From Asset Allocation to Risk Allocation

Transcription

1 EDHEC-Princeton Conference New-York City, April 3rd, 03 rom Asset Allocation to Risk Allocation Towards a Better Understanding of the True Meaning of Diversification Lionel Martellini Professor of inance, EDHEC Business School Scientific Director, EDHEC-Risk Institute Based on joint work with R. Deguest and A. Meucci

2 Outline Meanings of Diversification Measuring Diversification Effective Number of Bets Managing Diversification actor Risk Parity Portfolios Conclusion & Extensions

3 Meanings of Diversification Measuring Diversification Effective Number of Bets Managing Diversification actor Risk Parity Portfolios Conclusion & Extensions 3

4 Meanings of Diversification While the benefits of diversification are intuitively clear, what exactly is a well-diversified portfolio remains a bit mysterious. The most common intuitive explanation of diversification is that it is the practice of not putting all your eggs in one basket. Having eggs (dollars) spread across many baskets is, however, a rather loose prescription: true meaning of baskets (and many )? On the other hand, a fully unambiguous, yet not fully operational, definition of diversification has been provided by modern portfolio theory: a well-diversified portfolio is a high Sharpe ratio portfolio. Naïve and scientific diversification as non competing approaches: solutions to parameter uncertainty anchor scientific diversification to some (relevant?) notion of naive diversification (/N portfolio). 4

5 Meanings of Diversification Measuring Diversification Effective Number of Bets Managing Diversification actor Risk Parity Portfolios Conclusion & Externsions 5

6 True Meaning of Many Effective Number of Baskets Looking at the nominal number of constituents in a portfolio is a poor indication of how well-diversified a portfolio is. Introduce a measure for effective number of constituents (ENC) as the entropy of the weight distribution (maximized for EW portfolio): N ENC exp wi ln w i Max ENC wi i i N N ( w ),..., i 6

7 Shortcomings of ENC Measures The ENC measure can be deceiving when applied to assets with non homogenous risks: 50% in a % vol bond, 50% in (uncorrelated) 30% vol stock. Weights are highly diversified, but risk is highly concentrated. This is due to differences in vol levels: (50%) x(30%) >>(50%) x(%) Risk budgeting: measure the dispersion of risk (vs. $) contributions. It can also be deceiving for homogenous but correlated risks: 50%/50% in two bonds with similar duration and volatility. Weights highly diversified and risk homogenously distributed. Risk is still highly concentrated because of the high correlation. How to fix this additional problem? (*) or uncorrelated assets, p i is the contribution of asset i to portfolio risk; the focus is on the variance concentration curve as opposed to weight concentration curve. 7

8 True Meaning of Baskets Uncorrelated actors Decompose the portfolio return as the sum of N uncorrelated factors: 0 0 N N rp wi ri w r ' ' with 0 0 k k P w Σ w w Σ w Σ O i k 0 0 N Definition Introduce the effective number of (uncorrelated) bets (baskets), or ENB, as the dispersion (entropy) of the factor exposure distribution (Meucci (00)): N w k k ENB exp pk ln pk with pk k p where p k is the contribution of factor k to portfolio risk. 8

9 Commodity risk International RE risk equity risk Credit risk Equity risk Interest rate risks Illustration Consider a set of asset classes, and turn their correlated returns into uncorrelated factors using PCA. Asset Class P P P3 P4 P5 P6 P7 T-Bonds -.94% -0.0%.30% 4.59% 48.4% -4.89% 76.06% Corp. Bonds 0.0% 0.8% -0.66% 8.77% 56.3% % % US Equities 43.6% -.95% -3.73% -33.7% 53.74% 53.75% 0.36% Ex US Eq. 4.0% 6.67% % 70.8% -4.43% -7.3% 3.39% Private Eq % -4.3% -3.4% -53.5% % % 0.5% Real Estate 5.78% -6.89% 74.4% 9.4% 5.4% 9.7% 0.7% Commodities 5.6% 94.9% 8.86% -8.84% 3.68%.49% 0.9% Total Position in Risky Assets 4.9% 48.3% -5.86% 8.3% 5.99% % 6.49% Explained Variance 60.84% 0.46% 9.43% 4.94%.37%.8% 0.5% Cumulative 60.84% 8.30% 90.73% 95.67% 98.04% 99.85% 00% 9

10 Allocation in Terms of actor Exposure Asset Exposure T-Bonds 4% Corp. Bonds 6% US Equities 5% Ex US Eq. 5% Private Eq. 3% Real Estate 3% Commodities 4% Total Position in Risky Assets 00% Current portfolio is almost exclusively (96.69%!) exposed to factor (equity risk). As a result, the portfolio is not well diversified in terms of factor allocation: ENB.0/7 7.4% ENC 5.90/7 84.9% P P P3 P4 P5 P6 P7 actor exposure 36.0% -.84% -.97% 7.5% 6.79% -3.48% -6.% Variance decomposition 96.69% 0.0%.9% 0.34% 0.8% 0.03% 0.0% 0

11 Meanings of Diversification Measuring Diversification Effective Number of Bets Managing Diversification actor Risk Parity Portfolios Conclusion & Extensions

12 Managing Diversification RP The analysis can be used not only to measure, but also to manage, diversification within a policy portfolio. Define an factor risk parity (RP) portfolio by choosing factor exposures w k k (or equivalently weights w i) so as to maximize ENB, the effective number of bets (versus ENB), i.e., so that the contributions p k of all factors to portfolio risk are equal (to /N). Proposition the weight allocated to each factor k in a RP portfolio is proportional to the inverse of its volatility: ( w ) k k for each constituent k w M or w A M 443 w ' Σw N pk N N

13 Problem # with RP Relationship with MPT Modern portfolio theory stipulates that the optimal portfolio is the Maximum Sharpe ratio (MSR) portfolio.(*) The weight allocated to factor k is proportional to its Sharpe ratio divided by volatility: w Σ MSR Σ ( µ r ) λ MSR k k RP λ 0 O 0 w k N 0 0 λ l l N l Σ 0 0 w w if λ for all k µ ( r λ ) K λ N N k k k Proposition The RP and MSR portfolios coincide under the (agnostic) assumption that all factors have the same Sharpe ratio. (*) The MSR portfolio obtained from uncorrelated factors (subject to weight constraint) is identical to the MSR portfolio obtained from correlated asset returns (subject to the same constraints). 3

14 Relationship with MPT GMV Another agnostic (?) set of priors consists in assuming that all factors have the same expected returns, as opposed to the same Sharpe ratio; in this case, the MSR portfolios coincides with the GMV portfolio, which stipulates to choose factor exposures w k (or equivalently weights w i ) so as to minimize portfolio risk. The weight allocated to each factor k is proportional to the inverse of its variance (vs. proportional to the inverse of its volatility for RP portfolios): GMV w Σ 0 0 GMV k w k N Σ 0 0 O l 0 0 l N (*) It can be shown that the GMV portfolio obtained from factors (subject to weight constraints at the underlying asset class level) is identical to the GMV portfolio obtained from asset returns (subject to the same constraints). 4

15 Problem # with RP Unicity Assuming N uncorrelated assets (which implies AI), we obtain RP solutions (to be compared with GMV and MSR portfolios): RP w w :, + + RP w w :, GMV : w, w + + λ MSR : w, w λ + λ λ + λ λ Both RP solutions lead to p p /, which shows that both portfolios achieve the maximum ENB; on the other hand, they are not born equal in terms of variance ( RP < RP ), or Sharpe ratio (λ RP >λ RP ): λ + λ RP and λ RP ( + ) λ λ RP and λ RP ( ) λ + λ GMV and λ GMV + + λ + λ λ + λ MSR and λ MSR ( λ + λ ) λ + λ 5

16 Selecting the One RP Portfolio Two natural approaches to the selection of one amongst all ( N- ) possible RP portfolios: choose the one with lowest variance (RP GMV ), and/or the one with highest Sharpe ratio (RP MSR ). Proposition RP GMV : The sign of entry k is the same as the sign of the net exposure of factor k to the N asset classes (e.g., if factor k is net long in the constituents, then we choose +/ k ). RP MSR : The sign of entry k is the same as the sign of the Sharpe ratio of factor k (e.g. if factor k has a positive Sharpe ratio, then we choose +/ k ). Key insight: for the MSR portfolio subject to ENBN, one only needs to know the sign (and not absolute value) of factor premia. 6

17 Competing Strategies with Shortsale Constraints Current M EW GMV RP-GMV Av. Returns 7.8% 7.87% 6.57% 7.66% Volatility 3.3%.0% 4.3% 5.04% Sharpe ratio Max Drawdown 49.53% 44.59% 5.93%.00% VaR at 5%.7%.9% 0.78% 0.96% VaR at % 5.76% 4.48%.38%.69% Av. ENC Av. ENB On this sample period, RP portfolio shows higher ENB (<7 because of long-only constraints), improved risk levels with reasonably strong performance, resulting in attractive Sharpe ratio. 7

18 GMV Strategies under ENB Constraints EW GMV RP-GMV GMV-ENC4 GMV-ENB4 Av. Returns 7.87% 6.57% 7.66% 7.3% 7.04% Volatility.0% 4.3% 5.04% 5.50% 4.5% Sharpe ratio Max Drawdown 44.59% 5.93%.00%.% 6.05% VaR at 5%.9% 0.78% 0.96%.08% 0.79% VaR at % 4.48%.38%.69%.88%.4% Av. ENC Av. ENB We can perform mean-variance analysis (GMV, MSR) with a given target/constraint on ENB; this is somewhat similar to portfolio optimization with norm ENC constraints (DeMiguel et al. (009)), except that we use an arguably better measure of intuitive diversification (shrinkage interpretation for inputs -covariance matrix- also exists). 8

19 Meanings of Diversification Measuring Diversification Effective Number of Bets Managing Diversification actor Risk Parity Portfolios Conclusion & Extensions 9

20 Conclusion & Extensions It is being increasingly recognized that it is of high relevance to frame asset allocation decisions in terms of risk factors. Like many risk practitioners, ATP follows a portfolio construction methodology that focuses on fundamental economic risks [ ]. The strategic risk allocation is 35% equity risk, 5% inflation risk, 0% interest rate risk, 0% credit risk and 0% commodity risk. (H. G. Jepsen, CIO ATP, 0). These questions can be addressed within a formal framework, which can/should be extended in a number of directions: PCA is one possible approach to extracting factors but other approaches (explicit factors, minimal linear torsion) can be used to address the concern over factor interpretation and stability. Other possible improvements consist in (i) making the risk budgeting approach react to market conditions (e.g., bond yields) and (ii) extend to risk measures beyond portfolio volatility. 0