Estimation of Extreme Value at Risks Using

CAViaR Models

Midori Nagai

Graduate School of Economics, Hitotsubashi University

January 2016

Abstract

This paper presents a estimation method of extreme VaR by integrating aCAViaR model and the extreme value theory. We model the dynamics of VaRby a CAViaR model and estimate the parameters by applying the extreme valuetheory. We compared a performance of our method with estimating the pa-rameter by the conventional quantile regression. Our method outperforms thequantile regression method in the simulation in that it estimates the VaR moreaccurately, while in the empirical study, our method is inferior to the quantileregression method in the sense of a violation rate.

Contents

1 Introduction 2

2 Methods 4

2.1 CAViaR model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Extreme value theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.1 Basic theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.2 Quantile estimation . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Estimation of extreme VaR . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Simulation Study 13

4 Real Data Analysis 20

5 Conclusion 22

1 Introduction

Value-at-Risk (VaR) has been developed in the early 1990s in response to the market

crises occured around 1990, such as the stock market crash on Wall Steet in 1987 and

the break down of the European Monetary System in 1992. Nowadays VaR is a widely

used measure of market risk in risk management. VaR is a measure of the maximum

potential loss of a certain portfolio within a given time period for a given confidence

level. More specifically, conditional on the current information Ωt, VaR is defined as

the θ-quantile of the conditional return distribution, where 1 − θ ∈ (0, 1) represents

the confidence level associated with VaR. Let yt be the return of a portfolio. We are

then interested in estimating the VaR of yt, V aRt, defined by

P [ yt+1 < V aRt+1|Ωt] = θ.

Since VaR is conceptually simple, it has become a standard measure of market risk in

risk management.

Despite its conceptual simplicity, its estimation is a challenging statistical prob-

lem. So far various approachs have been developed to forecast VaR, but none of the

methods gives satisfactory solutions because typically the distribution of portfolio re-

turns changes over time. The most widely used approach is using fully parametric time

series models such as ARCH or GARCH models to capture the dynamics of volatili-

ties. The main weakness of this approach is necessity of assumptions on the shape of

the return distributions. The original GARCH and ARCH models assumed normal-

ity, which was soon realized to be inadequate. Its replacement with more fat-tailed

and possibly skewed distributions, such as Student-t distributions, is widely thought

to be effective. Nevertheless, there is still no answer which distribution to be assumed.

Regarding nomparametric methods, the most popupar approach is a historical simula-

tion. Using this method, we may obtain an estimate of VaR as an empirical quantile

of historical returns from a window of the most recent periods, implying simplicity in

computation. Despite the tractability in implementation, the VaR estimation could

be unstable especially in the extreme quantiles. On the other hand, other approaches

2

such as Conditional Autoregressive Value at Risk (CAViaR) models and applications of

extreme value theory, is known to have an advantage even in such extremal estimation.

The CAViaR model, introduced by Engle and Manganelli (2004), is based on a

quantile regression. It is a dynamic quantile model in that conditional quantiles follow

autoregressive processes. The autoregressive structure of the quantile is quite natural

since the series of financial returns empirically tend to exhibit volatility clustering

and the quantile of the distribution is tightly linked to the variance. The advantage

of this model is that no explicit distributional assumptions need to be made due to

modeling the quantile directly. Empirical evidence has shown that a CAViaR model is

competitive with other VaR models (Bao et al. 2006, Yu et al. 2010).

We can use quantile regression introduced by Koenker and Bassett (1978) to esti-

mate the parameters of a CAViaR model. However, the estimates of high quantiles by

a quantile regression tend to be unstable due to data sparsity in the tail areas. The

quantile level θ of VaR is often set as 0.01 or sometimes lower. In such cases, it is dif-

ficult to forecast VaR using a CAViaR model with parameters estimated by a quantile

regression without making distributional assumptions.

Extreme value theory (EVT) provides a solid framework to analyze rare events

and forcuses only on the tail of the distribution. EVT has been applied in many fields

including VaR estimation. A simple and widely used EVT approach for VaR estimation

is to fit generalized pareto distribution to the returns that exceed a particular threshold.

See, for example, Danielsson and DeVries (1997), Pownall and Koedijk (1999). On

the other hand, McNeil and Frey (2000) applied EVT to the residuals of Gaussian

AR(1)-GARCH(1,1) model, called conditional-EVT method. Bystrom (2004) extended

McNeil and Frey’s approach to block maxima method. For more details on applications

of EVT to VaR estimation, see McNeil et al. (2015).

In this paper, we present a method to estimate extreme VaR using both a CAViaR

model and EVT. We first estimate “not extreme” quantiles from CAViaR models by a

quantile regression, then extrapolate the extreme quantile from the estimated quantiles

via EVT. The extrapolation method is based on Wang et al. (2012) which gives a

3

method on conditional extreme quantile estimation. Although we can overcome the

unstableness of a quantile regression due to the application of EVT, we still do not need

to make extra assumptions on a return distribution which may lead to a misspecification

by integrating CAViaR and EVT.

The rest of the article is organized as follows. In Section 2, we briefly review

the CAViaR model and the extreme value theory. We also introduce the estimation

methods. In Section 3, we conduct a simulation study to assess the finite sample

performance of the proposed method. Section 4 presents an empirical application to

real data, and Section 5 conclude the paper.

2 Methods

2.1 CAViaR model

Engle and Manganelli (2004) introduced CAViaR to model conditional VaR. This ap-

proach models the quantile directly, instead of modeling the whole distribution. Em-

pirically it is well known that volatilities of stock market returns are usually autocor-

related. Since the quantile is tightly linked to the variance of the distribution, it is

natural to consider the VaR is also autocorrelated.

Engle and Manganelli (2004) proposed several CAViaR specifications. In this paper

we adopt the following three models. Let QYt(θ|Ωt−1) denote the θ-quantile of Yt

conditional on current information Ωt−1, i.e., QYt(θ|Ωt−1) := infy : FYt(y|Ωt−1) ≤ θ,where FYt(·|Ωt−1) is the conditional distribution function of the return Yt.

[Symmetric Absolute Value]

QYt+1(θ|Ωt) = β1 + β2QYt(θ|Ωt−1) + β3|yt|

[Asymmetric Slope]

QYt+1(θ|Ωt) = β1 + β2QYt(θ|Ωt−1) + β3max(yt, 0) + β4max(−yt, 0)

4

[Indirect GARCH]

QYt+1(θ|Ωt) = (β1 + β2QYt(θ|Ωt−1)2 + β3y

2t )

1/2

In the Symmetic Absolute Value model, the quantiles respond symmetrically to past

returns. The autoregressive parameter β2, is to be |β2| < 1, so long as the process is

mean-reverting. On the other hand, the symmetric assumption is relaxed in the Asym-

metric Slope model in that the quantile respond differently to positive and negative

returns.

The above CAViaR models are correctly specified if the data were generated from

following models.

yt = σtεt, εt ∼ i.i.d.(0, 1),

σt =

β∗1 + β∗

2σt−1 + β∗3 |yt−1|, · · · Symmtric Absolute Value

β∗1 + β∗

2σt−1 + β∗3 max(yt, 0) + β∗

4 max(−yt, 0), · · ·Asmmetric Slope

(β∗1 + β∗

2σ2t−1 + β∗

3y2t−1)

1/2. · · · Indirect GARCH

The parameters of CAViaR models can be estimated by a quantile regression. Con-

sider a quantile regression model

yt = ft(β) + εtθ

with Qθ(εtθ|Ωt) = 0, where Qε(·|Ωt) is the quantile function of the i.i.d.errors ε. Then

we can define the quantile regression estimator of β as;

β = argminβ

T∑i=1

ρθ(yt − ft(β))

where ρθ(u) := uθ − I(u < 0) is the so-called check function. For details about

quantile regression; see Koenker (2005).

When θ is a not extreme quantile level, we can appropreately estimate the quantile

by just minimizing the objective function. However, if the quantile level θ is extreme,

θ ≈0 or 1, quantile regression estimators tend to be unstable due to data sparseness in

the tail area, especially for heavy-tailed distributions. This motivate us to adopt EVT

to the estimation.

5

2.2 Extreme value theory

Extreme Value Theory (EVT) deals with extreme and rare events. In many cases, it

is difficult to get some inferences on such events, since the events could be beyond the

range of available data. EVT can be a strong tool in such situations. The pioneering

works on EVT are Fischer and Tippett (1928), Gnedenko (1943), and then they are

extended by Gumbel (1958). The theory has been applied in many fields, such as

hydorology, wind engineering, including finance and insurance from the viewpoint of

risk analysis. For the details of the theory; see Embrechts et al. (1997), de Haan and

Ferreira (2006).

2.2.1 Basic theory

Let X1, X2, . . . , Xn be i.i.d.random variables with a distribution function F . EVT is

concerned with the limit behavior of the sample maxima, Mn := max(X1, X2, . . . Xn),

while the central limit theorem is concerned with that of the partial sum, X1 +X2 +

· · ·+Xn. The distribution function of the sample maxima Mn is,

P (Mn ≤ x) = P (X1 < x,X2 < x, . . . , Xn ≤ x) = F n(x).

However, it has no practical value since F n(x) converges to zero for all x < x∗, and

converges to 1 for x ≥ x∗ as n → ∞, where x∗ is the right endpoint of F , i.e., x∗ :=

sup x : F (x) < 1. Therefore, in order to obtain a nondegenerate limit distribution,

we need to normalize the sample maxima Mn.

Under weak assumptions, it is known that there exists a sequence of constants

an > 0 and bn ∈ R(n = 1, 2, . . . ), such that

limn→∞

P

(Mn − bn

an≤ x

)= lim

n→∞F n(anx+ bn) = G(x), (1)

for some non-degenerate distribution G. If G exsists, it is known that G must be the

so-called generalized extreme value distribution:

Gγ(x) =

exp(−(1 + γx)−1/γ), γ = 0

exp(− exp(−x)), γ = 0

6

where 1+γx > 0. The shape parameter γ is called extreme value index which determins

the tail behavior. If the relation (1) holds for some G = Gγ, we say that the distribution

function F is in the domain of attraction of Gγ, we write F ∈ D(Gγ).

The class of extreme value distribution can be devided into three types of subclasses

according to the value of γ.

(a) For γ > 0, Gγ can be reparametrized as

Φα(x) :=

0, x ≤ 0,

exp(−x−α), x > 0,

where α = 1/γ. This class is often called the Frechet class of distributions. The

right end point of this distribution is infinity and the distribution has a heavy

tail. The right tail of the distribution decreases like a power law, and moments of

order grater than or equal to α do not exist. The Student-t distribution and the

Cauchy distribution have this type of extreme value distributions. In this paper,

we are interested in this type of distributions.

(b) For γ = 0,

G0(x) = exp(− exp(−x)).

This is often called Gumbel distribution. Though the right endpoint of the dis-

tribution is infinity, the distribution has a rather light tail in that it declines

exponentially and all moments exist. Normal and lognormal distributions have

this type of extreme value distributions.

(c) For γ < 0, with α = −1/γ, we can write Gγ as

Ψα(x) :=

exp(−(−x)α), x < 0,

1, x ≥ 0.

This class is called Weibull class of distributions. Since the right end point of

the distribution is −α < ∞, it has a short tail. The uniform distribution, for

example, has this type of extreme value distribution.

7

There is another limit distribution on extreme values. The limit relationship (1)

with G(x) = Gγ(x) = exp(−(1 + γx)−1/γ) is equivalent to,

limt→x∗

P

(X − t

f(t)

∣∣∣∣ X > t

)= lim

t→x∗

1− F (t+ xf(t))

1− F (t)= (1 + γx)−1/γ (2)

where f is some positive nondecreasing function and x∗ = sup x : F (x) < 1. This

means that the conditional distribution of (X − t)/f(t) given X > t, often referred to

as the excess distribution over threshold t, has the limit distribution,

Hγ(x) :=

1− (1 + γx)−1/γ γ = 0,

1− exp(−x) γ = 0,

where x ≥ 0 when γ ≥ 0, and 0 ≤ x ≤ −γ when γ < 0, as t → x∗. This class of

distribution functions is called the generalized Pareto distribution. The estimation of

extreme value distributions is often based on this type of limit formulas.

We have the same types of limiting distributions for the maxima of strictly station-

ary time series. Let (X1, X2, . . . , Xn) be a strictly stationary sequence with marginal

distributions F , (X1, X2, . . . , Xn) denote i.i.d. process with the same distribution func-

tion F , and Mn := max(X1, X2, . . . , Xn), Mn := max(X1, X2, . . . , Xn), denote maxima

of the stationary series and the i.i.d. series. Assume the i.i.d. process Xi is in the

maximum domain of attraction of Gγ, F ∈ D(Gγ), then for many processes Xi, there

exists θ ∈ (0, 1] such that

limn→∞

P

(Mn − bn

an≤ x

)= Gθ

γ(x). (3)

Note that the normalizing constants and the extreme value index are still the same

as the independent case. The condition for strictly stationaly process Xi to have the

extreme value distribution is known as the D(un) condition, obtained by Leadbet-

ter(1974). The condition D(un) is relatively weak, so that many processes including

linear processes, ARCH and GARCH processes, satisfy the condition. For the details

of the condition, see Leadbetter et al. (1983), Embrechts et al. (1997). The value

θ is called the extremal index. The extremal index equals to 1 in the case when the

series is i.i.d. or weakly dependent. In general, serial dependences leads to a clustering

8

of large values. Therefore, the maximum of a stationary time series is stochastically

smaller than that of an i.i.d. sequence with the same marginal distribution function.

The reciprocal of the extremal index 1/θ can be interpreted as the mean cluster size.

Together with (1) and (2), we have,

P

(Mnθ − bn

an≤ x

)∼ P

(Mn − bn

an≤ x

)

as n → ∞. This means that the distribution of the maximum of n observations from

strictly stationaly time series with the extremal index θ can be approximated by the

distribution of the maximum of nθ < n observations from i.i.d. series with the same

marginal distributions. Thus the convergence speed of the maximum from time series

to extreme value distribution becomes slower than that of the i.i.d. series.

2.2.2 Quantile estimation

Let xp := F−1(1− p) be the extreme quantile we want to estimate. The relation (2) is

equivalent to (Theorem 1.1.6 of de Haan and Ferreira (2006)),

limt→∞

U(tx)− U(t)

a(t)=

xγ − 1

γ

where U(t) = F−1(1 − 1/t), a(t) is a positive function with f(t) = a(1/(1 − F (t))).

This implies

U(tx) ≈ U(t) + a(t)xγ − 1

γ.

Since xp = U(1/p), by letting t = n/k, x = k/(np) with k → ∞, k/n → 0 as n → ∞,

we have

xp ≈ U(nk

)+ a

(nk

) ( knp

)γ− 1

γ.

Therefore, by replacing U(n/k), a(n/k), and γ with their suitable estimators, we have

an estimator of xp.

9

When the extreme value index γ is positive, a simpler version is available. The basic

assumption F ∈ D(Gγ>0) is equivalent to (Corollary 1.2.10 of de Haan and Ferreira

(2006)),

limt→∞

U(tx)

U(t)= xγ , t > 0.

Then similarly, we have

xpn ≈ U(nk

)( k

npn

)γ

. (4)

We can estimate U(n/k) by the empirical quantile Xn−k,n, where X1,n ≤ · · · ,≤ Xn,n

are the order statistics. Hence, the estimator of the quantile becomes

xpn := Xn−k,n

(k

npn

)γ

, (5)

where γ denotes a suitable estimator of the extreme value index γ depending only on

the k largest order statistics.

Several methods have been proposed to estimate the extreme value index γ. For

the case γ > 0, a simple and widely used estimator is the Hill estimator developed by

Hill (1975),

γ :=1

k

k∑i=1

logXn−i+1,n

Xn−k,n

. (6)

This estimator can be derived from several different methods, showing that the Hill

estimator is quite natural. One approach is based on the reformulation of the condition

F ∈ D(Gγ). We can rewrite the condition as,

limt→∞

1− F (tx)

1− F (t)= x−1/γ, x > 0.

Using partial integration, we have

limt→∞

∫∞t(log u− log t)dF (u)

1− F (t)= γ.

Then replacing F by the enpirical distribution function Fn, and t by the intermediate

order statistic Xn−k,n, we have the estimator (6). The replacement of t is motivated by

10

the fact that Xk,n → ∞ a.s., provided k → ∞ and k/n → 0 as n → ∞; see Proposition

4.1.14 of Embrechts et al. (1997).

The asymptotic properties of the Hill estimator have been investigated in various

models. The consistency was proved in quite general time series models, including

infinite order of moving averages and ARCH and GARCH processes, established by

Hsing (1991), Resnik and Startica (1995,1998). Asymptotic normality has also been

proved in many situations; see Resnik and Starica (1997) and Hill (2010).

The Hill estimator can be used only when the extreme value index is positive,

corresponding to the heavy-tailed situation. In the case γ ∈ R, we can use other

estimators such as the moment estimator, the maximum likelihood estimator, and the

Pickands estimator. However, since we focus on heavy tailed distributions in this paper,

we use the Hill estimator alone for the estimation.

To justify the quantile estimation (5), n/k should be large enough to justify the

approximation (4). On the other hand, k itself should also be sufficiently large so that

the intermediate order statistics Xn−k,n can estimate the intermediate quantile U(n/k)

well enough. Thus the following conditions are usually imposed in the extreme value

literature.

k → ∞, n/k → ∞, as n → ∞

In practice, the choise of k is a quite difficult problem. There is a trade-off between

bias and variance of γ. A smaller value of k leads to larger variance due to the lack

of sample, while larger k results in a severe bias because of including “not extreme

data” in the estimation. Theoretlically, we can choose the optimal k by minimizing

the mean squared error of the estimator. However, the optimal k depends on unknown

parameters which are difficult to be estimated in practice. Therefore, a commonly used

approach to choose k is to plot the estimator of γ versus k, and then choose the value

of k corresponding to the first stable part of the plot, which is often called as Hill plot.

11

2.3 Estimation of extreme VaR

To estimate the conditional extreme VaR by CAViaR, we adopt the estimation method

proposed by Wang et al. (2012). They integrated a quantile regression and EVT to

estimate the conditional extreme quantiles. The method estimates extreme conditional

quantiles by the following way. First, we use a quantile regression to estimate the

conditional quantiles for intermediate quantile levels. Then the estimated intermediate

conditional quantiles are used to estimate the extreme conditional quantiles. We should

note that although Wang et al.(2012) assumed the quantile regression model to be

linear, we use the method to nonlinear models.

We assume FYt(·|Ωt−1) is in the maximum domain of attraction of an extreme value

distribution Gγ(·), denoted by FYt(·|Ωt−1) ∈ D(Gγ). In this paper, the extreme value

index γ is restricted to be γ > 0, corresponding to heavy-tailed distributions, since

market returns generally show heavy-tailed structures.

Let QYt(θ|Ωt−1) = inf y : FYt(yt|Ωt−1) ≤ θ,denote the θ-th conditional quantile

of Yt given Ωt−1. Our main objective is to estimate the extreme conditional quantiles

QYt(θn|Ωt−1), with θn → 1 as n → ∞. We consider the following quantile regression

model:

QYt(θ|Ωt−1) = f(Ωt−1; βθ)

where Ωt is a vecter of information available at time t, and the parameter vector β

depends on θ.

First we estimate a sequence of intermediate conditional quantiles. Define a se-

quence of quantile levels 0 < θn−k < θn−k+1 < · · · < θm < 1, where m = n − [nη]

for some 0 < η < 1 and [u] denotes the integer part of u, and θj = j/(n + 1). We

assume k = kn → ∞ and k/n → 0 as n → ∞, and nη < k. We estimate these levels of

quantiles by a quantile regression;

βθj = argminβ

n∑i=1

ρθj(yi − f(Ωt−1; βθj))

where ρθ(u) = uθ − I(u < 0).

12

Forj = n − k, . . . ,m, define QYt(θj|Ωt−1) = f(Ωt−1; βθj). These can be roughly

regarded as the upper order statistics of a sample from FYt(·|Ωt−1). Since we assumed

FYt(·|Ωt−1) ∈ D(Gγ) with the extreme value index γ > 0, we can estimate γ by the Hill

estimator based on the pseudo order statistics QYt(θn−k|Ωt−1), · · · , QYt(θn−[nη ]|Ωt−1),

γ =1

k − [nη]

k∑j=1

logQYt(θn−[nη ]−j+1|Ωt−1)

QYt(θn−k|Ωt−1).

Consequently, QYt(θn|Ωt−1) can be estimated by

QYt(θn|Ωt−1) =

(1− θn−k

1− θn

)γ

QYt(θn−k|Ωt−1).

3 Simulation Study

In this section, we conduct a simulation study to investigate the comparative perfor-

mance of our EVT method and the conventional quantile regression method for esti-

mating CAViaR models. We consider the three CAViaR models mentioned in previous

section. The data are generated from

yt = σtεt, εt ∼ i.i.d.t∗(3), t = 1, . . . , T,

where t∗(3) is the standardized Student-t distribution with 3 degrees of freedom. Note

that the extreme value index γ of t(3) is 1/3 > 0 .

The standard error processes σt are specified as follows according to each of the

CAViaR models.

[Case 1] Symmetric Absolute Value model

σt = 0.1 + 0.8σt−1 + 0.03|yt−1|

[Case 2] Asymmetric Slope model

σt = 0.1 + 0.8σt−1 + 0.01max(yt−1, 0) + 0.03max(−yt−1, 0)

13

[Case 3] Indirect GARCH model

σ2t = 0.1 + 0.8σ2

t−1 + 0.1y2t−1

We consider three sample sizes T = 500, 1000 and 2000. We generate T observations

from these models, then estimate QYT+1(θ|ΩT ) for θ = 0.01, 0.005 and 0.001. The true

one-step-ahead quantile isQYT+1(θ|ΩT ) = σTT

∗−13 (θ), with T ∗−1

3 (θ) = T−13 (θ)/

√3 where

T−13 is the inverse Student-t cdf. Therefore, the true CAViaR models become,

[Case 1] Symmetric Absolute Value model

QyT+1(θ|ΩT ) = 0.1T ∗−1

3 (θ) + 0.8QyT (θ|ΩT−1) + 0.03T ∗−13 (θ)|yT |

[Case 2] Asymmetric Slope model

QyT+1(θ|ΩT ) = 0.1T ∗−1

3 (θ) + 0.8QyT (θ|ΩT−1)+

0.03T ∗−13 (θ)max(yT , 0) + 0.03T ∗−1

3 (θ)max(−yT , 0)

[Case 3] Indirect GARCH model

QyT+1(θ|ΩT ) = (0.1T ∗−1

3 (θ) + 0.8QYT(θ|ΩT−1)

2 + 0.1T ∗−13 (θ)y2T )

1/2

We calculate the true quantiles from these formulas. The number of replications is

500 for each scenario.

To optimize the objective function in the quantile regression estimation, we follow

Engle and Manganelli (2004). First, we generate 1000 parameter vectors from the

uniform distribution, U(0, 1), if the true parameter is positive, while U(−1, 0) when

the true parameter is negative. Specifically, β2 in all the models is generated from

U(0, 1) and the others from U(−1, 0). Then we compute the objective function for

each of these vectors and select the 15 vectors that produced the lowest value of the

objective function. The vectors are used as initial values for the optimization. For

each of these initial values, first we ran the Nelder-Mead simplex algorithm. Then we

let the optimal parameters as the initial parameters for the quasi-Newton algorithm

to get the new optimal parameters and run the simplex algorithm again with the new

14

initial values. We repeat this procedure until the convergence criterion is satisfied.

The convergence criterion is set to 10−3 for the parameters values. Finally, we select

the vector that produced the lowest value of the objective function among the 15

optimization procedures.

To estimate the quantile by the EVT method, we set the tuning parameters η = 0.2,

and k = [5T 1/3].

Hill estimator is defiened only when the data are positive. However, though the

true quantiles cannot be negative, the estimated quantiles sometimes could be negative.

Therefore we apply some shift to the data to create Hill estimators. The size of the shift

is set to the minimum of the data. More specifically, let x1, . . . , xn be the observed

data andMn denotes their maximum. We transform the data to x1+Mn, . . . , xn+Mn,

then get the quantile estimators of the transformed data, denoted by Q∗θn−k

, · · · , Q∗θn,

which are used to calculate the Hill estimator γ. Then we estimate the shifted quantile

by

Q∗n =

(1− θn−k

1− θn

)γ

Q∗n−k,

and the quantile estimate becomes

Q(θn|x) = Q∗n −Mn.

We should note that one of the main weaknesses of the Hill estimator is that it is not

shift invariant. Althogh a shift to the data does not affect the value of the extreme

value index γ, the convergence speed of the Hill estimator may change due to the shift.

To compare the performance, we focus on the mean absolute error (MAE) and

the root mean squared error (RMSE). The MAE is defined by 1n

∑ni=1 |q0i − qi|, and

RMSE is√

1n

∑ni=1(q

0i − qi)2, where q0i denotes the true quantile, qi is the estimated

quantile, and i denotes each replication. Meanwhile, the true conditional quantile

varies for each simulation, since it depends on time varying standard deviations of the

variables. Therefore, we also compare normalized versions of MAE and RMSE, defined

by 1n

∑ni=1 |(q0i − qi)/q

0i | and

√1n

∑ni=1((q

0i − qi)/q0i )

2.

15

Table 1: MAE and RMSE of QYT+1(θ|ΩT )

.

Symmetric Absolute Value Asymmetric Slope Indirect GARCHMAE RMSE MAE RMSE MAE RMSE

T = 2000QR

0.01 1.294 1.482 1.485 1.818 0.650 1.0890.005 1.751 2.063 1.979 2.312 0.889 1.5200.001 3.976 4.875 4.045 4.547 2.384 4.115

EVT0.01 0.375 0.496 0.961 1.148 0.693 1.0330.005 0.583 0.732 1.081 1.398 0.945 1.3830.001 1.603 1.809 1.287 2.085 2.186 2.946

T = 1000QR

0.01 1.367 1.658 1.396 1.572 0.683 1.0360.005 1.875 2.280 1.983 2.183 1.020 1.7100.001 3.859 4.823 4.034 4.358 3.213 5.082

EVT0.01 0.414 0.525 0.847 1.125 0.700 1.0180.005 0.647 0.781 0.923 1.319 1.000 1.4350.001 1.713 1.906 1.221 1.832 2.356 3.076

T = 500QR

0.01 1.544 2.133 1.375 1.580 0.913 1.4930.005 2.070 2.647 2.032 2.262 1.352 2.1690.001 3.659 3.952 4.022 4.292 3.411 5.277

EVT0.01 0.538 0.688 0.701 0.952 0.907 1.7010.005 0.806 0.985 0.805 1.143 1.313 2.3160.001 1.927 2.195 1.333 1.845 2.974 4.421

16

Table 2: Normalized versions of MAE and RMSE of QYT+1(θ|ΩT ).

Symmetric Absolute Value Asymmetric Slope Indirect GARCHMAE RMSE MAE RMSE MAE RMSE

T = 2000QR

0.01 0.596 0.672 0.628 0.699 0.169 0.2240.005 0.629 0.740 0.649 0.711 0.179 0.2400.001 0.809 0.968 0.763 0.800 0.284 0.428

EVT0.01 0.161 0.199 0.419 0.484 0.184 0.2500.005 0.198 0.234 0.363 0.450 0.195 0.2640.001 0.319 0.347 0.235 0.353 0.258 0.320

T = 1000QR

0.01 0.637 0.759 0.605 0.655 0.191 0.2510.005 0.677 0.806 0.671 0.714 0.227 0.3630.001 0.786 0.904 0.780 0.813 0.417 0.667

EVT0.01 0.184 0.219 0.370 0.453 0.197 0.2730.005 0.226 0.260 0.310 0.409 0.219 0.3020.001 0.348 0.376 0.228 0.313 0.297 0.373

T = 500QR

0.01 0.708 0.906 0.599 0.665 0.257 0.3940.005 0.741 0.916 0.689 0.745 0.296 0.4470.001 0.758 0.811 0.782 0.804 0.432 0.654

EVT0.01 0.243 0.285 0.311 0.417 0.248 0.4120.005 0.285 0.326 0.275 0.385 0.281 0.4390.001 0.393 0.430 0.255 0.341 0.369 0.490

17

0 100 200 300 400 500

−15

−10

−5

0

0 100 200 300 400 500

−15

−10

−5

0

0 100 200 300 400 500

−15

−10

−5

0

True EVTQR

Figure 1: Estimated and true quantiles of Symmmetric Absolute Value model withT = 500 and the quantile level θ = 0.01

Table 1 reports the MAE and RMSE, and Table 2 shows the normalized versions

of them. EVT denotes our EVT-based estimation method and QR denotes the con-

ventional quantile regression method.

Both tables show that the EVT-based estimation method is useful for forecasting

VaR using each CAViaR models. According to the Table 1, the EVT method outper-

forms the QR method for the Symmetric Absolute Value model and the Asymmetric

Slope model in every quantile level and sample size. In the Indirect GARCH case, we

cannot judge which method performs better when θ =0.01, 0.005. When it is more

extreme case, θ = 0.001, however, the EVT method clearly outperforms the quantile

18

0 100 200 300 400 500

−15

−10

−5

0

0 100 200 300 400 500

−15

−10

−5

0

0 100 200 300 400 500

−15

−10

−5

0

True EVTQR

Figure 2: Estimated and true quantiles of Symmmetric Absolute Value model withT = 500 and the quantile level θ = 0.001

regression method. Looking at the shifted versions of MAE and RMSE, in the Symmet-

ric Absolute model and the Asymmetric Slope model, the EVT method also performs

much better than the quantile regression method in all the situations. On the other

hand, in the Indirect GARCH with θ = 0.01, the quantile regression method performs

better than the EVT method. However, when θ = 0.001, the EVT method outper-

forms the quantile regression method in all the sample sizes. When θ = 0.005, the

EVT method is better when T = 500 and 1000, the sample size is rather small. Con-

sequently, the EVT method outperfoms the conventional quantile regression method

especially when the quantile level is more extreme and the sample size is small.

19

Figures 1 and 2 plot the estimated quantiles QYT+1(τ |ΩT ) by Symmetric Absolute

Value model and the true quantiles for each simulation, where T = 500. In Figure

1, θ = 0.01, and θ = 0.001 in Figure 2. Although both the QR and EVT method

underestimate the true quantiles, EVT estimates are closer to the true quantiles than

QR estimates. The EVT method seems to be quite stable especially when θ = 0.01.

On the other hand, the QR estimates are considerably unstable even when θ = 0.01.

4 Real Data Analysis

In this section, we apply the proposed method to empirical data to study the per-

formance of forcasting the VaR. We use the daily log return series of the Japanese

Nikkei 225 stock index from January 1990 to December 2014; the length of the data

is 5903. Table 3 provides the summary statistics on the return series. The standard

errors are in the parentheses, the JB denotes the Jarque-Bera test statistic. It shows

that the return series is negatively skewed, and has heavy-tails evidenced by that the

kurtosis is significantly positive. Consequently, the normality is strongly rejected on

the Jarque-Bera test.

Table 3: Summary statistics of the Nikkei 225 log-return seriesMean Variance Skewness Kurtosis JB−0.005 2.306 −0.209 5.047 6306.5(0.020) (0.032) (0.064)

We estimate the VaR for 3903 days, from February 2, 1999 to December 30, 2014.

We apply the rolling window method. To estimate the VaR of time t with window size

T , we use rt−T , . . . , rt−1 where rt denotes the return, to estimate the parameters of

the CAViaR models and then obtain the VaR estimates from the estimated models.

We examine all the three CAViaR models, and set the window size T = 500, 1000

and 2000. The estimated quantile levels are θ = 0.01, 0.005 and 0.001. For the EVT

method, k is set to 35 for T = 500, 45 for T = 1000, 50 for T = 2000. The data are

also shifted as the simulation in previous section to calculate the Hill estimator. The

20

Table 4: Violation rates of the Nikkei 225Symmetric Absolute Value Asymmetric Slope Indirect GARCH

QR EVT QR EVT QR EVTT =5000.01 0.01051 0.02101 0.00999 0.02000 0.01102 0.017420.005 0.00461 0.01486 0.00615 0.01256 0.00538 0.011020.001 0.00231 0.00692 0.00205 0.00410 0.00103 0.00436T = 10000.01 0.00999 0.02358 0.01256 0.02230 0.01128 0.020500.005 0.00641 0.01332 0.00743 0.01333 0.00615 0.011530.001 0.00333 0.00743 0.00333 0.00538 0.00205 0.00436T = 20000.01 0.01461 0.02870 0.01435 0.02665 0.01461 0.027680.005 0.00974 0.02127 0.00118 0.01691 0.01128 0.016400.001 0.00666 0.01153 0.00897 0.00743 0.00461 0.00641

size of the shift are set to the maximum return in the window.

The relative performance of each estimation method is calculated in terms of the

violation ratio. A violation occurs when a realized return fall below the estimated VaR.

The violation ratio is defined as the proportion of the number of violations to the total

number of VaR forcasts. If the VaR is estimated well, the violation ratio will be close

to the quantile level θ of the VaR.

The results are reported in Table 4. Our EVT-based VaR estimation method per-

forms poorer than the conventional quantile regression estimation method in every

model. The violation ratios of the EVT estimators are considerably higher than the

quantile levels, that means the VaR forcasts are too liberal. One reason of this result

might be the underestimation of the extreme value index caused by shifting the data,

leading to estimating the tail to be too light. As a result, the estimated quantiles tend

to be higher than the true quantiles.

In Figures 3 and 4, estimated VaR from the Symmetric Absolute Value model with

T = 500 are plotted with observed Nikkei 225 return series. Contrary to the simulation

results, EVT estimates are higher than QR estimates leading to the overestimation of

21

0 1000 2000 3000 4000

−0.

3−

0.2

−0.

10.

00.

1

0 1000 2000 3000 4000

−0.

3−

0.2

−0.

10.

00.

1

0 1000 2000 3000 4000

−0.

3−

0.2

−0.

10.

00.

1

Returns EVTQR

Figure 3: Nikkei 225 log-return series and orecasted VaR series from Symmetric Abso-lute Value model with T = 500, θ = 0.01

the quantiles. When the volatility of the returns seems to change, the EVT estimates

become considerably low compeared to the QR estimates. The range of fluctuation

of the forecasted VaR by EVT is smaller than that by QR, which is consistent to the

simulation results.

5 Conclusion

In this paper, we proposed a VaR estimation method by integrating a CAViaR model

and extreme value theory. We model the dynamics of quantiles by a CAViaR model

22

0 1000 2000 3000 4000

−0.

3−

0.2

−0.

10.

00.

1

0 1000 2000 3000 4000

−0.

3−

0.2

−0.

10.

00.

1

0 1000 2000 3000 4000

−0.

3−

0.2

−0.

10.

00.

1

Returns EVTQR

Figure 4: Nikkei 225 log-return series and orecasted VaR series from Symmetric Abso-lute Value model with T = 500, θ = 0.001

and estimate extreme quantiles by applying extreme value theory. We first estimate

intermediate quantile series by a quantile regression, then extrapolate high quantiles

with the estimated quantiles. Numerical studies showed that the extrapolation leads

to more accurate quantile estimation than the conventional quantile regression.

The empirical analysis on log-return series of a stock index, however, showed that

our estimation method performs worse than the conventional quantile regression es-

timation method, in terms of violation ratio. This result may be caused by shifting

the data for constructing the Hill estimator which is not shift invariant. Shifting the

data directly affects the value of the Hill estimator, of course also affect the estimated

23

quantiles. This is a main disadvantage of the Hill estimator.

For further study, using other estimators for the extreme value index, such as maxi-

mum likelihood estimator or moment estimator, is suggested. We can estimate general

γ ∈ R by adopting those estimators, leading to relaxing the heavy-taild assumption. In

addition, asymptotic properties of the Hill estimator and the quantile estimator based

on the pseudo order statistcs should be investigated. The i.i.d. assumption and linear

assumption on Wang et al.(2012) are violated in this paper.

Acknowledgements

I would like to express the deepest appreciation to Prof. Eiji Kurozumi helpful sug-

gestions and constructive comments. Without his guidance and persistent help this

paper would not have been possible. I would also like to express my appreciation to

the Center of Financial Engineering Education for supporting my research.

24

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