The Improved Value-at-Risk for Heteroscedastic Processes and Their Coverage Probability

A risk measure commonly used in financial risk management, namely, Value-at-Risk (VaR), is studied. In particular, we find a VaR forecast for heteroscedastic processes such that its (conditional) coverage probability is close to the nominal. To do so, we pay attention to the effect of estimator variability such as asymptotic bias and mean square error. Numerical analysis is carried out to illustrate this calculation for the Autoregressive Conditional Heteroscedastic (ARCH) model, an observable volatility type model. In comparison, we find VaR for the latent volatility model i.e., the Stochastic Volatility Autoregressive (SVAR) model. It is found that the effect of estimator variability is significant to obtain VaR forecast with better coverage. In addition, we may only be able to assess unconditional coverage probability for VaR forecast of the SVAR model. )is is due to the fact that the volatility process of the model is unobservable.


Introduction
Risk management and risk measures have become important topics to discuss among financial and actuarial practitioners as well as academia. ey have spent their efforts to seek better risk models and to apply these for real problems. One of the challenging topics in this area is finding a Value-at-Risk (VaR) and its assessment of accuracy. VaR may simply be defined as quantile of asset returns' distribution conditional on last observation, e.g., [1]. In fact, VaR is a value representing tolerated maximum loss of return or portfolio returns; thus, the calculation of VaR may be multiplied by the market value of portfolio, see [2]. is value is crucial for financial institution for capital reserving. It may also be used as an alarm to avoid the worse financial risk.
From statistical perspective, VaR forecast is an application of the concept of the prediction limit (upper) for future observations, given a collection of random variables for losses. Suppose that Y t is a sequence of random losses with probability distribution determined by the parameter vector θ. e available loss data are Y 1 , . . . , Y n . In this paper, we are concerned with a common risk measure in financial risk management, namely, Value-at-Risk (VaR). Specifically, we aim at finding VaR α n+1;θ such that it has coverage (conditional) probability equal to α, i.e., where F n previous information up to time n. For known θ, it is straightforward to find a VaR α n+1; θ that satisfies this condition. In practice, θ is unknown. An estimative VaR α n+1; θ forecast is obtained by replacing θ by an estimator θ. e coverage probability of such VaR α n+1; θ may not be adequate unless n is very large. It may be shown that the (conditional) coverage probability of VaR α n+1; θ differs from α by O(n − 1 ). We adopt the method of Kabaila and Syuhada [3,4] to modify the estimative VaR α n+1; θ forecast to obtain an improved + VaR α n+1; θ forecast with better coverage properties, i.e., it differs from α by O(n − 3/2 ). To find this improved + VaR α n+1 forecast, we need to evaluate the effect of estimator variability such as asymptotic bias and mean square error. When θ is either the maximum likelihood or conditional maximum likelihood estimator, this asymptotic (conditional) bias can be found using the elegant formula described in [5].
In this paper, we consider to find estimative and improved VaR forecasts for class of heteroscedastic processes. In particular, we deal with Autoregressive Conditional Heteroscedastic (ARCH) and Stochastic Volatility Autoregressive (SVAR) models. It is well known that the main difference of these two models lies on volatility function; the ARCH model has observable volatility function, whilst in the SVAR model the volatility function remains stochastic or latent. As a consequence, we obtain a VaR forecast which depends on last observation only for the ARCH model. Besides, the way of finding VaR forecast for the SVAR model is quite difference compared the one in the ARCH model. Furthermore, the estimative and improved VaR forecasts for the SVAR model may only be assessed through unconditional coverage probability. e remainder of this paper is organized as follows. Section 2 describes the theoretical background of estimative and improved VaR forecasts. Computation of VaR forecast for class of heteroscedastic processes and ARCH and SVAR models are given in Sections 3 and 4, respectively.

Description of the Estimative and Improved VaR Forecast
Value-at-Risk (VaR) is defined as a maximum loss that can be tolerated at a given level of significance. Let VaR α n+1; θ denote the estimative VaR forecast. e accuracy of this forecast can be calculated through its coverage (conditional) probability: where the expectation is with respect to the conditional distribution of Y n+1 , given previous information is up to time n. To calculate (2), we derive the following Taylor expansion: where θ r denotes the rth component of θ and the Einstein summation notation has been used. It can be seen that the effect of estimator variability contributes significantly to the coverage (conditional) probability of the VaR forecast. As noted by [5,6], where i(θ) is the expected information matrix. It follows from this that To obtain an improved VaR forecast, we follow the method of Kabaila and Syuhada [3]. Define us, to order n − 1 , d(θ) � c(θ). e improved VaR forecast, denoted as + VaR α t+1;θ , is with coverage (conditional) probability equal to α + O(n − 3/2 ). We begin to find an improved VaR forecast by finding by simulation. We do this for ARCH(1) and SVAR(1) processes in the next two sections. Our aim is to illustrate the different ways in obtaining the VaR forecast for these two processes. We may only be able to calculate unconditional coverage probability of the VaR forecast for the SVAR process.

Computation of the Estimative and Improved VaR Forecast for an ARCH(1) Process
for all integer t, where the ε t are independent and identically N(0, 1) distributed and θ � (a 0 , a 1 ). We demonstrate the calculation of the VaR forecast using the simulation algorithm of Kabaila [7] for computing some particular types of conditional expectations. e estimative VaR forecast is given by where (a 0 , a 1 ) is the conditional maximum likelihood estimator of (a 0 , a 1 ). To find the improved VaR forecast, we 2 Journal of Probability and Statistics need to find d(θ), for θ � θ and specified y n . We do this by first observing that and estimating this expectation by simulation using the method of Kabaila [7] as follows. Define X � (Y 1 , . . . , Y n− 1 ) and Y � Y n . e conditional expectation (11) is equal to where and y � y n . Let f(y | x) denote the probability density function of Y, conditional on X � x, evaluated at y. It may be shown that where ϕ(x; σ) denotes the probability density function of X ∼ N(0, σ 2 ), evaluated at x. e simulation used to estimate ϑ consists of M independent simulation runs, where the kth run consists of the following steps: Step 1: simulate an observation x k of X � (Y 1 , . . . , Y n− 1 ) Step 2: calculate and store f(y | x k ) and g(x k , y) (15) e standard error of this estimate is found using eorem 4.2 of Kabaila [7]. We carry out simulation to illustrate computation of the estimative and improved VaR forecasts. We take the following case: θ � (a 0 , a 1 ) � (0.15, 0.9), n � 200, and y n � 0.

VaR Forecast and Its Coverage Probability of a SVAR(1) Process
An alternative process for modeling volatility is Stochastic Volatility Autoregressive (SVAR) process. Unlike ARCH, the SVAR process has latent volatility function. Nonetheless, the SVAR(1) model is a good representation, from theoretical viewpoint, of the behavior of the returns in the real financial markets. e purpose of this section is to find estimative and improved VaR forecast for such SVAR process along with their coverage probability and, most importantly, to show how different in finding VaR forecast for SVAR model in comparison to the one of ARCH model. e desirability of having coverage probability α for the VaR forecast of the SVAR process has not been discussed by many authors. Most of the papers on the SVAR have investigated the parameter estimation method, estimating volatility and/or evaluating volatility prediction in the context of meansquared-error of forecasting.
We consider the SVAR(1) model which is developed as follows. Suppose that Y t is an asset return at time t, and we assume that the average return is zero. e distribution of Y t , conditional on the variance, is normal with mean zero and variance exp(V t ), where V t follows an autoregressive process of order one or AR(1) process. In other words, for t � 0, 1, . . ., where the ε t s are independent and identically distributed (i.i.d.) N(0, 1) and η t 's are i.i.d. N(0, σ 2 η ). e arrays of η t 's and ε t 's are independent. Let θ � (c, δ, σ 2 η ) be the parameter of the SVAR(1) model, where δ is the persistence parameter, whilst σ 2 η denotes the volatility of the volatility shock. Here, we restrict to the case that the SVAR(1) model is covariance stationary, i.e., the persistence parameter |δ| < 1.
e assumption of normality for ε t can be relaxed by using other distributions, see, for example, [8] who has shown the dominance of SVAR with heavy-tailed distributions compared to SVAR-normal. When the Gaussian distribution is employed for both ε t and η t , we call the model as a "Gaussian SVAR(1) model. " We begin to find the VaR forecast as follows. Consider the case of a stationary Gaussian SVAR(1) model, i.e., the persistence parameter δ < 1 and ε t and η t are N(0, 1) and where Φ(·) is the cumulative distribution function (cdf ) of the standard normal distribution and f V (· ; θ) denotes the unconditional probability density function (pdf ) of V. Define and also define VaR α n+1;θ to be the solution for z of For a specified estimator θ, we will obtain z � VaR α n+1;θ , which is the "estimative VaR forecast." e evaluation of the integral in the definition of H(z, θ, α)o analytically is not straightforward. Nonetheless, we can evaluate this numerically as follows. Firstly, we carry out a truncation of this integral. e lower and upper bound are set to be μ V − kσ V and μ V + kσ V , respectively, where k is a positive integer. us, the error of truncation is bounded above by (20) If we set k � 4, for instance, then the truncation error is bounded above by 6.33 × 10 − 5 . Alternatively, if we truncate the integral with lower bound μ V − 5σ V and upper bound μ V − 5σ V , i.e., k � 5, then we obtain the truncation error bounded above by 5.733 × 10 − 7 . e second step is conducting a numerical integration and find VaR α n+1;θ by numerical solution of H(z, θ, α) � 0. Now, in order to compute the improved VaR forecast, we first observe that and, as before, we estimate this (unconditional) expectation by simulation. Define and let f(· ; θ) be the probability density function of Y n+1 . e improved VaR forecast + VaR α n+1;θ is given by Monte Carlo simulation estimates to obtain the estimative and improved VaR forecasts are obtained as follows. We begin by running simulation data from a stationary Gaussian SVAR(1) model with sample size n � 100 and parameter θ � (0.01, 0.95, 0.17). e parameter estimation is conducted by applying the Maximum Likelihood-Efficient Important Sampling (ML-EIS) method, and we have used m � 50 simulation runs to obtain θ.
For a single run of the simulation described above, we obtain θ � (0.016, 0.962, 0.178

Concluding Remark
e computation of estimative and improved VaRs may be extended in two different directions. First, as pointed out by Dendramis et al. [9] and So and Yu [10], e.g., it is applied to volatility models with heavy-tailed distributions, such as t and GED, as well as models to capture leverage effect and asymmetric, such as Exponential Generalized ARCH or GJR Generalized ARCH. e second direction is by taking the adjusted VaR forecast that ensures or satisfies the coherent property. Some possible risk measures include modification of mean for observations beyond VaR or involving other dependent observations.

Data Availability
NASDAQ and NYSE data taken from Yahoo Finance were used to support the findings of this study.
Disclosure is paper has been presented in the 9th International Triennial Calcutta Symposium on Probability and Statistics, 2015.

Conflicts of Interest
e authors declare that they have no conflicts of interest.