Beyond the Bell Curve× Paul D.Kaplan, Ph.D., CFA Quantitative Research Director, Morningstar Europe, Ltd. © 2011 Morningst...
The Black Swan                 ×   An event that is inconsistent                     with past data but that              ...
The Black Turkey                   ×   “An event that is everywhere in                       in the data−it happens all th...
A Flock of Turkeys Nominal price return unless otherwise specified.Asset Class                                         Tim...
The Limitations of Mean-Variance Analysis× Fat tailsin returns not modeled× Covariation of returns assumed linear, cannot ...
Asset Returns Are Not Lognormally Distributed                                                                          18 ...
The Flaw of the Bell Shaped Curve                        Histogram of S&P 500 Monthly Returns – January 1926 to November 2...
The Flaw of the Bell Shaped Curve                        Histogram of S&P 500 Monthly Returns – January 1926 to November 2...
The Flaw of the Bell Shaped Curve                        Histogram of S&P 500 Monthly Returns – January 1926 to November 2...
Cracks in the Bell Curve: Continental Europe6432168                                                                       ...
Covariation of Returns: Linear or Nonlinear?      S&P 500 vs. EAFE, Monthly Total Returns: Jan. 1970 – Sep. 2010Source: Mo...
Alternative Return Distributions
Tame vs. Wild Randomness× Tame Randomness      × Image an auditorium   full of randomly selected        people.      × Wha...
Tame vs. Wild Randomness× Wild Randomness       × Image an auditorium   full of randomly selected         people.       × ...
Scalability              ×   A return distribution is                  scalable if changing the                  investmen...
Comparison of Asset Class Assumptions Models                 Lognormal    Johnson     Log-TLF      BootstrappingParametric...
Johnson Distribution: Continental Europe643216 8 42                                                        Lognormal1     ...
The Log-Stable Distribution                        Histogram of S&P 500 Monthly Returns – January 1926 to November 2008   ...
The Left Tail of the Log-Stable Distrubution                        Histogram of S&P 500 Monthly Returns – January 1926 to...
Log-Stable Distributions: Continental Europe643216 8 4 2 1                                                                ...
Log-TLF Distribution: Continental Europe643216 8 4 2 1                              Log-TLF(alpha=1.5: 97.7%)             ...
Bootstrap Distribution: Continental Europe64321684 21                                                                     ...
Comparing Distributions: Continental Europe6432168421            Log-TLF(alpha=1.5: 97.7%)                                ...
Modelling Covariation: Continental Europe    95% Confidence regions under alternative models                              ...
Measuring Long-Term Reward
Investment Horizon: One Period or Longer?Payout from $1 investment for 3 choices
Meet the Choices            A                                            B                   CSource: William Poundstone, ...
Meet the Choices     A             B   C
Meet the ChoicesKelly Criterion: Rank Alternatives by Geometric Mean        A                               B              C
Why the Kelly Criterion WorksCumulative Probability Distribution after Reinvesting 12 Times
Measuring Risk with VaR & CVaR× Value at Risk (VaR) describes the tail in terms of how much capital  can be lost over a gi...
Value-at-Risk (VaR)VaR identifies the return at a specific point (e.g. 1st or 5th percentile)        Worst 1st Percentile ...
Conditional Value-at-Risk (CVaR)CVaR identifies the probability weighted return of the entire tail                        ...
CVaR vs. VaRNotice that different return distributions can have the same VaRs,but different CVaRs                         ...
Markowitz 2.0
The Spirit of the Markowitz 2.0 Framework× Go beyond   traditional definition of good (expected return) and bad  (variance...
Building A Better OptimizerIssue                  Markowitz 1.0             Markowitz 2.0Return Distributions   Mean-Varia...
Markowitz 1.0 Inputs: Summary Statistics                                                     Correlation                  ...
Scenario Approach to Modeling Return DistributionsScenario #   Economic Conditions           Stock Market   Bond Market   ...
Scenarios Can be Added to Existing Models×   Tower Watson’s Extreme Risk Ranking at 30 June 2011    1. Depression         ...
Markowitz 2.0 Inputs: Scenarios                           4.5                            4                           3.5  ...
A Markowitz 2.0 Efficient Frontier
Read More About These and Other Ideas in December                            “The breadth and depth of the                ...
Asset Allocation:                             L’optimisation de portefeuilles dans Direct                                 ...
Qui utilise Morningstar Direct?                                  46
Dans quel but?                 47
Une nouvelle fonctionnalité intégrée à Morningstar Direct.
Quel est le process ?× Retenir un ensemble de classes d’actifs sur lesquelles nous  construirons une optimisation× Générer...
Paramétrer les classes d’actifs× Sélectionner   sur une base de 55000 indices les proxys dont les  historiques représenter...
Paramétrer les « inputs »Plusieurs lois de distribution de probabilité sont disponibles:×   Log-Normal distributions×   Tr...
La loi normale×   Distribution de probabilité par défaut×   Représentation graphique×   Choix entre différentes méthodolog...
Autres lois de distribution de probabilité× On peut aller plus loin que la loi normale et prendre en compte l’occurrence d...
Paramétrer les données sur lesquelles nous appliquerons la MVO
Calcul des volatilités:
Calcul de la matrice de corrélation:
Calcul des performances attendues:Plusieurs méthodologies sont disponibles:× Historique× CAPM× Building Blocks× Black Litt...
Paramétrage:CAPM :Le taux sans risque(historical risk free rate)Les Benchmarks:Domestic EquityMarket Portfolio
Black Litterman:Entrez vos propres prévisions en les pondérant par un degré de confiance:
Black Litterman (suite):Vos vues pourront être relatives:
Définir des contraintes:
Définir des contraintes (suite):
L’optimisation     Inclure différentes frontières d’efficience afin de comparer plusieurs                         modèles ...
L’optimisationTester son portefeuille:
L’optimisationQuel portefeuille pour la même volatilité ? :
Retrouvez des portefeuilles optimaux par profil d’investissement:
ResamplingMVO pure:
ResamplingMVO + Resampling:
ProjectionsRéaliser des projections de performances prenant en compte:× L’inflation× Les flux monétaires× Le rebalancement
ProjectionsParamétrage:
Projections  50% de probabilité de dépasser 500 K après 11 ans
Les avantages de la nouvelle fonctionnalité de Morningstar Direct : Asset Allocation             × Un accès à une base de ...
Beyond The Bell Curve
Beyond The Bell Curve
Upcoming SlideShare
Loading in …5
×

Beyond The Bell Curve

1,283
-1

Published on

Conférence sur « L’optimisation de l’allocation d’actifs : les nouvelles avancées »:

Markowitz 2.0 & New Distribution Models par Dr. Paul Kaplan

Published in: Business, Economy & Finance
0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total Views
1,283
On Slideshare
0
From Embeds
0
Number of Embeds
0
Actions
Shares
0
Downloads
35
Comments
0
Likes
0
Embeds 0
No embeds

No notes for slide

Beyond The Bell Curve

  1. 1. Beyond the Bell Curve× Paul D.Kaplan, Ph.D., CFA Quantitative Research Director, Morningstar Europe, Ltd. © 2011 Morningstar, Inc. All rights reserved. <#>
  2. 2. The Black Swan × An event that is inconsistent with past data but that happens anyway
  3. 3. The Black Turkey × “An event that is everywhere in in the data−it happens all the time−but to which one is willfully blind.” Source: Laurence B. Siegel, “Black Swan or Black Turkey? The State of Economic Knowledge and the Crash of 2007-2009,” Financial Analysts Journal, July/August 2010.
  4. 4. A Flock of Turkeys Nominal price return unless otherwise specified.Asset Class Time Period Peak to Trough DeclineU.S. stocks (real total return) 1911-1920 51%U.S. stocks (DJIA, daily) 1929-1932 89%Long U.S. Treasury bond (real 1941-1981 67%total return)U.S. stocks 1973-1974 49%U.K. stocks (real total return) 1972-1974 74%Gold 1980-1985 62%Oil 1980-1986 71%Japan stocks 1990-2009 82%U.S. stocks (S&P) 2000-2002 49%U.S. stocks (NASDAQ) 2000-2002 78%U.S. stocks (S&P) 2007-2009 57%Source: Laurence B. Siegel, “Black Swan or Black Turkey? The State of Economic Knowledge and the Crash of 2007-2009,”Financial Analysts Journal, July/August 2010.
  5. 5. The Limitations of Mean-Variance Analysis× Fat tailsin returns not modeled× Covariation of returns assumed linear, cannot handle optionality× Single period investment horizon (arithmetic mean)× Risk measured by volatility× These limitations largely due to the flaw of averages × Standard deviation is an average of squared deviations × Correlation in an average of comovements
  6. 6. Asset Returns Are Not Lognormally Distributed 18 UK (£) 16 14 12Excess Kurtosis UK (€) 10 8 6 4 Europe Ex UK(€) North America ($) Lognormal North America (£) 2 Far East (£) Europe Ex UK (£) Far East (€) Far East ($) North America (€) 0 -1.0 -0.5 0.0 0.5 1.0 1.5 Skewness
  7. 7. The Flaw of the Bell Shaped Curve Histogram of S&P 500 Monthly Returns – January 1926 to November 2008 Lognormal Distribution CurveNumber of Occurrences Returns Source: Paul D. Kaplan, “Déja Vu All Over Again,” in Morningstar Advisor , February/March 2009 Performance data shown represents past performance. Past performance is not indicative and not a guarantee of future results. Indices shown are unmanaged and not available for direct investment. Performance data does not factor in transaction costs or taxes.
  8. 8. The Flaw of the Bell Shaped Curve Histogram of S&P 500 Monthly Returns – January 1926 to November 2008 Lognormal Distribution CurveNumber of Occurrences Returns Source: Paul D. Kaplan, “Déja Vu All Over Again,” in Morningstar Advisor, February/March 2009 Performance data shown represents past performance. Past performance is not indicative and not a guarantee of future results. Indices shown are unmanaged and not available for direct investment. Performance data does not factor in transaction costs or taxes.
  9. 9. The Flaw of the Bell Shaped Curve Histogram of S&P 500 Monthly Returns – January 1926 to November 2008 Lognormal Distribution Curve S&P 500Number of Occurrences Mean less 3σ should occur about Mean less 3σ ≈ -15% once every 1000 observations In this time period, 10 of the 995 observations exceed -15% Returns Source: Paul D. Kaplan, “Déja Vu All Over Again,” in Morningstar Advisor, February/March 2009 Performance data shown represents past performance. Past performance is not indicative and not a guarantee of future results. Indices shown are unmanaged and not available for direct investment. Performance data does not factor in transaction costs or taxes.
  10. 10. Cracks in the Bell Curve: Continental Europe6432168 Lognormal4 2 1 -3 -30% -25% -20% -15% -10% -5% 0% 5% 10% 15% 20% Europe Ex UK(€) Bases on monthly on the MSCI Europe ex UK Gross Return index : January 1972 − August 2011 Source: Morningstar EnCorr, MSCI
  11. 11. Covariation of Returns: Linear or Nonlinear? S&P 500 vs. EAFE, Monthly Total Returns: Jan. 1970 – Sep. 2010Source: Morningstar® EnCorr ® Stocks, Bonds, Bills, and Inflation module, MSCI
  12. 12. Alternative Return Distributions
  13. 13. Tame vs. Wild Randomness× Tame Randomness × Image an auditorium full of randomly selected people. × What do you estimate the average weight to be? × Now image the largest person that you can think of enters. × How much does your estimate change?
  14. 14. Tame vs. Wild Randomness× Wild Randomness × Image an auditorium full of randomly selected people. × What do you estimate the average wealth to be? × Now image the wealthiest person that you can think of enters. × How much does your estimate change?
  15. 15. Scalability × A return distribution is scalable if changing the investment horizon preserves the shape. × Only the parameters need to be rescaled. × Allows the same model to be applied at any horizon
  16. 16. Comparison of Asset Class Assumptions Models Lognormal Johnson Log-TLF BootstrappingParametric Yes Yes Yes NoFlexible shape No Yes No YesScalable Yes No Yes NoRandomness Tame Tame Wild NACovariation Log-linear Gaussian Conditional Non-linear Copula Log-Linear
  17. 17. Johnson Distribution: Continental Europe643216 8 42 Lognormal1 Johnson -3 -30% -25% -20% -15% -10% -5% 0% 5% 10% 15% 20% Europe Ex UK(€) Bases on monthly on the MSCI Europe ex UK Gross Return index : January 1972 − August 2011 Source: Morningstar EnCorr, MSCI
  18. 18. The Log-Stable Distribution Histogram of S&P 500 Monthly Returns – January 1926 to November 2008 Log-stable Distribution CurveNumber of Occurrences Returns Source: Paul D. Kaplan, “Déja Vu All Over Again,” in Morningstar Advisor, February/March 2009 Performance data shown represents past performance. Past performance is not indicative and not a guarantee of future results. Indices shown are unmanaged and not available for direct investment. Performance data does not factor in transaction costs or taxes.
  19. 19. The Left Tail of the Log-Stable Distrubution Histogram of S&P 500 Monthly Returns – January 1926 to November 2008 Log-stable Distribution CurveNumber of Occurrences Returns Source: Paul D. Kaplan, “Déja Vu All Over Again,” in Morningstar Advisor, February/March 2009 Performance data shown represents past performance. Past performance is not indicative and not a guarantee of future results. Indices shown are unmanaged and not available for direct investment. Performance data does not factor in transaction costs or taxes.
  20. 20. Log-Stable Distributions: Continental Europe643216 8 4 2 1 -3 -30% -25% -20% -15% -10% -5% 0% 5% 10% 15% 20% Europe Ex UK(€) Bases on monthly on the MSCI Europe ex UK Gross Return index : January 1972 − August 2011 Source: Morningstar EnCorr, MSCI
  21. 21. Log-TLF Distribution: Continental Europe643216 8 4 2 1 Log-TLF(alpha=1.5: 97.7%) -3 -30% -25% -20% -15% -10% -5% 0% 5% 10% 15% 20% Europe Ex UK(€) Bases on monthly on the MSCI Europe ex UK Gross Return index : January 1972 − August 2011 Source: Morningstar EnCorr, MSCI
  22. 22. Bootstrap Distribution: Continental Europe64321684 21 -3 -30% -25% -20% -15% -10% -5% 0% 5% 10% 15% 20% Europe Ex UK(€) Bases on monthly on the MSCI Europe ex UK Gross Return index : January 1972 − August 2011 Source: Morningstar EnCorr, MSCI
  23. 23. Comparing Distributions: Continental Europe6432168421 Log-TLF(alpha=1.5: 97.7%) Johnson Bootstrap -3 -30% -25% -20% -15% -10% -5% 0% 5% 10% 15% 20% Europe Ex UK(€) Bases on monthly on the MSCI Europe ex UK Gross Return index : January 1972 − August 2011 Source: Morningstar EnCorr, MSCI
  24. 24. Modelling Covariation: Continental Europe 95% Confidence regions under alternative models 30% 20% 10%North America (€) Data Lognormal 0% Johnson -25% -20% -15% -10% -5% 0% 5% 10% 15% 20% 25% Log-Stable -10% -20% -30% Europe Ex UK(€) Bases on monthly on the MSCI Europe UK Gross Return index and MSCI North America Gross Return index converted at spot to EUR: January 1972 − August 2011. Source: Morningstar EnCorr, MSCI
  25. 25. Measuring Long-Term Reward
  26. 26. Investment Horizon: One Period or Longer?Payout from $1 investment for 3 choices
  27. 27. Meet the Choices A B CSource: William Poundstone, Fortune’s Formula, Hill and Wang 2005, p. 198.
  28. 28. Meet the Choices A B C
  29. 29. Meet the ChoicesKelly Criterion: Rank Alternatives by Geometric Mean A B C
  30. 30. Why the Kelly Criterion WorksCumulative Probability Distribution after Reinvesting 12 Times
  31. 31. Measuring Risk with VaR & CVaR× Value at Risk (VaR) describes the tail in terms of how much capital can be lost over a given period of time× A 5% VaR answers a question of the form × Having invested 10,000 euros, there is a 5% chance of losing X euros in T months. What is X?× Conditional Value at Risk (CVaR) is the expected loss of capital should VaR be breached× CVaR>VaR× VaR & CVaR depend on the investment horizon
  32. 32. Value-at-Risk (VaR)VaR identifies the return at a specific point (e.g. 1st or 5th percentile) Worst 1st Percentile Worst 5th Percentile 99% of all returns are better 95% of all returns are better 1% of all returns are worse 5% of all returns are worse
  33. 33. Conditional Value-at-Risk (CVaR)CVaR identifies the probability weighted return of the entire tail Worst 5th Percentile 95% of all returns are better 5% of all returns are worse
  34. 34. CVaR vs. VaRNotice that different return distributions can have the same VaRs,but different CVaRs Worst 5th Percentile 95% of all returns are better 5% of all returns are worse
  35. 35. Markowitz 2.0
  36. 36. The Spirit of the Markowitz 2.0 Framework× Go beyond traditional definition of good (expected return) and bad (variance)× Use any definition of good× Use any definition of bad× Use any distributional assumptions (parametric or non-parametric)
  37. 37. Building A Better OptimizerIssue Markowitz 1.0 Markowitz 2.0Return Distributions Mean-Variance Framework Scenarios+Smoothing (No fat tails) (Fat tails possible)Return Covariation Correlation Matrix Scenarios+Smoothing Linear Nonlinear (e.g. options)Investment Horizon Single Period Can use Multiperiod Kelly Criterion Arithmetic Mean Can use Geometric MeanRisk Measure Standard Deviation Can use Conditional Value at Risk and other risk measures
  38. 38. Markowitz 1.0 Inputs: Summary Statistics Correlation Expected StandardAsset Class Return Deviation 1 2 3 4A 5.00% 10.00% 1.00 0.34 0.32 0.32B 10.00% 20.00% 0.34 1.00 0.82 0.82C 15.00% 30.00% 0.32 0.82 1.00 0.71D 13.00% 30.00% 0.32 0.82 0.71 1.00
  39. 39. Scenario Approach to Modeling Return DistributionsScenario # Economic Conditions Stock Market Bond Market Real Estate 60/30/10 Return Return Return Mix1 Low Inflation, Low Growth 5% 4% 4% 4.6%2 Low Inflation, High Growth 15% 6% 11% 11.9%3 High Inflation, Low Growth -12% -8% -2% -9.8%4 High Inflation, High Growth 6% 0% 3% 3.9%In practice, 1,000 or more scenarios typical so that fat tailsand nonlinear covariations adequately modeled
  40. 40. Scenarios Can be Added to Existing Models× Tower Watson’s Extreme Risk Ranking at 30 June 2011 1. Depression 2. Sovereign default 3. Hyperinflation 4. Banking crisis 5. Currency crisis 6. Climate change 7. Political crisis 8. Insurance crisis 9. Protectionism 10. Euro break-up 11. Resource scarcity 12. Major war 13. End of fiat money 14. Infrastructure failure 15. Killer pandemic Source: Tim Hodgson, “Asset Allocation and Gray Swans,” Professional Investor, Autumn 2011.
  41. 41. Markowitz 2.0 Inputs: Scenarios 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0-60% -40% -20% 0% 20% 40% 60% 80% 100% 2.5 250% 200% 2 150% 1.5 100% 1 50% 0.5 0% -60% -40% -20% 0% 20% 40% 60% 80% 0 -50%-100% -50% 0% 50% 100% 150% 200% 250% -100% 1.4 350% 350% 1.2 300% 300% 250% 250% 1 200% 200% 0.8 150% 150% 0.6 100% 100% 0.4 50% 50% 0.2 0% 0% 0 -60% -40% -20% 0% 20% 40% 60% 80% -100% -50% 0% 50% 100% 150% 200% 250% -50% -50%-200% -100% 0% 100% 200% 300% 400% 500% -100% -100% 1.6 350% 350% 350% 1.4 300% 300% 300% 1.2 250% 250% 250% 1 200% 200% 200% 0.8 150% 150% 150% 0.6 100% 100% 100% 0.4 50% 50% 50% 0.2 0% 0% 0% -60% -40% -20% 0% 20% 40% 60% 80% -100% -50% 0% 50% 100% 150% 200% 250% -100% -50% 0% 50% 100% 150% 200% 250% 300% 350% 0 -50% -50% -50%-200% -100% 0% 100% 200% 300% 400% 500% -100% -100% -100%
  42. 42. A Markowitz 2.0 Efficient Frontier
  43. 43. Read More About These and Other Ideas in December “The breadth and depth of the articles in this book suggest that Paul Kaplan has been thinking about markets for about as long as markets have existed.” From the foreword
  44. 44. Asset Allocation: L’optimisation de portefeuilles dans Direct Johann Cayrouse CSC© 2011 Morningstar, Inc. All rights reserved.
  45. 45. Qui utilise Morningstar Direct? 46
  46. 46. Dans quel but? 47
  47. 47. Une nouvelle fonctionnalité intégrée à Morningstar Direct.
  48. 48. Quel est le process ?× Retenir un ensemble de classes d’actifs sur lesquelles nous construirons une optimisation× Générer une frontière d’efficience et identifier les portefeuilles optimaux× Construire des projections sur les portefeuilles obtenus: niveaux de probabilité de leur comportement
  49. 49. Paramétrer les classes d’actifs× Sélectionner sur une base de 55000 indices les proxys dont les historiques représenteront les classes d’actifs choisies× Utiliser l’un des groupes d’actifs prédéfinis par Morningstar ou paramétrer vos classes d’actifs.
  50. 50. Paramétrer les « inputs »Plusieurs lois de distribution de probabilité sont disponibles:× Log-Normal distributions× Truncated Lévy-Flight distributions× Log-T distributions× Johnson distributions× Bootstrap Historical Data
  51. 51. La loi normale× Distribution de probabilité par défaut× Représentation graphique× Choix entre différentes méthodologies pour estimer les performances attendues : Historique, CAPM, Black Litterman, Building blocks.
  52. 52. Autres lois de distribution de probabilité× On peut aller plus loin que la loi normale et prendre en compte l’occurrence d’événements extrêmes : Fat Tails, Skewness différent de 0 et Kurtosis (Excess Kurtosis) supérieur à 0.
  53. 53. Paramétrer les données sur lesquelles nous appliquerons la MVO
  54. 54. Calcul des volatilités:
  55. 55. Calcul de la matrice de corrélation:
  56. 56. Calcul des performances attendues:Plusieurs méthodologies sont disponibles:× Historique× CAPM× Building Blocks× Black Litterman
  57. 57. Paramétrage:CAPM :Le taux sans risque(historical risk free rate)Les Benchmarks:Domestic EquityMarket Portfolio
  58. 58. Black Litterman:Entrez vos propres prévisions en les pondérant par un degré de confiance:
  59. 59. Black Litterman (suite):Vos vues pourront être relatives:
  60. 60. Définir des contraintes:
  61. 61. Définir des contraintes (suite):
  62. 62. L’optimisation Inclure différentes frontières d’efficience afin de comparer plusieurs modèles de répartition d’actif..
  63. 63. L’optimisationTester son portefeuille:
  64. 64. L’optimisationQuel portefeuille pour la même volatilité ? :
  65. 65. Retrouvez des portefeuilles optimaux par profil d’investissement:
  66. 66. ResamplingMVO pure:
  67. 67. ResamplingMVO + Resampling:
  68. 68. ProjectionsRéaliser des projections de performances prenant en compte:× L’inflation× Les flux monétaires× Le rebalancement
  69. 69. ProjectionsParamétrage:
  70. 70. Projections 50% de probabilité de dépasser 500 K après 11 ans
  71. 71. Les avantages de la nouvelle fonctionnalité de Morningstar Direct : Asset Allocation × Un accès à une base de 55000 indices × Un outil basé sur internet × Un paramétrage souple et hautement personnalisable × Aller au-delà de la Loi Normale: Fat tails × Black Litterman : entrez vos vues× Resampling : envisager plusieurs scenarios sur votre optimisation
  1. A particular slide catching your eye?

    Clipping is a handy way to collect important slides you want to go back to later.

×