A Critical Review of Correlation-based Measures of Portfolio Diversification

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1 Stability Stability Investment Investment Solutions Solutions Diligence Diligence Client-centric A Critical Revie of Correlation-based Measures of Portfolio Diversification Presented to: orthfield 7 th Annual Research Conference October 7, 4 Presented by: Randy O Toole, CFA Federated Investors Senior Quantitative Analyst rotoole@federatedinv.com For Institutional/Investment Professional Use Only. ot for Distribution to the Public. Tracking number: -445

2 Overvie The Portfolio Diversification Index (PDI) and the Diversification Ratio (DR) both aim to () quantify diversification characteristics expressly related to correlations and () provide frameorks for constructing diversified portfolios The PDI and DR are in fact closely related to to very ell-knon riskbased portfolio construction approaches: Minimum Variance Portfolios (MVP) and Risk Parity Portfolios (RPP) We introduce the Minimum Correlation Portfolio (MCP) and sho ho it can be used to solve for the maximum value of the DR We clarify the properties of the PDI and sho that it quantifies diversification characteristics specific to so-called aïve Risk Parity portfolios We highlight a significant eakness in the PDI and sho that it can generate misleading estimates of diversification hen there are negative correlations among assets, in contrast to the DR hich clearly distinguishes the effects of positive and negative correlations on portfolio diversification For Institutional/Investment Professional Use Only. ot for Distribution to the Public.

3 Diversification and risk-based portfolio construction Risk-based portfolio construction approaches gre in popularity folloing the 8 global financial crisis, hen many portfolios and investment strategies thought to be ell-diversified become spectacularly and dramatically undiversified Risk-based methods esche expected returns and focus solely on volatilities and correlations, ith the goal of producing portfolios ith better diversification characteristics vis-à-vis mean-variance optimal (MVO) portfolios Several approaches to measuring diversification and constructing diversified portfolios have been proposed, hich raises an important question: What exactly do e mean by diversification? For Institutional/Investment Professional Use Only. ot for Distribution to the Public.

4 Defining and measuring diversification Diversification strives to smooth out unsystematic risk events in a portfolio so that the positive performance of some investments ill neutralize the negative performance of others. Therefore, the benefits of diversification ill hold only if the securities in the portfolio are not perfectly correlated. (investopedia.com) A ell-diversified portfolio is one that is expected to be immune against shocks created by a single or a fe assets. (Meucci, 9; Frahm and Weichers, ) Market portfolio; equal-eighted portfolio ( naïve diversification) umber of assets; portfolio eights; breadth of positions Information entropy Minimum variance and risk parity portfolios Lo co-movement across assets ( most diversified portfolio ) o definitive definition or unique measure of diversification For Institutional/Investment Professional Use Only. ot for Distribution to the Public. 3

5 Defining and calculating portfolio risk 4 Define the x covariance matrix of asset excess returns, V: For Institutional/Investment Professional Use Only. ot for Distribution to the Public. Define V(), the eighted covariance matrix associated ith a portfolio that has an x vector of portfolio eights, : SCS V ρ ρ ρ ρ ρ ρ ( ) WVW V The variance of this portfolio is ( ) ( ) ( ) ones is an xvector of l V l V l i j j i ij j i ρ

6 Contributions to portfolio risk Marginal contributions to portfolio risk (MCR): MCR ( ) ( ) V The MCR of an asset is the approximate increase or decrease in portfolio risk hen the eight of that asset is increased by one percentage point, here the increase is assumed to be financed using cash (as opposed to selling other assets in the portfolio) Total contributions to portfolio risk (TCR): TCR ( ) ( ) W TCRs are the amounts of portfolio risk contributed by each asset. The sum of the TCRs is equal to the volatility of the portfolio: i TCR i ( ) l WV V ( ) ( ) ( ) WV ( ) ( ) For Institutional/Investment Professional Use Only. ot for Distribution to the Public. 5

7 Minimum Variance Portfolio Minimum Variance Portfolio (MVP): min V + ( l) θ ( ) l V MVP ( l) V l, MVP V ( l V l) The MVP has the loest possible variance out of all fully invested portfolios A key feature of the MVP is that the marginal contributions to risk are the same for all assets: MCR ( ) MVP V MVP ( ) MVP The MVP is diversified in terms of marginal contributions to risk: an incremental addition to the eight of an asset ill increase the risk of the MVP by the same quantity as an identical incremental addition to the eight of any other asset MVP l MVP ( ) MVP For Institutional/Investment Professional Use Only. ot for Distribution to the Public. 6

8 Risk Parity Portfolio Risk Parity Portfolios (RPP) are characterized by equal total contributions to portfolio risk from each asset: TCR ( ) TCR ( ) i j i RPP j RPP, A RPP is diversified in terms of total contributions to risk: each asset contributes an equal amount to the volatility of the portfolio, and therefore gains and losses in the portfolio ill not be dominated by an individual position in any asset There is no closed-form solution to equalizing total risk contributions across assets unless all correlations are identical In the case of identical correlations, RPP eights are simply equal to the reciprocal of each asset s volatility (normalized to sum to one). This inversevolatility eighting scheme is commonly referred to as aïve Risk Parity (RP), and is often used in practice to approximate a RPP: l S RP ( l) S l For Institutional/Investment Professional Use Only. ot for Distribution to the Public. 7

9 Portfolio Diversification Index (PDI) Rudin and Morgan (6) apply principal component analysis (PCA) to a correlation matrix to calculate the Portfolio Diversification Index (PDI) PCA transforms correlated assets into a set of uncorrelated principal components (PCs) that are ordered according to ho much variation in the returns is retained by each PC PDI i RS i, RSi i Interpretation is that the set of correlated assets offers the same degree of diversification as PDI uncorrelated assets: The PDI is bound beteen one and PDI if all assets are perfectly uncorrelated ( ideal diversification) PDI < reflects more extensive co-movement across assets; more return variation is explained by the first fe PCs PDI ~ indicates diversification is effectively impossible j For Institutional/Investment Professional Use Only. ot for Distribution to the Public. λ i λ j λ i eigenvalue associated ith the i th PC RS i relative strength of the i th PC 8

10 PDI and Risk Parity 9 The PDI provides a summary statistic of diversification distinctly related to correlations as ell as a criterion for portfolio construction Implicit in using a correlation matrix is that volatilities have been standardized to the same level the PDI specifically quantifies the diversification characteristics of a aïve Risk Parity portfolio: The PDI does not measure the diversification of a given portfolio per se, but rather the diversification potential of a set of assets ere they to be combined into a RP portfolio Equal-eighted portfolios constructed using the PDI ill not reflect the RP diversification characteristics being measured by the PDI For Institutional/Investment Professional Use Only. ot for Distribution to the Public. ( ) C VW W V RP RP RP ρ ρ ρ ρ ρ ρ

11 Diversification Ratio (DR) Choueifaty and Coignard (8) quantify diversifying properties associated ith correlations using the Diversification Ratio (DR): DR ( ) V L V s ( ), V SCS, V L SLS, L ll, s Sl Unlike the PDI, the DR can be used to measure the diversification characteristics of a variety of portfolios For Institutional/Investment Professional Use Only. ot for Distribution to the Public.

12 Most Diversified Portfolio (MDP) The Most Diversified Portfolio (MDP) maximizes the DR The maximized DR quantifies the ultimate diversification potential of the assets that comprise a given portfolio or pool of securities, here diversification is no defined as the loest possible correlation across assets Solving for the MDP: min V + ( s) θ, s MDP ( s V s) V s ( l C l) S C l Sl ( ) V ( s V s) ( l C l) MDP MDP MDP The maximized value of the DR is DR MAX DR ( ) MDP MDP MDP V L V MDP MDP ( l C l) l C LC l( l C l) ( ) ( MDP ) l C l For Institutional/Investment Professional Use Only. ot for Distribution to the Public.

13 MDP and the Minimum Correlation Portfolio (MCP) ormalizing volatilities to equal one and solving for the MVP or the MDP produces the minimum correlation portfolio (MCP): min C + ( l) θ l C MCP ( l) C l ~ C l C l, ( ) ( ) ( ) MCP MCP MCP The normalized variance of the MCP is equal to the variance of the MDP MDP variance has the loest possible correlation across assets Alternate derivation of the MDP: () Standardize volatilities ( Choueifaty Synthetic Asset Transformation ) and solve for the MCP eights () Multiply the MCP eights by RP eights ( Choueifaty Synthetic Asset Back- Transformation ): ( s V s) V s ( l C MDP MDP MCP l) S C l S For Institutional/Investment Professional Use Only. ot for Distribution to the Public.

14 DR and the variance of the MDP (VMDP) The inverse of the square of the maximized DR is equal to the variance of the MDP: The variance of the MDP (VMDP) has convenient properties that make it a useful measure of diversification characteristics associated ith correlations: It is bound beteen zero and one ( ) ( l C l) MDP DR MAX /DR MAX / hen all correlations equal zero ( ideal diversification) /DR MAX > / indicates less diversification potential vis-à-vis ideal diversification; /DR MAX ~ signifies no possibility to diversify /DR MAX < / reflects hedging potential associated ith negative correlations; /DR MAX ~ indicates the existence of assets that are nearly perfectly negatively correlated From here on out e ill use /DR as our preferred diversification measure For Institutional/Investment Professional Use Only. ot for Distribution to the Public. 3

15 Comparing the DR and PDI The DR and PDI both quantify diversification in a single number; hoever, the results can be conflicting Example: scenario analysis for a global asset allocation manager using correlations and volatilities that are expected to prevail during periods of market turbulence, ith a particular interest in ho diversification opportunities are likely to be affected ormal Markets Correlations US Int'l Developed Emerging Markets Global Bonds Volatilities US 5% Int'l Developed 6% 8% Emerging Markets 5% 7% % Global Bonds 4% 3% 4% 8% Turbulent Markets Correlations US Int'l Developed Emerging Markets Global Bonds Volatilities US 3% Int'l Developed 9% 36% Emerging Markets 8% 85% 44% Global Bonds -% -% -% % For Institutional/Investment Professional Use Only. ot for Distribution to the Public. 4

16 Comparing the DR and PDI (con t) DR, MDP Weights and RP Weights for ormal and Turbulent Market Conditions 8% Most Diversified Portfolio (MDP) Weights 8% aive Risk Parity (RP) Weights 7% ormal 7% ormal 6% Turbulent 6% Turbulent 5% 5% 4% 3% ormal Turbulent /DR.6.4 4% 3% ormal Turbulent /DR.6.53 % % % % % US Int'l Dev Emg Mkt Global FI % US Int'l Dev Emg Mkt Global FI The DR indicates diversification potential is greater during turbulent periods for both the MDP and RP MDP and RP eights sho that diversification is increased by reducing exposure to equities and adding to bonds For Institutional/Investment Professional Use Only. ot for Distribution to the Public. 5

17 Comparing the DR and PDI (con t) PDI and Cumulative Percentage of Variation Explained ormal Turbulent PDI.8.84 % 9% 8% 7% 6% 5% 4% 3% % % % Cumulative % of Variation Explained ormal Turbulent st PC nd PC 3rd PC 4th PC The PDI indicates diversification opportunities are loer during times of turbulence The first to PCs account for much more variation in turbulent periods, hich is not surprising given the increase in stock market correlations For Institutional/Investment Professional Use Only. ot for Distribution to the Public. 6

18 Comparing the DR and PDI (con t) This example highlights a critical difference in these approaches to measuring diversification: The PDI accurately reflects the fact that more substantial co-movement exists among the majority of assets (i.e., the equity markets) during times of turbulence The maximized DR reflects the increase in diversification potential from the standpoint of a portfolio that is optimized to have the loest possible correlation across assets A stronger degree of positive correlation across most of the assets does not necessarily imply an overall reduction in the potential to diversify The PDI may belie portfolio diversification opportunities that are more readily apparent from the perspective of the DR/MDP For Institutional/Investment Professional Use Only. ot for Distribution to the Public. 7

19 egative correlations and hedging characteristics egative correlations are problematic for the PDI: MDP and PDI Diversification Profiles for To Assets.9 /DRMAX Diversification/Hedging Potential /PDI DR MAX PDI + ρ ρ Correlation Coefficient, ρ The PDI is symmetric ith respect to positive and negative correlations This is because the eigenvalues are the same regardless of the sign of ρ For Institutional/Investment Professional Use Only. ot for Distribution to the Public. 8

20 Example: three assets The to-asset example provides a simple illustration of the properties of both diversification measures; visualizing these properties becomes more complicated hen > : Higher dimensionality limits the number of possible assets The range of possible values for the correlation coefficients must be restricted in order to maintain a positive semi-definite correlation matrix With three assets a positive semi-definite matrix is guaranteed by defining one correlation as a function of () the other to correlations, and () a partial correlation coefficient (PCC) that holds fixed the asset the other to correlations have in common: ρ 3 ρ ρ3 + ρ ρ ρ Given a fixed value of the PCC, the diversification measures can be calculated for the entire constrained range of 3x3 correlation matrices and plotted as functions of the to free correlation coefficients 3 3 For Institutional/Investment Professional Use Only. ot for Distribution to the Public. 9

21 Conditional Correlations and Diversification/Hedging Profiles, ρ Conditional Correlations and Diversification/Hedging Profiles, ρ ρ 3 f(ρ,ρ 3 ;ρ ) MDP, long-only. ρ ρ ρ 3.99 /DR MAX ρ ρ 3.99 MDP, unconstrained /PDI /DR MAX ρ 3.99 /PDI ρ 3.99 ρ ρ For Institutional/Investment Professional Use Only. ot for Distribution to the Public.

22 Conditional Correlations and Diversification/Hedging Profiles, ρ Conditional Correlations and Diversification/Hedging Profiles, ρ ρ 3 f(ρ,ρ 3 ;ρ ) MDP, long-only. ρ ρ ρ 3.99 /DR MAX ρ ρ 3 MDP, unconstrained /PDI /DR MAX ρ ρ 3 /PDI ρ ρ 3 For Institutional/Investment Professional Use Only. ot for Distribution to the Public.

23 Conditional Correlations and Diversification/Hedging Profiles, ρ 3 Conditional Correlations and Diversification/Hedging Profiles, ρ 3 ρ 3 f(ρ,ρ 3 ;ρ 3 ) MDP, long-only..8 ρ ρ ρ 3.99 /DR MAX ρ ρ 3 MDP, unconstrained /PDI /DR MAX ρ ρ 3 /PDI ρ ρ 3 For Institutional/Investment Professional Use Only. ot for Distribution to the Public.

24 Conditional Correlations and Diversification/Hedging Profiles, ρ 3.75 Conditional Correlations and Diversification/Hedging Profiles, ρ 3.75 ρ 3 f(ρ,ρ 3 ;ρ 3.75) MDP, long-only.99 ρ ρ ρ 3 /DR MAX ρ ρ 3 MDP, unconstrained /PDI /DR MAX ρ ρ 3 /PDI ρ ρ 3 For Institutional/Investment Professional Use Only. ot for Distribution to the Public. 3

25 Conditional Correlations and Diversification/Hedging Profiles, ρ 3.99 Conditional Correlations and Diversification/Hedging Profiles, ρ 3.99 ρ 3 f(ρ,ρ 3 ;ρ 3.99) MDP, long-only.99 ρ ρ 3.99 ρ /DR MAX -.99 ρ ρ MDP, unconstrained /PDI.99 ρ /DR MAX.99 ρ /PDI -.99 ρ ρ For Institutional/Investment Professional Use Only. ot for Distribution to the Public. 4

26 Comparing the MDP and PDI The maximized DR profiles exhibit a number of distinctive characteristics: /DR MAX is close to its maximum value of one only hen all of the correlations are nearly equal to one lo diversification potential is due solely to relatively high positive correlations across all of the assets /DR MAX is close to zero hen negative correlations become relatively extreme significant hedging potential exists henever to of the assets are close to being perfectly negatively correlated Constraining the MDP eights to be non-negative reduces diversification opportunities but does not affect the fundamental properties of the maximized DR The no-shorting constraint also reflects more realistic investment possibilities as unconstrained MDP eights can take on large values that ould require extreme leverage to implement For Institutional/Investment Professional Use Only. ot for Distribution to the Public. 5

27 Comparing the MDP and PDI (con t) The profiles of the inverted PDI differ considerably from those of the MDP: The values of the PDI are symmetric ith respect to positive and negative correlations, ith the inverse of the PDI close to one hen all of the correlations are at extremes irrespective of sign The inverse of the PDI reaches its loer bound of / /3 hen all three correlations are equal to zero, hich occurs only in the case of ρ 3 The PDI can be vieed as operating on the absolute values of the correlations, and thus can produce misleading estimates of diversification potential For Institutional/Investment Professional Use Only. ot for Distribution to the Public. 6

28 Example: hedge fund investment styles We compare the DR and PDI using a universe consisting of the S&P 5 and nine Hedge Fund Research (HFR) investment style return indices: S&P 5 Short Bias Equity Market eutral Quantitative Directional Distressed/Restructuring Merger Arbitrage Macro FI Convertible Arbitrage FI Multi-Strategy FI Corporate For Institutional/Investment Professional Use Only. ot for Distribution to the Public. 7

29 DR and PDI ith and ithout Short Bias Correlations of S&P 5 and Hedge Fund Style Monthly Returns, January 99 - June 4 S&P 5 Short Bias Equity Market eutral Quantitative Directional Distressed/ Restructuring Merger Arbitrage Macro FI Convert Arbitrage FI Multi- Strategy FI Corporate Volatility (annualized) S&P % Short Bias -7% % Equity Market eutral 6% -8% % Quantitative Directional 79% -86% 33% % Distressed/Restructuring 53% -5% 39% 65% % Merger Arbitrage 53% -43% 9% 59% 57% % Macro 3% -35% 3% 55% 4% 33% % FI Convert Arbitrage 47% -34% 3% 44% 66% 48% 4% % FI Multi-Strategy 53% -46% 35% 6% 77% 46% 4% 78% % FI Corporate 53% -46% 6% 6% 83% 57% 38% 67% 8% 6.5% o Short Bias Ideal Diversification (/). /PDI.8 /DR MAX.47 With Short Bias..7.7 The maximized DR reflects the hedging potential afforded by the Short Bias Index The PDI indicates diversification potential is unchanged ith the inclusion of Short Bias For Institutional/Investment Professional Use Only. ot for Distribution to the Public. 8

30 DR and PDI ith and ithout Short Bias (con t) 36-month rolling correlations and volatilities, December 99 June PDI and Maximized DR, no Short Bias /PDI /DR / MAX PDI and Maximized DR, ith Short Bias /PDI /DR / MAX The maximized DR quantifies the hedging potential introduced by the Short Bias index: /DR MAX alays lies belo the ideal diversification value of / The PDI continually indicates varying degrees of less-than-ideal diversification potential for the universe that includes the Short Bias index For Institutional/Investment Professional Use Only. ot for Distribution to the Public. 9

31 Portfolio construction Methods that use the PDI as a tool to construct diversified portfolios have recently been introduced (for example, Crezee and Sinkels, ; Diyarbakirlioglu and Satman, 4) The approaches involve iterative procedures that combine assets until the maximum PDI is found for a portfolio ith a pre-specified number of holdings These maximized PDI rules are ubiquitously used to construct equally eighted portfolios, as opposed to aïve Risk Parity portfolios We use the maximum PDI to construct portfolios consisting of the S&P 5 combined ith the hedge fund style indices, ith and ithout the Short Bias Index Use full-sample volatilities and correlations Portfolios range from assets to all assets Compare equal-eighted portfolio, RP portfolios and long-only MDPs For Institutional/Investment Professional Use Only. ot for Distribution to the Public. 3

32 Equal-eighted portfolios Equal-eighted portfolio eights based on maximizing the PDI, no Short Bias FI Corporate FI Multi-Strategy FI Convert Arbitrage Macro Merger Arbitrage Distressed/Restructuring Quantitative Directional Equity Market eutral S&P umber of assets Equal-eighted portfolio eights based on maximizing the PDI, ith Short Bias FI Corporate FI Multi-Strategy FI Convert Arbitrage Macro Merger Arbitrage Distressed/Restructuring Quantitative Directional Equity Market eutral Short Bias S&P umber of assets For Institutional/Investment Professional Use Only. ot for Distribution to the Public. 3

33 aïve Risk Parity (RP) portfolios aive Risk Parity portfolio eights based on maximizing the PDI, no Short Bias FI Corporate FI Multi-Strategy FI Convert Arbitrage Macro Merger Arbitrage Distressed/Restructuring Quantitative Directional Equity Market eutral S&P umber of assets aive Risk Parity portfolio eights based on maximizing the PDI, ith Short Bias FI Corporate FI Multi-Strategy FI Convert Arbitrage Macro Merger Arbitrage Distressed/Restructuring Quantitative Directional Equity Market eutral Short Bias S&P umber of assets For Institutional/Investment Professional Use Only. ot for Distribution to the Public. 3

34 Most Diversified Portfolios (MDP) Most Diversified Portfolios, no Short Bias FI Corporate FI Multi-Strategy FI Convert Arbitrage Macro Merger Arbitrage Distressed/Restructuring Quantitative Directional Equity Market eutral S&P umber of assets Most Diversified Portfolios, ith Short Bias FI Corporate FI Multi-Strategy FI Convert Arbitrage Macro Merger Arbitrage Distressed/Restructuring Quantitative Directional Equity Market eutral Short Bias S&P umber of assets For Institutional/Investment Professional Use Only. ot for Distribution to the Public. 33

35 Portfolio construction -- summary Equal-eighted portfolios The Short Bias Index is not included for portfolios ith less than 6 assets The PDI indicates diversification is similar for portfolios ith 5 or more assets The DR and portfolio risk both fall significantly once the Short Bias Index is included aïve Risk Parity portfolios RP eights differ considerably from equal eights Risk is much loer for portfolios ith 5 or feer holdings, even though positions are more concentrated Compared ith equal eights, DRs are loer for portfolios ith < 6 but higher for 6 or more holdings hen the Short Bias Index is included Most Diversified Portfolios Long-only MDPs are very different from maximum PDI portfolios Maximum diversification potential is reached ith 6 assets ex-short Bias and just 4 assets including Short Bias All portfolios include the Short Bias Index hen it is available for portfolio selection For Institutional/Investment Professional Use Only. ot for Distribution to the Public. 34

36 Conclusion The PDI and the DR have become contenders in the quest for achieving diversification through risk-based portfolio construction methods Both approaches can be interpreted in terms of Minimum Variance and Risk Parity investment strategies The PDI quantifies diversification specific to aïve Risk Parity portfolios The maximized DR can be found using the Minimum Correlation Portfolio There are important caveats regarding the PDI: When used for portfolio construction, equal-eighted portfolios ill not reflect the aïve Risk Parity characteristics being measured by the PDI The PDI is unable to distinguish beteen correlations of different signs, and therefore can generate identical values for very different sets of correlations For Institutional/Investment Professional Use Only. ot for Distribution to the Public. 35

37 References Ang, A. H., and J. Chen. Asymmetric Correlations of Equity Portfolios. Journal of Financial Economics, Vol. 63, o. 3 (), pp Campbell, R., K. Koedijk, and P. Kofman. Increased Correlations in Bear Markets. Financial Analysts Journal, Vol. 58, o. (), pp Cappiello, L., R.F. Engle, and K. Sheppard. Asymmetric Dynamics in the Correlations of Global Equity and Bond Returns. Journal of Financial Economics, Vol. 4, o. 4 (6), pp Choueifaty, T. Methods and Systems for Providing an Anti-Benchmark Portfolio. Unites States Patent and Trademark Office, patent number US B (6). Choueifaty, Y., and Y. Coignard. Toard Maximum Diversification. The Journal of Portfolio Management, Vol. 35, o. (8), pp Chua, D. B., M. Kritzman, and S. Page. The Myth of Diversification. The Journal of Portfolio Management, Vol. 36, o. (Fall 9), pp Clarke, R., H. de Silva, and S. Thorley. Minimum Variance Portfolios in the U.S. Equity Market. The Journal of Portfolio Management, Vol. 33, o. (6), pp. -4. Clarke, R., H. de Silva, and S. Thorley. Risk Parity, Maximum Diversification, and Minimum Variance: An Analytic Perspective. The Journal of Portfolio Management, Vol. 39, o. 3 (3), pp Crezee, D. P., and L. A. P. Sinkels. High-conviction Equity Portfolio Optimization. Journal of Risk, Vol. 3, o. (Winter /), pp Diyarbakirlioglu, E., and M. H. Satman. The Maximum Diversification Index. Journal of Asset Management, Vol. 4, o. 6 (4), pp Fragkiskos, A. What is Portfolio Diversification? State Street White Paper (September 3). Frahm, G., and C. Wiechers. On the Diversification of Portfolios of Risky Assets. University of Cologne Discussion Paper in Statistics and Econometrics, o. / (January ). Goetzmann, W.., and A. Kumar. Equity Portfolio Diversification. Revie of Finance, Vol. (8), pp Goetzmann, W.., L. Li, and K.G. Rouenhorst. Long-Term Global Market Correlations. BER Working Paper o. 86 (ovember ). Gongloff, M. Hedge Funds Kiss Their Alpha Goodbye. Wall Street Journal MarketBeat (ovember, ). Jolliffe, I.T. Principal Component Analysis, nd ed. Springer-Verlag,. Kirchner, U., and C. Zunckel. Measuring Portfolio Diversification. arxiv.org Quantitative Finance Paper o..47 (February ). Kritzman, M., and Y. Li. Skulls, Financial Turbulence, and Risk Management. Financial Analysts Journal, Vol. 66, o. 5 (), pp For Institutional/Investment Professional Use Only. ot for Distribution to the Public. 36

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