4th Annual BSP‐UP Professorial Chair Lectures 21 – 23 February 2011

Bangko Sentral ng Pilipinas Malate, Manila

Lecture No. 6

Confidence Interval for Expected Shortfall Using Bootstrap Methods

by

Dr. Joselito Magadia BSP‐UP Centennial Professor

of Statistics

TITLE: Confidence Interval for Expected Shortfall Using Bootstrap Methods AUTHOR: Joselito Magadia, Ph.D. ABSTRACT: Expected shortfall (ES) and Value-at-Risk (VaR) are the two most common measures of risk, most especially in the context of market risk. ES complements VaR by giving the amount of loss in case VaR is exceeded. The financial literature presents different approaches to estimate VaR. Results from statistical theory could easily be applied to obtain standard errors for these different approaches to VaR estimation. However, the same could not be said of ES. This paper presents a class of methodologies to address the problem of estimating confidence intervals for ES in the light of these different approaches. Numerical results for the PSE Index are included. KEYWORDS: Backtesting 1. Introduction

Among risk managers, the two most popular risk measures are Value-at-Risk (VaR) and Expected Shortfall (ES). VaR measures the largest potential loss one can incur in a given time horizon at a given confidence level. This simple definition consists of three components: (1) a probability distribution, which could either be a density function f or a cumulative distribution function F, for the random variable L which represents loss ; (2) a time horizon ∆, usually ∆ = 1 day or ∆ = 10 days; and (3) a confidence level (1- α), typically 95% or 99%, that is, α is either 0.05 or 0.01. For this reason, VaR is often represented as . More formally, if F∆

−α1VaR L denotes the cumulative distribution function (cdf) of the random variable (rv) L representing loss incurred in the given time horizon ∆ , then

,1)(:inf1 αα −≥ℜ∈=∆− lFlVaR L (1)

i.e, VaR is simply the 100(1-α)th quantile of the loss cdf.

Second only to VaR in popularity is the ES. It answers the question: “In those rare instances when the loss does exceed the VaR, how much do we expect to lose?” A more technical definition is that ES is the conditional expectation of the rv L given that L is greater than VaR. Thus, we write

).|( 1∆−>= αVaRLLEES (2)

The most common example, though hardly applicable in practice, is the assumption of normally distributed loss. That is, if we assume that L ~ N(µ, σ2), then

1

VaR1-α = µ + σΦ-1(1-α) and α

αφσµ )1(( 1 −Φ+=

−

ES .

where φ and Φ denote, respectively, the probability density function (pdf) and cdf of the standard normal distribution.

Risk practitioners now realize that the normal distribution gives an unsatisfactory representation to the loss distribution owing to the fact that most loss distributions are characterized by fat tails. Consequently, the computed VaR and ES figures using a Gaussian distribution often underestimate these values. A number of techniques, though, abound in the literature for estimating these risk measures

Initially, the motivation for this paper is to present a small number of these techniques. The primary reason, though, is this: The abundance of these techniques gives risk analysts the means to arrive at different figures and, therefore, a lot of options to choose from to quantify the same conceptual risk. It would be helpful if one can attach a measure of reliability to these different estimates. Constructing confidence intervals around these estimates has long been recognized as a viable solution to this problem. VaR estimators do not present much of a problem since statistical theory has a long history of dealing with quantile estimators. The situation is different with ES, though. Even a simplifying assumption like having a continuous loss density function fL necessitates coming up an innovative approach to constructing a confidence interval (CI) around a complicated expression like

∫∞

−αα 1

)(1VaR L dxxf .

Backtesting provides a way of monitoring the performance of different methods and compare their relative performance. However, as pointed out by Christoffersen and Gonçalves (2005), confidence intervals (CIs) can also be used as measures of model and estimation risks. Analytic solutions to CI construction do exist, but oftentimes, under stringent conditions and/or assumptions. Recently, however, computer-intensive procedures are utilized to providing insights and at times, answers to problems which elude analytic solutions. Bootstrap methods consist one such class of procedures that have gained acceptance among statisticians and practitioners, in general. Section 2 provides a short background on ES estimation (§ 2.1) and bootstrap methodologies (§ 2.2).

2. Expected Shortfall and the Bootstrap

2.1 Expected Shortfall Estimators

We begin by considering a sequence of p-variate observations Ω =Zi : i = 1, …,

T of risk factors. We could think of the individual components of each vector Zi as

2

representing the (logarithm of the) value of a stock, the prevailing exchange rate, the index of a particular exchange, or just about any factor that could contribute to the loss or gain in our present position. All too often, one component is enough to represent the risk factor for one specific asset, however, there may be instances when more than one component have been used to represent the risk factors of an asset. Take the case of an option. We need at least five components of the vector Z to represent the different ways by which we can expect to lose (or gain) by dealing with a simple vanilla option.

Still, there would be times when it would be more convenient to deal with risk factor changes Xt = ∇Zt = Zt - Zt-1. We then obtain the values for the losses L by transforming these multivariate risk factor changes Xt via measurable function l: ℜp→ℜ into a univariate loss Lt , i.e., Lt = l(Xt ). Under this transformation, we now obtain the set Λ = Li : i =1,…, T which we use to estimate FL. Equations (1) and (2), applied to the estimator of FL provide estimates to VaR and ES.

As simple and straightforward the above procedure may be, the analyst still has to

contend with the decision as to how one should look at Ω. Two general pproaches exist: the unconditional approach and the conditional approach. The unconditional approach looks at Ω as a set of observations from a stable distribution, i.e., each observed vector Z is a realization of a random vector from a multivariate distribution FZ. On the other hand, the conditional approach sees Ω as a realization of a p-variate stochastic process with parameter space 1,…, T. As a consequence, the set Λ which was derived from Ω inherits the “(un)conditional” characteristic of Ω.

Within the realm of each approach, different estimation procedures for VaR have been proposed. And since, VaR and ES are closely linked, ES estimators vary as well. A short description follows each of these estimators:

Historical Simulation (HS)

A nonparametric (and perhaps, the simplest) approach to VaR-ES estimation is given by the procedure popularly known as HS. Here, VaR is estimated by the 100(1-α)th quantile of Λ and ES is given by the mean of all values that exceed the VaR estimate. In symbols, suppose L(1) < L(2) < … < L(T) represent the ordered values of Λ. Then VaRα is estimated by L([(1-α)T]+1)

1 and the ES estimator is given by

])1[(ˆ 1])1[(

)(

TT

LSE Ti

i

HS αα

−−=

∑+−> . (3)

Unconditional Variance-Covariance t-distribution

1 In the statistical literature, there exists at least nine different ways to define the quantile of a sample. This definition is what this paper will adopt. [⋅] refers to the greatest integer function.

3

Here, we assume that loss L is such that has a standard t distribution with ν degrees of freedom. McNeil etal.(2005) give the formula to estimate ES as

σµ /)(~−= LL

( ) ( )⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−−+−

+=−−

1)1()1( 21

,1

,,

ναν

αα

σµ ννν tttVC

FFfES , (4)

where ft,ν denotes the pdf and the cdf of a standard t-distribution with ν degrees of freedom.

ν,tF

Extreme Value Theory (EVT)

Extreme value theory relates to the asymptotic behavior of extreme observations of a random variable. It provides the fundamentals for the statistical modeling of rare events, and is used to compute tail risk measures. Two different but related methods may be applied in modeling extremes: block maxima models and threshold-exceedance models. The traditional block maxima models have largely been superseded by threshold exceedance models when it comes to financial data analysis so that the discussion will deal with the latter, in particular, the Hill method approach.2

Threshold exceedance models deal with large observations above some given

threshold u. The distributional model is given by

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

=−−

≠⎟⎠⎞

⎜⎝⎛ +−

=

−

0)/exp(1

011

)(

/1

,

ξσ

ξσξ ξ

σξ

ify

ify

yG

for 0 < y < (xF – u), and is known as the generalized Pareto distribution (GPD). ξ and σ are, respectively, called the shape and scale parameters. Under this model, given a threshold u, an estimator for the ES is given by Christoffersen and Gonçalves (2005) as

ξα

ξ

ˆ

ˆ1ˆ

−

⎟⎟⎠

⎞⎜⎜⎝

⎛

−=

uHill T

TuSE , (5)

where is an estimate of the shape parameter ξ and Tξ u is the number of observations that exceed the threshold u.

2 Another set of threshold-exceedance models are given by the POT point process models. See McNeil et. al (2005) and references therein for further discussion.

4

The Conditional Approach

The abovementioned unconditional approaches ignore the fact that there is an interrelationship among the elements of Λ, aside from the fact that the dynamic structure of the volatility was not even considered. The conditional approach tries to remedy this by modeling the dependency among the variables and at the same time address the issue of the time-varying nature of the variance of the observed series. The most common approach is to fit an ARMA-GARCH time series model to the set Λ.

This paper posits the ARMA(p,q)-GARCH(1,1) model for Λ expressed as

Lt = µt + σt εt, , t= 1,.., T (6)

σt2= ω +α(Lt-1 - µt-1)2 + βσt-1

2 (7)

where εt are iid with mean zero, variance one, and distribution function G and µt is generated by an ARMA(p,q) process.3

Expression for the ∆ time horizon ES under the given ARMA-GARCH specification (6) and (7) is generally given by McNeil et al. (2005) as

εσµ SESE TTTˆˆˆˆ

∆+∆+∆+ += (8)

where ∆+Tµ and ∆+Tσ are, respectively, the ARMA and GARCH forecasts at time T + ∆, and is an estimate of the ES based on the distribution of εεSE ˆ t.

Assumptions on the distribution of εt may vary. This paper will pursue only one of the many that exists: that the distribution of εt is given by its empirical cdf . In order too obtain an estimate of ES under this assumption, let us denote by tε the residuals of

the fitted ARMA-GARCH model and let ηt = ∑=

−t

tt T 1

ˆ1ˆ εεT

be the centered residuals.

Then the estimator of ESε (known as the filtered historical simulation (FHS) estimator) is given by the expression

∑∑ >>

>

=q

t

qq tt

ttt

ISE

ηη

ηε η

1ˆ ,

where q = η([αT]+1) is the ([αT] + 1) order statistic of η1, …, ηT.

3 The ARMA(p,q) process is given by Lt = µ + φ1Lt-1 + … + φpLt-p + εt + θ1εt-1 + … + θqεt-q , where εt ~ iid with mean zero and constant finite variance η.

5

2.2 Bootstrap Methods

A sample y1, y2, …, yn representing the rvs Y1, Y2, … Yn is taken from a distribution F. The objective is to estimate a quantity T whose value is determined if the whole population (or all of its value) is completely specified. Oftentimes, this quantification comes in the form of an algorithm, a procedure, or a formula which is applied to F, i.e., we could think of T as a functional on F and write T = T(F). An estimator t(Y1, Y2,…, Yn) is used to approximate T which yields a value of the estimate t(y1, y2, …, yn). Statistical theory, then, is used to derive the sampling distribution of t if one wishes to characterize t in terms of its bias, standard error, etc. If the distribution F is unknown, or if the sampling distribution of t does not easily yield to analytical methods, bootstrap methods could be applied to approximate the sampling distribution of t.

Perhaps, the closest term that one can use synonymously with bootstrapping is resampling. When we speak of bootstrap procedures, we usually refer to the various ways to obtain a “resample”. Some of these approaches are discussed below.

The Basic Bootstrap and the Parametric Bootstrap

To perform the basic bootstrap, the available data y1, y2, …, yn is resampled (with replacement) resulting in one bootstrap sample b = y1*, y2*, …,yn*. This resampling procedure is done B times giving us bi : i = 1, …, B. The statistic t is applied to each bi which yields ti = t(bi): i = 1, …, B.

On the other hand, if we wish to apply a parametric bootstrap, we first assume that the sampley1, y2, …, yn was taken from a known distribution FY(θ). If θ is unknown, then it is estimated by y(θθ = ˆˆ

ˆ1, y2, …, yn ). A bootstrap sample b is generated

by taking a random sample of size n from FY(θ) or FY( ). A total of B such datasets are generated and the statistic t applied to each dataset to give t

θi = t(bi): i = 1, …, B.

In either case of the abovementioned procedures, an estimator for the sampling distribution Ft of the statistic t is given by the empirical cumulative distribution function (ecdf)

∞<<∞−≤= ∑ xxtIB

xFi

it ,)(1)(ˆ

where I(A) denotes the indicator function of the set A.

6

Bootstrapping Time Series Data

Bootstrap methods for time series data can be classified into two broad classes: model-based resampling and block resampling.

Model–based resampling relies on the fact that most time series model specifications involve a so-called innovation process. Innovation processes are usually white noise processes, i.e. a series of uncorrelated, identically distributed random variables. A bootstrap sample can then be generated by sampling from a set of model residuals or from an assumed distribution of the innovation process, given some starting values that will initialize the generation of the series according to the given model. Christoffersen and Gonçalves (2005) presented a comprehensive bootstrap algorithm for GARCH-based measures of risk for the model Lt = σtεt with σt given by eqn. (7) and their results apply to a broad range of GARCH models. Shimizu (2010) also gave a model-based bootstrap resampling scheme for an ARMA-GARCH process and results presented therein also give some theoretical results on the OLS estimators of process’ parameters.

The other approach for handling time series data is known as block resampling. To perform this procedure, the original series is first subdivided into blocks of contiguous values. The blocks are sampled with replacement. A bootstrap resample is then obtained by piecing back the resampled blocks together in the order they were drawn. A number of variations exist. The most common is known as the fixed block bootstrap Here, the time series is supposed to be subdivided into non-overlapping blocks of equal length, say l. If the number of observations T is not divisible by l, then the last block will not contain l values. In this case, the data is “wrapped around” itself so that the last block will be augmented by the values at the beginning of the series. Other schemes use overlapping blocks to address the issue. In general, the time series generated becomes nonstationary. Another variation, known as the stationary block bootstrap, uses a variable block length scheme. The length of each block is supposed to come from a geometric distribution with mean l. The guesswork on trying to determine an appropriate value for l can be eliminated due largely to the results presented by Politis and White (2004).

A combination of block and model-based resampling is done by a procedure known as post-blackening. This is done by first pre-whitening the series by fitting a model. The resulting residual series is then block bootstrapped to obtain a “sample residual series”. A bootstrap sample is generated by applying the model with the “sample residual series” serving as the innovation process.

Confidence Intervals

To assess the uncertainty about ES, confidence intervals (CIs) with a given coverage probability γ can be constructed. The coverage probability is the relative frequency that the CI will include the true ES if all possible samples were taken from the data generating mechanism.

7

A two-sided confidence interval with coverage probability is defined using the quantities and such that P[ < ES < ] = γ. Oftentimes, we take p

1pθ ˆ ˆ ˆ21 p−θ

1pθ 21 p−θ 1 = p2 = (1-γ)/2. Three such sets of limits will be considered here:

Normal approximation: RpRp zSEzSE varˆ,varˆ2/1

*2/1

*21 γγ θθ −− +=−= ,

where z1-γ/2 = Φ-1(1- γ/2),

∑∑==

−−

==R

iiR

R

ii SEES

RES

RSE

1

2**

1

** )(1

1var,1 , and is the ES computed from

each bootstrap sample b

*iES

i.

Basic Bootstrap CI: *)])2/)(1([(

*)])2/1)(1([(

ˆ2ˆ,ˆ2ˆ21 γγ θθ +−+ −=−= RpRp ESSEESSE

Efron’s Percentile CI: *)])2/1)(1([(

*)])2/)(1([( 21

ˆ,ˆγγ θθ +++ == RpRp ESES

2.3 Review of Literature

The ES models presented above are, by no means, exhaustive. Multivariate distributions which are closed under affine transformations could also be considered as possible candidates to fit risk factor changes. Other times series methodologies like exponential smoothing, nonlinear models, etc. could also be considered. McNeil et.al (2005), Christoffersen (2006) and Jorion (2006) and the references therein present a wide array of techniques for VaR (and by extension, ES) estimation. A more recent result uses regressors to estimate ES. (Tanase, 2010).

The same could also be said about bootstrap and bootstrap CI methodologies.

Volumes could be written on the different methodologies involved. Davison and Hinkley (1997) provide a comprehensive discussion on most of the techniques applied in this paper. Chernick (2008) provides a survey of the most recent developments. Bootstrap variants like the Bayesian bootstrap, Sieve bootstrap etc. are prominently mentioned.

Several authors have attempted to tackle the problem of constructing CI for ES.

Manistre and Hancock (2005) derived the variance of the HS estimator assuming a Pareto distribution for the losses utilizing the delta method to derive its variance. A number of biased sampling procedures were also proposed to reduce variance. Lan etal (2010) proposed a two-level simulation approach. Their methodology calls for nested simulation, whereby risk factors are sampled at an outer level of simulation, while the inner level of simulation provides estimates of loss given each realization of the risk factors. Christoffersen and Gonçalves (2005) provide a comprehensive approach to CI construction using bootstrap methods via model-based resampling. Their results show that when assuming independent returns the bootstrap intervals work well for HS. However, when GARCH effects were taken into consideration, the resulting intervals

8

were much too narrow. They also used five risk models and a nonGaussian GARCH in their analysis.

Most references emphasize that bootstrapping does not always work. Chernick (2005) devoted a whole chapter to address this concern together with some ways to remedy some problematic situations. Shimizu (2010) also showed that a model-based procedure called the wild bootstrap need not work in estimating VaR.

3. An Empirical Example The preceding discussion will now be applied to the PSE Index data from January

2, 1995 to February 26, 2010. If we denote the value of the index at time t as Pt , then the loss Lt incurred by assuming a long position is given by Lt = - (log Pt – log Pt-1). (See Figures 1 and 2.)

ES estimates for 1-α = 0.99 and ∆ = 1 will be obtained using some of the

approaches described above. The resulting 95% CIs from the different bootstrap procedures are presented as well. Figure 1. PSE Index from Jan. 2, 1995 to Feb. 26, 2010

800

1,200

1,600

2,000

2,400

2,800

3,200

3,600

4,000

96 98 00 02 04 06 08

PSE Index

Backtesting unconditional ES estimates rely on the identity

( ) [ ] 011

111 =− ∆

−+ >=∆−+

αα VaRLt

tIESLE .

9

Essentially, this suggests that the discrepancy on the days that the VaR is violated should come from a distribution with mean 0. A way to do this is to consider a bootstrap procedure wherein the discrepancies are resampled and their means computed. CIs are then constructed. We can consider the null hypothesis of zero mean rejected if the CI does not contain 0. Figure 3 shows the percentile bootstrap CIs for the means of the discrepancies for Historical Simulation, Variance-Covariance-t and GPD approaches: all of them contain the point 0 so that we cannot reject the null hypothesis of a zero mean. That is, as far as the point estimates of ES are concerned, all of them pass the backtesting procedure. Results for the parametric bootstraps are almost similar

SELtˆ

1 −+

Figure 2. Loss incurred by assuming a long position

-.20

-.15

-.10

-.05

.00

.05

.10

.15

96 98 00 02 04 06 08

LOSS

The process of backtesting conditional ES estimators, although stated differently (see McNeil et al.(2005)) is implemented similarly to that of the unconditional approach. Unlike the unconditional approach wherein only one value for VaR and ES are used, the conditional approach first gives VaR and ES estimates for each time period. From this point on, the procedure is similar to the unconditional approach.

10

Figure 3. 95% Percentile bootstrap CIs (B = 999) for the HS, VC-t and GPD. (top to bottom) All generated CIs contain the point x = 0 implying that we cannot reject the hypothesis that discrepancies observed during those times that VaR was violated come from a distribution with zero mean.

Results: Basic Bootstrap

Figure 4 summarizes the results of an analysis using the basic bootstrap. It shows the kernel density estimate of the sampling distribution of the ES using historical simulation (HS), the variance-covariance t (VC-t) and the generalized Pareto distribution (GPD) approaches. Table 1 gives the CIs generated by the basic bootstrap applied to estimators. A graphical representation is provided by Figure 5. Each of the three approaches is represented by two CIs: the basic CI and the percentile CI. Although the tests for normality that were applied clearly rejected the null hypothesis of a Gaussian distribution, the resulting normal CIs (represented by the dashed segments) are not too much different from those of the other two. The bottommost CI in figure 3 is the one obtained using maximum likelihood estimation of the parameters of the generalized Pareto distribution.

11

Figure 3. Kernel density estimates for the sampling distribution of the ES under HS (solid line), VC-t (dashed line) and GPD (dotted line) using the basic bootstrap. (B=999)

4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0

0.0

0.2

0.4

0.6

0.8

1.0

Sampling Distribution of ES

Table 1. VaR.99 and ES.99 point estimates together with the corresponding 95% CI for the ES using basic bootstrap

Approach VaR0.99 ES0.99 Lower CL Upper CL

Basic 5.299 7.133

Percentile 5.456 7.290 Historical Simulation 4.64527 6.294868

Normal 5.385 7.228

12

Basic 5.538 7.574

Percentile 5.734 7.770 VC-t 4.385194 6.687399

Normal 5.629 7.707

Basic 5.595 7.594

Percentile 5.714 7.770

Normal 5.649 7.492

Generalized Pareto

Distribution 4.544709 6.54055

ML 5.802 8.038 Figure 5. 95% CIs obtained from basic bootstrap. Legend: Vertical Lines represent estimated ES. Horizontal lines represent the 95% CI for each of the 3 approaches under the basic bootstrap. Black (HS), red (VC-t) and blue (GPD)

5.0 5.5 6.0 6.5 7.0 7.5 8.0

95% Bootstrap CI for ES (Unconditional Approaches)

ES

13

Results: Parametric Bootstrap Figure 6. Histogram of losses.

0

200

400

600

800

1,000

1,200

-0.15 -0.10 -0.05 0.00 0.05 0.10 0.15

Series: LOSSSample 1/02/1995 2/26/2010Observations 3696

Mean -2.28e-05Median 0.000193Maximum 0.142100Minimum -0.163760Std. Dev. 0.016219Skewness -0.276077Kurtosis 15.81689

Jarque-Bera 25344.95Probability 0.000000

To perform a parametric bootstrap, we first observe from the table accompanying figure 5 that the value of the sample kurtosis suggests that the losses come from a heavy-tailed distribution. A large family of distribution to choose from is the (univariate) generalized hyperbolic distribution with pdf given by

( )( )

( ) [ ]

( ) λ

γµλ

λ

λ

λλ

γψµχ

γψµχ

χψχψπ

γψψλψχµ −

−−

−

+−+

+−++= )2/1(

22

)(22)2/1(

)2/1(2

))()((

))()((

)((2),,,;(

x

exK

Kxf

x

,

where Kλ is the modified Bessel function of the third kind with index λ.

Table 2 gives the results of fitting some subclasses of the generalized hyperbolic distribution, These subclasses are the NIG, hyperbolic and t models. Using, as a criterion, the maximum value of the log likelihood the best model among these subclasses is the symmetric t distribution with 3 degrees of freedom. The standard error of the parameter estimate of the degrees of freedom ν is 0.18. It should be noted that results from the symmetric and asymmetric fits within each subclass are quite similar which suggests that the distribution is a symmetric one. A more formal test using the log likelihood ratio (p-values > 0.2 for both subfamilies) confirm that there is no sufficient evidence to reject the null hypothesis that the symmetric and asymmetric fits are similar.

Figures 7 and 8, together with Table 3, summarize the results for the parametric bootstrap. A notable result here is that the normal approximation CI for the VC-t

14

Table 2. Comparison of univariate models in the generalized hyperbolic family, showing estimates of selected parameters and the value of the log likelihood at the maximum.

Model Parameter estimate(s) Log Likelihood

t model ν = 3.137 -6522.035

NIG (symmetric) χψ = 0.548 -6531.322

NIG (asymmetric) χψ = 0.548 -6531.317

Hyperbolic(symm) χψ = 0.2958 -6563.200

Hyperbolic(asymm) χψ = 0.2896 -6563.025

Figure 7. Kernel density estimates for the sampling distribution under HS (solid line), VC-t (dashed line) and GPD (dotted line) using the parametric bootstrap. (B=999)

5 6 7 8 9

0.0

0.2

0.4

0.6

0.8

1.0

Sampling Distribution of ES

ES

15

Table 3. VaR.99 and ES.99 point estimates together with the corresponding 95% CI for the ES using parametric bootstrap

VaR.99 ES.99 Lower CL Upper CL

Basic 4.386 7.114

Percentile 5.476 8.204 Historical Simulation 4.64527 6.294868

Normal 4.591 7.342

Basic 5.957 7.725

Percentile 5.650 7.417 VC-t 4.385194 6.687399

Normal 3.091 10.937

Basic 5.675 7.870

Percentile 5.438 7.633 Generalized

Pareto Distribution

4.544709 6.54055

Normal 5.896 8.000 Figure 8. 95% CIs obtained from the parametric bootstrap

4 6 8 10

95% Bootstrap CI for ES (Unconditional Approaches)

ES

16

Results from a Conditional Block Resampling Approach: The Filtered Historical Simulation Table 4 shows the results from performing block resampling approach to the set Λ. A block length of size 60 was chosen, both for the fixed length approach and for the stationary block bootstrap (wherein the block lengths vary randomly according to a geometric distribution with mean 60.) The choice of block was determined by practical consideration. It must not be too large so as to have a small number of blocks to sample independently and it must not be too small so as to lose any interrelationships between any two consecutive values. (The block size recommended by Politis and White (2004) was too small.) At a bootstrap sample size of B = 999 (the number used for the unconditional approaches), no useful information can be inferred. There are lower confidence limits which are negative and upper class limits that are too large. This could be explained by nature we obtain a bootstrap sample. If a block was cut while the series is experiencing a period of high volatility and will be appended to another block undergoing a behavior which entirely different, then the reconstituted series will have a different dynamics from the original. A model fitted to the reconstituted series could be entirely different from the original.

A not-so-unexpected result, however, was obtained when the number of bootstrap sample increased to 13000 (by accident.) Statistical theory states that the estimator approach that of the true sampling distribution as the sample size gets large. It does not say, however, how large is large. Table 4.

ES.99 Lower CL Upper CL

Basic -19.393 9.99407

Percentile 3.2877 32.675 Fixed Length (B = 999)

Normal -23.968 30.3154

Basic 4.820 7.451

Percentile 5.139 7.770 Fixed Length (B = 13000)

Normal 5.002 7.646

Basic -20.719 9.95857

Percentile 3.323 34.001

Filtered Historical Simulation

6.640912

Geometric (Stationary

Block Bootstrap) (B = 999) Normal -18.7324 25.4136

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Conclusions With so many estimators to choose from, one faces the dilemma on which is the proper choice. Confidence intervals have long been used by practitioners and analysts to measure the reliability of an estimator, which can be in the form of a formula, an algorithm, or a procedure. At the pace that researches and collaborations are going on, one is faced with new techniques and approaches from every field of specialization trying to find solutions to problems, both new and old. One hardly has the time to find a rigorous proof or analysis of a promising ad-hoc methodology. Bootstrapping offers a way to deal with this dilemma. Computation-intensive it may be, implementation is simple. All one needs to do is repeat the procedure over and over a large number of times. One has to bear in mind, though, the caveat that BOOTSTRAPPING DOES NOT ALWAYS WORK. The bootstrap, just like any statistical procedure, is data-specific, i.e., the data should conform to what the bootstrap procedure was originally intended for. If it is cross-sectional data, then there are bootstrap procedures for cross-sectional. If it is time series data, then there are bootstrap procedures for time series data. One should not force the issue of applying a non-appropriate procedure to data which it was not originally meant for. Acknowledgement: The author wishes to acknowledge the financial support provided by the Bangko Sentral ng Pilipinas . References: Andreev, Andriy and Kanto, Antti, Conditional Value-at-risk Estimation Using Non-

integer Values of Degrees of Freedom in Students t-distribution. Journal of Risk, Vol. 7, No. 2, pp. 55-62, Winter 2004/05.

Chernik, Michael R. (2008) Bootstrap Methods: A Guide for Researchers and

Practitioners 2nd Ed. John Wiley & Sons, New Jersey. Christofferson, Peter (2006) Value-at-Risk Models. (available at

http://www.christoffersen.com/CHRISTOP/research/christoffersenspringer20sept2006.pdf

Christoffersen, Peter and Silvia Gonçalves (2005), Estimation Risk in Financial Risk

Management. Journal of Risk, Vol. 7, No. 3, pp. 1-28. Davison, A. C. and Hinkley, D,V. (1997) Bootstrap Methods and Their Applications.

Cambridge University Press, UK.

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Jayasuriya, S. A., and R. Rossiter (2008) “Asymmetric Volatility, Risk and Filtered Historical Simulation” Journal of Academy of Business and Economics, Vol. 8(3), 56-67.

Jorion, P. (2006) Value-at-Risk, 3rd edition. McGraw-Hill, New York. Lan Hai, Barry L. Nelson, and Jeremy Staum. (2010) “A Confidence Interval Procedure

for Expected Shortfall Risk Measurement via Two-Level Simulation” Operations Research, Vol. 58, No. 5, September-October 2010, pp. 1481-1490.

Maasoumi E, Racine JS (2009). A robust entropy-based test of asymmetry for discrete and continuous processes." Econometric Reviews, 28, 246-261. Manistre, B. John and Geoffrey H. Hancock (2005) “Variance of the CTE Estimator”

North American Actuarial Journal, Vol. 9, No. 2, pp. 129-156. McNeil, Alexander J., Rüdiger Frey, and Paul Embrechts (2005) Quantitative Risk

Management: Concepts, Techniques, and Tools. Princeton University Press, New Jersey.

Politis, D.N. and H. White (2004), “Automatic block-length selection for the dependent bootstrap”, Econometric Reviews 23(1), 53-70. Shimizu, Kenneth.(2010) Bootstrapping Stationary ARMA-GARCH Models, Vieweg

Teubner. Statistical Science, Vol.18, Issue 2. May, 2003. (Special Issue on the bootstrap) Tanase, Andrei Valentin (2010) Expected Shortfall estimators and their use in asset allocation. PhD dissertation: Tor Vergata University, Rome.(available at http://dspace.uniroma2.it/dspace/bitstream/2108/1317/1/Tanase_TesiDottorato_16062010.pdf). Yamai, Y., and T. Yoshiba (2002) “Comparative Analyses of Expected Shortfall and Value-at-Risk: Their Estimation Error, Decomposition, and Optimization,” Monetary and Economic Studies, 20 (1), Institute for Monetary and Economic Studies, Bank of Japan, 2002, pp. 57-86 (available at http://www.imes.boj.or.jp/english/publication/mes/2002/me20-1-3.pdf)

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