Attilio Meucci
Managing Diversification
COMMON MEASURES OF DIVERSIFICATION
DIVERSIFICATION DISTRIBUTION
MEAN-DIVERSIFICATION FRONTIER
CONDITIONAL ANALYSIS
REFERENCES
A. MEUCCI - Managing Diversification
returns of securities (stocks, bonds, options, structured products, …)
portfolio return
portfolio weights
A. MEUCCI - Managing Diversification Common Measures of Diversification
weight-based definitions
A. MEUCCI - Managing Diversification Common Measures of Diversification
portfolio weights
weight-based definitions
1w 3w2w
5w1
0
portfolio weights
- positive
- sum to one
distribution
security number
A. MEUCCI - Managing Diversification Common Measures of Diversification
distributionentropy ⇐
1w 3w2w
5w1
0
security number
weight-based definitions
A. MEUCCI - Managing Diversification Common Measures of Diversification
portfolio weights
- positive
- sum to one
distributionentropy ⇐
1w 3w2w
5w1
0
security number
weight-based definitions
A. MEUCCI - Managing Diversification Common Measures of Diversification
portfolio weights
- positive
- sum to one
weight-based definitions
A. MEUCCI - Managing Diversification Common Measures of Diversification
weight-based definitions
risk-based definitionsreturns correlation matrix
A. MEUCCI - Managing Diversification Common Measures of Diversification
weight-based definitions
risk-based definitions
returns covariance matrix
returns standard deviations
A. MEUCCI - Managing Diversification Common Measures of Diversification
factor-based definitionweight-based definitions
risk-based definitions
A. MEUCCI - Managing Diversification Common Measures of Diversification
portfolio return due to “idiosyncratic”
factor-based definitionweight-based definitions
risk-based definitions These definitions apply in specific circumstances and or under
restrictive hypotheses
A. MEUCCI - Managing Diversification Common Measures of Diversification
COMMON MEASURES OF DIVERSIFICATION
DIVERSIFICATION DISTRIBUTION
MEAN-DIVERSIFICATION FRONTIER
CONDITIONAL ANALYSIS
REFERENCES
A. MEUCCI - Managing Diversification
A. MEUCCI - Managing Diversification Diversification Distribution
Example: portfolio of two securities
- one bond
- one stock
1 50%w =
2 50%w =
{ } ( )21Var 1%R =
{ } ( )22Var 30%R =
A. MEUCCI - Managing Diversification Diversification Distribution
if correlations = 0
Example: portfolio of two securities
- one bond
- one stock
1 50%w =
2 50%w =
{ } ( )21Var 1%R =
{ } ( )22Var 30%R =
A. MEUCCI - Managing Diversification Diversification Distribution
security number
weighs highly diversified
risk highly concentrated
if correlations = 0
if correlations = 0
A. MEUCCI - Managing Diversification Diversification Distribution
A. MEUCCI - Managing Diversification Diversification Distribution
Example: portfolio of two government bonds in same duration bucket
Bond 1
Bond 2
1 50%w =
2 50%w =
{ } ( )21Var 1%R =
{ } ( )22Var 1%R =
if correlations = 0
Example: portfolio of two government bonds in same duration bucket
Bond 1
Bond 2
1 50%w =
2 50%w =
{ } ( )21Var 1%R =
{ } ( )22Var 1%R =
weighs highly diversified
volatility homegeneous
A. MEUCCI - Managing Diversification Diversification Distribution
high concentration due to correlations: full exposure to first principal component
{ }Cov≡ RΣ
PCA
eigenvectors
eigenvalues
principal portfolios
principal variances
1R
2R
principal portfolio 2
principal portfolio 1
A. MEUCCI - Managing Diversification Diversification Distribution
return of principal portfolios
{ }Cov≡ RΣ
A. MEUCCI - Managing Diversification Diversification Distribution
weights of original portfolio on principal portfolios
return of principal portfolios
{ }Cov≡ RΣ
A. MEUCCI - Managing Diversification Diversification Distribution
weights of original portfolio on principal portfolios
return of principal portfolios
{ }Cov≡ RΣ
A. MEUCCI - Managing Diversification Diversification Distribution
weights of original portfolio on principal portfolios
return of principal portfolios
variance concentration curvecontribution to original portfolio variance from n-th principal portfolio:
total variance variance concentration curve
principal portfolio number
A. MEUCCI - Managing Diversification Diversification Distribution
weights of original portfolio on principal portfolios
return of principal portfolios
variance concentration curvecontribution to original portfolio variance from n-th principal portfolio:
Example: portfolio of two government bonds in same duration bucket
Bond 1
Bond 2
1 50%w =
2 50%w =
{ } ( )21Var 1%R =
{ } ( )22Var 1%R =
weighs highly diversified
volatility homegeneous
variance concentration curve loads on one principal portfolio
A. MEUCCI - Managing Diversification Diversification Distribution
weights of original portfolio on principal portfolios
return of principal portfolios
variance concentration curve
volatility concentration curvecontribution to original portfolio volatility from n-th principal portfolio: “hot spots”
total volatility volatility concentration curve
principal portfolio number
A. MEUCCI - Managing Diversification Diversification Distribution
weights of original portfolio on principal portfolios
return of principal portfolios
variance concentration curve
volatility concentration curve
diversification distributioncontribution to original portfolio r-square from n-th principal portfolio
1
0
diversification distribution
principal portfolio number
A. MEUCCI - Managing Diversification Diversification Distribution
weights of original portfolio on principal portfolios
return of principal portfolios
variance concentration curve
volatility concentration curve
diversification distribution
A. MEUCCI - Managing Diversification Diversification Distribution
weights of original portfolio on principal portfolios
return of principal portfolios
variance concentration curve
volatility / tracking error concentration curve
diversification distribution
Example: management with benchmark
portfolios weights
benchmark weights
relative weights
weights of original portfolio on principal portfolios
return of principal portfolios
variance concentration curve
diversification distribution
relative weights
volatility / tracking error concentration curve
Example: management with benchmark
weights of original portfolio on principal portfolios
return of principal portfolios
variance concentration curve
diversification distribution
relative weights
volatility / tracking error concentration curve
Example: management with benchmark
COMMON MEASURES OF DIVERSIFICATION
DIVERSIFICATION DISTRIBUTION
MEAN-DIVERSIFICATION FRONTIER
CONDITIONAL ANALYSIS
REFERENCES
A. MEUCCI - Managing Diversification
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
A. MEUCCI - Managing Diversification Mean-Diversification Frontier
weights of original portfolio on principal portfolios
return of principal portfolios
1
0
diversification distribution
principal portfolio number
A. MEUCCI - Managing Diversification Mean-Diversification Frontier
variance concentration curve
volatility concentration curve
diversification distribution: “probability mass”
diversification index ?
weights of original portfolio on principal portfolios
return of principal portfolios
1
0
diversification distribution
principal portfolio number
A. MEUCCI - Managing Diversification Mean-Diversification Frontier
variance concentration curve
volatility concentration curve
diversification distribution: “probability mass”
diversification index
entropy
A. MEUCCI - Managing Diversification Mean-Diversification Frontier
Effective number of betsdiversification
index
entropy
weights
diversification distribution: “probability mass”
A. MEUCCI - Managing Diversification Mean-Diversification Frontier
full concentrationEffective number of bets
full concentration
full diversification
weights
weights
diversification distribution: “probability mass”
A. MEUCCI - Managing Diversification Mean-Diversification Frontier
Effective number of bets
weights
weights
Mean-diversification frontier
A. MEUCCI - Managing Diversification Mean-Diversification Frontier
full concentration
full diversification
Effective number of bets
weights
A. MEUCCI - Managing Diversification Mean-Diversification Frontier
full concentration
full diversification
Effective number of bets
Mean-diversification frontier
Allocation in terms of original portfolio weights
not principal portfolios
A. MEUCCI - Managing Diversification Mean-Diversification Frontier
full concentration
full diversification
Effective number of bets
weights
Transaction costs Mean-diversification frontier
Non linear, non-continuous function of current and
target portfolio
A. MEUCCI - Managing Diversification Mean-Diversification Frontier
full concentration
full diversification
Effective number of bets
weights
Transaction costs adjusted mean-diversification frontier
A. MEUCCI - Managing Diversification Mean-Diversification Frontier
full concentration
full diversification
Effective number of bets
weights
Transaction costs adjusted mean-diversification frontier
Effective number of bets
Expected return
COMMON MEASURES OF DIVERSIFICATION
DIVERSIFICATION DISTRIBUTION
MEAN-DIVERSIFICATION FRONTIER
CONDITIONAL ANALYSIS
REFERENCES
A. MEUCCI - Managing Diversification
Constraints feasible set
A. MEUCCI - Managing Diversification Conditional Analysisfeasible reallocations
current portfolio
Feasible trades
such that
Conditional PCA
Constraints
conditional principal portfoliosfeasible
feasible reallocations
current portfolio
feasible set
A. MEUCCI - Managing Diversification Conditional Analysis
Feasible trades Complementary, unfeasible trades
conditional principal portfoliosfeasible
conditional principal portfolioscomplementary
feasible set
such thatsuch that
A. MEUCCI - Managing Diversification Conditional Analysis
Conditional PCA
Constraintsfeasible reallocations
current portfolio
COMMON MEASURES OF DIVERSIFICATION
DIVERSIFICATION DISTRIBUTION
MEAN-DIVERSIFICATION FRONTIER
CONDITIONAL ANALYSIS
REFERENCES
A. MEUCCI - Managing Diversification
Article:Attilio Meucci, “Managing Diversification”Risk - May 2009 extended version available at http://ssrn.com/abstract=1358533
MATLAB examples:MATLAB Central Files Exchange (see above article)
This presentation:www.symmys.com > Teaching > Talks
A. MEUCCI - Managing Diversification References