A description of the new "post-modern" risk management paradigm

1. Post-Modern
Portfolio Theory
August 2009
Marc Gross
Managing Director, FinAnalytica
Confidential and Proprietary, for
use only by express permission

2. Nassim Taleb quote…
“MPT produces measures such as “sigmas”,
“betas”, “Sharpe ratios”, “correlation”, “value
at risk”, “optimal portfolios” and “capital asset
pricing model” that are incompatible with the
possibility of those consequential rare events
I call “black swans” (owing to their rarity, as
most swans are white). ”
Confidential and Proprietary, for
use only by express permission

3. Where Do We Go Next ?…
• Ignore the quantitative metrics
OR
• Adapt them to reflect the market realities
AND
• Connect them with the necessary education,
understanding and processes to use them
correctly
Confidential and Proprietary, for
use only by express permission

4. Post Modern Portfolio Theory
Quick-Fix Reaction or
Enduring Paradigm Shift?
Confidential and Proprietary, for
use only by express permission

5. A warning from the past…
• “Risk models have to be based on market realities, since the converse is
unlikely to happen. This will enable financial institutions to come up with
both better risk mitigation strategies and internal incentive structures for
more decentralized risk management processes.
• “Regulators and policy makers should become more sensitive to the
inadequacy of current risk modeling approaches. Their misleading risk
assessment may not only jeopardize individual financial institutions but,
due to the institutions’ synchronization of misjudgment, will also be a
destabilizing factor in national and international financial systems.”
Dr. Svetlozar Rachev & Dr. Stefan Mittnik
University of Karlsrhue, January 11, 2006
Published interview www.risiko-manager.com
New Approaches for Portfolio Optimization: Parting with the Bell Curve
Confidential and Proprietary, for
use only by express permission

6. Shifting our Risk Paradigm
• MPT Assumptions Are So Deeply Ingrained in
Our Market Thinking That We Are Shocked
When Market Behaviours Contradict Them
• Can We Really Be Having ANOTHER Ten Sigma
Event?
• How Many Can I Reasonably Expect To See In
My Lifetime?
Confidential and Proprietary, for
use only by express permission

7. MPT “Translation” For The Real World
Old World Real World
Normal (Gaussian) Fat-tailed Distributions
Distributions
– Correlation Tail & Asymmetric Dependence
– Sigmas Expected Tail Loss
– Sharpe Ratios STARR Performance
– BS Option pricing Tempered-Stable Option Pricing
– Markowitz Optimal Fat-tail ETL Optimal Portfolios
Portfolios
Confidential and Proprietary, for
use only by express permission

8. Phenomena of Primary Market Drivers
• Univariate level
– Fat-tails
– Asymmetry
– Time-varying volatility
– Complex Dependence (Asymmetric Tail)
DJ Daily returns
Confidential and Proprietary, for
use only by express permission

9. Stable Family
Rich history in probability theory
Kolmogorov and Levy (1930-1950), Feller (1960’s)
Long known to be useful model for heavy-tailed returns
Mandelbrot (1963) and Fama (1965)
Positive skewed densities Symmetric densities
(  1.5) (   0)
Confidential and Proprietary, for
use only by express permission

10. “On the days when no new
information is available,
trading is slow and the price
process evolves slowly. On
days when new information
violates old expectations,
trading is brisk, and the price
process evolves much faster”.
Clark (1973)
Subordinator (g(W)) < 1

Confidential and Proprietary, for
Emp.
use only by express permission Fat-tailed
Fat-tailed

11. Fat-Tails Leave Open the Possibility
of Extreme Events
Subordinator > 1
Confidential and Proprietary, for
use only by express permission

12. 17,000 Market Factor Backtest
Factors Tested Number Percentage
Equities 8346 48.5%
CDS Spreads 7803 45.3%
Interest Rates 528 3.1%
Implied Volatilities 518 3.0%
Currencies 12 0.1%
Total 17207 100.00%
Confidential and Proprietary, for
use only by express permission

13. 17,000 Market Factor Backtest
88% Require Fat-tailed Models 93% Require Fat-tailed Models
May 2007 Dec 2008
6% 3% 7%0%
14%
4%
90%
76%
Normal Vol Clust Enhanced Normal Normal Vol Clust Enhanced Normal
Stable Vol Clust Enhanced Stable
Stable Vol Clust Enhanced Stable
85%, 95%, 97.5%, and 99% VaR tested
Confidential and Proprietary, for
use only by express permission

14. Tail Parameter is Different
Across Assets & Time
• Important to:
– Distinguish tail risk contributors and diversifiers
– Changes in the market extreme risk
S&P 500 alpha
after removing GARCH
2
1.95
1.9
1.85
1.8
1.75
1.7
1.65
1.6
1.55
1.5
15/06/2000 15/06/2001 15/06/2002 15/06/2003 15/06/2004 15/06/2005 15/06/2006 15/06/2007 15/06/2008
Confidential and Proprietary, for
use only by express permission

15. Tail Fatness Parameter is a Leading Indicator
of Market Stress
• Helps to Forecast Market Regime Switch and Shift Portfolio
Toward Less Risky Assets
– Like a Foreshock in Earthquake Prediction
S&P 500 alpha
after removing GARCH
2
1.95
1.9
1.85
1.8
1.75
1.7
1.65
1.6
1.55
1.5
15/06/2000 15/06/2001 15/06/2002 15/06/2003 15/06/2004 15/06/2005 15/06/2006 15/06/2007 15/06/2008
Confidential and Proprietary, for
use only by express permission

16. MSCI Germany EUR
DE DAX
MSCI Hong Kong HKD
US RUSSELL 2000
FR CAC 40
MSCI India INR
MSCI Russia USD
MSCI China CNY
IN BSE SENSEX 30
JP NIKKEI 225
US S&P 500
MSCI United Kingdom GBP
US DOW JONES INDUS. AVG Tail parameter Alpha for
UK FT SE 100
MSCI France EUR 41 indices after
S&P GSCI Energy Index
MSCI WRLD/Energy USD
removing GARCH effect
US NASDAQ COMPOSITE
RU RTS INDEX
/May 15th 2009/
HK HANG SENG
MSCI Japan JPY
MSCI Germany EUR
DE DAX
MSCI Hong Kong HKD
US RUSSELL 2000
FR CAC 40
MSCI India INR
MSCI Russia USD
MSCI China CNY
There is NO
IN BSE SENSEX 30
JP NIKKEI 225
universal tail index!
US S&P 500
MSCI United Kingdom GBP
US DOW JONES INDUS. AVG
UK FT SE 100
MSCI France EUR
S&P GSCI Energy Index
MSCI WRLD/Energy USD
US NASDAQ COMPOSITE
RU RTS INDEX
HK HANG SENG
MSCI Japan JPY
Confidential and Proprietary, for1.75
1.6 1.65 1.7 1.8 1.85 1.9 1.95 2
use only by express permission

17. Why Do Fat-Tails Matter?
Confidential and Proprietary, for
use only by express permission

18. Daily Return: S&P 500 Index
Confidential and Proprietary, for
use only by express permission

19. Crash Probability : Black Monday
On October 19 (Monday), 1987 the S&P 500 index dropped by 23%. Fitting
the models to a data series of 2490 daily observations ending with October
16 (Friday), 1987 yields the following results:
Confidential and Proprietary, for
use only by express permission

20. Crash Probability: U.S. Financial Crisis
On the September 29 (Monday), 2008 the S&P 500 index dropped by 9%. Fitting
the models to a data series of 2505 daily observations ending with the September
26 (Friday), 2008 yields the following results:
Once Per Twenty Trillion Years Vs.
Once Per Year and a Half
Confidential and Proprietary, for
use only by express permission

21. Modified VaR & Cornish-Fisher Expansion
Advantages
• Relies on the Tailor expansion of the PDF
• Represents the derivatives as a function of higher moments
• Can accommodate for skewness and kurtosis to some extent
• Easy to compute
Pitfalls
• Local approximation starting from the Normal distribution
• Very High Estimation Error at Low Data Frequencies
• Becomes more inaccurate going further in the tail
• Multivariate expansion needs estimates of all third and fourth
co-moments, which are very unstable
Confidential and Proprietary, for
use only by express permission

22. Cornish-Fisher Expansion
MSCI Emerging Markets
Montly Data
Daily Data
Modified VaR
Understates Risk by 50%+
At 99% Confidence
Confidential and Proprietary, for
use only by express permission

23. MPT “Translation” For The Real World
Old World Real World
Normal (Gaussian) Fat-tailed Distributions
Distributions
– Correlation Tail & Asymmetric Dependence
– Sigmas Expected Tail Loss
– Sharpe Ratios STARR Performance
– BS Option pricing Tempered-Stable Option Pricing
– Markowitz Optimal Fat-tail ETL Optimal Portfolios
Portfolios
Confidential and Proprietary, for
use only by express permission

24. Classical Correlation
• Assumes Linear Dependence
• Assumes Symmetrical Dependence (Same in
Up or Down Markets)
• Assumes Dependence Structure Remains
Static in a Market Crisis
• Wrongly Presumes Diversification Effects Will
Help Us When We Need Them Most
Confidential and Proprietary, for
use only by express permission

25. Copula Models
• Copulas – functions describing dependence structure
• Gaussian Copulas
– Assumes Tail Events Are Independent
• Skewed Student’s t Copula:
– Dynamic Dependence Changes Between Normal and
Extreme Market Conditions
– Dependence is Often Highly Asymmetric
Confidential and Proprietary, for
use only by express permission

26. Post-Modern Methods
Modeling of Extreme Dependency in market crashes is critical
for making the correct investment decisions
Weaker Upside Dependence
Much Stronger Downside Dependence
Confidential and Proprietary, for
use only by express permission

27. Copula Model Features
• Produces tail dependent scenarios
• Capable of handling skewness in the
dependence structure
• Can be applied in high dimensional cases – up
to 20,000 risk variables
• Adaptive across different frequencies and
market conditions
• Computationally efficient scenario generation
and parameter estimation
Confidential and Proprietary, for
use only by express permission

28. Credit Crunch in Aug 2007
Confidential and Proprietary, for
use only by express permission

29. The meltdown in Oct 2008
Confidential and Proprietary, for
use only by express permission

30. MPT “Translation” For The Real World
Old World Real World
Normal (Gaussian) Fat-tailed Distributions
Distributions
– Correlation Tail & Asymmetric Dependence
– Sigmas Expected Tail Loss
– Sharpe Ratios STARR Performance
– BS Option pricing Tempered-Stable Option Pricing
– Markowitz Optimal Fat-tail ETL Optimal Portfolios
Portfolios
Confidential and Proprietary, for
use only by express permission

31. Sigma Vs. Downside Risk Measures
(VAR & ETL)
• Sigma Assumes a Normal Distribution
• Sigma Assumes Symmetry of Risk
• Sigma Penalises Extreme Positive Returns
• Downside Risk Measures Are Better Aligned
With Investor Preferences
• ETL is a More Informative Downside Risk
Measure (Based on Expected Shortfall)
Confidential and Proprietary, for
use only by express permission

32. VaR vs ETL: Better Information
• VaR does not provide any information about the expected
losses beyond the “normal market conditions”:
• Two funds: equal upside but clearly different downside!
• However: VaR (Fund_X) = 1.46 & VaR (Fund_Y) = 1.46
Confidential and Proprietary, for
use only by express permission

33. ETL vs. VaR: Example
ETL vs VaR - 10 Lowest Returns
0
10 9 8 7 6 5 4 3 2 1
-1
VaR (X) = VaR (Y) -2
Returns
VaR (X) = VaR (Y) -3
ETL (X) << ETL (Y)
-4
ETL (X) << ETL (Y)
-5
-6
Fund_X Fund_Y
Return Rank Fund_X Fund_Y
92 -0.85 -0.85
P(r  qr ( ))   93
94
-0.88
-1.14
-0.88
-1.14
95 -1.26 -1.26
VaRr (1   )   qr ( ) 96
97
-1.46
-1.63
-1.46
-3.26
98 -1.64 -3.28
ETL(1   )  E(r | r  VaR (1   )) 99
100
-1.96 -3.92
-4.16
-2.08
101 -2.4 -4.8
Confidential and Proprietary, for
ETL -1.942 -3.884
use only by express permission

34. Why not normal ETL?
1% STABLE ETL vs. NORMAL VAR AND ETL: $1M OVERNIGHT
30
25
STABLE DENSITY
NORMAL DENSITY
20
15
Normal VaR = $47K
Normal ETL = $51K
10
Stable ETL = $147K
5
0
-0.2 -0.1 0.0 0.1 0.2
OXM DAILY RETURNS
Confidential and Proprietary, for
use only by express permission

35. Post-Modern
Risk Adjusted Performance Measures
  rf Symmetric Risk Penalty Based on
SHARPE 
 Normal Distribution Assumptions
ETL   E (r | r  VaR ) ETR1  E (r | r  q1 )
  rf
STARR  Fat-Tailed Downside Risk Penalty
ETL
ETR Asymmetric, Fat-Tailed Downside
R  Ratio  Risk Penalty and Upside Reward
ETL
Confidential and Proprietary, for
use only by express permission

36. Advanced Asset Selection
Leveraging Risk Asymmetry
• Traditional ranking methods are based standard
deviation (volatility) and Sharpe ratio
– Penalize upside potential
• Advanced methods based on accurate skewed
fat-tail models
– Better rankings
– Better targeting of due diligence resources
Confidential and Proprietary, for
use only by express permission

37. Manager Ranking – ETL vs St.Dev.
• If returns are not symmetrically distributed, ETL and σ give
different rankings:
σ
Order by ETL
Confidential and Proprietary, for
use only by express permission

38. Asset Ranking – ETL vs St.Dev.
•St.Dev. not distinguish between upside and downside:
F_31 F_2
Confidential and Proprietary, for
use only by express permission

39. Asset Ranking – STARR vs. Sharpe
• Rankings by STARR and Sharpe are also different:
Ranking by STARR
Ranking by Sharpe
• STARR is a downside risk-adjusted return measure.
• Sharpe Ratio penalizes upside potential.
Confidential and Proprietary, for
use only by express permission

40. Asset Ranking – Rachev Ratio
• The Rachev Ratio compares upside potential to downside risk:
F_19
Confidential and Proprietary, for
use only by express permission

41. Putting it All Together
• Tail-Risk Budgeting
Confidential and Proprietary, for
use only by express permission

42. Portfolio Risk Budgeting
• Marginal Contribution to Risk
Standard Approach: St Dev
(Ωw )i cov(ri , rP )
MCTRi  
P P
 P w Ωw
 wi  MCTRi  w  w     P
i P
ETL
The expression for marginal contribution to ETL is
ETL
MCETL i   E  ri | rp  VaR rp 
wi
and the resulting risk decomposition:
 w  MCETL   w E  r | r
i
i i
i
i i p  VaR rp   ETL rp 
Confidential and Proprietary, for
use only by express permission

43. Tail-Risk Decomposition
Identify Extreme Risk Hotspots
How Much Do You Lose When You Exceed VaR?
See Risk Contribution
From any Factor Node Diversification Opportunities
Using Fat-tailed and Skewed
Risk Measures
Your Own
View of Risk
Point & Click Drilldown Reports
Confidential and Proprietary, for
use only by express permission

44. Implied Return Fundamentals
• Implied returns represent forecasts of the
expected returns under which the current
portfolio has a maximal reward-risk ratio
• How can we improve the STARR ratio?
– Calculate IR of all positions
– If µIR,i > ERi then decrease wi with a small amount
– If µIR,i < ERi then increase wi with a small amount
– The larger the difference (µIR,i -ERi), the stronger
the impact
Confidential and Proprietary, for
use only by express permission

45. Implied Returns Based on Tail Risk
• The same analysis is valid if we use ETL instead of
standard deviation in which case we use the STARR
ratio.
• The input required is generated scenarios for the
positions.
Confidential and Proprietary, for
use only by express permission

46. Tail Risk Budgeting
Gain Allocation Consensus Interactively
Risk Management as a Profit Center
What is the Hurdle
Rate a Manager
Should Deliver to
Tactical Rebalancing
Justify Their
Opportunities.
Contribution to Risk?
Is That Consistent
With Explicit
Investment
Insufficient Return to Committee
Justify Tail Exposure. Expectations?
Confidential and Proprietary, for
use only by express permission

47. Marginal Contribution to Tail-Risk
Vs. Return
Does the Reward
Justify the
Extreme Risk?
Confidential and Proprietary, for
use only by express permission

48. Opt. Portfolio Performance over Time
Confidential and Proprietary, for
use only by express permission

49. Post-modern Risk Analysis
• Higher accuracy using skewed fat-tailed distribution
models and extreme correlations (copula) models
• Reliable identification of factor drivers of portfolio
risk
• Complete tail risk budgeting framework
PAYOFF:
• Better allocation decisions
• More reliable risk management
• Improved communication with investors and
regulators
Confidential and Proprietary, for
use only by express permission

50. Q&A…
Thank you!
Additional Questions?
Marc_gross@yahoo.com
Confidential and Proprietary, for
use only by express permission

51. References
Confidential and Proprietary, for
use only by express permission