Meucci Managing Diversification - Operations researchieor.columbia.edu/files/seasdepts/industrial-engineering...entropy distribution 8 w1 w3 w2 1 w5 0 security number weight-based

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