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The Journal of Risk FinanceThe Effect of Asymmetries on Stock Index Return Value-at-Risk Estimates

CHRIS BROOKS GITA PERSAND

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The Effect of Asymmetries

on Stock Index Return

Value-at-Risk Estimates

C H R I S B R O O K S A N D G I T A P E R S A N D

T

here is widespread ag reem ent in the

relevant literature that equity return

volatility rises more following neg

ative than positive shocks. Pagan and

Schwert [1990], Nelson [1991], Campb ell and

Hentschel [1992] , Engle and N g [1993] ,

Glosten e t al. [1993], and H en ry [1998], for

example, all demonstrate the existence of asym

me tric effects in stock index return s. The re are

broadly two potential explanations for such

asymmetry in variance that have been suggested

in this litera ture . T he first is the levera ge

effect, usually associated w ith Black [1976] and

Chris tie [1982], which posits that if equity

values fall, the firm's debt-to-equity ratio will

rise,

thus inducing equityholders to perceive

the stream of future income accruing to their

positions as being relatively more risky than

previously. A second possible explanation for

observed asymm etries is term ed the volatility

feedback hypothesis. Assuming constant cash

flows, if expected returns increase when stock

price volati l i ty increases, then s tock prices

should fall when volatility rises.

An entirely separate l ine of academic

enquiry has centered around the determina

tion of wh at is term ed an insti tutions value

at risk (VaR). Va R is a calculation of the likely

losses that might occur from changes in the

mark et prices of a particular securities or po rt

fo l io pos i t ion . The minimum capi ta l r i sk

requirement (M C R R ) or position r isk require

me nt (P RR ) is then def ined as the minim um

amount of capital required to absorb all but a

prespecif ied propo r t ion of expec ted fu ture

losses. Dim son and M arsh [1995, 1997] argue

that portfolio-based approaches to determining

P R R s are mo re eff icient than a l ternat ive

approaches, since the former allow fully and

directly for the risk-reduction benefits from

having a diversified book.

Th e nu mb er of s tudies in existence that

seek to determine appropr ia te methods for

calcu la t ing and evaluat ing value-a t - r isk

methodologies has increased substantially in

the past five

years.

Jackson et al. [1998] assess

the empirical performan ce o f various mode ls

for value at risk using historical returns from

the actual portfolio of a large investment ban k.

Alexander and Leigh [1997] offer an analysis

of the re la t ive per fo rma nce of equal ly

we igh ted , exponen t i a l ly we igh ted mov ing

ave rage (EWMA) , and GAR C H mode l f o r e

casts of volatility, evaluated using traditional

s tatis t ical and operational adequacy criteria.

Th e G A R C H mod el is found to be preferable

to E W M A in t e rms o f min im iz ing the

number of exceedances in a backtest, although

the s imple unw eigh ted average is supe rior to

both. The issue of sample length is discussed

in Hoppe [1998], who argues that, for all asset

classes and holding periods tested, the use of

(unweighted) shorter samples of data yields

more accurate VaRs than longer runs. Kupiec

[1995] , on the other hand, argues that long

samples are always required in order to eval

uate the effectiveness of those estimates using

exception tests . Berkowitz [2001] also exam-

CHRIS

BROOKS

is professor of finance at the

ISMA

Centre,

University

of

Reading,

in the UK.

[email protected]

GITA PERSAND

is a

lecturer

in

finance

at the

Department

o f

Economics,

University

of Bristol, in the UK .

[email protected]

W I N T E R 2003

THE J OUR NAL OF RISK FINANCE 2 9

mailto:[email protected]:[email protected]:[email protected]:[email protected]

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ines the frequency and the ma gnitu de of exc ept ion s

(days w he n th e VaR is insufficient to cover actual trad ing

losses), although it is in the spirit of VaR modeling as

per the Bas le recomme ndat ions that on ly the num ber of

exceptions is considered and not their sizes. Other recent

studies that compare different models for computing VaR

include those o f Brooks and Persand [2000a, 2000b] ,

Brooks et al . [2000], Longin [2000], Vlaar [2000], and

Lopez and Walter [2001]. Mu ch research has relied upo n

an assumption of normally dis tr ibuted returns, while in

practice almost all asset return distributions are fat tailed.

Two studies that have proposed methods to deal with

leptokurtosis are Huisman et al . [1998] and Hull and

White [1998] .

This article extends recent research and considers th e

effect of any asymmetries that may be present in the data

on the evaluation and accuracy of value-at-risk estimates.

A number of recent s tudies have examined the use of

extreme value dis tr ibutions for computing value-at-r isk

estimates. Such models can explicitly allow for lepto kurtic

return distributions, but can also account for asymmetries

in the unconditional distributions by fitting separate models

for the upper and lower tails. McNeil and Frey [2000], for

example, propose a new method for estimating VaR based

on a combinat ion of G A R C H m odel ing wi th ex treme

value dis tr ibutions. Other s tudies employing EVT have

included Longin [2000], Neftci [2000], and Brooks et al.

[2002].

However, since the extreme value theory (EVT)

approach to com puting position risk requirements has been

considered elsewhere, we do not discuss it further here.

Instead, we examine a number of alternative models that

may be used to capture asymmetries in asset return distri

butions. Our analysis is conducted in the context of the

stock markets of five Southeast Asian economies, and the

S&P 500 index, which is employed as a benchmark. We

compare the performance of various models, some of which

do, and some of which do not, allow for asymmetries. The

models are evaluated in the context of the Basle Com

mittee rules , and their proposed method for determining

whether models for the calculation of VaR are adequate

using a holdout sample. To anticipate our main finding,

we conclude that allowing for asymmetries can lead to

improved V aR estimates, and that a simple asymmetric risk

measure proves to be the most stable and reliable method

for calculating VaR.

The remainder of this article is organized into six

sections. Section 2 presents and describes the data em ployed,

while the various symmetric and asymmetric volati l i ty

models uti l ized are displayed in section 3. Section 4

describes the methodology employed for the calculation of

value at risk, while the value-at-risk estimates from the

various models are presented and evaluated in section 5.

Finally, section 6 offers some concluding remarks.

D A T A

Th e analysis unde rtaken in this article is based on daily

closing prices of five Southeast Asian stock market indices:

the Ha ng Seng Price Index, N ikkei 225 Stock Average Price

Index, Singapore Straits Times Price Index, South Korea

SE Composite Price Index, and Bangkok Book Club Price

Index. We also employ the U.S. S&P 500 composite index

E X H I B I T 1

Sum mary Statist ics: Stock Inde x Returns for the Period January 1, 1985-April 29, 1999

(3,737 Observations)

Ho ng Ko ng

Japan

Sing a po re

South Korea

Thai land

Portfolio

S P 500

M ea n

0.00077

0.00019

0.00037

0.00056

0.00042

0.00046

0.00024

Variance

0.00030

0.00018

0.00020

0.00027

0.00030

0.00009

0.00010

S k e w n e s s

- 2 . 2 7 6 1 3 *

0.10919*

-1 .18975*

0.42596*

0.15243*

- 0 . 6 2 7 9 2 *

- 3 . 6 2 4 9 3 *

Kurtosis

51 .96710*

10.03701*

44 .72603*

5.55209*

9.85702*

14.59893*

80.59497*

Normali ty

Test**

422368*

15643*

311360*

4897*

15095*

33324*

1019595*

* denotes significance

at the 1% level.

** Bera-Jarque Normality Test.

Th e

portfolio

is an equally

weighted combination

of the five Southeast Asian market returns.

3 0 TH E EFFECT OF ASYMMETRIES ON STOCK INDEX RET UR N VALU E-AT-RISK ESTIMATES

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returns as a bench mark for comparison. T he data, obtained

from Primark Datastream, run from January 1, 1985, to

April 29, 1999, giving a total of 3,737 observations. All sub

sequent analysis is performed on the daily log returns, with

the sum ma ry statistics being given in E xhib it 1. All six returns

series exhibit the standard property of asset return data that

they have fat-tailed distribution s as indicated by the sig

nificant coefficient of excess kurtosis. These characteristics

are also shown by the highly significant Jarque-Bera nor

mality test statistics. All series are also are, eit her significantly

skewed to the left (U.S., Hong Kong, and Singapore) or to

the right (Japan, South Korea, and Thailand). Unconditional

skewness is an arguably important but neglected feature of

many asset return series (see, for example, Harvey and Sid-

dique [1999]), which if ignored could lead to misspecified

risk manage men t models. An illustration of where skewness

may arise is given by the view that equity price movem ents

go up the stairs and dow n in the elevator, imply ing that

upward movements are smaller but more frequent than

downward movements,

ceteris

paribus. T his being the case, a

symmetrical measure such as variance will no longer b e an

appropriate measure of the total risk of an asset.

We also form, and perform subsequent analysis on,

an equally weighte d portfolio consisting of the five So uth

east Asian market indices listed above. The advantage of

diversification is clearly recognized in this case: the vari

ance of the portfolio is only around one-half of the vari

ance of even the least volatile of the individual com pon ent

series—see the penultimate colum n o f Exhibit 1.

On

the

other hand, the distr ibution of portfolio returns is st i l l

skewed (to the left) and is leptokurtic . T he skewness in all

five series of returns data, and in the portfolio returns (albeit

in different directions), is one manifestation of an asym

metry— in othe r words, this is prima acie evidence that the

unconditional distribution of returns is not symmetric. We

shall retu rn to this feature of the data in the following sec

tion in the context of value-at-risk estimation.

S Y M M E T R I C A N D A S Y M M E T R I C

V O L A T IL I TY M O D E L S

There exist a number of different methods for deter

mining an insti tution 's value at r isk. The most popular

me tho ds can be usefully classified as be ing eit her p ara

metric or non-parametric. In the former category comes

the volatilities and correlations app roach popularize d by

J.P. M orga n [1996]; this me tho d involves the estimation of

a volatility parameter, and, conditional upon an assump

tion of normality, the volatility estimate is multiplied by the

appropriate critical value from the no rma l distribution and

by the value of the asset or portfolio, to o btain an estimate

of the VaR in money terms. There are broadly two ways

that VaR could be ca lcu la ted under the parametr ic

umb rella. First , one could estimate a mo del for ret urn

volatility, and, conditioned upon this, employ the appro

priate critical value from a relevant distribution. Second,

one could, rather than estimating a conditional parametric

model, employ the unconditional distr ibution of returns,

fitting one of many other available parametric distribu

tions. Press [1967] and Kon [1984], for example, both sug

gest the use of a mixture of norma l distributions to explain

the observed kurtosis and skewness in the distribution of

stock returns. Madan and Seneta [1990], on the o ther hand,

propose an entirely new class of models, known as variance-

gamma processes, as a model for stock prices.

Non-parametr ic approaches to the ca lcu la t ion of

value at risk can take many forms, bu t the simplest is derived

from an estimate of the unconditional density of the sample

of returns. So, for example, the VaR expressed as a pro

por tion of the initial value of the asset or portfolio, wh ich

is required to cover 99% of expected losses, would simply

be given by the absolute value of the first percen tile of th e

return distribution. Jorio n [1995] shows that the param etric

approach to VaR can be preferable, even in situations where

the returns are not normally distributed. In any case, non-

parametric approaches are beyond the scope of this article

and are thus not considered further (but see, for example,

Jorion [1996] or Dowd [1998] for extended descriptions

of the various methods available under the non-parametric

umbrella). Under the parametric approach, the normal dis

tribution is employed almost universally, and the VaR (in

money terms) for model i is given by

wh er e α ( N

5%

) is the relevant value from the standard

normal tables, a. is the volatility estimate, and V is the

value of the portfolio. VaR is also commonly expressed

as a proportion of the asset or portfolio value, and this

convention will also be adopted in this study. Once a view

is taken that a parametric approach will be em ployed for

the calculation of VaR, the issue simply boils down to

estimation of the volatility parameter that describes the

asset or portfolio;

1

we now present a variety of models

which can be used for estimating and modeling σ

i

.

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T h e Un co nd i t io na l Va r ia nce

a n d t h e E W M A M o d e l

An estimate of the unconditional variance provides

the simplest method for forecasting future volatility, also

known as an equally weighted moving average model, as

forecasts are constructed by simply calculating the standard

deviation from the most recent three years of historical

data. This estimated historical standard deviation is then

the prediction of the standard deviation over the evalua

tion period. The sample period is then rolled forward by

60 observations (one trading quarter), σ

i

re-estimated, and

so on. To ensure consistency and a fair comparison, we

also use the same framework (a three-year rolling sample

updated every 60 observations) for the estimation of the

parameters for all other approaches employed.

We similar ly employ the exponent ia l ly weighted

mo ving average (EW M A) m ode l , popu lar ized by J .P.

Morgan, and which has been found to produce accurate

volatil i ty forecasts. U nd er the E W M A specif ication, t he

fit ted variance from the model, which becomes the fore

cast for multi-step ahead forecasts, is an expo nentially

declining function of previous squared values. The decay

factor, λ , is set to 0.94 following the

J.P.

Mo r g an r ec

ommendation and previous research in this area.

T he GARCH( 1 ,1 ) M o de l

Recent research (see Alexander and Leigh [1997],

for example) has suggested tha t GARCH-type models

may be preferable for modeling volatility in a risk man

agement context, and hence our conditional model anal

y sis co m men ces w i th th e p la in v ani ll a GA R C H( 1 ,1 )

model, given as follows:

wh er e x

t

= Log(P

t

/P

t-1

) wi th P

t

being the value of the

stock index at time t and ε

t

~ N(0, h

t

) .

2 , 3

The volatility

forecasts a re const ruc ted by i te ra t ing the condi t ional

expectations operator in the usual fashion.

To determine whether an asymmetr ic condi t ional

volatility model is necessary and to ensure that the model

does not und erpre dict or overpredict volatil i ty in periods

where there are large innovations in returns, the stan

dardized residuals from the GARCH(1,1) specif ication

are examined for sign and size bias, which is carried out

using the tests of Engle a nd N g [1993]. T he test for asym

metry in return volatility is given as follows:

E X H I B I T 2

Engle and

N g

[1993] Test for the GAR CH(1,1) Mo del: Stock Index Returns

for the P eriod Janu ary 1, 1985-Ap ril 29, 1999

(3,737 Observations)

Hong Kong

Japan

Sing a po re

South Korea

Thailand

Portfolio

S P 500

Φ

0

0.913

(0.138)**

0 937

(0.112)**

0 705

(0 .145)**

0 979

(0 .075)**

0 846

(0.113)**

1.085

(0.110)**

0 960

(0.115)**

Φ

1

0.189

(0.176)

-0.042

(0.148)

0.159

(0.188)

0 082

(0.100)

0 322

(0.108)**

-0.168

(0.151)

0 037

(0.151)

Φ

2

-0.128

(0.103)

-0.304

(0.095)**

-0.429

(0.119)**

0.061

(0.072)

0 085

(0.100)

-0.173

(0.101)

-0.222

(0.093)*

Φ

3

-0.088

(0.142)

-0.151

(0.117)

0.149

(0.146)

-0.037

(0.070)

-0.018

(0.108)

-0.179

(0.115)

-0.178

(0.122)

χ

2

3)

10.141*

19.723**

20.342**

1.991

10.222*

10.992*

17.590**

Standard

errors

are in parentheses; * and

**

indic te signific nce

at the 5% and 1%

levels respectively.

Th e

portfolio

is an equally

weighted combin tion

of

the

five Southeast Asian market returns.

3 2 TH E EFFECT OF ASYMMETRIES ON STOCK INDEX RE TU RN VALU E-AT-RISK ESTIMATES

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wh er e

ω

t

is a

wh i t e n o i se d i s tu r b an ce t e r m .

Z

t-1

is

defined as an indicator dummy that takes the value 1 if

ε

t

/√ h

t

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wh er e C

0

, A

1

, and B

1

are parameter matr ices to be esti

mated , ε

t-1

is a vector of lagged errors, and

The BEKK parameterization requires estimation of

only 55 free parameters in the conditional v ariance-cov ari-

ance s t ruc ture (compared wi th 255 in the comple te ly

unrestr icted

vec

model) , and guarantees H

t

positive defi

nite. Simple modifications to (6) can be made to allow for

asymmetr ies under the GJR and EG A R CH formula tions.

The modif ied model inc lud ing asymmetry te rms in a

quadratic form could be written

where ξ

j,t

= min{ε

t

, 0} and D

1

is a 5

× 5 parameter matrix

including all of the asymmetry coefficients. In all cases,

the estimated asymmetry terms are significant at the 5%

level or better.

T h e S e m i - V a r i a n c e

Modern Portfolio Theory (MPT), which underlies

many theoretical models in f inance, carries with i t the

explicit assu mp tion that asset retu rns are joi ntl y elliptically

dis tr ibuted. Such an assumption m ay be considered u nd e

sirable since it rules ou t the possibility of asym metric retu rn

distributions. Risk in the M P T framew ork is measured in

term s of surpr ises rath er than in term s of losses or failure

to achieve an expected or benchmark re turn . Such a

description of risk does not tie in well with m ost investors'

notions of what constitutes a risk.

Another approach to estimating a volati l i ty param

eter, which may be used in the present context, and which

has been given broad theoretical consideration in an asset

allocation framew ork, is the co nce pt of the lower partial

momen t (LPM)— s ee , f o r example , C hoob ineh and

Bran t ing [1986] , Ha r low [1991] , and Grootveld and

Hallerbach [1999]. T he lower partial m om en t of order a

around

τ is defined as

wh ere F(X) is the cum ulative distr ibution function of the

return X. Setting α = 2, and τ = (I in Equation (8) above

defines the semi-variance of a random variable X with

mean μ .

Fo l lo win g J .P . Mo r g an R i sk M et r i c s an d man y

empirical studies, assuming

4

that μ = 0, an asym ptotically

unbiased and strongly consistent estimator of the semi-

varian ce for a sam ple of size T is given by (see Jos eph y and

Aczel [1993])

5

whe re T denotes the num ber of te rms for which x

t

< 0,

and the scaled semi-standard deviation which we employ

for the com pu tati on of value at risk is given by th e square

root of (9).

The square root of 9) can thus replace directly the

usual standard deviation in (1). The scaled semi-standard

deviation ( the square root of (9)) can account for the

uncondi t ional skewness in the d is t r ibu t ion of re turns .

Expressed in this way, we can still employ the standard

normal critical value since we now implicitly assume that

x

t

< 0 follows a half-n orm al d istribu tion. If desired, the

formula in (9) could be trivially modified to estimate the

upper scaled semi-standard deviation, which would be of

interest for calculating the VaR of a short posit ion.

M E T H O D O L O G I E S F O R C A L C U L A T I N G

A N D E V A L U A T I N G T H E V A L UE AT R I S K

Th e regulatory environment under the Basle Co m

mittee Rules requires that VaR should be calculated as

the higher of 1) the firm's previous day's value at risk m ea

sured according to the parameters given below and 2) an

average of the daily VaR measures on each of the pre

ceding 60 business days, with the latter subjected to a

multiplication factor. Value at risk is to be computed on

a daily basis over a mi nim um holding period of 10 days.

However, shorter holding periods can be used, but they

have to be scaled up to 10 days, and in general this is

achieved using the square root of t ime rule. Moreover,

VaR has to be estimated at the 99% probability level, using

daily data over a minimum length of one year (250 trading

days) , with the estimates being updated at least every


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quarter. T he rules do leave the ban k a broad degree of flex

ibility in ho w the VaR is actually calculated. Fo r exam ple,

the M C R R es timates can be updated more f requent ly

than quarterly, a longer run of data than one trading year

can be employed, and the BIS does not s tipulate which

model should be employed for the calculations. Changes

in any of these factors could potentially result in large

changes in the calculated M C R R , so it is imp ortant that

all candidate models for the calculation of VaR be thor

oughly evaluated.

The mul t ip l ica t ion fac tor , which has a minimum

value of

3 ,

dep ends on the regulator 's view of the quality

of the bank's risk management system, and more precisely

on the backtesting results of the models. Unsatisfactory

results might see an increase in the multiplication factor

of

3,

up to a maximum of 4. The regulator performs an

assessment of the soundness of the bank's procedure in

the fo l lowing way. Un derp redi c t io n of losses by VaR

models (that is, the days on which the bank's calculated

value at risk is insufficient to cover the actual realized

losses in i ts t rad ing book ) are term ed exce pt ion s .

Between zero and four exceptions over the previous 250

days places the bank in the Green Zone; between five

and nine, i t is in the Yellow Zo ne ; and wh en 10 or m ore

exceptions are noted, the bank is in the Re d Z one . W he n

the bank is in the Yellow Zone, one would almost cer

tainly expect the regulatory body to increase the multi

plication factor, while if the firm falls into th e R ed Zo ne ,

it is likely to be no longer permitted to use the internal

modeling approach. I t will instead be required to revert

back to the building bloc k approach, wh ich does not

include a reductio n in the M C R R for diversified books

and whi ch w ill almost certainly yield a mu ch h igher c ap

ital charge. It is thus important to the securities firm or

bank, as well as its regulators, that its risk measurement

procedures are sound.

The VaR for each individual index was estimated,

using the s imple 5% del ta-n orm al approach proposed

in the literature, i.e.,

VaR

=

Marked position of asset

×

sensitivity to price movement

×

Adverse price movement per day

(10)

Th e sensitivity to price mo vem ents is taken to be 1

since we s tudy equities wh ich are linear instruments; the

adverse price move per day is equal to (1.645 × 3 × a)

where σ is the estimated standard deviation of the asset

returns over the sample period. Note that we multiply

the VaRs by the regulatory scaling factor of 3 , and we use

the 5% one-sided normal cr it ical value. While the Basle

rules require the use of a 1% VaR , we use a 5% VaR since

the use of 1% together with the scaling factor results in a

VaR that is so large as to re nder the m odels virtually ind is

tinguishable from one another. σ is calculated on a length

of three years of data (based on the above seven men

tioned models) and the sample is rolled over after each

quarter (60 days). For our selected data sample, this leads

to 46 separa te subsamples and out-of -sample per iods

wh ich can be used to evaluate the adequacy of the valu e-

at-risk estimates.

Each model for VaR outlined above is estimated for

the six series under investigation and for the portfolio of

Southeast Asian equities. In the cases wh ere a p aram etric

mo del for the cond itional co variances is no t specified (the

variance, the semi-variance, all univariate models) , we

estimate the portfolio returns and determine the VaR for

the portfolio as a single series. Such an approach may use

fully be term ed the full valu ation approa ch, since i t

involves a calculation o f the return s, and therefore the full

value of the portfolio, for each time period. These results

for the portfolio calculated from a single column of port

fo lio re turns wi l l be d isplayed und er the un iva r ia te

mo del heading . For the mul t ivar ia te G A R C H m odels ,

we use a slightly different app roach, w hic h m ay be view ed

as a mod ified version of the volatilities and co rrela tions

method. This approach makes use of Modern Portfolio

Theory, whereby for an N-asset portfolio, the value at

risk can be calculated by using the following formula:

wh e r e Va R is the value at risk of the portfolio , α

i

are the

weights given to each of the assets in the portfolio, VaR.

are the values at risk of the individual series A and B, and

p

ij

is the estimated correlation between the returns to i

and

j

Since under a mul t ivar ia te GARCH approach , we

have a specification, and can therefore generate forecasts,

for the conditional covariances as well as the conditional

variances, we make use of these in Equation (11) above.

Thus the VaRs for the individual assets and the correla

tions are estimated using the forecasts obtained from the

m u l t i v a r i a t e G A R C H , E G A R C H , a n d G J R m o d e l s .

Then the equally weighted portfolio VaR is constructed

using (11) and labeled as M G A R C H , M GJ R, and

M E G A R C H fo r t h e m u lt iv a ri at e G A R C H , G J R , a nd

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E X H I B I T

3

Value-at-Risk Estimates: Stock Index Retu rns for the Period January 1, 1985-April 29, 1999 (3,737 Observations)

Hong Kong Japan

Singapore S. Korea

Thailand

Portfolio

S&P 500

Panel A: Symmetric Approaches

Variance

Mean

Std. Deviation

EWMA

Mean

Std. Deviation

G RCH

Mean

Std. Deviation

MG RCH

Mean

Std. Deviation

0.07411

0.01356

0.07483

0.04858

0.07317

0.03670

0.07330

0.02089

0.06019

0.01878

0.06254

0.03006

0.05534

0.02792

0.05904

0.01911

0.05165

0.01489

0.05851

0.03903

0.04908

0.02672

0.05046

0.01630

0.07002

0.02584

0.07463

0.03579

0.06793

0.04318

0.06728

0.02191

0.07486

0.01676

0.08288

0.04316

0.07152

0.04619

0.07031

0.02862

0.04005

0.00991

0.04345

0.02333

0.03208

0.04023

0.03536

0.02259

0.02010

0.00714

0.01841

0.01174

0.05273

0.03867

-

Panel B: Asymmetric Approaches

GJR

Mean

Std. Deviation

EGARCH

Mean

Std. Deviation

Semi-Variance

Mean

Std. Deviation

MGJR

Mean

Std. Deviation

MEG RCH

Mean

Std. Deviation

0.07335

0.03716

0.07399

0.05252

0.07461

0.01316

0.07397

0.02714

0.07401

.

0.04860

0.05618

0.02766

0.05923

0.02709

0.06025

0.01142

0.06004

0.02265

0.06014

0.02610

0.05062

0.02749

0.05082

0.08298

0.05166

0.01253

0.05094

0.01752

0.05100

0.02718

0.06805

0.04842

0.06953

0.05331

0.07009

0.01963

0.06818

0.02354

0.07011

0.03047

0.07289

0.04887

0.07351

0.05271

0.07500

0.01570

0.07349

0.03045

0.07414

0.03283

0.03493

0.03931

0.03823

0.07184

0.04030

0.00742

0.03696

0.02809

0.03920

0.02881

0.02025

0.00718

0.01935

0.00836

0.01800

0.00520

-

-

Note:

Model acronyms

re

as

follows:

variance denotes VaR calculated using the historical standard deviation of

returns;

EWMA denotes the VaR calculated usin

the exponentially weighted moving average

method;

MGJR and MEGARCH denote the multivariate GJR and EGARCH models respectively; semi variance denot

a VaR calculated using the scaled semi standard deviation of (9). Cell entries refer to the average and standard deviation of the VaR across the 46 out of sample wi

dows; VaR is expressed as a proportion of the initial value of the position. The portfolio is an equally weighted combination of the five Southeast Asian market returns.

EGARCH models respectively, and these results are dis

played unde r the multivariate model heading.

RESULTS OF V A R ESTIM TION

N D EV LU TION

The VaR estimates for each asset and for each model

are presented in Exhibit 3. Comparing the VaR estimates

across Southeast Asian countries, they are highest for Hong

Kong and Thailand, and lowest for the U.S., Singapore,

and the portfolio; unsurprisingly, the countries with the

highest VaR estimates were also those which had the highest

unconditional variance of returns. For most models, the

country with the least stable VaRs over time, shown by

the larger standard deviation of VaR across the 46 win

dows, is South Korea, which did not have one of the highest

average VaRs. Interestingly, the average of the VaR estimates

for the S&P is approximately half that of the portfolio of

five Southeast Asian stock indices, and about one-quarter

to one-third of that of the individual component indices.

Comparing the VaR estimates across models, on the

whole the average VaRs are fairly similar. The uncondi

tional variance and the semi-variance-based estimates and

the multivariate EGARCH model seem to give the largest

VaRs,

while the simple GARCH models (both univariate

and multivariate), which do not allow for asymmetries, give

3 6 THE EFFECT OF ASYMMETRIES ON STOCK INDEX RET URN VALUE-AT-RISK ESTIMATES

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E X H I B I T 4

VaR for the Portfolio in 6 Ro l l ing Sa m ples as a Proportion of Initial Value,

Est imated Using GARCH

and

Semi-Variance

the lowest average VaRs across the 46 windows for all coun

tries, except for the U.S., where the univariate symmetric

G A R C H V a R s are uncharacteristically high. For example,

in the case of the equally w eighted portfolio, the univariate

GARCH(1,1) model yields an average daily VaR of aro und

3.2%, while the semi-variance-based estimator provides an

average VaR of ust above 4%. The least stable VaRs, evi

denced

by the

highest standard deviation across

the 46

w i n

dows, arise from the E G A R C H m o d el s, in bo th their

univariate and multivariate forms. By far the most stable

VaRs over time for all countries and the portfolio are those

calculated using the semi-variance. The semi-var iance-

based VaRs have a standard deviation across windows of

the order of one-half that of other mod els, except for those

based on the (symmetric) variance, which have only slightly

higher variabilities across windows. In the context of the

S&P index returns, the semi-variance gives a mean VaR of

1.8%, the

lowest of any mod el.

An

example of the stability

of the VaR based on the semi-variance com pared w ith its

competitors can be gleaned from Exhibit 4, which plots

the estimated VaRs

for the

portfolio using

the

G A R C H

model and the semi-variance. Both

the

high er average value-

at-r isk estimate under the la t ter model , and its relative

smoothness over time, are apparent. The stability of the

semi-variance VaR estimates relative to those calculated

using the variance is quite surprising. If the returns were iid

normally distributed, the semi-variance VaR should have

a larger sampling va riability since it employs approximately

half as ma ny observation s as for the variance (and the trailing

samples used for the variance and semi-variance cover the

same three-year period).

Yet the

semi-variance VaRs

are

equally stable

for

H o n g K o n g

and are

more stable

for all

other series. Clearly, however, the return distributions are

non-normal and in particular are asymm etric. It also appe ars

to

be the

case that

the

lower halves of the distributions that

are picked ou t by the sem i-variance are m ore stable over time

than the upper tails, which are included in the variance cal

culations but not in the semi-variance.

Finally, comparing the univariate models with their

corresponding multivariate counterparts ,

we

note that

in

all cases, that

is, for

returns

to the

individual indices

and

fo r the portfolio, allowing for t ime-va ry ing co -mov e

ments between

the

series leads

to

slightly higher average

VaRs which are considerably more stable over time.

O ne may suggest at first blush tha t an optimal m odel

is one that gives the lowest average and the most stable

VaR estimate over t ime. A low VaR wou ld be deemed

preferable for the obvious reason that a lower VaR implies

that the firm should be required to tie up less of its capital

in an unprofitable, liquid form, while a stable VaR would

be appealing on the grounds that a highly variable VaR

would make it difficult to assess the riskiness of the secu

rities firm over

the

long term. However,

a

more appro

priate test

of

the adequacy

of

the

VaR

models

is

under

an

assessment of how they actually perform wh en used on a

holdout sample ( backtests in the Basle Comm ittee ter

minology). Regulators impose severe penalties

on

firms

whose models generate more than an acceptable num ber

of exceedances or except ions (see section 4 above),

and for this reason as well as to minimize the p ossibility of

financial distress, it is in the firm's interests to ensure that

its VaR mo dels perf orm satisfactorily in backtests. Back-

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E X H I B I T 5

Backtesting (i.e. Out-of-Sample Tests) of the VaR for Stock Index Returns January 1985-April 29, 1999

(3,737 Ob servations)

Hong Kong

J a p a n S i n g a p o r e

S . Korea Tha i l and Por t fo l i o

S&P 500

P a n e l A : S y m m e t r i c A p p r o a c h e s

V a r i a n c e

Mean

Std. Deviat ion

Green

Yellow

Red

E W M A

M e a n

Std. Deviat ion

Green

Yellow

Red

G A R C H

M e a n

Std. Deviat ion

Green

Yellow

Red

M G A R C H

M e a n

Std. Deviat ion

Green

Yellow

Red

1.10870

1.43338

4 3

3

0

1.65217

2 . 0 8 3 1

4 2

4

0

1.84783

2 . 2 4 0 7 1

3 9

6

1

1.43478

1.77203

4 1

5

0

0 . 3 9 1 3 0

0 . 8 8 1 3 7

4 6

0

0

1.78261

5 . 9 0 6 3 5

4 3

1

2

1.04348

2 . 4 6 7 1 8

4 1

4

1

0 . 5 2 1 7 4

1.41011

4 4

2

0

0 . 7 8 2 6 1

1.28085

4 4

2

0

1.34783

2 . 0 7 8 6 5

4 2

4

0

1.43478

2 . 2 8 6 7 0

4 1

3

2

1.13043

1.52911

4 3

3

0

0 . 6 0 8 7 0

1.02151

4 6

0

0

0 . 7 6 0 8 1

2 . 6 2 6 3

4 4

1

1

1.00070

1.98889

4 3

3

0

0 . 5 4 3 4 8

1.14904

4 4

2

0

1.36957

1.27120

4 5

1

0

1.43478

2 . 1 6 6 9 5

4 2

3

1

1.69565

3 . 8 2 8 9 5

4 2

2

2

1.30435

1.77476

4 2

4

0

1.06522

1.10357

4 6

0

0

1.69565

2 . 4 5 7 3 7

4 2

3

1

1.00000

1.57762

4 3

3

0

0 . 1 0 8 7 0

0 . 7 3 7 2 1

4 5

1

0

0 . 3 2 6 0 9

0 . 7 0 0 9 3

4 6

0

0

0 . 6 3 0 4 3

1.10270

4 6

0

0

0 . 5 6 5 2 2

0 . 7 7 8 9 5

4 6

0

0

-

P a n e l B : A s y m m e t r i c A p p r o a c h e s

GJR

M e a n

Std. Deviat ion

Green

Yellow

Red

E G A R C H

Mean

Std. Deviat ion

Green

Yellow

Red

S e m i - V a r i a n c e

M e a n

Std. Deviat ion

Green

Yellow

Red

M G JR

M e a n

Std. Deviat ion

Green

Yellow

Red

M E G A R C H

M e a n

Std. Deviat ion

Green

Yellow

Red

1.73913

2 . 1 2 3 2 6

4 1

5

0

0 . 8 6 9 5 7

1.80873

4 0

6

0

1.00000

1.07497

4 6

0

0

1.36957

1.70436

4 2

4

0

0 . 5 6 5 2 2

1.40874

4 4

2

0

0 . 8 2 6 0 9

2 . 0 1 4 4 4

4 3

3

0

0 . 8 4 7 8 3

2 . 2 3 0 7 7

4 1

5

0

0 . 2 6 0 8 7

0 . 4 9 1 4 7

4 6

0

0

0 . 6 0 8 7 0

1.86760

4 3

3

0

0 . 1 3 0 4 3

0 . 8 8 4 6 5

4 5

1

0

1.17391

1.62350

4 1

5

0

0 . 9 3 4 7 8

1.59725

4 2

4

0

0 . 9 5 6 5 2

1.26415

4 6

0

0

1.06522

1.43608

4 3

3

0

0 . 1 9 5 6 5

1.18546

4 5

1

0

1.00000

1.98886

4 3

3

0

0 . 9 2 1 7 4

1.89626

4 4

2

0

0 . 5 2 1 7 4

0 . 9 6 0 0 7

4 6

0

0

0 . 1 7 3 9 1

1.03932

4 5

1

0

0 . 1 3 0 4 3

0 . 7 4 8 5 9

4 5

1

0

1.21739

1.56224

4 4

2

0

0 . 3 2 6 0 9

0 . 8 7 0 6 2

4 5

1

0

1.17391

1.23476

4 6

0

0

1.21739

1.93118

- 4 2

4

0

0 . 3 6 9 5 7

1.08236

4 5

1

0

0 . 9 9 8 6 4

1.21100

4 4

2

0

0 . 3 2 6 0 9

1.24819

4 4

2

0

0 . 9 5 6 5 2

1.09456

4 6

0

0

0 . 1 7 3 9 1

0 . 7 6 8 9 6

4 5

1

0

0 . 3 0 4 3 5

1.05134

4 5

1

0

0 . 3 6 9 5 7

0 . 7 4 1 1 3

4 6

0

0

0 . 5 2 1 7 4

0 . 8 8 7 9 2

4 6

0

0

0 . 6 3 0 4 1

1.10270

4 6

0

0

-

-

Note: Model

acronyms

are as ollows:

variance denotes

V aR

calculated

using the

historical standard deviation

of returns; EWMA

denotes

the VaR

calculated

using

the exponentially

weighted

moving

average

method; MCJR and MEGARC H

denote

the multivariate GJR and EGARC H

models respectively; semi variance

denotes a VaR

calculated

using the

scaled

semi-standard deviation of

(9).

Cell

entries refer

to the

average

and standard deviation of the number of

exceedances

of

the VaR

across

the 46 out-of-sample windows, and the number of times that

such

a number of

exceedances

would have

placed

the irm in the Green, Yellow, and

Red Zones; VaR is

expressed

as a

proportion

of

the

initial value of

the

position. The portfolio is an equally weighted

combination

of

the

five Southeast Asian

market returns.


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E X H I B I T

6

N u m b e r

of

Exceedances

for the

Portfolio

in 6

Ro l l ing Sa m ples

of VaR

Calculated

Using GARCH

and

Semi-Variance

test results

for

each model

and for

each asset series

are

p re

sented in Exhibit 5. The mean and standard deviation of

the percentage of days

for

which there

was an

exceedance

in each

of the 46

rolling samples

of

length

250

observa

tions are given, together w ith the number of t imes where

the firm's

VaR was

insufficient

on

enough days

in the

sample

to

place

it in the

Yellow

or Red

Zone. Again, first

considering the results across coun trie s, in general the esti

mated VaR models for Hong Kong and Singapore seem to

be

the

least adequa te, while those

for the

U.S., South Korea,

an d

for the

portfolio seem

to

yield

the

fewest exceptions.

On ly the univariate GARCH model would have ever led

a securities firm with positions

in

these equities into

the

Red Zone, with consequent disallowance

of

the internal

mode l and increased capital charges. Of the 46 sample

periods,

the

models

for

four

of

the five countries would

have been

in the Red

Z o n e

at

least once— with

the

worst

being Singapore and Thailand, wh ere the firm wo uld have

been

in the Red

Z o n e

for two

periods.

C ompar ing

the

G A R C H

and

mul tivar ia te G A R CH

models with their asymmetric counterparts,

the

latter seem

to

do

somewhat bet ter .

The

average propor t ion

of

exceedances is lower for the GJR and E G A R C H m o d el s,

especially

the

latter, than those conditional variance models

that

do not

allow

for

asymmetries ,

for all

countries .

For

example, the average percentage of exceedances in the case

of Thailand is approximately

1.7

using

the

GA R C H m odel ,

but only

1.2 and 0.3 for the GJR and

E G A R C H m o de ls

respectively. Also, the multivariate models tend to fare

better than their univariate counterparts , irrespective

of

whethe r

the

models

are

symm etr ic

or

asymmetric. This

is

particularly tru e in the portfolio context, w here the model

estimation approach uses

the

time-v arying forecasts of the

conditional covariances

as

well

as the

volatilities.

For

example, the univar ia te G A R C H m odel has an average of

exactly

1%

exceedances, placing

the

firm

in the

Yellow

Z o n e

on

three occasions;

the

multivariate GA R C H model,

on the other hand, has an average percentage of exceedances

of only 0 .1% , placing the firm in the Yellow Z on e only

once.

In the

case of the

U.S.

stock re turns,

all

models

are

deemed safe, with

no

ventures into either

the

Yellow

or

Red Zones . This appears to arise from a fall in S&P

volatility d urin g

the

out-of-sample periods compared with

the start

of

the sample.

For

example, splitting

the

whole

sample exactly in half, the S&P variance in the second half

is approximately 50%

of

that

in the

first

half.

In terms

of

the Basle Committee criteria,

the

clear

winne r is the semi-variance-based estimator, whic h, if it

had been employed

by a

securities firm with long posi

tions

in any

of these marke t indices

or

a portfolio of them ,

would never have been

out

of the Green Z one . W hile this

improved performance would

not

come costlessly

to the

firm, in

the

sense that

the VaR

from

the

semi-variance-

based model

is

almost always

the

highest,

the

improvement

in perform ance is considerable relative to the models which

genera ted

the

next highest

VaR

levels

(the

asymmetr ic

multivariate models).

The

usefulness of the measure based

on the semi-variance is also shown in Exhib i t 6, which

presents

the

number

of

exceedances

for the

portfolio

of

stock returns, using this method

and

also

the

univariate

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G A R C H m o d e l . As can be seen, the semi-variance pro

duces, for most w indow s, a smaller num ber o f exceedances

in the 250-day holdou t samples than the G A R C H m o d e l.

Moreover, comparing Exhibits 4 and 6, the numbers of

G A R C H m o d e l e x c e e d a n c e s are greater when the

GARCH model VaR calcu la t ions are considerably below

those of the sem i-variance e stimator. This seems to sug

gest that the G A R C H m o de l at times underestimates the

VaR relative to the semi-variance m odel and relative to

the actual out- turn.

C O N C L U S I O N S

This study sought to consider the effect of two classes

of asymmetry on the size and adequacy of market-based

value-at-risk estimates in the context of

five

Southeast Asian

stock market indices, and an equally weighted portfolio

comprising these indices, with S&P 500 index returns exam

ined for comparison. We found significant statistical ev i

dence that both uncond itional skewness and a conditionally

asymmetric response of volatility to positive and negative

returns were present in the data. We then examined a number

of symmetric and asymmetric models for the determina

tion of a securities firm's position risk requirement. Our pri

mary f inding was that the semi-variance mod el , which

explicitly allows for asymmetry, leads to more stable VaRs

which would be deem ed m ore accurate under the Basle

Com mittee rules than models which do not allow for such

asymmetries. In particular, VaR estimates based upon a simple

modification to the usual semi-variance estimator were the

only ones whic h w ould have left

the

firm w ith

a

margin

of

safety during every time period and for every asset.

Put another way, models that do not allow for asym

metries either

in the

uncon ditional retu rn dis tr ibution

or

in the response of volatility to the sign of returns, lead

to inappropriately small VaRs. Altho ugh the cost of

c a p

ital is on ave rage l% -20% h ighe r for the as ymmet r i c

models , we conjecture that the additional m argin of safety

is necessary and makes the addit ional cost w or thw hi le .

Firms w hich fall into the Yel low Z on e are likely to have

their multiplication factor raised from 3 to 4 or 5, an

increase of at least

3 3%

in the required VaR. Firms w hich

trip into

the Red

Z o n e w o u l d

be

forbidden from using

an internal model, and wou ld be required to revert to the

building block approach which specif ies a flat charge of

8% for equi t ies . This would represent an approximate

doubl ing of the required capital for the por tfo l ios co m

pared with that reported for the asymm etr ic models in

Exhib i t 3.

Skewness is a hitherto vir tually neglected feature of

financial asset return series, and it seems plausible that

better value-at-r isk estimates should arise from method

ologies which are able to capture all of the stylized fea

tures that are undeniably present in the data. We propose

that future research may seek to form a mode l wh ich can

capture the unco ndition al skewness in the data as well as

volatility clustering, leverage effects, and uncondi t ional

kurtosis ; one such mode l is the autoregressive cond itional

skewness formulation, recently proposed by Harvey and

Sidd ique [1999 ] , wh ich , due to its infancy, is as yet

untested

in the

risk man agem ent arena.

E N D N O T E S

1

The models employed here are widely used and hence

only brief model descriptions are given, although see Brails-

ford and Faff [1996] or Brooks [1998] for thorough discussions

of alternative models

for

prediction of volatility and their rel

ative forecasting performances under standard statistical evalu

ation metrics.

2

The method

of

maximum likelihood using

a

Gaussian

density

and

employing

the

BFGS algorithm

is

employed

for

estimation of the parameters of all models from the G A R C H

family, including the multivariate and asymmetric models.

3

The model coefficient estimates for each mo del are not

presented due to space constraints, although they are available

upon request from the authors.

4

In fact, this assumption can be made witho ut loss of gen

erality, for a zero mean series

x

t

:

t

=

1, . . ., T can be constructed

by subtracting the mean of the series from each observation t.

5

In (9), this is scaled by T_/ T_ - 1) 1 as T_ ∞ o

obtain a consistent (asymptotically unbiased) es timator.

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