12.1.1 History of VaR

In the last few decades, risk managers have truly experienced a revolution. The rapid increase in the usage of risk management techniques has spread well beyond derivatives and is totally changing the way institutions approach their financial risk. In response to the financial disasters of the early 1990s, a new method called Value at Risk (VaR) was developed as a simple method to quantify market risk (in recent years, VaR has been used in many other areas of risk including credit risk and operational risk). Some of the financial disasters of the early 1990s are the following:

• Figure 12.1 shows the effect of Black Monday, which occurred on October 19, 1987. In a single day, the Dow Jones stock index (DJIA) crashed down by 22.6% (by 508 points), causing a negative knock-on effect on other stock markets worldwide. Overall the stock market lost $0.5 trillion.
• The Japanese stock price bubble, creating a $2.7 trillion loss in capital; see Figure 12.2. According to this website, “the Nikkei Index after the Japanese bubble burst in the final days of 1989. Again, the market showed a substantial recovery for several months in mid-1990 before sliding to new lows.”
• Figure 12.3 describes the dot-com bubble. During 1999 and 2000, the NASDAQ rose at a dramatic rate with all technology stocks booming. However, on March 10, 2000, the bubble finally burst, because of sudden simultaneous sell orders in big technology companies (Dell, IBM, and Cisco) on the NASDAQ. After a peak at $5048.62 on that day, the NASDAQ fell back down and has never since been recovered.
• Figure 12.4 describes the 1997 Asian financial crisis. It first occurred at the beginning of July 1997. During that period a lot of Asia got affected by this financial crisis, leading to a pandemic spread of fear to a worldwide economic meltdown. The crisis was first triggered when the Thai baht (Thailand currency) was cut from being pegged to the US dollars and the government floated the baht. In addition, at the time Thailand was effectively bankrupt from the burden of foreign debt it acquired. A later period saw a contagious spread of the crisis to Japan and to South Asia, causing a slump in asset prices, stock market, and currencies.
• The Black Wednesday, resulting in £800 million losses; see Figure 12.5; According to http://en.wikipedia.org/wiki/Black_Wednesday, Black Wednesday “refers to the events of September 16, 1992 when the British Conservative government was forced to withdraw the pound sterling from the European exchange rate mechanism (ERM) after they were unable to keep it above its agreed lower limit.”
• The infamous financial disasters of Orange County, Barings, Metallgesellschaft, Daiwa, and so many more.

12.1.2 Definition of VaR

Till Guldimann is widely credited as the creator of value at risk (VaR) in the late 1980s. He was then the head of global research at J.P. Morgan. VaR is a method that uses standard statistical techniques to assess risk. The VaR “measures the worst average loss over a given horizon under normal market conditions at a given confidence level” (Jorion, 2001, p. xxii). The value of VaR can provide users with information in two ways: as a summary measure of market risk or an aggregate view of a portfolio's risk. Overall VaR is a forward-looking risk measure and used by financial institutions, regulators, nonfinancial corporations, and asset management exposed to financial risk. The most important use of VaR has been for capital adequacy regulation under Basel II and later revisions.

Let denote a stationary financial series with marginal cumulative distribution function (cdf) and marginal probability density function (pdf) . The VaR for a given probability is defined mathematically as

12.1

That is, VaR is the quantile of exceeded with probability . Figure 12.6 illustrates the definition given by (12.1).

Sometimes, VaR is defined for log returns of the original time series. That is, if , are the log returns for some with marginal cdf and then VaR is defined by (12.1). If and denote the mean and standard deviation of the log returns, then one can write

12.2

where denotes the quantile function of the standardized log returns .

12.1.3 Applications of VaR

Applications of VaR can be classified as:

• Information reporting—it measures aggregate risk and corporation risk in a nontechnical way for easy understanding.
• Controlling risk—setting position limits for traders and business units, so they can compare diverse market risky activities.
• Managing risk—reallocating of capital across traders, products, business units, and whole institutions.

Applications of VaR have been extensive. Some recent applications and application areas have included estimation of highly parallel architectures (Dixon et al., 2012), estimation for crude oil markets (He et al., 2012a), multiresolution analysis-based methodology in metal markets (He et al., 2012b), estimation of optimal hedging strategy under bivariate regime switching ARCH framework (Chang, 2011b), energy markets (Cheong, 2011), Malaysian sectoral markets (Cheong and Isa, 2011), downside residential market risk (Jin and Ziobrowski, 2011), hazardous material transportation (Kwon, 2011), operational risk in Chinese commercial banks (Lu, 2011), longevity and mortality (Plat, 2011), analysis of credit default swaps (Raunig et al., 2011), exploring oil-exporting country portfolio (Sun et al., 2011), Asia-focused hedge funds (Weng and Trueck, 2011), measure for waiting time in simulations of hospital units (Dehlendorff et al., 2010), financial risk in pension funds (Fedor, 2010), catastrophic event modeling in the Gulf of Mexico (Kaiser et al., 2010), estimating the South African equity market (Milwidsky and Mare, 2010), estimating natural disaster risks (Mondlane, 2010), wholesale price for supply chain coordination (Wang, 2010), US movie box office earnings (Bi and Giles, 2009), stock market index portfolio in South Africa (Bonga-Bonga and Mutema, 2009), multiperiod supply inventory coordination (Cai et al., 2009), Toronto stock exchange (Dionne et al., 2009), modeling volatility clustering in electricity price return series (Karandikar et al., 2009), calculation for heterogeneous loan portfolios (Puzanova et al., 2009), measurement of HIS stock index futures market risk (Yan and Gong, 2009), stock index futures market risk (Gong and Li, 2008), estimation of real estate values (He et al., 2008), foreign exchange rates (Ku and Wang, 2008), artificial neural network (Lin and Chen, 2008), criterion for management of stormwater (Piantadosi et al., 2008), inventory control in supply chains (Yiu et al., 2008), layers of protection analysis (Fang et al., 2007), project finance transactions (Gatti et al., 2007), storms in the Gulf of Mexico (Kaiser et al., 2007), midterm generation operation planning in electricity market environment (Lu et al., 2007), Hong Kong's fiscal policy (Porter, 2007), bakery procurement (Wilson et al., 2007), newsvendor models (Xu and Chen, 2007), optimal allocation of uncertain water supplies (Yamout et al., 2007), futures floor trading (Lee and Locke, 2006), estimating a listed firm in China (Liu et al., 2006), Asian Pacific stock market (Su and Knowles, 2006), Polish power exchange (Trzpiot and Ganczarek, 2006), single loss approximation to VaR (Böcker and Klüppelberg, 2005), real options in complex engineered systems (Hassan et al., 2005), effects of bank technical sophistication and learning over time (Liu et al., 2004), risk analysis of the aerospace sector (Mattedi et al., 2004), Chinese securities market (Li et al., 2002), risk management of investment-linked household property insurance (Zhu and Gao, 2002), project risk measurement (Feng and Chen, 2001), long-term capital management for property/casualty insurers (Panning, 1999), structure-dependent securities and FX derivatives (Singh, 1997), and mortgage-backed securities (Jakobsen, 1996).

12.1.4 Aims

The aim of this chapter is to review known methods for estimating VaR given by (12.1). The review of methods is divided as follows: general properties (Section 12.2), parametric methods (Section 12.3), nonparametric methods (Section 12.4), semiparametric methods (Section 12.5), and computer software (Section 12.6). For each estimation method, we give the main formulas for computing VaR. We have avoided giving full details for each estimation method (e.g., interpretation, asymptotic properties, finite sample properties, finite sample bias, sensitivity to outliers, quality of approximations, comparison with competing estimators, advantages, disadvantages, and application areas) because of space concerns. These details can be read from the cited references.

12.1.5 Further Material

The review of value of risk presented here is not complete, but we believe we have covered most of the developments in recent years. For a fuller account of the theory and applications of value risk, we refer the readers to the following books: Bouchaud and Potters (2000, Chapter 3), Delbaen (2000, Chapter 3), Moix (2001, Chapter 6), Voit (2001, Chapter 7), Dupacova et al. (2002, Part 2), Dash (2004, Part IV), Franke et al. (2004), Tapiero (2004, Chapter 10), Meucci (2005), Pflug and Romisch (2007, Chapter 12), Resnick (2007), Ardia (2008, Chapter 6), Franke et al. (2008), Klugman et al. (2008), Lai and Xing (2008, Chapter 12), Taniguchi et al. (2008), Janssen et al. (2009, Chapter 18), Sriboonchitta et al. (2010, Chapter 4), Tsay (2010),Capinski and Zastawniak (2011), Jorion (2001), and Ruppert (2011, Chapter 19).

12.2.1 Ordering Properties

Pflug (2000) and Jadhav and Ramanathan (2009) establish several ordering properties of . Given random variables , , , and a constant , some of the properties given by Pflug (2000) and Jadhav and Ramanathan (2009) are the following:

1. is translation equivariant, that is, .
2. is positively homogeneous, that is, for .
3. .
4. is monotonic with respect to stochastic dominance of order 1 (a random variable is less than a random variable with respect to stochastic dominance of order 1 if for all monotonic integrable functions ); that is, is less than a random variable with respect to stochastic dominance of order 1 and then .
5. is comonotone additive, that is, if and are comonotone, then . Two random variables and defined on the same probability space are said to be comonotone if for all , almost surely.
6. if then .
7. is monotonic, that is, if , then .

Let denote the joint cdf of with marginal cdfs and . Write to mean , where is known as the copula (Nelsen, 1999), a joint cdf of uniform marginals. Let have the joint cdf and have the joint cdf , , and . Then, Tsafack (2009) shows that if is stochastically less than , then for .

12.2.2 Upper Comonotonicity

If two or more assets are comonotonic, then their values (whether they be small, medium, large, etc.) move in the same direction simultaneously. In the real world, this may be too strong of a relation. A more realistic relation is to say that the assets move in the same direction if their values are extremely large. This weaker relation is known as upper comonotonicity (Cheung, 2009).

Let denote the loss of the th asset. Let with joint cdf . Let . Suppose all random variables are defined on the probability space . Then, a simple formula for the VaR of in terms of values at risk of can be established if is upper comonotonic.

We now define what is meant by upper comonotonicity. A subset is said to be comonotonic if for all and whenever and belong to . The random vector is said to be comonotonic if it has a comonotonic support.

Let denote the collection of all zero probability sets in the probability space. Let . For a given , let denote the upper quadrant of and let denote the lower quadrant of . Let .

Then, the random vector is said to be upper comonotonic if there exist and a zero probability set such that

1. is a comonotonic subset of ,
2. ,
3. is an empty set.

If these three conditions are satisfied, then the VaR of can be expressed as

12.3

for and , a comonotonic threshold as constructed in Lemma 2 of Cheung (2009).

12.2.3 Multivariate Extension

In this chapter, we shall focus mainly on univariate VaR estimation. Multivariate VaR is a much more recent topic.

Let be a random vector in with joint cdf . Prékopa (2012) gives the following definition of multivariate VaR:

12.4

Note that MVaR may not be a single vector. It will often take the form of a set of vectors.

Prékopa (2012) gives the following motivation for multivariate VaR: “A finance company generally faces the problem of constructing different portfolios that they can sell to customers. Each portfolio produces a random total return and it is the objective of the company to have them above given levels, simultaneously, with large probability. Equivalently, the losses should be below given levels, with large probability. In order to ensure it we look at the total losses as components of a random vector and find a multivariate -quantile or MVaR to know what are those points in the -dimensional space ( being the number of portfolios), that should surpass the vector of total losses, to guarantee the given reliability.”

Cousin and Bernardinoy (2011) provide another definition of multivariate VaR:

or equivalently

where is the boundary of the set .

Cousin and Bernardinoy (2011) establish various properties of MVaR similar to those in the univariate case. For instance,

1. the translation equivariant property holds, that is,
   
2. the positively homogeneous property holds, that is,
   
3. if is quasi-concave (Nelsen, 1999) then

   

   for , where denotes the th component of ;

4. if is a comonotone nonnegative random vector and if is quasi-concave (Nelsen, 1999), then
   

   for ;

5. if in distribution for every , then
   

   for all ;

6. if is stochastically less than for every , then
   

   for all .

Bivariate VaR in the context of a bivariate normal distribution has been considered much earlier by Arbia (2002).

A matrix variate extension of VaR and its application for power supply networks are discussed in Chang (2011a).

12.2.4 Risk Concentration

Let denote future losses, assumed to be nonnegative independent random variables with common cdf and survival function . Degen et al. (2010) define risk concentration as

If is regularly varying with index , (Bingham et al., 1989), meaning that as , then it is shown that

12.5

as . Degen et al. (2010) also study the rate of convergence in (12.5).

Suppose , are regularly varying with index , . According to Jang and Jho (2007), for ,

for all for some . This property is referred to as subadditivity. If holds as , then the property is referred to as asymptotic subadditivity. For ,

as . This property is referred to as asymptotic comonotonicity. For ,

for all for some . If holds as then the property is referred to as asymptotic superadditivity.

Let denote a counting process independent of with for . According to Jang and Jho (2007), in the case of subadditivity,

for all for some . In the case of asymptotic comonotonicity,

as . In the case of superadditivity,

for all for some .

Suppose is multivariate regularly varying with index according to Definition 2.2 in Embrechts et al. (2009a). If is a measurable function such that

then it is shown that

see Lemma 2.3 in Embrechts et al. (2009b).

12.2.5 Hürlimann's Inequalities

Let denote a random variable defined over , with mean and variance . Hürlimann (2002) provides various upper bounds for : for ,

for ,

for ,

12.6

The equality in (12.6) holds if and only if .

Now suppose is a random variable defined over , with mean , variance , skewness , and kurtosis . In this case, Hürlimann (2002) provides the following upper bound for :

where is the percentile of the standardized Chebyshev–Markov maximal distribution. The latter is defined as the root of

if and as the root of

if , where

where , , , and .

12.2.6 Ibragimov and Walden's Inequalities

Let denote a portfolio return made up of asset returns, , and the nonnegative weights . Ibragimov (2009) provides various inequalities for the VaR of . They suppose that are independent and identically distributed and belong to either , the class of distributions that are convolutions of symmetric stable distributions with and , or , convolutions of distributions from the class of symmetric log-concave distributions and the class of distributions that are convolutions of symmetric stable distributions with and .

Here, denotes a stable distribution specified by its characteristic function

where , , , , and . The stable distribution contains as particular cases the Gaussian distribution for , the Cauchy distribution for and , the Lévy distribution for and , the Landau distribution for and , and the dirac delta distribution for and .

Furthermore, let . Write to mean that for and , where and denote the components of and in descending order. Let and .

With these notations, Ibragimov (2009) provides the following inequalities for . Suppose first that and belong to . Then,

1. (i)  if ;
2. (i)  for all .

Suppose now that and belong to . Then,

1. (i)  if ;
2. (i)  for all .

Further inequalities for VaR are provided in Ibragimov and Walden (2011) when a portfolio return, say, , is made up of a two-dimensional array of asset returns, say, . That is,

where are referred to as “row effects,” are referred to as “column effects,” and are referred to as “idiosyncratic components.”

Let , , , , , and .

With these notations, Ibragimov and Walden (2011) provide the following inequalities for :

1. if belong to , then for all .
2. if belong to , then for all .
3. if belong to , then for all .
4. if belong to , then for all .
5. if belong to , then for all .
6. if belong to , then for all .
7. if belong to , then for all .
8. if belong to , then for all .

Ibragimov and Walden (2011, Section 12.4) discuss an application of these inequalities to portfolio component VaR analysis.

12.2.7 Denis et al.'s Inequalities

Let denote prices of financial assets. The process could be modeled by

where is a Brownian motion; is a compound Poisson process independent of ; are jump times for ; is an adapted integrable process; and , are certain random variables.

Denis et al. (2009) derive various bounds for the VaR of the process

The following assumptions are made:

1. For all , .
2. Jumps of the compound Poisson process are nonnegative and is not identically equal to zero.
3. The process for is well defined and integrable.
4. The jumps have a Laplace transform, , where is a positive constant.
5. There exists such that almost surely for all .
6. There exists and such that

   

   almost everywhere for all . In this case, let

   

   for .

With these assumptions, Denis et al. (2009) show that

For , Denis et al. (2009) show that

If the jumps follow a simple Poisson process, Denis et al. (2009) show that

If the jumps follow an exponential distribution with parameter , Denis et al. (2009) show that

About the issue of continuity/discontinuity of the market with jumps, see Walter (2015).

12.2.8 Jaworski's Inequalities

Jaworski (2007, 2008) considers the following situation: suppose , are the quotients of the currency rates at the end and at the beginning of an investment; suppose that the joint cdf of is , where is a copula (Nelsen, 1999) and is the marginal cdf of ; suppose is the part of the capital invested in the th currency, where are nonnegative and sum to one. Then, the final investment value is

where . Jaworski (2007, 2008) defines the value of risk for a given and a probability as

Jaworski (2007) shows this VaR can be bounded as

for portfolios consisting of only one currency, where and .

12.2.9 Mesfioui and Quessy's Inequalities

Suppose a portfolio is made up of assets and let denote the losses for the assets. Suppose also that the joint cdf of is , where is a copula (Nelsen, 1999) and is the marginal cdf of . Furthermore, define the dual of a given copula (Definition 2.4, Mesfioui and Quessy, 2005) as

With these notations, Mesfioui and Quessy (2005) derive various inequalities for the VaR of . If is such that and for some copulas and , then

where

and

If are identical random variables with common cdf and if is such that is nonincreasing for , then it is shown under certain conditions that

where is the diagonal section of .

Mesfioui and Quessy (2005) also show that if is a random variable with mean and variance , then

where

and

where . If , have means , and variances , , then it is shown that

where and .

12.2.10 Slim et al.'s Inequalities

Suppose a portfolio is made up of assets. Let denote the losses for the assets. Let and denote the cdf and the pdf of . Let denote the value for which is nonincreasing for all . Given this notation, the total portfolio loss can be expressed as for some nonnegative weights summing to one. Slim et al. (2012) show that the VaR of can be bounded as follows:

where

and

for . The use of the earlier results allows easy computation for explicit VaR bounds for possibly dependent risks.

12.3.1 Gaussian Distribution

If are observations from a Gaussian distribution with mean and variance , then VaR can be estimated by

12.7

where is the sample mean and is the sample variance

12.8

The estimator in (12.7) is biased and consistent. If the in (12.8) is replaced by , then (12.7) becomes unbiased and consistent.

12.3.2 Student's Distribution

If are observations from a Student's distribution with degrees of freedom, then VaR can be estimated by (Arneric et al., 2008)

where is the excess sample kurtosis and is the percentile of a Student's random variable with degrees of freedom.

12.3.3 Pareto-Positive Stable Distribution

Sarabia and Prieto (2009) and Guillen et al. (2011) introduce the Pareto-positive stable distribution specified by the cdf

12.9

for , , and . Here, and are shape parameters and is a scale parameter. The Pareto distribution is the particular case of (12.9) for .

The Pareto-positive stable distribution has been applied to risk management; see, for example, Guillen et al. (2011). If is a random variable having the cdf (12.9), then it is easy to see that

for . So, if are maximum likelihood estimators of , then

for .

12.3.4 Log-Folded Distribution

Brazauskas and Kleefeld (2011) introduce the log-folded distribution specified by the quantile function

for , where is a scale parameter, is a shape parameter, and denotes the quantile function of a Student's random variable with degrees of freedom. Brazauskas and Kleefeld (2011) also provide an application of this distribution to risk management.

Suppose is a random sample from the log-folded distribution with order statistics . Brazauskas and Kleefeld (2011) show that the VaR can be estimated by

where

or

where

where and are integers such that and as , where and are trimming proportions with .

12.3.5 Variance–Covariance Method

Suppose the portfolio return, say, , is made up of asset returns, , , as

where are nonnegative weights summing to one. Suppose also E, Var, and Cov. The variance–covariance method suggests that the VaR of can be approximated by

An estimator can be obtained by replacing the parameters , , and by their maximum likelihood estimators.

12.3.6 Gaussian Mixture Distribution

Let denote the financial asset prices and let denote the log return corresponding to the original financial series. Zhang and Cheng (2005) consider the model that have a Gaussian mixture distribution specified by the pdf

for , where the mixing coefficients sum to one. Let denote the VaR corresponding to the th component, that is,

Let denote the VaR corresponding to the mixture model, that is,

Then, Theorem 1 in Zhang and Cheng (2005) shows that

always holds.

Furthermore, let denote the significance level of VaR corresponding to the th component, that is,

Let denote the significance level of VaR corresponding to the mixture model, that is,

Then, Theorem 2 in Zhang and Cheng (2005) shows that

always holds.

12.3.7 Generalized Hyperbolic Distribution

Suppose the log returns, , follow the model

where is the volatility process and are independent and identical random variables with zero mean and unit variance. Let denote the corresponding VaR. Suppose are independent and identical and have the generalized hyperbolic distribution specified by the pdf

where is a location parameter, is a shape parameter, is an asymmetry parameter, is a scale parameter, , , and denotes the modified Bessel function of order .

Tian and Chan (2010) propose a method based on saddlepoint approximation for computing . It can be described as follows:

1. Estimate by

   

   for , where are some nonnegative weights summing to one.

2. Compute as the root of ,where is defined in Step 3.
3. Compute as the root of
   

   where

   
4. Estimate by .

12.3.8 Fourier Transformation Method

Siven et al. (2009) suggest a method for computing VaR by approximating the cdf by a Fourier series. The approximation is given by the following result due to Hughett (1998): suppose

1. that there exist constants and such that and for all ,
2. that there exist constants and such that for all , where denotes the characteristic function corresponding to .

Then, for constants , and , the cdf can be approximated as

where , denotes the real part, and

An estimator for is obtained by solving the equation

for .

12.3.9 Principal Components Method

Brummelhuis et al. (2002) use an approximation based on the principal component method to compute VaR. If is a vector of risk factors over time and if is a random variable, they define VaR to be

12.10

This equation is too general to be solved. So, Brummelhuis et al. (2002) consider the quadratic approximation

and assume that is normally distributed with mean and covariance matrix . Under this approximation, we can rewrite (12.10) as

Let denote the Cholesky decomposition and let

Also let denote the principal components decomposition of , , and . With these notations, Brummelhuis et al. (2002) show that VaR can be approximated by

where is the root of

12.3.10 Quadratic Forms

Suppose the financial series are realizations of a quadratic form

where is a standard normal vector, , and . Examples include nonlinear positions like options in finance or the modeling of bond prices in terms of interest rates (duration and convexity). Here, s are the eigenvalues sorted in ascending order. Suppose there are distinct eigenvalues. Let denote the highest index of the th distinct eigenvalue with multiplicity . For , let

Let denote the moment-generating function of evaluated at . With this notation, Jaschke et al. (2004) derive various approximations for VaR. The first of these applicable for is

where denotes the percentile of a noncentral chi-square random variable with degrees of freedom and noncentrality parameter . The second of the approximations applicable for and is

where

The third of the approximations applicable for and is

where

12.3.11 Elliptical Distribution

Suppose a portfolio return, say, , is made up of asset returns, say, , , as , where are nonnegative weights summing to one, and . Kamdem (2005) derives various expressions for the VaR of by supposing that has an elliptically symmetric distribution.

If has the joint pdf , where is the mean vector, is the variance–covariance matrix, and is a continuous and integrable function over , then it is shown that

where is the root of

where

12.11

If follows a mixture of elliptical pdfs given by

where is the mean vector for the th elliptical pdf, is the variance–covariance matrix for the th elliptical pdf, and are nonnegative weights summing to one, then it is shown that the VaR of is the root of

where is defined as in (12.11).

12.3.12 Copula Method

Suppose a portfolio return, say, , is made up of two asset returns, and , as , where is the portfolio weight for asset 1 and is the portfolio weight for asset 2. Huang et al. (2009) consider computation of VaR for this situation by supposing that the joint cdf of is , where is a copula (Nelsen, 1999), is the marginal cdf of , and is the marginal pdf of . Then, the cdf of is

where is the copula pdf. So, can be computed by solving the equation

In general, this equation will have to be solved numerically or by simulation.

Franke et al. (2011) consider the more general case that the portfolio return is made up of asset returns, , ; that is,

for some nonnegative weights summing to one. Suppose as in the preceding text that the joint cdf of is , where is the marginal cdf of and is the marginal pdf of . Then, the cdf of is

where

and

So, can be computed by solving the equation

Again, this equation will have to be computed by numerical integration or simulation.

12.3.13 Gram–Charlier Approximation

Simonato (2011) suggests a number of approximations for computing (12.2). The first of these is based on Gram–Charlier expansion.

Let denote the skewness coefficient and the kurtosis coefficient of the standardized log returns. Simonato (2011) suggests the approximation

where is the inverse function of

where denotes the standard normal cdf and denotes the standard normal pdf.

12.3.14 Delta–Gamma Approximation

Let denote a vector of returns normally distributed with zero means and covariate matrix . Suppose the return of an associated portfolio takes the general form . It will be difficult to find the value of risk of for general . Some approximations are desirable. The delta–gamma approximation is a commonly used approximation (Feuerverger and Wong, 2000).

Suppose we can approximate for a vector and a matrix. Let denote the Cholesky decomposition. Let and denote the eigenvalues and eigenvectors of . Let denote the entries of , where . Then, the delta–gamma approximation is that

12.12

where are independent standard normal random variables. The value of risk can be obtained by inverting the distribution of the right-hand side of (12.12).

12.3.15 Cornish–Fisher Approximation

Another approximation suggested by Simonato (2011) is based on Cornish–Fisher expansion. With the notation as in Section 12.3.13, the approximation is

where is the inverse function of

where denotes the standard normal quantile function.

12.3.16 Johnson Family Method

A third approximation suggested by Simonato (2011) is based on the Johnson family of distributions due to Johnson (1949).

Let denote a standard normal random variable. A Johnson random variable can be expressed as

where

Here, , , , and are unknown parameters determined, for example, by the method of moments; see Hill et al. (1976).

With the notation as in the preceding text, the approximation is

where

where denotes the standard normal quantile function.

12.3.17 Tukey Method

Jiménez and Arunachalam (2011) present a method for approximating VaR based on Tukey's and family of distributions.

Let denote a standard normal random variable. A Tukey and random variable can be expressed as

for and . The family of lognormal distributions is contained as the particular case for . The family of Tukey's distribution is contained as the limiting case for .

With the notation as in Section 12.3.13, the approximation suggested by Jiménez and Arunachalam (2011) is

where and are location and scale parameters. For and , is a normal random variable with mean and standard deviation , so and . For and , is an exponential random variable with parameter , so and . For and , is a Student's random variable with ten degrees of freedom, so and .

12.3.18 Asymmetric Laplace Distribution

Trindade and Zhu (2007) consider the case that the log returns of are a random sample from the asymmetric Laplace distribution given by the pdf

for , , and . The maximum likelihood estimator of is derived as

where are the maximum likelihood estimators of . Trindade and Zhu (2007) show further that

in distribution as , where and .

12.3.19 Asymmetric Power Distribution

Komunjer (2007) introduces the asymmetric power distribution as a model for risk management. A random variable, say, , is said to have this distribution if its pdf is

12.13

for , where , and . Note that is a shape parameter and is a scale parameter. The cdf corresponding to (12.13) is shown to be (Lemma 1, Komunjer, 2007)

12.14

where . Inverting (12.14) as in Lemma 2 of Komunjer (2007), we can express as

12.15

where denotes the inverse function of . An estimator of can be obtained by replacing the parameters in (12.15) by their maximum likelihood estimators; see Proposition 2 in Komunjer (2007).

12.3.20 Weibull Distribution

Gebizlioglu et al. (2011) consider estimation of VaR based on the Weibull distribution. Suppose is a random sample from a Weibull distribution with the cdf specified by for , and . Then, the estimator for VaR is

Gebizlioglu et al. (2011) consider various methods for obtaining the estimators and . By the method of maximum likelihood, and are the simultaneous solutions of

and

where is the sample mean and is the sample variance. By Cohen and Whitten (1982)'s modified method of maximum likelihood, and are the simultaneous solutions of

and

where are the order statistics in ascending order. By Tiku (1967 and 1968) and Tiku and Akkaya (2004)'s modified method of maximum likelihood,

where

By the least squares method, and are those minimizing

with respect to and . By the weighted least squares method, and are those minimizing

with respect to and . By the percentile method, and are those minimizing

with respect to and .

12.3.21 ARCH Models

ARCH models are popular in finance. Suppose the log returns, say, , of follow the ARCH model specified by

where are independent and identical random variables with zero mean, unit variance, pdf , and cdf , and is an unknown parameter vector satisfying and , . If are the maximum likelihood estimators, then the residuals are

where

Taniai and Taniguchi (2008) show that VaR for this ARCH model can be approximated by

where

where , , , , , , and , .

12.3.22 GARCH Models

Suppose the financial returns, say, , satisfy the model

12.16

where are independent and identical standard normal random variables, is the return at time , denotes the lag operator satisfying , is the polynomial , is the polynomial , is the conditional variance, and are independent and identical residuals with zero means and unit variances. One popular specification for is

12.17

This corresponds to the GARCH model.

For the model given by (12.16) and (12.17), Chan (2009b) proposes the following algorithm for computing VaR:

1. Estimate the maximum likelihood estimates of the parameters in (12.16) and (12.17).
2. Using the parameter estimates, compute the standardized residuals .
3. Compute the first sample moments for .
4. Compute

   

   The parameters are determined from the sample moments of Step 3 in a way explained in Chan (2009a) and Rockinger and Jondeau (2002).

5. Compute as the root of the equation
   

12.3.23 GARCH Model with Heavy Tails

Chan et al. (2007) consider the case that financial returns, say, , come from a GARCH specified by

where is strictly stationary with , and are zero mean, unit variance, independent, and identical random variables independent of . Further, Chan et al. (2007) assume that have heavy tails, that is, their cdf, say, , satisfies

for all , where and . Chan et al. (2007) show that the VaR for this model given by

can be estimated by

where

where and as , , are the order statistics of , and and as . Chan et al. (2007) also establish asymptotic normality of .

12.3.24 ARMA–GARCH Model

Suppose the financial returns, say, , , satisfy the ARMA –GARCH model specified by

where are independent standard normal random variables. For this model, Hartz et al. (2006) show that the -step ahead forecast of VaR can be estimated by

where

The parameter estimators required can be obtained, for example, by the method of maximum likelihood.

12.3.25 Markov Switching ARCH Model

Suppose the financial returns, say, , , satisfy the Markov switching ARCH model specified by

where are standard normal random variables, is an unobservable random variable assumed to follow a first-order Markov process, and is a typical ARCH process. This model is due to Bollerslev (1986). An estimator of the VaR at time can be obtained by inverting the cdf of with its parameters replaced by their maximum likelihood estimators.

12.3.26 Fractionally Integrated GARCH Model

Suppose the financial returns, say, , , satisfy the fractionally integrated GARCH model specified by

where are random variables with zero means and unit variances. This model is due to Baillie et al. (1996). An estimator of the VaR at time can be obtained by inverting the cdf of with its parameters replaced by their maximum likelihood estimators. This of course depends on the distribution of . If, for example, are normally distributed, then , where may be the maximum likelihood estimator of .

12.3.27 RiskMetrics Model

Suppose are the log returns of and let denote the information up to time . The RiskMetrics model (RiskMetrics Group, 1996) is specified by

The VaR for this model can be computed by inverting

with the parameters, and , replaced by their maximum likelihood estimators.

12.3.28 Capital Asset Pricing Model

Let denote the return on asset , let denote the “risk-free rate,” and let denote the “return on the market portfolio.” With this notation, Fernandez (2006) considers the capital asset pricing model given by

for , where are independent random variables with Var and Var. It is easy to see that

Fernandez (2006) shows that the VaR of the portfolio of assets can be expressed as

12.18

where is a vector of portfolio weights, is the initial value of the portfolio, , and diag . An estimator of (12.18) can be obtained by replacing the parameters by their maximum likelihood estimators.

12.3.29 Dagum Distribution

The Dagum distribution is due to Dagum (1977 and 1980). It has the pdf and cdf specified by

and

respectively, for , , , and . Domma and Perri (2009) discuss an application of this distribution for VaR estimation. They show that

where are maximum likelihood estimators of based on being a random sample coming from the Dagum distribution. Domma and Perri (2009) show further that

in distribution as , where and

Here, is the expected information matrix of . An explicit expression for the matrix is given in the appendix of Domma and Perri (2009).

12.3.30 Location-Scale Distributions

Suppose is a random sample from a location-scale family with cdf and pdf . Then,

12.19

where . The point estimator for VaR is

where

and

Bae and Iscoe (2012) propose various confidence intervals for VaR. Based on and asymptotic normality, Bae and Iscoe (2012) propose the interval

12.20

where is the confidence level, is the kurtosis of , and is the skewness of . Based on Bahadur (1966)'s almost sure representation of the sample quantile of a sequence of independent random variables, Bae and Iscoe (2012) propose the interval

where is the th quantile and is its sample counterpart.

Sometimes the financial series of interest is strictly positive. In this case, if is a random sample from a log location-scale family with cdf , then (12.19) and (12.20) generalize to

and

respectively, as noted by Bae and Iscoe (2012).

12.3.31 Discrete Distributions

Göb (2011) considers VaR estimation for the three most common discrete distributions: Poisson, binomial, and negative binomial. Let

Then, the VaR for the Poisson distribution is

Letting

the VaR for the binomial distribution is

Letting

the VaR for the negative binomial distribution is

Göb (2011) derives various properties of these VaR measures in terms of their parameters. For the Poisson distribution, the following properties were derived:

1. For fixed , is increasing in with . There are values , , such that, for , on the interval and for . In particular, .
2. For fixed , , let . Then, for , for .

For the binomial distribution, the following properties were derived:

1. For fixed , is increasing in . There are values such that, for , on the interval and for . In particular, and .
2. For fixed , , let . Then, for , for .

For the negative binomial distribution, the following properties were derived:

1. For fixed , is decreasing in . There are values , , such that, for , on the interval and for . In particular, .
2. For fixed , , let . Then, for , for .

Empirical estimation of the three VaR measures can be based on asymptotic normality.

12.3.32 Quantile Regression Method

Quantile regressions have been used to estimate VaR; see Koenker and Bassett (1978), Koenker and Portnoy (1997), Chernozhukov and Umantsev (2001), and Engle and Manganelli (2004). The idea is to regress the VaR on some known covariates. Let at time denote the financial variable, let denote a vector of covariates at time , let denote a vector of regression coefficients, and let denote the corresponding VaR. Then, the quantile regression model can be rewritten as

12.21

In the linear case, (12.21) could take the form

The parameters in (12.21) can be estimated by least squares as in standard regression.

12.3.33 Brownian Motion Method

Cakir and Raei (2007) describe simulation schemes for computing VaR for single-asset and multiple-asset portfolios. Let denote the price at time , let denote a holding period divided into small intervals of equal length , let denote the change in over , let denote a standard normal shock, let denote the mean of returns over the holding period , and let denote the standard deviation of returns over the holding period . With these notations, Cakir and Raei (2007) suggest the model

12.22

Under this model, the VaR for single-asset portfolios can be computed as follows:

1. Starting with , simulate using (12.22).
2. Repeat Step (i) 10, 000 times.
3. Compute the empirical cdf over the holding period.
4. Compute as percentile of the empirical cdf.

The VaR for multiple-asset portfolios can be computed as follows:

1. Suppose the price at time for the th asset follows
   12.23

   for , where is the number of assets and the notation is the same as that for single-asset portfolios. The standard normal shocks, , need not be correlated.

2. Starting with , , simulate , using (12.23).
3. Compute the portfolio price for the holding period as the weighted sum of the individual asset prices.
4. Repeat Steps (ii) and (iii) 10,000 times.
5. Compute the empirical cdf of the portfolio price over the holding period.
6. Compute as percentile of the empirical cdf.

12.3.34 Bayesian Method

Pollard (2007) defines a Bayesian VaR. Let denote the financial variable of interest at time . Let denote the posterior pdf of given some parameters and “state” variables . Pollard (2007) defines the Bayesian VaR at time as

12.24

The “state” variables are assumed to follow a transition pdf .

Pollard (2007) also proposes several methods for estimating (12.24). One of them is the following:

1. Use Markov chain Monte Carlo to simulate samples , from the joint conditional posterior pdf of given .
2. For from 1 to , simulate from the conditional posterior pdf of given and .
3. For from 1 to , simulate from the conditional posterior pdf of given and .
4. Compute the empirical cdf
   12.25
5. Estimate VaR as .

12.3.35 Rachev et al.'s Method

Let denote a portfolio return made up of asset returns, , and the nonnegative weights summing to one. Suppose are independent random variables. Then, it can be shown that (Rachev et al., 2003) , where

and

Hence, the value of risk of can be estimated by the following algorithm due to Rachev et al. (2003):

• Estimate and (to obtain, say, and ) using possible data on the th asset return.
• Estimate and by
  

  and

  

  respectively.

• Estimate as the th quantile of .

12.4.1 Historical Method

Let denote the order statistics in ascending order corresponding to the original financial series . The historical method suggests to estimate VaR by

for .

12.4.2 Filtered Historical Method

Suppose the log returns, , follow the model, , discussed before, where is the volatility process and are independent and identical random variables with zero means. Let denote the order statistics of . The filtered historical method suggests to estimate VaR by

for , where denotes an estimator of at time . This method is due to Hull and White (1998) and Barone-Adesi et al. (1999).

12.4.3 Importance Sampling Method

Suppose is the empirical cdf of . As seen in Section 12.4.1, an estimator for VaR is . This estimator is asymptotically normal with variance equal to

This can be large if is closer to zero or one. There are several methods for variance reduction. One popular method is importance sampling. Suppose is another cdf and let and

Hong (2011) shows that under certain conditions can provide estimators for VaR with smaller variance.

12.4.4 Bootstrap Method

Suppose is the empirical cdf of . The bootstrap method can be described as follows:

1. Simulate independent sample from .
2. For each sample estimate , say, for , using the historical method.
3. Take the estimate of VaR as the mean or the median of for .

One can also construct confidence intervals for VaR based on the bootstrapped estimates , .

12.4.5 Kernel Method

Kernels are commonly used to estimate pdfs. Let denote a symmetric kernel, that is, a symmetric pdf. The kernel estimator of can be given by

12.26

where is a smoothing bandwidth and

A variable width version of (12.26) is

12.27

where is the distance of from its th nearest neighbor among the remaining data points and . The kernel estimator of VaR, say, , is then the root of the equation

12.28

for , where is given by (12.26) or (12.27). According to Sheather and Marron (1990), could also be estimated by

where is given by (12.26) or (12.27) and are the ascending order statistics of .

The estimator in (12.28) is due to Gourieroux et al. (2000). Its properties have been studied by many authors. For instance, Chen and Tang (2005) show under certain regularity conditions that

in distribution as , where

Here, denotes the indicator function.

12.4.6 Chang et al.'s Estimators

Chang et al. (2003) propose several nonparametric estimators for the VaR of log returns, say, with pdf . The first of these is , where and , where denotes the greatest integer less than or equal to . This estimator is shown to have the asymptotic distribution

in distribution as . It is sometimes referred to as the historical simulation estimator. The second of the proposed estimators is

This estimator is shown to have the asymptotic distribution

in distribution as . The third of the proposed estimators is

where

This estimator is shown to have the asymptotic distribution

in distribution as .

12.4.7 Jadhav and Ramanathan's Method

Jadhav and Ramanathan (2009) provide a collection of nonparametric estimators for . Let denote the order statistics in ascending order corresponding to . For given , define , , , , , and . The collection provided is

where

where denote the incomplete beta function ratio defined by

The last of the estimators in the collection is due to Kaigh and Lachenbruch (1982). The second last is due to Harrell and Davis (1982).

12.4.8 Jeong and Kang's Method

Suppose the log returns, , follow the model, , discussed before. Let denote the corresponding VaR. Jeong and Kang (2009) propose a fully nonparametric estimator for the defined by

where is the -field generated by . Let

and

for some kernel function with bandwidth . With this notation, Jeong and Kang (2009) propose the estimator

where

and

Here, can be determined using a recursive algorithm presented in Section 12.2.1 of Jeong and Kang (2009).

12.5.1 Extreme Value Theory Method

Let denote the maximum of financial returns. Extreme value theory says that under suitable conditions there exist norming constants and such that

in distribution as . The parameter is known as the extreme value index. It controls the tail behavior of the extremes.

There are several estimators proposed for . One of the earliest estimators due to Hill (1975) is

12.29

where are the order statistics in descending order. Another earliest estimator due to Pickands (1975) is

12.30

The tails of for most situations in finance take the Pareto form, that is,

12.31

for some constant . Embrechts et al. (1997, p. 334) propose estimating by .

Combining (12.29) and (12.31), Odening and Hinrichs (2003) propose estimating VaR by

12.32

This estimator is actually due to Weissman (1978).

An alternative approach is to suppose that the maximum of financial returns follows the generalized extreme value cdf (Fisher and Tippett, 1928) given by

12.33

for , , , and . In this case, the VaR can be estimated by

where are the maximum likelihood estimators of . Prescott and Walden (1990) provide details of maximum likelihood estimation for the generalized extreme value distribution.

The Gumbel distribution is the particular case of (12.33) for . It has the cdf specified by

for and . If the maximum of financial returns follows this cdf, then the VaR can be estimated by

where are the maximum likelihood estimators of .

For more on extreme value theory, estimation of the tail index, and applications, we refer the readers to Longin (1996, 2000), Beirlant et al. (2017), Fraga Alves and Neves (2017), and Gomes et al. (2015).

12.5.2 Generalized Pareto Distribution

The Pareto distribution is a popular model in finance. Suppose the log return, say, , of comes from the generalized Pareto distribution with cdf specified by

for , , and , where is some threshold and is the number of observed exceedances above .

For this model, several estimators are available for the VaR. Let denote the order statistics in ascending order. The first estimator due to Pickands (1975) is

where

for . The second estimator due to Dekkers et al. (1989) is

where

Suppose now that the returns are from the alternative generalized Pareto distribution with cdf specified by

for . Then, the VaR is

12.34

If and are the maximum likelihood estimators of and , respectively, then the maximum likelihood estimator of VaR is

There are several methods for constructing confidence intervals for (12.34). One popular method is the bias-corrected method due to Efron and Tibshirani (1993). This method based on bootstrapping can be described as follows:

1. Given a random sample , calculate the maximum likelihood estimate and , the maximum likelihood estimate with the th data point, , removed.
2. Simulate from the generalized Pareto distribution with parameters .
3. Compute the maximum likelihood estimate, say, , for the sample simulated in Step 2.
4. Repeat Steps 2 and 3, times.
5. Compute

   

   and

   

   where

   

   and

   

   where is the mean of .

6. Compute the bias-corrected confidence interval for VaR as
   

   where is the percentile of .

Note that and are the bootstrap replicates of and VaR, respectively.

12.5.3 Matthys et al.'s Method

Several improvements have been proposed on (12.32). The one due to Matthys et al. (2004) takes account of censoring. Suppose only of the are actually observed; the remaining are considered to be censored or missing. In this case, Matthys et al. (2004) show that VaR can be estimated by

where

Here, is a tuning parameter and takes values in the unit interval. Among other properties, Matthys et al. (2004) establish asymptotic normality of .

12.5.4 Araújo Santos et al.'s Method

The improvement of (12.32) due to Araújo Santos et al. (2006) takes the expression

where and

12.5.5 Gomes and Pestana's Method

The improvement of (12.32) due to Gomes and Pestana (2007) takes the expression

where

Here, is a tuning parameter. Under suitable conditions, Gomes and Pestana (2007) show further that

in distribution as .

12.5.6 Beirlant et al.'s Method

The improvement of (12.32) due to Beirlant et al. (2008) takes the expression

where is as given by Section 12.5.5, and

This estimator is shown to be consistent.

12.5.7 Caeiro and Gomes's Method

Caeiro and Gomes (2008 and 2009) propose several improvements on (12.32). The first of these takes the expression

where and are as given by Section 12.5.5, and is as given by Section 12.5.6. The second of these takes the expression

where and are as given by Section 12.5.5, and is as given by Section 12.5.6. The third of these takes the expression

where and are as given by Section 12.5.5, is as given by Section 12.5.6, and denotes the incomplete beta function defined by

The fourth of these takes the expression

where , , and are as given in Section 12.5.5. All of these estimators are shown to be consistent and asymptotically normal.

12.5.8 Figueiredo et al.'s Method

The latest improvement of (12.32) is due to Figueiredo et al. (2012). It takes the expression

where and

with as defined in Section 12.5.5 and as defined in Section 12.5.4.

12.5.9 Li et al.'s Method

Let be such that and as . Li et al. (2010) derive estimators for for large . They give the estimator

where

and

where and are the simultaneous solutions of the equations

and

where

and

Li et al. (2010) show under suitable conditions that

in distribution as .

12.5.10 Gomes et al.'s Method

Gomes et al. (2011) propose a bootstrap-based method for computing VaR. The method can be described as follows:

1. For an observed sample, , compute as in Section 12.5.5 for and .
2. Compute the median of , say, , for . Also compute

   

   for . Choose the tuning parameter, , as zero if and as one otherwise.

3. Compute and using the formulas in Section 12.5.5 and the chosen tuning parameter.
4. Compute , in Section 12.5.5 with the estimates and in Step 3.
5. Set and .
6. Generate bootstrap samples and from the empirical cdf of .
7. Compute for the bootstrap samples in Step 6. Let , denote the estimates for the bootstrap samples of size . Let , denote the estimates for the bootstrap samples of size .
8. Compute
   

   and

   

   for and .

9. Compute
   

   for .

10. Compute
    
11. Compute with the estimates and in Step 3.
12. Compute
    
13. Finally, estimate as
    

12.5.11 Wang's Method

Wang (2010) combined the historical method in Section 12.4.1 with the generalized Pareto model in Section 12.5.2 to suggest the following estimator for VaR:

where and are the maximum likelihood estimators of and , respectively, and is an appropriately chosen threshold.

12.5.12 -Estimation Method

Iqbal and Mukherjee (2012) provide an -estimator for VaR. They consider a GARCH (1, 1) model for returns specified by

where

and are independent and identical random variables symmetric about zero. The unknown parameters are , and they belong to the parameter space, the set of all with , and . The -estimator, say, , is obtained by solving the equation

where

and

where for some skew-symmetric function and denotes the derivative of . Iqbal and Mukherjee (2012) propose that can be estimated by multiplied by the th order statistic of .

12.5.13 Generalized Champernowne Distribution

Generalized Champernowne distribution was introduced by Buch-Larsen et al. (2005) as a model for insurance claims. A random variable, say, , is said to have this distribution if its cdf is

12.35

for , where , , and is the median. Charpentier and Oulidi (2010) provide estimators of based on beta kernel quantile estimators. They suggest the following algorithm for estimating :

• Suppose is a random sample from (12.35).
• Let denote the estimators of the parameters ; if the method of maximum likelihood is used, then the estimators can be obtained by maximizing the log likelihood given by

  

• Transform , where is given by (12.35) with replaced by .
• Estimate the cdf of as
  

  where is given by either

  

  or

  

  where .

• Solve for by using some Newton algorithm.
• Estimate by .