Multiple-Period Market Risk Prediction under
Long Memory:
When VaR is Higher than Expected∗
Harald Kinateder†
Niklas Wagner‡
Version: January 2014
∗
We would like to thank Axel Buchner, Wolfgang Kürsten, Hato Schmeiser, Jochen Wilhelm,
two anonymous referees as well as participants at the Actuarial Approach for Financial
Risk Meeting, the Center for Quantitative Risk Analysis Conference, the NYU–University
of Florence International Risk Management Conference, the Financial Risks International
Forum, the International Conference on Operations Research and the Karlsruhe Symposium
on Finance, Banking, and Insurance for helpful comments. All errors and omissions remain
with the authors.
†
Department of Business and Economics, University of Passau.
‡
Correspondence: Niklas Wagner, Department of Business and Economics, University of
Passau, 94030 Passau, Germany. Phone: +49 851 509 3240, Fax: +49 851 509 3242, E-mail:
[email protected].
Multiple-Period Market Risk Prediction
under Long Memory:
When VaR is Higher than Expected
Abstract
Several authors, including Andersen and Bollerslev (1998), stress the
importance of long-term volatility dependence for value-at-risk (VaR) prediction. The present paper addresses multiple-period market risk forecasts
under long memory persistence in market volatility. To this aim, we propose volatility forecasts based on a combination of the GARCH(1,1)-model
with potentially fat-tailed and skewed innovations and a long memory
specification of the slowly declining influence of past volatility shocks.
As the square-root-of-time rule is known to be mis-specified, we use the
GARCH setting of Drost and Nijman (1993) as the benchmark forecasting
model. Our empirical study of equity market risk is based on daily index
returns during the period January 1975 to December 2010. We study
the out-of-sample accuracy of VaR predictions for five, ten, 20 and 60
trading days and document that our approach remarkably improves VaR
forecasts for the longer horizons. The result is only in part due to higher
predicted risk levels. Ex-post calibration to equal unconditional risk levels
illustrates that our approach also enhances efficiency in allocating VaR
capital through time.
Keywords: multiple-period value-at-risk, volatility scaling, long memory,
GARCH, Hurst exponent, square-root-of-time rule
JEL Classification: C22
Various periods of financial market stress—including for example the market
crash of October 1987, the burst of the internet bubble at the beginning of
the new millennium and the recent financial crisis which began in 2007—have
increased both the regulatory as well as the industry demand for effective risk
management, where value-at-risk (VaR) has become a standard in determining
the level of risk capital. While financial institutions have to assure that their
capital cushion is adequate in advance of a stress period such that institutions do
not run into liquidity demands during a downturn, too conservative levels of risk
capital lead to unnecessarily high and hence unproductive capital provisions.1
In the present paper we address market volatility prediction, which plays
a central role in the above management trade-off. Two prominent concepts in
volatility prediction include models of autoregressive conditional heteroskedasticity (ARCH) (see Engle (1982) and Bollerslev (1986)) as well as the long memory
(LM) property, which attests a slower than exponential decay in the autocorrelation function of absolute returns (see Ding et al. (1993), Ding and Granger
(1996), Andersen and Bollerslev (1997) and Cont (2001), for example).2 Based
on this evidence, Andersen and Bollerslev (1998) stress the importance of longterm volatility dependencies when risk management applications address longer
1
Berkowitz and O’Brien (2002) investigate the internal VaR estimates of six U.S. banks
during January 1998 to March 2000. They report that banks VaR levels tend to be higher
than those obtained from a non-linear time-series model which is fitted from an external risk
management perspective.
2
Poterba and Summers (1986) are among the first to study the persistence of stock
market volatility. They document that Standard & Poor’s Composite volatility persistence
is characterized by half-lives of up to six months.
1
forecast horizons. Despite this, the Basel capital adequacy framework allows risk
managers to derive multiple-day risk forecasts from daily risk measures via the
square-root-of-time rule thereby implicitly assuming independent and identically
distributed (i.i.d.) asset returns (see Basel Committee on Banking Supervision
(2004) and Basel Committee on Banking Supervision (1996)). Furthermore,
Basel rules require risk managers in banks to calculate VaR for holding periods
of at least ten days, while other financial institutions—such as pension funds and
insurance companies—typically even face longer horizons.
Given these institutional requirements, we study straightforward approaches
to multiple-period market risk prediction. Our estimates of conditional volatility are derived from the asymmetric generalized autoregressive conditional heteroskedastic (GARCH) model of Glosten et al. (1993), GJR-GARCH(1,1). We
account for long memory by using a scaling approach to multiple-period volatility.
We thereby introduce a weight function, which models the slowly declining
relation between past volatility shocks and future volatility based on hyperbolical
rather then exponential shape.3 As a benchmark model for multiple-period
volatility, we use the GARCH(1,1) setting of Drost and Nijman (1993). As
in Drost and Nijman (1993), we model single-period returns (e.g. at the daily
frequency) and then use a scaling approach to derive multiple-period return
volatility rather than choosing the simpler approach of modeling lower-frequency
3
The use of a GARCH model and our weight function is the major difference of our
model to the status quo of VaR construction based on exponential weightings as given by
the “RiskMetrics” approach (see e.g. Longerstaey and Spencer (1996)).
2
(i.e. aggregate) multiple-period returns for volatility forecasting via a standard
GARCH model. This volatility prediction approach has two main advantages.
First, it allows us to derive a consistent set of volatility predictions for various
horizons.
Second, the use of higher-frequency return observations tends to
increase forecasting ability. As such, Wong and So (2003) stress that, as far
as parameter estimation accuracy is concerned, a reduction in the observation
frequency appears to be undesirable. Ghysels et al. (2009) find that modeling
aggregate multiple-period returns for volatility forecasting has almost as poor
properties as the naive square-root-of-time scaling method. One reason for this
finding appears to be that the multiple-period ahead volatility forecast of the
GARCH(1,1) model appears to converge too fast to its unconditional mean.
While predictions based on the square-root-of-time rule were shown to perform poorly especially for long horizons (see Diebold et al. (1997), Dowd et al.
(2004) and Danı́elsson and Zigrand (2006)), focus in the literature has so far
been mostly on single-period ahead VaR forecasts. Notable exceptions include
Dowd et al. (2004), Brummelhuis and Kaufmann (2007), Bali et al. (2009),
Ghysels et al. (2009), Ederington and Guan (2010), Brownlees et al. (2012)
and Alexander et al. (2013), for example.4 However, none of the studies so far
considers long memory in the context of multiple-period VaR forecasting. The
4
Brummelhuis and Kaufmann (2007) and Ghysels et al. (2009) include studies of the longterm behavior of traditional scaling methods. Ederington and Guan (2010) and Brownlees et
al. (2012) analyze volatility forecasting over longer horizons for ARCH models without long
memory. Alexander et al. (2013) consider higher conditional moments in GARCH based VaR
forecasts. Bali et al. (2009) investigate the positive intertemporal relation between VaR and
expected returns.
3
one-period fractionally integrated GARCH (FIGARCH) model of Baillie et al.
(1996) offers a setting in which short-term and long memory volatility dependencies are jointly captured. Despite its general suitability, it appears that the
FIGARCH setting has some shortcomings, which are relevant to multiple-period
VaR forecasting applications. First, FIGARCH implies a return process that is
not covariance stationary. Second, while FIGARCH is strictly stationary under
normally distributed innovations, technical conditions on the expectation of the
innovations imply that FIGARCH is not stationary under Student-t innovations
(see e.g. Caporin (2003)). However, Student-t innovations are particularly useful
in modeling fat-tailed return behavior beyond GARCH (see Bollerslev (1987)
and e.g. Lin and Shen (2006)). Finally, FIGARCH applications report mixed
results on model performance. For example, Belratti and Morana (1999) compare
GARCH and FIGARCH multiple-period high-frequency volatility forecasts for
horizons up to ten days and conclude that there appears to be no particular need
for the FIGARCH model as forecasts in their setting appear fairly similar. As an
alternative to FIGARCH, Zumbach (2004) introduces a new weighting approach,
which is based on historical volatilities from different time horizons.
In proposing a novel estimator for the multiple-period ahead volatility of
GJR-GARCH(1,1), we aim at an improved scaling approach to multiple-period
volatility while maintaining most of the well-known and convenient properties
of the standard GARCH model. In order to test the forecasting ability of our
approach, we study the out-of-sample accuracy of multiple-period VaR predic-
4
tions for 5, 10, 20 and 60 trading day forecast horizons. Innovation quantiles are
either based on the popular Cornish-Fisher approximation or on the generalized
Student-t distribution. Our results show that the VaR forecasts can be enhanced
by the consideration of long memory properties, also leading to better coverage
during stress periods. The result is only in part due to the higher levels of
conditional risk forecasts. We perform a robustness check with respect to each
model’s efficiency in allocating VaR capital through time. Even after controlling
for the unconditional VaR levels of all approaches, our long memory GJRGARCH(1,1) model delivers results which are not dominated by the alternative
models. Hence, our approach allows for overall forecast improvements, and (as
expected) the effect is stronger for the longer forecast horizons. We conclude
that conventional approaches are to be treated with caution as they tend to
underestimate longer-term levels of market risk.
The remainder of this paper is organized as follows. Section 1 provides a brief
overview of the long memory property and introduces the Hurst exponent. In
Section 2 we present our approach to multiple-period volatility prediction under
long memory and outline competing standard VaR models. Section 3 contains
an empirical study of the models’ respective forecasting ability in four major
equity markets. Section 4 concludes.
5
1
Long Memory Review
Long memory (LM, or long-range dependence) is a stylized property of the timeseries behavior of financial market volatility. In this section we provide a brief
review.
1.1
Concept
LM refers to a slow decay of the autocorrelation function (ACF) of a stationary
time series process. More precisely, a stationary process exhibits long memory,
given that its ACF, ρ(τ ) with lag τ ≥ 0, behaves asymptotically as a power
law for τ → ∞, i.e. ρ(τ ) ∼ c τ −δ , with constant c > 0 and coefficient δ ∈
(0, 1). Introducing the Hurst coefficient H, which characterizes the long memory
behavior of the time series for
1
2
< H < 1, the relation can be rewritten as
ρ(τ ) ∼ c τ 2H−2 .
(H)
Fractional Brownian motion, Bt
with
1
2
< H < 1, as discussed e.g.
in
Mandelbrot and van Ness (1968), is an example of a process where the stationary
(H)
increments, ∆Bt,τ
(H)
= Bt
(H)
− Bt−τ , τ > 0, which are normally distributed
(H)
with mean zero and variance E[(∆Bt,τ )2 ] = τ 2H σ 2 , exhibit long memory. The
boundary case H =
1
2
(0.5)
implies that Bt
is a Brownian motion with i.i.d.
increments with mean zero and variance equal to τ σ 2 .
6
1.2
Evidence
While empirical evidence suggests that the autocorrelation of asset returns is
typically close to zero, dependence in absolute and squared returns behaves
remarkably different.
Ding et al. (1993) and Ding and Granger (1996) analyze daily logarithmic
returns of the Standard & Poor’s 500 index from January 3, 1928 to August
30, 1991 and document the slow decay of the ACF in squared and absolute
returns. Even for extremely large lags, the authors document ACFs which
are significantly different from zero. Andersen and Bollerslev (1997), Bollerslev
and Mikkelsen (1996), Cont (2001), Ding et al. (1993), and Ding and Granger
(1996) stress that the influence of volatility shocks on future volatility does not
decline exponentially but rather hyperbolically. As Zumbach (2004) remarks,
the correlation magnitudes are rather small (of order 2 to 15 percent, depending
on the time lag and the asset class). Hence, while volatility prediction remains
subject to substantial error, consideration of the long memory property may
prove to be useful for long-term volatility prediction. Andersen and Bollerslev
(1997) point out that long memory is a salient feature of the return generating
process, rather than an artifact due to potential regime shifts.
7
2
2.1
Forecasting Market Risk
Single-Period Returns
Based on a probability space (Ω, F, P), we model financial returns as a stochastic
process {Rt }1≤t≤T on Ω, which represents possible states of the world, and Ft
is the σ-algebra reflecting information up to time t, such that Fs ⊆ Ft for 0 ≤
s ≤ t ≤ T . We define the single-period return as the change in the logarithmic
prices Pt of a financial asset, Rt ≡ log Pt −log Pt−1 . We now impose the following
commonly used return process assumptions.
Assumption 2.1 The return process {Rt }1≤t≤T belongs to a location-scale family of probability distributions of the form
Rt = µt + σt Zt , Zt ∼ F (0, 1).
(1)
In Assumption 2.1, σt > 0 is independent of Zt . The Zt ’s are standardized
i.i.d. innovations with mean zero and unit variance, which are drawn from the
stationary distribution function F with existing inverse F ← . The period-t return
innovations are given by
t = σt Zt .
The location µt and the scale σt are both Ft−1 -measurable, i.e. they denote
conditional mean µt = E(Rt |Ft−1 ) and conditional variance σt2 = Var(Rt |Ft−1 ),
respectively.
8
Assumption 2.2 For simplicity of exposition, assume that µt = µ in equation
(1). As σt is independent of Zt , we have E(σt Zt ) = 0 and with E(Rt ) =
E(Rt−τ ), ∀τ > 0, the assumption is a sufficient condition for:
E[(Rt − µt )(Rt−τ − µt−τ )] = 0, ∀τ > 0.
Hence, single-period returns {Rt }1≤t≤T are serially uncorrelated.
Assumption 2.2 represents a convenient simplification in model (1), which allows
us to focus on nonlinear dependence in the following derivations. While the
assumption is supported by empirical evidence (see e.g. Cont (2001)), it may
easily be relaxed by introducing linear time series dependence of the autoregressive moving average (ARMA)-type (see e.g. Taylor (2005)).5
Assumption 2.1 involves various specifications of the distribution function F
in model (1). A common choice is the standard normal distribution, Zt ∼ N (0, 1).
In order to account for skewness and excess kurtosis (i.e. so-called fat-tails) in
the innovations, the standardized skewed Student-t distribution offers a more
flexible choice, which nests the standard normal distribution.
Assumption 2.3 The distribution function F in equation (1) is given by the
5
We follow this route in our empirical investigation and introduce an AR(1) process for
the returns. All following statements about the returns then equivalently hold for the AR(1)
return innovations.
9
standardized skewed Student-t distribution. For ν > 2, its density function is
f (z|ξ, ν) =

2



ξ+
2



 ξ+
1
ξ
1
ξ
s(ξz|ν) if
z s |ν if
ξ
z<0
(2)
z ≥ 0,
where ξ > 0 refers to the asymmetry parameter, ν accounts for the tail thickness
and s(·|ν) denotes the symmetric Student-t density with ν degrees of freedom
− ν+1
2
2
Γ( ν+1
)
z
s(z|ν) = ν p 2
1−
ν−2
Γ( 2 ) ν(ν − 2)
and Γ(ν) =
R∞
0
e−x xν−1 dx is the gamma function.
The standardized skewed Student-t distribution nests the symmetric Student-t
distribution for ξ = 1 and the standard normal distribution for ν → ∞ and
ξ = 1.
2.2
Multiple-Period Returns
Given the returns of Section 2.1 above and a discrete time horizon h ≥ 1, the
multiple-period return (or h-period return), Rt,h = log Pt+h−1 − log Pt−1 , is given
P
by Rt,h = hτ=1 Rt+τ −1 . Single-period returns follow with h = 1, i.e. Rt,1 = Rt .
Further, let Ft,h (r) denote the time-(t−1) conditional distribution of the multipleperiod return, i.e. Ft,h (r) = P(Rt,h ≤ r|Ft−1 ).
Based on Assumption 2.1, we have µt,h = E(Rt,h |Ft−1 ) =
10
Ph
τ =1
µt+τ −1 . Under
Assumption 2.1 and 2.2 above, it follows that
µt,h = E(Rt,h |Ft−1 ) = hµ
(3)
and that
2
σt,h
h
X
= Var Rt,h |Ft−1 =
Var t+τ −1 |Ft−1 .
(4)
τ =1
The above equation describes the conditional variance of returns of various
horizons h, all conditional on information Ft−1 , and thereby defines the term
structure of volatility.
2.3
Multiple-Period Value-at-Risk
Given some probability 0 < α < 1, the Ft−1 -measurable value-at-risk for the
α
h-period-ahead time interval (t − 1, t + h − 1], V aRt,h
, is defined as
α
V aRt,h
= − inf {r : Ft,h (r) ≥ α} ,
(5)
α
which yields P(Rt,h ≤ −V aRt,h
|Ft−1 ) = α. It turns out that the conditional
distribution of the multiple-period return, Ft,h (r), is generally analytically inα
tractable when h is large. While the evaluation of V aRt,h
therefore in general
involves high-dimensional integration, model equation (1) in Assumption 2.1
11
allows for the well-known convenient analytical VaR expression
α
V aRt,h
= − µt,h + σt,h F ← (α) ,
(6)
which will be used in the following.
2.4
Multiple-Period Volatility
Multiple-period (or h-period) volatility σt,h , as given by the positive square
root of equation (4), represents the most important variable in the above VaR
representation (6). Given our setting of Section 2.2, we focus on multiple-period
volatility prediction. In detail, we consider i.i.d. returns, self-affine returns,
GARCH(1,1) and a long memory scaling approach.
Independent and Identically Distributed Returns
Given that returns {Rt }1≤t≤T are assumed to be i.i.d., the so-called square-rootof-time rule results. With σt = σ and (4) the h-period volatility is
√
σt,h =
h σ.
(7)
As pointed out in Section 1, financial returns are typically stochastically dependent as, for example, absolute and squared returns tend to be highly positively
correlated. In accordance with this, numerous studies indicate that the perfor-
12
mance of the rule in risk management applications is very weak (see Diebold et al.
(1997), Danı́elsson and Zigrand (2006) and Ghysels et al. (2009), for example).
Self-Affine Returns
A self-affine return process {Rt }1≤t≤T satisfies the scaling property, Rt,h ∼ hH Rt ,
for h > 0 and 0 < H < 1. Given that returns are self-affine, a straightforward
generalization of the square-root-of-time rule (7) follows from the definition of
self-affinity. We have:
σt,h = hH σ.
(8)
The choice of the Hurst coefficient H allows for anti-persistent behavior with
0 < H < 12 , LM persistence with
1
2
< H < 1 and includes the i.i.d. case for
H = 21 .
GARCH(1,1) Returns
Given that the return process {Rt }1≤t≤T follows the GARCH(1,1) model of
Bollerslev (1986) with
2
σt2 = ω + α2t−1 + βσt−1
,
(9)
parameters ω > 0, α ≥ 0, β ≥ 0, α + β < 0 and unconditional return variance,
σ 2 = ω/(1 − α − β). It follows that the τ -step ahead (spot) volatility forecast
13
conditional on Ft−1 is
σt+τ −1
q
= σ 2 + (σt2 − σ 2 )(α + β)τ −1 .
(10)
The method of temporal volatility aggregation of Drost and Nijman (1993) allows
to analytically derive the corresponding volatility of h-period returns. Drost and
Nijman show that, under given regularity conditions, h-period return volatility
is
σt,h
q
2
= ω(h) + α(h) 2t−h,h + β(h) σt−h,h
(11)
where the low frequency parameters ω(h) , α(h) and β(h) are functions of the high
frequency parameters of the single-period model; see the results in Drost and
Nijman (1993), p. 915-916.6
Long Memory Scaling
In violation to the assumptions of the standard class of GARCH models, long
memory in return volatility (see Section 1.2) will affect the volatility decay. In
the GARCH(1,1) model this decay is characterized by the weight (α + β)τ −1 as
given in equation (10). In order to refine the influence of past volatility shocks
on future volatility, we introduce an alternative scaling-based weight function,
which measures volatility persistence. The function includes the autocorrelation
6
Note that the flow variable results in Drost and Nijman (1993) correspond to the above
definition of multiple-period returns. Equation (11) requires an initial estimate of the low
2
frequency volatility σt−h,h
.
14
of absolute returns and their Hurst exponent as arguments. In detail, we make
the following assumptions.
Assumption 2.4 The absolute return process |Rt |1≤t≤T has Hurst coefficient
0 < H < 1 and its ACF at lag τ is denoted as ρ(τ ). Assume further that the
following condition holds: 0 < ρ(τ ) < H, ∀τ > 0.
Hence, absolute return autocorrelations for all lags are strictly positive and
dominated by the Hurst coefficient. For each given lag, τ = 1, ..., h, we next
define the weight function
g[ρ(τ ), H] = ρ(τ )(H−ρ(τ )) .
(12)
Given Assumption 2.4, we have 0 < g[·] < 1. The partial derivatives of (12) with
respect to each argument
∂g[·]
= ρ(τ )(H−ρ(τ )) [H/ρ(τ ) − log(ρ(τ )) − 1] > 0,
∂ρ(τ )
∂g[·]
= ρ(τ )(H−ρ(τ )) log(ρ(τ )) < 0,
∂H
imply that the weights are monotone and increasing in the level of the lag-τ
autocorrelations while decreasing in the level of the Hurst coefficient. Figure
1 illustrates the behavior of the weight function g[·] for given levels of H and
ρ(·) under varying lags τ . A uniform percentage increase in the autocorrelation
15
coefficients is shown to yield an increase in the weight function level and vice
versa. A higher Hurst coefficient in contrast yields higher levels of the weight
function and vice versa. As financial time series with high volatility persistence
tend to produce excess returns in future periods of market stress, incorporating
H and ρ(·) into VaR models, might be very helpful to have higher risk weights
which prevent financial institutions from being undercapitalized in stress periods.
VaR models without long memory using exponential weightings (e.g. RiskMetrics
approach) may underpredict VaR (particularly in advance of a financial turmoil)
as the effect of past volatility shocks on future volatility dies out too fast when
using exponential weightings.
[Insert Figure 1 about here]
Conditional on Ft−1 , the τ -step ahead volatility forecast is now given as
σt+τ −1 =
q
σ 2 + (σt2 − σ 2 ) g[ρ(τ − 1), H].
(13)
In equation (13), the weight function g[·] characterizes volatility persistence and
0 < g[·] < 1 assures that spot volatility remains bounded. Under the given
assumptions of Section 2.1 and the volatility term structure relation (4), it follows
that the volatility of h-period returns is
σt,h
v
u h
uX
=t
σ2
t+τ −1
v
u
h
u
X
= thσ 2 + (σt2 − σ 2 )
g[ρ(τ − 1), H].
τ =1
τ =1
16
(14)
Two limiting cases in equation (14) are worth mentioning. Under i.i.d. returns,
i.e. when H =
1
2
and all autocorrelation coefficients ρ(τ ) approach zero, the
weight function g[·] is zero for all lags and therefore the square-root-of-time rule
based on the unconditional variance, σ 2 , results. In the limit where g[·] would
approach one for all lags, the square-root-of-time rule based on the conditional
variance, σt2 , results.7
2.5
Alternative Multiple-Period VaR Models
Our multiple-period volatility results in Section 2.4 allow for a variety of competing multiple-period VaR models. All models derive VaR from equation (6),
α
V aRt,h
= − µt,h + σt,h F ← (α) ,
while they differ in their derivation of multiple-period volatility, σt,h . Given
Section 2.4, we present a hybrid forecast model that relies on GARCH as a model
which captures important features of conditional return volatility (see e.g. Engle
and Patton (2001)) and combines them with a volatility scaling approach that
allows for long memory effects in volatility. In detail, we consider the following
alternative models.
7
In the special case h = 1, i.e. τ = 1, the function g[·] takes the value one. This yields
the ordinary one-day ahead volatility forecast, σt,1 . For h > 1, i.e. τ > 1, we typically have
g[·] < 1, as ρ(τ − 1) is usually below one.
17
Model M0: Drost/Nijman Benchmark Model
The multiple-period volatility, σt,h , is derived from the Drost/Nijman equation
(11).
Potentially fat-tailed and/or skewed innovations are modeled via the
asymmetric Student-t distribution F as given in Assumption 2.3. The model
considers GARCH(1,1) volatility term structure effects, but does not incorporate
potential long memory effects. Hence, it serves as our benchmark model of
volatility scaling without long memory and is denoted as model M0.
Model M1: Long Memory Scaling
The multiple-period volatility, σt,h , is derived from the scaling relation in equation (8). In this case, the α-quantile F ← (α) is given as the quantile of the
unconditional return distribution F . The model serves as a simple and convenient extension of the square-root-of-time rule, which allows for fat-tails and for
potential long memory effects in volatility. It does not consider volatility term
structure effects, since no spot volatilities, σt+τ −1 , are calculated. As a result,
volatility term structure effects besides the simple scaling relation in (8) cannot
be considered. It is denoted as model M1.
Model M2: Long Memory Scaling with GARCH-σt
The GARCH(1,1) model allows for an adaption to volatility clusters. It provides
a conditional one-period ahead volatility forecast and a flexible model of the
volatility term structure. Potentially fat-tailed and/or skewed innovations are
18
modeled via the asymmetric Student-t distribution F as given in Assumption
2.3. We use our volatility scaling approach, which considers long memory effects
via the weight function g. Hence, σt,h is given by equation (14). The model is
denoted as model M2.
Model M3: Long Memory Scaling with Asymmetric GARCH-σt
In this model, the GARCH component of model M2 is represented by the
asymmetric GARCH model of Glosten et al. (1993), GJR-GARCH(1,1). I.e.,
the return innovations t in equation (1) have conditional variance:
2
σt2 = ω + α2t−1 + γ2t−1 1{t−1 <0} + βσt−1
.
(15)
In addition to the GARCH(1,1) parameter restrictions, α+γ ≥ 0 and α+γ +β <
1 are imposed (see e.g. Taylor (2005), p. 221). The indicator function 1{t−1 <0}
allows for a possible asymmetric response of conditional variance to lagged return
innovations. Using GJR-GARCH(1,1) in model M2, we arrive at model M3.8
3
Empirical Analysis
This section considers the predictive performance of the alternative VaR estimation models M0, M1, M2 and M3, which were introduced in Section 2.5. We
Assuming for simplicity that E(1{t−1 <0} ) = 21 , where a sufficient condition is that the
distribution function F is symmetric, the τ -step ahead volatility forecast of GJR-GARCH(1,1)
follows directly from equation (10).
8
19
study the models’ performance in the assessment of equity market risk. We
additionally consider the robustness of our results in several directions.
3.1
3.1.1
The Data and Preliminary Analysis
Dataset and Descriptive Statistics
Our return data are obtained from Thomson One Banker Datastream. We
choose the Standard and Poor’s 500 (S&P 500), the Dow Jones Industrials (DOW
JONES), the NASDAQ Composite and the German DAX index as representative
equity market indices with sufficient daily price history. The data comprise
respective close index levels Pt during the 36-year period from January 2, 1975 to
December 31, 2010. Based on index closing levels, we calculate daily continuously
compounded percentage returns where h = 1. Summing up daily returns, we
further derive non-overlapping returns for holding period horizons h > 1, namely
for h = 5 (weekly), h = 10 (biweekly), h = 20 (monthly) and h = 60 (quarterly)
return horizons. Table 1 presents summary statistics of the four index return
series. All return distributions exhibit negative sample skewness. The kurtosis
estimates for increasing h indicate declining degrees of fat-tailedness for lowerfrequency returns.
[Insert Table 1 about here]
20
3.1.2
Long Range Dependence
We next consider long memory effects in the index returns. Various statistical
methods are available for an estimation of the Hurst coefficient H. We use the
variance of residuals approach of Peng et al. (1994), which is an improvement of
the original rescaled-range, or so-called R/S, technique.9
Table 2 provides the estimated Hurst exponents for our index returns, including absolute and squared returns. The table includes t-values for tests of the
null hypothesis of independence, H0 : H = 1/2.
[Insert Table 2 about here]
Table 2 illustrates that the null hypothesis has to be rejected for all return
transformations (identity, squared value and absolute value) and for all indices.
While the S&P 500 and the DOW JONES exhibit mild anti-persistence, NASDAQ and DAX exhibit persistence in daily returns. For squared and absolute
returns, the estimated Hurst exponents for all indices lie within the persistent region and all estimates are highly significant. Persistence is strongest
for absolute returns, where the indices also display high autocorrelations as
illustrated in Appendix C. As is known (see e.g. Andersen and Bollerslev (1997);
9
Taqqu et al. (1995) study the reliability of several methods for estimating the Hurst
exponent for simulated sequences of fractional Gaussian noise and fractional autoregressive
integrated moving average processes. The authors find that the asymptotically unbiased Peng
et al. (1994)-approach provides superior results. We provide an illustrative Monte Carlo study
in Appendix A. The statistical literature documents some earlier methods for the estimation of
H, see e.g. Beran (1994). For further readings concerning Hurst estimation issues, see e.g. Lo
(1991), Peters (1992) and Nawrocki (1995).
21
Cont (2001)), these findings contradict the random walk hypothesis of asset
returns, which would imply that (arbitrary, possibly non-linear) functions of Rt
yield independent variables. As to be expected, none of the transformations of
simulated Brownian motion increments, BM(9050), leads to a rejection of the
null in Table 2.
3.1.3
Model Estimation
Estimation of the GARCH models with skewed Student-t innovations is carried
out by a maximum likelihood (see e.g. Davidson and MacKinnon (2004), Chapter 10). We account for autocorrelation in the daily returns Rt by using an
AR(1) term in the conditional mean equation (1), such that the GARCH model
innovations satisfy the uncorrelatedness Assumption 2.2.10
The time-t one-step-ahead predicted volatilities, σt , for all VaR models (M0,
M1, M2 and M3) are based on rolling 250 trading day estimates of the model
parameters. The moving window is updated every h trading days, which yields
c estimates. As a starting volatility level we use the unconditional mean
b T −250
h
(see e.g. Diebold et al. (1997)). Estimates of the autocorrelation structure and
Peng et al. (1994)-estimates of the Hurst exponent for absolute returns are used in
all models except the benchmark model M0. We point out that Assumption 2.4 is
well satisfied for all our given point estimates. The highest autocorrelation point
estimate of 0.39 for the NASDAQ is considerably lower than the corresponding
10
Hence, in equation (1), µt = ϕRt−1 . The imposed parameter restriction |ϕ| < 1 ensures
stationarity.
22
NASDAQ Hurst estimate of 0.858 (see Table 2).
3.2
Out-of-Sample VaR Forecasts
In order to examine out-of-sample model performance, we calculate the 1 percent
0.01
for all horizons h. The time-t VaR forecasts are updated every h trading
V aRt,h
days which results in b T −250
c successive VaR predictions. Christoffersen (1998)
h
LR-test statistics for VaR forecast performance evaluation are reported in the
following; see Appendix B for a detailed explanation.11
[Insert Tables 3 & 4 about here]
The out-of-sample VaR forecast results are summarized in Tables 3 and 4.
For each index and forecast horizon, we provide the average value of the VaR
estimates, V aR and the empirical coverage rate α
b. The LRuc statistic tests
for the correct violation level, which is α = 1% in our setting. We consider a
violation of a backtesting statistic to occur under a corresponding p-value of the
respective test statistic below 5 percent. VaR forecasts may be correct on average,
but produce violation clusters, a phenomenon ignored by unconditional coverage.
The LRind statistic tests for independence of the VaR violations. The conditional
11
The models are validated based on non-overlapping h-period returns in order to provide
independent out-of-sample tests. In practical risk management applications, returns Rt,h and
VaR forecasts may be updated on each trading day t with the effect that overlapping returns
imply dependent test observations. In this case, the backtesting results are based on smoothed
observations and have to be treated with extreme care as they might signal superior model
performance.
23
coverage statistic, LRcc , combines both concepts. We therefore consider it as our
most important backtesting statistic.
Testing model performance in our setting involves four different horizons and
for four different markets. Given an overall of 16 backtesting results in Tables 3
and 4, we find that models M2 and M3 achieve higher p-values for the conditional
coverage statistic than model M0 in 13 out of 16 cases. Model M1 achieves better
conditional coverage than model M0 in four out of 16 cases. While model M3
shows no violation of the conditional coverage statistic, model M2 causes one
violation, benchmark model M0 causes two and model M1 causes four.
The predictions of the models M0 and M2 for the 60-day VaR horizon as
well as 60-day returns are plotted in Figure 2, those of models M1 and M3 in
Figure 3. The plots illustrate that the VaR predictions of the two models are
relatively smooth and that they represent valid alternatives to the Drost-Nijman
benchmark. In the following, we analyze the models’ performance in more detail.
3.2.1
Unconditional Coverage versus Conditional Coverage
Tables 3 and 4 show results and p-values of unconditional, independence and
conditional coverage tests. Generally, a VaR model does not work well, when
several backtesting statistics have p-values below 0.05. The unconditional coverage test indicates, that for the 5-day horizon of NASDAQ and DAX, and for the
20-day horizon of NASDAQ, the predicted VaR of model M0 causes excessive
VaR exceedances leading to insufficient conditional coverage. For the 60-day
24
horizon of the DOW JONES the VaR of model M1 violates the unconditional
coverage statistics, because the VaR level is too low. Also, model M1 creates a
violation of LRuc and LRcc for the 20-day horizon of DOW JONES, NASDAQ
and DAX. Model M2 passes all violation tests with p-values above 0.05 for the
S&P 500, DOW JONES and DAX. Regarding the 5-day horizon of NASDAQ,
model M2 violates the conditional coverage test. Model M3 passes all violation
tests with p-values above 0.05 for all indices. All models pass the independence
test indicating that there is no severe clustering of VaR violations over time. As
a result, backtesting violations result due to excessive VaR exceedances.
3.2.2
Short-term versus Long-term Forecasts
Concerning short-term, i.e. five and ten day forecasting results, it is striking that
the predicted VaR of NASDAQ and DAX shows several violations of our test
statistics. For the 5-day horizon, M2 and M3 achieve always better conditional
coverage than M0. The 5-day horizon of the NASDAQ appears to be a difficult
forecasting scenario. In the longer-term environment, the results of model M1
deteriorate. The unconditional coverage test indicates that for the 20-day horizon
of DOW JONES, NASDAQ and DAX, the predicted VaR of model M1 yields
an excessive number of VaR exceedances. For the 60-day horizon of the DOW
JONES, the predicted VaR of model M1 is too low. The poorer performance of
model M1 for the 20 and 60-day horizon is supposedly driven by the fact that it
does not account for volatility term structure effects besides the simple scaling
25
relation (8). In all long-term scenarios models M2 and M3 always dominate
model M1. Considering model M0, we find that it never dominates models M2
and M3. Also, model M0 creates a violation of LRuc and LRcc for the 20day horizon of NASDAQ. For the 60-day horizon, models M2 and M3 achieve
the best conditional coverage for the S&P 500, DOW JONES, and the DAX.
Regarding the NASDAQ, the conditional coverage of models M0, M2, and M3
is roughly identical. The standard GARCH benchmark (model M0) will yield
underestimation in required capital levels. For the 60-day horizon, the predicted
VaR leads to a percentage average capital shortfall of about 1 percent for the
S&P 500, DOW JONES, and the DAX relative to the predictions of model M3.
[Insert Figures 2 & 3 about here]
3.2.3
Symmetric versus Asymmetric GARCH Variance
Models M2 and M3 use different estimates of one-day ahead variance, σt2 . We
find that, for all markets and all forecast horizons, model M3 produces higher
average capital levels than model M2. The highest percentage difference in the
average predicted capital level occurs for the 5-day horizon. Beyond the 5day horizon, the percentage differences in the average capital levels appear to
be monotonically decreasing. Moreover, model M3 passes all backtesting tests
with p-values above 0.05, whereas model M2 shows a violation of the conditional
coverage statistic (p-value = 0.029) for the 5-day horizon of NASDAQ (see Tables
3 and 4). In sum, our results underpin that using an asymmetric GARCH model
26
does not harm but rather improve forecasting results.
3.3
Robustness Tests
In order to examine whether the backtesting performance of models M2 and M3
in the above section is merely achieved due to higher average levels of predicted
required capital or potentially also due to a better adaption to varying market
conditions, we first perform an additional robustness check under normalized
unconditional VaR. Second we test for the robustness of our results with respect
to the choice of the error distribution F using the Cornish-Fisher approach as
an alternative.
3.3.1
VaR Forecasts under Normalized Unconditional VaR
In this section we study how our models perform after calibration to an identical
average capital employed, i.e. after fixing an average unconditional VaR level. We
choose the average capital of model M0, V aRM 0 , as the benchmark unconditional
VaR level. The normalized conditional VaR of models M1, M2, and M3 is then
achieved by the following normalization
Normalized-V aRM i =
V aRM 0
V aRM i ,
V aRM i
where V aRM i is the predicted conditional VaR (6) and V aRM i denotes the
average predicted VaR of model M1, M2 or M3. As a result, the average sample
27
VaR is the same for all models, but the backtesting results differ due to different
allocations of VaR capital through time. We point out that this calibration of all
models to the same unconditional VaR level represents an ex-post examination.
It serves as a robustness check with respect to a model’s efficiency in allocating
VaR capital.
[Insert Tables 5 & 6 about here]
Our results for VaR forecasts under normalized M0 unconditional VaR are
provided in Tables 5 and 6, with the setting as in Section 3.2. Our results
demonstrate that—even under normalized unconditional VaR—model M2 dominates model M0 in eleven out of 16 and model M3 dominates the benchmark
in seven out of 16 cases. Models M2 and M3 only exhibit one violation of the
conditional coverage statistic. In contrast, model M1 causes eight and model M0
two violations. Starting from the previous results of Tables 3 and 4, it appears
that models M1 and M3 maintain their empirical coverage levels for 60-day VaR
predictions. Model M2 also maintains its long-term conditional coverage, except
for the 60-day horizon of the DOW JONES. The findings underpin the robustness
of the long-term forecasting power of models M2 and M3. The overall evidence
confirms the hypothesis that differences in VaR forecasting performance are not
only due to different unconditional VaR levels, but also due to differences in the
models’ adaption to market conditions.
28
3.3.2
Cornish-Fisher Quantiles
As an alternative to the skewed Student-t distribution of Assumption 2.3, we
consider the Cornish-Fisher expansion as a distributional assumption for the
model innovations. We then test the performance of the resulting VaR forecasts
for the benchmark model M0.12
Based on the Cornish-Fisher expansion, the quantiles of a distribution (which
is close to the Normal) are defined as:
−1
FCF
(α) = N −1 (α) + q(α, δ, κ)
with
1 −1
1 −1
(N (α))2 − 1 δ +
(N (α))3 − 3N −1 (α) κ
6
24
1
−1
3
−1
−
2(N (α)) − 5N (α) δ 2 ,
36
q(α, δ, κ) =
where N −1 (α) is the α-quantile of a standard normal distribution, δ is the
standardized skewness and κ is the excess kurtosis.
Table 7 contains model M0 VaR performance results with Cornish-Fisher
quantiles. As the sample kurtosis results in Table 1 indicate, the empirical
distribution of high frequency returns show larger tail deviations from the normal
distribution than those of lower frequency returns. In line with this, the Cornish12
We provide the results of model M0 only as the results for the other models are similar
and available by the authors upon request.
29
Fisher quantiles become more accurate under longer forecast horizons h. Overall,
the use of the Cornish-Fisher quantiles is less suited to capture fat-tails than
our Student-t assumption. As such the Cornish-Fisher approach yields lower
risk levels represented by lower average predicted VaR. Insufficient coverage is
most obvious for the five and ten day prediction horizons as the Cornish-Fisher
quantiles underestimate the tail risk.
[Insert Table 7 about here]
3.4
Forecasting Illustrations
3.4.1
Predicted VaR during a Crisis Period
We illustrate the out-of-sample behavior of our models during the 2008 financial
crisis. As long-term risk is particularly critical to predict and the effect of large
moves in prices is more apparent, we use the 60-day, i.e. quarterly, forecast
horizon.
[Insert Figure 4 about here]
Figure 4 shows the evolution of the 60-day VaR during the 3rd quarter of 2007
to the 4th quarter of 2009, which includes the collapse of Lehman Brothers on
September 15, 2008. The plots illustrate that all indices are hit by the Lehman
shock. However, there are differences in the model responses during and after
the shock. All models forecast an increased and adequate 4th quarter 2008
30
capital requirement for the DOW JONES and the DAX. Model M0 fails to do
so for the S&P 500 and the NASDAQ, however, models M1, M2 and M3 achieve
adequate capital requirements. Models M2 and M3 forecast increased VaR levels
after the shock, while model M1 is less conservative and thereby closer to actual
market developments during 2009. The increased capital levels after the Lehman
shock of models M2 and M3 are a lagged response to significantly increased
volatility clustering in the 4th quarter of 2008. It is also important to note that
immediately before the Lehman shock the long memory models (especially M2
and M3) predict higher capital requirements for the U.S. indices as model M0.
3.4.2
Predicted Market Volatility at Different Horizons
We illustrate market volatility prediction at different horizons based on model
M3 and compare them to the standard GJR-GARCH(1,1) model as well as the
simple square-root-of-time rule. We choose two forecast dates as example cases.
The first forecast date, March 27, 2001, is within the burst of the internet bubble
and represents a high volatility setting. The second forecast is of September 25,
1996, a date which represents a rather tranquil market period.
Figure 5 illustrates the respective volatility forecasts, namely multiple-period
volatility σt,h as well as τ -step ahead spot volatility σt+τ −1 for time horizons
h, τ ∈ [1, 60]. All forecasts are based on parameter estimates which are derived
from the preceding 250 trading day returns. March 27, 2001 results are given
in panels (a) and (c), while September 25, 1996 results are in panels (b) and
31
(d). In comparison to model M3 predictions (dashed line) and standard GJRGARCH(1,1) predictions (dotted line), the simple square-root-of-time rule based
on an unconditional volatility estimate (solid line) tends to underestimate future
market volatility during a stress period.
[Insert Figure 5 about here]
A comparison between the volatility predictions based on model M3 and
the standard GJR-GARCH(1,1) approach in panels (c) and (d) illustrates that
the long memory component of model M3 implies slower volatility mean reversion. Standard GARCH approaches assume that past volatility shocks decline
exponentially leading to large volatility responses for short horizons which than
decline quickly. On the other hand, model M3 implies that the effect of past
volatility shocks persists over longer horizons. The standard GJR-GARCH(1,1)
approach shows a smooth decay for increasing lags which is based on estimated
GARCH model parameters. In contrast, the weight function g in model M3
is based on estimates of lag-τ autocorrelation coefficients as well as the Hurst
coefficient, which induces noise as visible in Figure 5. This noise is smoothed via
summation in our calculation of the multiple-period volatility in equation (14).
Overall, the predicted τ -step ahead spot volatility of model M3 shows a more
realistic decay in volatility.
32
4
Conclusion
The prediction of multiple-period market risk relies on adequate volatility forecasts. While GARCH models perform quite well for one-day ahead volatility
forecasts even during the recent 2007-2009 financial crisis (see e.g. Brownlees et
al. (2012)), longer term volatility forecasts are more demanding. When risk
forecasts are extrapolated to longer prediction horizons, the square-root-of-time
rule is inappropriate for obvious reasons. We propose a straightforward asymmetric GARCH approach which addresses long memory in volatility as well as
skewness and fat-tails in the return innovations. We obtain superior forecast
results as compared to other models of multiple-period volatility forecasting.
While our results confirm the importance of long memory for VaR prediction, we
would suggest that other refinements may of course also prove to be important
in adequate market risk prediction (see e.g. robust estimation as in Mancini
and Trojani (2011), extreme value statistics as in Harmantzis et al. (2006) or
nonparametric kernel estimation as in Huang and Tseng (2009)).
In sum we would suggest that financial institutions are well advised to implement more accurate risk prediction metrics and that improvements of the current
Basel regulative framework should consider longer holding period requirements
(see also e.g. Flannery (2013)). As traditional methods such as the square-rootof-time rule tend to underestimate capital levels, this would not only help to
enhance financial institutions’ equity cushions in advance of periods of market
33
stress but also provide incentives for longer term risk management strategies. It
would thereby also help to mitigate the well-known problem of procyclicality in
risk capital allocation (see e.g. Gordy and Howells (2006)).
References
Alexander, Carol, Emese Lazar, and Silvia Stanescu (2013): Forecasting VaR
using analytic higher moments for GARCH processes, International Review of
Financial Analysis 30: 36–45.
Andersen, Torben G. and Tim Bollerslev (1997): Heterogeneous information
arrivals and return volatility dynamics: Uncovering the long-run in high
frequency returns, Journal of Finance 52: 975–1005.
Andersen, Torben G. and Tim Bollerslev (1998): Answering the skeptics:
Yes, standard volatility models do provide accurate forecasts, International
Economic Review 39: 885–905.
Baillie, Richard T., Tim Bollerslev, and Hans O. Mikkelsen (1996): Fractionally
integrated generalized autoregressive conditional heteroskedasticity, Journal
of Econometrics 74: 3–30.
Bali, Turan G., K. Ozgur Demirtas, and Haim Levy (2009): Is there an
intertemporal relation between downside risk and expected returns?, Journal
of Financial and Quantitative Analysis 44: 883–909.
34
Basel Committee on Banking Supervision (1996): Amendment to the Basel
Capital Accord to incorporate market risk, BIS, Basel.
Basel Committee on Banking Supervision (2004): International convergence of
capital measurement and capital standards: A revised framework, BIS, Basel.
Belratti, Andrea and Claudio Morana (1999): Computing value at risk with high
frequency data, Journal of Empirical Finance 6: 431–455.
Beran, Jan (1994):
Statistics for Long-Memory Processes, Chapman &
Hall/CRC, Boca Raton, London, New York.
Berkowitz, Jeremy and James O’Brien (2002): How accurate are value-at-risk
models at commercial banks?, Journal of Finance 57: 1093–1111.
Bollerslev, Tim (1986): Generalized autoregressive conditional heteroskedasticity, Journal of Econometrics 31: 307–327.
Bollerslev, Tim (1987): A conditionally heteroskedastic time series model for
speculative prices and rates of return, Review of Economics and Statistics 69:
542–547.
Bollerslev, Tim and Hans O. Mikkelsen (1996): Modeling and pricing long
memory in stock market volatility, Journal of Econometrics 73: 151–184.
Brownlees, Christian, Robert Engle, and Bryan Kelly (2012): A Practical guide
to volatility forecasting through calm and storm, Journal of Risk 14: 3–22.
35
Brummelhuis, Raymond and Roger Kaufmann (2007): Time-scaling of value-atrisk in GARCH(1,1) and AR(1)-GARCH(1,1) processes, Journal of Risk 9:
39–94.
Caporin, Massimiliano (2003): Stationarity, memory, and parameter estimation
of FIGARCH models, Greta Working Paper, Venice.
Christoffersen, Peter (1998):
Evaluating interval forecasts, International
Economic Review 39: 841–862.
Cont, Rama (2001): Empirical properties of asset returns: Stylized facts and
statistical issues, Quantitative Finance 1: 223–236.
Danı́elsson, Jón and Jean-Pierre Zigrand (2006): On time-scaling of risk and the
square-root-of-time rule, Journal of Banking and Finance 30: 2701–2713.
Davidson, Russell and James G. MacKinnon (2004): Econometric theory and
methods, Oxford University Press, New York.
Diebold, Francis X., Andrew Hickman, Atsushi Inoue, and Til Schuermann
√
(1997): Converting 1-day volatility into h-day volatility: Scaling by h is
worse than you think, Working Paper, Wharton Financial Institutions Center.
Ding, Zhuanxin, Clive W. J. Granger, and Robert F. Engle (1993): A long
memory property of stock market returns and a new model, Journal of
Empirical Finance 1: 83–106.
36
Ding, Zhuanxin and Clive W.J. Granger (1996): Modeling volatility persistence
of speculative returns: A new approach, Journal of Econometrics 73: 185–215.
Dowd, Kevin, David Blake, and Andrew Cairns (2004): Long-Term Value at
Risk, Journal of Risk Finance 5: 52–57.
Drost, Feike C. and Theo E. Nijman (1993): Temporal aggregation of GARCH
processes, Econometrica 61: 909–927.
Ederington, Louis H. and Wei Guan (2010): Longer-term time series volatility
forecasts, Journal of Financial and Quantitative Analysis 45: 1055–1076.
Engle, Robert F. (1982): Autoregressive conditional heteroskedasticity with
estimates of the variance of United Kingdom inflation, Econometrica 50: 987–
1008.
Engle, Robert F. and Patton Andrew J. (2001): What good is a volatility model?,
Quantitative Finance 1: 237–245.
Flannery, Mark J. (2013): Maintaining Adequate Bank Capital, Journal of
Money, Credit and Banking, Forthcoming.
Ghysels, Eric, Antonio Rubia, and Rossen Valkanov (2009): Multi-period
forecasts of volatility: Direct, iterated, and mixed-data approaches, Working
Paper, University of North Carolina.
37
Glosten, Lawrence R., Ravi Jagannathan, and David E. Runkle (1993): On the
relation between the expected value and the volatility of the nominal excess
return on stocks, Journal of Finance 48: 1779–1801.
Gordy, Michael B. and Bradley Howells (2006): Procyclicality in Basel II:
Can we treat the disease without killing the patient?, Journal of Financial
Intermediation 15: 395–417.
Harmantzis, Fotios C., Linyan Miao, and Yifan Chien (2006): Empirical study of
value-at-risk and expected shortfall models with heavy tails, Journal of Risk
Finance 7: 117–135.
Huang, Alex Y.-H. and Tsung-Wei Tseng (2009): Forecast of value at risk for
equity indices: an analysis from developed and emerging markets, Journal of
Risk Finance 10: 393–409.
Lin, Chu-Hsiung and Shan-Shan Shen (2006): Can the student-t distribution
provide accurate value at risk?, Journal of Risk Finance 7: 292–300.
Lo, Andrew W. (1991): Long-term memory in stock market prices, Econometrica
59: 1279–1313.
Longerstaey, Jacques and Martin Spencer (1996): RiskMetricsTM–Technical
Document, Morgan Guaranty Trust Company of New York, New York.
Mancini, Loriano and Fabio Trojani (2011): Robust value-at-risk prediction,
Journal of Financial Econometrics 9: 281–313.
38
Mandelbrot, Benoit B. and J. W. Van Ness (1968): Fractional Brownian motion,
fractional noises and applications, SIAM Review 10: 422–437.
Nawrocki, David (1995): R/S analysis and long term dependence in stock market
indices, Managerial Finance 21: 78–91.
Peng, Chuang-Kang, Sergey V. Buldyrev, Shlomo Havlin, Michael Simons,
H. Eugene Stanley, and Ary L. Goldberger (1994): Mosaic organization of
DNA nucleotides, Physical Review E 49: 1685–1689.
Peters, Edgar E. (1992): R/S analysis using logarithmic returns, Financial
Analysts Journal 48: 81–82.
Poterba, James M. and Lawrence H. Summers (1986): The persistence of
volatility and stock market fluctuations, American Economic Review 76: 1142–
1151.
Taqqu, Murad S., Teverovsky Vadim, and Willinger Walter (1995): Estimators
for long-range dependence: An empirical study, Fractals 4: 785–798.
Taylor, Stephen (2005):
Asset price dynamics, volatility and prediction,
Princeton University Press, Princeton.
Wong, Chi-Ming and Mike K. P. So (2003): On conditional moments of GARCH
models, with applications to multiple period value at risk estimation, Statistica
Sinica 13: 1015–1044.
39
Zumbach, Gilles (2004): Volatility processes and volatility forecast with long
memory, Quantitative Finance 4: 70–86.
40
Appendix
A
R/S versus Variance of Residuals Approach
This section reports the results of a Monte Carlo simulation of the small sample
properties of the R/S versus the variance of residuals approach.
The true
parameter is based on a discretized standard Brownian motion with H =
1
2
and
8,609 simulated observations. The results are illustrated in Figure 6 below. The
R/S approach is positively biased, with a mean estimate of 0.569. In contrast,
the variance of residuals approach provides an average estimate of 0.490. The
q
b − H)2 ], of the variance of residuals
root mean squared error, RM SE = E[(H
technique is 0.0165 as opposed to 0.1034 for the R/S approach.
Figure 6: Parameter estimation densities based on 10,000 estimations via the R/S and the
variance of residuals approach. The dashed line refers to the true parameter, H = 0.5, of the
underlying simulated discretized Brownian motion.
B
Backtesting VaR
Market risk models predict VaR with error. The validity of a VaR prediction
model is measured based on predicted versus actual loss levels. To evaluate
the out-of-sample performance of the proposed models we follow the concept
α }
of Christoffersen (1998). The indicator (or hit) function It = 1{Rt,h <−V aRt,h
represents the history of observations, t = 1, ..., T , for which losses in excess of
the predicted VaR occur.
B.1
Unconditional Coverage
When a VaR model is designed perfectly, the number of observations that fall
outside the predicted VaR should be exactly in line with the given VaR level,
such that E(It |Ft ) = α holds. Hence, the test of unconditional coverage is
H0 : E(It |Ft ) = α vs. H1 : E(It |Ft ) 6= α.
Under the null hypothesis, the likelihood-ratio (LR) test statistic follows as
LRuc = −2 ln[L(α)/L(b
α)] ∼ χ2 (1),
where L(α) is the binomial likelihood with parameter α and α
b =
the maximum likelihood estimator of α.
(16)
1
T
PT
t=1 It
is
B.2
Independence
Besides the above requirement VaR violations should be independent, which
requires an additional test. Let nij denote the number of observations for which
It = j occurred following It−1 = i and assume that {It } is a first-order Markov
chain with transition probabilities πij = P(It = j|It−1 = i). This yields the
likelihood
n01
n11
L(Π) = (1 − π01 )n00 π01
(1 − π11 )n10 π11
.
Maximum likelihood estimators for the transition probabilities are:
π
b01 =
n11
n01
, and π
b11 =
.
n00 + n01
n10 + n11
Under the null hypothesis of independence, P(It = 0) = π0 = π01 = π11 , which
implies
L(π0 ) = (1 − π0 )n00 +n10 π0n01 +n11 .
The maximum likelihood estimate for π
b0 is
π
b0 =
n01 + n11
.
n00 + n10 + n01 + n11
b the independence LR test statistic is
Based on π
b0 and Π,
b ∼ χ2 (1).
LRind = −2 ln[L(b
π0 )/L(Π)]
(17)
B.3
Conditional Coverage
The LRind statistic (17) tests for independence, but it does not take coverage
into account. Christoffersen (1998) therefore proposes a combined test statistic:
LRcc = LRuc + LRind
b ∼ χ2 (2).
= −2 ln[L(α)/L(Π)]
(18)
C.1
ACFs of absolute index returns
Figure 7: ACFs of absolute index returns. Sample period: January 2, 1975 to December 31,
2010.
C.2
ACFs of squared index returns
Figure 8: ACFs of squared index returns. Sample period: January 2, 1975 to December 31,
2010.
D
Tables
Table 1: Index Return Summary Statistics
Summary statistics of index returns for various aggregation levels h. For h = 1, standard
Phillips-Perron unit root test statistics indicate that the null has to be rejected in favor of the
stationarity alternative for all indices. Reported time series lengths, T , for the DAX are not
equal to those for the U.S. markets due to differences in national holidays. Sample period:
January 2, 1975 to December 31, 2010.
Index
S&P 500
T
9085
DOW JONES
9085
NASDAQ
9085
DAX
9050
h
1
5
10
20
60
1
5
10
20
60
1
5
10
20
60
1
5
10
20
60
Mean
0.0318
0.1588
0.3177
0.6354
1.8788
0.0320
0.1600
0.3202
0.6404
1.9030
0.0416
0.2079
0.4165
0.8331
2.4736
0.0312
0.1559
0.3119
0.6276
1.8428
Std. Dev.
1.1023
2.3584
3.2060
4.5205
7.7401
1.0950
2.3516
3.1962
4.4428
7.6572
1.2946
2.9658
4.2449
6.1857
11.7570
1.2981
2.8416
3.9498
5.4896
10.8364
Skewness
-1.17
-1.38
-1.30
-1.23
-0.56
-1.48
-1.63
-1.30
-1.24
-0.48
-0.28
-0.99
-1.40
-1.08
-0.66
-0.32
-0.54
-0.88
-0.80
-0.88
Kurtosis
31.30
18.64
12.87
9.94
5.99
43.04
23.51
13.73
11.08
5.64
12.85
11.21
12.04
7.69
4.34
10.77
6.24
7.05
4.59
5.14
Table 2: Estimates of the Hurst Exponent
Empirical estimates of the Hurst exponent H for daily index data including three return
transformations (identity, squared value and absolute value). Estimates are based on the
variance of residuals approach of Peng et al. (1994). BM(9050) denotes an average estimate
for 10000 replications of a simulated discretized Brownian motion with 9050 observations. *
denotes rejection of the null at the 5 percent significance level. Sample period: January 2,
1975 to December 31, 2010.
Index
S&P 500
DOW JONES
NASDAQ
DAX
BM(9050)
Rt
0.477
0.471
0.533
0.521
0.490
t-value
-6.55*
-8.12*
-5.49*
-4.94*
-0.76*
Rt2
0.674
0.643
0.814
0.797
0.495
t-value
15.11*
16.40*
30.53*
26.85*
-0.22-
|Rt |
0.813
0.802
0.858
0.840
0.495
t-value
*21.81*
*22.67*
*23.18*
*22.54*
*.-0.22*
60
days
20
days
10
days
5
days
LRcc
LRind
V aR
α
b
Size
LRuc
LRcc
LRind
V aR
α
b
Size
LRuc
LRcc
LRind
V aR
α
b
Size
LRuc
LRcc
LRind
V aR
α
b
Size
LRuc
M0
7.12
0.57
1760
3.93*
[0.048]
0.11
[0.735]
4.04
[0.133]
9.93
1.48
880
1.77
[0.184]
0.39
[0.532]
2.16
[0.340]
14.25
2.05
440
3.73
[0.053]
0.38
[0.540]
4.11
[0.128]
24.24
2.05
146
1.26
[0.262]
0.13
[0.723]
1.39
[0.501]
M1
7.42
1.14
1760
0.32
[0.574]
0.46
[0.498]
0.78
[0.678]
10.36
1.48
880
1.77
[0.184]
1.79
[0.181]
3.56
[0.169]
14.17
2.73
440
9.01**
[0.003]
0.67
[0.412]
9.68**
[0.008]
24.13
2.05
146
1.26
[0.262]
0.13
[0.723]
1.39
[0.501]
DAX
M2
7.28
0.91
1760
0.15
[0.697]
0.29
[0.588]
0.44
[0.801]
10.30
1.02
880
0.00
[0.946]
0.19
[0.666]
0.19
[0.909]
14.56
1.82
440
2.40
[0.122]
0.30
[0.586]
2.70
[0.260]
25.29
1.37
146
0.18
[0.671]
0.06
[0.814]
0.24
[0.889]
M3
7.67
0.68
1760
2.03
[0.155]
0.16
[0.685]
2.19
[0.334]
10.78
0.91
880
0.08
[0.783]
0.15
[0.702]
0.23
[0.895]
15.05
1.59
440
1.32
[0.251]
0.23
[0.634]
1.55
[0.463]
25.73
1.37
146
0.18
[0.671]
0.06
[0.814]
0.24
[0.889]
M0
5.79
1.13
1767
0.30
[0.585]
1.49
[0.223]
1.79
[0.410]
8.02
0.91
883
0.08
[0.775]
0.15
[0.702]
0.23
[0.892]
11.29
1.36
441
0.52
[0.471]
0.17
[0.684]
0.69
[0.710]
18.73
2.04
147
1.24
[0.266]
0.13
[0.724]
1.37
[0.506]
M1
6.12
1.36
1767
2.06
[0.151]
0.66
[0.416]
2.72
[0.257]
8.38
1.25
883
0.50
[0.480]
0.28
[0.598]
0.78
[0.678]
11.76
2.72
441
8.98**
[0.003]
0.67
[0.413]
9.65**
[0.008]
18.41
3.40
147
5.27*
[0.022]
0.35
[0.553]
5.62
[0.060]
DOW
M2
6.00
1.08
1767
0.10
[0.753]
0.41
[0.520]
0.51
[0.774]
8.51
0.57
883
1.99
[0.158]
0.06
[0.811]
2.05
[0.359]
11.87
0.91
441
0.04
[0.842]
0.07
[0.787]
0.11
[0.945]
20.53
0.68
147
0.17
[0.679]
0.01
[0.907]
0.18
[0.912]
M3
6.30
1.02
1767
0.01
[0.937]
0.37
[0.543]
0.38
[0.828]
8.81
0.45
883
3.35
[0.067]
0.04
[0.849]
3.39
[0.184]
12.41
0.91
441
0.04
[0.842]
0.07
[0.787]
0.11
[0.945]
20.96
0.68
147
0.17
[0.679]
0.01
[0.907]
0.18
[0.912]
Results of 1%-VaR predictions for holding periods of 5, 10, 20 and 60 days. V aR denotes average sample VaR. LR-statistics are as defined
in Appendix B, where * and ** denote rejection of the null at the 5 and 1 percent significance levels, respectively (corresponding p-values in
parenthesis). VaR forecasts are based on a moving 250 trading day window. The first forecast date of the DOW JONES is December 30, 1975.
The first forecast date of the DAX is January 7, 1976. Sample period: January 2, 1975 to December 31, 2010.
Table 3: VaR Forecasts: DAX and DOW
60
days
20
days
10
days
5
days
LRcc
LRind
V aR
α
b
Size
LRuc
LRcc
LRind
V aR
α
b
Size
LRuc
LRcc
LRind
V aR
α
b
Size
LRuc
LRcc
LRind
V aR
α
b
Size
LRuc
M0
6.82
1.81
1767
9.47**
[0.002]
0.26
[0.609]
9.73**
[0.008]
9.71
1.59
883
2.60
[0.107]
1.56
[0.214]
4.16
[0.126]
14.10
2.72
441
8.98**
[0.003]
0.67
[0.413]
9.65**
[0.008]
24.09
2.04
147
1.24
[0.266]
0.13
[0.724]
1.37
[0.506]
NASDAQ
M1
M2
7.83
6.94
0.85
1.47
1767
1767
0.43
3.46
[0.512]
[0.063]
0.26
3.59
[0.612]
[0.058]
0.69
7.05*
[0.710]
[0.029]
11.30
9.83
0.79
1.25
883
883
0.41
0.50
[0.521]
[0.480]
0.11
0.28
[0.738]
[0.598]
0.52
0.78
[0.769]
[0.678]
16.21
13.92
2.72
2.04
441
441
8.98**
3.71
[0.003]
[0.054]
0.67
0.38
[0.413]
[0.540]
9.65**
4.09
[0.008]
[0.130]
29.27
24.28
2.72
2.04
147
147
2.99
1.24
[0.084]
[0.266]
0.22
0.13
[0.636]
[0.724]
3.21
1.37
[0.200]
[0.506]
M3
7.36
1.30
1767
1.48
[0.223]
1.06
[0.304]
2.54
[0.281]
10.32
1.13
883
0.15
[0.698]
0.23
[0.632]
0.38
[0.827]
14.38
2.04
441
3.71
[0.054]
0.38
[0.540]
4.09
[0.130]
24.85
2.04
147
1.24
[0.266]
0.13
[0.724]
1.37
[0.506]
M0
5.75
1.19
1767
0.60
[0.439]
1.33
[0.249]
1.93
[0.381]
7.96
0.91
883
0.08
[0.775]
0.15
[0.702]
0.23
[0.892]
11.25
1.59
441
1.30
[0.254]
0.23
[0.635]
1.53
[0.465]
18.88
2.04
147
1.24
[0.266]
0.13
[0.724]
1.37
[0.506]
S&P 500
M1
M2
6.10
5.98
1.25
0.96
1767
1767
1.00
0.03
[0.318]
[0.874]
0.56
0.33
[0.456]
[0.565]
1.56
0.36
[0.460]
[0.837]
8.47
8.48
1.36
0.68
883
883
1.03
1.03
[0.309]
[0.310]
0.33
0.08
[0.565]
[0.775]
1.36
1.11
[0.506]
[0.573]
11.69
11.92
2.49
1.13
441
441
7.03**
0.08
[0.008]
[0.782]
0.56
0.11
[0.453]
[0.735]
7.59*
0.19
[0.022]
[0.909]
18.67
20.54
2.72
1.36
147
147
2.99
0.17
[0.084]
[0.677]
0.22
0.06
[0.636]
[0.814]
3.21
0.23
[0.200]
[0.892]
M3
6.32
0.85
1767
0.43
[0.514]
0.26
[0.612]
0.69
[0.711]
8.90
0.57
883
1.99
[0.158]
0.06
[0.811]
2.05
[0.359]
12.41
1.13
441
0.08
[0.782]
0.11
[0.735]
0.19
[0.909]
21.05
0.68
147
0.17
[0.679]
0.01
[0.907]
0.18
[0.912]
Results of 1%-VaR predictions for holding periods of 5, 10, 20 and 60 days. V aR denotes average sample VaR. LR-statistics are as defined
in Appendix B, where * and ** denote rejection of the null at the 5 and 1 percent significance levels, respectively (corresponding p-values
in parenthesis). VaR forecasts are based on a moving 250 trading day window. The first forecast date of the S&P 500 and the NASDAQ is
December 30, 1975. Sample period: January 2, 1975 to December 31, 2010.
Table 4: VaR Forecasts: NASDAQ and S&P 500
60
days
20
days
10
days
5
days
LRcc
LRind
V aR
α
b
Size
LRuc
LRcc
LRind
V aR
α
b
Size
LRuc
LRcc
LRind
V aR
α
b
Size
LRuc
LRcc
LRind
V aR
α
b
Size
LRuc
M0
7.12
0.57
1760
3.92*
[0.048]
0.11
[0.735]
4.03
[0.133]
9.93
1.48
880
1.78
[0.183]
0.39
[0.532]
2.17
[0.339]
14.26
2.05
440
3.75
[0.053]
0.38
[0.539]
4.13
[0.127]
24.24
2.05
146
1.26
[0.262]
0.13
[0.723]
1.39
[0.501]
M1
7.12
1.31
1760
1.53
[0.216]
0.61
[0.435]
2.14
[0.343]
9.93
1.59
880
2.64
[0.104]
1.54
[0.215]
4.18
[0.124]
14.26
2.73
440
9.05**
[0.003]
0.67
[0.412]
9.72**
[0.008]
24.24
2.05
146
1.26
[0.262]
0.13
[0.723]
1.39
[0.501]
DAX
M2
7.12
0.97
1760
0.02
[0.887]
0.33
[0.565]
0.35
[0.839]
9.93
1.14
880
0.16
[0.688]
0.23
[0.631]
0.39
[0.822]
14.26
2.05
440
3.75
[0.053]
0.38
[0.539]
4.13
[0.127]
24.24
1.37
146
0.18
[0.671]
0.06
[0.814]
0.24
[0.889]
M3
7.12
0.97
1760
0.02
[0.887]
2.03
[0.155]
2.05
[0.340]
9.93
1.48
880
1.78
[0.183]
0.39
[0.532]
2.17
[0.339]
14.26
2.05
440
3.75
[0.053]
0.38
[0.539]
4.13
[0.127]
24.24
1.36
146
0.18
[0.671]
0.06
[0.814]
0.24
[0.889]
M0
5.79
1.13
1767
0.30
[0.584]
1.49
[0.223]
1.79
[0.410]
8.02
0.91
883
0.08
[0.775]
0.15
[0.702]
0.23
[0.892]
11.29
1.36
441
0.52
[0.471]
0.17
[0.684]
0.69
[0.710]
18.73
2.04
147
1.24
[0.266]
0.13
[0.724]
1.37
[0.506]
DOW
M1
5.79
1.64
1767
6.16*
[0.013]
0.97
[0.325]
7.13*
[0.028]
8.02
1.59
883
2.60
[0.107]
0.45
[0.502]
3.05
[0.218]
11.29
2.95
441
11.10**
[0.001]
0.79
[0.374]
11.89**
[0.003]
18.73
2.72
147
2.99
[0.084]
0.22
[0.636]
3.21
[0.200]
M2
5.79
1.08
1767
0.10
[0.752]
0.41
[0.520]
0.51
[0.774]
8.02
0.79
883
0.41
[0.521]
0.11
[0.738]
0.52
[0.769]
11.29
1.59
441
1.30
[0.254]
0.23
[0.635]
1.53
[0.465]
18.73
1.36
147
0.17
[0.677]
0.06
[0.814]
0.23
[0.892]
M3
5.79
1.30
1767
1.49
[0.222]
0.61
[0.436]
2.10
[0.351]
8.02
1.25
883
0.50
[0.480]
0.28
[0.598]
0.78
[0.678]
11.29
1.36
441
0.52
[0.471]
0.17
[0.684]
0.69
[0.710]
18.73
0.68
147
0.17
[0.679]
0.01
[0.907]
0.18
[0.912]
Results of 1%-VaR predictions for holding periods of 5, 10, 20 and 60 days. V aR denotes average sample VaR. LR-statistics are as defined
in Appendix B, where * and ** denote rejection of the null at the 5 and 1 percent significance levels, respectively (corresponding p-values in
parenthesis). VaR forecasts are based on a moving 250 trading day window. The first forecast date of the DOW JONES is December 30, 1975.
The first forecast date of the DAX is January 7, 1976. Sample period: January 2, 1975 to December 31, 2010.
Table 5: Normalized VaR Forecasts: DAX and DOW
60
days
20
days
10
days
5
days
LRcc
LRind
V aR
α
b
Size
LRuc
LRcc
LRind
V aR
α
b
Size
LRuc
LRcc
LRind
V aR
α
b
Size
LRuc
LRcc
LRind
V aR
α
b
Size
LRuc
M0
6.82
1.81
1767
9.48**
[0.002]
0.26
[0.609]
9.74**
[0.008]
9.71
1.59
883
2.60
[0.107]
1.56
[0.214]
4.16
[0.126]
14.10
2.72
441
8.98**
[0.003]
0.67
[0.413]
9.65**
[0.008]
24.09
2.04
147
1.24
[0.266]
0.13
[0.724]
1.37
[0.506]
NASDAQ
M1
M2
6.82
6.82
1.64*
1.47
1767
1767
6.16
3.47
[0.013]
[0.062]
0.46
3.59
[0.500]
[0.058]
6.62*
7.06*
[0.037]
[0.029]
9.71
9.71
1.93
1.36
883
883
6.01*
1.03
[0.014]
[0.309]
0.67
0.33
[0.414]
[0.565]
6.68*
1.36
[0.036]
[0.506]
14.10
14.10
3.63
2.04
441
441
18.37**
3.71
[0.000]
[0.054]
1.21
0.38
[0.272]
[0.540]
19.58**
4.09
[0.000]
[0.130]
24.09
24.09
2.72
2.04
147
147
2.99
1.24
[0.084]
[0.266]
0.22
0.13
[0.636]
[0.724]
3.21
1.37
[0.200]
[0.506]
M3
6.82
1.47
1767
3.47
[0.062]
3.59
[0.058]
7.06*
[0.029]
9.71
1.47
883
1.74
[0.188]
1.80
[0.180]
3.54
[0.171]
14.10
2.04
441
3.71
[0.054]
0.38
[0.540]
4.09
[0.130]
24.09
2.04
147
1.24
[0.266]
0.13
[0.724]
1.37
[0.506]
M0
5.75
1.19
1767
0.60
[0.438]
1.33
[0.249]
1.93
[0.381]
7.96
0.91
883
0.08
[0.775]
0.15
[0.702]
0.23
[0.892]
11.25
1.59
441
1.30
[0.254]
0.23
[0.635]
1.53
[0.465]
18.88
2.04
147
1.24
[0.266]
0.13
[0.724]
1.37
[0.506]
S&P 500
M1
M2
5.75
5.75
1.59
1.08
1767
1767
5.19*
0.10
[0.023]
[0.752]
0.90
0.41
[0.342]
[0.520]
6.09*
0.51
[0.048]
[0.774]
7.96
7.96
1.47
0.79
883
883
1.74
0.41
[0.188]
[0.521]
0.39
0.11
[0.533]
[0.738]
2.13
0.52
[0.356]
[0.769]
11.25
11.25
2.72
1.36
441
441
8.97**
0.52
[0.003]
[0.471]
0.67
0.17
[0.413]
[0.684]
9.64**
0.69
[0.008]
[0.710]
18.88
18.88
2.72
1.32
147
147
2.99
0.14
[0.084]
[0.709]
0.22
0.05
[0.363]
[0.817]
3.21
0.19
[0.200]
[0.908]
M3
5.75
1.36
1767
2.07
[0.151]
0.66
[0.416]
2.73
[0.256]
7.96
1.13
883
0.15
[0.698]
0.23
[0.632]
0.38
[0.827]
11.25
1.59
441
1.30
[0.254]
0.23
[0.635]
1.53
[0.465]
18.88
1.32
147
0.14
[0.709]
0.05
[0.817]
0.19
[0.908]
Results of 1%-VaR predictions for holding periods of 5, 10, 20 and 60 days. V aR denotes average sample VaR. LR-statistics are as defined
in Appendix B, where * and ** denote rejection of the null at the 5 and 1 percent significance levels, respectively (corresponding p-values
in parenthesis). VaR forecasts are based on a moving 250 trading day window. The first forecast date of the S&P 500 and the NASDAQ is
December 30, 1975. Sample period: January 2, 1975 to December 31, 2010.
Table 6: Normalized VaR Forecasts: NASDAQ and S&P 500
Table 7: VaR Forecasts: Model M0 with Cornish-Fisher Quantiles
The table displays results of 1%-VaR prediction for holding periods of 5, 10, 20 and 60 days
of model M0 equipped with Cornish-Fisher quantiles. V aR denotes average sample VaR. LRstatistics are as defined in Appendix B, where * and ** denotes rejection of the null at the
5 and 1 percent level, respectively (corresponding p-values are given in parenthesis). VaR
forecasts are based on a moving 250 trading day window. Sample period: January 2, 1975 to
December 31, 2010.
V aR
α
b
LRuc
5
days
LRind
LRcc
V aR
α
b
LRuc
10
days
LRind
LRcc
V aR
α
b
LRuc
20
days
LRind
LRcc
V aR
α
b
LRuc
60
days
LRind
LRcc
DAX
6.15
0.80
0.80
[0.371]
0.22
[0.636]
1.02
[0.600]
8.57
2.05
7.46**
[0.006]
3.82
[0.051]
11.28**
[0.004]
12.31
2.95
11.14**
[0.001]
0.79
[0.374]
11.93**
[0.003]
20.92
2.05
1.26
[0.262]
0.13
[0.723]
1.39
[0.501]
DOW
5.23
1.64
6.16*
[0.013]
0.46
[0.500]
6.60*
[0.037]
7.25
2.27
10.51**
[0.001]
0.52
[0.470]
11.03**
[0.004]
10.18
2.49
7.03**
[0.008]
0.56
[0.453]
7.59*
[0.022]
16.95
2.72
2.99
[0.084]
0.22
[0.636]
3.21
[0.200]
NASDAQ
5.86
3.06
48.79**
[0.000]
7.54**
[0.006]
56.33**
[0.000]
8.33
3.40
31.56**
[0.000]
0.00
[0.984]
31.56**
[0.000]
12.09
3.62
18.37**
[0.000]
1.21
[0.272]
19.57**
[0.000]
20.90
5.44
14.34**
[0.000]
0.92
[0.337]
15.26**
[0.000]
S&P 500
5.22
1.64
6.16*
[0.013]
0.46
[0.500]
6.62*
[0.037]
7.22
1.92
6.01*
[0.014]
0.67
[0.414]
6.68*
[0.036]
10.19
2.72
8.98**
[0.003]
0.67
[0.413]
9.65**
[0.008]
17.15
2.04
1.24
[0.266]
0.13
[0.724]
1.37
[0.506]
E
Figures
(a)
(b)
Figure 1: Weight function g for given H and ρ(·) under varying lags τ . The base
case autocorrelation coefficients and the Hurst exponent are estimated from absolute daily
logarithmic returns of the S&P 500 index during the period January 2, 1975 to December 31,
2010. Panel (a) shows g for different Hurst exponents H, (0.813, 0.913 and 0.713). Panel (b)
shows g for different levels of the autocorrelation coefficients ρ(τ ), (sample autocorrelations
and plus/minus 10 percent uniform in levels).
Figure 2: 60-day ahead VaR forecasts (6) for Drost-Nijman M0 (dashed line) and model M2
(solid line). All out-of-sample forecasts are based on a moving window of 250 trading days.
The dots represent the non-overlapping 60-day index return of the respective market index.
The forecasting period of the S&P 500, the DOW JONES and the NASDAQ is December 30,
1975 to December 31, 2010. The forecasting period of the DAX is January 7, 1976 to December
30, 2010.
Figure 3: 60-day ahead VaR forecasts (6) for long memory scaling M1 (dashed line) and model
M3 (solid line). All out-of-sample forecasts are based on a moving window of 250 trading days.
The dots represent the non-overlapping 60-day index return of the respective market index.
The forecasting period of the S&P 500, the DOW JONES and the NASDAQ is December 30,
1975 to December 31, 2010. The forecasting period of the DAX is January 7, 1976 to December
30, 2010.
Figure 4: 60-day ahead VaR forecasts (6) for models M0 (dotted-dashed line), M1 (solid line),
M2 (dashed line), and M3 (dotted line) from Q3 2007 to Q4 2009. The dots represent the nonoverlapping 60-day index return of the respective market index. All out-of-sample forecasts
are based on a moving window of 250 trading days.
(a) March 27, 2001
(b) September 25, 1996
(c) March 27, 2001
(d) September 25, 1996
Figure 5: Out-of-sample S&P 500 volatility forecasts for time horizons h ∈ [1, 60] based on
the preceding 250 trading days on March 27, 2001 (panels (a), (c)) and on September 25,
1996 (panels (b), (d)). Multiple-period volatility σt,h forecasts are given in panels (a) and
(b), τ -step ahead spot volatility σt+τ −1 forecasts are given in panel (c) and (d), respectively.
The dashed line represents Model M3 predictions. The dotted line represents the standard
GJR-GARCH(1,1)
prediction. The solid line is the unconditional volatility, which is scaled
√
with h in panels (a) and (b).