Meucci Managing Diversification · A. MEUCCI - Managing Diversification Mean-Diversification...

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


portfolio weights


1w 3w2w

5w1

0

portfolio weights

- positive

- sum to one

distribution

security number


distributionentropy ⇐

1w 3w2w

5w1

0

security number



portfolio weights

- positive

- sum to one


risk-based definitionsreturns correlation matrix



risk-based definitions

returns covariance matrix

returns standard deviations


factor-based definitionweight-based definitions

risk-based definitions


portfolio return due to “idiosyncratic”

factor-based definitionweight-based definitions

risk-based definitions These definitions apply in specific circumstances and or under

restrictive hypotheses






REFERENCES


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 =


if correlations = 0

Example: portfolio of two securities

- one bond

- one stock

1 50%w =

2 50%w =

{ } ( )21Var 1%R =

{ } ( )22Var 30%R =


security number

weighs highly diversified

risk highly concentrated

if correlations = 0

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 =

if correlations = 0


Bond 1

Bond 2

1 50%w =

2 50%w =

{ } ( )21Var 1%R =

{ } ( )22Var 1%R =


volatility homegeneous


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


return of principal portfolios

{ }Cov≡ RΣ


weights of original portfolio on principal portfolios


{ }Cov≡ RΣ




variance concentration curvecontribution to original portfolio variance from n-th principal portfolio:

total variance variance concentration curve

principal portfolio number




variance concentration curvecontribution to original portfolio variance from n-th principal portfolio:


Bond 1

Bond 2

1 50%w =

2 50%w =

{ } ( )21Var 1%R =

{ } ( )22Var 1%R =


volatility homegeneous

variance concentration curve loads on one principal portfolio




variance concentration curve

volatility concentration curvecontribution to original portfolio volatility from n-th principal portfolio: “hot spots”

total volatility volatility concentration curve






volatility concentration curve

diversification distributioncontribution to original portfolio r-square from n-th principal portfolio

1

0

diversification distribution






volatility / tracking error concentration curve


Example: management with benchmark

portfolios weights

benchmark weights

relative weights





relative weights

volatility / tracking error concentration curve

Example: management with benchmark





REFERENCES






diversification distribution: “probability mass”

1

0



A. MEUCCI - Managing Diversification Mean-Diversification Frontier



1

0







diversification index ?



1

0







diversification index

entropy


Effective number of betsdiversification

index

entropy

weights



full concentrationEffective number of bets

full concentration

full diversification

weights

weights



Effective number of bets

weights

weights

Mean-diversification frontier


full concentration



weights


full concentration



Mean-diversification frontier

Allocation in terms of original portfolio weights

not principal portfolios


full concentration



weights

Transaction costs Mean-diversification frontier

Non linear, non-continuous function of current and

target portfolio


full concentration



weights

Transaction costs adjusted mean-diversification frontier


full concentration



weights

Transaction costs adjusted mean-diversification frontier


Expected return





REFERENCES


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





REFERENCES


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

Meucci Managing Diversification · A. MEUCCI - Managing Diversification Mean-Diversification...

Documents

Transcript of Meucci Managing Diversification · A. MEUCCI - Managing Diversification Mean-Diversification...