The expected shortfall is a risk measure which has been mostly used among actuaries and insurance companies. The expected shortfall on a portfolio of financial assets is the conditional expected loss given that the loss is greater than a high quantile named as value at risk (VaR). While the expected shortfall and VaR are two popular risk measures, the expected shortfall is becoming increasingly important due to its better properties. For example, the expected shortfall satisfies the sub-additive property, whereas the VaR does not. Sub-additivity, which is one of the four requirements of coherent risk measures, implies in the context of risk management that the total risk on a portfolio should not be greater than the sum of the individual risks. See  and  for details.
Let be a sequence of stationary random variables with common distribution function F that describes the negative profit and loss distribution, where denotes the set of all integers. For a given positive value p close to zero, the confidence level VaR, denoted by , is defined as the -th high quantile of the loss function F. That is
VaR is defined as the loss of a financial position over a time horizon that would be exceeded with small probability p. See    for more discussions on VaR. The expected shortfall associated with a confidence level on a portfolio, denoted by , is a conditional expectation defined as follows
As can be seen from the definitions above, the expected shortfall gives the potential size of the loss that exceeds it, whereas VaR does not. Moreover, as a risk measure, in addition to being coherent, the expected shortfall gives weights to all quantiles greater than the high quantile , whereas the VaR gives all its weight to a single quantile and tells nothing about the size of the loss that exceeds it. Thus, as a risk measure, the expected shortfall is more applicable and produces better incentive for traders than VaR.
In the past, the estimation of expected shortfall has been mainly developed for the identically independently distributed (IID) observations based on the extreme value theory in a parametric or semi-parametric framework. We refer the reader to  and  for details. However, many empirical studies showed that financial data are often weakly dependent with heavy tails. It is also challenging to build a parametric model that can capture the tail behavior for calculating risk measures since the data are generally sparse in the tail part of the loss distribution. As a result, nonparametric based methods can play an important role in diverse problems in risk managements.
For the weak dependence case, under suitable mixing conditions, first proposed a nonparametric kernel estimator for the expected shortfall in the context of portfolio allocation and derived its asymptotic properties. Reference  compared the performance of the sample estimator and the kernel smoothed estimator of the expected shortfall and showed that extra smoothing does not result in more accurate approximation for the expected shortfall.
Although properties of expected shortfall are well studied in the literature, no work seems to be available on the properties of bootstrap approximations for the expected shortfall. Our main contribution in this research is to provide a theoretical foundation to the practical applications of the moving block bootstrap for the expected shortfall.
In this paper, we investigate the asymptotic properties of nonparametric block bootstrap methods for estimating the sampling distribution and the asymptotic variance of expected shortfall under weak dependence setting.
The rest of the paper is organized as follows. In Section 2, we introduce some background material, including a brief description of the moving block bootstrap (MBB) method. In Section 3, we state the main results of this paper. Technical details and proofs will be presented in Section 4.
We first define the sample estimator of expected shortfall. For a sample , of stationary random variables with common distribution function F, let denote the corresponding empirical distribution function, putting mass 1/n on each , i.e.,
where denotes the indicator function which is defined as follows
Then, is the sample estimator of VaR at confidence level . The sample estimator of the expected shortfall, denoted by , can be defined as
where denotes the largest integer not exceeding x for , i.e.,
Then, the normalized expected shortfall, denoted by , is defined as below:
Our goal in this paper is to investigate the asymptotic properties of bootstrap approximations to the distribution and the variance of the normalized expected shortfall, .
Reference  first introduced the bootstrap method into the statistical world. The bootstrap is a flexible method that can be applied to a variety of problems. It can be used to approximate the quantities of interest such as distribution, bias, variance, significance level in a nonparametric framework.
However, Efron’s bootstrap method fails when the data are not independent   . Block bootstrap methods for dependent data have been put forward by several authors, notably by  -  . See  and references therein for a detailed account of results on bootstrap methods (for smooth functions of the data) in the dependent case.
It is worth mentioning that although there has been a considerable amount of work on properties of block bootstrap methods for smooth functionals of weakly dependent data, not many theoretical results seem to be available on properties of the moving block bootstrap (MBB) and other block bootstrap methods in case of nonsmooth functionals. Reference  first showed that blocking bootstrap methods provide a valid approximation to the distribution and its asymptotic variance of a non-smooth function of data, the normalized sample quantile. Recently, due to its applications in financial times series data analysis, quantile based methods (for nonsmooth functions of data) are becoming increasingly attractive, such as expected shortfall  , quantile hedging  , risk management  , among others.
In this paper, we investigate the blocking bootstrap approximation to the expected shortfall based on time series data. For definiteness and conciseness, we shall exclusively concentrate on the MBB method that was independently proposed by  and  .
For the sake of integrity, we briefly describe the MBB method for estimating the sampling distributions of statistics based on weakly dependent observations. Let be a sample from the stationary process . For , a positive integer between 1 and n, we define the overlapping blocks of size as:
Let be a random sample of blocks from , where , that is, are independently and identically distributed as .
The observations in the resampled block are denoted by . The MBB sample consists of , where . Let
be a random quantity of interest that is a function of the random variables and of some unknown population parameter . Then, the MBB version of is defined as
where is a sensible estimator of based on . The MBB estimator of the distribution of can be defined as the conditional distribution of , given . Note that Efron’s bootstrap method is a special case of the MBB method with block length .
An alternative definition of the MBB version of of (2.2) is given by resampling blocks from , and using the first n out of the -many resampled values. However, the difference between the two versions is asymptotically negligible. To simplify the proofs of the main results, here we shall use the version given by (2.3) based on b complete resampled blocks.
Throughout this paper, we use , , and to denote, respectively, the conditional probability, the conditional expectation, and the conditional variance, given .
Now, we are in a position to define the MBB version of the normalized expected shortfall, , for a given . Let denote the MBB empirical distribution function, i.e., . Then, the MBB version of the sample VaR, , is defined as . Similarly, the MBB version of the expected shortfall can be defined as below
where , and . The MBB version of the normalized expected shortfall, , is given by
Note that in the definition of the MBB version of , we center by . As in the case of the sample mean  , and the sample quantile  , this helps center constant for the MBB sample expected shortfall.
denote the distribution function of . Then, the MBB estimator of is given by the conditional distribution of , i.e., by
We conclude this section with an introduction of some standard dependence condition on the ’s. Suppose that the random variables are defined on a common probability space . Let be the s-field generated by random variables , . For , we define
The sequence is called strongly mixing or a-mixing if as . Strong mixing is a fairly non-restrictive dependence assumption. Empirical studies have showed that many log financial returns are strong mixing with exponential decay coefficients.
As a convention, we assume throughout this paper that unless otherwise specified, limits are taken as . Next we state the main results of the paper.
3. Main Results
The first result asserts consistency of the MBB approximation for the distribution function of .
Theorem 1. Assume that with p close to zero and that F has a positive and continuous derivative f in a neighborhood of with . Also, suppose that for some and , and that and . Then, under the moment condition
Theorem 1 shows that the MBB method provides a valid approximation to the distribution of the centered and scaled sample expected shortfall under geometric mixing and under a mild condition on the block length .
The next result proves the consistency of the MBB variance estimators under the same conditions as Theorem 1.
Theorem 2. Under the conditions of Theorem 1,
Theorem 2 shows that under some mild conditions, the MBB estimator of the asymptotic variance of the centered and scaled expected shortfall at confidence level converges in probability.
Remark 1. Note that in addition to the regularity conditions, we require the moment condition (3.1) for both main results. The consistency of the MBB approximation to the distributions of some quantities, as in the cases of sample mean and sample quantile, does not require moment condition although consistency of the MBB variance estimator in general needs some moment condition. The moment condition of Theorem 1 may be relaxed.
We now introduce some basic notation. Let C, denote generic constants in that depend on their arguments (if any) but not on the variables n and x. For real numbers x and y, write , .
For any real number x, , we introduce the following
Note that and , , are block averages. Then,
which implies that
For a random variable X and a real number q, we define
Recall that unless otherwise indicated, limits are taken by letting n tend to infinity. The first lemma is Theorem 1 of  . It states the asymptotic normality of the centered and scaled sample estimator of expected shortfall, , for a given close to zero. We include this result here for the sake of completeness.
Lemma 1. Suppose that F is differentiable at with a positive derivative and that for some and . Then,
where is as defined in (3.3).
The next lemma is a consistency result of the MBB variance estimator of , the asymptotic variance of the normalized expected shortfall.
Lemma 2. Under conditions of Theorem 2, we have
Proof: By Lemma 5.4 (iii) of  , for , we have,
where is a constant. Let
Then, for , ,
since is continuous (and hence bounded) in a neighborhood of . Because the mixing coefficient decays exponentially, it is easily seen that, for ,
Hence, applying Lemma 3.2 of Lahiri (2003) with , and , we obtain
which together with the Markov’s inequality lead to
Now, we investigate .
Theorem 3.1 of Lahiri (2003) implies that
Next, we show that
Using (4.6), (4.7), we get
Hence, Equation (4.11) is proved. By (4.10), (4.11), and Cauchy-Schwartz inequality, we have,
We complete the proof of (4.12). Combining (4.9)-(4.12), we obtain
Here we used the condition on the block length . We complete the proof of Lemma 2. ,
Lemma 3 below gives a convergence result of the third moment of the MBB block average.
Lemma 3. Under the conditions of Theorem 2,
Proof: It can be verified by using (4.5) with ,
which leads to
Using (4.6), (4.13), and the fact that
Note that for , , and ,
Then, Lemma 3.2 of Lahiri (2003) implies,
Finally, combining (4.14), (4.15), and (4.17) gives,
Thus, we obtain,
Lemma 3 is proved. ,
Proof of Theorem 1: By the definitions of , , , , , and the fact that are IID, we get
Then, for any ,
which together with the Berry-Esseen Theorem gives
Lemma 2 and Lemma 3 imply, respectively,
Then, by (4.20)-(4.22), we have
uniformly in . This together with Lemma 1 yields
That is, converges in probability to . We complete the proof of Theorem 1. ,
Proof of Theorem 2: Theorem 2 is a consequence of Lemma 2 and Equation (4.19).
since are IID, which implies that
The proof of Theorem 2 is completed. ,
In this paper, we establish the asymptotic properties of the blocking bootstrap estimators of the expected shortfall. We prove that the MBB method provides a valid approximation to the distribution of the centered and scaled sample estimator of the expected shortfall and show that under mild regularity conditions, the MBB variance estimator is consistent.
As in many situations where the block bootstrap methodology is involved, the performance of the block bootstrap distribution function and variance estimators of critically depends on the block size . Although there have been some theoretical works on the choice of the optimal block length in the literature, the optimal block lengths and/or the optimal rates of convergence in Theorems 1 and 2 are unknown at this stage.
It is also of interest to conduct both empirical studies and simulations to investigate the performance of the MBB estimators and compare the obtained results with those in  .
We thank the reviewer for his/her helpful comments and suggestions which improved the manuscript.