Alternative Risk Measures for Alternative Investments - Jean-Paul

A way to understand optimal portfolios structure. Decomposition of risk measures ... Approximating algorithm Larsen, Mausser & Uryasev. ○ Based on Expected ...
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Alternative Risk Measures for Alternative Investments

A. Chabaane

Y. Malevergne

BNP Paribas ACA Consulting

ISFA Actuarial School Lyon

JP. Laurent ISFA Actuarial School Lyon BNP Paribas

F. Turpin BNP Paribas email : [email protected]

http://laurent.jeanpaul.free.fr/

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Outline ! Optimizing under VaR constraints "

Estimation techniques

"

VaR analytics and efficient portfolios comparison

! Optimizing under alternative risk constraints "

Expected Shortfall, Downside Risk measure,…

"

Risk measures analytics and efficient portfolios comparison

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Main result

Decomposition of risk measures " " "

VaR Expected Shortfall Downside Risk

A way to understand optimal portfolios structure

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Data set ! 16 hedge funds

! Data structure " "

monthly data 139 observations

! High skewness and kurtosis ! Low (or negative) correlations "

diversification potentiality

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Value at Risk estimation techniques

! Empirical quantile ! Weighted average of quantiles: “L-estimator” (Granger & Silvapulle (2001)) ! Kernel smoothing: (Gourieroux, Laurent & Scaillet (2000) ) ! Gaussian VaR : computed under Gaussian assumption

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Contribution of rank statistics (1)

! We denote by (a’r)1:n ≤…≤ (a’r)n:n the rank statistics of the portfolio allocation a ! VaR estimators depend only on the rank statistics ! VaR estimators are differentiable and positively homogeneous of degree one (with respect to the rank statistics) Thus, we can decompose VaR using Euler ’s equality :

∂VaR(a' R) VaR(a' R) = ∑ (a' r ) i:n i =1 ∂( a ' r ) i:n n

see J-P. Laurent [2003]

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Contribution of rank statistics (2) ! Weights associated with the rank statistics for the different VaR estimators Partial Partial derivatives derivatives zoom zoom on on the the left left skew skew 0,2 0,2 00 00

22

44

66

88

10 10

12 12

14 14

16 16

18 18

20 20

-0,2 -0,2 -0,4 -0,4 -0,6 -0,6 -0,8 -0,8 -1 -1 -1,2 -1,2 Granger Granger VaR VaR

! !

Gaussian Gaussian VaR VaR

Empirical Empirical VaR VaR

GLS GLS VaR VaR

Empirical VaR is concentrated on a single point Granger VaR is distributed around empirical VaR

! GLS VaR : smoother weighting scheme ! Gaussian VaR involves an even smoother pattern AFFI December 2003

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Mean VaR optimization ! A non-standard optimization program "

VaR is not a convex function

"

VaR is not differentiable in the general case (with respect to allocation)

"

Local minima are often encountered

! Genetic algorithms (see P.-A. Barès, R.Gibson and S. Gyger [2002]) "

Derived from the natural selection process

"

Time consuming: slow convergence

! Approximating algorithm Larsen, Mausser & Uryasev "

Based on Expected Shortfall (see below) optimization program

"

We get a sub-optimal solution

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Mean VaR efficient frontier 1,6%

1,5%

Expected return / Empirical VaR 1,4%

1,3%

1,2%

1,1%

1,0%

0,9% -0,2%

0,0%

0,2%

0,4%

0,6%

0,8%

1,0%

1,2%

1,4%

1,6%

1,8%

Emp i ri c a l V a R M ean / S&M VaR (GA)

M ean / Emp irical VaR (GA)

M ean / emp irical VaR (Lars en)

M ean / Variance

M ean / Kernel VaR (GA)

! VaR efficient frontiers are close ! Far from the mean-Gaussian VaR efficient frontier ! Larsen & al. approximating algorithm performs poorly AFFI December 2003

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Mean VaR efficient portfolios (1) 11

Efficient Efficient portfolios portfolios according according to to empirical empirical VaR VaR (GA) (GA)

11

0.9 0.9

0.9 0.9

0.8 0.8

0.8 0.8

0.7 0.7

0.7 0.7

0.6 0.6

0.6 0.6

0.5 0.5

0.5 0.5

0.4 0.4

0.4 0.4

0.3 0.3

0.3 0.3

0.2 0.2

0.2 0.2

0.1 0.1

0.1 0.1

00 0.86% 0.86% 0.94% 0.94% 1.01% 1.01% 1.09% 1.09% 1.17% 1.17% 1.24% 1.24% 1.32% 1.32% 1.40% 1.40% 1.47% 1.47% 1.55% 1.55% 1.63% 1.63% 1.70% 1.70% 1.78% 1.78% Return Return 11

Efficient Efficient portfolios portfolios according according to to Kernel Kernel VaR VaR (GA) (GA)

00 0.86% 0.86% 0.94% 0.94% 1.01% 1.01% 1.09% 1.09% 1.17% 1.17% 1.24% 1.24% 1.32% 1.32% 1.40% 1.40% 1.47% 1.47% 1.55% 1.55% 1.63% 1.63% 1.70% 1.70% 1.78% 1.78% Return Return 11

0.9 0.9

0,9 0,9

0.8 0.8

0,8 0,8

0.7 0.7

0,7 0,7

0.6 0.6

0,6 0,6

0.5 0.5

Efficient Efficient portfolios portfolios according according to to Gaussian Gaussian VaR VaR

0,5 0,5

0.4 0.4

0,4 0,4

0.3 0.3

0,3 0,3

0.2 0.2

0,2 0,2

0.1 0.1 00 0.86% 0.86% 0.94% 0.94% 1.01% 1.01% 1.09% 1.09% 1.17% 1.17% 1.24% 1.24% 1.32% 1.32% 1.40% 1.40% 1.47% 1.47% 1.55% 1.55% 1.63% 1.63% 1.70% 1.70% 1.78% 1.78%

Return

AFFI December

Efficient Efficient portfolios portfolios according according to to Granger Granger VaR VaR (GA) (GA)

AXA AXA Rosenberg Rosenberg Market Market Neutral Neutral Strategy Strategy LP LP Bennett Bennett Restructuring Restructuring Fund Fund LP LP Genesis Genesis Emerging Emerging Markets Markets Fund Fund Ltd Ltd Blue Rock Rock Capital Capital Fund Fund LP LP 2003 Blue Aquila International Fund Ltd Aquila International Fund Ltd Red Red Oak Oak Commodity Commodity Advisors Advisors Inc Inc

0,1 0,1 00 0,88% 0,88% 0,96% 0,96% 1,03% 1,03% 1,11% 1,11% 1,19% 1,19% 1,26% 1,26% 1,34% 1,34% 1,42% 1,42% 1,49% 1,49% 1,57% 1,57% 1,65% 1,65% 1,72% 1,72% 1,80% 1,80% Re Re turn turn

Discovery Discovery MasterFund MasterFund Ltd Ltd Calamos Calamos Convertible Convertible Hedge Hedge Fund Fund LP LP RXR RXR Secured Secured Participating Participating Note Note Dean Dean Witter Witter Cornerstone Cornerstone Fund Fund IV IV LP LP Bay Bay Capital Capital Management Management

Aetos Aetos Corporation Corporation Sage Sage Capital Capital Limited Limited Partnership Partnership Arrowsmith Arrowsmith Fund Fund Ltd Ltd GAMut GAMut Investments Investments Inc Inc Blenheim Blenheim Investments Investments LP LP (Composite) (Composite)

10

Mean VaR optimal portfolios (2) ! Interpretations of the previous graphs " "

Empirical VaR leads to portfolio allocations that change quickly with the return objectives since it is based on a single rank statistic As expected (according to the decomposition) GLS VaR leads to smooth changes in the efficient allocations 2,5% 2,5%

Efficient Efficient frontiers frontiers in in aa Mean-Em Mean-Empirical pirical VaR VaR diagram diagram

"

"

VaR is not sub-additive but…

Arrowsmit ArrowsmithhFund FundLt Ltdd

2,0% 2,0%

GAMut GAMut Inve Invest stme ment ntssInc Inc

…we find a surprisingly strong diversification effect Taking into account Hedge Funds indexes leads to different result

1,5% 1,5% Re Redd Oa OakkCommodit CommodityyAdvisors AdvisorsInc Inc Be Benne nnetttt Re Rest struc ructturing uringFund FundLP LP

1,0% 1,0%

(see Y. Malevergne and D. Sornette [2002], H. Geman and C. Kharoubi [2003])

Ae Aettos osCorpora Corporattion ion RXR ic ipa tting Not RXRSSeeccure ureddPPaart rt ing Noteert Ca mos Conve Caicla laipa mos Conve rtible ible He Hedge dge Fund Fund SSaage ge Ca Capit pitaallLimit LimiteeddPPaart rtne nership rship Blue Blue Roc Rockk Ca Capit pitaallFund FundLP LP

0,5% 0,5%

-2% -2%

0,0% 0,0% 0% 0%

LP LP

Ba Bayy Ca Capit pitaallMa Mana nage geme ment nt

AXA AXARose Rosenbe nberg rgMa Marke rkett Ne Neut utra rall SSttra ratteegy gyLP LP

2% 2%

Mean Mean -- Granger Granger VaR VaR

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Ble Blenhe nheim im Inve Invest stme ment ntssLP LP (Composit ee)) De IV (Composit Deaann Wit Wittteerr Corne Cornerst rstone one Fund Fund IV LP LP

4% 4%

6% 6%

Mean Mean -- Empirical Empirical VaR VaR

Ge Gene nesis sisEme Emerging rging Ma Marke rkettssFund Fund Lt Ltdd Aquila Aquila Int Inteerna rnattiona ionallFund FundLt Ltdd

Disc Discove overy ry Ma Mast steerFund rFund Lt Ltdd

8% 8%

10% 10%

Mean Mean -- GLS GLS VaR VaR

12% 12%

14% 14%

Mean Mean -- Gaussian Gaussian VaR VaR

11

Comparison of efficient portfolios !

Comparison of efficient portfolios under VaR constraints for a given 1.2% level of expected return

Optimal Optimal portfolios portfolios for for 1% 1% level level of of return return Emprical Emprical Granger VaR GLS GLS VaR VaR Granger VaR VaR VaR constrainted constrainted constrainted constrainted constrainted constrainted 3,9% 2,2% 1,0% AXA 3,9% 2,2% 1,0% AXA Rosenberg Rosenberg Market Market Neutral Neutral Strategy Strategy LP LP 0,3% 2,0% 1,2% Discovery 0,3% 2,0% 1,2% Discovery MasterFund MasterFund Ltd Ltd 4,5% 2,4% 0,6% Aetos 4,5% 2,4% 0,6% Aetos Corporation Corporation 21,0% 26,0% 25,4% Bennett 21,0% 26,0% 25,4% Bennett Restructuring Restructuring Fund Fund LP LP 0,1% 0,0% 0,0% Calamos 0,1% 0,0% 0,0% Calamos Convertible Convertible Hedge Hedge Fund Fund LP LP 28,6% 30,4% 29,8% Sage 28,6% 30,4% 29,8% Sage Capital Capital Limited Limited Partnership Partnership 0,2% 0,0% 0,0% Genesis 0,2% 0,0% 0,0% Genesis Emerging Emerging Markets Markets Fund Fund Ltd Ltd 2,1% 8,8% 6,5% RXR Secured Participating Note 2,1% 8,8% 6,5% RXR Secured Participating Note 2,0% 2,2% 1,2% Arrow 2,0% 2,2% 1,2% Arrow smith smith Fund Fund Ltd Ltd 23,6% 17,6% 25,1% Blue 23,6% 17,6% 25,1% Blue Rock Rock Capital Capital Fund Fund LP LP 0,1% 0,0% 0,0% Dean 0,1% 0,0% 0,0% Dean Witter Witter Cornerstone Cornerstone Fund Fund IV IV LP LP 9,9% 6,9% 9,1% GAMut 9,9% 6,9% 9,1% GAMut Investments Investments Inc Inc 0,0% 0,0% 0,0% Aquila 0,0% 0,0% 0,0% Aquila International International Fund Fund Ltd Ltd 2,6% 1,6% 0,0% Bay 2,6% 1,6% 0,0% Bay Capital Capital Management Management 0,3% 0,0% 0,0% Blenheim 0,3% 0,0% 0,0% Blenheim Investments Investments LP LP (Composite) (Composite) 0,9% 0,0% 0,0% Red 0,9% 0,0% 0,0% Red Oak Oak Commodity Commodity Advisors Advisors Inc Inc

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Aetos Aetos Corporation Corporation RXR RXR Secured Secured Participating Participating Note Note Blue Blue Rock Rock Capital Capital Fund Fund LP LP

-0,35 -0,35

-0,25 -0,25

-0,15 -0,15

-0,05 -0,05

0,05 0,05

0,15 0,15

0,25 0,25

12

Behavior of left tails !

Comparison of efficient portfolios under VaR constraints for a given 1.2% level of expected return Optimal Optimal portfolios portfolios for for 1% 1% level level of of return return

Emprical Emprical Granger VaR GLS GLS VaR VaR Granger VaR VaR VaR constrainted constrainted constrainted constrainted constrainted constrainted 3,9% 2,2% 1,0% AXA 3,9% 2,2% 1,0% AXA Rosenberg Rosenberg Market Market Neutral Neutral Strategy Strategy LP LP 0,3% 2,0% 1,2% Discovery 0,3% 2,0% 1,2% Discovery MasterFund MasterFund Ltd Ltd 4,5% 2,4% 0,6% Aetos 4,5% 2,4% 0,6% Aetos Corporation Corporation 21,0% 26,0% 25,4% Bennett Restructuring Fund LP 21,0% 26,0% 25,4% Bennett Restructuring Fund LP 0,1% 0,0% 0,0% Calamos 0,1% 0,0% 0,0% Calamos Convertible Convertible Hedge Hedge Fund Fund LP LP 28,6% 30,4% 29,8% Sage Capital Limited Partnership 28,6% 30,4% 29,8% Sage Capital Limited Partnership 0,2% 0,0% 0,0% Genesis 0,2% 0,0% 0,0% Genesis Emerging Emerging Markets Markets Fund Fund Ltd Ltd 2,1% 8,8% 6,5% RXR 2,1% 8,8% 6,5% RXR Secured Secured Participating Participating Note Note 2,0% 2,2% 1,2% Arrow 2,0% 2,2% 1,2% Arrow smith smith Fund Fund Ltd Ltd 23,6% 17,6% 25,1% Blue 23,6% 17,6% 25,1% Blue Rock Rock Capital Capital Fund Fund LP LP 0,1% 0,0% 0,0% Dean 0,1% 0,0% 0,0% Dean Witter Witter Cornerstone Cornerstone Fund Fund IV IV LP LP 9,9% 6,9% 9,1% GAMut 9,9% 6,9% 9,1% GAMut Investments Investments Inc Inc 0,0% 0,0% 0,0% Aquila 0,0% 0,0% 0,0% Aquila International International Fund Fund Ltd Ltd 2,6% 1,6% 0,0% Bay 2,6% 1,6% 0,0% Bay Capital Capital Management Management 0,3% 0,0% 0,0% Blenheim 0,3% 0,0% 0,0% Blenheim Investments Investments LP LP (Composite) (Composite) 0,9% 0,0% 0,0% Red 0,9% 0,0% 0,0% Red Oak Oak Commodity Commodity Advisors Advisors Inc Inc

!

Aetos Aetos Corporation Corporation RXR RXR Secured Secured Participating Participating Note Note Blue Blue Rock Rock Capital Capital Fund Fund LP LP 10 10

00 -0,15 -0,15

-0,13 -0,13

-0,11 -0,11

-0,09 -0,09

-0,07 -0,07

-0,05 -0,05

-0,03 -0,03

-0,01 -0,01

Analysis of 3 asset distributions and resulting allocations " " " " " "

Aetos C. has a fat tail : penalized by Granger VaR and even more by GLS VaR RXR S. has a thin tail : favored by Granger VaR and GLS VaR Blue R. has a thin extreme tail which quickly thickens : penalized by Granger VaR

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Alternative Risk Measures

14

Alternative risk measures ! Recent works about risk measures properties (P. Artzner, F.Delbaen, J-M. Eber and D. Heath [1999], D. Tasche [2002], C. Acerbi [2002], H. Föllmer and A. Schied [2002]) "

widens the risk measure choice range

! Some choice criteria "

coherence properties

"

numerical tractability

! Properties of optimal portfolios analysis "

comparison of different optimal portfolios

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Expected shortfall ! Definition: mean of “losses “ beyond the Value at Risk ! Properties "

Coherent measure of risk

"

Spectral representation

#

optimal portfolio may be very sensitive to extreme events if α is very low

! Algorithm "

Linear optimization algorithms (see R.T Rockafellar & S. Uryasev [2000]) #

"

may be based on the simplex optimization program

Quick computation

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Downside risk ! Definitions "

Let x1, x2, …xn be the values of a portfolio (historical or simulated)

"

The downside risk is defined as follows

[

1 n + SV ( X ) = ∑ (x − xi ) n i =1

] −x 2

! Properties "

Coherent measure of risk (see T. Fischer [2001])

"

No spectral representation #

"

fails to be comonotonic additive

Could be a good candidate to take into account the investors positive return preference

! Algorithm "

Derived from the variance optimization #

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the Athayde’s recursive algorithm 17

Contribution of rank statistics ! Decomposition of the risk measures as for the VaR case Partial derivatives zoom on the left skew

0,05 0 0

5

10

15

20

-0,05 -0,1 -0,15 -0,2 -0,25 -0,3 Granger VaR

DSR

ES

STDV

! VaR and ES weights are concentrated on extreme rank statistics ! Variance and Downside risk weights exhibit a smoother weighting scheme AFFI December 2003

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Optimal portfolios 11

Efficient Efficient portfolios portfolios according according to to standard standard deviation deviation 11

0.9 0.9

0.9 0.9

0.8 0.8

0.8 0.8

0.7 0.7

0.7 0.7

0.6 0.6

0.6 0.6

0.5 0.5

0.5 0.5

0.4 0.4

0.4 0.4

0.3 0.3

0.3 0.3

0.2 0.2

0.2 0.2

0.1 0.1

0.1 0.1

00 0.88% 0.88% 0.96% 0.96% 1.03% 1.03% 1.11% 1.11% 1.19% 1.19% 1.26% 1.26% 1.34% 1.34% 1.42% 1.42% 1.49% 1.49% 1.57% 1.57% 1.65% 1.65% 1.72% 1.72% 1.80% 1.80% Return Return

11

Efficient Efficient portfolios portfolios according according to to sem semi-variance i-variance

00 0.86% 0.86% 0.94% 0.94% 1.01% 1.01% 1.09% 1.09% 1.17% 1.17% 1.24% 1.24% 1.32% 1.32% 1.40% 1.40% 1.47% 1.47% 1.55% 1.55% 1.63% 1.63% 1.70% 1.70% 1.78% 1.78% Return Return

11

0.9 0.9

0.9 0.9

0.8 0.8

0.8 0.8

0.7 0.7

0.7 0.7

0.6 0.6

0.6 0.6

0.5 0.5

0.5 0.5

0.4 0.4

0.4 0.4

0.3 0.3

0.3 0.3

0.2 0.2

0.2 0.2

Efficient Efficient portfolio portfolio according according to to ES ES (Uryasev) (Uryasev)

0.1 0.1

0.1 0.1

00

00 0.86% 1.32% 0.86% 0.94% 0.94% 1.01% 1.01% 1.09% 1.09% 1.17% 1.17% 1.24% 1.24% 1.32% 1.40% 1.40% 1.47% 1.47% 1.55% 1.55% 1.63% 1.63% 1.70% 1.70% 1.78% 1.78% Return Return

AFFI December 2003

Efficient Efficient portfolios portfolios according according to to Granger Granger VaR VaR (GA) (GA)

AXA AXA Rosenberg Rosenberg Market Market Neutral Neutral Strategy Strategy LP LP Bennett Bennett Restructuring Restructuring Fund Fund LP LP Genesis Genesis Emerging Emerging Markets Markets Fund Fund Ltd Ltd Blue Blue Rock Rock Capital Capital Fund Fund LP LP Aquila Aquila International International Fund Fund Ltd Ltd Red Red Oak Oak Commodity Commodity Advisors Advisors Inc Inc

0.86% 0.86% 0.94% 0.94% 1.01% 1.01% 1.09% 1.09% 1.17% 1.17% 1.24% 1.24% 1.32% 1.32% 1.40% 1.40% 1.47% 1.47% 1.55% 1.55% 1.63% 1.63% 1.70% 1.70% 1.78% 1.78% Return Return

Discovery Discovery MasterFund MasterFund Ltd Ltd Calamos Calamos Convertible Convertible Hedge Hedge Fund Fund LP LP RXR RXR Secured Secured Participating Participating Note Note Dean Dean Witter Witter Cornerstone Cornerstone Fund Fund IV IV LP LP Bay Bay Capital Capital Management Management

Aetos Aetos Corporation Corporation Sage Sage Capital Capital Limited Limited Partnership Partnership Arrowsmith Arrowsmith Fund Fund Ltd Ltd GAMut GAMut Investments Investments Inc Inc Blenheim Blenheim Investments Investments LP LP (Composite) (Composite)

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Comparison of efficient portfolios !

Comparison of efficient portfolios under risk measure constraints for a given 1% level of expected return

Optimal portfolios for 1% level of return ES VaR DSR constrainted constrainted constrainted 12,9% 2,2% 1,5% AXA Rosenberg Market Neutral Strategy LP 0,4% 2,0% 0,8% Discovery MasterFund Ltd 0,0% 2,4% 2,3% Aetos Corporation 13,1% 26,0% 18,1% Bennett Restructuring Fund LP 0,0% 0,0% 0,0% Calamos Convertible Hedge Fund LP 20,7% 30,4% 30,0% Sage Capital Limited Partnership 0,0% 0,0% 0,0% Genesis Emerging Markets Fund Ltd 11,5% 8,8% 9,5% RXR Secured Participating Note 2,9% 2,2% 1,8% Arrow smith Fund Ltd 22,2% 17,6% 24,6% Blue Rock Capital Fund LP 0,0% 0,0% 0,1% Dean Witter Cornerstone Fund IV LP 14,8% 6,9% 10,9% GAMut Investments Inc 0,0% 0,0% 0,0% Aquila International Fund Ltd 1,6% 1,6% 0,4% Bay Capital Management 0,0% 0,0% 0,0% Blenheim Investments LP (Composite) 0,0% 0,0% 0,0% Red Oak Commodity Advisors Inc

AFFI December 2003

Arrowsmith Arrowsmith Fund Fund AXA AXA Rosenberg Rosenberg Market Market Neutral Neutral Strategy Strategy LP LP Blue Blue Rock Rock Capital Capital Fund Fund LP LP

-0,35 -0,35

-0,25 -0,25

-0,15 -0,15

-0,05 -0,05

0,05 0,05

0,15 0,15

0,25 0,25

20

Behavior of left tails !

Comparison of efficient portfolios under risk measure constraints for a given 1% level of expected return

Optimal portfolios for 1% level of return ES VaR DSR constrainted constrainted constrainted 12,9% 2,2% 1,5% AXA Rosenberg Market Neutral Strategy LP 0,4% 2,0% 0,8% Discovery MasterFund Ltd 0,0% 2,4% 2,3% Aetos Corporation 13,1% 26,0% 18,1% Bennett Restructuring Fund LP 0,0% 0,0% 0,0% Calamos Convertible Hedge Fund LP 20,7% 30,4% 30,0% Sage Capital Limited Partnership 0,0% 0,0% 0,0% Genesis Emerging Markets Fund Ltd 11,5% 8,8% 9,5% RXR Secured Participating Note 2,9% 2,2% 1,8% Arrow smith Fund Ltd 22,2% 17,6% 24,6% Blue Rock Capital Fund LP 0,0% 0,0% 0,1% Dean Witter Cornerstone Fund IV LP 14,8% 6,9% 10,9% GAMut Investments Inc 0,0% 0,0% 0,0% Aquila International Fund Ltd 1,6% 1,6% 0,4% Bay Capital Management 0,0% 0,0% 0,0% Blenheim Investments LP (Composite) 0,0% 0,0% 0,0% Red Oak Commodity Advisors Inc

!

Arrowsmith Arrowsmith Fund Fund Ltd Ltd Blue Blue Rock Rock Capital Capital Fund Fund LP LP AXA AXA Rosenberg Rosenberg Market Market Neutral Neutral Strategy LP Strategy LP

-0,35 -0,35

-0,3 -0,3

-0,25 -0,25

-0,2 -0,2

-0,15 -0,15

-0,1 -0,1

-0,05 -0,05

00

Analysis of 3 asset distributions and resulting allocations " " " " " "

Axa R. has very few points in the extreme tail : favored by Expected Shortfall Arrowsmith. has an extreme low return : penalized by Downside Risk Blue R. has a thin extreme tail which quickly thickens : penalized by VaR

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