Smart Beta: Managing Diversification of Minimum Variance Portfolios

Transcription

1 Smart Beta: Managing Diversification of Minimum Variance Portfolios Jean-Charles Richard and Thierry Roncalli Lyxor Asset Management 1, France University of Évry, France Risk Based and Factor Investing Conference Imperial College, London, UK November 5, The opinions expressed in this presentation are those of the authors and are not meant to represent the opinions or official positions of Lyxor Asset Management. Jean-Charles Richard and Thierry Roncalli Smart Beta: Managing Diversification of MV Portfolios 1 / 20

2 Summary Main result The difference in ex-post performance is mainly explained by the ex-ante level of volatility reduction targeted by smart beta portfolios. The choice of the diversification metric is marginal. Two consequences: 1 Management report 2 Performance attribution Jean-Charles Richard and Thierry Roncalli Smart Beta: Managing Diversification of MV Portfolios 2 / 20

3 Risk-based portfolios Main objective Risk-based portfolios Diversification profile of risk-based portfolios The EW portfolio x i = x j Weights are equal. The ERC portfolio RC i = RC j Risk contributions are equal. The GMV portfolio min 1 2 x Σx Minimize the volatility. The MDP portfolio max x σ x Σx Maximize the diversification ratio. Jean-Charles Richard and Thierry Roncalli Smart Beta: Managing Diversification of MV Portfolios 3 / 20

4 Risk-based portfolios GMV optimization program Risk-based portfolios Diversification profile of risk-based portfolios arg min 1 2 x Σx 1 x = 1 u.c. n i=1 x2 i c 1 x 0 Euro Stoxx 50 Index One-year empirical covariance matrix February Jean-Charles Richard and Thierry Roncalli Smart Beta: Managing Diversification of MV Portfolios 4 / 20

5 Risk-based portfolios ERC optimization program Risk-based portfolios Diversification profile of risk-based portfolios arg min 1 2 x Σx 1 x = 1 u.c. n i=1 x2 i c 1 n i=1 lnx i c 2 x 0 Euro Stoxx 50 Index One-year empirical covariance matrix February Jean-Charles Richard and Thierry Roncalli Smart Beta: Managing Diversification of MV Portfolios 5 / 20

6 Risk-based portfolios MDP optimization program Risk-based portfolios Diversification profile of risk-based portfolios arg min 1 2 x Σx 1 x = 1 u.c. n i=1 x2 i c 1 n i=1 x i σ i c 3 x 0 Euro Stoxx 50 Index One-year empirical covariance matrix February Jean-Charles Richard and Thierry Roncalli Smart Beta: Managing Diversification of MV Portfolios 6 / 20

7 Risk-based portfolios Diversification profile of risk-based portfolios Diversification profile of risk-based portfolios Figure: The case of Euro Stoxx 50 Index in February 2013 Jean-Charles Richard and Thierry Roncalli Smart Beta: Managing Diversification of MV Portfolios 7 / 20

8 Mixing the constraints Mixing the constraints A unified optimization framework Families of well-defined risk-based portfolios Each risk-based portfolio is a minimum variance portfolio under a specific constraint: 1 x = 1 (GMV) n i=1 x i 2 c 1 (EW) n i=1 lnx i c 2 (ERC) n i=1 x iσ i c 3 (MDP) Mixing the constraints We can combine these different constraints to obtain better diversified risk-based portfolios. The first and fourth constraints allow the GMV portfolio and the MDP respectively to be obtained. The second and third constraints manage the diversification in terms of weights and risk contributions. Jean-Charles Richard and Thierry Roncalli Smart Beta: Managing Diversification of MV Portfolios 8 / 20

9 Mixing the constraints An example (EW MDP) Mixing the constraints A unified optimization framework Families of well-defined risk-based portfolios arg min 1 2 x Σx 1 x = 1 u.c. n i=1 x2 i c 1 n i=1 x i σ i c 3 x 0 Euro Stoxx 50 Index One-year empirical covariance matrix February Jean-Charles Richard and Thierry Roncalli Smart Beta: Managing Diversification of MV Portfolios 9 / 20

10 A unified optimization framework Mixing the constraints A unified optimization framework Families of well-defined risk-based portfolios We can write the constrained problem using Lagrange multipliers: Remark x = argmin 1 2 x Σx (1) ) λ gmv ( n i=1 λ erc ( n u.c. x 0 i=1 xi 2 i=1 x i ) + λ h ( n lnx i ) λ mdp ( n i=1 x i σ i ) The previous framework can be extended by replacing the variance minimization problem by the tracking error minimization problem. In this case, Problem (1) must include a new penalty function which is equal to: λ te ( n i=1 x i (Σx cw ) i ) = λ te β (x x cw )σ 2 (x cw ) Jean-Charles Richard and Thierry Roncalli Smart Beta: Managing Diversification of MV Portfolios 10 / 20

11 A unified optimization framework The first-order condition is: Mixing the constraints A unified optimization framework Families of well-defined risk-based portfolios L (x) x i = (Σx) i λ gmv + 2λ h x i λ erc x i λ mdp σ i λ te (Σx cw ) i = 0 The solution is the positive root of the second degree (convex) equation: x 2 i ) ( ) σ 2 i + 2λ h + xi (σ i x j ρ i,j σ j λ gmv λ mdp σ i λ te (Σx cw ) i λ erc = 0 j i We finally obtain the following CCD numerical solution: xi = λ gmv + λ mdp σ i + λ te (Σx cw ) i σ i j i x j ρ i,j σ j 2 ( ) σi 2 + 2λ h (σi ) 2 ( ) j i x j ρ i,j σ j λ gmv λ mdp σ i λ te (Σx cw ) i + 4 σ 2 i + 2λ h λerc + 2 ( ) σi 2 + 2λ h Jean-Charles Richard and Thierry Roncalli Smart Beta: Managing Diversification of MV Portfolios 11 / 20

12 A unified optimization framework Mixing the constraints A unified optimization framework Families of well-defined risk-based portfolios It is not possible to match all the diversification constraints Only a subset of Lagrange multipliers is interesting from a mathematical (and financial) point of view This is equivalent to imposing the following constrained structure: x = argmin 1 2 x Σx D (x;γ) c 1 u.c. B (x;δ) = c 2 x 0 where D (x;γ) and B (x;δ) are the diversification and budget constraints: D (x;γ) = γ B (x;δ) = δ n i=1 n i=1 lnx i (1 γ) x i + (1 δ) n i=1 n i=1 x 2 i x i σ i (ERC / EW) (GMV / MDP) Jean-Charles Richard and Thierry Roncalli Smart Beta: Managing Diversification of MV Portfolios 12 / 20

13 Mixing the constraints A unified optimization framework Families of well-defined risk-based portfolios Families of well-defined risk-based portfolios The parameter γ [0,1] controls the trade-off between weight and risk diversification whereas the parameter δ [0, 1] controls the budget allocation. We can then restrict (c 1,c 2 ) by considering this optimization problem: x (λ,γ,δ) = argmin 1 2 x Σx λd (x;γ) + (λ 1)B (x;δ) (2) u.c. x 0 where λ 0 controls the impact on the diversification. Parameters GMV EW ERC MDP RP BP λ γ 0/ δ Extension to the tracking-error volatility ( B (x;δ)). Jean-Charles Richard and Thierry Roncalli Smart Beta: Managing Diversification of MV Portfolios 13 / 20

14 Examples Mixing the constraints A unified optimization framework Families of well-defined risk-based portfolios Figure: New families of smart beta portfolios Jean-Charles Richard and Thierry Roncalli Smart Beta: Managing Diversification of MV Portfolios 14 / 20

15 No free lunch in smart beta No free lunch in smart beta Volatility reduction Ex-ante volatility reduction explains ex-post behavior Performance of smart beta portfolios Rule 1 There is no free lunch in smart beta. In particular, it is not possible to target a high volatility reduction, to be highly diversified and to take low beta risk. Figure: Relationship between the volatility reduction and the beta Jean-Charles Richard and Thierry Roncalli Smart Beta: Managing Diversification of MV Portfolios 15 / 20

16 Volatility reduction No free lunch in smart beta Volatility reduction Ex-ante volatility reduction explains ex-post behavior Performance of smart beta portfolios Rule 2 The smart beta portfolios have a time-varying objective of volatility reduction and tracking error. Figure: Boxplot of the volatility reduction (in %) Jean-Charles Richard and Thierry Roncalli Smart Beta: Managing Diversification of MV Portfolios 16 / 20

17 No free lunch in smart beta Volatility reduction Ex-ante volatility reduction explains ex-post behavior Performance of smart beta portfolios Ex-ante volatility reduction explains ex-post behavior Rule 3 When we impose the same objective of volatility reduction η, smart beta portfolios become comparable. Table: Average correlation between risk-based portfolios (in %) Index η VR TE β D w D rc D ρ R t 5% SX5E 10% % % TPX100 10% % % SPX 10% % % MXEF 10% % Average Jean-Charles Richard and Thierry Roncalli Smart Beta: Managing Diversification of MV Portfolios 17 / 20

18 No free lunch in smart beta Volatility reduction Ex-ante volatility reduction explains ex-post behavior Performance of smart beta portfolios Relationship between volatility reduction and excess return Rule 4 The performance of smart beta portfolios depends on the market risk premium. Figure: Jul Feb Figure: Mar Dec Jean-Charles Richard and Thierry Roncalli Smart Beta: Managing Diversification of MV Portfolios 18 / 20

19 Managing the trade-off between volatility reduction and diversification Dynamic smart beta strategy Managing the trade-off between volatility reduction and diversification The previous rules can be used to build dynamic smart beta strategies. When risk is perceived as high/low, we expect a lower/higher risk premium: 1 High level of volatility reduction; 2 High level of risk diversification. We link the parameter λ in Problem (2) to the market sentiment, which is approximated by the cross-section (CS) volatility: λ = 1 φ σ t cs σ + t σt σt and we impose that γ = 1 (ERC) and δ = 1 (GMV). Jean-Charles Richard and Thierry Roncalli Smart Beta: Managing Diversification of MV Portfolios 19 / 20

20 Empirical results Managing the trade-off between volatility reduction and diversification Dynamic smart beta strategy Risk-off: High σt cs λ = 0 GMV / Risk-on: Low σt cs D #1 corresponds to the case φ = 1 and λ [0,1]. D #2 corresponds to the case φ = 0.85 and λ [0.15,1]. λ = 1 ERC. Table: Comparing GMV, ERC and dynamic smart beta strategies ( ) CW GMV ERC D #1 D #2 CW GMV ERC D #1 D #2 SX5E TPX100 µ (x) (in %) σ (x) (in %) SR(x) DD (x) (in %) τ (x) SPX MXEF µ (x) (in %) σ (x) (in %) SR(x) DD (x) (in %) τ (x) Jean-Charles Richard and Thierry Roncalli Smart Beta: Managing Diversification of MV Portfolios 20 / 20