on the market risk measurement and basel accords€¦ · on the market risk measurement and basel...

Market Risk VaR Models A Regulatory Discussion

On the market risk measurement and BaselAccords

Jean-Paul Laurent Hassan Omidi Firouzi

Université Paris I, Panthéon Sorbonne LabexRefi 1

September 03 2015

Advanced Methods in Mathematical FinanceAngers

1. Laboratory of Excellence for Financial Regulation

1 Market Risk

2 VaR Models

3 A Regulatory Discussion

Market risk

Market risk is the risk that the value of a trading portfolio of a bankwill decrease (loss of a bank) due to the change in value of thefollowing risk factors. (FRTB October 2013)

Equity risk : the risk of changing in the equity prices.

Foreign exchange (FX) risk : the risk of changing in FXrates.

Interest rate risk : the risk of changing interest rate change.

Commodity risk : the risk of changing in commodity prices(crude oil price, silver, etc.).

Credit spread risk : the risk due to changes in credit spread.

Market risk

Market risk and Basel III

Approaches to risk measurement :

1 Standardized approach.2 Internal model-based approach (IMB).

Two main changes under Basel III framework :1 Moving from Value at Risk (VaR) to Expected Shortfall (ES).2 Stressed calibration. Using the most severe 12-month stress

period in computing ES.

Approaches to risk measurement :1 Standardized approach.

2 Internal model-based approach (IMB).

Approaches to risk measurement :1 Standardized approach.2 Internal model-based approach (IMB).

Two main changes under Basel III framework :

1 Moving from Value at Risk (VaR) to Expected Shortfall (ES).2 Stressed calibration. Using the most severe 12-month stress

Two main changes under Basel III framework :1 Moving from Value at Risk (VaR) to Expected Shortfall (ES).

2 Stressed calibration. Using the most severe 12-month stressperiod in computing ES.

Determining the eligibility of trading activities for the IMBapproach (FRTB October 2013)

1 Overall assessment of the banks’ firm-wide internal risk capitalmodel which would be based on both qualitative and quantitativefactors (if pass go to the step 2 otherwise go to the standardizedapproach).

2 Banks nominate which trading desks are in-scope for modelapproval and which are out-of-scope (if pass go to the step 3otherwise go to the standardized approach).

3 Assessment of trading desk-level model performance againstquantitative criteria (if pass go to the step 4 otherwise go to thestandardized approach).

P&L attribution,backtesting.

4 Computing Expected Shortfall for both modellable andnon-modellable risk factors.

P&L attribution,

backtesting.

VaR models

There are different approaches to estimate value at risk in practice(Alexander, 2008).

1 Parametric approach. Variance-covariance and Monte Carlomethods.

2 non parametric approaches. Historical simulation (HS) andBRW methods (Boudoukh et al. 1998).

3 semi parametric approaches. Filter historical simulation (FHS)(Barone et al., 1999 and Gurrola-Perez et al., 2015) andvolatility-weighted historical simulation (VWHS) models (Hulland White, 1998).

VaR models

Parametric approach :

1 enables us to come up with an analytic formula for VaR,2 assumes that the returns of the risk factors are i.i.d. random

vectors, ! !3 mostly assume the returns for risk factors are normal

distribution (or generally elliptical distributions). ! ! (SeeChicheportiche 2012)

VaR models

Parametric approach :1 enables us to come up with an analytic formula for VaR,

2 assumes that the returns of the risk factors are i.i.d. randomvectors, ! !

3 mostly assume the returns for risk factors are normaldistribution (or generally elliptical distributions). ! ! (SeeChicheportiche 2012)

VaR models

Parametric approach :1 enables us to come up with an analytic formula for VaR,2 assumes that the returns of the risk factors are i.i.d. random

vectors, ! !

3 mostly assume the returns for risk factors are normaldistribution (or generally elliptical distributions). ! ! (SeeChicheportiche 2012)

VaR models

Parametric approach :1 enables us to come up with an analytic formula for VaR,2 assumes that the returns of the risk factors are i.i.d. random

vectors, ! !3 mostly assume the returns for risk factors are normal

distribution (or generally elliptical distributions). ! ! (SeeChicheportiche 2012)

VaR models

In non-parametric approach :

1 both HS and BRW do not assume any assumption about thedistribution of risk factors.

2 HS assigns equal weights to all historical return which results inhaving i.i.d. returns through time. This violates the fact that thevolatility of returns is not fixed and tends to fluctuate through time.

3 BRW method relaxes previous drawback of HS by assigning moreprobability weights to recent events than those happened in the farpast.

4 Both HS and BRW do not consider upper tail of returns (Pritsker,2006).

VaR models

In semi-parametric approach :

1 we fit historical data with a model tracking the time varyingbehavior of volatility of returns (EWMA and GARCH).

2 The applied model is faced with different model risks (e.g.estimation error and model inaccuracy).

The combination of EWMA model with VWHS or variance-covariancemethod (RiskMetrics approach) are the most used models betweenbanks for computing market risk. (Alexander 2008)

VaR models

In semi-parametric approach :1 we fit historical data with a model tracking the time varying

behavior of volatility of returns (EWMA and GARCH).

2 The applied model is faced with different model risks (e.g.estimation error and model inaccuracy).

VaR models

behavior of volatility of returns (EWMA and GARCH).2 The applied model is faced with different model risks (e.g.

estimation error and model inaccuracy).

VaR models

behavior of volatility of returns (EWMA and GARCH).2 The applied model is faced with different model risks (e.g.

estimation error and model inaccuracy).

Challenges related to the EWMA model

In VWHS method, we adjust returns in the following way :

r̃t =σT

σtrt, 1 ≤ t ≤ T, (1)

where σT is the volatility of the most recent data in the data set andσt is the volatility of rt.

Volatilitites are estimated using the EWMA model :

σ2t+1 = λσ

2t + (1 − λ)r2

t , (2)

where λ ∈ (0, 1) is called decay factor and has to be estimated fromhistorical data.

r̃t =σT

σtrt, 1 ≤ t ≤ T, (1)

σ2t+1 = λσ

2t + (1 − λ)r2

t , (2)

r̃t =σT

σtrt, 1 ≤ t ≤ T, (1)

σ2t+1 = λσ

2t + (1 − λ)r2

t , (2)

r̃t =σT

σtrt, 1 ≤ t ≤ T, (1)

σ2t+1 = λσ

2t + (1 − λ)r2

t , (2)

Example

In the RiskMetrics of J.P. Morgan (1996) approach, the decay factoris fixed and λ = 0.94 for daily returns and λ = 0.97 for monthlyreturns. Moreover, a normality assumption of the return process isassumed.

The EWMA estimator is based on the maximum likelihood estima-tor of the variance of the normal distribution, and is therefore opti-mal when returns are conditionally normal (see Nelson and Foster,1996).

Example

In the RiskMetrics of J.P. Morgan (1996) approach, the decay factoris fixed and λ = 0.94 for daily returns and λ = 0.97 for monthlyreturns. Moreover, a normality assumption of the return process isassumed.

The EWMA estimator is based on the maximum likelihood estima-tor of the variance of the normal distribution, and is therefore opti-mal when returns are conditionally normal (see Nelson and Foster,1996).

Questions :

1 Decay factors should be fixed for all risk factors, assetclasses, and over time ?

2 When model (2) is optimal ?3 Which method is robust in estimating decay factors ?4 The effect of choosing different estimation methods for

estimating decay factors on VaR backtesting ?5 In the case of rejection at desk level, what is the cost for falling

back to the standardized approach ?

Questions :1 Decay factors should be fixed for all risk factors, asset

classes, and over time ?

2 When model (2) is optimal ?3 Which method is robust in estimating decay factors ?4 The effect of choosing different estimation methods for

classes, and over time ?2 When model (2) is optimal ?

3 Which method is robust in estimating decay factors ?4 The effect of choosing different estimation methods for

classes, and over time ?2 When model (2) is optimal ?3 Which method is robust in estimating decay factors ?

4 The effect of choosing different estimation methods forestimating decay factors on VaR backtesting ?

5 In the case of rejection at desk level, what is the cost for fallingback to the standardized approach ?

classes, and over time ?2 When model (2) is optimal ?3 Which method is robust in estimating decay factors ?4 The effect of choosing different estimation methods for

estimating decay factors on VaR backtesting ?

5 In the case of rejection at desk level, what is the cost for fallingback to the standardized approach ?

classes, and over time ?2 When model (2) is optimal ?3 Which method is robust in estimating decay factors ?4 The effect of choosing different estimation methods for

Figure: Fluctuations of the estimated decay factors for 3 different assetsand for the desk composed of linear combinations of these 3 assets for asample from 17/12/2012 to 31/12/2013.

Pritsker in his article (Pritsker, 2006) discusses about drawbacksof tests for correct conditional coverage (independence of VaRexceptions) property for exceptions and shows that the power oftests for correct conditional coverage is very low.

Example : Consider the VaR model (1). In the case of an increasingin volatility, assume that lambda parameter in (2) is not well estima-ted, then it is very likely that there will be a clustering of exceptions,due to high persistence of volatility. In this case, if number of his-torical data window will be short enough, then it is likely to haveconditional coverage property while in practice there is autocorrela-tions between exceptions.

Pritsker in his article (Pritsker, 2006) discusses about drawbacksof tests for correct conditional coverage (independence of VaRexceptions) property for exceptions and shows that the power oftests for correct conditional coverage is very low.

Example : Consider the VaR model (1). In the case of an increasingin volatility, assume that lambda parameter in (2) is not well estima-ted, then it is very likely that there will be a clustering of exceptions,due to high persistence of volatility. In this case, if number of his-torical data window will be short enough, then it is likely to haveconditional coverage property while in practice there is autocorrela-tions between exceptions.

Challenges related to backtesting EWMA model

The conditional distribution of short horizon asset returns is oftenfound to have fatter than those for the normal distribution (see Bol-lerslev et al., 1992). Therefore, the EWMA estimator of the varianceis no longer optimal ! !

How about choosing an alternative EWMA estimator which is robustto non normality in returns ? (See Harris and Guermat, 2002).

σ2t+1 = {λσt + (1 − λ)

√2|rt|}

2, (3)

σ2t+1 = {λσt + (1 − λ)

√2|rt|}

2, (3)

σ2t+1 = {λσt + (1 − λ)

√2|rt|}

2, (3)

In RiskMetrics approach, the decay factor is recommended to beestimated using minimizing the squared error loss function. i.e.,

λ̂ = arg minλ∈(0,1)

T∑i=1

[σ2i (λ) − r2

i ]2, (4)

where σ2i (λ) follows equation (2).

Our evidence suggests that the estimator (4) is not robust ! !

In RiskMetrics approach, the decay factor is recommended to beestimated using minimizing the squared error loss function. i.e.,

T∑i=1

[σ2i (λ) − r2

i ]2, (4)

where σ2i (λ) follows equation (2).

Our evidence suggests that the estimator (4) is not robust ! !

There are other alternatives to equation (4) to estimate decay fac-tors.

In (Fan and Gu, 2003), the authors propose an estimationmethod for decay factors based on pseudo-likelihood method.

λ̂ = − arg minλ∈(0,1)

T∑i=1

(log(σ2i ) +

), (5)

where ri is the ith return in historical data window and σ2i is

obtained from equation (2).

λ̂ = − arg minλ∈(0,1)

T∑i=1

(log(σ2i ) +

), (5)

λ̂ = − arg minλ∈(0,1)

T∑i=1

(log(σ2i ) +

), (5)

In (González-Rivera et al., 2007), the authors estimatesoptimal decay factors by minimizing the check loss function(scoring function).

T∑i=1

ρα(ei), (6)

where ei = ri − qα,i, and ρα(ei) = (α − 1{ei<0})ei for α ∈ (0, 1). qα,i isthe quantile function at confidence level α.

T∑i=1

ρα(ei), (6)

T∑i=1

ρα(ei), (6)

Backtesting

Each VaR method results in a different estimation.

Question : How can one study the accuracy of the used VaR me-thod ?Backtesting ? !

is a method based on statistical tools which compares actual profitsand losses with VaR estimates.

Backtesting

Question : How can one study the accuracy of the used VaR me-thod ?

Backtesting ? !

Backtesting

Christoffersen,P., (Christoffersen, 1998) has shown that a VaR mo-del is valid if and only if the following two conditions are satisfied :

1 the unconditional coverage property. i.e, E(It) = α, whereIt = 1{rt<−VaRα(rt)},

2 the independence condition. i.e, for all s , t, Is and It areindependent,

where rt is the return or P&L of a risk factor.

Backtesting

The most important backtest methods include :

1 the traffic light approach propose by the Basel Committee(BCBS, 1996)

2 Christoffersen’s interval forecast test for independence test(Christofferson, 1998)

3 the mixed Kupiec-test for independence test (Haas, 2001)4 the test based on the univariate Ljung-Box type for both

unconditional and independence tests (Berkowitz andO’Brien, 2002)

5 regression quantile based test (Engle and Manganelli, 2004)and

6 the geometric-VaR test (Pelletier, 2014).

Backtesting

The most important backtest methods include :1 the traffic light approach propose by the Basel Committee

(BCBS, 1996)

2 Christoffersen’s interval forecast test for independence test(Christofferson, 1998)

Backtesting

(BCBS, 1996)2 Christoffersen’s interval forecast test for independence test

(Christofferson, 1998)

Backtesting

(Christofferson, 1998)3 the mixed Kupiec-test for independence test (Haas, 2001)

4 the test based on the univariate Ljung-Box type for bothunconditional and independence tests (Berkowitz andO’Brien, 2002)

Backtesting

(Christofferson, 1998)3 the mixed Kupiec-test for independence test (Haas, 2001)4 the test based on the univariate Ljung-Box type for both

Backtesting

Backtesting method proposed by Berkowitz et al., (2002)

In Berkowitz et al., the authors proposed a univariate test of theLjung-Box type that considers the nullity of the first K autocorrela-tions for the exceptions sequence.

If we assume ρk the univariate exceptions sequence autocorrelationof order k, then to test if ρk = 0 holds for the first K autocorrelations,we have

Q = 63000K∑

250 − k. (7)

Under null hypothesis, ρk = 0 holds for the first K autocorrelations,Q follows a χ2

Q = 63000K∑

250 − k. (7)

Q = 63000K∑

250 − k. (7)

Traffic light approach used by BCBS

The regulatory backtesting process is carried out by comparing thelast 250 daily 97.5% and 99% VaR estimates with correspondingdaily trading outcomes at desk and bank-wide levels.

If a model is accurate, then the violations (exceptions) distributionshould follow Bin(250, α) for α = 0.025 and α = 0, 01.

P(X = k) =(250

)αk(1 − α)250−k, (8)

where k is the number of exceptions.

P(X = k) =(250

)αk(1 − α)250−k, (8)

P(X = k) =(250

)αk(1 − α)250−k, (8)

Based on this, the Committee has classified outcomes for the back-testing of the firm-wide model into three categories.

1 Green zone : model is likely to be correct. Maximum 4exception out of 250 days (BCBS 1998).

2 Yellow zone : The range from 5 to 9 exceptions constitutes theyellow zone. Outcomes in this range are plausible for bothaccurate and inaccurate models (BCBS 1998).

3 Red zone : Outcomes in the red zone (10 or more exceptions)should generally lead to an automatic presumption that aproblem exists with a bank’s model (BCBS 1998).

Backtesting at desk level under Basel III framework

"Backtesting requirements are based on comparing each desk’s 1-day static value-at-risk measure at both the 97.5th percentile andthe 99th percentile, using at least one year of current observationsof the desk’s one-day P&L" (FRTB October 2013).

"If any given desk experiences either more than [12] exceptions atthe 99th percentile or [30] exceptions at the 97.5th percentile in themost recent 12-month period, all of its positions must be capitalizedusing the standardized approach" (FRTB October 2013).

Backtesting at desk level under Basel III framework

"Backtesting requirements are based on comparing each desk’s 1-day static value-at-risk measure at both the 97.5th percentile andthe 99th percentile, using at least one year of current observationsof the desk’s one-day P&L" (FRTB October 2013).

"If any given desk experiences either more than [12] exceptions atthe 99th percentile or [30] exceptions at the 97.5th percentile in themost recent 12-month period, all of its positions must be capitalizedusing the standardized approach" (FRTB October 2013).

P&L Attribution, FRTB July 2015

The P&L attribution requirements are based on two metrics :

1 The mean of the difference between the risk-theoretical andhypothetical P&L (unexplained P&L) divided by the standarddeviation of the hypothetical P&L should be in [−10%, 10%] ;and

2 The variance of the unexplained P&L divided by the varianceof the hypothetical P&L should be less than 20%.

Questions

Thanks for your attention !

References

Alexander, C. Market Risk Analysis, Volume IV : Value at RiskModels. Wiley, 2008.Assessment of Chicago Mercantile Exchange Inc. against theFinancial Stability Standards for Central Counterparties.September 2014.Barone-Adesi G, Giannopoulos, K, Vosper, L. VaR WithoutCorrelations for non-linear Portfolios. Journal of FuturesMarkets, 583-602, 19, August, 1999.Basel Committee on Banking Supervision (BCBS). Impactstudy on the proposed frameworks for market risk and CVArisk, July 2015.BCBS. Consultative Document, Fundamental review of thetrading book : A revised market risk framework, October 2013.BCBS. Supervisory Framework For The Use of "Back-testing"in Conjunction With The Internal Models Approach to MarketRisk Capital Requirements, 1996.

References

Berkowitz, J, O’Brien, J. How Accurate are Value-at-RiskModels at Commercial Banks ?, Journal of Finance, VOL.LVII, NO. 3, June 2002.

Bollerslev, T, Chou, R, Kroner, K. ARCH Modelling in Finance,Journal of Econometrics 52, 5-59, 1992.

Boudoukh, J, Richardson, M, Whitelaw, R. The Best of BothWorlds. Risk, 11(May), 64-67, 1998.

Chicheportiche, R, Bouchaud, J. P. The joint distribution ofstock returns is not elliptical. International Journal ofTheoretical and Applied Finance, 15(03), 2012.

Christofferson, P. Evaluating Interval Forecasts. InternationalEconomic Review, 39 , 841-862, 1998.

Christoffersen, P. Elements of Financial Risk Management,academic Press 2003.

References

Engle, R. F, Manganelli, S. CAViaR : ConditionalAutoregressive Value at Risk by Regression Quantiles.Journal of Business & Economic Statistics, vol. 22, 2004,367-381.Fan, J, Gu, J. Semiparametric estimation of Value at Risk.Econometrics Journal, volume 6, pp. 261-290, 2003.González-Rivera, G, Lee, T-H, Yoldas, E. Optimality of theRiskMetrics VaR model. Finance Research Letters 4,137-145, 2007.Gurrola-Perez, P, Murphy, D. Filtered historical simulationValue-at-Risk models and their competitors. Working PaperNo. 525, Bank of England, 2015.Guermat, C, Harris, R.D.F. Robust Conditional VarianceEstimation and Value-at-Risk. The Journal of Risk 4 : 25-41,2002.

References

Haas, M. New Methods in Backtesting. Financial Engineering,Research Center Caesar, Bonn, 2001.

Hull, J, White, A. Incorporating volatility updating into thehistorical simulation method for Value-at-Risk. Journal of Risk,Volume 1, 1998.

Kupiec, P. Techniques for Verifying the Accuracy of RiskManagement Models. Journal of Derivatives 3 :73-84, 1995.

Nelson, D., Foster, D. Asymptotic Filtering Theory forUnivariate ARCH Models, Econometrica 62, 1-41, 1996.

Pelletier, D. The Geometric-VaR Backtesting Method. 2014.

Pritsker, M. The Hidden Dangers of Historical Simulation. TheJournal of Banking and Finance. 30 (2),pp. 561-582, 2006.

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