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

Authors: Khreshna Syuhada
Journal: Journal of Probability and Statistics
 (2020)

---

## Abstract

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. This is due to the fact that the volatility process of the model is unobservable.


---

## Body

## 1. Introduction

Risk management and risk measures have become important topics to discuss among financial and actuarial practitioners as well as academia. They 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]. This 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 thatYt is a sequence of random losses with probability distribution determined by the parameter vector θ. The available loss data are Y1,…,Yn. 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 VaRn+1;θα such that it has coverage (conditional) probability equal to α, i.e., (1)PθYn+1≤VaRn+1;θαFn=α,∀θ,where Fn previous information up to time n. For known θ, it is straightforward to find a VaRn+1;θα that satisfies this condition. In practice, θ is unknown. An estimative VaRn+1;θ^α forecast is obtained by replacing θ by an estimator θ^. The coverage probability of such VaRn+1;θ^α may not be adequate unless n is very large. It may be shown that the (conditional) coverage probability of VaRn+1;θ^α differs from α by On−1.We adopt the method of Kabaila and Syuhada [3, 4] to modify the estimative VaRn+1;θ^α forecast to obtain an improved V+aRn+1;θ^α forecast with better coverage properties, i.e., it differs from α by On−3/2. To find this improved V+aRn+1α 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.The remainder of this paper is organized as follows. Section2 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.


## 2. 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. LetVaRn+1;θ^α denote the estimative VaR forecast. The accuracy of this forecast can be calculated through its coverage (conditional) probability:(2)PθYn+1≤VaRn+1;θ^αFn=EθFθVaRn+1;θ^αFnFn,where the expectation is with respect to the conditional distribution of Yn+1, given previous information is up to time n. To calculate (2), we derive the following Taylor expansion:(3)FθVaRn+1;θ^αFn=FθVaRn+1;θαFn+∂FθVaRn+1;θ˜αFn∂θ˜rθ˜=θθ^r−θr+12∂2FθVaRn+1;θ˜αFn∂θ˜r∂θ˜sθ˜=θθ^r−θrθ^s−θs+⋯,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],(4)Eθθ^−θFn=n−1bθ+⋯,Eθθ^−θθ^−θTFn=i−1θ+⋯,where iθ is the expected information matrix. It follows from this that(5)PθYn+1≤VaRn+1;θ^αFn=α+cθn−1+⋯.To obtain an improved VaR forecast, we follow the method of Kabaila and Syuhada [3]. Define(6)dθ=PθYn+1≤VaRn+1;θ^αFn−α.Thus, to ordern−1, dθ=cθ. The improved VaR forecast, denoted as V+aRt+1;θ^α, is(7)V+aRn+1;θ^α=VaRn+1;θ^α−dθ^fVaRn+1;θ^α;θ^Fn,with coverage (conditional) probability equal to α+On−3/2. We begin to find an improved VaR forecast by finding(8)PθYn+1≤VaRn+1;θ^αFn,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.


## 3. Computation of the Estimative and Improved VaR Forecast for an ARCH(1) Process

Consider an ARCH(1) processYt satisfying(9)Yt=σtεt,σt2=a0+a1Yt−12,for all integer t, where the εt are independent and identically N0,1 distributed and θ=a0,a1. We demonstrate the calculation of the VaR forecast using the simulation algorithm of Kabaila [7] for computing some particular types of conditional expectations.The estimative VaR forecast is given by(10)VaRn+1;θ^α=Φ−11−αa^0+a^1yn2,where a^0,a^1 is the conditional maximum likelihood estimator of a0,a1. To find the improved VaR forecast, we need to find dθ, for θ=θ^ and specified yn. We do this by first observing that(11)PθYn+1≤VaRn+1;θ^αYn=yn=PθYn+1≤Φ−11−αa^0+a^1Yn2Yn=yn=Pθεn+1≤Φ−11−αa^0+a^1yn2a0+a1yn2Yn=yn=EθΦΦ−11−αa^0+a^1yn2a0+a1yn2Yn=yn,and estimating this expectation by simulation using the method of Kabaila [7] as follows.DefineX=Y1,…,Yn−1 and Y=Yn. The conditional expectation (11) is equal to(12)ϑ=EθgX,yY=y,where(13)gX,y=ΦΦ−11−αa^0+a^1yn2a0+a1yn2,and y=yn. Let fyx denote the probability density function of Y, conditional on X=x, evaluated at y. It may be shown that(14)fyx=ϕyn;a0+a1yn2,where ϕx;σ denotes the probability density function of X∼N0,σ2, evaluated at x.The simulation used to estimateϑ consists of M independent simulation runs, where the kth run consists of the following steps:Step 1: simulate an observationxk of X=Y1,…,Yn−1Step 2: calculate and storefyxk and gxk,yThe estimate ofϑ is(15)∑k=1Mfyxkgxk,y∑ℓ=1Mfyxℓ.The standard error of this estimate is found using Theorem 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 yn=0.1. The estimative 0.95 VaR forecast VaRn+1;θ^0.95=0.6861. The conditional coverage probability of estimative VaR forecast is calculated by using M=10,000 simulation runs. The resulting estimate of (11) is 0.9423, with standard error 0.00019. The improved 0.95 VaR forecast V+aRn+1;θ^0.95 is therefore estimated to be 0.7173. The standard error of the estimate of (11) indicates that this estimate leads to a sufficiently accurate estimate of the improved VaR forecast. Note that computation is performed using MATLAB and the MATLAB statistics toolbox.


## 4. 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.The 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. The desirability of having coverage probabilityα 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 mean-squared-error of forecasting.We consider the SVAR(1) model which is developed as follows. Suppose thatYt is an asset return at time t, and we assume that the average return is zero. The distribution of Yt, conditional on the variance, is normal with mean zero and variance expVt, where Vt follows an autoregressive process of order one or AR(1) process. In other words,(16)Yt=expVt2εt,Vt=γ+δVt−1+ηt,for t=0,1,…, where the εts are independent and identically distributed (i.i.d.) N0,1 and ηt’s are i.i.d. N0,ση2. The arrays of ηt’s and εt’s are independent. Let θ=γ,δ,ση2 be the parameter of the SVAR(1) model, where δ is the persistence parameter, whilst ση2 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. The assumption of normality for εt 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 εt and ηt, 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 parameterδ<1 and εt and ηt are N0,1 and N0,ση2 distributed, respectively. The unconditional distribution of Vt is NμV,σV2, where μV=γ/1−δo and σV2=ση2/1−δ2. We observe that(17)PθYn+1≤z=PθexpVn+12εn+1≤z=EθPθexpVn+12εn+1≤zVn+1=EθΦzexpVn+1/2=∫−∞∞Φzexpv/2fVv;θdv,where Φ⋅ is the cumulative distribution function (cdf) of the standard normal distribution and fV⋅;θ denotes the unconditional probability density function (pdf) of V.Define(18)Hz,θ,α=∫−∞∞Φzexpv/2fVv;θdv−α,and also define VaRn+1;θα to be the solution for z of PθYn+1≤z=α, i.e.,(19)Hz,θ,α=0.For a specified estimatorθ^, we will obtain z=VaRn+1;θ^α, which is the “estimative VaR forecast.”The evaluation of the integral in the definition ofHz,θ,αo analytically is not straightforward. Nonetheless, we can evaluate this numerically as follows. Firstly, we carry out a truncation of this integral. The lower and upper bound are set to be μV−kσV and μV+kσV, respectively, where k is a positive integer. Thus, the error of truncation is bounded above by(20)∫−∞μV−kσVΦzexpv/2fv;θdv+∫μV+kσV∞Φzexpv/2fv;θdv≤2∫μV+kσV∞fv;θdv=21−Φk.If we setk=4, for instance, then the truncation error is bounded above by 6.33×10−5. Alternatively, if we truncate the integral with lower bound μV−5σV and upper bound μV−5σV, i.e., k=5, then we obtain the truncation error bounded above by 5.733×10−7. The second step is conducting a numerical integration and find VaRn+1;θα by numerical solution of Hz,θ,α=0.Now, in order to compute the improved VaR forecast, we first observe that(21)PθYn+1≤VaRn+1;θ^α=Pθεn+1≤VaRn+1;θ^αexpVn+1/2=EθPθεn+1≤VaRn+1;θ^αexpVn+1/2Y1,…,Yn,Vn+1=EθΦVaRn+1;θ^αexpVn+1/2,and, as before, we estimate this (unconditional) expectation by simulation.Define(22)dθ=PθYn+1≤VaRn+1;θ^α−α,and let f⋅;θ be the probability density function of Yn+1. The improved VaR forecast V+aRn+1;θ^α is given by(23)V+aRn+1;θ^α=VaRn+1;θ^α−dθ^fVaRn+1;θ^α;θ^.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 sizen=100 and parameter θ=0.01,0.95,0.17. The 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 obtainθ^=0.016,0.962,0.178. The corresponding estimative 0.95 VaR forecast VaRn+1;θ^0.95 is obtained by evaluating Hz,θ^,0.95 via numerical integration in which the VaRn+1;θ^α is computed by numerical solution of Hz,θ^,0.95=0 for z. The resulting estimative 0.95 VaR forecast VaRn+1;θ^0.95 is 2.28. Note that when evaluating the integral we have chosen k=4 so that the truncated integral has error bounded above by 6.33×10−5, which is reasonably small.We find the improved VaR forecast that corresponds to this estimative one by usingθ^ as the “true parameter value” in the simulation. We carry out M=1000 simulation runs. These simulations are used to estimate dθ^ in the obvious way. The improved 0.95 VaR forecast V+aRn+1;θ^0.95=2.511, which differs significantly from the estimative 0.95 VaR forecast VaRn+1;θ^0.95.


### 4.1. Numerical Illustration

In order to describe clearly in real application setting for the abovementioned VaR forecast methods, we provide a numerical study of VaR forecasts especially for the case Generalized ARCH and SVAR models (and some extensions of volatility models i.e., GJR Generalized ARCH and Threshold Generalized ARCH) with some distributions assumptions. We have used NASDAQ and NYSE data (YahooFinance) for duration of 1 January 2007 till 10 December 2016 that comprises of 2516 observations. The choice of volatility models may not be easy with regard to different results as the effect of involving features attached to the models (Table1).Table 1
The VaR forecast (and improvement) for Generalized ARCH and SVAR models and some extensions of volatility models.
Volatility models
VaR
n
+
1
;
θ
^
0.99
VaR
n
+
1
;
θ
^
0.95
GARCH-n (1, 1)
0.0210 (0.0274)
0.0148 (0.0218)
GARCH-t (1, 1)
0.0236 (0.0289)
0.0167 (0.0229)
SVAR-n (1)
0.0240 (0.0330)
0.0200 (0.0258)
SVAR-t (1)
0.0245 (0.0311)
0.0216 (0.0277)
GJR-GARCH-n (1, 1)
0.0228 (0.0341)
0.0162 (0.0268)
GJR-GARCH-t (1, 1)
0.0249 (0.0367)
0.0176 (0.0278)
TGARCH-n (1, 1)
0.0228 (0.0266)
0.0195 (0.0217)
TGARCH-t (1, 1)
0.0243 (0.0299)
0.0162 (0.0265)


## 4.1. Numerical Illustration

In order to describe clearly in real application setting for the abovementioned VaR forecast methods, we provide a numerical study of VaR forecasts especially for the case Generalized ARCH and SVAR models (and some extensions of volatility models i.e., GJR Generalized ARCH and Threshold Generalized ARCH) with some distributions assumptions. We have used NASDAQ and NYSE data (YahooFinance) for duration of 1 January 2007 till 10 December 2016 that comprises of 2516 observations. The choice of volatility models may not be easy with regard to different results as the effect of involving features attached to the models (Table1).Table 1
The VaR forecast (and improvement) for Generalized ARCH and SVAR models and some extensions of volatility models.
Volatility models
VaR
n
+
1
;
θ
^
0.99
VaR
n
+
1
;
θ
^
0.95
GARCH-n (1, 1)
0.0210 (0.0274)
0.0148 (0.0218)
GARCH-t (1, 1)
0.0236 (0.0289)
0.0167 (0.0229)
SVAR-n (1)
0.0240 (0.0330)
0.0200 (0.0258)
SVAR-t (1)
0.0245 (0.0311)
0.0216 (0.0277)
GJR-GARCH-n (1, 1)
0.0228 (0.0341)
0.0162 (0.0268)
GJR-GARCH-t (1, 1)
0.0249 (0.0367)
0.0176 (0.0278)
TGARCH-n (1, 1)
0.0228 (0.0266)
0.0195 (0.0217)
TGARCH-t (1, 1)
0.0243 (0.0299)
0.0162 (0.0265)


## 5. Concluding Remark

The 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. The 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.



---

*Source: 7638517-2020-03-10.xml*