#### 1.1. Literature Review on VaR and CVaR

According to the branch of application at which the concept of risk is situated, risk can be defined and expressed in several ways each has its own target (Reference [

1], Chapters 14–15).

As given in the well-known Basel Committee on Banking Supervision (BCBS), a finance institute is forced to fulfill capital needs to cover all possible losses because of risk sources during normal operations: i.e., operational risk, market risk and credit risk [

2]. A finance firm may need to figure out possible losses to its portfolios so as to better allocate its funds and plan for payments to investors, see also the discussions in Reference [

3].

In the field of mathematical finance, one of the pioneer measures for doing such a thing is the Value-at-Risk (VaR). VaR was first given by Markowitz in Reference [

4] and Roy in Reference [

5] separately to estimate the portfolio’s VaR and actually for optimizing profit for the associated specific risk levels. This measure is defined as follows [

6]:

wherein

X is a random variable,

$\alpha $ is the pre-determined tail level,

${F}_{X}(\xb7)$ stands for the cumulative distribution function (CDF), and the function min is sometimes replaced by inf. Note that both cases of lower tails or losses (e.g.,

$\alpha $ equal to 1%) or upper tails or profits (e.g.,

$\alpha $ equal to 99%) can be used by (

1) when employing

$\alpha $ or

$1-\alpha $ as tail levels. However, in this work we focus on profits, though everything is exchangeable.

The approaches to VaR could be investigated in three categories [

7]: (i) fully parametric models approach based on a volatility models; (ii) non-parametric approaches based on the Historical Simulation (HS) methods and (iii) Extreme Value Theory approach based on modeling the tails of the return distribution. It is known that VaR is used for obtaining the loss in the entity being evaluated and the occurrence probability for the loss defined. The VaR measure is normally employed at the institutional portfolios by banks to understand the occurrence and extent ratio of possible losses [

8]. Note also that a fundamental factor in optimization of a portfolio is to find a right measure of risk to scalarize the portfolio’s risk.

On the other hand, as long as an investment has given an (almost) stable behavior along time, then the VaR can be enough for the management of risk in the portfolio including the investment [

9]. Nonetheless, by losing the stability, the chance that this risk measure gives a full picture of involved risks gets decreased since it is not that sensitive to anything after its own boundary (threshold). To illustrate further, lately as an instance, the oil sector has reflected instability in international oil prices, which have been more representative since 2004 and respond to various existing factors.

Conditional Value at Risk (CVaR), which is also sometimes called the AVaR, or expected tail loss (ETL), is actually the average loss of the specified distribution in the extreme tail area [

10]. Hence, it is an alternative measure of risk which could overcome on some of the drawbacks of VaR and always yields a higher magnitude for the risk in contrast to the VaR (Reference [

1], chapter 15). The mechanism of CVaR is based on the conditional expectation of the loss that exceeds VaR, viz,

CVaR is constructed from the VaR for a portfolio or investment and is employed in the optimization of a portfolio for efficient risk allocation. Risk allocation is the process of identifying risk and determining how and to what extent they should be shared. Most owners understand that risk is an inherent part of the construction process and cannot be eliminated. Accurate risk identification and the assignment of risk to the party make the stock-holders or financial managers to carefully step ahead and minimize the risk of losing the capital. This could improve quality, reduce delays and resolve disputes efficiently [

11].

The definition (

2) is also called the lower CVaR (Tail VaR) as long as the inequality in the conditional expectation is with equality. But once the inequality be strict (without the equality sign), it will be called the upper CVaR definition (or expected shortfall (ES)) [

12]. Some stock markets, such as NYSE, have required their accepted companies to estimate and report their risk employing ES, which makes this measure an important and useful risk measure.

Another definition of CVaR for continuous probability distributions is [

13]:

wherein the

$\alpha $-tail distribution is defined by

By considering the examined time horizon (yearly, monthly, weekly, or even daily), VaR could be computed showing the suffering probability for a loss while the mixture of a specific investment portfolio is given. Although this is a simple and widely-used measure for financial traders and managers, it has its own limitations which restricts its application and reliably in many of the practical cases. Some of these such downfalls are discontinuity for some discrete distributions, lack of convexity, and, mainly, its inability to quantify risk in the extreme tail region after the boundary of VaR. And because of this, CVaR was introduced. In the recent document published by the Basel Committee on Banking Supervision in Reference [

14], both the value-at-risk and the expected shortfall have been considered and discussed in detail.

Another limitation in risk allocation and managing portfolios is the assumption of normal distribution for the underlying factors. Anyhow, it was revealed that the normal assumption is not the best consideration for finance instruments, yielding in the application of alternative distributions. Now, by encountering a volatile and uncertain scenario, the question is how we could find efficient models that facilitate better risk management?

This paper discusses closed and explicit formulations for the measures VaR and CVaR under the extreme value distribution (EVD). This is important since history has revealed that millions of dollars may be lost in a short piece of time because of failure in handling the financial risks in market. Within this circumstance, we will concentrate this paper on VaR and CVaR explicit formulations, comparing the precision of both values and taking into account the EVD as an alternative to the well-known normal distribution for risk management. To support the discussions, the contribution of this work as an application is considered and simulated from time series in forecasting the prices/returns of several well-known stocks in different period of times under the generalized autoregressive conditional heteroskedasticity (GARCH) model [

15].

#### 1.2. Garch Model

Several works (see Reference [

16] and the references cited therein) have revealed that the future variance prediction through modern GARCH-type models is necessary for managing portfolio risk efficiently, due to the presence of the effects of heteroskedastic (i.e., the volatility of the under studying process is basically not constant.)

It is known that the volatility is the square root of the conditional variance of the log return process given its previous values using

${\mathcal{F}}_{t-1}$ as the

$\sigma $-algebra generated by

${x}_{0},{x}_{1},\dots ,{x}_{t-1}$. This means, as long as

${p}_{t}$ (price of a stock in stick exchange at time

t) is the time series computed at time

t, we can then define the log returns as follows [

17]:

and then we can also define

Since we will illustrate the usefulness of the proposed explicit risk measures under the EVD by the GARCH model, it is now requisite to recall the GARCH model, which is in fact an autoregressive moving average model (ARMA) model for the variance of error [

17]. Among the general parametric case of GARCH(

p,

q) process at which

p is the order of the GARCH terms

${\sigma}^{2}$; and

q is the order of the ARCH terms

${\u03f5}^{2}$, we here use the GARCH(1,1) as follows:

wherein

${z}_{t}$ is a stochastic piece (independent and identically distributed (i.i.d.) innovations having unit variance and vanishing mean). Here,

$w>0$,

${r}_{t}$ is the actual return,

$\varrho $ is the expected return,

${\sigma}_{t}$ is the volatility of the returns on day

t, and finally

The condition

$\lambda +\beta <1$ in (

8) shows to have a stationary solution of the GARCH model.

The model (

7) has become employed in economic time series modeling and is programmed in several of the econometric and statistics packages [

18]. It is favored over other models of stochastic volatility (SV) by many practitioners due to its straightforward programming. Since it is in fact given by stochastic difference equations in discrete time, the likelihood function is easier to tackle than models with continuous-time feature, and also because finance information is basically given at discrete intervals [

19]. To see some background on the relation between the GARCH-EVT model and the GARCH-VaR or GARCH-CVaR, one may refer to Reference [

20].

A fruitful feature of the GARCH-type models is that they capture the fat-tailedness along with volatility clustering. Following the result in Reference [

21], the stationary solution of GARCH(1,1) process follows a heavy-tailed distribution. Therefore, the GARCH-type models turn to be an effective instrument in risk management.

#### 1.3. Motivation and Article’S Plan

The motivation behind choosing the EVD for risk management does not only lie in the fact that no closed form measures for this distribution are existed in literature, but also in the point that it has fatter right tail in contrast to the commonly used normal distribution [

22]. It must be noticed that the selection of distribution for computing VaR and CVaR impacts the approximation of the quartiles that determine the risk. Besides, better adjustment of the empirical data to a specific distribution type enables construction of functions that more efficiently estimate the risk provided the conditions of volatility and uncertainty.

It is necessary to state that very recently Norton et al. in Reference [

23] investigated the CVaR for some common distributions and more specifically derived a generic expression for GEV distribution, at which this paper deals with a special case of 0 as the free parameter. Here, the way of deriving the closed formulation for the VaR and CVaR under the EVD is different and mainly we then focus to employ the obtained VaR/CVaR formulas to the historical returns of several U.S. stocks using the GARCH process. Furthermore, the methodology of this work follows the footsteps of the discussions of [

24], but somewhat different since neither no closed forms for VaR and CVaR under EVD is given nor a direct application on the GARCH process is given in Reference [

24] for forecasting the risk on the tickers considered in this paper. For more, an interested reader may refer to Reference [

25].

The remaining sections of this work are organized in what follows.

Section 2 is devoted to the basic notions of the EVD and obtaining a closed formulation for the VaR measure under this distribution. Distribution of the innovations plays a key role when analyzing unconditional and conditional downside risk. Next, in

Section 3, the CVaR is given. Then,

Section 4 provides an application of the new closed formulations for computing the VaR and CVaR risk measures in a portfolio having a stock based on the GARCH(1,1) process; one may refer to Reference [

26] for further discussions on this. In time series for finance and economy, specific characteristics, like the volatility clustering and being fat-tailed, yield challenges in handling downside risk evaluation. However, the model (

7) captures such characteristics regardless of the distributional assumptions on the innovation process. The paper ends, in

Section 5, by providing several comments and outlooks for future works.