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Attilio Meucci
(Re)Defining and Managing Diversification
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A. Meucci – Factor-based Portfolio Management
Standard approach: Modern Portfolio Theory
Optimal Portfolio = optimal mean-variance weights under constraints
Modern Portfolio Theory
Standard approach: Modern Portfolio Theory
Portfolio Return = weighted average of asset returns
Optimal Portfolio = optimal mean-variance weights under constraints
Modern Portfolio Theory A. Meucci – Factor-based Portfolio Management
Standard approach: Modern Portfolio Theory
Portfolio Return = weighted average of asset returns
Optimal Portfolio = optimal mean-variance weights under constraints
vector of expected asset returns matrix of asset covariances
Modern Portfolio Theory A. Meucci – Factor-based Portfolio Management
Trend 1: From asset-based allocation to factor-based allocation
Optimal Portfolio = optimal mean-variance
Factors and Premia A. Meucci – Factor-based Portfolio Management
Trend 1: From asset-based allocation to factor-based allocation
Portfolio Return = Linear factor model (factors: momentum, value, …,PCA,…
strategies,…)
Factors and Premia
Optimal Portfolio = optimal mean-variance
A. Meucci – Factor-based Portfolio Management
Trend 1: From asset-based allocation to factor-based allocation
Portfolio Return = Linear factor model (factors: momentum, value, …,PCA,…
strategies,…)
Optimal Portfolio = optimal mean-variance exposures under constraints
Factors and Premia A. Meucci – Factor-based Portfolio Management
Trend 1: From asset-based allocation to factor-based allocation
Portfolio Return = Linear factor model (factors: momentum, value, …,PCA,…
strategies,…)
vector of factor premia matrix of factor covariances
Optimal Portfolio = optimal mean-variance exposures under constraints
Factors and Premia A. Meucci – Factor-based Portfolio Management
Trend 2: from mean-variance to risk parity, or diversification management
Portfolio Return = weighted average of asset returns
“contributions” to risk
Diversification: risk “contributions” A. Meucci – Factor-based Portfolio Management
Trend 2: from mean-variance to risk parity, or diversification management
Optimal Portfolio = equal “contributions” to risk
matrix of asset covariances
“contributions” to risk
Diversification: risk “contributions”
Portfolio Return = weighted average of asset returns
A. Meucci – Factor-based Portfolio Management
Diversification: Effective Number of Bets
Portfolio Return = weighted average of asset returns
A. Meucci – Factor-based Portfolio Management
if correlations = 0
security number
Diversification: Effective Number of Bets A. Meucci – Factor-based Portfolio Management
if correlations = 0
Diversification: Effective Number of Bets A. Meucci – Factor-based Portfolio Management
{ }Cov≡ RΣ
PCA
eigenvectors
eigenvalues
Diversification: Effective Number of Bets A. Meucci – Factor-based Portfolio Management
{ }Cov≡ RΣ
PCA
eigenvectors
eigenvalues
uncorrelated, maximum variance portfolios
Diversification: Effective Number of Bets A. Meucci – Factor-based Portfolio Management
{ }Cov≡ RΣ
PCA
eigenvectors
eigenvalues
uncorrelated, maximum variance portfolios
variances of uncorrelated, maximum variance portfolios
Diversification: Effective Number of Bets A. Meucci – Factor-based Portfolio Management
{ }Cov≡ RΣ
PCA
eigenvectors
eigenvalues
principal portfolios
principal variances
Diversification: Effective Number of Bets A. Meucci – Factor-based Portfolio Management
principal portfolios
principal variances
{ }Cov≡ RΣ
PCA
Diversification: Effective Number of Bets A. Meucci – Factor-based Portfolio Management
return of principal portfolios
{ }Cov≡ RΣ
Diversification: Effective Number of Bets A. Meucci – Factor-based Portfolio Management
weights of original portfolio on principal portfolios
return of principal portfolios
{ }Cov≡ RΣ
Diversification: Effective Number of Bets A. Meucci – Factor-based Portfolio Management
weights of original portfolio on principal portfolios
return of principal portfolios
{ }Cov≡ RΣ
Diversification: Effective Number of Bets A. Meucci – Factor-based Portfolio Management
weights of original portfolio on principal portfolios
return of principal portfolios
variance concentration curve contribution to original portfolio variance from n-th principal portfolio:
total variance variance concentration curve
principal portfolio number
Diversification: Effective Number of Bets A. Meucci – Factor-based Portfolio Management
weights of original portfolio on principal portfolios
return of principal portfolios
variance concentration curve
volatility concentration curve contribution to original portfolio volatility from n-th principal portfolio: “hot spots”
total volatility volatility concentration curve
principal portfolio number
Diversification: Effective Number of Bets A. Meucci – Factor-based Portfolio Management
weights of original portfolio on principal portfolios
return of principal portfolios
variance concentration curve
volatility concentration curve
diversification distribution contribution to original portfolio r-square from n-th principal portfolio
1
0
diversification distribution
principal portfolio number
Diversification: Effective Number of Bets A. Meucci – Factor-based Portfolio Management
weights of original portfolio on principal portfolios
return of principal portfolios
variance concentration curve
volatility concentration curve
diversification distribution
Diversification: Effective Number of Bets A. Meucci – Factor-based Portfolio Management
weights of original portfolio on principal portfolios
return of principal portfolios
variance concentration curve
volatility concentration curve
diversification distribution: “probability mass”
1
0
diversification distribution
principal portfolio number
Diversification: Effective Number of Bets A. Meucci – Factor-based Portfolio Management
diversification
weights of original portfolio on principal portfolios
return of principal portfolios
variance concentration curve
volatility concentration curve
diversification distribution: “probability mass”
effective number of bets
Diversification: Effective Number of Bets A. Meucci – Factor-based Portfolio Management
full concentration
weights
diversification distribution: “probability mass”
effective number of bets
Diversification: Effective Number of Bets A. Meucci – Factor-based Portfolio Management
full concentration
full diversification
weights
weights
diversification distribution: “probability mass”
effective number of bets
Diversification: Effective Number of Bets A. Meucci – Factor-based Portfolio Management
full concentration
full diversification
weights
weights
mean-diversification frontier
effective number of bets
Diversification management A. Meucci – Factor-based Portfolio Management
Next Steps: Minimal Torsion Bets A. Meucci – Factor-based Portfolio Management
Next Steps: Minimal Torsion Bets A. Meucci – Factor-based Portfolio Management
Next Steps: Minimal Torsion Bets A. Meucci – Factor-based Portfolio Management
???
Next Steps: Minimal Torsion Bets
Factor-based Risk Parity
A. Meucci – Factor-based Portfolio Management
Next Steps: Minimal Torsion Bets A. Meucci – Factor-based Portfolio Management
Next Steps: Minimal Torsion Bets A. Meucci – Factor-based Portfolio Management
i) Factors on Demand: best portfolio-specific linear factor model
i) Factors on Demand: best portfolio-specific linear factor model
Next Steps: Minimal Torsion Bets A. Meucci – Factor-based Portfolio Management
Original Factors Minimum-Torsion Bets Principal Components Bets
ii) Minimal Torsion Bets: , uncorrelated factors closest to factors
i) Factors on Demand: best portfolio-specific linear factor model
ii) Minimal Torsion Bets: , uncorrelated factors closest to factors
Next Steps: Minimal Torsion Bets A. Meucci – Factor-based Portfolio Management
Riccati root of correlation
i) Factors on Demand: best portfolio-specific linear factor model
ii) Minimal Torsion Bets: , uncorrelated factors closest to factors
Next Steps: Minimal Torsion Bets A. Meucci – Factor-based Portfolio Management
Riccati root of correlation
ii) Diversification Distribution and Effective Number of Minimum Torsion Bets
Minimum-Torsion Diversification Distribution
Portfolio Weights
Effective Number of Bets (normalized)
1
Wei
ghts
/ P
roba
bilit
ies
A. Meucci – Factor-based Portfolio Management
Minimal Torsion Bets: Case Study
The Effective Number of PCA Bets in the S&P500 is close to 1, since the first PCA factor loadings are similar to the weights of the stocks in the S&P500
The Effective Number of Minimal Torsion Bets in the S&P500 yields intuitive results
A. Meucci – Factor-based Portfolio Management N
orm
alize
d Ef
fect
ive
Num
ber o
f Bet
s = E
NB/
N
Principal Components
Minimum-Torsion
Marginal Contributions
Portfolio Weights
Wei
ghts
/ P
roba
bilit
ies
A. Meucci – Factor-based Portfolio Management
Next Steps: Minimal Torsion Bets A. Meucci – Factor-based Portfolio Management
References
Effective Number of Bets: http://symmys.com/node/199
Factors on Demand: http://symmys.com/node/164
Minimal Torsion Bets: http://symmys.com/node/599
A. Meucci – Factor-based Portfolio Management
