Conférence sur « L’optimisation de l’allocation d’actifs : les nouvelles avancées »: Markowitz 2.0 & New Distribution Models par Dr. Paul Kaplan

1. Beyond the Bell Curve
× Paul D.Kaplan, Ph.D., CFA
Quantitative Research Director, Morningstar Europe, Ltd.
© 2011 Morningstar, Inc. All rights reserved.
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2. The Black Swan
× An event that is inconsistent
with past data but that
happens anyway

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. A Flock of Turkeys
Nominal price return unless otherwise specified.
Asset Class Time Period Peak to Trough Decline
U.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. 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. Asset Returns Are Not Lognormally Distributed
18
UK (£)
16
14
12
Excess 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. The Flaw of the Bell Shaped Curve
Histogram of S&P 500 Monthly Returns – January 1926 to November 2008
Lognormal Distribution Curve
Number 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. The Flaw of the Bell Shaped Curve
Histogram of S&P 500 Monthly Returns – January 1926 to November 2008
Lognormal Distribution Curve
Number 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. The Flaw of the Bell Shaped Curve
Histogram of S&P 500 Monthly Returns – January 1926 to November 2008
Lognormal Distribution Curve
S&P 500
Number 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. Cracks in the Bell Curve: Continental Europe
64
32
16
8 Lognormal
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

11. Covariation of Returns: Linear or Nonlinear?
S&P 500 vs. EAFE, Monthly Total Returns: Jan. 1970 – Sep. 2010
Source: Morningstar® EnCorr ® Stocks, Bonds, Bills, and Inflation module, MSCI

12. Alternative Return Distributions

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. 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. 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. Comparison of Asset Class Assumptions Models
Lognormal Johnson Log-TLF Bootstrapping
Parametric Yes Yes Yes No
Flexible shape No Yes No Yes
Scalable Yes No Yes No
Randomness Tame Tame Wild NA
Covariation Log-linear Gaussian Conditional Non-linear
Copula Log-Linear

17. Johnson Distribution: Continental Europe
64
32
16
8
4
2
Lognormal
1
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. The Log-Stable Distribution
Histogram of S&P 500 Monthly Returns – January 1926 to November 2008
Log-stable Distribution Curve
Number 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. The Left Tail of the Log-Stable Distrubution
Histogram of S&P 500 Monthly Returns – January 1926 to November 2008
Log-stable Distribution Curve
Number 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. Log-Stable Distributions: Continental Europe
64
32
16
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. Log-TLF Distribution: Continental Europe
64
32
16
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. Bootstrap Distribution: Continental Europe
64
32
16
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

23. Comparing Distributions: Continental Europe
64
32
16
8
4
2
1
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. 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. Measuring Long-Term Reward

26. Investment Horizon: One Period or Longer?
Payout from $1 investment for 3 choices

27. Meet the Choices
A B C
Source: William Poundstone, Fortune’s Formula, Hill and Wang 2005, p. 198.

28. Meet the Choices
A B C

29. Meet the Choices
Kelly Criterion: Rank Alternatives by Geometric Mean
A B C

30. Why the Kelly Criterion Works
Cumulative Probability Distribution after Reinvesting 12 Times

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. 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. 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. CVaR vs. VaR
Notice 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. Markowitz 2.0

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. Building A Better Optimizer
Issue Markowitz 1.0 Markowitz 2.0
Return 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 Mean
Risk Measure Standard Deviation Can use Conditional Value at Risk and
other risk measures

38. Markowitz 1.0 Inputs: Summary Statistics
Correlation
Expected Standard
Asset Class Return Deviation 1 2 3 4
A 5.00% 10.00% 1.00 0.34 0.32 0.32
B 10.00% 20.00% 0.34 1.00 0.82 0.82
C 15.00% 30.00% 0.32 0.82 1.00 0.71
D 13.00% 30.00% 0.32 0.82 0.71 1.00

39. Scenario Approach to Modeling Return Distributions
Scenario # Economic Conditions Stock Market Bond Market Real Estate 60/30/10
Return Return Return Mix
1 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 tails
and nonlinear covariations adequately modeled

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. 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. A Markowitz 2.0 Efficient Frontier

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

45. Asset Allocation:
L’optimisation de portefeuilles dans Direct
Johann Cayrouse CSC
© 2011 Morningstar, Inc. All rights reserved.

46. Qui utilise Morningstar Direct?
46

47. Dans quel but?
47

48. Une nouvelle fonctionnalité intégrée à Morningstar Direct.

49. 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

50. 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.

51. 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

52. 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.

53. 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.

54. Paramétrer les données sur lesquelles nous appliquerons la MVO

55. Calcul des volatilités:

56. Calcul de la matrice de corrélation:

57. Calcul des performances attendues:
Plusieurs méthodologies sont disponibles:
× Historique
× CAPM
× Building Blocks
× Black Litterman

58. Paramétrage:
CAPM :
Le taux sans risque
(historical risk free rate)
Les Benchmarks:
Domestic Equity
Market Portfolio

59. Black Litterman:
Entrez vos propres prévisions en les pondérant par un degré de confiance:

60. Black Litterman (suite):
Vos vues pourront être relatives:

61. Définir des contraintes:

62. Définir des contraintes (suite):

63. L’optimisation
Inclure différentes frontières d’efficience afin de comparer plusieurs
modèles de répartition d’actif.
.

64. L’optimisation
Tester son portefeuille:

65. L’optimisation
Quel portefeuille pour la même volatilité ? :

66. Retrouvez des portefeuilles optimaux par profil d’investissement:

67. Resampling
MVO pure:

68. Resampling
MVO + Resampling:

69. Projections
Réaliser des projections de performances prenant en compte:
× L’inflation
× Les flux monétaires
× Le rebalancement

70. Projections
Paramétrage:

71. Projections
50% de probabilité de dépasser 500 K après 11 ans

72. 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