Post-Modern
                                Portfolio Theory
                               August 2009
                  ...
Nassim Taleb quote…


         “MPT produces measures such as “sigmas”,
         “betas”, “Sharpe ratios”, “correlation”, ...
Where Do We Go Next ?…


    • Ignore the quantitative metrics
      OR
    • Adapt them to reflect the market realities
 ...
Post Modern Portfolio Theory


                                Quick-Fix Reaction or
                              Endurin...
A warning from the past…


    • “Risk models have to be based on market realities, since the converse is
      unlikely t...
Shifting our Risk Paradigm


    • MPT Assumptions Are So Deeply Ingrained in
      Our Market Thinking That We Are Shocke...
MPT “Translation” For The Real World


         Old World                          Real World
    Normal (Gaussian)       ...
Phenomena of Primary Market Drivers
    • Univariate level
          –   Fat-tails
          –   Asymmetry
          –   T...
Stable Family
     Rich history in probability theory
         Kolmogorov and Levy (1930-1950), Feller (1960’s)

     Long...
“On the days when no new
                                                     information is available,
                  ...
Fat-Tails Leave Open the Possibility
                                 of Extreme Events

                                 ...
17,000 Market Factor Backtest


              Factors Tested         Number       Percentage

    Equities                ...
17,000 Market Factor Backtest

             88% Require Fat-tailed Models            93% Require Fat-tailed Models
       ...
Tail Parameter is Different
                                   Across Assets & Time
   • Important to:
          – Disting...
Tail Fatness Parameter is a Leading Indicator
                              of Market Stress
        • Helps to Forecast M...
MSCI Germany EUR

                  DE DAX

      MSCI Hong Kong HKD

        US RUSSELL 2000

               FR CAC 40

 ...
Why Do Fat-Tails Matter?




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




Confidential and Proprietary, for
use only by express permission
Crash Probability : Black Monday

   On October 19 (Monday), 1987 the S&P 500 index dropped by 23%. Fitting
   the models ...
Crash Probability: U.S. Financial Crisis

   On the September 29 (Monday), 2008 the S&P 500 index dropped by 9%. Fitting
 ...
Modified VaR & Cornish-Fisher Expansion

    Advantages
    •    Relies on the Tailor expansion of the PDF
    •    Repres...
Cornish-Fisher Expansion
                                             MSCI Emerging Markets




                          ...
MPT “Translation” For The Real World


         Old World                          Real World
   Normal (Gaussian)        ...
Classical Correlation


    • Assumes Linear Dependence
    • Assumes Symmetrical Dependence (Same in
      Up or Down Mar...
Copula Models
    • Copulas – functions describing dependence structure

    • Gaussian Copulas
          – Assumes Tail E...
Post-Modern Methods
Modeling of Extreme Dependency in market crashes is critical
for making the correct investment decisio...
Copula Model Features


    • Produces tail dependent scenarios
    • Capable of handling skewness in the
      dependence...
Credit Crunch in Aug 2007




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




Confidential and Proprietary, for
use only by express permission
MPT “Translation” For The Real World


         Old World                          Real World
   Normal (Gaussian)        ...
Sigma Vs. Downside Risk Measures
                                 (VAR & ETL)

    • Sigma Assumes a Normal Distribution
 ...
VaR vs ETL: Better Information

   • VaR does not provide any information about the expected
     losses beyond the “norma...
ETL vs. VaR: Example
                                    ETL vs VaR - 10 Lowest Returns

                                 ...
Why not normal ETL?
              1% STABLE ETL vs. NORMAL VAR AND ETL: $1M OVERNIGHT

             30
             25



...
Post-Modern
                  Risk Adjusted Performance Measures
                           rf       Symmetric Risk Pena...
Advanced Asset Selection
                                Leveraging Risk Asymmetry

    • Traditional ranking methods are ...
Manager Ranking – ETL vs St.Dev.
        • If returns are not symmetrically distributed, ETL and σ give
                  ...
Asset Ranking – ETL vs St.Dev.
   •St.Dev. not distinguish between upside and downside:
      F_31                        ...
Asset Ranking – STARR vs. Sharpe

   • Rankings by STARR and Sharpe are also different:
           Ranking by STARR
      ...
Asset Ranking – Rachev Ratio
   • The Rachev Ratio compares upside potential to downside risk:




 F_19




Confidential ...
Putting it All Together




                           • Tail-Risk Budgeting




Confidential and Proprietary, for
use onl...
Portfolio Risk Budgeting
    • Marginal Contribution to Risk
    Standard Approach: St Dev
                               ...
Tail-Risk Decomposition
                              Identify Extreme Risk Hotspots
                    How Much Do You L...
Implied Return Fundamentals
    • Implied returns represent forecasts of the
      expected returns under which the curren...
Implied Returns Based on Tail Risk

    • The same analysis is valid if we use ETL instead of
      standard deviation in ...
Tail Risk Budgeting
           Gain Allocation Consensus Interactively
                      Risk Management as a Profit C...
Marginal Contribution to Tail-Risk
                             Vs. Return




                                           ...
Opt. Portfolio Performance over Time




Confidential and Proprietary, for
use only by express permission
Post-modern Risk Analysis

    • Higher accuracy using skewed fat-tailed distribution
      models and extreme correlation...
Q&A…



                                    Thank you!

                       Additional Questions?

                   M...
References




Confidential and Proprietary, for
use only by express permission
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Post Modern Portfolio Theory

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Post Modern Portfolio Theory

  1. 1. Post-Modern Portfolio Theory August 2009 Marc Gross Managing Director, FinAnalytica Confidential and Proprietary, for use only by express permission
  2. 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. 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. 4. Post Modern Portfolio Theory Quick-Fix Reaction or Enduring Paradigm Shift? Confidential and Proprietary, for use only by express permission
  5. 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. 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. 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. 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. 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. 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. 11. Fat-Tails Leave Open the Possibility of Extreme Events Subordinator > 1 Confidential and Proprietary, for use only by express permission
  12. 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. 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. 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. 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. 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. 17. Why Do Fat-Tails Matter? Confidential and Proprietary, for use only by express permission
  18. 18. Daily Return: S&P 500 Index Confidential and Proprietary, for use only by express permission
  19. 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. 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. 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. 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. 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. 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. 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. 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. 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. 28. Credit Crunch in Aug 2007 Confidential and Proprietary, for use only by express permission
  29. 29. The meltdown in Oct 2008 Confidential and Proprietary, for use only by express permission
  30. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 41. Putting it All Together • Tail-Risk Budgeting Confidential and Proprietary, for use only by express permission
  42. 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. 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. 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. 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. 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. 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. 48. Opt. Portfolio Performance over Time Confidential and Proprietary, for use only by express permission
  49. 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. 50. Q&A… Thank you! Additional Questions? Marc_gross@yahoo.com Confidential and Proprietary, for use only by express permission
  51. 51. References Confidential and Proprietary, for use only by express permission

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