note the difference between an extreme value (i.e. the value of the outcome of the random variable) and the excess value, or the amount by which the extreme value exceeds a certain threshold. The Pickands-Balkema-de Haan theorem uses the generalized Pareto distribution, which models the excess values of a random variable and is defined below. Definition 2.2.2. The generalized Pareto distribution (GPD) is characterized by its cumulative distribution function:
Gξ,σ(x) = ( 1 − 1 + ξx/σ−1/ξ if ξ 6= 0; 1 − exp − x/σ if ξ = 0, (2.3)
where σ > 0 and x ≥ 0 if ξ ≥ 0 and 0 ≤ x ≤ −σ/ξ if ξ < 0.
Similar to the GEV, the shape parameter ξ divides the GPD in three sub-families. When ξ > 0 the GPD is heavy-tailed and also known as the Pareto distribution with shape parameter α = 1/ξ. When ξ < 0 the distribution is called a type II Pareto distribution and when ξ = 0 the GPD is a regular exponential distribution. The location parameter µ can be introduced to define Gξ,µ,σ(x) = Gξ,σ(x − µ) (McNeil, 1997).
With the GPD defined, an analogous theorem to the Fisher-Tippett theorem for ex-cess values can be introduced. The Pickands-Balkema-de Haan theorem gives a limiting distribution for the extreme values above a certain threshold.
Definition 2.2.3. Define a high threshold u < x0, where x0 is the right endpoint of a distribution F(x), i.e. x0= sup{x ∈ R : F(x) < 1} ≤ ∞. Let the excess value y = x − u, then
Fu(y) := P X − u ≤ y | X > u =
F(y + u) − F(u)
1 − F(u) , (2.4)
for 0 ≤ y < x0− u.
Using this definition, Balkema and De Haan (1974) and Pickands (1975) give the following theorem.
Theorem 2.2.2. If F(x) ∈ MDA Hξ(x), ∃σ(u) > 0 such that lim
u→x0 0≤y<x0sup−u|Fu(y) − Gξ,σ(u)(y)| = 0. (2.5) This essentially states that under MDA conditions, there exists a σ as a function of threshold u, such that if u is chosen to be large, Fu(y) will be uniformly close to Gξ,σ(y) for some ξ and σ. This also gives insight into the trade-off dilemma for choosing the appropriate threshold u. A very high threshold will result in a good theoretical approximation, but also one with a large confidence interval since not many data points can be used to get statistical significance. On the other hand, a low threshold will render a lot of data points, but the convergence theory might not be applicable anymore (McNeil, 1999). To estimate the conditional extreme values instead of only the excess values, the distribution function Gξ,σ(x − u) = Gξ,u,σ(x) can be used (McNeil,1997).
2.2.3 Estimating the unconditional tail distribution
The Pickands-Balkema-de Haan theorem gives a method for estimating the distribution function of excess values, conditional on them exceeding a predefined threshold. But this is not yet a good estimation for the tail of the unconditional distribution function F(x) of the extreme values. The goal is to find an estimator bF(x) for F(x) to use for predicting the size and probabilities of extreme values occurring. Since x = y + u, rewriting equation 2.4gives:
Fu(y) =
F(x) − F(u)
1 − F(u) ⇔
F(x) − F(u) = 1 − F(u)Fu(x − u) ⇔
F(x) = 1 − F(u)Fu(x − u) + F(u), (2.6)
for some unknown value F(u) and distribution Fu(x − u). The results from 2.5 can be used to estimate Fu(x − u) by cFu(x − u) = Gξ,ˆˆσ(x − u), but for F(u) = P(X ≤ u) another method is needed. McNeil (1999) suggests to use historical simulation (HS) to estimate the value of F(u), or in other words, the probability that the outcome of a random variable is smaller than some given threshold u. With HS this probability is just estimated as the fraction of these occurrences in the past relative to the total number of observations n. More formally this can be written as:
b
F(u) = n − Nu
n ,
where Nuis the number of threshold exceedances in the sample. Replacing F(u) by bF(u) and Fu(x − u) by Gξ,ˆˆσ(x − u) in2.6, gives the estimator
b F(x) = 1 − bF(u)Gξ,ˆˆσ(x − u) + bF(u) = 1 −n − Nu n Gξ,ˆˆσ(x − u) + n − Nu n = 1 −Nu n 1 + ˆ ξ(x − u) ˆ σ !−1/ ˆξ , (2.7)
for some ˆξ 6= 0, ˆσ > 0 and, since it is constructed from the unconditional distribution Fu(y), x > u. In other words, this distribution function can be used to estimate the tail of the original distribution of x, where the tail is defined as all values of x > u. It would be possible to use this method for separately estimating both tails of an asymmetrical distribution and use a more basic approach to estimate the distribution between both tails, as set out for example by Wang et al. (2012). However, since the focus of this paper is to estimate quantile-based risk measures for large losses (and not large gains), it is sufficient to estimate just the tail corresponding to large losses. As a convention, losses are denoted as positive numbers, so the right tail will be estimated.
2.3
The GARCH model
As mentioned earlier, the EVT requires the outcomes of the random variables to be independent and identically distributed. The iid assumption equivalence for time series is stationarity. A time series is called stationary if its statistical properties remain constant over time (Heij et al.,2004). However, this is often not the case (McNeil and Frey,2000). The first step is to transform the data to reflect daily relative losses. This is done by a transformation of the original price series into negative daily log returns.
Estimating risk measures using EVT-GARCH — Joran Houtman 7
Definition 2.3.1. Define the original price series to be st, then the daily relative losses are given by the return series
rt= − log st st−1 ≈ st−1− st st−1 .
The difficulty lies in estimating a model for the series rt to replicate the data gen-erating process as good as possible. The general assumption holds that the outcomes of rt are the sum of a predictable process and a random disturbance, also known as an innovation process t. Denote all previous outcomes of rt as the information set Rt−1= {rt−1, rt−2, ...}. The process rt can then be described as
rt= µt+ t, (2.8)
where µt = E [rt|Rt−1] is the expected value of rt given all previous information and ideally E [t] = 0, Et2 = σ2 for all t and E [it] = 0 for all i 6= t. This corresponds to an innovation process with zero mean, constant variance over time and all autocor-relations zero. These assumptions can be unrealistic, especially if taking into account the clustering of volatility in the markets. An alternative is the use of a generalized au-toregressive conditional heteroscedasticity (GARCH) model for the innovation process in combination with an autoregressive model for the level of the return series rt. This model can be described by
rt= φ0+ φ1rt−1+ t, (2.9)
t|Rt−1∼ N 0, σt2 ,
σt2= α0+ α12t−1+ α2σ2t−1 (2.10)
= α0+ α1(rt−1− α − φrt−2)2+ α2σt−12 ,
where α0, α1, α2 > 0, α1 + α2 < 1 and |φ| < 1. This model is also known as an AR(1)-GARCH(1,1) model. As suggested by McNeil and Frey (2000), the parameters θ = {φ0, φ1, α0, α1, α2} will be estimated by bθ = {cφ0, cφ1,αb0,αb1,αb2} using maximum likelihood estimation (MLE). The conditional normality assumption of the error terms explains why a plain GARCH model (i.e. without EVT) for estimation of the extreme losses is insufficient; the normal distribution does not take into account the possibility of heavy tails. Therefore, the standardized residuals xt= et/σbtmust be used for further processing in EVT. Here, et are the residuals of the regression after estimation, i.e. et= rt− cφ0− cφ1rt−1. When the model is correctly specified, the standardized residuals are approximately uncorrelated and normally distributed with a constant conditional variance (Heij et al.,2004;McNeil and Frey,2000).
2.4
Risk measures
To limit the downside risk when investing in a certain asset, risk measures for the returns on that asset need to be estimated.Artzner et al.(1999) define risk as an investor’s future net worth. According to this definition, a risk measure is a function that assigns a value to the future net worth of an investor. They provide a set of axioms to restrict risk measures to functions with intuitively desired properties and define such risk measures as coherent risk measures.
Definition 2.4.1. A coherent risk measure is a risk measure that satisfies the proper-ties of the axioms translation invariance, subadditivity, positive homogeneity and mono-tonicity, as defined byArtzner et al. (1999).
2.4.1 Choosing the right risk measures
Theoretically, only coherent risk measures should be considered, but in practice more properties could play a role in choosing the most appropriate risk measure. Examples of these are the ease of calculation and interpretation and even simply because a regulator demands reporting of certain risk measures. Moreover, a risk manager should not focus on one single risk measure, but should use as much information as available to base decisions on. Currently the most widely used risk measure for financial institutions is value-at-risk (VaR), first publicized by JP Morgan after the 1987 crash (Rocco,2014). A detailed overview of VaR is given byDuffie and Pan(1997).
Definition 2.4.2. Value-at-Risk is a quantile-based risk measure. For a random variable X, define VaRα(X) at the α · 100% level as
VaRα(X) = inf {x ∈ R : P (X ≤ x) ≥ α} = F−1X (α), where α ∈ [0, 1].
This can be interpreted as the maximum amount you will lose with a probability of α · 100%. In practice α is usually chosen to be close to 1 to reflect a high confidence. The main advantage of VaR is that it captures the occurrence of large losses, because it marks the start of the tail of the distribution (Rocco,2014). Note the difference between the earlier defined threshold u and VaR: u marks the start of the empirical distribution, which is chosen with a conservative margin to make sure the later to be estimated VaRα(X) > u, where X is the random variable representing the residual series from the GARCH model. Discarding data points xt > VaRα(X) for fitting the GPD would severely bias the estimated VaR itself. Since VaR is simply the inverse of the estimated distribution function at a chosen confidence level, it is also fairly straightforward to calculate. Unfortunately, VaR also has some major drawbacks. One of the disadvantages is that it is not a coherent risk measure as defined byArtzner et al. (1999), since it does not satisfy the property of subadditivity. This can be proved by a counter example given by Kaas et al. (2008).
Proof. Define the random variables X, Y ∼ Pareto(1, 1) and X, Y independent, then for all α ∈ (0, 1),
VaRα(X + Y ) > VaRα(X) + VaRα(Y ), therefore, VaR is not always subadditive.
For estimating the VaR of a single asset at a time, as is the case in this paper, this is obviously not a problem. However, when a portfolio needs to be estimated with modeled dependencies, this result can be very counterintuitive. It means in certain cases instead of diminishing risk when a portfolio is diversified, the risk could actually increase.
A second disadvantage of VaR is that given an outcome is larger than the VaR, it gives no information about the potential magnitude of the loss. This problem is where the danger of misinterpreting the VaR lies; one could easily be tricked into thinking the VaR is the maximum loss. Moreover, the VaR does not fully utilize the information of the shape of the distribution in the tail, but a risk measure that does is the expected shortfall (McNeil,1999).
Estimating risk measures using EVT-GARCH — Joran Houtman 9
Definition 2.4.3. For a random variable X, define the expected shortfall1 ESα(X) at the α · 100% level as the expected loss given the loss exceeds the VaRα(X), that is
ESα(X) = E [X | X > VaRα(X)]
= VaRα(X) + E [X − VaRα(X) | X > VaRα(X)]
= 1 1 − α Z 1 α VaRr(X) dr, where α ∈ [0, 1].
From the definition, it can be seen that ES captures exactly what happens in the tail and according to Artzner et al.(1999) it is also a coherent risk measure. Since the VaR is such a common risk measure and expected shortfall possesses some characteristics that VaR does not, both risk measures will be estimated. It turns out the VaR has to be calculated anyway when estimating the ES.
2.4.2 Estimation theory of risk measures
Assuming the successful and significant estimation of the distribution of the stationary residual data from the GARCH model using the EVT, it is a small step to give an unconditional estimator for the VaR and ES. Note that (un)conditional in this section refers to the (in)dependence of the estimator on the stochastic volatility. For a static risk measure this is sufficient, but the aim of this paper is to give dynamic risk measures, it therefore incorporates the estimated conditional volatility by using the GARCH model. The unconditional VaR can be calculated by inverting the estimated distribution bF(x) from 2.7(McNeil and Frey,2000):
d VaRα(X) = u + ˆ σ ˆ ξ " n Nu · (1 − α) − ˆξ − 1 # . (2.11)
This estimate can subsequently be used for the calculation of the expected shortfall, where c ESα(X) = 1 1 − α Z 1 α d VaRr(X) dr = d VaRα(X) 1 − ˆξ + ˆ σ − ˆξu 1 − ˆξ . (2.12)
Note that the bσ in these equations is the estimated scale parameter of the GPD dis-tribution and not the bσt representing the current stochastic volatility estimated by the GARCH model.
To transform the unconditional estimators into conditional 1-day forecast estimators, the expected next-day return and volatility are needed. Define the random variable Zt as a white noise process, i.e. a normally distributed process with mean zero and a standardized variance, and rewrite equation2.8, then
rt= µt+ σtZt, where Zt∼ N(0, 1).
From this equation, combined with the AR-GARCH model from2.9and 2.10, the one-1There are multiple definitions for expected shortfall in the literature. While the writers opinion is
that shortfall indicates ’the loss not accounted for’ (that is xt− VaRα), the majority of recent papers
accept the terminology ’shortfall’ as the total loss given it is larger than VaRα. This convention is
day forecast estimates of the conditional mean bµt+1 and variance bσ 2 t+1 can be deduced: b µt+1= E [rt+1|Rt] = E [φ0+ φ1rt+ t+1|Rt] = cφ0+ cφ1rt, (2.13) b σt+12 = Eα0+ α12t + α2σt2|Rt =αb0+αb1bt 2+ b α2σbt 2, (2.14)
where bt = rt−µbt = rt− cφ0 − cφ1rt−1. Here bt is simply the earlier defined residual et from the AR-GARCH model, because at time t both rt and rt−1 are known. The final step in the process is to use these estimations to estimate the one-day forecast of the conditional value-at-risk dVaRtα(R) and expected shortfall cEStα(R), where R is the random variable representing the return series (McNeil and Frey,2000). Using 2.11and
2.12, multiplying these with the expected conditional standard deviation and adding the expected conditional mean to rescale and relocate the distribution, the result is
d
VaRtα(R) =µbt+1+σbt+1· dVaRα(X), (2.15)
c
Chapter 3
Data and methods
One of the objectives of this research is to find out how to reduce the risk when investing in the MINT countries. In this section, the data that is used as an input for the models will be discussed, followed by a detailed discussion of the method, for reproducing the results.
3.1
The MINT countries
On first inspection, the MINT countries - Mexico, Indonesia, Nigeria and Turkey - do not have a lot in common. Geographically, they are located in very different regions. They are all in different continents, with Mexico being next to a stable economic giant while Nigeria is located in a very unstable region in Africa. Even looking at the basic economic indicators in Table3.1, the similarities are not so obvious. The largest country has around 3.5 times more inhabitants than the smallest and when comparing the nominal GDP per capita, the highest (Mexico) is around 7 times larger than the lowest (Nigeria). In contrast, the estimated GDP growth in 2013 and especially the forecast GDP growth for 2014 are all reasonably high compared to the the world GDP growth rates (2.4 % and 3.2 % for 2013 and 2014 respectively). However, these numbers do not give an indication of the risk of investing in these countries. For this, more information is needed about the type of investment.
Table 3.1: MINT countries demographics. (World Bank)
Population GDP (nom.) GDP/cap. ∆GDP ∆GDP 2012 Trade
(MM) (BN) USD 2013 USD 2013(e) 2014(f) (BN) USD
Mexico 118.3 1 327 11 224 1.4 % 3.4 % 741.7
Indonesia 247.6 868 3 498 5.6 % 5.3 % 365.5
Nigeria 174.5 292 1 673 6.7 % 6.7 % 149.0
Turkey 73.7 822 10 744 4.3 % 3.5 % 392.0
When an investor decides to allocate a part of his assets to investments in a certain country, there are several options. For example, real estate in the specific country can be bought, government bonds could be acquired or shares of a particular firm could be acquired. This all results in exposure to a certain sector or asset group. Since O’Neill
(2013) did not specify any of these asset classes, a good reflection of the general economic condition of a country would be its MSCI index. These country specific indices are maintained by Morgan Stanley Capital International and give a weighted average of all
listed securities in that particular country. The diversification has the added advantage that no specific company or asset type risk is overweighted and mainly the market risk determines the risk of the index.
Choosing a good time frame is a trade off between having a lot of data points and incorporating irrelevant statistical properties. For the peaks-over-threshold method a lot of raw data points are needed, since the goal is to model infrequent losses. It is assumed that the data prior to 2006 can not give reliable estimates for current situations. Therefore the last 2000 daily observations will be used to make predictions for the next-day VaR and ES. This is equivalent to approximately the last eight years of available data.
3.2
Exploratory data analysis
When collecting data from an external source, one unavoidably needs to prepare it for use. The data from MSCI are gross daily price levels in local currency to include dividends and avoid influence of the exchange rates. The exchange rate risk should be modeled as a separate risk, depending on the country of residence of the investor. The number of data points during a eight year period might differ between countries due to bank holidays, missing data and closed markets. For this reason, from this eight year period, the last 1200 data points will be extracted to make the samples the same size for ease of comparison. The series will be normalized to the value 100 for the last data point. This will make the results better comparable, but will not have any mathematical consequences. These series are referred to as the price level series st, with t = 1200 being the most recent data point.
To benchmark the results against a reliable index for economic developments, the S&P500 index is used. This index is weighted according to the market capitalization of the 500 largest US companies and therefore assumed to give a general indication on the health of the economy. Data on the S&P500 is widely available (e.g. on Yahoo Finance) and the same process as for the MINT countries is used to prepare the data.
To estimate losses, the data is transformed to negative log returns. This return series rt will have positive values for percentage losses and negative values for percentage gains. Because the natural logarithm is taken, 100 · rt% is approximately equal to the daily percentage loss of the price level series. The most important statistics for the return series of the MINT countries and the S&P500 index are given in Table 3.2. For
Table 3.2: Return series statistics
Mexico Indonesia Nigeria Turkey S&P500
Mean -0.0004 -0.0005 -0.0006 -0.0004 -0.0005 Median -0.0002 -0.0004 -0.0000 -0.0004 -0.0008 Minimum -0.0407 -0.0730 -0.0434 -0.0685 -0.0463 Maximum 0.0573 0.1102 0.0557 0.1076 0.0690 St. dev. 0.0096 0.0145 0.0112 0.0157 0.0105 Obs. (n) 1999 1999 1999 1999 1999
example, from this table it can be seen that the largest loss for Indonesia during the last eight years was approximately 11 % on a single day. Also note that the maximum
Estimating risk measures using EVT-GARCH — Joran Houtman 13
loss in Mexico and Nigeria were smaller than the maximum loss of the S&P500 and that the standard deviation of Mexico is also the lowest of the five return series. This could mean two things: the S&P500 has a higher risk for large losses than Mexico, or the maximum loss and standard deviation are not reliable measures of risk. It could even be both. One of the aims of this paper is to make a statement about risk measures, and value-at-risk and expected shortfall in particular.
The next step is to examine the statistical properties of all return series. This is necessary to determine if the series appear to show trends, autocorrelation, volatility clustering and heavy tails. If this is the case, the model needs to be adjusted according to the theory to produce approximately independent and identical distributed residuals, which in turn can be used for extreme value theory.
Generally, the proposed AR-GARCH model introduced in the theory section will be able to model trends, autocorrelation and volatility clustering, but modeling these effects when they are not present in the dataset will unnecessarily increase estimation errors. The easiest way to determine if these statistical features are present in the data is by using graphs. Statistical tests could also be used, but if the graphs are clear, statistical tests are not necessary. To ultimately determine if the fitted model gives a good estimation of the underlying process, it is preferred to use a combination of visual confirmation and statistical tests.
For each country and for the S&P500 index, four graphs are shown. These can be found in appendix A. The first is a plot of the logarithm of the price levels against time to get a visual indication of a (linear) trend in the data. Secondly, a plot of the logarithm of the returns against time is shown to confirm volatility clustering. Thirdly a quantile-quantile plot of the empirical distribution against the normal distribution is used to show if the tails of the empirical distribution are heavier than what one would expect if the data was generated from a normal distribution. In the latter case, the data points would approximately all lie on a straight line. Lastly, a histogram of the density of the returns with a fitted normal distribution is shown to indicate the (non-)normality of the distribution. Thus only the order autocorrelation remains to be determined. This is best done by using the autocorrelation function (ACF) and partial autocorrelation function (PACF), but since these are generally unknown, the sample ACF and sample PACF will be used. The ACF gives the correlation between lagged values, while the partial ACF gives the correlation between the lagged value not explained by smaller lags. For each country and for the benchmark, the graphs will be discussed below. The sample ACF and PACF can also be found in appendix A.
3.2.1 Mexico exploratory data analysis
When looking at the graphs in Figure3.1, some statistical properties are hard to ignore. From the left graph it can be seen that there is an obvious upward trend in the data, although this trend is not so obvious for the last year. The plot of the log-returns gives clear evidence for the necessity of a GARCH-model. The volatility clustering can be seen as a relative calm period between observations 250 and 500, followed by a very volatile period around observation 500, and subsequently slowly decaying volatily in the next 100 observations.
The left plot in Figure 3.2 shows that the empirical distribution is definitely not normally distributed by showing very heavy tails compared to the normal distribution. The same conclusion can be drawn from the right graph, where a normal distribution is fitted to the data. This does not seem like a good fit, since the density around the
0 500 1000 1500 2000
3.6
4.0
4.4
log price levels
Observations Pr ice inde x 0 500 1000 1500 2000 −0.10 0.00 0.05 log−returns Observations log−retur ns
Figure 3.1: Mexico price levels and log-returns
mean is a lot higher than expected and the frequency of extreme outcomes is also higher than expected. These two graphs also indicate an asymmetrical empirical distribution, resulting in the necessity of either fitting a skewed distribution or separate modeling of the tails. The latter could be easily done with EVT after conditioning the data with an GARCH model, but in this research the main interest is in modeling the losses and not the gains. −3 −2 −1 0 1 2 3 −0.10 0.00 0.05 Q−Q plot
Normal distribution quantiles
Sample Quantiles
Negative log−returns
Daily percentage loss
Density −0.10 −0.05 0.00 0.05 0 10 30 50
Figure 3.2: Mexico quantile-quantile plot and return histogram
Finally, the presence or absence of autocorrelation needs to be determined. The sample ACF and sample PACF give a good indication of this. From Figure 3.3 it can be seen that there is little significant autocorrelation, although at lag 3 there seems to be some. This is rather small and it could be that taking one lag in the autoregressive part of the model and using GARCH reduces the remaining autocorrelation enough to avoid having to take up multiple lags in the model.
3.2.2 Indonesia exploratory data analysis
The graphs for all other countries can be found in appendix A. Looking at the top left graph in Figure5.5, it is clear that the index for Indonesia also has an upward trend over the entire observation period. The plot of the log-returns also clearly shows volatility clustering is present, possibly even more so than Mexico. A very calm period is seen between the 700th and 900th observations, followed by a period of much higher volatil-ity. The quantile-quantile plot indicates that the density in the tails of the empirical distribution is much higher than the normal density, rejecting the normal distribution as a good fit for the Indonesian index. This is confirmed by the histogram and the fitted normal distribution, where the density is observed to be much higher around the mean
Estimating risk measures using EVT-GARCH — Joran Houtman 15 −0.06 −0.02 0.02 0.06 Lag Sample A CF Sample ACF 5 10 15 20 25 −0.06 −0.02 0.02 0.06 Lag Sample P A CF Sample PACF 5 10 15 20 25
Figure 3.3: Mexico (partial) autocorrelation functions
than for the normal distribution. This calls for separate modeling of the tails, when approximately i.i.d. residuals are obtained from the AR-GARCH model.
The sample ACF and PACF from Indonesia in Figure3.4shows, that there is signifi-cant autocorrelation for the third and fourth lags, whereas there is little to no correlation for the first and second lags. Incorporating three lags in the autoregressive part of the model, instead of just one, is necessary to make the residuals stationary when this is not captured by the GARCH model.
−0.05 0.00 0.05 Lag Sample A CF Sample ACF 5 10 15 20 25 −0.05 0.00 0.05 Lag Sample P A CF Sample PACF 5 10 15 20 25
Figure 3.4: Indonesia (partial) autocorrelation functions
3.2.3 Nigeria exploratory data analysis
Nigeria seems to have different characteristics when observing its statistical properties. Figure 5.9 shows an upward trend in the price level series, but like Mexico the trend is not so obvious at the beginning and end of the sample period. There is also less obvious volatility clustering, although some periods could be seen as more volatile than others. It could be that the volatility decays slower, thus making it appear to be more evenly distributed across the sample period, or the series is just not so dependent on volatility in previous periods. In any of the cases, the GARCH model will condition for any significant heteroscedasticity. The quantile-quantile plot again indicates heavy tails, though more moderate than the other series and even the benchmark (note the smaller scale of the y-axis). However, the histogram again indicates a normal assumption for the data distribution is unrealistic. The sample ACF and PCAF in Figure 5.10 show that a rather large and significant autocorrelation of order one is present. Possibly the second order autocorrelation is also significant, but this must be investigated further when fitting the AR-GARCH model.
3.2.4 Turkey exploratory data analysis
The price level graph in Figure 5.13 does show a positive trend over the entire sample period, but in the second and last year of the sample period, it has an obvious downward trend. This might result in an insignificant trend-parameter estimate, but it would be beyond the scope of this paper to model more seasonalities and discontinuities in sample periods. The extra gain from modeling these properties would most likely be very small, since the purpose is to model the large single day losses, whereas a trend by definition is a shift over a longer period.
The plot of the log-returns does show high peaks, followed by decaying volatility, justifying the use of a GARCH model. The data again exhibits heavy and asymmetric tails, as can be seen from the quantile-quantile plot. This is confirmed by the histogram, which shows that the normal distribution is not a good assumption for modeling the returns of the Turkish index. The sample ACF and PACF in Figure 5.14 show that there is very little autocorrelation present.
3.2.5 Benchmark exploratory data analysis
The S&P500 benchmark index has all the signs of predictive statistical properties. Look-ing at the price level plot in Figure 5.17, a very clear positive trend is visible, with not many large losses in the second half of the sample period. This confirms the choice of using at least 1200 data points for estimation, rather than a shorter time frame. How-ever, even with this series, the volatility clustering is visible in the log-returns plot. The period between the 500th and 600th observations is a clear example of a high volatility period. The quantile-quantile plot shows heavy tails, with the empirical distribution moving smoothly away from the normal distribution quantiles. The histogram shows that even for the S&P500 index, a normal assumption is very unrealistic. In Figure
5.18, the sample ACF and PACF show significant autocorrelation for the first, third and fifth orders. Further investigation when fitting the AR-GARCH model is necessary to determine the right order of lags to include.
3.3
Obtaining stationary residuals
For the remainder of this chapter, the theory as explained in Chapter 2will be closely followed. From the exploratory data analysis, it has become clear that for all series an AR-GARCH model needs to be fitted to generate approximately independent and identically distributed residuals, which can be used for fitting the generalized Pareto distribution. When the residuals are i.i.d., this implies they are also stationary. Following general accepted theory (McNeil and Frey,2000), an AR(1)-GARCH(1,1) model will be estimated, as explained in Paragraph2.3. If necessary this model can be expanded to an AR(3)-GARCH(1,1) model for cases with higher order autocorrelation. The coefficients produced by these models will be used to predict the next-day expected return and expected volatility, which in turn will be used for estimating the next-day VaR and ES. For the estimation of the GARCH models, the R package ruGARCH for univariate GARCH models will be used (Ghalanos,2014).
3.4
Selecting a threshold
After obtaining the i.i.d. residuals, a decision must be made about the definition of extreme outcomes, in other words: where to cut off the tail from the main distribution.
Estimating risk measures using EVT-GARCH — Joran Houtman 17
This point is defined in the theory section in definition3.4as the threshold u. The most common way to determine a correct threshold is by using the mean excess function
en(u) = E [X − u | X > u] .
It describes the excess value of an outcome, given it exceeds the threshold. For a GPD distribution to be a good fit, it should follow a straight line above a certain u (McNeil,
1997). However, selection according to this method is still subjective, as it can be very hard to determine the point where the function becomes linear. Therefore, for this research the threshold u will be chosen such that the data series have at least k = 100 values above the threshold. The minimum number of exceedances are for Nigeria with a total of 102 values above a threshold of u0 = 1.6, and the maximum is for the S&P500 with 135 exceedances. The chosen value of k is justified byMcNeil and Frey(2000), who find that for their dataset of 1000 observations, the value u is very robust for values of 80 < k < 140. This ensures the number of observations is large enough to keep the variance of the estimators low, and small enough to make sure the observations are still part of the tail of the distribution. By using 2000 observations, apposed to 1000 observations by McNeil and Frey (2000), the variance of the estimators will be lower, while the fraction of exceedances is still high enough to estimate the 97.5th quantile.
3.5
Estimating the risk measures
Having chosen a suitable threshold and using the results from the GARCH model, all the information needed to fit the GPD is present. Using 2.3 and 2.7 from the theory section, the parameters of the distribution are estimated using maximum likelihood estimation. In this case, the R package POT (Ribatet, 2006) is used to estimate the distribution with the function fitgpd(). For a certain confidence level α, the estimated parameters of the GPD can be used for values of dVaRα(X) and cESα(X), as defined in
2.11 and 2.12 respectively. Separately, the GARCH estimates are used to give values for the next day expected conditional mean µbt+1 and variance σb
2
t+1 as defined in 2.13 and 2.14. Finally these are all combined to give the one-day forecast of the conditional value-at-risk dVaRtα(R) and expected shortfall cEStα(R) from2.15 and 2.16.
Results and discussion
In this section the main results of the application of the EVT-GARCH approach to the MINT countries and the benchmark will be presented. The structure of this section follows the order of processing the data, so first the GARCH model results will be given, then the GPD estimation results and lastly the risk measure estimates will be presented. To conclude this chapter, the results will be compared and analyzed in the discussion section.
4.1
GARCH model results
The estimation of the GARCH model serves two purposes: the first is to generate in-dependent and identically distributed residual series to reliably estimate the GPD, and the second purpose is to give one-day forecasts of the mean and variation of the return series. As explained in3.2.2, the sample partial autocorrelation function can be used to check for the absence of autocorrelation in the residual series. The AR(1)-GARCH(1,1) model gives satisfactory results for all data series, except for Indonesia. As can be seen in Figure 4.1, the model with one AR-term still has significant autocorrelation at the third lag, while this problem is solved when using three AR-terms. A return series plot
−0.06 −0.02 0.02 0.06 Lag Sample P A CF SPACF AR(1) 5 10 15 20 25 −0.06 −0.02 0.02 0.06 Lag Sample P A CF SPACF AR(3) 5 10 15 20 25
Figure 4.1: Indonesia sample PACF for AR(1) and AR(3) models.
can be used to visually determine if a series is iid. The effect of applying a GARCH model for Mexico can clearly be seen in Figure 4.2. In the left graph high and low volatility periods can be distinguished, while in the right graph this is more difficult. This indicates the residuals generated by the GARCH model are identically distributed and thus suitable for estimating a generalized Pareto distribution.
In appendix B the results are shown of the Q-statistics test on residuals and squared 18
Estimating risk measures using EVT-GARCH — Joran Houtman 19 0 500 1000 1500 2000 −0.10 −0.05 0.00 0.05 Before GARCH Observations log−retur ns 0 500 1000 1500 2000 −2 0 2 4 6 After GARCH Observations log−retur ns residuals
Figure 4.2: Mexico return series before and after GARCH
residuals, the ARCH LM test, the Nyblom stability test, and the Pearson goodness-of-fit test. Both Q-statistics tests give an indication of serial correlation after GARCH with H0 : no serial correlation. The ARCH LM test indicates if the conditional variance is correctly specified with H0 : correctly specified conditional variance. The Nyblom test gives an indication of the stability of the parameters over time and a joint statistic larger than the asymptotic critical value indicates stable parameters over time. The Pearson goodness-of-fit test has H0 : residuals are normal distributed. The p-values of the Pearson test are all very close to zero, confirming the necessity for modeling the tails with a GPD. The Nyblom test for Indonesia shows the necessity of using an extra autoregressive term to make the parameters stable over time. The tests confirm the conclusion from the visual data that all residuals are approximately iid for all MINT countries. The GARCH model for the S&P500 could be further improved, but this is beyond the scope of this research.
One could ask why the residual series cannot be used directly to estimate quan-tiles such as Value-at-Risk, instead of applying the extreme value theory to estimate the GPD. One reason is that EVT gives the possibility to model the two tails of a distribution separately. A second reason is that a normal distribution is assumed for estimating the residuals. The exploratory data analysis in chapter 3.2 indicated that the data series are heavier tailed than the normal distribution. Evidence of both these characteristics can still be seen after applying the GARCH model for all data series, justifying the use of EVT. As an illustration the Q-Q plot of the residual series against a normal distribution and its corresponding histogram of Mexico are shown in Figure
4.3. For graphs of the sample PACF, residual plots, Q-Q plots, and histograms for all residual series after GARCH, the reader is referred to appendix A.
In Table 4.1, the descriptive statistics of all standardized residual series are shown. As expected, the mean of all series is close to zero and the standard deviation is close to one. A mean close to zero is expected because the estimation technique MLE maximizes the likelihood of the sample outcome for a given distribution. The standard deviation is expected to be close to one because the residuals are divided by the daily estimated stan-dard deviation. The values for skewness indicate all residual series are asymmetrically distributed, since a skewness of zero indicates a perfect symmetric sample distribution. A kurtosis of three indicates a normal distributed tail, the excess kurtosis gives the estimated values above three. A positive value for all residual series indicates these are still heavy tailed. This information is consistent with the theory presented in chapter
Figure 4.3: Mexico residuals after GARCH −3 −2 −1 0 1 2 3 −2 0 2 4 6 Q−Q plot
Normal distribution quantiles
Residual quantiles Histogram residuals Residuals Density −2 0 2 4 6 0.0 0.2 0.4 0.6 0.8
Table 4.1: Descriptive statistics of residual series.
Mexico Indonesia Nigeria Turkey S&P500
Mean 0.0278 0.0268 0.0139 0.0429 0.0447 Median 0.0212 0.0090 0.0562 0.0584 -0.0222 Minimum -3.3053 -5.0010 -4.6789 -4.5068 -3.1279 Maximum 6.4373 6.3639 5.1916 6.6959 6.3873 St. dev. 0.9995 0.9970 0.9999 0.9985 0.9985 Skewness 0.4418 0.4343 -0.1484 0.1084 0.5434 Exc. Kurt. 1.7023 2.7076 1.3488 1.9098 1.5809
2.3and with the visual evidence presented above, and further reiterates the importance of modeling the tails separately with a generalized Pareto distribution.
The coefficient estimates of the AR-GARCH models for all return series are given in Table 4.2. A p-value below 0.05 indicates a coefficient is statistically significant dif-ferent from zero. This information can be used to track the estimation error for the risk measures, however in this thesis only point estimates will be computed. The φ-coefficients are estimates of the autoregressive terms of the model and the α-φ-coefficients are estimates of the GARCH terms of the model. The unconditional trend is given by φ0, which is negative for all return series. Since the returns indicate losses, this means there is a slight upward unconditional trend during the observed sample period. The dependence of todays return on yesterday’s outcome is given by φ1. For Nigeria, this term is very significant and shows a loss yesterday is often paired with a loss today. Indonesia however has autoregressive terms up to order three to take into account the significant negative dependence on returns three days ago. This was already shown to be relevant in the graph of the partial autocorrelation function (PACF) in Figure 4.1.
The trend in the GARCH terms is zero for all series, which is in line with intuition, since there is no reason to believe volatility should always increase or decrease over time. The dependence of current volatility on the lagged residuals is given by α1 and is significant for all series. The dependence of the current volatility on the lagged volatility itself is given by α2 and is very significant for all series. The stationarity condition for a GARCH model requires that α1+ α2 < 1, which is the case for all MINT countries and the S&P500. Therefore according to these results, the residuals can be used as input for the GPD estimation.
Estimating risk measures using EVT-GARCH — Joran Houtman 21
Table 4.2: AR-GARCH model output (p-values included in brackets).
Mexico Indonesia Nigeria Turkey S&P500
φ0 -0.0005 -0.0010 -0.0006 -0.0013 -0.0007 (0.02) (0.00) (0.05) (0.00) (0.00) φ1 0.0340 0.0471 0.3425 -0.0009 -0.0667 (0.15) (0.05) (0.00) (0.97) (0.01) φ2 NA -0.0361 NA NA NA (0.13) φ3 NA -0.0684 NA NA NA (0.01) α0 0.0000 0.0000 0.0000 0.0000 0.0000 (0.45) (0.24) (0.00) (0.00) (0.06) α1 0.0747 0.1071 0.1606 0.0974 0.0994 (0.00) (0.00) (0.00) (0.00) (0.00) α2 0.9194 0.8780 0.7821 0.8795 0.8857 (0.00) (0.00) (0.00) (0.00) (0.00)
4.2
GPD estimation results
In line with the theory and the earlier described approach in chapter3.4, the GPD will be fitted to the data using maximum likelihood estimation and a threshold of u0 = 1.6. The results are shown in Table4.3, where bσ is the scale parameter,σbseis the standard error of the scale parameter, bξ is the shape parameter, bξse is the standard error of the shape parameter, and Nu is the total number of observations above the threshold. The
Table 4.3: GPD estimation results.
Mexico Indonesia Nigeria Turkey S&P500
b σ 0.6265 0.6746 0.5099 0.4750 0.7494 b σse 0.0829 0.0950 0.0745 0.0661 0.0766 b ξ 0.0588 0.0740 0.0631 0.1442 -0.0650 b ξse 0.0966 0.1064 0.1075 0.1044 0.0555 Nu 122 117 102 116 135
standard error of the scale parameter σbse is relatively small compared to the shape parameterσ for all data series. This means subsequent estimations based on this valueb can be made with a high confidence. However, the shape parameter standard errors bξse are relatively large compared to the estimate bξ. This is a consequence of the trade-off mentioned in paragraph 3.4: a lower threshold results in more exceedances and thus a lower variation in the estimate bξ, while a higher threshold results in a better theoretical approximation, as stated in Theorem2.2.2.
The positive values for bξ for all MINT countries indicate that the GPD fitted to the residual series is heavy-tailed. Turkey in particular has a relatively high value for the shape parameter, while the S&P500 seems to have a short tailed fitted GPD. This is the
2 3 4 5 2 3 4 5 6 Q−Q plot Mexico Fitted GPD distribution Empir ical distr ib ution − − −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− −−−− − − − − − −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− −−−−− −−− − − − 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 2 3 4 5 6 Q−Q plot S&P500 Fitted GPD distribution Empir ical distr ib ution −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− −−−−−−−−− − − − − −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− −−−−−−− −−−− − − − − −
Figure 4.4: Mexico and S&P500 Q-Q plots for fitted GPD.
first indication of the relatively high risk for large losses when investing in the MINT countries, compared to the benchmark.
A quantile-quantile plot can be used to conclude whether or not the fitted GPD distribution is an appropriate distribution for the tails of the losses. All Q-Q plots of the sample data against the fitted data can be found in the appendix. As an illustration the Q-Q plots of Mexico and the S&P500 are given in Figure 4.4. For Mexico the GPD can be said to be a reasonable fit for the observed data, certainly rendering much better results than an assumed normal distribution, before or even after GARCH (see Figures
3.2and 4.3, respectively).
The quantiles of the S&P500 sample data are systematically overestimated by the GPD quantiles, suggesting the GPD is too heavy tailed to be a good fit for the S&P500 data. This corresponds with the conclusion drawn above from Table4.3. However, when estimating large losses, it is better to be too conservative (i.e. estimate heavier tails) than too optimistic (i.e. underestimating the frequency and magnitude of large losses). As can be seen from the corresponding Q-Q plots of Indonesia, Nigeria, and Turkey in the appendix, the GPD fits the observed data for these countries very well. However, this information is not a sufficient indicator for the higher risk for the MINT countries, which is why the risk measures need to be estimated.
4.3
Risk measure estimates
The risk measures that are estimated are Value-at-Risk (VaR) and Expected Shortfall (ES). In words, VaR is the worst expected loss with a certain degree of confidence α. ES is the average expected loss, given the loss is larger than the VaR, with the same degree of confidence α. For readability, the VaR and ES in this paragraph are transformed to represent the return of a price index, given the current price is 100 units. The same transformation is done for the expected value µ and the standard deviation σ.
The GARCH model applied above gives the opportunity to estimate both an un-conditional Value-at-Risk dVaRα(S) and Expected Shortfall cESα(S), and a conditional
d
VaRtα(S) and cEStα(S). The term unconditional means that the risk measure does not depend on the current volatility, but on the unconditional volatility. The conditional risk measures do take into account the current volatility and are thus dynamic risk measures, hence the superscript t.
In Table4.4, the unconditional risk measures are given. These risk measures have no predictive value, but describe how high the one-day risk of large losses were on average
Estimating risk measures using EVT-GARCH — Joran Houtman 23
over the observed period. Given the current price of 100 units, the unconditional mean b
µs gives the expected next-day return and the unconditional standard deviation b σs is the expected standard deviation of the next-day return. Both values can be extracted as a by-product from the GARCH model. Please note that in this context, expected means on average and not predicted.
Table 4.4: Unconditional risk measures estimations.
Mexico Indonesia Nigeria Turkey S&P500
d VaR0.975(S) 3.04 4.20 2.36 3.74 2.69 d VaR0.990(S) 3.91 5.50 2.97 4.73 3.42 d VaR0.999(S) 6.26 9.09 4.65 7.71 5.05 c ES0.975(S) 4.02 5.68 3.05 4.90 3.46 c ES0.990(S) 4.93 7.06 3.70 6.02 4.14 c ES0.999(S) 7.40 10.86 5.48 9.38 5.66 b µs 0.05 0.10 0.06 0.12 0.07 b σs 1.43 1.99 1.24 1.91 1.20
The unconditional risk measures are used to interpret the risk over a longer period. As immediately becomes clear, all time series have a positive expected return, although the expected return for Indonesia and Turkey is significantly higher than the expected return for Mexico and Nigeria. In contrast to what was expected, Nigeria seems to have very similar characteristics as the S&P500, with a relative low VaR and ES compared to the other MINT countries. Note that a relative low VaR and ES indicate a relative low probability of large losses. Based on the unconditional expected returns, VaR, and ES of the MINT countries, it would be more natural to group Mexico and Nigeria or Indonesia and Turkey. The resulting pair of Mexico and Nigeria would be similar in risk and expected return as the S&P500, on an individual basis. Indonesia and Turkey would be considerably more risky in terms of large losses, but they also have a higher expected return. However, taking into account the current volatility completely changes the position of Nigeria.
The conditional risk measures are given in Table4.5. Because these risk measures are the results of a rolling model, in practice these numbers would be re-estimated every day by adding the latest observations and dropping the first observations. An unexpected shock would therefore heavily influence the next-day predicted VaR and ES. The data used for this thesis ends on 19-05-2014, thus the prediction would be for 20-05-2014, therefore any conclusions are valid for that date only. Monte Carlo simulation can be used to extend the forecast period beyond a single day, such asMcNeil and Frey(2000) do, but multiple-day forecasting is beyond the scope of this research.
Again focusing on Table 4.5, the value of µbst+1 is the predicted next-day expected return, taking into account the current volatility. The value ofbσt+1s denotes the standard deviation, or volatility, ofµb
s
t+1. It is clear that for all data series, the current volatility is much lower than the average volatility over the sample period. The expected return for the MINT countries is similar to the unconditional expected return, except for Nigeria, where it is a little higher. The expected return for the S&P500 is considerably lower than average, but this is combined with a very low volatility. This confirms the reputation of the S&P500 as a relative low volatility equity index. This is also motivated by the
Table 4.5: Conditional dynamic risk measures forecasts for 20-05-2014
Mexico Indonesia Nigeria Turkey S&P500
d VaRt0.975(S) 0.25 0.43 0.51 0.47 0.37 d VaRt0.990(S) 0.34 0.59 0.66 0.63 0.49 d VaRt0.999(S) 0.57 1.05 1.08 1.11 0.74 c ESt0.975(S) 0.35 0.61 0.68 0.66 0.49 c ESt0.990(S) 0.44 0.79 0.84 0.84 0.60 c ESt0.999(S) 0.69 1.28 1.29 1.38 0.84 b µst+1 0.05 0.10 0.09 0.12 0.04 b σst+1 0.14 0.24 0.30 0.29 0.18
low values for ES and VaR, where a loss of more than 0.74 % the next day is deemed highly unlikely (less than 0.1 % probability). Even so, if the loss would be higher, it is expected to be 0.84 %.
As in the unconditional case, Mexico seems to have very similar characteristics as the S&P500, for this particular day it is even predicted to have a higher expected return and a lower probability of large losses. In contrast to Mexico, the current volatility seems to affect Nigeria differently than the other indices. The expected return is significantly higher, more in line with Indonesia and Turkey than Mexico or the S&P500. Looking at the VaR and ES of Nigeria, this is now almost the same as Indonesia and Turkey. Thus, at first Nigeria seemed to be a relative safe country to invest in, but on closer inspection it is just as risky as Indonesia and Turkey. Moreover, on a marginal level, Indonesia outperforms Nigeria on all levels. Indonesia has a higher expected return, yet a lower VaR and ES for all confidence levels than Nigeria. A probable cause of the current high dynamic risk estimates for Nigeria, could be the investor nervousness caused by recent Boko Haram attacks and abductions reported widely in international media.
Chapter 5
Conclusion
The main purpose of this research was to find a method to quantify large losses and estimate the probability of them occuring. This purpose served the goal of the risk man-ager who wants to limit his or her downside risk when investing in the MINT countries. It has been succesfully shown that dynamically estimating the Value-at-Risk and Ex-pected Shortfall using a combination of extreme value theory and GARCH provides a framework for a risk manager to base investment decisions on. Applying this framework also showed that based on current levels of risk, Mexico seems to be a misfit in the MINT countries. It has an expected return which is more in line with the benchmark instead of the other MINT countries, and its VaR and ES are also much lower than the other MINT countries. Moreover, it even outperforms the S&P500 in both areas.
Leaving out any of the steps mentioned above would result in underestimating the risk of large losses in the case of the MINT countries. Estimating a GARCH-model is necessary to account for the volatility clustering which is shown to be present in all investigated time series. Incorporating one or more autoregressive terms for the mean is useful to condition for autocorrelation. Since a normal distribution is assumed for the residuals of the GARCH model, it is also necessary to estimate the tails of the residual series separately to account for the leptokurtosis. Using the generalized Pareto distribution, justified by extreme value theory, the need to make an assumption about the underlying distribution of the tails can be avoided. The graphic results confirm that the fitted GPD on the GARCH residuals is a reasonably good fit for all data series considered.
An added advantage of using GARCH modeling is it predictive capabilities. The framework presented allows for the forecasting of the next-day values of the VaR and ES for different levels of confidence. The importance of taking into account the current volatility, as opposed to the unconditional volatility, has been demonstrated by means of an illustrative comparison. In practice, the model would have to be re-estimated every day by the risk manager. The results of the dynamic model estimated in this research showed the current volatility was significantly lower than the unconditional volatility. As a consequence, the estimated VaR and ES were much lower compared to the unconditional case. This would give the risk manager the confidence to safely allocate a larger fraction of the assets to any of the MINT countries.
The model even showed small arbitrage opportunities between Indonesia and Nigeria since Indonesia has a higher predicted return, yet a lower predicted ES shortfall for all confidence levels. However, to exploit these, one would have to do more research and also take into account more factors, such as dependencies between the two countries, estimation errors, and transaction costs.
