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Value at Risk: A Comparative Analysis. Filip Iorgulescu. Introduction. Why is VaR a challenging subject for me?. - scientific but also practical - easy to understood, difficult to determine - a benchmark with certain shortcomings. - PowerPoint PPT Presentation
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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
De
nsity
-4 -2 0 2 4
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Histogram of portln_norm
portln_norm
De
nsity
-4 -2 0 2 4
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Histogram of portln_norm
portln_norm
De
nsity
-4 -2 0 2 4
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
2 5 10
5e
-05
2e
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Tail of Underlying Distribution
x [log scale]
1-F
(x)
[log
sca
le]
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
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 % 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%
Avg capital Avg capital coveragecoverage
-- -- -- -- 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
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 % 1.791.79%%
1.22%1.22% 0.94%0.94% 0.94%0.94% 0.19%0.19% 0.94%0.94% 0.28%0.28%
Avg capital Avg capital coveragecoverage
-- -- 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
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 % 1.411.41%%
1.22%1.22% 0.94%0.94% 1.03%1.03% 0.19%0.19% 1.03%1.03% 0.38%0.38%
Avg capital Avg capital coveragecoverage
-- -- 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
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 % 1.411.41%%
1.32%1.32% 0.94%0.94% 0.94%0.94% 0.19%0.19% 1.03%1.03% 0.38%0.38%
Avg capital Avg capital coveragecoverage
-- -- 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
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
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!