Value-at-Risk models and Basel Capital Charges
Evidence from Emerging and Frontier Stock Markets Adrián F. Rossignolo University of Leicester Meryem D. Fethi University of Leicester Mohamed Shaban University of Leicester
Abstract In the wake of the subprime crisis of 2007 which uncovered shortfalls in capital levels of most financial institutions, the Basel Committee planned to strengthen current regulations contained in Basel II. While maintaining the Internal Model approach based on Value-at-Risk, a stressed VaR calculated over highly strung periods is to be added to present directives to constitute Minimum Capital Requirements. Consequently, the adoption of the appropriate VaR specification remains a subject of paramount importance as it determines the financial condition of the firm. In this article I explore the performance of several models to compute MCR in the context of Emerging and Frontier stock markets within the present and proposed capital structures. Considering the evidence gathered, two major contributions arise: a) heavy-tailed distributions -particularly Extreme Value (EV) ones-, reveal as the most accurate technique to model market risks, hence preventing huge capital deficits under current measures; b) the application of such methods could allow slight modifications to present mandate and simultaneously avoid sVaR or at least reduce its scope, thus mitigating the impact regarding the enhancement of capital base. Therefore, I suggest that the inclusion of EV in planned supervisory accords should reduce development costs and foster healthier financial structures.
Keywords: Value at Risk, Extreme Value Theory, Emerging and Frontier markets, Capital Requirements, stressed VaR.
JEL classification: C3, G7.
1. Introduction The world financial system has undergone one of its most severe crisis in 2007-2008. Several factors acted simultaneously to ignite turmoil with devastating consequences felt all over the globe as a consequence of the interconnectedness of the national economies. One of the relevant effects brought about by the catastrophe has been represented by the inability of the banks to meet market losses: capital was insufficiently constituted to provide coverage for unexpected adverse events. The shortage was of such an extent that many institutions had to be bailed out by governments at the expense of the taxpayers’ resources; although this action averted a complete deadlock in capital markets, it simultaneously introduced elements like moral hazard and subsidies. The current framework contained in Basel II Capital Accord has established Value-at-Risk (VaR) as the official measure of market risk and enforced it to constitute the central point to the determination of capital charges. Moreover, as the Basel Committee on Banking Supervision (BCBS) has not hitherto recommended a particular VaR methodology, the adoption of the most appropriate VaR approach becomes a matter of the utmost importance to be decided purely on empirical grounds. However, the magnitude of the plight prompted the BCBS to put forward a proposal to increase the Minimum Capital Requirements (MCR) for market risks in accordance with the opinion of national regulators1. The intended scheme plans the introduction of a stressed VaR (sVaR) which ought to be added to the base VaR (cVaR) in order to form the new MCR in an attempt to curb the procyclicality of the measure in force. The aforementioned context highlights the significance of developing a precise VaR model to cover market losses and simultaneously build a capital buffer high enough as to allow institutions to distribute dividends in light of a further BCBS directive which restricts the dividend payout unless the capital level exceeds MCR by a quantity called Capital Conservation Buffer equivalent to 2.5% of the amount of the risk weighted assets. Besides the traditional reluctance on the part of the academics to study emerging and frontier markets, BCBS’s Consultative Documents have mostly been submitted to developed nations and it is unlikely that these proposals should be evaluated and its impact assessed in the spheres of emerging and frontier markets. This study aims at filling that empirical void, as I analyse the accuracy of several VaR 1
“Capital required against trading activities should be increased significantly (e.g., several times)”. (Financial Services Authority (2009:7)).
1
specifications and gauge its effect on capital charges from the perspective of nondeveloped stock markets under the current and proposed regulations. The article unfolds as follows. Section 2 briefly synthesises the basic concepts and models to be tried; Section 3 states the current and proposed theoretical frameworks; Section 4 delineates topics regarding the Data and Methodology; Section 5 describes the empirical results from the exercise in connection with BCBS directives while Section 6 offers the concluding statements. Finally, Section 7 stages a sensitivity analysis gauging the impact of BCBS’s planned mandate whereas Section 8 presents some overall closing remarks regarding the new sVaR approach.
2. Theoretical background. Concepts and definitions 2.1. Definition of Value-at-Risk VaR is a statistical risk metric that expresses the maximum loss in the value of exposures due to adverse market movements that a company is reasonably confident will not be exceeded if its positions are maintained static during a certain period of time t2. Losses are associated with confidence levels (α): those losses greater than VaR are only suffered with a specific small probability (1 – α) (McNeil, Frey and Embrechts (2005) and Linsmeier and Pearson (1996)). Therefore, for some confidence level α ∈ (0;1), VaR at the confidence level α is the smallest number l such that the probability that the loss L exceeds l is smaller or equal than (1 – α):
VaR (α ) = inf {l ∈ R : P ( L > l ) ≤ (1 − α )} = inf {l ∈ R : FL (l ) ≥ α } 3
[1]
where FL denotes the loss distribution function4. Given that Pr(Losst+1>VaRt+1)=α, or equivalently in relative terms or returns5 Pr(rt+1 0; α, β ≥ 0; α + β < 1. A high β (persistence) means that volatility takes a long time to fade after a crisis episode, whereas a high α (error) indicates promptness to react to market movements. GARCH models were introduced by Bollerslev (1986) and Taylor (1986), since then arguably erecting as one of the most popular among financial community13 as they enable to seize the volatility clustering effect (although in a symmetric fashion) and treat variance as a persistent phenomenon. GARCH power might also be enhanced applying its flexibility to depart from Gaussianity: this article will make use of the heavy-tailed Student-t in addition to the typical GARCH-Normal specification.
2.2.4.2. EGARCH(1;1) rt +1 = σ t +1 zt +1
ln(σ t2+1 ) = w + β ln(σ t2 ) + α
rt
σt
+γ
rt
σt
[8]
with γ < 0, α > 0 and zt+1 ~ iid D(0;1) (D(0;1) retains the meaning expressed in [7]). EGARCH representations were first proposed by Nelson (1991) and have become an interesting option as long as forecasts of conditional variance are guaranteed to be nonnegative upon them being expressed in log form14. Its structure allows the term γ to account for the leverage effect when, as it is the common case observed throughout the markets, γ < 0 and α > 0 drive ln(σ2t+1) to react more rapidly to falls than to
12
It is acknowledged that the list lies very far away from sophistication or completeness: it is merely indicative and illustrative of the wide range of possible alternatives. 13 For a more detailed explanation about GARCH models, see Bollerslev (1986), Bollerslev, Engle and Nelson (1994), for example. 14 Nelson (1991) and Engle, Bollerslev and Nelson (1994) treat EGARCH exhaustively.
5
corresponding rises15. Like GARCH previously, EGARCH may incorporate several distributions for D(0;1), thus Normal and t cases will be evaluated. For pragmatic reasons16, and in view of the negligible marginal gains obtained extending the quantity of lags, the study will restrict GARCH-EGARCH applications to first order lags, backed by McNeil, Frey and Embrechts (2005) who advocate the use of lower order models citing parsimony reasons. Consequently, VaR (α ) t +1 = σ t +1 Φ −1 (α )
VaR(α ) t +1 = σ t +1
[9]
and
(d − 2) d −1 t d−1 (α )
[10]
where σt+1 represents the volatility forecast derived from GARCH/EGARCH model and the rest of the symbols stand as before.
2.2.5. Extreme Value Theory (EVT) EVT represents an alternative to former specifications as it emphasises the tails of the series allowing right and left ends to be modelled separately and, most importantly, recognising the heavy-tailed nature of empirical distributions. On the grounds of efficiency and practicality, the study will develop within the Peaks-Over-Threshold (POT) variant disregarding the Block Maxima Method (BMM) (Coles (2001)). This section only enunciates the necessary phases to apply EVT-POT to VaR estimation; for a detailed treatment and theoretical aspects the interested reader may refer to Embrechts, Klüppelberg and Mikosch (1997), McNeil, Frey and Embrechts (2005), or Reiss and Thomas (2007). 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 threshold u; d) Calculating excesses over the threshold yt = Xt – u (Xt > u); e) Applying the Balkema & de Haan (1974)-Pickands (1975) theorem to fit the two-parameter limiting Generalised Pareto Distribution (GPD) Gξ,σ(y) for the excesses y above the threshold u. 15
Cho and Engle (1999) and Chopra, Lakonishok and Ritter (1992) document the leverage effect in developed markets. 16 Eventually avoiding the curse of dimensionality. Gujarati (1997) recommends that econometric models should follow the parsimony principle.
6
Accordingly [11], 1 − (1 + ξ y σ )
−1 / ξ
if ξ ≠ 0
Gξ,σ(y) =
1 − exp(− y σ )
if ξ = 0
where ξ
: tail index parameter
σ
: scale parameter
as u increases, where σ > 0, and y ≥ 0 when ξ ≥ 0 and 0 ≤ y ≤ -σ/ξ when ξ < 0. For finance applications ξ > 0 ought to be verified, then Gξ,σ(y) becomes the classic Pareto or Fréchet distribution picturing heavy tails. Reiss and Thomas (2007) remark that GPD precision could be enhanced affixing a location parameter µ, thus making Gξ,σ(y – µ). Christoffersen (2003) points that the main flaw of POT methodology resides in the identification of the threshold u. Regardless of some efforts to find an appropriate mechanism17 there is not a reliable method to calculate it, hence the current study favours a technique based on the analysis of an array of elements such as the sample Mean Excess Function (MEF), QQ plots, sample Kernel Density and sample Quantile Function, following Reiss and Thomas (2007) and McNeil and Saladin (1997). The threshold u would denote the value from where the MEF exhibits a positive gradient (excesses follow a GPD with ξ > 0), provided the estimated parameters exhibit stability within a range of the selected u (Coles (2001)). After some algebraic operations, the POT-quantile reads18: ∧
G
−1
(α ) = u +
σ 1 − α −ξ ∧ ξ w n
∧
− 1 [12]
where w represents the number of observations above the threshold u and the rest of the symbols conserve their meaning. Following Quasi-Maximum-Likelihood (QML)19 as in Bollerslev and Wooldridge (1992) and McNeil and Frey (2000), POT-quantile [12] is 17
Reiss and Thomas (2007) or Beirlant, Vynckier and Teugels (1996) explain methods which seem to work reasonably well but should be handled with care, as they may eventually result in the selection of a very high number of upper order statistics. Danielsson, Hartmann and de Vries (1998) and Coronel-Brizio and Hernandez-Montoya (2004) introduce interesting approaches to this subject. Christoffersen (2003) presents a “rule of thumb” and Neftci (2000) followed by Bekiros and Georgoutsos (2003) estimates u = 1.176σn where σn is the sample standard deviation and 1.176 = F-1(0.10) = 1.44 (d − 2 ) d assuming F ~ Student-t(6). 18 19
McNeil, Frey and Embrechts (2005), McNeil and Saladin (1997) and Fernandez (2003). Also called Pseudo-Maximum-Likelihood (PML).
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calculated using the iid standardised residuals after pre-whitening the data employing a GARCH-Normal model. Therefore, the final VaR expression for EVT-POT becomes: VaR (α ) t +1 = σ t +1G −1 (α ) [13]
where: σt+1
: volatility forecast derived from GARCH-Normal model;
G-1(α)
: inverse of the cumulative density function of the GPD distribution (αquantile of G)
3. Regulatory framework 3.1. Basel II Capital Accord In 1996 the BCBS issued an Amendment to incorporate a specific treatment for market risks, largely overlooked in Basel I Capital Accord and eventually included in Basel II Capital Accord20. This adjustment allows institutions to employ the Internal Model Approach (IMA) to have their market risk Minimum Capital Requirements (MCR) determined by their own VaR estimates, that in turn derive from their respective VaR models. Risk-capital charges result from21: m MCRt +1 = max c 60
60
∑VaR(99%) i =1
t −i +1
; VaR(99%)t [14]
i.e., the maximum between the previous day’s VaR and the average of the last 60 daily VaRs increased by the multiplier22 mc = 3(1+k) and k Є [0; 1] according to the result of Backtesting23. BCBS demands VaR estimation to observe the following quantitative requirements: a) Daily-basis estimation24; b) Confidence level α set at 99%; b) One-year minimum sample extension with quarterly or more frequent updates25; c) No specific models prescribed: banks are free to adopt their own schemes; d) Regular Backtesting and Stress Testing programme for validation purposes26. 20
BCBS (1996, 2004). BCBS demands the use of a 10-day holding period through the square-root-of-time rule. However, the present research will omit the specification and work with a 1-day holding period instead. See Section 4 Methodology. 22 mc will be, at minimum, 3. Although BCBS does not enlighten its derivation, Stahl (1997) and Danielsson (1998) provide a statistical explanation. This value can sometimes be so conservative that any incentives to develop an accurate model by achieving k=0 might be quickly overshadowed. 23 Section 3.1.1. 24 Footnote 19. 25 Section 4. 21
8
3.1.1. Backtesting It constitutes a statistical technique to assess the quality of the risk measurement specifications which involves the comparison between the daily VaR forecast with the actual losses27. Albeit the assumptions behind VaR calculations may be labelled incoherent (Artzner et. al. (1997, 1999), Acerbi and Tasche (2002)), it is useful to evaluate whether the model is capable of capturing the trading volatility. Backtesting procedure entails counting the number of times that losses exceed VaR estimates in approximately 250 trading days. McNeil, Frey and Embrechts (2005) use the indicator function: 1
if Lt+1 > VaRt+1(α)
0
otherwise
I t +1 = I {Lt +1 > VaRt +1 (α )} =
where: It
: indicator function accumulating the excessions, exceptions or violations behaving like outcomes of iid Bernoulli trials with success probability 1-α;
Lt+1
: realised loss for period t+1;
VaRt+1(α) : conditional VaR estimates for t+1 BCBS proposes a three-zone -Green, Yellow and Red- layout to classify VaR models. Consequently28: Zone Green
Definition Outcomes consistent with low probability of Error II
Characteristics - Number of exceptions between 0 and 4 - No capital surcharge, k = 0
Yellow
Results uncertain and compatible with either precise or inaccurate models
Red
Presumption model
- Number of exceptions between 5 and 9 - Strong suggestion of imprecise specifications, particularly as number of exceptions grow - Capital penalties increase with number of violations - Capital charges determined to return model to a 99% coverage - Encourages sharpness to keep penalties low - Number of excessions equal or greater than 10 - k increased to 1 immediately - Subsequent model invalidation
of
inaccurate
26 The focus of the present article is restricted to Backtesting. BCBS (1996, 2004, 2009), Jorion (1996), Penza and Bansal (2001), Christoffersen (2003), Dowd (1998, 2005), RiskMetrics Technical Document (1996) and Osteirrischische Nationalbank (1999), just to name a few, cater for basic concepts and extensive treatment of stress testing. 27 “…the backtesting framework …involves the use of risk measures calibrated to a one-day holding period”. (BCBS (2006:312). 28 This categorisation is designed to compromise the probabilities of Error I: erroneous rejection of accurate models and Error II: incorrect acceptance of inaccurate models. For a detailed statistical treatment of foundations of Backtesting recur to BCBS (2006).
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Backtesting results determine the extent of capital surcharge through the value of the scaling factor k as the quantity of exceptions in a sample of 250 trading days is transformed into a number indicating the increase in the multiplier to be applied to mc (Chart N°1). BCBS establishes that the beginning of each zone is marked by the points where the cumulative probability of a Bernoulli distribution with 99% success probability reaches 95% for the Yellow Zone (5 exceptions) and 99.99% for the Red one (10 or more violations) respectively. Given that five excessions represent a 98% coverage for 250 observations, it would be necessary to enhance the multiplier in 40% to restore the coverage to the 99% demanded (supposing that returns follow a Normal distribution and the scaling factor mc=3)29,30. However, although it is possible for k in [14] to achieve nullity (specifications belonging to Green Zone), Danielsson, Hartmann and de Vries (1998) remark that mc assuming a minimum of 3 in [14] conspires against the development of accurate models. Given that mc is calculated as in [14] in the current regulatory framework, the present article will unfold according to the respective procedures.
3.2. Basel III Capital Accord The trading sessions following Lehman Brothers’ bankruptcy in September 2009 triggered unusual losses of such magnitude that financial institutions found their capital buffers unquestionably insufficient to match those deficits. Though these market movements were of a weird nature, BCBS (2009) partly blamed the previous Amendment of 1996 for failing to grab some key extreme risks31 verified in the turmoil. Additionally, some national regulators increased the pressure on BCBS demanding tougher measures to avoid further embarrassing bailouts at the expense of taxpayers. The Financial Services Authority (2009) issued an influential report which highlighted some deficiencies of the VaR approach that may have provoked, among other equally important reasons, the insolvency of several firms. In particular, it is mentioned that most VaR models are unable to capture fat-tail risks: “Short-term observation periods
29
The quotient between the 99% and 98% cumulative normal distribution amounts to 1.14 which, for a scaling factor of 3represents a 40% increase in the base level. In effect, for five exceptions 1-5/250 = 0.98 and 3*Ф-1(0.99)/Ф-1(0.98) -3 ~ 0.40 assuming normality. Hence, k = 0.40 or 40%. 30 Chart N°1. 31 The emphasis is also laid on default and migration risk, among others (BCBS (2009)). These measures lie beyond the scope of the present study, which will be restricted to the introduction of the stressed VaR.
10
plus assumption of Normal distribution can lead to large underestimation of probability of extreme losses” (FSA, (2009:23))32. To tackle this specific issue, while maintaining the Basel II methodology33, BCBS proposed the introduction of a stressed VaR (sVaR) metric to increase MCR. Its calculation complies with the same guidelines that current VaR (cVaR) (Section 3.1) though the dataset must belong to a “…continuous 12-month period of significant financial stress…” (BCBS (2009:14)), i.e., when market movements would have inflicted great losses on the banks. The stricter daily capital demands reflect in sVaR added to cVaR: m MCRt +1 = max c 60
m cVaR(99%) t −i+1 ; cVaR(99%) t + max s i =1 60 60
∑
60
∑sVaR(99%) i =1
t −i +1
; sVaR(99%) t [15]
where: - cVaR(99%) t : 99% cVaR for day t; - mc
: multiplier for cVaR (Section 3.1);
- sVaR (99%) t : 99% sVaR for day t; - ms
: multiplier for sVaR
with ms = 3(1+k) and k arises from Backtesting results for cVaR (not for sVaR). As k Є [0; 1] institutions are encouraged to develop precise VaR models in order to keep k≈0 and avoid penalties to establish MCR. Besides strengthening MCR by means of the sVaR component in MCR formula, Basel III focuses on reinforcing the protection against periods of acute economic and financial strain through two additional layers to be placed on top of MCR in the following order34: - Capital Conservation Buffer (CCB): banks are obliged to build a 2.5% cushion over MCR in order to avoid its deterioration during stress periods. This safeguard is intended to serve as a first line of defence once heavy losses are recorded. BCBS (2010) indicates that this shortfall absorber must be restored to the original value restraining earnings distribution: the closer the bank moves to MCR (the greater deterioration of CCB), the smaller the rate at which profits are handed out or dividends paid until the buffer is fully rebuilt to the starting level; 32
FSA highlights the procyclicality that emerges using observation periods as short as one year: falls in confidence raise volatilities, which vanish liquidity and increase volatility even more. (FSA (2009)). 33 Some slight variations regarding the data updating scheme are also put forward. (BCBS (2009)). 34 Andersen (2011) finds substantial evidence of the cyclicality in Basel II and suggests a different riskweighting scheme to circumscribe that shortfall.
11
- Countercyclical Capital Buffer (CyCB): in practice, this is an extension of CCB, although it operates in a range between 0% and 2.5% of the exposures determined at the discretion of national regulators according to the point of the business cycle when risks mount up as a result of excess credit growth35. Synthetically, Basel III capital structure could be depicted as follows: Capital requirements and buffers Capital concept Percentage of exposure Observations Minimum Capital Requirement 8.0 Minimum quantity (MCR) Calculated via SA (8%) or IMA Capital Conservation Buffer 2.5 Summoned in event of crisis (CCB) Restored through earnings retention Countercyclical Capital Buffer 0 – 2.5 At the discretion of national regulators (CyCB) (excessive credit growth) Note: Exposures are expressed as a percentage in terms of total capital.
Technically speaking, Basel III innovations stem from the introduction of the sVaR to calculate MCR, as both CCB and CyCB constitute supplementary capital layers with fixed and externally determined proportions respectively. Therefore, the scope of the present article will be limited to the performance of VaR models.
4. Methodology The primary data are formed by univariate price series belonging to a sample of ten stock market blue-chip indices: six belong to emerging markets (Brazil, Hungary, India, Czech Republic, Indonesia and Malaysia) and four to frontier markets (Argentina, Lithuania, Tunisia and Croatia)36 retrieved from the corresponding stock exchanges websites. Series were converted into a string of (logarithmic) returns to achieve stationarity and ergodicity (Bowerman and O’Connell (1993) and Alexander (2008b)). As in Hansen and Lunde (2005) and Mapa (2003), the time series of returns were separated into two periods for the purposes of parameter estimation and evaluation of forecasts respectively. Although as rule-of-thumb Christoffersen (2003) ascertains that models should utilise at least the last 1000 observations, it is acknowledged that time series should be as long as possible (Dowd 2005). On the other hand, the Forecast period contains the financial crisis which unravelled in September-October 2008. This event represents an
35
Stolz and Wedow (2011) illustrate the performance of countercyclical capital buffers for Germany. In order to categorise the markets the study follows the FTSE Global Equity Index Series Country Classification, September 2009 update. (FTSE (2009)). 36
12
interesting example of a stress test37 to determine the real time performance of the models. For the indices aforementioned:
Stock Exchange
Stock Index Bovespa
Estimation Number of period data points Sao Paulo 03/01/2000 1976 Brazil 28/12/2007 Budapest Cetop20 30/01/2002 1494 Hungary 28/12/2007 Mumbai Sensex 01/07/1997 2600 India 31/12/2007 Prague Px 05/01/1995 3243 Czech Republic 28/12/2007 Jakarta Jkse 04/01/2000 1925 Indonesia 28/12/2007 Kuala Lumpur Klse 04/01/1999 2215 Malaysia 31/12/2007 Buenos Aires Merval 09/10/1996 2777 Argentina 31/12/2007 Vilnius Omx 03/01/2000 2045 Lithuania 28/12/2007 Tunisia Tunindex 31/12/1997 2496 Tunisia 31/12/2007 Zagreb Crobex 04/01/1999 2247 Croatia 31/12/2007 Note: Emerging markets above solid line, Frontier markets below.
Forecast Period 02/01/2008 30/12/2008 02/01/2008 31/12/2008 02/01/2008 31/12/2008 02/01/2008 30/12/2008 02/01/2008 30/12/2008 02/01/2008 28/12/2008 02/01/2008 31/12/2008 02/01/2008 30/12/2008 02/01/2008 31/12/2008 02/01/2008 31/12/2008
Number of data points 249 251 245 253 243 248 249 244 247 250
VaR estimation and evaluation will adopt BCBS’s mandate. In effect:
4.1. General issues - Daily time horizon. The ten-day holding period is to be excluded due to the inconsistencies of the ‘square root of time’ rule (Danielsson (2004) and Danielsson and Zigrand (2006)) and the possibility of masking the inaccuracy of the models by increasing insufficient daily VaR using extrinsic multiples38; - One-tailed VaR estimations (left tail, i.e., long positions) performed with confidence level α anchored at 99%.
4.2. VaR specifications - Linear: standard deviation calculated using a rolling window of 1000 days complemented by Normal and t distributions (Section 2.2.3. [5] and [6]);
37
Footnote 24.
38
Danielsson (1998) stresses the excessive conservatism attained using
13
10 factor to augment VaR.
- Historical Simulation: estimations employ a moving sample of the most recent 1000 points (Section 2.2.1. [3]); - Filtered HS: GARCH and EGARCH models applying Maximum Likelihood (ML), both featuring Normal and t distributions (Section 2.2.2. [4]); - Conditional: GARCH and EGARCH obtained via ML. Normal and t variants applied to the distribution of standardised residuals rt/σt, (Section 2.2.4., [7] to [10]); - EVT: POT format through methodology described in Section 2.2.5. [12] to [13].
4.3. VaR validation The accuracy of VaR models will be gauged employing Backtesting, in accordance with the stipulations laid out in Section 3.1.1.
4.4. Stressed VaR The calculation of sVaR is performed adhering to the instructions stated in BCBS (2009) explained in Section 3.2. The periods of heavy losses for the indices selected become: Stock Exchange Stock Index Stress period Sao Paulo Bovespa 25/09/2000 Brazil 28/08/2001 Budapest Cetop20 01/03/2002 Hungary 28/02/2003 Mumbai Sensex 01/10/2000 India 30/09/2001 Prague Px 01/09/2000 Czech Republic 31/08/2001 Jakarta Jkse 01/04/2000 Indonesia 31/03/2001 Kuala Lumpur Klse 01/06/2000 Malaysia 31/05/2001 Buenos Aires Merval 01/10/1997 Argentina 30/09/1998 Vilnius Omx 01/09/2000 Lithuania 31/08/2001 Tunisia Tunindex 01/03/2002 Tunisia 28/02/2003 Zagreb Crobex 01/04/2002 Croatia 31/03/2003 Note: Emerging markets above solid line, Frontier markets below.
4.5. Minimum Capital Requirements (MCR)
14
Loss posted 39.32% 17.03% 39.17% 41.87% 42.02% 45.96% 86.36% 29.39% 20.08% 19.73%
Capital charges are estimated firstly on the grounds of [14] and Backtesting (Section 3.1.1.) to comply with current directives and secondly applying [15] to cover the proposed framework.
5. Results 5.1. Stylised facts about data Statistics deployed in Chart N°2 depict the common patterns of daily financial time series. Overall, the mean return is not significantly different from zero, hence enabling driftless GARCH-family representations (Sections 2.2.2., 2.2.4. and 2.2.5.). An indicator to assess the behaviour of the tails of the distributions –skewness- assumes negative values for all series save for Tunisia and Croatia, meaning that left tails are longer than its right counterparts and viceversa for the two exceptions. Extreme values appear more concentrated on the negative (positive) region of the distribution, bolstering the idea that in general market crashes provoke asymmetry (Jondeau and Rockinger (1999)) notwithstanding which the alleged superiority of EGARCH schemes over GARCH remains a matter of empirical concern. Large Jarque-Bera statistics and kurtosis figures beyond 3 confirm the departure from normality, fact that coupled with the empirical quantiles of the standardised return series overcoming the theoretical Gaussian ones hint at leptokurtic distributions. Values of Box-Ljung portmanteau Q(20) denote that it is not possible to reject the null of independence of linear returns at 95% and 99%. Furthermore, the very high numbers for Q2(20) attained for squared returns provide reasonable evidence of heteroskedasticity and volatility clustering, thus paving the way for conditional volatility models.
5.2. VaR forecasts, Backtesting, bank capital and Basel regulations 5.2.1. VaR forecasts, Backtesting and bank capital Rooting in the outcomes exposed in Charts N°3 (number of exceptions) and N°4 (Basel Zones and increase in multiplication factor), the following reflections apply39: a) Linear models prove to be inadequate as rejection -irrespective of Normal or t distributions- is obtained in every stock exchange without distinction between Emerging and Frontier markets;
39
Appendix B.
15
b) HS reveals unquestionably inaccurate, being discarded in all markets (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, with Normal and t distributions working in Brazil, Hungary and India, Normal in Malaysia and t in Czech Republic while EGARCH filter featuring both Normal and t distributions only progresses in Hungary. However, advances are very scarce as they alternate between 75% and 85% of additional capital required (between 8 and 9 exceptions); d) Conditional models represent a significant leap forward, both for GARCH and EGARCH techniques. The t distribution clearly works better than the Normal one for both models as it avoids the Red Zone in every market, either Emerging or Frontier (except Lithuania in EGARCH setting). The Normal distribution gives mixed results, failing in one Emerging Market (Indonesia) and two Frontier (Lithuania and Croatia) for GARCH and two Emerging (Hungary and Indonesia) and two Frontier (Lithuania and Croatia) employing EGARCH. However, Normal distribution cannot avoid the Yellow Zone and demands surcharges ranging from 50% (Czech Republic, Malaysia and Argentina for GARCH and India and Malaysia using EGARCH) to 85% in Brazil (EGARCH). The t distribution relieves the pressure on shareholders as they escape constituting extra capital in India and Malaysia (GARCH and EGARCH), Argentina (EGARCH) and Croatia (GARCH): GARCH allows it in two Emerging Markets (India and Malaysia) and one Frontier (Croatia), while EGARCH follows suit in two Emerging Markets (India and Malaysia) and one Frontier (Argentina), while falling in the Red Zone in Lithuania; e) EVT is undoubtedly the best performer. It does not require auxiliary capital in any market, fact reflected in the highest VaR for every Emerging and Frontier market for 01-Jan-09 among those models that avoid the Red Zone and consequent reestimation. Moreover, its levels could not be considered excessive or insufficient as the proportion of VaR excessions stays close to the stipulated value (only Hungary-1.59% and India-1.23% exceed the standard 1%, percentages not enough to claim capital reinforcement in Chart N°3).
5.2.2. VaR outcomes and Basel regulations 5.2.2.1. Current regulations 16
The procyclical behaviour of conditional VaR prompted the BCBS to demand the calculation of the average 60-day VaR enhanced by the multiplication or hysteria40 factor currently set at 3, supplemented by an add-on coefficient which level depends on Backtesting results (Chart N°4). Among those models lying in the Green or Yellow Zones, FHS delivers the highest values for k (Section 3.1.) in most Emerging Markets (GARCH-Normal in Brazil, India and Malaysia, GARCH-t in Hungary -shared with both EGARCH-, India and Czech Republic) but this circumstance is largely explained by the poor Backtesting performance (additional 85% demanded in every market except Malaysia, 75%). The same situation arises with conditional GARCH-Normal and EGARCH-Normal in Tunisia (both 75%), EGARCH-Normal in Argentina (65%), GARCH-t in Lithuania (75%) and EGARCH-t in Croatia (50%) and HS in Indonesia (85%). The most consistent capital charges across Emerging and Frontier markets appear to be delivered from GARCH-t and EVT, with similar levels. However, it is noticeable that GARCH-t can only match EVT at the expense of the increases stipulated for the Yellow Zone (all stock exchanges except India, Malaysia and Croatia) and the subsequent scrutiny on the part of the regulators. One of the major concerns regarding the employment of EVT residing in the high amount of capital demanded could be averted as they do not exceed quantities given by other heavy-tailed specifications like GARCH-t or EGARCH-t with the further advantage of falling within the Green Zone, this way avoiding periodic revisions demanded by the Yellow Zone (Chart N°5).
5.2.2.2. Proposed regulations The calculation of the sVaR coincides with that of the base VaR, except for the point that it must be carried out over a 12-month continuous term wreaking havoc on the financial position of the company. The outcome in Chart N°6 shows that, excluding disqualified models in Red Zone in Chart N°4 (bold letters), EVT delivers the highest sVaR values for highly strung periods for either Emerging or Frontier Markets. Nevertheless, after applying the former Backtesting results (Chart N°7), the balance switches to the rest of the schemes on the grounds of the 60-day average and, most importantly, the penalties envisaged for inaccuracy (between brackets): HS (Indonesia85%-), GARCH-Normal (India-75%), GARCH-t (Argentina-40%-), EGARCH-Normal
40
Jorion (1996) and Dowd and Hutchinson (2010), for example, employ the name “hysteria factor” to refer to the “multiplication factor”.
17
(Czech Republic-75%- and Tunisia-75%-), EGARCH-t (Brazil-85%- and Hungary75%-) and EVT retaining three markets (Malaysia, Lithuania and Croatia). The new context proposed by the BCBS brings a more levelled panorama with HS, GARCH-Normal, GARCH-t, EGARCH-Normal, EGARCH-t and EVT claiming top spots, even though for all models but EVT the result is largely due to the add-in factor verified in Backtesting (Chart N°7). Interestingly enough, EVT presents the lowest total amounts in three Emerging markets and stays near the bottom in the remaining ones. Chart N°8 conveys the picture of BCBS’s planned increase in the capital base with the addition of capital buffers based on sVaR: the total capital charge exhibits three markets where GARCH-Normal delivers the highest VaR values (Brazil, India and Tunisia), two for FHS: GARCH-t in Hungary and Czech Republic, one for EGARCH-Normal (Argentina) and EVT in Indonesia, Malaysia, Lithuania and Croatia, all of them except EVT stemming from high multiples of k (Chart N°4). Furthermore, EVT presents the lowest total amounts in three cases (Brazil, Hungary and Czech Republic), edging the lower end in the rest of the markets (except Malaysia).
6. Conclusions: VaR models and Basel regulations The evidence collected appears enough to discard Linear models, HS and FHS. Although the standard deviation for Linear models is allowed to change in time, it is unable to capture the dynamics of the latent volatility. No improvement is virtually recorded employing a heavy-tailed t distribution instead of the Normal one as the underlying risk measure is inherently flawed. HS reaffirms all the disadvantages mentioned in Section 2.1.1., for example, last VaR overcoming 60-day VaR average in Hungary reveals the presence of the Ghost or Shadow effect. Usage of the empirical distribution of residuals in FHS delivers only slight gains in some Emerging markets (t distribution), albeit in overall it remains ineffective. The significant progress derived from conditional models highlights the need to resort to these schemes. The results suggest that the EGARCH technique brings no significant advantage over GARCH, in turn meaning that the leverage effect marked by Chopra, Lakonischok and Ritter (1992) and Cho and Engle (1999) is hardly noticeable given that the adjustments brought about by EGARCH-t are equal or greater than their GARCH-t counterparts (except Argentina). Moreover, the density assumption exerts dominance over the particular specification: GARCH-t and EGARCH-t improve the
18
performance of their Normal counterparts, therefore making unavoidable the usage of heavy-tailed distributions. Findings are highly supportive of the EVT approach in comparison with its competitors. Its application would have shielded institutions from huge losses produced in the event of the 2008 crisis and prevented the constitution of extra-capital while simultaneously building up a capital base not excessive in relation to the rest of the specifications.
7. The impact of Basel proposed directives: a sensitivity analysis Grounding on the results of Section 5.2, this chapter applies the most accurate modelEVT- to evaluate the impact of the BCBS’s proposed regulations to toughen the required minimum capital levels. According to the present directives, MCR would provide enough coverage for losses bigger than twice the size of the maximum deficit posted in the forecast period (except in Argentina, which coefficient is 1.36) (Chart N°9-Column [1]). This assertion means that the maximum daily loss dealt with by the model peaks 42.36% in Czech Republic (minimum 28.49% in Malaysia) in Emerging markets, and 26.94% (Lithuania) and 10.88% (Tunisia) in Frontier exchanges (Chart N°9-Column [2]). Given that in general higher volatility and higher losses translate into more elevated capital cushions (Brooks, Clare and Persand (2000)), there is evidence that Emerging markets observed an upsurge of volatility steeper than Frontier ones (Hungary and Czech Republic more than doubled sample values, Chart N°10-Columns [1]-[3]) and of maximum daily losses as well (50% and 12% respectively, average values in Chart N°10-Columns [4]-[6]). The sVaR modification seems to achieve the objectives by driving the total coverage to more than four times the maximum loss of the forecast period in both Emerging and Frontier markets (Chart N°11-Column [1]). These figures represent an average increase of approximately 50% and 86% in Emerging and Frontier markets respectively (Chart N°11-Column [2]) -with Brazil topping the table with 59% in the former and Argentina reaching 51% in the latter in terms of the MCR41- (Chart N°11Column [3]). Potential deficits determined by new MCR would appear relatively high for both groups compared with the shortfalls recorded in times of crisis, hence making
41
Proposed regulations mean an increase in the MCR of 190% in Argentina and 100% in Malaysia, for example.
19
institutions put by excessively unproductive capital levels which could otherwise be directed to credit purposes. Acknowledging that proposed regulations will surely lead to an increase in MCR of a remarkable quantity, a sensitivity analysis considering different scenarios regarding the level of the multipliers mc and ms and their respective capital level (i.e., greatest daily shortfall) was performed in Chart N°1242. The assessment of the outcomes discloses interesting implications displayed in Chart N°13. In the first place it is judged that as national regulators ought to approve the election of the stress period to calculate sVaR they could also dictate the level of the multiplier ms to be applied to their respective markets, and there seems to be proof that this level might well be set below 3 and still provide considerable coverage. For instance, if the factor ms were fixed at 1.5, 1.0 or 0.5 –Chart N°13-Columns [1]-[6], the maximum daily loss covered in terms of the greatest shortfall in the forecast period would still edge 3.50-3.70 (on average) in Emerging markets and 3.20-3.40 in Frontier ones (values which represent daily deficits beyond 40% for the former and 25% for the latter, maximums and minimums for each category being Brazil-51% and Malaysia-36%, Lithuania-38% and Tunisia-12%). Even though those amounts embody an average increase from current directives of more than 17% in Emerging markets and 36% in Frontier stock exchanges (Chart N°14-Columns [1], [3] and [5]), they do not place so heavy a burden on institutions as in the case of sVaR (decrease between 15%-21% in the former and 14%-23% in the latter) (Chart N°14-Columns [2], [4] and [6]). The second observation indicates that the goals intended by the BCBS could also be accomplished omitting sVaR43 as a certain degree of additional conservatism may be attained augmenting the multiplicative factor mc for the base VaR and simultaneously phasing out sVaR. As the results hint, if that constant44 were to be recalibrated at 3.5 or 4, capital levels would grow at 17% and 33% respectively with reference to present directives (Chart N°14-Columns [7] and [9]), at the same time declining (average percentages) 20% and 9% in Emerging markets45 and 32% and 22% in Frontier ones evaluated against the sVaR proposition (Chart N°14-Columns [8] and [10]). Capitals set 42
MCR equal the maximum daily loss, as capital levels are built to match shortfalls on portfolios. 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 corresponding figure employed in the sensitivity analysis. Analogously, ms=3(1+k)=3 or the respective constant stated. The former reflection applies to Chart N°12. 45 It is acknowledged, however, that Hungary and Czech Republic observe an enhancement of MCR of approximately 3% and 1% respectively were mc to be set at 4. 43
20
in this fashion would allow banks to endure average losses of almost or more than three times the maximum daily loss of the forecast period for both markets (mc=3.5 delivers 49% for Czech Republic and 31% for Lithuania, whereas mc=4 matches 56% and 36% respectively). For Emerging stock exchanges, multiples of 3.5 and 4 protect against a 42% and 48% daily losses while for Frontier ones, shortfalls amount to 24% and 27% respectively in average values (Chart N°13-Columns [7]-[10]). A quick glance at Argentina may be useful to mark that the planned modification of MCR should confer national regulators enough flexibility to rule over sVaR. If authorities deem a capital level providing institutions coverage below twice the size of the maximum daily loss in the forecast period as insufficient (no sVaR calculated), they could demand the constitution of capital cushions using sVaR in order to push this amount to 2.39 (ms=1 and 0.5) (Chart N°13-Columns [3] and [5]). Concurrently, Lithuania, for example, could well scrap sVaR and increase mc to 3.5 to sufficiently face a daily loss of 31% (Chart N°13-Column [8]).
8. Final conclusions on Basel proposed directives The evidences collected attempt to provide quantitative information about the impact of the BCBS’s proposal to increase MCR constituted by financial institutions in Emerging and Frontier stock markets. To begin with, supra-national regulators should demand usage of techniques capable of dealing with large fluctuations in the future -particularly EVT- hence discouraging or banning the employment of traditional methodologies which provide capital buffers only for common market variations (most notably Linear, HS, FHS or Normal specifications). For every market researched, the intended variation characterised by sVaR applying a base multiple of ms=3 in addition to the current MCR appears somewhat excessive and immobilises funds unnecessarily. Therefore, both factors could be dissociated and calculated independently at the discretion of national supervisors in view of the particular patterns of the respective markets. In this sense, one viable alternative may involve working out a combination between current MCR and a lighter sVaR value. Another potential version would see sVaR term scrapped and the base multiplication mc factor lifted to some number between 3 and 4 (for example), at the discretion of the corresponding domestic controller. This last suggestion echoes the Japanese position (2008) regarding the BCBS’s proposition concerning market risk demands. It is believed that the adoption of the
21
planned stringent measures will surely give birth to disincentives to devise accurate VaR methodologies, as the shortcomings of the models will be compensated by the combination of add-in factors and penalty constants. In contrast, the implementation of sound heavy-tailed techniques like EVT could provide extensive coverage, avoid the building of superfluous capital buffers and at the same time allow the institutions to match huge future losses without incurring in high development costs to estimate sVaR.
22
Chart N°1 Backtesting: The Three-Zone Approach Number of exceptions – Increase in scaling factor Zone
Number of exceptions
Increase in scaling Cumulative factor k probability 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. For other quantities the chart should be reworked. (2): Probabilities obtained using a Bernoulli distribution with probability of success 99%. (2) The Yellow Zone begins at that number of exceptions where the probability of obtaining at a maximum that quantity equals or exceeds 95%. (3): The Red Zone starts where the probability of obtaining that quantity or fewer violations is at least 99.99%.
Chart N°2 Stylised facts about return series Parameter Index Observ. Mean
Brazil
Hungary
Bovespa Cetop20 1976 1494 0.00067 0.00082 (0.10) (0.01) Median 0.00128 0.0012 Maximum 0.07336 0.04035 Minimum -0.07539 -0.05553 Std. Dev. 0.01816 0.01232 Skewness -0.21315 -0.45307 Kurtosis 3.7020 4.5937 JarqueBera 55.5327 209.2272 (0.00) (0.00) Q(20) 30.35 28.39 (0.06) (0.10) Q2(20) 164.20 270.88 (0.00) (0.00) q(0.01) -4.1320 -4.5432 q(1.00) -2.5198 -3.0770 q(2.50) -2.0972 -2.1934 q(5.00) -1.7387 -1.6190 q(10.00) -1.2576 -1.1831 q(90.00) 1.1825 1.1983 q(95.00) 1.5646 1.5663 q(97.50) 1.8529 1.8159 q(99.00) 2.3919 2.4801 q(99.99) 3.9221 3.2069 Note: p-values in brackets.
India Sensex 2600 0.0006 (0.06) 0.00126 0.08592 -0.11809 0.01605 -0.33473 6.3879 1291.993 (0.00) 74.81 (0.00) 616.75 (0.00) -6.6995 -2.7853 -2.1932 -1.5806 -1.1831 1.0948 1.5302 1.8976 2.2987 5.2104
Czech Rep. Px 3243 0.00036 (0.09) 0.00056 0.07048 -0.07077 0.01192 -0.28789 5.6667 1005.655 (0.00) 81.5 (0.00) 1100.1 (0.00) -5.7068 -2.8245 -2.1823 -2.3010 -1.1366 1.1440 1.2624 1.9397 2.4556 5.5463
Indonesia
Malaysia
Argentina
Jkse 1925 0.00071 (0.02) 0.00122 0.06733 -0.10933 0.01391 -0.72075 7.6550 1904.667 (0.00) 42.24 (0.00) 154.49 (0.00) -7.4724 -2.9102 -2.1103 -1.6203 -1.1319 1.0947 1.5083 1.9186 2.3766 4.5893
Klse 2215 0.00043 (0.05) 0.00048 0.05851 -0.06342 0.01044 -0.21373 8.5905 2901.295 (0.00) 107.96 (0.00) 656.12 (0.00) -6.0699 -3.0014 -1.9295 -1.4132 -0.9744 1.0664 1.4864 2.1111 2.9018 5.5170
Merval 2777 0.00047 (0.05) 0.00104 0.16117 -0.14765 0.02216 -0.16568 8.6782 3743.358 (0.00) 40.72 (0.00) 1475.9 (0.00) -6.3172 -3.1031 -1.9803 -1.4396 -0.9980 1.1578 1.5977 2.2520 3.0802 5.8195
23
Lithuania Omx Vilnius 2045 0.0008 (0.00) 0.00076 0.0458 -0.10216 0.00905 -0.72195 13.5705 9698.399 (0.00) 114.54 (0.00) 30.63 (0.06) -10.0735 -2.9112 -1.9949 -1.5710 -1.0523 1.0888 1.5511 1.9414 2.8569 4.9595
Tunisia
Croatia
Tunindex 2496 0.00038 (0.00) 0.00016 0.03041 -0.02125 0.00443 0.40024 6.1949 1128.208 (0.00) 601.49 (0.00) 1618.7 (0.00) -4.6805 -2.5043 -1.9290 -1.5631 -1.0950 1.1499 1.6334 2.1418 2.8418 6.4453
Crobex 2247 0.00089 (0.00) 0.00045 0.12697 -0.09029 0.01435 0.28738 12.9047 9215.895 (0.00) 17.18 (0.64) 431.97 (0.00) -6.3293 -2.8458 -1.9576 -1.4467 -0.9298 0.9498 1.4450 2.0612 3.0378 8.5865
Chart N°3 Backtesting Quantity and proportion of exceptions in forecast period Model Index Brazil
Exp. Num. 2
Lin. Norm. 18
Lin. t 16
HS
1%
7.23%
6.43%
Hungary India Czech Republic Indonesia Malaysia Argentina
Lithuania Tunisia
3 1% 2 1% 3 1% 2 1% 2 1% 2 1% 2 1% 2
Croatia
FHS E-N 12
FHS E-t 11
CV G-N 8
CV G-t 6
CV E-N 9
CV E-t 9
6.43%
3.61%
3.21%
4.82%
4.42%
3.21%
2.41%
3.61%
3.61%
0.80%
2
23
18
15
8
9
9
9
7
5
10
8
4
9.16%
7.17%
5.98%
3.19%
3.59%
3.59%
3.59%
2.79%
1.99%
3.98%
3.19%
1.59%
23
17
14
9
9
18
18
8
3
6
3
3
9.43%
6.97%
5.74%
3.69%
3.69%
7.38%
7.38%
3.28%
1.23%
2.46%
1.23%
1.23%
27
20
16
10
9
12
12
6
5
8
6
0
10.71%
7.94%
6.35%
3.97%
3.57%
4.76%
4.76%
2.38%
1.98%
3.17%
2.38%
0.00%
24 9.92% 16
17 7.02% 12
9
18
15
23
22
13
7
12
8
1
3.72%
7.44%
6.20%
9.50%
9.09%
5.37%
2.89%
4.96%
3.31%
0.41%
10
8
10
10
11
6
4
6
4
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%
22
17 6.85% 26
13
13
13
13
6
5
7
2
1
5.24%
5.24%
5.24%
5.24%
2.42%
2.02%
2.82%
0.81%
0.40%
23
19 7.66% 20
29
30
30
16
8
18
10
10
0
9.47%
8.23%
10.70%
11.93%
12.35%
12.35%
6.58%
3.29%
7.41%
4.12%
4.12%
0.00%
8.87%
1%
FHS G-t 8
EVT
16
FHS G-N 9
14
10
16
17
17
19
18
8
6
8
6
1
5.69%
4.07%
6.50%
6.91%
6.91%
7.72%
7.32%
3.25%
2.44%
3.25%
2.44%
0.41%
2
30
20
19
11
26
13
20
11
4
11
6
0
1%
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 N°4 Backtesting The Three Zone Approach – Increase in scaling factor k Model Index Brazil Hungary India Czech Republic Indonesia Malaysia Argentina
Lithuania Tunisia Croatia
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
Red 100% Red 100% Red 100% Red 100% Red 100% Red 100% Red 100% Red 100% Red 100% Red 100%
Red 100% Red 100% Red 100% Red 100% Red 100% Red 100% Red 100% Red 100% Red 100% Red 100%
Red 100% Red 100% Red 100% Red 100% Yellow 85% Red 100% Red 100% Red 100% Red 100% Red 100%
Yellow 85% Yellow 75% Yellow 85% Red 100% Red 100% Yellow 75% Red 100% Red 100% Red 100% Red 100%
Yellow 75% Yellow 85% Yellow 85% Yellow 85% Red 100% Yellow 75% Red 100% Red 100% Red 100% Red 100%
Red 100% Yellow 85% Red 100% Red 100% Red 100% Red 100% Red 100% Red 100% Red 100% Red 100%
Red 100% Yellow 85% Red 100% Red 100% Red 100% Red 100% Red 100% Red 100% Red 100% Red 100%
Yellow 75% Yellow 65% Yellow 75% Yellow 50% Red 100% Yellow 50% Yellow 50% Red 100% Yellow 75% Red 100%
Yellow 50% Yellow 40% Green 0.00% Yellow 40% Yellow 65% Green 0.00% Yellow 40% Yellow 75% Yellow 50% Green 0.00%
Yellow 85% Red 100% Yellow 50% Yellow 75% Red 100% Yellow 50% Yellow 65% Red 100% Yellow 75% Red 100%
Yellow 85% Yellow 75% Green 0.00% Yellow 50% Yellow 75% Green 0.00% Green 0.00% Red 100% Yellow 50% Yellow 50%
Green 0.00% Green 0.00% Green 0.00% Green 0.00% Green 0.00% Green 0.00% Green 0.00% Green 0.00% Green 0.00% Green 0.00%
Note:
Increase in scaling factor k pictured in second row of respective stock exchange. Emerging markets above solid line, Frontier markets below.
24
Chart N°5 Minimum Capital Requirements VaR MCR(VaR) – Current directives Market / Model Linear-N Linear-t Hist. Sim. 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
Hungary
India
28.54% 34.25% 36.04% 51.62% 50.76% 40.37% 40.85% 51.59% 47.23% 41.67% 43.50% 41.80%
24.30% 29.23% 33.14% 49.36% 52.41% 38.84% 38.88% 45.48% 41.42% 40.23% 37.61% 39.08%
25.25% 29.06% 30.07% 41.35% 41.76% 31.53% 33.41% 42.92% 27.49% 31.57% 23.84% 28.56%
Czech Republic 23.71% 28.10% 32.50% 58.54% 55.80% 46.22% 48.43% 48.69% 51.58% 43.32% 44.50% 42.36%
Indonesia
Malaysia
Argentina
Lithuania
Tunisia
Croatia
22.50% 27.69% 28.28% 32.28% 37.26% 24.02% 26.12% 38.64% 36.67% 31.11% 31.95% 36.96%
12.81% 15.18% 18.35% 17.02% 18.36% 17.34% 17.33% 15.83% 11.84% 15.81% 11.92% 28.49%
25.45% 28.79% 29.58% 20.71% 20.89% 20.06% 20.04% 19.05% 20.06% 21.32% 14.74% 17.60%
16.05% 18.76% 21.04% 18.00% 18.61% 15.61% 16.26% 22.57% 27.26% 19.64% 24.82% 26.94%
7.76% 9.30% 8.22% 8.81% 8.88% 7.80% 7.88% 11.03% 10.13% 9.23% 8.53% 10.83%
23.46% 28.24% 33.56% 29.41% 20.97% 30.18% 24.31% 29.05% 18.02% 30.75% 17.53% 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 N°6 The stressed VaR (sVaR) proposal – sVaR values Market / Model Linear-N Linear-t Hist. Sim. 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
Hungary
India
4.71% 5.00% 5.21% 6.00% 6.02% 5.67% 5.73% 6.22% 6.46% 6.54% 6.79% 8.82%
2.07% 3.34% 3.61% 2.29% 2.34% 2.70% 2.73% 2.27% 2.48% 2.74% 2.96% 3.22%
5.98% 4.87% 5.34% 5.48% 5.49% 3.85% 4.07% 6.27% 7.01% 5.20% 5.98% 7.31%
Czech Republic 3.35% 3.48% 3.54% 2.27% 2.25% 2.41% 2.42% 2.68% 2.95% 2.96% 3.31% 3.49%
Indonesia
Malaysia
Argentina
Lithuania
Tunisia
Croatia
3.62% 3.96% 4.39% 2.85% 3.42% 2.29% 2.65% 3.38% 4.17% 3.07% 3.79% 6.46%
3.57% 4.02% 4.54% 2.39% 2.16% 1.77% 1.78% 2.64% 2.76% 2.14% 2.44% 7.12%
5.41% 6.62% 6.68% 9.14% 9.73% 8.02% 8.35% 9.66% 11.30% 8.67% 10.31% 13.39%
2.12% 2.46% 1.99% 3.20% 3.35% 3.40% 3.54% 4.54% 6.04% 5.33% 6.45% 10.83%
1.10% 1.26% 1.14% 0.49% 0.49% 0.54% 0.53% 0.72% 0.76% 0.75% 0.79% 1.24%
2.25% 5.10% 5.12% 2.46% 2.01% 2.41% 2.45% 2.45% 3.28% 2.48% 3.47% 4.38%
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
25
Chart N°7 Minimum Capital Requirements sVaR: MCR(sVaR) – Proposed directives Market / Model Linear-N Linear-t Hist. Sim. 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
Hungary
India
27.32% 28.91% 29.14% 21.03% 19.79% 23.78% 23.83% 21.11% 18.72% 26.05% 26.84% 17.10%
16.08% 20.72% 22.95% 14.56% 15.62% 16.54% 16.58% 13.42% 12.34% 18.07% 16.99% 11.53%
17.03% 29.08% 29.54% 16.30% 16.20% 16.51% 16.41% 17.77% 11.20% 16.83% 12.12% 11.83%
Czech Republic 19.93% 20.86% 21.15% 17.42% 16.09% 16.42% 16.67% 15.48% 16.09% 17.98% 17.46% 13.47%
Indonesia
Malaysia
Argentina
Lithuania
Tunisia
Croatia
21.28% 23.95% 24.42% 18.21% 20.33% 16.15% 17.92% 21.32% 20.05% 21.36% 22.11% 20.39%
21.36% 24.18% 25.82% 16.69% 18.65% 16.45% 16.45% 15.82% 11.92% 15.20% 11.44% 28.47%
28.58% 34.81% 36.36% 45.07% 46.23% 41.48% 42.60% 36.27% 38.24% 38.10% 27.18% 33.50%
12.64% 14.65% 11.85% 8.71% 7.88% 8.29% 8.14% 12.41% 12.49% 13.20% 14.89% 14.81%
6.63% 7.57% 6.94% 4.01% 4.01% 4.35% 4.32% 5.15% 4.67% 5.26% 4.77% 5.06%
12.76% 30.95% 30.95% 15.23% 12.73% 13.87% 13.70% 15.14% 10.39% 14.24% 9.70% 13.54%
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 N°8 Total Minimum Capital Requirements MCR = MCR (VaR) + MCR (sVaR) Market / Model Linear-N Linear-t Hist. Sim. 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
Hungary
India
55.86% 63.16% 65.19% 72.65% 70.55% 64.16% 64.68% 72.70% 65.96% 67.72% 70.34% 58.90%
40.37% 49.95% 56.10% 63.92% 68.03% 55.38% 55.46% 58.89% 53.77% 58.30% 54.59% 50.60%
42.28% 58.14% 59.61% 57.65% 57.96% 48.05% 49.82% 60.69% 38.69% 48.40% 35.96% 40.39%
Czech Republic 43.64% 48.96% 53.65% 75.96% 71.89% 62.65% 65.10% 64.18% 67.66% 61.30% 61.95% 55.83%
Indonesia
Malaysia
Argentina
Lithuania
Tunisia
Croatia
43.78% 51.64% 52.70% 50.49% 57.59% 40.17% 44.04% 59.96% 56.72% 52.46% 54.06% 57.35%
34.16% 39.36% 44.17% 33.70% 37.01% 33.79% 33.78% 31.65% 23.76% 31.02% 23.36% 56.96%
54.04% 63.59% 65.94% 65.79% 67.12% 61.54% 62.64% 55.32% 58.30% 59.42% 41.93% 51.11%
28.69% 33.42% 32.89% 26.71% 26.50% 23.89% 24.40% 34.99% 39.75% 32.84% 39.71% 41.75%
14.39% 16.86% 15.16% 12.82% 12.89% 12.15% 12.21% 16.18% 14.79% 14.49% 13.30% 15.89%
36.22% 59.19% 64.51% 44.63% 33.70% 44.05% 38.01% 44.19% 28.41% 44.99% 27.23% 39.50%
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
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Chart N°9 Current MCR: Loss Coverage and Maximum daily loss Market / Index
Loss Coverage [1] Brazil 3.46 Hungary 3.03 India 2.46 Czech Republic 2.62 Indonesia 3.37 Malaysia 2.85 Average Emerging 2.97 Argentina 1.36 Lithuania 3.82 Tunisia 2.16 Croatia 2.41 Average Frontier 2.44 Note: Loss Coverage = MCR(VaR) / Maximum Loss Forecast Period
Maximum daily loss [2] 41.80% 39.08% 28.56% 42.36% 36.96% 28.49% 36.21% 17.60% 26.94% 10.83% 25.97% 20.33%
Chart N°10 Standard Deviation and Maximum daily loss – Sample period vs Forecast period Market
Brazil Hungary India Czech Rep. Indonesia Malaysia Avg.Emerg. Argentina Lithuania Tunisia Croatia Avg.Frontier
Standard Deviation Sample [1] 1.82% 1.23% 1.60% 1.19% 1.39% 1.05%
Standard Deviation Forecast [2] 3.29% 2.90% 2.86% 3.04% 2.18% 1.37%
Standard Deviation Variation [3] 81.28% 135.72% 78.53% 155.18% 56.47% 31.06%
2.22% 1.00% 0.44% 1.43%
2.86% 2.01% 0.51% 2.62%
29.07% 121.58% 14.40% 82.38%
Max. Daily Loss Sample [4] 7.54% 5.55% 11.81% 7.08% 10.93% 6.34% 8.21% 14.76% 5.87% 2.12% 9.03% 7.95%
Max. Daily Loss Forecast [5] 12.10% 12.89% 11.60% 16.19% 10.95% 9.98% 12.28% 12.95% 7.05% 5.00% 10.76% 8.94%
Max. Daily Loss Variation [6] 60.45% 132.11% -1.73% 128.70% 0.19% 57.34% 49.65% -12.28% 19.95% 135.51% 19.21% 12.49%
Chart N°11 Proposed MCR-Loss Coverage and Maximum daily loss-Variation over current MCR Market / Index
Loss Coverage
Variation over present MCR [1] [2] Brazil 4.87 40.92% Hungary 3.93 29.50% India 3.48 41.40% Czech Republic 3.45 31.80% Indonesia 5.24 55.16% Malaysia 5.71 99.94% Average Emerging 4.44 49.79% Argentina 3.95 190.37% Lithuania 5.93 54.99% Tunisia 3.17 46.70% Croatia 3.67 52.12% Average Frontier 4.18 86.05% Note: Loss Coverage = MCR(sVaR) / Maximum Loss Forecast Period
27
MCR = Maximum Daily Loss [3] 58.90% 50.60% 40.39% 55.83% 57.35% 56.96% 53.34% 51.11% 41.75% 15.89% 39.50% 37.06%
Chart N°12 Sensitivity Analysis - Total MCR with varying scaling factors mc and ms Case N° 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
Multiples mc – ms mc=3 mc=3/ms=0 mc=3/ms=0.5 mc=3/ms=1 mc=3/ms=1.5 mc=3/ms=2 mc=3/ms=2.5 mc=3/ms=3 mc=3.5 mc=3.5/ms=0 mc=3.5/ms=0.5 mc=3.5/ms=1 mc=3.5/ms=1.5 mc=3.5/ms=2 mc=3.5/ms=2.5 mc=3.5/ms=3 mc=4 mc=4/ms=0 mc=4/ms=0.5 mc=4/ms=1 mc=4/ms=1.5 mc=4/ms=2 mc=4/ms=2.5 mc=4/ms=3 mc=4.5 mc=4.5/ms=0 mc=4.5/ms=0.5 mc=4.5/ms=1 mc=4.5/ms=1.5 mc=4.5/ms=2 mc=4.5/ms=2.5 mc=4.5/ms=3 mc=5 mc=5/ms=0 mc=5/ms=0.5 mc=5/ms=1 mc=5/ms=1.5 mc=5/ms=2 mc=5/ms=2.5 mc=5/ms=3
Brazil
Hungary
India
41.80% 50.61% 50.61% 50.61% 50.61% 53.20% 56.05% 58.90% 48.76% 57.58% 57.58% 57.58% 57.58% 60.16% 63.01% 65.86% 55.73% 64.54% 64.54% 64.54% 64.54% 67.13% 69.98% 72.83% 62.69% 71.51% 71.51% 71.51% 71.51% 74.09% 76.94% 79.79% 69.66% 78.48% 78.48% 78.48% 78.48% 81.06% 83.91% 86.76%
39.08% 42.30% 42.30% 42.92% 44.84% 46.76% 48.68% 50.60% 45.59% 48.81% 48.81% 49.43% 51.35% 53.27% 55.20% 57.12% 52.10% 55.32% 55.32% 55.94% 57.87% 59.79% 61.71% 63.63% 58.61% 61.83% 61.83% 62.46% 64.38% 66.30% 68.22% 70.14% 65.13% 68.35% 68.35% 68.97% 70.89% 72.81% 74.73% 76.65%
28.56% 35.87% 35.87% 35.87% 35.87% 36.45% 38.42% 40.39% 33.32% 40.63% 40.63% 40.63% 40.63% 41.21% 43.18% 45.15% 38.09% 45.39% 45.39% 45.39% 45.39% 45.97% 47.94% 49.91% 42.85% 50.15% 50.15% 50.15% 50.15% 50.73% 52.70% 54.67% 47.61% 54.91% 54.91% 54.91% 54.91% 55.49% 57.46% 59.43%
Czech Republic 42.36% 45.85% 45.85% 46.85% 49.09% 51.34% 53.58% 55.83% 49.42% 52.91% 52.91% 53.91% 56.15% 58.40% 60.64% 62.89% 56.48% 59.97% 59.97% 60.97% 63.21% 65.46% 67.70% 69.94% 63.54% 67.03% 67.03% 68.03% 70.27% 72.52% 74.76% 77.00% 70.60% 74.09% 74.09% 75.09% 77.33% 79.57% 81.82% 84.06%
Indonesia
Malaysia
Argentina
Lithuania
Tunisia
Croatia
36.96% 43.42% 43.42% 43.76% 47.16% 50.55% 53.95% 57.35% 43.12% 49.58% 49.58% 49.92% 53.32% 56.71% 60.11% 63.51% 49.28% 55.75% 55.75% 56.08% 59.48% 62.87% 66.27% 69.67% 55.44% 61.91% 61.91% 62.24% 65.64% 69.03% 72.43% 75.83% 61.60% 68.07% 68.07% 68.40% 71.80% 75.19% 78.59% 81.99%
28.49% 35.61% 35.61% 37.98% 42.72% 47.47% 52.21% 56.96% 33.24% 40.35% 40.35% 42.73% 47.47% 52.22% 56.96% 61.71% 37.98% 45.10% 45.10% 47.47% 52.22% 56.97% 61.71% 66.46% 42.73% 49.85% 49.85% 52.22% 56.97% 61.71% 66.46% 71.20% 47.48% 54.60% 54.60% 56.97% 61.72% 66.46% 71.21% 75.95%
17.60% 30.99% 30.99% 30.99% 34.35% 39.94% 45.52% 51.11% 20.53% 33.92% 33.92% 33.92% 37.29% 42.87% 48.45% 54.04% 23.47% 36.86% 36.86% 36.86% 40.22% 45.80% 51.39% 56.97% 26.40% 39.79% 39.79% 39.79% 43.15% 48.74% 54.32% 59.91% 29.33% 42.72% 42.72% 42.72% 46.09% 51.67% 57.25% 62.84%
26.94% 37.76% 37.76% 37.76% 37.76% 37.76% 39.28% 41.75% 31.43% 42.25% 42.25% 42.25% 42.25% 42.25% 43.77% 46.24% 35.91% 46.74% 46.74% 46.74% 46.74% 46.74% 48.26% 50.73% 40.40% 51.23% 51.23% 51.23% 51.23% 51.23% 52.75% 55.22% 44.89% 55.72% 55.72% 55.72% 55.72% 55.72% 57.24% 59.71%
10.83% 12.07% 12.07% 12.51% 13.36% 14.20% 15.04% 15.89% 12.70% 13.88% 13.88% 14.32% 15.16% 16.01% 16.85% 17.69% 14.44% 15.68% 15.68% 16.12% 16.97% 17.81% 18.65% 19.50% 16.24% 17.49% 17.49% 17.93% 18.77% 19.61% 20.46% 21.30% 18.05% 19.29% 19.29% 19.73% 20.58% 21.42% 22.26% 23.11%
25.97% 30.35% 30.35% 30.48% 32.74% 34.99% 37.25% 39.50% 30.30% 34.68% 34.68% 34.81% 37.06% 39.32% 41.58% 43.83% 34.63% 39.01% 39.01% 39.14% 41.39% 43.65% 45.90% 48.16% 38.95% 43.33% 43.33% 43.47% 45.72% 47.98% 50.23% 52.49% 43.28% 47.66% 47.66% 47.79% 50.05% 52.30% 54.56% 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). Case 8: Proposed Basel mandate (2009) Cases 3, 4 and 5: Lessened ms values: 1.5, 1 and 0.5. Cases 9 and 17: Augmented mc values: 3.5 and 4, no sVaR computed.
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Chart N°13 Sensitivity analysis – Selected examples. Loss Coverage and Maximum daily loss Market Example
Brazil Hungary India Czech Rp. Indonesia Malaysia Avg.Emg. Argentina Lithuania Tunisia Croatia Avg.Ftier
mc=3 ms=1.5 Loss coverage [1] 4.18 3.48 3.09 3.03 4.30 4.28 3.73 2.65 5.36 2.67 3.04 3.43
mc=3 ms=1.5 MCR= Max. daily loss [2] 50.61% 44.84% 35.87% 49.09% 47.16% 42.72% 45.05% 34.35% 37.76% 13.36% 32.74% 29.55%
mc=3 ms=1 Loss coverage [3] 4.18 3.33 3.09 2.89 3.99 3.81 3.55 2.39 5.36 2.50 2.83 3.27
mc=3 ms=1 MCR= Max. daily loss [4] 50.61% 42.92% 35.87% 46.85% 43.76% 37.98% 43.00% 30.99% 37.76% 12.51% 30.48% 27.94%
mc=3 ms=0.5 Loss coverage [5] 4.18 3.28 3.09 2.83 3.96 3.57 3.49 2.39 5.36 2.41 2.82 3.25
mc=3 ms=0.5 MCR= Max. daily loss [6] 50.61% 42.30% 35.87% 45.85% 43.42% 35.61% 42.28% 30.99% 37.76% 12.07% 30.35% 27.79%
mc=3.5
mc=3.5
mc=4
mc=4
Loss coverage
MCR= Max. daily loss [8] 48.76% 45.59% 33.32% 49.42% 43.12% 33.24% 42.24% 20.53% 31.43% 12.70% 30.30% 23.74%
Loss coverage
MCR= Max. daily loss [10] 55.73% 52.10% 38.09% 56.48% 49.28% 37.98% 48.28% 23.47% 35.91% 14.44% 34.63% 27.11%
[7] 4.03 3.54 2.87 3.05 3.94 3.33 3.46 1.59 4.46 2.54 2.81 2.85
[9] 4.61 4.04 3.28 3.49 4.50 3.81 3.95 1.81 5.10 2.89 3.22 3.25
Note: Loss Coverage [1],[3],[5] = MCR(sVaR) / Maximum Loss Forecast Period - Loss Coverage [7],[9] = MCR(VaR) / Maximum Loss Forecast Period
Chart N°14 Sensitivity analysis: Selected examples – Variation over Current and Proposed directives Market Example
Brazil Hungary India Czech Rp. Indonesia Malaysia Avg.Emg. Argentina Lithuania Tunisia Croatia Avg.Ftier
mc=3 ms=1.5 Variation over current directives [1] 21.10% 14.75% 25.59% 15.90% 27.58% 49.97% 25.81% 95.18% 40.20% 22.72% 26.06% 46.04%
mc=3 ms=1.5 Variation over proposed directives [2] -14.07% -11.39% -11.18% -12.06% -17.78% -24.99% -15.25% -32.78% -9.55% -16.35% -17.13% -18.95%
mc=3 ms=1 Variation over current directives [3] 21.10% 9.83% 25.59% 10.60% 18.39% 33.31% 19.80% 76.07% 40.20% 14.97% 17.37% 37.15%
mc=3 ms=1 Variation over proposed directives [4] -14.07% -15.19% -11.18% -16.08% -23.70% -33.32% -18.92% -39.36% -9.55% -21.63% -22.84% -23.34%
mc=3 ms=0.5 Variation over current directives [5] 21.10% 8.24% 25.59% 8.24% 17.49% 24.98% 17.61% 76.07% 40.20% 10.90% 16.87% 36.01%
mc=3 ms=0.5 Variation over proposed directives [6] -14.07% -16.42% -11.18% -17.87% -24.28% -37.49% -20.22% 1.00% -9.55% -24.40% -23.17% -14.03%
mc=3.5
mc=3.5
mc=4
mc=4
Variation over current directives [7] 16.67% 16.67% 16.67% 16.67% 16.67% 16.67% 16.67% 16.67% 16.67% 16.67% 16.67% 16.67%
Variation over proposed directives [8] -17.21% -9.91% -17.49% -11.48% -24.81% -41.65% -20.42% -59.82% -24.73% -20.47% -23.31% -32.08%
Variation over current directives [9] 33.33% 33.33% 33.33% 33.33% 33.33% 33.33% 33.33% 33.33% 33.33% 32.65% 33.33% 33.16%
Variation over proposed directives [10] -5.38% 2.96% -5.70% 1.17% -14.07% -33.31% -9.06% -54.08% -13.97% -9.58% -12.35% -22.50%
Note: Loss Coverage [1],[3],[5] = MCR(sVaR) / Maximum Loss Forecast Period - Loss Coverage [7],[9] = MCR(VaR) / Maximum Loss Forecast Period
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Appendix A: Insights into VaR definition
McNeil, Frey and Embrechts (2005:38-40) provide a concise explanation to express the VaR formula in terms of quantiles. Recalling that VaR is defined by:
VaR (α ) = inf {l ∈ R : P ( L > l ) ≤ (1 − α )} = inf {l ∈ R : FL (l ) ≥ α } where FL (or simply F46) denotes the loss distribution, VaR represents in probabilistic terms a quantile of the loss distribution function F. Acknowledging that the generalised inverse function of an increasing function G: ℜ → ℜ is defined as G ← ( y ) := inf {x ∈ℜ : G ( x) ≥ y} , then, for some distribution function F, the generalised inverse F← is called the quantile function of F. As α ϵ (0;1) the α-quantile of F surges from: qα ( F ) := F ← (α ) = inf {x ∈ℜ : F ( x ) ≥ α } , noting that qα ( F ) := qα ( X ) for a random variable X with distribution function F. Supposing that F is continuous and strictly increasing, qα ( F ) := F −1 (α ) , F-1 being the inverse of the distribution function F. Furthermore, the authors emphasise that a general point x0 such that x0 ϵ ℜ is the α-quantile of any distribution function F provided that: a) F(x0) ≥ α, and b) F(x) < α -for all x < x0- are both met. Assuming that the loss distribution function F follows a Normal distribution with mean µ and variance σ2, it is possible to write47 VaR(α) = µ + σ Ф-1(α) -α ϵ (0;1)-, where Ф represents the distribution function of a standardised Normal and Ф-1(α) the αquantile of Ф. McNeil, Frey and Embrechts (2005) also prove the quantile definition showing that as the loss distribution function F is a strictly increasing function, F(VaRα)=α. Consequently, L−µ P ( L ≤ VaRα ) = P ≤ Φ −1 (α ) = Φ Φ −1 (α ) = α σ
[
]
The expression VaR(α) = µ + σ Ф-1(α), adequate for losses following a standard Normal distribution, could be extended to other location-scale families like the Studentt or the Generalised Pareto Distribution (GPD). For the former example, if losses L ~ t(d, µ, σ2), i.e., losses follow a t distribution with d degrees of freedom, the VaR formula becomes VaR(α ) = µ + σ (d − 2) d −1 t d−1 (α ) , with t symbolising the distribution function
46
Footnote 4 in main text. It is important to bear in mind that these examples portray VaR formulas using the mean µ of the corresponding loss distribution. In the case of driftless series like those pictured in the body of the article, µ = 0 and its value would be omitted from the respective VaR formulas. 47
30
of the standard t distribution and σ
(d − 2) d −1 its variance. Finally, in the latter case, if
losses beyond the appropriate threshold u are distributed according to a GPD Gξ,σ(y), the ∧ σ VaR would be given by VaR(α ) = µ + σ u + ∧ ξ
∧ ∧ 1 − α −ξ∧ − 1 , with µ, σ, u, σ , ξ , w n
w and n keeping the respective meanings indicated in the main text and ∧ σ G (α ) = u + ∧ ξ −1
1 − α −ξ∧ − 1 denoting the α-quantile of the GPD. w n
The above procedure requires (at least) the practical estimation of the mean and variance of the loss distribution -µ and σ2-, which usually roots in the historical change in the risk-factor data (returns). Both parameters can be calculated following two general alternative routes: a) Unconditionally: mean and variance are simply obtained using the sample mean and variance under the supposition that the series of returns belong to a stationary process; b) Conditionally: the data are considered as time series where the subscript t+1, either in µ t+1 or σt+12, indicate the conditional mean and variance employing information up to time t. Estimates of the statistical moments allowing the computation of both parameters are obtained by means of the formal estimation of any time series model, like those belonging to the GARCH family analysed in the main text.
The present study is focused on the comparison among the most widespread VaR models and, in this vein, it is assumed that: a) It is always possible to calculate the historical change in the risk-factor data, i.e., returns always exist; b) At least the first four moments of all time series exist and are finite; c) The processes dealt with are at a minimum weakly, second-order or covariance stationary. Hence, the behaviour of the time series is similar at any time an observation is carried out irrespective of the time span between two observations.
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Appendix B: Some reflections on the behaviour of the models across markets
The behaviour of the models in the Backtesting period could be somewhat inferred from the characteristics of the distribution of returns during that time span. Chart B displays the common statistics recorded in the year 2008 for every index as well as selected variations comparing the sample and prediction terms These outcomes must be assessed in connection with Chart N°2 and Chart N°3 in the main text which depict the stylised facts about asset return series and the quantity of excessions recorded in Backtesting48. It is not surprising that linear models irrespective of the distribution applied are unable to cope with the significant hike in volatility (with peaks of 155.18% in Czech Republic and 121.58% in Lithuania for Emerging and Frontier markets respectively): the inherent flaws of the standard deviation as a risk measure chiefly characterised by the equal weighting structure do not allow the clusters of high variations typical of crisis times (Messina (2005)). The only performances (if any) worth of being singled out are Malaysia and Tunisia, i.e., those stock exchanges with the lowest increase in standard deviation (31.06% and 14.40% respectively); although the Student-t scheme delivers more accurate results, it nevertheless remains insufficient to drive all the models out of the Red Zone. HS gives practically uniform results across countries no matter their location, in a way similar to those of the Linear specifications: the dramatic changes observed in the shape of the distributions pose difficulties for the method at the time of tracking the volatility spikes. Indonesia represents the solitary instance where HS avoids the regulatory disqualification: 9 exceptions (Yellow Zone with 85% surcharge) most likely due to the relatively similar kurtosis and 1% quantile value. However, the examples of India and Lithuania (mainly) cast some doubt on the former explanation, given that the kurtosis and the cut-off points for the 1% quantile are reduced. Additional research into the length of the window used to calculate HS values may shed some light on this method, although it is an established fact throughout the financial literature that its snags largely overcome its advantages hence making HS an inadequate and unreliable method to calculate VaR. The introduction of the conditional modelling in FHS helps HS to enhance its accuracy though the improvement fails to report less than 8 exceptions (high Yellow Zone) considering all countries in both categories. In general terms the GARCH variant
48
Chart B contains all the figures enunciated in the present Appendix.
32
exhibits better performance than its exponential counterpart, especially in those markets where the skewness reverted from negative to positive (Brazil) or decreased significantly (India and Lithuania49, 80.28% and 76.66% respectively). Finally, the application of the Student-t likelihood function either for GARCH or EGARCH models falls short of achieving a considerable impact for Backtesting purposes: it may be surmised that the changes verified in the distribution of returns between the sample and forecast periods hamper the precision that both specifications (irrespective of the likelihood function employed to derive them) may be supposed to achieve. Remarkable advances are attained with conditional volatility models resorting to a pre-specified statistical distribution, either Normal or Student-t. In effect, comparing the distributional assumptions of both models (i.e., Normal variants against Student-t ones for GARCH and EGARCH separately) it may be appreciated that the Student-t representations avoid Red Zone for every stock market (except in Lithuania) and always report less exceptions than its Gaussian counterparts; the leptokurtic characteristics of the distribution of returns (i.e., the kurtosis and the cut-off points of the extreme quantiles) in the forecast period broadly provide an appropriate explanation about the prevalence of the heavy-tailed distribution. The Normal distribution, using GARCH or EGARCH specifications, struggle to drive the models to the Green Zone; most results situate in the Yellow Zone with scattered appearances in the Red region (Hungary, Indonesia, Lithuania and Croatia for EGARCH and Indonesia and Croatia for GARCH). For those stock exchanges where the models avoid being disqualified, the failure of the Normal distribution may also be assessed in terms of the reduced values of the d(hat)50 in the prediction term especially in Brazil, Hungary, Indonesia, Lithuania and Croatia. The joint assessment of the d(hat) and the values of the degrees of freedom delivered by the respective ML optimisation appears useful to understand the difference in the performance between the conditional volatility specifications51. The figures belonging to the sample and forecast periods convey that the majority of the Emerging markets suffered a drop in d(hat) (except India and Indonesia) while in Frontier ones the situation reverts (excluding Tunisia). Furthermore, those countries that experienced an increase in d(hat) in the forecast period compared to the sample one also had its kurtosis 49
The EGARCH-t specification achieves the smallest quantity of violations, though still in the Red Zone. d(hat) stands for the estímate of the degrees of freedom of a Student-t(d) distribution affixed to the distribution of returns (Alexander (2008a)). 51 The failure of the EGARCH scheme to clearly perform better than the GARCH model hints at the absence or the presence of a weak version of the leverage effect cited by Chopra, Lakonishok and Ritter (1992) and Cho and Engle (1999)). 50
33
decreased and viceversa. Recalling that the higher the degrees of freedom in a Student-t distribution the closer its resemblance to the Normal one (with kurtosis edging near 3) (Da Costa Lewis (2003)), the stock exchanges with the steepest decline in the approximate value of d(hat) and its corresponding enhancement in the kurtosis would pose the greatest challenge to the models, like Brazil and Hungary (52.24% and 32.67% respectively). It is also important to bear in mind that the performance of the models could also be dictated by the value of the degrees of freedom d delivered in the ML optimisation process in comparison with the estimate of the degrees of freedom of the distribution d(hat). Provided the outcomes are evaluated in terms of the exceptions recorded in Backtesting, the GARCH specification performs clearly better than the EGARCH one in Brazil (6 against 9) and Hungary (5 vs 8), either country featuring sudden falls in the d(hat) and values of GARCH-d in the ML process closer to the estimate of the forecast period than those given by the EGARCH variant (17.96 vs 24.50 and 9.87 against 10.64). The outstanding results of both models in India and Indonesia could be put down to the reduction in the kurtosis and the fact that the distributions do not display excessive concentration in the 0.01% and 1% quantiles; under this framework, Czech Republic presents the inverse example. Analogous considerations may be applied to Frontier markets. For instance, in Argentina, even though there is a 17% decrease in kurtosis values, the negative asymmetry rising by 288% and the negative returns clustering in the 1% quantile (q(0.01) = -3.37 in the forecast period and q(0.01) = -3.02 in the sample term), contribute to explain the superiority of the EGARCH-t (2 exceptions) technique over its symmetric GARCH-t counterpart (5 violations). No sizable difference appears in Tunisia, where the abrupt variation in sign and absolute value of the skewness (from positive to negative) alongside the kurtosis augmenting in 92% seem to have affected both specifications alike. Croatia records similar values of the d(hat) and a concentration on the 1% quantile in the prediction term akin to that of the sample observations, but the decrease in kurtosis and the enlargement of the positive asymmetry might deliver a slight advantage to the GARCH-t scheme to the detriment of its exponential peer. However, there seems to be no apparent explanation for the imprecision in Lithuania besides the 121.58% leap in standard deviation: models should have performed better. Finally, it is necessary to underline the robustness of the EVT scheme as it manages the Green Zone in every market despite some dramatic changes in the distribution of returns (e.g., Hungary). 34
Chart B Forecast period: Basic statistics and selected variations over sample period – Emerging markets Index/ Brazil Brazil Hungary Period Parameter Forecast Variation Forecast Mean -0.0021 CS -0.0030 Median 0.0013 CS -0.0015 Maximum 0.1368 86.45% 0.1038 Minimum -0.1210 60.45% -0.1289 Std. Dev. 0.0329 81.28% 0.0290 Skewness 0.2141 CS -0.4081 Kurtosis 6.0540 63.32% 7.9694 q(0.01) -3.5989 -10.5186 q(1.00) -2.6641 -8.2961 q(2.50) -2.0986 -5.6385 q(5.00) -1.5841 -4.3916 q(10.00) -1.0765 -2.5657 q(90.00) 0.9567 1.8798 q(95.00) 1.4714 2.8236 q(97.50) 2.2898 4.3958 q(99.00) 2.8177 7.3298 q(99.99) 4.2108 8.3519 d(hat) 5.9646 -52.24% 5.2074 Note: “CS” stands for “Change of sign”
Hungary
India
India
Variation CS CS 157.32% 132.11% -10.35% 72.99% 131.52% -32.67%
Forecast -0.0030 -0.0034 0.0790 -0.1160 0.0286 -0.0660 3.8726 -3.9116 -2.3785 -1.9723 -1.6117 -1.2110 1.2433 1.9230 2.0514 2.2663 2.8581 10.8760
Variation CS CS -8.04% -1.73% 78.53% -80.28% -39.46% 88.61%
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Czech Republic Forecast -0.0029 -0.0023 0.1236 -0.1619 0.0304 -0.2895 9.5137 -13.5269 -8.2298 -5.3746 -4.8777 -2.6493 2.0700 2.5074 4.1453 8.3505 10.3114 4.9211
Czech Republic Variation CS CS 75.42% 128.70% 155.18% 0.74% 67.94% -21.28%
Indonesia
Indonesia
Malaysia
Malaysia
Forecast -0.0002 0.0007 0.0762 -0.1095 0.0218 -0.4761 6.9340 -5.0114 -2.8134 -2.1435 -1.7318 -1.0475 0.9133 1.4765 2.0911 2.8274 3.5040 5.5252
Variation CS -39.70% 13.22% 0.19% 56.47% -33.41% -9.35% 4.44%
Forecast -0.0020 -0.0018 0.0406 -0.0998 0.0137 -1.3972 13.3380 -7.0129 -2.5290 -2.1057 -1.3903 -0.9906 1.0874 1.5895 1.8732 2.2148 3.0891 4.5804
Variation CS CS -30.69% 57.34% 31.06% 488.64% 54.98% -9.66%
Chart B (cont.) Forecast period: Basic statistics and selected variations over sample period – Frontier markets Index/ Argentina Argentina Lithuania Period Parameter Forecast Variation Forecast Mean -0.0028 CS -0.0045 Median 0.0000 -99.06% -0.0027 Maximum 0.1043 -35.27% 0.1100 Minimum -0.1295 -12.28% -0.0911 Std. Dev. 0.0286 29.07% 0.0201 Skewness -0.6433 288.06% -0.1687 Kurtosis 7.1952 -17.21% 11.2690 q(0.01) -4.4196 -4.2838 q(1.00) -3.3711 -1.0517 q(2.50) -2.3319 -0.6379 q(5.00) -2.1824 -0.4466 q(10.00) -1.0754 -0.2126 q(90.00) 0.9481 0.7537 q(95.00) 1.0455 0.9623 q(97.50) 1.7460 1.1385 q(99.00) 2.7071 1.5516 q(99.99) 3.7384 2.5005 d(hat) 5.4302 7.44% 4.7256 Note: “CS” stands for “Change of sign”
Lithuania
Tunisia
Tunisia
Croatia
Croatia
Variation CS CS 140.20% -10.82% 121.58% -76.66% -17.13% 3.49%
Forecast 0.0004 0.0002 0.0361 -0.0500 0.0051 -0.2513 11.9022 -9.2443 -2.4846 -1.8983 -1.4600 -1.0590 1.1042 1.5497 1.9665 2.5715 6.7644 4.6740
Variation -9.08% 7.88% 18.84% 135.51% 14.40% CS 91.86% -20.41%
Forecast -0.0045 -0.0019 0.1478 -0.1076 0.0262 0.3261 9.1376 -3.9192 -2.8429 -2.3368 -2.1366 -0.9604 0.9018 1.0393 1.5874 2.9968 5.7536 4.9776
Variation CS CS 16.39% 19.21% 82.38% 13.33% -29.36% 8.12%
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