Value-at-Risk models and Basel capital charges: Evidence from Emerging and Frontier stock markets

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Journal of Financial Stability 8 (2012) 303– 319

Contents lists available at SciVerse ScienceDirect

Journal of Financial Stability

journal homepage: www.elsevier.com/locate/jfstabil

alue-at-Risk models and Basel capital chargesvidence from Emerging and Frontier stock markets

drian F. Rossignoloa,∗, Meryem Duygun Fethib, Mohamed Shabanb

School of Management, University of Leicester, Santo Tome 2870, C1417GFL Buenos Aires, ArgentinaSchool of Management, University of Leicester, Leicester LE1 7RH, United Kingdom

r t i c l e i n f o

rticle history:eceived 6 November 2010eceived in revised form 9 November 2011ccepted 28 November 2011vailable online 7 December 2011

EL classification:37

a b s t r a c t

In the wake of the subprime crisis of 2007 which uncovered shortfalls in capital levels of most financialinstitutions, the Basel Committee planned to strengthen current regulations contained in Basel II. Whilemaintaining the Internal Model Approach based on Value-at-Risk, a stressed VaR calculated over highlystrung periods is to be added to present directives to constitute Minimum Capital Requirements. Conse-quently, the adoption of the appropriate VaR specification remains a subject of paramount importance asit determines the financial condition of the firm. In this article I explore the performance of several modelsto compute MCR in the context of Emerging and Frontier stock markets within the present and proposedcapital structures. Considering the evidence gathered, two major contributions arise: (a) heavy-tailed

eywords:alue-at-Riskxtreme Value Theorymerging and Frontier marketsapital Requirements

distributions – particularly Extreme Value (EV) ones-, reveal as the most accurate technique to modelmarket risks, hence preventing huge capital deficits under current measures; (b) the application of suchmethods could allow slight modifications to present mandate and simultaneously avoid sVaR or at leastreduce its scope, thus mitigating the impact regarding the enhancement of capital base. Therefore, I sug-gest that the inclusion of EV in planned supervisory accords should reduce development costs and fosterhealthier financial structures.

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tressed VaR

. Introduction

The world financial system has undergone one of its most severerisis in 2007–2008. Several factors acted simultaneously to igniteurmoil with devastating consequences felt all over the globe as aonsequence of the interconnectedness of the national economies.ne of the relevant effects brought about by the catastrophe haseen represented by the inability of the banks to meet market

osses: capital was insufficiently constituted to provide coverageor unexpected adverse events. The shortage was of such an extenthat many institutions had to be bailed out by governments at thexpense of the taxpayers’ resources; although this action averted aomplete deadlock in capital markets, it simultaneously introducedlements like moral hazard and subsidies.

The current framework contained in Basel II Capital Accord hasstablished Value-at-Risk (VaR) as the official measure of market
isk and enforced it to constitute the central point to the deter-ination of capital charges. Moreover, as the Basel Committee
n Banking Supervision (BCBS) has not hitherto recommended a

∗ Corresponding author. Tel.: +54 11 45033632.E-mail addresses: [email protected] (A.F. Rossignolo), [email protected]

M.D. Fethi), [email protected] (M. Shaban).

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572-3089/$ – see front matter © 2011 Elsevier B.V. All rights reserved.oi:10.1016/j.jfs.2011.11.003

© 2011 Elsevier B.V. All rights reserved.

articular VaR methodology, the adoption of the most appropri-te VaR approach becomes a matter of the utmost importanceo be decided purely on empirical grounds. However, the magni-ude of the plight prompted the BCBS to put forward a proposalo increase the Minimum Capital Requirements (MCR) for marketisks in accordance with the opinion of national regulators.1 Thentended scheme plans the introduction of a stressed VaR (sVaR)

hich ought to be added to the base VaR (cVaR) in order to form theew MCR in an attempt to curb the procyclicality of the measure

n force.The aforementioned context highlights the significance of

eveloping a precise VaR model to cover market losses and simulta-eously build a capital buffer high enough as to allow institutionso distribute dividends in light of a further BCBS directive whichestricts the dividend payout unless the capital level exceeds MCRy a quantity called Capital Conservation Buffer equivalent to.5% of the amount of the risk weighted assets. Besides the tra-
itional reluctance on the part of the academics to study Emergingnd Frontier markets, BCBS’s Consultative Documents have mostlyeen submitted to developed nations and it is unlikely that these
1 “Capital required against trading activities should be increased significantlye.g., several times)” (Financial Services Authority, 2009, p, 7).

dx.doi.org/10.1016/j.jfs.2011.11.003

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dx.doi.org/10.1016/j.jfs.2011.11.003

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2Stylised facts present in financial time series like autocorrela-

tion of squared returns and volatility clustering (Christoffersen,2003; Dowd, 2005; McNeil et al., 2005) pave the way for devel-oping time-varying variance models. Postulating the dependence

04 A.F. Rossignolo et al. / Journal of

roposals should be evaluated and its impact assessed in thepheres of Emerging and Frontier markets. This study aims at fillinghat empirical void, as I analyse the accuracy of several VaR specifi-ations and gauge its effect on capital charges from the perspectivef non-developed stock markets under the current and proposedegulations.

The article unfolds as follows. Section 2 briefly synthesises theasic concepts and models to be tried; Section 3 states the cur-ent and proposed theoretical frameworks; Section 4 delineatesopics regarding the Data and Methodology; Section 5 describeshe empirical results from the exercise in connection with BCBSirectives while Section 6 offers the concluding statements. Finally,ection 7 stages a sensitivity analysis gauging the impact of BCBS’slanned mandate whereas Section 8 presents some overall closingemarks regarding the new sVaR approach.

. Theoretical background. Concepts and definitions

.1. Definition of Value-at-Risk

VaR is a statistical risk metric that expresses the maximumoss in the value of exposures due to adverse market movementshat a company is reasonably confident will not be exceeded ifts positions are maintained static during a certain period of time.2 Losses are associated with confidence levels (˛): those lossesreater than VaR are only suffered with a specific small probability1 – ˛) (McNeil et al., 2005; Linsmeier and Pearson, 1996). There-ore, for some confidence level ̨∈ (0;1), VaR at the confidence level

is the smallest number l such that the probability that the loss Lxceeds l is smaller or equal than (1 – ˛)3:

aR (˛) = inf{l ∈ R : P(L > l) ≤ (1 − ˛)} = inf{l ∈ R : FL(l) ≥ ˛} (1)

here FL denotes the loss distribution function.4 Given thatr(Losst+1 > VaRt+1) = ˛, or equivalently in relative terms or returns5

r(rt+1 < VaRt+1) = ˛, VaR could be characterised as

aR(˛)t+1 = �t+1F−1(˛) (2)

ith the following symbols meaning: �t+1 is the volatility of the lossistribution function F (measured by the standard deviation) and−1(˛) is the inverse of the loss distribution function, i.e., ˛-quantilef F.

.2. Value-at-Risk models

A synopsis of the methods to be used in this research is statedelow.

.2.1. Historical Simulation (HS)Arguably the simplest and most popular route to VaR, it only

equires the estimation of the appropriate quantile employing theuantiles of a window of past sample returns:

aR(˛)t+1 = Q˛(rt; rt−1; . . . ; rt−n+1) (3)

espite the dimensionality problem is reduced to univariate cate-ory and conceptual simplicity and easiness of implementation arechieved, it is a logically inconsistent approach.6 Its flaws ground

2 Hence, VaR requires the estimation of a quantile of the distribution of profitsnd losses, i.e., the distribution of returns.3 McNeil et al. (2005).4 The subscript L will hereafter be dropped from FL for simplicity reasons.5 Appendix A.6 Manganelli and Engle (2004).

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cial Stability 8 (2012) 303– 319

undamentally on the absence of assumptions about the depen-ence structure of returns (Manganelli and Engle, 2004), alongsidehe equal weighting structure and the extrapolation of the sam-le distribution to any forecasted term (Dowd, 2005), the lengthf the ‘past’,7 because it must satisfy variance and bias constraintsimultaneously and the likely presence of ghost or shadow effects.8

.2.2. Filtered Historical Simulation (FHS)Barone-Adesi et al. (1998) devised one significant improve-

ent to HS by combining conditional volatility modelling9 withhe empirical distribution of returns, hence retaining the patternsf HS and simultaneously admitting ways to alter the unrealisticuppositions of HS (Dowd, 2005). FHS demands fitting some modele.g. a GARCH-family technique) to the sample data to account forhe empirical patterns in order to obtain the volatility predictionshich are in turn employed to generate a set of iid standardised

eturns.10 The ˛-quantile of the series of standardised returns ishen multiplied by the volatility forecast to obtain the FHS VaR:

aR(˛)t+1 = �t+1F−1(˛), (4)

here �t+1 is the volatility forecast derived from any (GARCH-amily) volatility model and F−1(˛) is the inverse of the cumulativeensity function of the empirical distribution of residuals, i.e., ˛-uantile of F.

Although FHS takes into account the changing market con-itions by blending conditional volatility modelling with thempirical distribution (Dowd, 2005), Pritsker (2001) affirms thatHS VaR is still unable to capture extreme events.

.2.3. Linear specificationsLinear techniques only require the estimation of the standard

eviation of the portfolio by means of the sample estimate of vari-nce appropriately increased by the quantile of a pre-specifiedistribution, typically Normal or Student-t. Formally,

aR(˛)t+1 = �t+1˚−1(˛) (5)

aR(˛)t+1 = �t+1

√(d − 2)d−1t−1

d(˛) (6)

here �t+1 is the standard deviation of the sample of returns;−1(˛) is the inverse of the cumulative density function of the stan-

ard Normal distribution (˛-quantile of Ф); t−1d(˛) is the inverse

f td, the distribution function of a standardised t (˛-quantile of t);nd d is the degrees of freedom of the t distribution.

Their simplicity appears overshadowed by their disadvantages,hich derive both from the limitations of the standard deviation

s a measure of risk – mainly the equal weighting structure andhe inaccuracy beyond the mean11 – and the inadequacy of theppended distribution.

.2.4. Models of conditional volatility

7 Manganelli and Engle (2004) recommend between six months and 2 years,uester et al. (2006) propose four years and Christoffersen (2003) suggests betweenne and four years, for example.8 Penza and Bansal (2001).9 Section 2.2.3. The selection of the GARCH model is not restricted to GARCH. The

resent research will also apply EGARCH specifications with Normal and t likelihoodunctions.10 Standardised returns are obtained dividing the realised returns by the respectiveARCH volatility forecast: εt = rt( �̂t )

−1.11 Balzer (2005).

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f current volatility on past volatilities and returns, they appearapable of capturing the unconditional non-normality of the datand mimicking the volatility clouding effect.

Pursuing strictly practical motivations, this research will unfoldithin the boundaries of two simple representations12: Generalutoregressive Conditional Heteroskedastic (GARCH) and Expo-ential GARCH (EGARCH).

.2.4.1. GARCH(1;1).

t+1 = �t+1zt+1

2t+1 = w + ˛r2

t + ˇ�2t (7)

ith zt+1∼ iid D(0;1) (where D(0;1) denotes the conditional distri-ution of the GARCH process, which presents null mean and unitariance) and w > 0; ˛, ̌≥ 0; ̨ + ̌ < 1.

A high ̌ (persistence) means that volatility takes a long timeo fade after a crisis episode, whereas a high ̨ (error) indicatesromptness to react to market movements. GARCH models were

ntroduced by Bollerslev (1986) and Taylor (1986), since thenrguably erecting as one of the most popular among financialommunity13 as they enable to seize the volatility clustering effectalthough in a symmetric fashion) and treat variance as a persistenthenomenon. GARCH power might also be enhanced applying itsexibility to depart from Gaussianity: this article will make usef the heavy-tailed Student-t in addition to the typical GARCH-ormal specification.

.2.4.2. EGARCH(1;1).

t+1 = �t+1zt+1

n(�2t+1) = w + ̌ ln(�2

t ) + ˛∣∣∣ rt

�t

∣∣∣+ �rt

�t(8)

ith � < 0, ̨ > 0 and zt+1∼ iid D(0;1) (D(0;1) retains the meaningxpressed in (7)).

EGARCH representations were first proposed by Nelson (1991)nd have become an interesting option as long as forecasts of condi-ional variance are guaranteed to be nonnegative upon them beingxpressed in log form.14 Its structure allows the term � to accountor the leverage effect when, as it is the common case observedhroughout the markets, � < 0 and ̨ > 0 drive ln(�2

t+1) to react moreapidly to falls than to corresponding rises.15 Like GARCH previ-usly, EGARCH may incorporate several distributions for D(0;1),hus Normal and t cases will be evaluated.

For pragmatic reasons,16 and in view of the negligible marginalains obtained extending the quantity of lags, the study will restrictARCH–EGARCH applications to first order lags, backed by McNeilt al. (2005) who advocate the use of lower order models citing
arsimony reasons.
Consequently,

aR(˛)t+1 = �t+1˚−1(˛) (9)

12 It is acknowledged that the list lies very far away from sophistication or com-leteness: it is merely indicative and illustrative of the wide range of possiblelternatives.13 For a more detailed explanation about GARCH models, see Bollerslev (1986),ollerslev et al. (1994), for example.14 Nelson (1991) and Engle, Bollerslev and Nelson (1994) treat EGARCH exhaus-ively.15 Cho and Engle (1999) and Chopra et al. (1992) document the leverage effect ineveloped markets.16 Eventually avoiding the curse of dimensionality. Gujarati (1997) recommendshat econometric models should follow the parsimony principle.

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nd

aR(˛)t+1 = �t+1

√(d − 2)d−1t−1

d(˛) (10)

here �t+1 represents the volatility forecast derived fromARCH/EGARCH model and the rest of the symbols stand as before.

.2.5. Extreme Value Theory (EVT)EVT represents an alternative to former specifications as it

mphasises the tails of the series allowing right and left ends to beodelled separately and, most importantly, recognising the heavy-

ailed nature of empirical distributions. On the grounds of efficiencynd practicality, the study will develop within the Peaks-Over-hreshold (POT) variant disregarding the Block Maxima MethodBMM) (Coles, 2001). This section only enunciates the necessaryhases to apply EVT-POT to VaR estimation; for a detailed treat-ent and theoretical aspects the interested reader may refer to

mbrechts et al. (1997), McNeil et al. (2005), or Reiss and Thomas2007).

POT process synthetically requires the following steps:

a) Finding a sequence of iid returns or random variables: X1,X2, . . ., Xn;

b) Selecting a sufficiently high threshold u;(c) Defining extremes as values of Xt exceeding some high thresh-

old u;d) Calculating excesses over the threshold yt = Xt− u (Xt > u);e) Applying the Balkema and De Haan (1974) and Pickands (1975)

theorem to fit the two-parameter limiting Generalised ParetoDistribution (GPD) G�,�(y) for the excesses y above the thresholdu.

Accordingly (11),

�,�(y) =

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1 −(

1 + �y

�

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if � /= 0

1 − exp(−y

�

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(11)

here � is the tail index parameter and � is the scale parameter,s u increases, where � > 0, and y ≥ 0 when � ≥ 0 and 0 ≤ y ≤ −�/�,hen � < 0. For finance applications � > 0 ought to be verified, then

�,�(y) becomes the classic Pareto or Fréchet distribution picturingeavy tails. Reiss and Thomas (2007) remark that GPD precisionould be enhanced affixing a location parameter �, thus making

�,�(y – �).Christoffersen (2003) points that the main flaw of POT method-

logy resides in the identification of the threshold u. Regardlessf some efforts to find an appropriate mechanism17 there is not

reliable method to calculate it, hence the current study favours technique based on the analysis of an array of elements such ashe sample Mean Excess Function (MEF), QQ plots, sample Ker-el Density and sample Quantile Function, following Reiss and
homas (2007) and McNeil and Saladin (1997). The threshold uould denote the value from where the MEF exhibits a positive
radient (excesses follow a GPD with � > 0), provided the estimated

17 Reiss and Thomas (2007) or Beirlant et al. (1996) explain methods which seemo work reasonably well but should be handled with care, as they may even-ually result in the selection of a very high number of upper order statistics.anielsson et al. (1998) and Coronel-Brinzio and Hernandez-Montoya (2004) intro-uce interesting approaches to this subject. Christoffersen (2003) presents a “rulef thumb” and Neftci (2000) followed by Bekiros and Georgoutsos (2005) estimates

= 1.176�n, where �n is the sample standard deviation and 1.176 = F−1(0.10) =.44√

(d − 2)/d assuming F ∼ Student-t(6).

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arameters exhibit stability within a range of the selected u (Coles,001).

After some algebraic operations, the POT-quantile reads18:

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here w represents the number of observations above the thresh-ld u and the rest of the symbols conserve their meaning.ollowing Quasi-Maximum-Likelihood (QML)19 as in Bollerslevnd Wooldridge (1992) and McNeil and Frey (2000), POT-quantile12) is calculated using the iid standardised residuals after pre-hitening the data employing a GARCH-Normal model. Therefore,

he final VaR expression for EVT-POT becomes:

aR(˛)t+1 = �t+1G−1(˛) (13)

here �t+1 is the volatility forecast derived from GARCH-Normalodel and G−1(˛) is the inverse of the cumulative density function

f the GPD distribution (˛-quantile of G).

. Regulatory framework

.1. Basel II Capital Accord

In 1996 the BCBS issued an Amendment to incorporate a specificreatment for market risks, largely overlooked in Basel I Capi-al Accord and eventually included in Basel II Capital Accord.20

his adjustment allows institutions to employ the Internal Modelpproach (IMA) to have their market risk Minimum Capitalequirements (MCR) determined by their own VaR estimates,hat in turn derive from their respective VaR models. Risk-capitalharges result from21:

CRt+1 = max

(mc

60

60∑i=1

VaR(99%)t−i+1; VaR(99%)t

)(14)

.e., the maximum between the previous day’s VaR and the averagef the last 60 daily VaRs increased by the multiplier22 mc = 3(1 + k)nd k ∈ [0; 1] according to the result of Backtesting.23 BCBSemands VaR estimation to observe the following quantitativeequirements:

a) Daily-basis estimation24;b) Confidence level ̨ set at 99%;

(c) One-year minimum sample extension with quarterly or morefrequent updates25;

d) No specific models prescribed: banks are free to adopt theirown schemes;

e) Regular Backtesting and Stress Testing programme for valida-tion purposes.26

18 McNeil et al. (2005), McNeil and Saladin (1997) and Fernandez (2003).19 Also called Pseudo-Maximum-Likelihood (PML).20 BCBS (1996, 2004).21 BCBS demands the use of a 10-day holding period through the square-root-of-ime rule. However, the present research will omit the specification and work with

1-day holding period instead. See Section 4.22 mc will be, at minimum, 3. Although BCBS does not enlighten its derivation, Stahl1997) and Danielsson et al. (1998) provide a statistical explanation. This value canometimes be so conservative that any incentives to develop an accurate model bychieving k=0 might be quickly overshadowed.23 Section 3.1.1.24 Footnote 19.25 Section 4.26 The focus of the present article is restricted to Backtesting. BCBS (1996,004, 2009a,b), Jorion (1996), Penza and Bansal (2001), Christoffersen (2003),

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.1.1. BacktestingIt constitutes a statistical technique to assess the quality of the

isk measurement specifications which involves the comparisonetween the daily VaR forecast with the actual losses.27 Albeit thessumptions behind VaR calculations may be labelled incoherentArtzner et al., 1997, 1999; Acerbi and Tasche, 2002), it is usefulo evaluate whether the model is capable of capturing the trad-ng volatility. Backtesting procedure entails counting the numberf times that losses exceed VaR estimates in approximately 250rading days. McNeil et al. (2005) use the indicator function:

t+1 = I{Lt+1>VaRt+1(˛)} ={

1 if Lt+1 > VaRt+1(˛)0 otherwise

here It is the indicator function accumulating the excessions,xceptions or violations behaving like outcomes of iid Bernoulli tri-ls with success probability 1 − ˛; Lt+1 is the realised loss for period

+ 1; and VaRt+1(˛) is the conditional VaR estimates for t + 1.BCBS proposes a three-zone – Green, Yellow and Red-layout to

lassify VaR models. Consequently28:

Zone Definition Characteristics

Green Outcomes consistentwith low probabilityof Error II

– Number of exceptions between 0 and4– No capital surcharge, k = 0

Yellow Results uncertainand compatible witheither precise orinaccurate models

– Number of exceptions between 5 and9– Strong suggestion of imprecisespecifications, particularly as numberof exceptions grow– Capital penalties increase withnumber of violations– Capital charges determined to returnmodel to a 99% coverage– Encourages sharpness to keeppenalties low

Red Presumption ofinaccurate model

– Number of excessions equal orgreater than 10– k increased to 1 immediately– Subsequent model invalidation

Backtesting results determine the extent of capital surchargehrough the value of the scaling factor k as the quantity of excep-ions in a sample of 250 trading days is transformed into a numberndicating the increase in the multiplier to be applied to mc (Chart). BCBS establishes that the beginning of each zone is marked byhe points where the cumulative probability of a Bernoulli distri-ution with 99% success probability reaches 95% for the Yellowone (5 exceptions) and 99.99% for the Red one (10 or more vio-ations), respectively. Given that five excessions represent a 98%overage for 250 observations, it would be necessary to enhancehe multiplier in 40% to restore the coverage to the 99% demanded
supposing that returns follow a Normal distribution and the scal-ng factor mc = 3).29,30 However, although it is possible for k in14) to achieve nullity (specifications belonging to Green Zone),
owd (1998, 2005), RiskMetrics Technical Document (1996) and Osterreischischeationalbank (1999), just to name a few, cater for basic concepts and extensive

reatment of stress testing.27 “. . . the backtesting framework . . .involves the use of risk measures calibratedo a one-day holding period” (BCBS, 2006, p. 312).28 This categorisation is designed to compromise the probabilities of Error I: erro-eous rejection of accurate models and Error II: incorrect acceptance of inaccurateodels. For a detailed statistical treatment of foundations of Backtesting recur to

CBS (2006).29 The quotient between the 99% and 98% cumulative normal distribution amountso 1.14 which, for a scaling factor of 3represents a 40% increase in the base level.n effect, for five exceptions 1 − 5/250 = 0.98 and 3 * Ф−1(0.99)/Ф−1(0.98) − 3 ∼ 0.40ssuming normality. Hence, k = 0.40% or 40%.30 Chart No. 1.

A.F. Rossignolo et al. / Journal of Financial Stability 8 (2012) 303– 319 307

Chart 1Backtesting: the three-zone approach. Number of exceptions – increase in scalingfactor.

Zone Number of exceptions Increase inscaling factor k

Cumulativeprobability

Green Zone 0 0.00 8.11%1 0.00 28.58%2 0.00 54.32%3 0.00 75.81%4 0.00 89.22%

Yellow Zone 5 0.40 95.88%6 0.50 98.63%7 0.65 99.60%8 0.75 99.89%9 0.85 99.97%

Red Zone 10 or more 1.00 99.99%

Notes:(1) Values correspond to a forecast period of 250 independent observations. Forother quantities the chart should be reworked.(2) Probabilities obtained using a Bernoulli distribution with probability of success99%.(3) The Yellow Zone begins at that number of exceptions where the probability ofo(v

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Capital Requirements and buffers.

Capital concept Percentage ofexposure

Observations

Minimum CapitalRequirement (MCR)

8.0 Minimum quantityCalculated via SA (8%) or IMA

Capital ConservationBuffer (CCB)

2.5 Summoned in event of crisisRestored through earningsretention

Countercyclical CapitalBuffer (CyCB)

0–2.5 At the discretion of nationalregulators (excessive credit

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btaining at a maximum that quantity equals or exceeds 95%.4) The Red Zone starts where the probability of obtaining that quantity or feweriolations is at least 99.99%.

anielsson et al. (1998) remark that mc assuming a minimum of 3 in14) conspires against the development of accurate models. Givenhat mc is calculated as in (14) in the current regulatory frame-ork, the present article will unfold according to the respectiverocedures.

.2. Basel III Capital Accord

The trading sessions following Lehman Brothers’ bankruptcy ineptember 2009 triggered unusual losses of such magnitude thatnancial institutions found their capital buffers unquestionably

nsufficient to match those deficits. Though these market move-ents were of a weird nature, BCBS (2009a,b) partly blamed the

revious Amendment of 1996 for failing to grab some key extremeisks31 verified in the turmoil. Additionally, some national regula-ors increased the pressure on BCBS demanding tougher measureso avoid further embarrassing bailouts at the expense of taxpayers.he Financial Services Authority (2009) issued an influential reporthich highlighted some deficiencies of the VaR approach that mayave provoked, among other equally important reasons, the insol-ency of several firms. In particular, it is mentioned that most VaRodels are unable to capture fat-tail risks: “Short-term observa-

ion periods plus assumption of Normal distribution can lead toarge underestimation of probability of extreme losses” (FSA, 2009,. 23).32

To tackle this specific issue, while maintaining the Basel IIethodology,33 BCBS proposed the introduction of a stressed VaR

sVaR) metric to increase MCR. Its calculation complies with theame guidelines that current VaR (cVaR) (Section 3.1) though theataset must belong to a “. . . continuous 12-month period of sig-ificant financial stress. . .” (BCBS, 2009a p. 14)), i.e., when marketovements would have inflicted great losses on the banks.

31 The emphasis is also laid on default and migration risk, among others (BCBS,009b). These measures lie beyond the scope of the present study, which will beestricted to the introduction of the stressed VaR.32 FSA highlights the procyclicality that emerges using observation periods as shorts one year: falls in confidence raise volatilities, which vanish liquidity and increaseolatility even more (FSA, 2009).33 Some slight variations regarding the data updating scheme are also put forwardBCBS, 2009a).

4

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b

growth)

ote: Exposures are expressed as a percentage in terms of total capital.

The stricter daily capital demands reflect in sVaR added to cVaR:

CRt+1 = max

(mc

60

60∑i=1

cVaR(99%)t−i+1; cVaR(99%)t

)

+ max

(ms

60

60∑i=1

sVaR(99%)t−i+1 sVaR(99%)t;

)(15)

here cVaR(99%)t is the 99% cVaR for day t; mc is the multiplieror cVaR (Section 3.1); sVaR(99%)t is the 99% sVaR for day t; and

s is the multiplier for sVaR, with ms = 3(1 + k) and k arises fromacktesting results for cVaR (not for sVaR). As k ∈ [0; 1] institutionsre encouraged to develop precise VaR models in order to keep

≈ 0 and avoid penalties to establish MCR.Besides strengthening MCR by means of the sVaR component in

CR formula, Basel III focuses on reinforcing the protection againsteriods of acute economic and financial strain through two addi-ional layers to be placed on top of MCR in the following order34:

Capital Conservation Buffer (CCB): banks are obliged to build a2.5% cushion over MCR in order to avoid its deterioration duringstress periods. This safeguard is intended to serve as a first lineof defence once heavy losses are recorded. BCBS (2010) indicatesthat this shortfall absorber must be restored to the original valuerestraining earnings distribution: the closer the bank moves toMCR (the greater deterioration of CCB), the smaller the rate atwhich profits are handed out or dividends paid until the bufferis fully rebuilt to the starting level;

Countercyclical Capital Buffer (CyCB): in practice, this is an exten-sion of CCB, although it operates in a range between 0% and 2.5%of the exposures determined at the discretion of national reg-ulators according to the point of the business cycle when risksmount up as a result of excess credit growth.35

Synthetically, Basel III capital structure could be depicted asollows:

Technically speaking, Basel III innovations stem from the intro-uction of the sVaR to calculate MCR, as both CCB and CyCBonstitute supplementary capital layers with fixed and externallyetermined proportions, respectively. Therefore, the scope of theresent article will be limited to the performance of VaR models.

. Methodology

The primary data are formed by univariate price series belong-ng to a sample of 10 stock market blue-chip indices: six belong

34 Andersen (2011) finds substantial evidence of the cyclicality in Basel II anduggests a different risk-weighting scheme to circumscribe that shortfall.35 Stolz and Wedow (2011) illustrate the performance of countercyclical capitaluffers for Germany.

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o emerging markets (Brazil, Hungary, India, Czech Republic,ndonesia and Malaysia) and four to Frontier markets (Argentina,ithuania, Tunisia and Croatia)36 retrieved from the correspond-ng stock exchanges websites. Series were converted into a stringf (logarithmic) returns to achieve stationarity and ergodicityBowerman and O’Connell, 1993; Alexander, 2008b). As in Hansennd Lunde (2005) and Mapa (2003), the time series of returns wereeparated into two periods for the purposes of parameter estima-ion and evaluation of forecasts, respectively.

Although as rule-of-thumb Christoffersen (2003) ascertains thatodels should utilise at least the last 1000 observations, it is

cknowledged that time series should be as long as possible (Dowd,005). On the other hand, the Forecast period contains the finan-ial crisis which unravelled in September–October 2008. This eventepresents an interesting example of a stress test37 to determinehe real time performance of the models. For the indices aforemen-ioned:

Stockexchange

Stock index Estimationperiod

Numberof datapoints

Forecastperiod

Numberof datapoints

Sao Paulo Bovespa 03/01/2000 1976 02/01/2008 249Brazil 28/12/2007 30/12/2008Budapest Cetop20 30/01/2002 1494 02/01/2008 251Hungary 28/12/2007 31/12/2008Mumbai Sensex 01/07/1997 2600 02/01/2008 245India 31/12/2007 31/12/2008Prague Px 05/01/1995 3243 02/01/2008 253Czech Republic 28/12/2007 30/12/2008Jakarta Jkse 04/01/2000 1925 02/01/2008 243Indonesia 28/12/2007 30/12/2008Kuala Lumpur Klse 04/01/1999 2215 02/01/2008 248Malaysia 31/12/2007 28/12/2008

Buenos Aires Merval 09/10/1996 2777 02/01/2008 249Argentina 31/12/2007 31/12/2008Vilnius Omx 03/01/2000 2045 02/01/2008 244Lithuania 28/12/2007 30/12/2008Tunisia Tunindex 31/12/1997 2496 02/01/2008 247Tunisia 31/12/2007 31/12/2008Zagreb Crobex 04/01/1999 2247 02/01/2008 250Croatia 31/12/2007 31/12/2008

ote: Emerging markets above solid line, Frontier markets below.

VaR estimation and evaluation will adopt BCBS mandates in theollowing way.

.1. General issues

Daily time horizon. The 10-day holding period is to be excludeddue to the inconsistencies of the ‘square root of time’ rule(Danielsson, 2004; Danielsson and Zigrand, 2006) and the pos-sibility of masking the inaccuracy of the models by increasinginsufficient daily VaR using extrinsic multiples38;

One-tailed VaR estimations (left tail, i.e., long positions) per-formed with confidence level ̨ anchored at 99%.

.2. VaR specifications

Linear: standard deviation calculated using a rolling window of1000 days complemented by Normal and t distributions (Section2.2.3, (5) and (6));

36 In order to categorise the markets the study follows the FTSE Global Equity Indexeries Country Classification, September 2009 update (FTSE, 2009).37 Footnote 24.38 Danielsson et al. (1998) stresses the excessive conservatism attained using

√10

actor to augment VaR.

a–tacbmatfi


Historical Simulation: estimations employ a moving sample of themost recent 1000 points (Section 2.2.1, (3));

Filtered HS: GARCH and EGARCH models applying MaximumLikelihood (ML), both featuring Normal and t distributions (Sec-tion 2.2.2, (4));

Conditional: GARCH and EGARCH obtained via ML. Normal andt variants applied to the distribution of standardised residualsrt/�t (Section 2.2.4, (7)–(10));EVT: POT format through methodology described in Section 2.2.5,(12) and (13).

.3. VaR validation

The accuracy of VaR models will be gauged employing Backtest-ng, in accordance with the stipulations laid out in Section 3.1.1.

.4. Stressed VaR

The calculation of sVaR is performed adhering to the instruc-ions stated in BCBS (2009a,b) explained in Section 3.2. The periodsf heavy losses for the indices selected become:

Stock exchange Stock index Stress period Loss posted

Sao Paulo Bovespa 25/09/2000 39.32%Brazil 28/08/2001Budapest Cetop20 01/03/2002 17.03%Hungary 28/02/2003Mumbai Sensex 01/10/2000 39.17%India 30/09/2001Prague Px 01/09/2000 41.87%Czech Republic 31/08/2001Jakarta Jkse 01/04/2000 42.02%Indonesia 31/03/2001Kuala Lumpur Klse 01/06/2000 45.96%Malaysia 31/05/2001

Buenos Aires Merval 01/10/1997 86.36%Argentina 30/09/1998Vilnius Omx 01/09/2000 29.39%Lithuania 31/08/2001Tunisia Tunindex 01/03/2002 20.08%Tunisia 28/02/2003Zagreb Crobex 01/04/2002 19.73%Croatia 31/03/2003

ote: Emerging markets above solid line, Frontier markets below.

.5. Minimum Capital Requirements (MCR)

Capital charges are estimated firstly on the grounds of (14) andacktesting (Section 3.1.1.) to comply with current directives andecondly applying (15) to cover the proposed framework.

. Results

.1. Stylised facts about data

Statistics deployed in Chart 2 depict the common patterns ofaily financial time series. Overall, the mean return is not signifi-antly different from zero, hence enabling driftless GARCH-familyepresentations (Sections 2.2.2, 2.2.4 and 2.2.5). An indicator tossess the behaviour of the tails of the distributions – skewness

assumes negative values for all series save for Tunisia and Croa-ia, meaning that left tails are longer than its right counterpartsnd vice versa for the two exceptions. Extreme Values appear moreoncentrated on the negative (positive) region of the distribution,olstering the idea that in general market crashes provoke asym-
etry (Jondeau and Rockinger, 1999) notwithstanding which the
lleged superiority of EGARCH schemes over GARCH remains a mat-er of empirical concern. Large Jarque–Bera statistics and kurtosisgures beyond 3 confirm the departure from normality, fact that


Chart 2Stylised facts about return series.

Parameter Brazil Hungary India Czech Rep. Indonesia Malaysia Argentina Lithuania Tunisia CroatiaIndex Bovespa Cetop20 Sensex Px Jkse Klse Merval Omx Vilnius Tunindex Crobex

Observ. 1976 1494 2600 3243 1925 2215 2777 2045 2496 2247Mean 0.00067 0.00082 0.0006 0.00036 0.00071 0.00043 0.00047 0.0008 0.00038 0.00089

(0.10) (0.01) (0.06) (0.09) (0.02) (0.05) (0.05) (0.00) (0.00) (0.00)Median 0.00128 0.0012 0.00126 0.00056 0.00122 0.00048 0.00104 0.00076 0.00016 0.00045Maximum 0.07336 0.04035 0.08592 0.07048 0.06733 0.05851 0.16117 0.0458 0.03041 0.12697Minimum −0.07539 −0.05553 −0.11809 −0.07077 −0.10933 −0.06342 −0.14765 −0.10216 −0.02125 −0.09029Std. dev. 0.01816 0.01232 0.01605 0.01192 0.01391 0.01044 0.02216 0.00905 0.00443 0.01435Skewness −0.21315 −0.45307 −0.33473 −0.28789 −0.72075 −0.21373 −0.16568 −0.72195 0.40024 0.28738Kurtosis 3.7020 4.5937 6.3879 5.6667 7.6550 8.5905 8.6782 13.5705 6.1949 12.9047Jarque–Bera 55.5327 209.2272 1291.993 1005.655 1904.667 2901.295 3743.358 9698.399 1128.208 9215.895

(0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00)Q(20) 30.35 28.39 74.81 81.5 42.24 107.96 40.72 114.54 601.49 17.18

(0.06) (0.10) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.64)Q2(20) 164.20 270.88 616.75 1100.1 154.49 656.12 1475.9 30.63 1618.7 431.97

(0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.06) (0.00) (0.00)q(0.01) −4.1320 −4.5432 −6.6995 −5.7068 −7.4724 −6.0699 −6.3172 −10.0735 −4.6805 −6.3293q(1.00) −2.5198 −3.0770 −2.7853 −2.8245 −2.9102 −3.0014 −3.1031 −2.9112 −2.5043 −2.8458q(2.50) −2.0972 −2.1934 −2.1932 −2.1823 −2.1103 −1.9295 −1.9803 −1.9949 −1.9290 −1.9576q(5.00) −1.7387 −1.6190 −1.5806 −2.3010 −1.6203 −1.4132 −1.4396 −1.5710 −1.5631 −1.4467q(10.00) −1.2576 −1.1831 −1.1831 −1.1366 −1.1319 −0.9744 −0.9980 −1.0523 −1.0950 −0.9298q(90.00) 1.1825 1.1983 1.0948 1.1440 1.0947 1.0664 1.1578 1.0888 1.1499 0.9498q(95.00) 1.5646 1.5663 1.5302 1.2624 1.5083 1.4864 1.5977 1.5511 1.6334 1.4450q(97.50) 1.8529 1.8159 1.8976 1.9397 1.9186 2.1111 2.2520 1.9414 2.1418 2.0612

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q(99.00) 2.3919 2.4801 2.2987 2.4556

q(99.99) 3.9221 3.2069 5.2104 5.5463

ote: p-Values in brackets.

oupled with the empirical quantiles of the standardised returneries overcoming the theoretical Gaussian ones hint at leptokurticistributions.

Values of Box–Ljung portmanteau Q(20) denote that it is notossible to reject the null of independence of linear returns at 95%nd 99%. Furthermore, the very high numbers for Q2(20) attainedor squared returns provide reasonable evidence of heteroskedas-icity and volatility clustering, thus paving the way for conditionalolatility models.

.2. VaR forecasts, Backtesting, bank capital and Basel regulations

.2.1. VaR forecasts, Backtesting and bank capitalRooting in the outcomes exposed in Charts 3 (number of

xceptions) and 4 (Basel Zones and increase in multiplication fac-or), the following reflections apply39:

a) Linear models prove to be inadequate as rejection – irrespec-tive of Normal or t distributions – is obtained in every stockexchange without distinction between Emerging and Frontiermarkets;

b) HS reveals unquestionably inaccurate, being discarded in allmarkets (except Indonesia on the brink of rejection);

(c) FHS delivers marginal improvements only in emerging markets.All FHS specifications in Frontier markets are disqualified too.Some scattered gains are observed using the GARCH filter, withNormal and t distributions working in Brazil, Hungary and India,Normal in Malaysia and t in Czech Republic while EGARCH filterfeaturing both Normal and t distributions only progresses inHungary. However, advances are very scarce as they alternatebetween 75% and 85% of additional capital required (between8 and 9 exceptions);

d) Conditional models represent a significant leap forward, bothfor GARCH and EGARCH techniques. The t distribution clearlyworks better than the Normal one for both models as it avoids

39 Appendix B.

wt

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6 2.9018 3.0802 2.8569 2.8418 3.03783 5.5170 5.8195 4.9595 6.4453 8.5865

the Red Zone in every market, either Emerging or Frontier(except Lithuania in EGARCH setting). The Normal distributiongives mixed results, failing in one Emerging Market (Indonesia)and two Frontier (Lithuania and Croatia) for GARCH and twoEmerging (Hungary and Indonesia) and two Frontier (Lithuaniaand Croatia) employing EGARCH. However, Normal distribu-tion cannot avoid the Yellow Zone and demands surchargesranging from 50% (Czech Republic, Malaysia and Argentinafor GARCH and India and Malaysia using EGARCH) to 85% inBrazil (EGARCH). The t distribution relieves the pressure onshareholders as they escape constituting extra capital in Indiaand Malaysia (GARCH and EGARCH), Argentina (EGARCH) andCroatia (GARCH): GARCH allows it in two Emerging Markets(India and Malaysia) and one Frontier (Croatia), while EGARCHfollows suit in two Emerging Markets (India and Malaysia)and one Frontier (Argentina), while falling in the Red Zone inLithuania;

e) EVT is undoubtedly the best performer. It does not require aux-iliary capital in any market, fact reflected in the highest VaR forevery Emerging and Frontier market for 01-January-09 amongthose models that avoid the Red Zone and consequent reesti-mation. Moreover, its levels could not be considered excessiveor insufficient as the proportion of VaR excessions stays closeto the stipulated value (only Hungary – 1.59% and India – 1.23%exceed the standard 1%, percentages not enough to claim capitalreinforcement in Chart 3).

.2.2. VaR outcomes and Basel regulations

.2.2.1. Current regulations. The procyclical behaviour of condi-ional VaR prompted the BCBS to demand the calculation of theverage 60-day VaR enhanced by the multiplication or hysteria40

hich level depends on Backtesting results (Chart 4). Amonghose models lying in the Green or Yellow Zones, FHS delivers

40 Jorion (1996) and Dowd and Hutchinson (2010), for example, employ the namehysteria factor” to refer to the “multiplication factor”.

310 A.F. Rossignolo et al. / Journal of Financial Stability 8 (2012) 303– 319

Chart 3Backtesting quantity and proportion of exceptions in forecast period.

Model index Exp. num. Lin. norm. Lin. t HS FHS G-N FHS G-t FHS E-N FHS E-t CV G-N CV G-t CV E-N CV E-t EVT

Brazil 2 18 16 16 9 8 12 11 8 6 9 9 21% 7.23% 6.43% 6.43% 3.61% 3.21% 4.82% 4.42% 3.21% 2.41% 3.61% 3.61% 0.80%

Hungary 3 23 18 15 8 9 9 9 7 5 10 8 41% 9.16% 7.17% 5.98% 3.19% 3.59% 3.59% 3.59% 2.79% 1.99% 3.98% 3.19% 1.59%

India 2 23 17 14 9 9 18 18 8 3 6 3 3Czech 1% 9.43% 6.97% 5.74% 3.69% 3.69% 7.38% 7.38% 3.28% 1.23% 2.46% 1.23% 1.23%Republic 3 27 20 16 10 9 12 12 6 5 8 6 0

1% 10.71% 7.94% 6.35% 3.97% 3.57% 4.76% 4.76% 2.38% 1.98% 3.17% 2.38% 0.00%Indonesia 2 24 17 9 18 15 23 22 13 7 12 8 1

1% 9.92% 7.02% 3.72% 7.44% 6.20% 9.50% 9.09% 5.37% 2.89% 4.96% 3.31% 0.41%Malaysia 2 16 12 10 8 10 10 11 6 4 6 4 1

1% 6.48% 4.86% 4.05% 3.24% 4.05% 4.05% 4.45% 2.43% 1.62% 2.43% 1.62% 0.40%

Argentina 2 22 19 17 13 13 13 13 6 5 7 2 11% 8.87% 7.66% 6.85% 5.24% 5.24% 5.24% 5.24% 2.42% 2.02% 2.82% 0.81% 0.40%

Lithuania 2 23 20 26 29 30 30 16 8 18 10 10 01% 9.47% 8.23% 10.70% 11.93% 12.35% 12.35% 6.58% 3.29% 7.41% 4.12% 4.12% 0.00%

Tunisia 2 14 10 16 17 17 19 18 8 6 8 6 11% 5.69% 4.07% 6.50% 6.91% 6.91% 7.72% 7.32% 3.25% 2.44% 3.25% 2.44% 0.41%

Croatia 2 30 20 19 11 26 13 20 11 4 11 6 01% 12.05% 8.03% 7.63% 4.42% 10.44% 5.22% 8.03% 4.42% 1.61% 4.42% 2.41% 0.00%

Notes: Emerging markets above solid line, Frontier markets below.List of abbreviations: Exp. Num., expected number; Lin., linear; G-N, GARCH-Normal; G-t, GARCH-t; E-N, EGARCH-Normal; E-t, EGARCH-t; CV: conditional volatility.

Chart 4Backtesting The three zone approach – increase in scaling factor k.

Model index Lin. Norm. Lin. t HS FHS G-N FHS G-t FHS E-N FHS E-t CV G-N CV G-t CV E-N CV E-t EVT

Brazil Red Red Red Yellow Yellow Red Red Yellow Yellow Yellow Yellow Green100% 100% 100% 85% 75% 100% 100% 75% 50% 85% 85% 0.00%

Hungary Red Red Red Yellow Yellow Yellow Yellow Yellow Yellow Red Yellow Green100% 100% 100% 75% 85% 85% 85% 65% 40% 100% 75% 0.00%

India Red Red Red Yellow Yellow Red Red Yellow Green Yellow Green Green100% 100% 100% 85% 85% 100% 100% 75% 0.00% 50% 0.00% 0.00%

Czech Red Red Red Red Yellow Red Red Yellow Yellow Yellow Yellow GreenRepublic 100% 100% 100% 100% 85% 100% 100% 50% 40% 75% 50% 0.00%Indonesia Red Red Yellow Red Red Red Red Red Yellow Red Yellow Green

100% 100% 85% 100% 100% 100% 100% 100% 65% 100% 75% 0.00%Malaysia Red Red Red Yellow Yellow Red Red Yellow Green Yellow Green Green

100% 100% 100% 75% 75% 100% 100% 50% 0.00% 50% 0.00% 0.00%

Argentina Red Red Red Red Red Red Red Yellow Yellow Yellow Green Green100% 100% 100% 100% 100% 100% 100% 50% 40% 65% 0.00% 0.00%

Lithuania Red Red Red Red Red Red Red Red Yellow Red Red Green100% 100% 100% 100% 100% 100% 100% 100% 75% 100% 100% 0.00%

Tunisia Red Red Red Red Red Red Red Yellow Yellow Yellow Yellow Green100% 100% 100% 100% 100% 100% 100% 75% 50% 75% 50% 0.00%

Croatia Red Red Red Red Red Red Red Red Green Red Yellow Green100% 100% 100% 100% 100% 100% 100% 100% 0.00% 100% 50% 0.00%

Note: Increase in scaling factor k pictured in second row of respective stock exchange.Emerging markets above solid line, Frontier markets below.

Chart 5Minimum Capital Requirements VaR MCR(VaR) – current directives.

Market/model Brazil Hungary India Czech Republic Indonesia Malaysia Argentina Lithuania Tunisia Croatia

Linear-N 28.54% 24.30% 25.25% 23.71% 22.50% 12.81% 25.45% 16.05% 7.76% 23.46%Linear-t 34.25% 29.23% 29.06% 28.10% 27.69% 15.18% 28.79% 18.76% 9.30% 28.24%Hist. Sim. 36.04% 33.14% 30.07% 32.50% 28.28% 18.35% 29.58% 21.04% 8.22% 33.56%FHS-G-N 51.62% 49.36% 41.35% 58.54% 32.28% 17.02% 20.71% 18.00% 8.81% 29.41%FHS-G-t 50.76% 52.41% 41.76% 55.80% 37.26% 18.36% 20.89% 18.61% 8.88% 20.97%FHS-E-N 40.37% 38.84% 31.53% 46.22% 24.02% 17.34% 20.06% 15.61% 7.80% 30.18%FHS-E-t 40.85% 38.88% 33.41% 48.43% 26.12% 17.33% 20.04% 16.26% 7.88% 24.31%CV-G-N 51.59% 45.48% 42.92% 48.69% 38.64% 15.83% 19.05% 22.57% 11.03% 29.05%CV-G-t 47.23% 41.42% 27.49% 51.58% 36.67% 11.84% 20.06% 27.26% 10.13% 18.02%CV-E-N 41.67% 40.23% 31.57% 43.32% 31.11% 15.81% 21.32% 19.64% 9.23% 30.75%CV-E-t 43.50% 37.61% 23.84% 44.50% 31.95% 11.92% 14.74% 24.82% 8.53% 17.53%EVT 41.80% 39.08% 28.56% 42.36% 36.96% 28.49% 17.60% 26.94% 10.83% 25.97%

Notes: Values in bold letters indicate specifications belonging to the Red Zone to be eventually excluded by regulators.List of abbreviations: Exp. Num., expected number; Lin., linear; G-N, GARCH-Normal; G-t, GARCH-t; E-N, EGARCH-Normal; E-t, EGARCH-t; CV: conditional volatility.


Chart 6The stressed VaR (sVaR) proposal – sVaR values.

Market/Model Brazil Hungary India Czech Republic Indonesia Malaysia Argentina Lithuania Tunisia Croatia


Notes: Values in bold letters indicate specifications belonging to the Red Zone to be eventually excluded by regulators.List of abbreviations: Exp. Num., expected number; Lin., linear; G-N, GARCH-Normal; G-t, GARCH-t; E-N, EGARCH-Normal; E-t, EGARCH-t; CV: conditional volatility.

Chart 7Minimum Capital Requirements sVaR: MCR(sVaR) – Proposed directives.

Market/model Brazil Hungary India Czech Republic Indonesia Malaysia Argentina Lithuania Tunisia Croatia


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otes: Values in bold letters indicate specifications belonging to the Red Zone to beist of abbreviations: Exp. Num., expected number; Lin., linear; G-N, GARCH-Norma

he highest values for k (Section 3.1) in most Emerging MarketsGARCH-Normal in Brazil, India and Malaysia, GARCH-t in Hungary

shared with both EGARCH-, India and Czech Republic) but thisircumstance is largely explained by the poor Backtesting perfor-ance (additional 85% demanded in every market except Malaysia,

5%). The same situation arises with conditional GARCH-Normalnd EGARCH-Normal in Tunisia (both 75%), EGARCH-Normal inrgentina (65%), GARCH-t in Lithuania (75%) and EGARCH-t in Croa-

ia (50%) and HS in Indonesia (85%). The most consistent capital
harges across Emerging and Frontier markets appear to be deliv-red from GARCH-t and EVT, with similar levels. However, it isoticeable that GARCH-t can only match EVT at the expense ofhe increases stipulated for the Yellow Zone (all stock exchanges
5woc

hart 8otal Minimum Capital Requirements MCR = MCR (VaR) + MCR (sVaR).

Market/model Brazil Hungary India Czech Republic Indon

Linear-N 55.86% 40.37% 42.28% 43.64% 43.78Linear-t 63.16% 49.95% 58.14% 48.96% 51.64Hist. Sim. 65.19% 56.10% 59.61% 53.65% 52.70FHS-G-N 72.65% 63.92% 57.65% 75.96% 50.49FHS-G-t 70.55% 68.03% 57.96% 71.89% 57.59FHS-E-N 64.16% 55.38% 48.05% 62.65% 40.17FHS-E-t 64.68% 55.46% 49.82% 65.10% 44.04CV-G-N 72.70% 58.89% 60.69% 64.18% 59.96CV-G-t 65.96% 53.77% 38.69% 67.66% 56.72CV-E-N 67.72% 58.30% 48.40% 61.30% 52.46CV-E-t 70.34% 54.59% 35.96% 61.95% 54.06EVT 58.90% 50.60% 40.39% 55.83% 57.35

otes: Values in bold letters indicate specifications belonging to the Red Zone to be eventist of abbreviations: Exp. Num., expected number; Lin., linear; G-N, GARCH-Normal; G-t,

ually excluded by regulators.GARCH-t; E-N, EGARCH-Normal; E-t, EGARCH-t; CV: conditional volatility.

xcept India, Malaysia and Croatia) and the subsequent scrutinyn the part of the regulators. One of the major concerns regard-ng the employment of EVT residing in the high amount of capitalemanded could be averted as they do not exceed quantities giveny other heavy-tailed specifications like GARCH-t or EGARCH-tith the further advantage of falling within the Green Zone, thisay avoiding periodic revisions demanded by the Yellow Zone

Chart 5).

.2.2.2. Proposed regulations. The calculation of the sVaR coincidesith that of the base VaR, except for the point that it must be carried

ut over a 12-month continuous term wreaking havoc on the finan-ial position of the company. The outcome in Chart 6 shows that,

esia Malaysia Argentina Lithuania Tunisia Croatia

% 34.16% 54.04% 28.69% 14.39% 36.22%% 39.36% 63.59% 33.42% 16.86% 59.19%% 44.17% 65.94% 32.89% 15.16% 64.51%% 33.70% 65.79% 26.71% 12.82% 44.63%% 37.01% 67.12% 26.50% 12.89% 33.70%% 33.79% 61.54% 23.89% 12.15% 44.05%% 33.78% 62.64% 24.40% 12.21% 38.01%% 31.65% 55.32% 34.99% 16.18% 44.19%% 23.76% 58.30% 39.75% 14.79% 28.41%% 31.02% 59.42% 32.84% 14.49% 44.99%% 23.36% 41.93% 39.71% 13.30% 27.23%% 56.96% 51.11% 41.75% 15.89% 39.50%

ually excluded by regulators.GARCH-t; E-N, EGARCH-Normal; E-t, EGARCH-t; CV: conditional volatility.

312 A.F. Rossignolo et al. / Journal of Financial Stability 8 (2012) 303– 319

Chart 9Current MCR: Loss coverage and Maximum Daily Loss.

Market/index Loss coverage [1] Maximum daily loss [2]

Brazil 3.46 41.80%Hungary 3.03 39.08%India 2.46 28.56%Czech Republic 2.62 42.36%Indonesia 3.37 36.96%Malaysia 2.85 28.49%

Average Emerging 2.97 36.21%

Argentina 1.36 17.60%Lithuania 3.82 26.94%Tunisia 2.16 10.83%Croatia 2.41 25.97%

Average Frontier 2.44 20.33%

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Chart 11Proposed MCR-Loss coverage and Maximum daily loss-variation over current MCR.

Market/index Loss coverage [1] Variation overpresent MCR [2]

MCR = Maximumdaily loss [3]

Brazil 4.87 40.92% 58.90%Hungary 3.93 29.50% 50.60%India 3.48 41.40% 40.39%Czech Republic 3.45 31.80% 55.83%Indonesia 5.24 55.16% 57.35%Malaysia 5.71 99.94% 56.96%

Average Emerging 4.44 49.79% 53.34%

Argentina 3.95 190.37% 51.11%Lithuania 5.93 54.99% 41.75%Tunisia 3.17 46.70% 15.89%Croatia 3.67 52.12% 39.50%

Average Frontier 4.18 86.05% 37.06%

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ote: Loss coverage = MCR(VaR)/maximum loss forecast period.

xcluding disqualified models in Red Zone in Chart 4 (bold letters),VT delivers the highest sVaR values for highly strung periods forither Emerging or Frontier Markets. Nevertheless, after applyinghe former Backtesting results (Chart 7), the balance switches tohe rest of the schemes on the grounds of the 60-day average and,

ost importantly, the penalties envisaged for inaccuracy (betweenrackets): HS (Indonesia – 85%–), GARCH-Normal (India – 75%),ARCH-t (Argentina – 40%–), EGARCH-Normal (Czech Republic –5%– and Tunisia – 75%–), EGARCH-t (Brazil – 85%– and Hungary75%–) and EVT retaining three markets (Malaysia, Lithuania androatia).

The new context proposed by the BCBS brings a more levelledanorama with HS, GARCH-Normal, GARCH-t, EGARCH-Normal,GARCH-t and EVT claiming top spots, even though for all mod-ls but EVT the result is largely due to the add-in factor verifiedn Backtesting (Chart 7). Interestingly enough, EVT presents theowest total amounts in three Emerging markets and stays nearhe bottom in the remaining ones. Chart 8 conveys the picturef BCBS’s planned increase in the capital base with the additionf capital buffers based on sVaR: the total capital charge exhibitshree markets where GARCH-Normal delivers the highest VaR val-es (Brazil, India and Tunisia), two for FHS: GARCH-t in Hungarynd Czech Republic, one for EGARCH-Normal (Argentina) and EVTn Indonesia, Malaysia, Lithuania and Croatia, all of them exceptVT stemming from high multiples of k (Chart 4). Furthermore, EVTresents the lowest total amounts in three cases (Brazil, Hungary
nd Czech Republic), edging the lower end in the rest of the marketsexcept Malaysia).
pi2

hart 10tandard deviation and Maximum daily loss – sample period vs. forecast period.

Market Standard deviationsample [1]

Standard deviationforecast [2]

Standard devivariation [3]

Brazil 1.82% 3.29% 81.28%

Hungary 1.23% 2.90% 135.72%

India 1.60% 2.86% 78.53%

Czech Rep. 1.19% 3.04% 155.18%

Indonesia 1.39% 2.18% 56.47%

Malaysia 1.05% 1.37% 31.06%

Average Emerging

Argentina 2.22% 2.86% 29.07%

Lithuania 1.00% 2.01% 121.58%

Tunisia 0.44% 0.51% 14.40%

Croatia 1.43% 2.62% 82.38%

Average Frontier

ote: Loss coverage = MCR(sVaR)/maximum loss forecast period.

. Conclusions: VaR models and Basel regulations

The evidence collected appears enough to discard Linear mod-ls, HS and FHS. Although the standard deviation for Linear modelss allowed to change in time, it is unable to capture the dynamics ofhe latent volatility. No improvement is virtually recorded employ-ng a heavy-tailed t distribution instead of the Normal one as thenderlying risk measure is inherently flawed. HS reaffirms all theisadvantages mentioned in Section 2.2.1, for example, last VaRvercoming 60-day VaR average in Hungary reveals the presencef the Ghost or Shadow effect. Usage of the empirical distributionf residuals in FHS delivers only slight gains in some Emergingarkets (t distribution), albeit in overall it remains ineffective.The significant progress derived from conditional models high-

ights the need to resort to these schemes. The results suggesthat the EGARCH technique brings no significant advantage overARCH, in turn meaning that the leverage effect marked by Choprat al. (1992) and Cho and Engle (1999) is hardly noticeable givenhat the adjustments brought about by EGARCH-t are equal orreater than their GARCH-t counterparts (except Argentina). More-ver, the density assumption exerts dominance over the particularpecification: GARCH-t and EGARCH-t improve the performancef their Normal counterparts, therefore making unavoidable thesage of heavy-tailed distributions.

Findings are highly supportive of the EVT approach in com-
arison with its competitors. Its application would have shielded
nstitutions from huge losses produced in the event of the008 crisis and prevented the constitution of extra-capital while

ation Maximum dailyloss sample [4]

Maximum dailyloss forecast [5]

Maximum dailyloss variation [6]

7.54% 12.10% 60.45%5.55% 12.89% 132.11%

11.81% 11.60% −1.73%7.08% 16.19% 128.70%

10.93% 10.95% 0.19%6.34% 9.98% 57.34%

8.21% 12.28% 49.65%

14.76% 12.95% −12.28%5.87% 7.05% 19.95%2.12% 5.00% 135.51%9.03% 10.76% 19.21%

7.95% 8.94% 12.49%


Chart 12Sensitivity analysis – total MCR with varying scaling factors mc and ms .

Case no. Multiplesmc − ms Brazil Hungary India Czech Republic Indonesia Malaysia Argentina Lithuania Tunisia Croatia

1 mc = 3 41.80% 39.08% 28.56% 42.36% 36.96% 28.49% 17.60% 26.94% 10.83% 25.97%2 mc = 3/ms = 0 50.61% 42.30% 35.87% 45.85% 43.42% 35.61% 30.99% 37.76% 12.07% 30.35%3 mc = 3/ms = 0.5 50.61% 42.30% 35.87% 45.85% 43.42% 35.61% 30.99% 37.76% 12.07% 30.35%4 mc = 3/ms = 1 50.61% 42.92% 35.87% 46.85% 43.76% 37.98% 30.99% 37.76% 12.51% 30.48%5 mc = 3/ms = 1.5 50.61% 44.84% 35.87% 49.09% 47.16% 42.72% 34.35% 37.76% 13.36% 32.74%6 mc = 3/ms = 2 53.20% 46.76% 36.45% 51.34% 50.55% 47.47% 39.94% 37.76% 14.20% 34.99%7 mc = 3/ms = 2.5 56.05% 48.68% 38.42% 53.58% 53.95% 52.21% 45.52% 39.28% 15.04% 37.25%8 mc = 3/ms = 3 58.90% 50.60% 40.39% 55.83% 57.35% 56.96% 51.11% 41.75% 15.89% 39.50%9 mc = 3.5 48.76% 45.59% 33.32% 49.42% 43.12% 33.24% 20.53% 31.43% 12.70% 30.30%

10 mc = 3.5/ms = 0 57.58% 48.81% 40.63% 52.91% 49.58% 40.35% 33.92% 42.25% 13.88% 34.68%11 mc = 3.5/ms = 0.5 57.58% 48.81% 40.63% 52.91% 49.58% 40.35% 33.92% 42.25% 13.88% 34.68%12 mc = 3.5/ms = 1 57.58% 49.43% 40.63% 53.91% 49.92% 42.73% 33.92% 42.25% 14.32% 34.81%13 mc = 3.5/ms = 1.5 57.58% 51.35% 40.63% 56.15% 53.32% 47.47% 37.29% 42.25% 15.16% 37.06%14 mc = 3.5/ms = 2 60.16% 53.27% 41.21% 58.40% 56.71% 52.22% 42.87% 42.25% 16.01% 39.32%15 mc = 3.5/ms = 2.5 63.01% 55.20% 43.18% 60.64% 60.11% 56.96% 48.45% 43.77% 16.85% 41.58%16 mc = 3.5/ms = 3 65.86% 57.12% 45.15% 62.89% 63.51% 61.71% 54.04% 46.24% 17.69% 43.83%17 mc = 4 55.73% 52.10% 38.09% 56.48% 49.28% 37.98% 23.47% 35.91% 14.44% 34.63%18 mc = 4/ms = 0 64.54% 55.32% 45.39% 59.97% 55.75% 45.10% 36.86% 46.74% 15.68% 39.01%19 mc = 4/ms = 0.5 64.54% 55.32% 45.39% 59.97% 55.75% 45.10% 36.86% 46.74% 15.68% 39.01%20 mc = 4/ms = 1 64.54% 55.94% 45.39% 60.97% 56.08% 47.47% 36.86% 46.74% 16.12% 39.14%21 mc = 4/ms = 1.5 64.54% 57.87% 45.39% 63.21% 59.48% 52.22% 40.22% 46.74% 16.97% 41.39%22 mc = 4/ms = 2 67.13% 59.79% 45.97% 65.46% 62.87% 56.97% 45.80% 46.74% 17.81% 43.65%23 mc = 4/ms = 2.5 69.98% 61.71% 47.94% 67.70% 66.27% 61.71% 51.39% 48.26% 18.65% 45.90%24 mc = 4/ms = 3 72.83% 63.63% 49.91% 69.94% 69.67% 66.46% 56.97% 50.73% 19.50% 48.16%25 mc = 4.5 62.69% 58.61% 42.85% 63.54% 55.44% 42.73% 26.40% 40.40% 16.24% 38.95%26 mc = 4.5/ms = 0 71.51% 61.83% 50.15% 67.03% 61.91% 49.85% 39.79% 51.23% 17.49% 43.33%27 mc = 4.5/ms = 0.5 71.51% 61.83% 50.15% 67.03% 61.91% 49.85% 39.79% 51.23% 17.49% 43.33%28 mc = 4.5/ms = 1 71.51% 62.46% 50.15% 68.03% 62.24% 52.22% 39.79% 51.23% 17.93% 43.47%29 mc = 4.5/ms = 1.5 71.51% 64.38% 50.15% 70.27% 65.64% 56.97% 43.15% 51.23% 18.77% 45.72%30 mc = 4.5/ms = 2 74.09% 66.30% 50.73% 72.52% 69.03% 61.71% 48.74% 51.23% 19.61% 47.98%31 mc = 4.5/ms = 2.5 76.94% 68.22% 52.70% 74.76% 72.43% 66.46% 54.32% 52.75% 20.46% 50.23%32 mc = 4.5/ms = 3 79.79% 70.14% 54.67% 77.00% 75.83% 71.20% 59.91% 55.22% 21.30% 52.49%33 mc = 5 69.66% 65.13% 47.61% 70.60% 61.60% 47.48% 29.33% 44.89% 18.05% 43.28%34 mc = 5/ms = 0 78.48% 68.35% 54.91% 74.09% 68.07% 54.60% 42.72% 55.72% 19.29% 47.66%35 mc = 5/ms = 0.5 78.48% 68.35% 54.91% 74.09% 68.07% 54.60% 42.72% 55.72% 19.29% 47.66%36 mc = 5/ms = 1 78.48% 68.97% 54.91% 75.09% 68.40% 56.97% 42.72% 55.72% 19.73% 47.79%37 mc = 5/ms = 1.5 78.48% 70.89% 54.91% 77.33% 71.80% 61.72% 46.09% 55.72% 20.58% 50.05%38 mc = 5/ms = 2 81.06% 72.81% 55.49% 79.57% 75.19% 66.46% 51.67% 55.72% 21.42% 52.30%39 mc = 5/ms = 2.5 83.91% 74.73% 57.46% 81.82% 78.59% 71.21% 57.25% 57.24% 22.26% 54.56%40 mc = 5/ms = 3 86.76% 76.65% 59.43% 84.06% 81.99% 75.95% 62.84% 59.71% 23.11% 56.82%

Notes: Cases 1, 3, 4, 5, 8, 9 and 17 are highlighted in bold letters as referenced in main text.Case 1: Current Basel II directives (no sVaR).CCC

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Acknowledging that proposed regulations will surely lead toan increase in MCR of a remarkable quantity, a sensitivity anal-ysis considering different scenarios regarding the level of the

ase 8: Proposed Basel mandate (2009).ases 3, 4 and 5: Lessened ms values: 1.5, 1 and 0.5.ases 9 and 17: Augmented mc values: 3.5 and 4, no sVaR computed.

imultaneously building up a capital base not excessive in relationo the rest of the specifications.

. The impact of Basel proposed directives: a sensitivitynalysis

Grounding on the results of Section 5.2, this chapter applies theost accurate model-EVT – to evaluate the impact of the BCBS’s

roposed regulations to toughen the required minimum capitalevels.

According to the present directives, MCR would provide enoughoverage for losses bigger than twice the size of the maximumeficit posted in the forecast period (except in Argentina, whichoefficient is 1.36) (Chart 9, Column [1]). This assertion means thathe maximum daily loss dealt with by the model peaks 42.36%n Czech Republic (minimum 28.49% in Malaysia) in Emerging

arkets, and 26.94% (Lithuania) and 10.88% (Tunisia) in Frontierxchanges (Chart 9, Column [2]). Given that in general higher
olatility and higher losses translate into more elevated capitalushions (Brooks et al., 2000), there is evidence that Emerging mar-ets observed an upsurge of volatility steeper than Frontier onesHungary and Czech Republic more than doubled sample values, 1
hart 10, Columns [1]–[3]) and of maximum daily losses as well50% and 12%, respectively, average values in Chart 10, Columns4]–[6]).

The sVaR modification seems to achieve the objectives by driv-ng the total coverage to more than four times the maximumoss of the forecast period in both Emerging and Frontier mar-ets (Chart 11, Column [1]). These figures represent an averagencrease of approximately 50% and 86% in Emerging and Frontier

arkets, respectively (Chart 11, Column [2]) – with Brazil toppinghe table with 59% in the former and Argentina reaching 51% inhe latter in terms of the MCR41 – (Chart 11, Column [3]). Poten-ial deficits determined by new MCR would appear relatively highor both groups compared with the shortfalls recorded in timesf crisis, hence making institutions put by excessively unproduc-ive capital levels which could otherwise be directed to crediturposes.

41 Proposed regulations mean an increase in the MCR of 190% in Argentina and00% in Malaysia, for example.

3 Financial Stability 8 (2012) 303– 319

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ultipliers mc and ms and their respective capital level (i.e., great-st daily shortfall) was performed in Chart 12.42 The assessment ofhe outcomes discloses interesting implications displayed in Chart3. In the first place it is judged that as national regulators oughto approve the election of the stress period to calculate sVaR theyould also dictate the level of the multiplier ms to be applied toheir respective markets, and there seems to be proof that thisevel might well be set below 3 and still provide considerable cov-rage. For instance, if the factor ms were fixed at 1.5, 1.0 or 0.5

Chart 13, Columns [1]–[6], the maximum daily loss covered inerms of the greatest shortfall in the forecast period would stilldge 3.50–3.70 (on average) in Emerging markets and 3.20–3.40 inrontier ones (values which represent daily deficits beyond 40% forhe former and 25% for the latter, maximums and minimums forach category being Brazil-51% and Malaysia-36%, Lithuania-38%nd Tunisia-12%). Even though those amounts embody an aver-ge increase from current directives of more than 17% in Emergingarkets and 36% in Frontier stock exchanges (Chart 14, Columns

1], [3] and [5]), they do not place so heavy a burden on institutionss in the case of sVaR (decrease between 15% and 21% in the formernd 14–23% in the latter) (Chart 14, Columns [2], [4] and [6]).

The second observation indicates that the goals intended byhe BCBS could also be accomplished omitting sVaR43 as a certainegree of additional conservatism may be attained augmentinghe multiplicative factor mc for the base VaR and simultaneouslyhasing out sVaR. As the results hint, if that constant44 were toe recalibrated at 3.5 or 4, capital levels would grow at 17% and3%, respectively, with reference to present directives (Chart 14,olumns [7] and [9]), at the same time declining (average per-entages) 20% and 9% in Emerging markets45 and 32% and 22% inrontier ones evaluated against the sVaR proposition (Chart 14,olumns [8] and [10]). Capitals set in this fashion would allowanks to endure average losses of almost or more than three timeshe maximum daily loss of the forecast period for both marketsmc = 3.5 delivers 49% for Czech Republic and 31% for Lithuania,hereas mc = 4 matches 56% and 36%, respectively). For Emerg-

ng stock exchanges, multiples of 3.5 and 4 protect against a 42%nd 48% daily losses while for Frontier ones, shortfalls amount to4% and 27%, respectively, in average values (Chart 13, Columns7]–[10]).

A quick glance at Argentina may be useful to mark that thelanned modification of MCR should confer national regulatorsnough flexibility to rule over sVaR. If authorities deem a capitalevel providing institutions coverage below twice the size of the

aximum daily loss in the forecast period as insufficient (no sVaRalculated), they could demand the constitution of capital cushionssing sVaR in order to push this amount to 2.39 (ms = 1 and 0.5)Chart 13, Columns [3] and [5]). Concurrently, Lithuania, for exam-le, could well scrap sVaR and increase mc to 3.5 to sufficiently face

daily loss of 31% (Chart 13, Column [8]).

. Final conclusions on Basel proposed directives

The evidences collected attempt to provide quantitative infor-ation about the impact of the BCBS’s proposal to increase MCR

42 MCR equal the maximum daily loss, as capital levels are built to match shortfallsn portfolios.43 Provided a heavy-tailed approach to estimate the base VaR (cVaR) is employed.44 Given that k = 0 for EVT models in all markets, mc = 3(1 + k) = 3 or the correspond-ng figure employed in the sensitivity analysis. Analogously, ms = 3(1 + k) = 3 or theespective constant stated. The former reflection applies to Chart 12.45 It is acknowledged, however, that Hungary and Czech Republic observe annhancement of MCR of approximately 3% and 1%, respectively, where mc to beet at 4. C

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cial Stability 8 (2012) 303– 319 315

onstituted by financial institutions in Emerging and Frontier stockarkets. To begin with, supra-national regulators should demand

sage of techniques capable of dealing with large fluctuations inhe future – particularly EVT – hence discouraging or banning themployment of traditional methodologies which provide capitaluffers only for common market variations (most notably Linear,S, FHS or Normal specifications). For every market researched, the

ntended variation characterised by sVaR applying a base multiplef ms = 3 in addition to the current MCR appears somewhat exces-ive and immobilises funds unnecessarily. Therefore, both factorsould be dissociated and calculated independently at the discre-ion of national supervisors in view of the particular patterns of theespective markets. In this sense, one viable alternative may involveorking out a combination between current MCR and a lighter sVaR

alue. Another potential version would see sVaR term scrapped andhe base multiplication mc factor lifted to some number between 3nd 4 (for example), at the discretion of the corresponding domesticontroller.

This last suggestion echoes the Japanese position (2008) regard-ng the BCBS’s proposition concerning market risk demands. It iselieved that the adoption of the planned stringent measures willurely give birth to disincentives to devise accurate VaR method-logies, as the shortcomings of the models will be compensated byhe combination of add-in factors and penalty constants. In con-rast, the implementation of sound heavy-tailed techniques likeVT could provide extensive coverage, avoid the building of super-uous capital buffers and at the same time allow the institutions toatch huge future losses without incurring in high development

osts to estimate sVaR.

cknowledgment

Meryem Duygun Fethi would like to acknowledge the supportf the University of Leicester’s sabbatical scheme.

ppendix A. Insights into VaR definition

McNeil et al. (2005, pp. 38–40) provide a concise explanation toxpress the VaR formula in terms of quantiles. Recalling that VaRs defined by:

aR (˛) = inf{l ∈ R : P(L > l) ≤ (1 − ˛)} = inf{l ∈ R : FL(l) ≥ ˛}here FL (or simply F46) denotes the loss distribution, VaR rep-

esents in probabilistic terms a quantile of the loss distributionunction F.

Acknowledging that the generalised inverse func-ion of an increasing function G : → is defined as←(y) := inf {x ∈ : G(x) ≥ y}, then, for some distribution func-

ion F, the generalised inverse F← is called the quantileunction of F. As ̨∈ (0;1) the ˛-quantile of F surges from:˛(F) := F←(˛) = inf {x ∈ : F(x) ≥ ˛}, noting that q˛(F) := q˛(X) for aandom variable X with distribution function F. Supposing that Fs continuous and strictly increasing, q˛(F) := F−1(˛), F−1 being thenverse of the distribution function F. Furthermore, the authorsmphasise that a general point x0 such that x0∈ is the ˛-quantilef any distribution function F provided that: (a) F(x0) ≥ ˛, and (b)
(x) < ̨ – for all x < x0 – are both met.
Assuming that the loss distribution function F follows a Normalistribution with mean � and variance �2, it is possible to write47

46 Footnote 4.47 It is important to bear in mind that these examples portray VaR formulas usinghe mean � of the corresponding loss distribution. In the case of driftless series likehose pictured in the body of the article, � = 0 and its value would be omitted fromhe respective VaR formulas.

3 Finan

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ieattmttoadsresults situate in the Yellow Zone with scattered appearances in theRed region (Hungary, Indonesia, Lithuania and Croatia for EGARCHand Indonesia and Croatia for GARCH). For those stock exchanges


aR(˛) = � + � Ф−1(˛) − ̨∈ (0;1), where Ф represents the distribu-ion function of a standardised Normal and Ф−1(˛) the ˛-quantilef Ф. McNeil et al. (2005) also prove the quantile definition show-ng that as the loss distribution function F is a strictly increasingunction, F(VaR˛) = ˛. Consequently,

(L ≤ VaR˛) = P[

L − �

�≤ ˚−1(˛)

]= ˚[˚−1(˛)] = ˛

The expression VaR(˛) = � + � Ф−1(˛), adequate for losses fol-owing a standard Normal distribution, could be extended to otherocation-scale families like the Student-t or the Generalised Paretoistribution (GPD). For the former example, if losses L ∼ t(d, �, �2),

.e., losses follow a t distribution with d degrees of freedom, theaR formula becomes VaR(˛) = � + �

√(d − 2)d−1t−1

d(˛), with t

ymbolising the distribution function of the standard t distributionnd �

√(d − 2)d−1 its variance. Finally, in the latter case, if losses

eyond the appropriate threshold u are distributed according to aPD G�,�(y), the VaR would be given by

aR(˛) = � + �

{u + �̂

�̂

[(1 − ˛

w/n

)−�̂

− 1

]},

ith �, �, u, �̂, �̂, w and n keeping the respective meanings indicated

n the main text andG−1(˛) ={

u + �̂

�̂

[(1−˛w/n

)−�̂

− 1

]}denoting

he ˛-quantile of the GPD.The above procedure requires (at least) the practical estima-

ion of the mean and variance of the loss distribution −� and �2,hich usually roots in the historical change in the risk-factor data

returns). Both parameters can be calculated following two generallternative routes:

(a) Unconditionally: mean and variance are simply obtained usingthe sample mean and variance under the supposition that theseries of returns belong to a stationary process;

b) Conditionally: the data are considered as time series where thesubscript t + 1, either in �t+1 or �t+1

2, indicate the conditionalmean and variance employing information up to time t. Esti-mates of the statistical moments allowing the computation ofboth parameters are obtained by means of the formal estima-tion of any time series model, like those belonging to the GARCHfamily analysed in the main text.

The present study is focused on the comparison among the mostidespread VaR models and, in this vein, it is assumed that:

a) It is always possible to calculate the historical change in therisk-factor data, i.e., returns always exist;

b) At least the first four moments of all time series exist and arefinite;

c) The processes dealt with are at a minimum weakly, second-order or covariance stationary. Hence, the behaviour of the timeseries is similar at any time an observation is carried out irre-spective of the time span between two observations.

ppendix B. Some reflections on the behaviour of theodels across markets

The behaviour of the models in the Backtesting period could be
omewhat inferred from the characteristics of the distribution ofeturns during that time span. Chart B1 displays the common statis-ics recorded in the year 2008 for every index as well as selectedariations comparing the sample and prediction terms These s

utcomes must be assessed in connection with Charts 2 and 3 in theain text which depict the stylised facts about asset return series

nd the quantity of excessions recorded in Backtesting.48

It is not surprising that linear models irrespective of the distribu-ion applied are unable to cope with the significant hike in volatilitywith peaks of 155.18% in Czech Republic and 121.58% in Lithua-ia for Emerging and Frontier markets, respectively): the inherentaws of the standard deviation as a risk measure chiefly charac-erised by the equal weighting structure do not allow the clustersf high variations typical of crisis times (Messina, 2005). The onlyerformances (if any) worth of being singled out are Malaysia andunisia, i.e., those stock exchanges with the lowest increase intandard deviation (31.06% and 14.40%, respectively); although thetudent-t scheme delivers more accurate results, it neverthelessemains insufficient to drive all the models out of the Red Zone. HSives practically uniform results across countries no matter theirocation, in a way similar to those of the Linear specifications: theramatic changes observed in the shape of the distributions poseifficulties for the method at the time of tracking the volatilitypikes. Indonesia represents the solitary instance where HS avoidshe regulatory disqualification: 9 exceptions (Yellow Zone with 85%urcharge) most likely due to the relatively similar kurtosis and% quantile value. However, the examples of India and Lithuaniamainly) cast some doubt on the former explanation, given thathe kurtosis and the cut-off points for the 1% quantile are reduced.dditional research into the length of the window used to calcu-

ate HS values may shed some light on this method, although it isn established fact throughout the financial literature that its snagsargely overcome its advantages hence making HS an inadequatend unreliable method to calculate VaR.

The introduction of the conditional modelling in FHS helps HSo enhance its accuracy though the improvement fails to report lesshan 8 exceptions (high Yellow Zone) considering all countries inoth categories. In general terms the GARCH variant exhibits bettererformance than its exponential counterpart, especially in thosearkets where the skewness reverted from negative to positive

Brazil) or decreased significantly (India and Lithuania,49 80.28%nd 76.66%, respectively). Finally, the application of the Student-

likelihood function either for GARCH or EGARCH models fallshort of achieving a considerable impact for Backtesting purposes:t may be surmised that the changes verified in the distribution ofeturns between the sample and forecast periods hamper the preci-ion that both specifications (irrespective of the likelihood functionmployed to derive them) may be supposed to achieve.

Remarkable advances are attained with conditional volatil-ty models resorting to a pre-specified statistical distribution,ither Normal or Student-t. In effect, comparing the distributionalssumptions of both models (i.e., Normal variants against Student-

ones for GARCH and EGARCH separately) it may be appreciatedhat the Student-t representations avoid Red Zone for every stock

arket (except in Lithuania) and always report less exceptionshan its Gaussian counterparts; the leptokurtic characteristics ofhe distribution of returns (i.e., the kurtosis and the cut-off pointsf the extreme quantiles) in the forecast period broadly provide anppropriate explanation about the prevalence of the heavy-tailedistribution. The Normal distribution, using GARCH or EGARCHpecifications, struggle to drive the models to the Green Zone; most

48 Chart B contains all the figures enunciated in the present Appendix B.49 The EGARCH-t specification achieves the smallest quantity of violations, thoughtill in the Red Zone.

A.F.

Rossignolo

et al.

/ Journal

of Financial

Stability 8 (2012) 303– 319

317

Chart B1Forecast period: Basic statistics and selected variations over sample period.

Index/periodparameter

BrazilForecast

BrazilVariation

HungaryForecast

HungaryVariation

IndiaForecast

ndiaVariation

Czech RepublicForecast

Czech RepublicVariation

IndonesiaForecast

IndonesiaVariation

MalaysiaForecast

MalaysiaVariation

Emerging marketsMean −0.0021 CS −0.0030 CS −0.0030 CS −0.0029 CS −0.0002 CS −0.0020 CSMedian 0.0013 CS −0.0015 CS −0.0034 CS −0.0023 CS 0.0007 −39.70% −0.0018 CSMaximum 0.1368 86.45% 0.1038 157.32% 0.0790 −8.04% 0.1236 75.42% 0.0762 13.22% 0.0406 −30.69%Minimum −0.1210 60.45% −0.1289 132.11% −0.1160 −1.73% −0.1619 128.70% −0.1095 0.19% −0.0998 57.34%Std. Dev. 0.0329 81.28% 0.0290 −10.35% 0.0286 78.53% 0.0304 155.18% 0.0218 56.47% 0.0137 31.06%Skewness 0.2141 CS −0.4081 72.99% −0.0660 −80.28% −0.2895 0.74% −0.4761 −33.41% −1.3972 488.64%Kurtosis 6.0540 63.32% 7.9694 131.52% 3.8726 −39.46% 9.5137 67.94% 6.9340 −9.35% 13.3380 54.98%q(0.01) −3.5989 – −10.5186 – −3.9116 – −13.5269 – −5.0114 – −7.0129 –q(1.00) −2.6641 – −8.2961 – −2.3785 – −8.2298 – −2.8134 – −2.5290 –q(2.50) −2.0986 – −5.6385 – −1.9723 – −5.3746 – −2.1435 – −2.1057 –q(5.00) −1.5841 – −4.3916 – −1.6117 – −4.8777 – −1.7318 – −1.3903 –q(10.00) −1.0765 – −2.5657 – −1.2110 – −2.6493 – −1.0475 – −0.9906 –q(90.00) 0.9567 – 1.8798 – 1.2433 – 2.0700 – 0.9133 – 1.0874 –q(95.00) 1.4714 – 2.8236 – 1.9230 – 2.5074 – 1.4765 – 1.5895 –q(97.50) 2.2898 – 4.3958 – 2.0514 – 4.1453 – 2.0911 – 1.8732 –q(99.00) 2.8177 – 7.3298 – 2.2663 – 8.3505 – 2.8274 – 2.2148 –q(99.99) 4.2108 – 8.3519 – 2.8581 – 10.3114 – 3.5040 – 3.0891 –d(hat) 5.9646 −52.24% 5.2074 −32.67% 10.8760 88.61% 4.9211 −21.28% 5.5252 4.44% 4.5804 −9.66%

Index/period parameter ArgentinaForecast

ArgentinaVariation

LithuaniaForecast

LithuaniaVariation

TunisiaForecast

TunisiaVariation

CroatiaForecast

CroatiaVariation

Frontier marketsMean −0.0028 CS −0.0045 CS 0.0004 −9.08% −0.0045 CSMedian 0.0000 −99.06% −0.0027 CS 0.0002 7.88% −0.0019 CSMaximum 0.1043 −35.27% 0.1100 140.20% 0.0361 18.84% 0.1478 16.39%Minimum −0.1295 −12.28% −0.0911 −10.82% −0.0500 135.51% −0.1076 19.21%Std. Dev. 0.0286 29.07% 0.0201 121.58% 0.0051 14.40% 0.0262 82.38%Skewness −0.6433 288.06% −0.1687 −76.66% −0.2513 CS 0.3261 13.33%Kurtosis 7.1952 −17.21% 11.2690 −17.13% 11.9022 91.86% 9.1376 −29.36%q(0.01) −4.4196 – −4.2838 – −9.2443 – −3.9192 –q(1.00) −3.3711 – −1.0517 – −2.4846 – −2.8429 –q(2.50) −2.3319 – −0.6379 – −1.8983 – −2.3368 –q(5.00) −2.1824 – −0.4466 – −1.4600 – −2.1366 –q(10.00) −1.0754 – −0.2126 – −1.0590 – −0.9604 –q(90.00) 0.9481 – 0.7537 – 1.1042 – 0.9018 –q(95.00) 1.0455 – 0.9623 – 1.5497 – 1.0393 –q(97.50) 1.7460 – 1.1385 – 1.9665 – 1.5874 –q(99.00) 2.7071 – 1.5516 – 2.5715 – 2.9968 –q(99.99) 3.7384 – 2.5005 – 6.7644 – 5.7536 –d(hat) 5.4302 7.44% 4.7256 3.49% 4.6740 −20.41% 4.9776 8.12%

Note: “CS” stands for “Change of sign”

3 Finan

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here the models avoid being disqualified, the failure of the Normalistribution may also be assessed in terms of the reduced values ofhe d(hat)50 in the prediction term especially in Brazil, Hungary,ndonesia, Lithuania and Croatia.

The joint assessment of the d(hat) and the values of the degreesf freedom delivered by the respective ML optimisation appearsseful to understand the difference in the performance between theonditional volatility specifications.51 The figures belonging to theample and forecast periods convey that the majority of the Emerg-ng markets suffered a drop in d(hat) (except India and Indonesia)

hile in Frontier ones the situation reverts (excluding Tunisia). Fur-hermore, those countries that experienced an increase in d(hat)n the forecast period compared to the sample one also had itsurtosis decreased and vice versa. Recalling that the higher theegrees of freedom in a Student-t distribution the closer its resem-lance to the Normal one (with kurtosis edging near 3) (Da Costaewis, 2003), the stock exchanges with the steepest decline in thepproximate value of d(hat) and its corresponding enhancementn the kurtosis would pose the greatest challenge to the models,ike Brazil and Hungary (52.24% and 32.67%, respectively). It is alsomportant to bear in mind that the performance of the models couldlso be dictated by the value of the degrees of freedom d deliveredn the ML optimisation process in comparison with the estimatef the degrees of freedom of the distribution d(hat). Provided theutcomes are evaluated in terms of the exceptions recorded inacktesting, the GARCH specification performs clearly better thanhe EGARCH one in Brazil (6 against 9) and Hungary (5 vs. 8), eitherountry featuring sudden falls in the d(hat) and values of GARCH-

in the ML process closer to the estimate of the forecast periodhan those given by the EGARCH variant (17.96 vs. 24.50 and 9.87gainst 10.64). The outstanding results of both models in India andndonesia could be put down to the reduction in the kurtosis and theact that the distributions do not display excessive concentrationn the 0.01% and 1% quantiles; under this framework, Czech Repub-ic presents the inverse example. Analogous considerations may bepplied to Frontier markets. For instance, in Argentina, even thoughhere is a 17% decrease in kurtosis values, the negative asymmetryising by 288% and the negative returns clustering in the 1% quan-ile (q(0.01) = −3.37 in the forecast period and q(0.01) = −3.02 in theample term), contribute to explain the superiority of the EGARCH-

(2 exceptions) technique over its symmetric GARCH-t counterpart5 violations). No sizable difference appears in Tunisia, where thebrupt variation in sign and absolute value of the skewness (fromositive to negative) alongside the kurtosis augmenting in 92%eem to have affected both specifications alike. Croatia records sim-lar values of the d(hat) and a concentration on the 1% quantile inhe prediction term akin to that of the sample observations, but theecrease in kurtosis and the enlargement of the positive asymme-ry might deliver a slight advantage to the GARCH-t scheme to theetriment of its exponential peer. However, there seems to be nopparent explanation for the imprecision in Lithuania besides the21.58% leap in standard deviation: models should have performedetter.

Finally, it is necessary to underline the robustness of the EVTcheme as it manages the Green Zone in every market despite someramatic changes in the distribution of returns (e.g. Hungary).

50 d(hat) stands for the estimate of the degrees of freedom of a Student-t(d) distri-ution affixed to the distribution of returns (Alexander, 2008a).51 The failure of the EGARCH scheme to clearly perform better than the GARCHodel hints at the absence or the presence of a weak version of the leverage effect

ited by Chopra et al. (1992) and Cho and Engle (1999).

D

D

D


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