Attilio Meucci. Managing Diversification
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1 Attilio Meucci Managing Diversification
2 A. MEUCCI - Managing Diversification COMMON MEASURES OF DIVERSIFICATION DIVERSIFICATION DISTRIBUTION MEAN-DIVERSIFICATION FRONTIER CONDITIONAL ANALYSIS REFERENCES
3 A. MEUCCI - Managing Diversification Common Measures of Diversification portfolio return returns of securities (stocks, bonds, options, structured products, ) portfolio weights
4 A. MEUCCI - Managing Diversification Common Measures of Diversification weight-based definitions portfolio weights
5 A. MEUCCI - Managing Diversification Common Measures of Diversification weight-based definitions distribution portfolio weights - positive - sum to one 1 w 5 w 3 0 w 1 w 2 security number
6 A. MEUCCI - Managing Diversification Common Measures of Diversification weight-based definitions entropy distribution portfolio weights - positive - sum to one 1 w 5 w 3 0 w 1 w 2 security number
7 A. MEUCCI - Managing Diversification Common Measures of Diversification weight-based definitions entropy distribution portfolio weights - positive - sum to one 1 w 5 w 3 0 w 1 w 2 security number
8 A. MEUCCI - Managing Diversification Common Measures of Diversification weight-based definitions
9 A. MEUCCI - Managing Diversification Common Measures of Diversification weight-based definitions risk-based definitions returns correlation matrix
10 A. MEUCCI - Managing Diversification Common Measures of Diversification weight-based definitions risk-based definitions returns standard deviations returns covariance matrix
11 A. MEUCCI - Managing Diversification Common Measures of Diversification weight-based definitions factor-based definition portfolio return due to idiosyncratic risk-based definitions
12 A. MEUCCI - Managing Diversification Common Measures of Diversification weight-based definitions factor-based definition risk-based definitions These definitions apply in specific circumstances and or under restrictive hypotheses
13 A. MEUCCI - Managing Diversification COMMON MEASURES OF DIVERSIFICATION DIVERSIFICATION DISTRIBUTION MEAN-DIVERSIFICATION FRONTIER CONDITIONAL ANALYSIS REFERENCES
14 A. MEUCCI - Managing Diversification Diversification Distribution
15 A. MEUCCI - Managing Diversification Diversification Distribution if correlations = 0 Example: portfolio of two securities - one bond - one stock w 1 = 50% w 2 = 50% { R } = ( ) 2 Var 1% 1 { R } = ( ) 2 Var 30% 2
16 A. MEUCCI - Managing Diversification Diversification Distribution if correlations = 0 Example: portfolio of two securities - one bond - one stock w 1 = 50% w 2 = 50% { R } = ( ) 2 Var 1% 1 { R } = ( ) 2 Var 30% 2 weighs highly diversified risk highly concentrated security number
17 A. MEUCCI - Managing Diversification Diversification Distribution if correlations = 0
18 A. MEUCCI - Managing Diversification Diversification Distribution if correlations = 0 Example: portfolio of two government bonds in same duration bucket Bond 1 w 1 = 50% { R } = ( ) 2 Var 1% 1 Bond 2 w 2 = 50% { R } = ( ) 2 Var 1% 2
19 A. MEUCCI - Managing Diversification Diversification Distribution Example: portfolio of two government bonds in same duration bucket Bond 1 w 1 = 50% { R } = ( ) 2 Var 1% 1 Bond 2 w 2 = 50% { R } = ( ) 2 Var 1% 2 weighs highly diversified volatility homegeneous high concentration due to correlations: full exposure to first principal component
20 A. MEUCCI - Managing Diversification Diversification Distribution Σ Cov{ R} R 2 PCA principal portfolio 2 R 1 principal portfolio 1 eigenvectors principal portfolios eigenvalues principal variances
21 A. MEUCCI - Managing Diversification Diversification Distribution Σ Cov{ R} return of principal portfolios
22 A. MEUCCI - Managing Diversification Diversification Distribution Σ Cov{ R} return of principal portfolios weights of original portfolio on principal portfolios
23 A. MEUCCI - Managing Diversification Diversification Distribution Σ Cov{ R} return of principal portfolios weights of original portfolio on principal portfolios
24 A. MEUCCI - Managing Diversification Diversification Distribution total variance variance concentration curve principal portfolio number return of principal portfolios weights of original portfolio on principal portfolios variance concentration curve contribution to original portfolio variance from n-th principal portfolio:
25 A. MEUCCI - Managing Diversification Diversification Distribution Example: portfolio of two government bonds in same duration bucket Bond 1 Bond 2 w 1 = 50% w 2 = 50% weighs highly diversified { R } ( ) 2 1 = { R } = ( ) 2 Var 1% Var 1% 2 volatility homegeneous variance concentration curve loads on one principal portfolio return of principal portfolios weights of original portfolio on principal portfolios variance concentration curve contribution to original portfolio variance from n-th principal portfolio:
26 A. MEUCCI - Managing Diversification Diversification Distribution total volatility volatility concentration curve principal portfolio number return of principal portfolios weights of original portfolio on principal portfolios variance concentration curve volatility concentration curve contribution to original portfolio volatility from n-th principal portfolio: hot spots
27 A. MEUCCI - Managing Diversification Diversification Distribution 1 diversification distribution 0 principal portfolio number return of principal portfolios weights of original portfolio on principal portfolios variance concentration curve volatility concentration curve diversification distribution contribution to original portfolio r-square from n-th principal portfolio
28 A. MEUCCI - Managing Diversification Diversification Distribution return of principal portfolios weights of original portfolio on principal portfolios variance concentration curve volatility concentration curve diversification distribution
29 Example: management with benchmark portfolios weights benchmark weights relative weights return of principal portfolios weights of original portfolio on principal portfolios variance concentration curve volatility / tracking error concentration curve diversification distribution
30 Example: management with benchmark relative weights return of principal portfolios weights of original portfolio on principal portfolios variance concentration curve volatility / tracking error concentration curve diversification distribution
31 Example: management with benchmark relative weights return of principal portfolios weights of original portfolio on principal portfolios variance concentration curve volatility / tracking error concentration curve diversification distribution
32 A. MEUCCI - Managing Diversification COMMON MEASURES OF DIVERSIFICATION DIVERSIFICATION DISTRIBUTION MEAN-DIVERSIFICATION FRONTIER CONDITIONAL ANALYSIS REFERENCES
33 A. MEUCCI - Managing Diversification Mean-Diversification Frontier 1 diversification distribution 0 principal portfolio number return of principal portfolios weights of original portfolio on principal portfolios variance concentration curve volatility concentration curve diversification distribution: probability mass
34 A. MEUCCI - Managing Diversification Mean-Diversification Frontier diversification index? 1 diversification distribution 0 principal portfolio number return of principal portfolios weights of original portfolio on principal portfolios variance concentration curve volatility concentration curve diversification distribution: probability mass
35 A. MEUCCI - Managing Diversification Mean-Diversification Frontier diversification index entropy 1 diversification distribution 0 principal portfolio number return of principal portfolios weights of original portfolio on principal portfolios variance concentration curve volatility concentration curve diversification distribution: probability mass
36 A. MEUCCI - Managing Diversification Mean-Diversification Frontier Effective number of bets diversification index entropy
37 A. MEUCCI - Managing Diversification Mean-Diversification Frontier Effective number of bets full concentration weights diversification distribution: probability mass
38 A. MEUCCI - Managing Diversification Mean-Diversification Frontier Effective number of bets full concentration full diversification weights weights diversification distribution: probability mass
39 A. MEUCCI - Managing Diversification Mean-Diversification Frontier Effective number of bets full concentration full diversification weights Mean-diversification frontier weights
40 A. MEUCCI - Managing Diversification Mean-Diversification Frontier Effective number of bets full concentration full diversification Mean-diversification frontier weights Allocation in terms of original portfolio weights not principal portfolios
41 A. MEUCCI - Managing Diversification Mean-Diversification Frontier Effective number of bets full concentration full diversification Transaction costs weights Mean-diversification frontier Non linear, non-continuous function of current and target portfolio
42 A. MEUCCI - Managing Diversification Mean-Diversification Frontier Effective number of bets full concentration weights full diversification Transaction costs adjusted mean-diversification frontier
43 A. MEUCCI - Managing Diversification Mean-Diversification Frontier Effective number of bets full concentration weights full diversification Transaction costs adjusted mean-diversification frontier Effective number of bets Expected return
44 A. MEUCCI - Managing Diversification COMMON MEASURES OF DIVERSIFICATION DIVERSIFICATION DISTRIBUTION MEAN-DIVERSIFICATION FRONTIER CONDITIONAL ANALYSIS REFERENCES
45 A. MEUCCI - Managing Diversification Conditional Analysis Constraints feasible set feasible reallocations current portfolio
46 A. MEUCCI - Managing Diversification Conditional Analysis Constraints feasible set feasible reallocations current portfolio Conditional PCA conditional principal portfolios feasible Feasible trades such that
47 A. MEUCCI - Managing Diversification Conditional Analysis Constraints feasible set conditional principal portfolios complementary feasible reallocations current portfolio Conditional PCA conditional principal portfolios feasible Feasible trades Complementary, unfeasible trades such that such that
48 A. MEUCCI - Managing Diversification COMMON MEASURES OF DIVERSIFICATION DIVERSIFICATION DISTRIBUTION MEAN-DIVERSIFICATION FRONTIER CONDITIONAL ANALYSIS REFERENCES
49 A. MEUCCI - Managing Diversification References Article: Attilio Meucci, Managing Diversification Risk - May 2009 extended version available at MATLAB examples: MATLAB Central Files Exchange (see above article) This presentation: > Teaching > Talks
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