Value at Risk: A Comparative Analysis

Value at Risk:Value at Risk: A A ComparativeComparative

AnalysisAnalysis

Filip IorgulescuFilip Iorgulescu

Introduction Introduction

VaRVaR - the number that measures risk, so - the number that measures risk, so popular it may not need an introductionpopular it may not need an introduction

The objective:The objective: to develop a comparison to develop a comparison between different approaches to VaR by the between different approaches to VaR by the means of a portfolio consisting of three stocks means of a portfolio consisting of three stocks traded at BSEtraded at BSE

Why is VaR a challenging subject for me?Why is VaR a challenging subject for me?

- scientific but also practical - scientific but also practical

- easy to understood, difficult to - easy to understood, difficult to determinedetermine

- a benchmark with certain - a benchmark with certain shortcomings shortcomings

The Methodology The Methodology

Mainly based on Christoffersen (2002) and Mainly based on Christoffersen (2002) and Codirlasu (2007)Codirlasu (2007)

The analysis focuses on two aspects that are The analysis focuses on two aspects that are taken into account when computing VaR:taken into account when computing VaR:

- the volatility models- the volatility models

- the distributional approaches- the distributional approaches

It was considered that T = 1 day and p = 1% It was considered that T = 1 day and p = 1%

Therefore, I set out from 1-day, 1% VaR Therefore, I set out from 1-day, 1% VaR formulaformula

VaRVaR1%1% = - ( = - (QQ1%1%σσ + + μμ)S )S

The following approaches were considered:The following approaches were considered:

Historical volatilityHistorical volatility EWMA volatility EWMA volatility model for the three model for the three stocksstocks GARCH volatility GARCH volatility model for the portfoliomodel for the portfolio GARCH volatility GARCH volatility models for the three models for the three stocksstocks GARCH volatility GARCH volatility models for the stocks models for the stocks along with a dynamic along with a dynamic conditional correlation conditional correlation modelmodel

Standard N Standard N distributiondistribution Historical quantile Historical quantile (last 750 days)(last 750 days) t-Student, NIG and t-Student, NIG and GH distributionsGH distributions CF approximationCF approximation Extreme Value Extreme Value Theory. ES measure Theory. ES measure was also consideredwas also considered p

ttttpt VaRRREES 1111

Welcome to the Championship Welcome to the Championship of VaR measures!!! of VaR measures!!!

The computed VaR measures were examined The computed VaR measures were examined according to the following criteria:according to the following criteria:- precision- precision- level of capital coverage- level of capital coverage- calculation requirements- calculation requirements

The pitch is ready… all the tickets are sold… The pitch is ready… all the tickets are sold…

Meet the players – The Data Meet the players – The Data A portfolio consisting of three stocks with A portfolio consisting of three stocks with equal weights: ATB, AZO and TLV. Arguments:equal weights: ATB, AZO and TLV. Arguments:- quoted at Category 1 at BSEquoted at Category 1 at BSE- diversification across industriesdiversification across industries

Period: 5 Jan 2001 – 9 May 2008Period: 5 Jan 2001 – 9 May 2008

Main features:Main features:- Volatility clustering => conditional volatility - Volatility clustering => conditional volatility models are recommendedmodels are recommended- Non-normality => other distributional - Non-normality => other distributional approaches are neededapproaches are needed- No unit roots- No unit roots

Group 1 – Historical volatility Group 1 – Historical volatility Histogram of portln_norm

portln_norm

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nsity

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Histogram of portln_norm

portln_norm

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nsity

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Histogram of portln_norm

portln_norm

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nsity

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Tail of Underlying Distribution

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1-day, 1-day, 1% VaR 1% VaR graphsgraphs

HistoricHistorical al

volatilitvolatility y

approacapproachh

VaR TypeVaR TypeVaR VaR N N

StdStd

Hist Hist VaRVaR

VaR VaR NIGNIG

VaR VaR GHDGHD

VaR VaR CFCF

VaR VaR EVTEVT

ES ES

EVTEVT

Errors %Errors % 1.6%1.6% 1.13%1.13% 1.03%1.03% 0.94%0.94% 0.19%0.19% 0.85%0.85% 0.38%0.38%

Avg capital Avg capital coveragecoverage

-- -- -- 5.91%5.91% 12.71%12.71% 6.2%6.2% 9.65%9.65%

VaR GHD seems to be the most appropriate VaR GHD seems to be the most appropriate choice under historical volatility approach.choice under historical volatility approach.

AdvantagesAdvantages Simplest volatility Simplest volatility approachapproach Good precision Good precision resultsresults

DisadvantagesDisadvantages Strictly dependent on Strictly dependent on these portfolio these portfolio allocationsallocations Rather high capital Rather high capital coverage levelscoverage levels


StdStd

Hist Hist VaRVaR

VaR VaR NIGNIG

VaR VaR GHDGHD

VaR VaR CFCF

VaR VaR EVTEVT

ES ES

EVTEVT

Errors %Errors % 2.262.26%%

1.41%1.41% 1.03%1.03% 1.13%1.13% 0.19%0.19% 1.13%1.13% 0.19%0.19%


-- -- -- -- 10.37%10.37% -- 9.29%9.29%

ES EVT seems to be the most appropriate ES EVT seems to be the most appropriate choice under EWMA volatility approach.choice under EWMA volatility approach.

AdvantagesAdvantages Lower capital Lower capital coveragecoverage No estimations for No estimations for λλ No strict No strict dependence on dependence on current allocationscurrent allocations

DisadvantagesDisadvantages Very weak precision Very weak precision resultsresults Carries the Carries the limitations of EWMA limitations of EWMA modelmodel

Group 2 – EWMA volatility Group 2 – EWMA volatility


StdStd

Hist Hist VaRVaR

VaR VaR NIGNIG

VaR VaR GHDGHD

VaR VaR CFCF

VaR VaR EVTEVT

ES ES

EVTEVT


1.22%1.22% 0.94%0.94% 0.94%0.94% 0.19%0.19% 0.94%0.94% 0.28%0.28%


-- -- 5.37%5.37% 5.29%5.29% 13.23%13.23% 5.36%5.36% 9.42%9.42%

VaR GHD seems to be the most appropriate VaR GHD seems to be the most appropriate choice under GARCH (portfolio) volatility choice under GARCH (portfolio) volatility approach.approach.

AdvantagesAdvantages Lower capital Lower capital coveragecoverage Very good precisionVery good precision No difficult No difficult calculations requiredcalculations required

DisadvantagesDisadvantages Strictly dependent on Strictly dependent on these portfolio these portfolio allocationsallocations

Group 3 – GARCH for the Group 3 – GARCH for the portfolio portfolio


StdStd

Hist Hist VaRVaR

VaR VaR NIGNIG

VaR VaR GHDGHD

VaR VaR CFCF

VaR VaR EVTEVT

ES ES

EVTEVT


1.22%1.22% 0.94%0.94% 1.03%1.03% 0.19%0.19% 1.03%1.03% 0.38%0.38%


-- -- 5.21%5.21% -- 11.54%11.54% -- 9%9%

VaR NIG seems to be the most appropriate VaR NIG seems to be the most appropriate choice under GARCH (stocks) volatility choice under GARCH (stocks) volatility approach.approach.

AdvantagesAdvantages Lower capital Lower capital coveragecoverage No strict No strict dependence on dependence on current allocationscurrent allocations

DisadvantagesDisadvantages Weak precision Weak precision resultsresults For large portfolios For large portfolios many estimations are many estimations are neededneeded

Group 4 – GARCH for the stocks Group 4 – GARCH for the stocks


StdStd

Hist Hist VaRVaR

VaR VaR NIGNIG

VaR VaR GHDGHD

VaR VaR CFCF

VaR VaR EVTEVT

ES ES

EVTEVT


1.32%1.32% 0.94%0.94% 0.94%0.94% 0.19%0.19% 1.03%1.03% 0.38%0.38%


-- -- 5.15%5.15% 5.07%5.07% 11.83%11.83% -- 8.76%8.76%

VaR GHD seems to be the most appropriate VaR GHD seems to be the most appropriate choice under GARCH DCC volatility approach.choice under GARCH DCC volatility approach.

AdvantagesAdvantages Lower capital Lower capital coveragecoverage Good precision Good precision resultsresults No strict No strict dependence on dependence on current allocationscurrent allocations

DisadvantagesDisadvantages For large portfolios For large portfolios the estimation of the the estimation of the variance and variance and correlation models is correlation models is very difficultvery difficult

Group 5 – GARCH DCCGroup 5 – GARCH DCC

Volatility Volatility approachapproach

HistoricHistoricalal

EWMEWMAA

GARCH GARCH portfoliportfoli

oo

GARCH GARCH stocksstocks

GARCH GARCH DCCDCC

Aggregate Aggregate precision precision resultsresults

44 22 55 33 44

Complex approaches to VaR are more Complex approaches to VaR are more available to the practitioners. However, are available to the practitioners. However, are the results worth the effort?the results worth the effort?

GARCH portfolio has the advantage of GARCH portfolio has the advantage of precision and smaller calculation effort, but is precision and smaller calculation effort, but is strictly dependent on current portfolio strictly dependent on current portfolio allocations.allocations. GARCH DCC has the advantage of lower GARCH DCC has the advantage of lower capital coverage but requires many capital coverage but requires many estimationsestimations

Conclusions – The SemifinalsConclusions – The Semifinals

Taking into account the non-normality of the Taking into account the non-normality of the data proved usefuldata proved useful VaR CF and ES EVT: highest precision results VaR CF and ES EVT: highest precision results – highest levels of capital coverage– highest levels of capital coverage VaR NIG and VaR GHD may prove to be VaR NIG and VaR GHD may prove to be more appropriate risk measuresmore appropriate risk measures


StdStd

Hist Hist VaRVaR

VaR VaR NIGNIG

VaR VaR GHDGHD

VaR VaR CFCF

VaR VaR EVTEVT

ES ES

EVTEVT

Agg Agg precision precision resultsresults

00 00 33 33 55 22 55

0.5%0.5% 0.1%0.1% 0.05%0.05% 0.01%0.01%

VaRVaR ESES VaRVaR ESES VaRVaR ESES VaRVaR ESES

11 55 11 55 22 55 55 55

VaR GHD under GARCH DCC volatility VaR GHD under GARCH DCC volatility approach may be considered the “best” risk approach may be considered the “best” risk measure for the analyzed portfolio and the measure for the analyzed portfolio and the chosen period. Yet, is the effort of estimating a chosen period. Yet, is the effort of estimating a DCC model worth the decrease from 5.21% to DCC model worth the decrease from 5.21% to 5.07%?5.07%?

Conclusions – The FinalConclusions – The Final

VaR TypeVaR Type

VaR VaR GHD GHD

HistoricHistorical al

volatilitvolatilityy

ES ES EVT EVT

EWMAEWMA

VaR VaR GHD GHD

GARCH GARCH portport

VaR NIG VaR NIG GARCH GARCH stocksstocks

VaR GHD VaR GHD GARCH GARCH

DCCDCC

Errors %Errors % 0.94%0.94% 0.19%0.19% 0.94%0.94% 0.94%0.94% 0.94%0.94%

Avg Avg capital capital

coveragecoverage5.91%5.91% 9.29%9.29% 5.29%5.29% 5.21%5.21% 5.07%5.07%

Using more improved volatility models tends Using more improved volatility models tends to reduce the level of capital coverageto reduce the level of capital coverage

It is recommended to use other distributional It is recommended to use other distributional approaches to deal with the non-normality of approaches to deal with the non-normality of the returnsthe returns CF approximation tends to overestimate risk CF approximation tends to overestimate risk leading to very high levels of capital coverageleading to very high levels of capital coverage NIG and GHD seem to be reasonable NIG and GHD seem to be reasonable distributional approachesdistributional approaches VaR under the EVT approach does not VaR under the EVT approach does not perform very well, while ES leads to nice perform very well, while ES leads to nice precision results but at the cost of high levels precision results but at the cost of high levels of capital coverage. Other threshold values of capital coverage. Other threshold values should be also considered.should be also considered.Thank you for attending this Thank you for attending this

presentation!presentation!

Value at Risk: A Comparative Analysis

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