Equally-weighted Risk contributions: a new method to build risk balanced diversified portfolios S´ eminaire ??? ome Teiletche ∗ S´ ebastien Maillard, Thierry Roncalli and J´ erˆ September 2008 † ∗ The respective affiliations are SGAM AI, University of Evry and SGAM AI, and University of Paris Dauphine and LODH. † These slides and the corresponding working paper may be downloaded from http://www.thierry-roncalli.com.
Agenda • The asset allocation problem • Heuristic solutions: minimum variance and 1/n portfolios • A new method: the Equal Risk Contributions (ERC) portfolio • Properties of the ERC portfolio • Some practical applications Equally-weighted risk contributions portfolios 1
1 The asset allocation problem • The classical mean-variance model • Two heuristic solutions • The minimum variance portfolio • The 1/n portfolio Equally-weighted risk contributions portfolios The asset allocation problem 1-1
1.1 The limitations of the classical mean-variance model The classical asset-allocation problem is: x ⋆ ( φ ) x ⊤ Σ x − φx ⊤ µ = arg min x u.c. 1 ⊤ x = 1 and 0 ≤ x ≤ 1 ⇒ Main difficulty is the sensitivity of the solution to small changes in the inputs. Equally-weighted risk contributions portfolios The asset allocation problem 1-2
An illustrative example (0) ( µ i , σ i ) are respectively equal to (8% , 12%), (7% , 10%), (7 . 5% , 11%), (8 . 5% , 13%) and (8% , 12%). The correlation matrix is C 5 ( ρ ) with ρ = 60%. (a) ρ = 50% (b) µ 2 = 8% (c) ρ = 50% and µ 2 = 8% (d) µ 3 = 10% ⇒ We compute optimal portfolios x ∗ such that σ ( x ∗ ) = 10% : Case (0) (a) (b) (c) (d) x 1 23.96 26.46 6.87 6.88 21.64 x 2 6.43 0.00 44.47 33.61 0.00 x 3 16.92 6.97 0.00 0.00 22.77 x 4 28.73 40.11 41.79 52.63 33.95 23.96 26.46 6.87 6.88 21.64 x 5 Equally-weighted risk contributions portfolios The asset allocation problem 1-3
Some equivalent portfolios Let us consider the case (0) and the optimal portfolio x ∗ with ( µ ( x ∗ ) , σ ( x ∗ )) = (7 . 99% , 10%). Here are some portfolios which are closed to the optimal portfolio: x 1 23.96 5 5 35 35 50 5 5 10 x 2 6.43 25 25 10 25 10 30 25 x 3 16.92 5 40 10 5 15 45 10 x 4 28.73 35 20 30 5 35 10 35 20 45 23.96 35 35 40 40 15 30 30 10 x 5 µ ( x ) 7.99 7.90 7.90 7.90 7.88 7.90 7.88 7.88 7.88 7.93 σ ( x ) 10.00 10.07 10.06 10.07 10.01 10.07 10.03 10.00 10.03 10.10 ⇒ These portfolios have very different compositions, but lead to very close mean-variance features. ⇒ Some of these portfolios appear more balanced and, in some sense, more diversified than the optimal potfolio. Equally-weighted risk contributions portfolios The asset allocation problem 1-4
Equally-weighted risk contributions portfolios The asset allocation problem 1-5
1.2 Some solutions • Portfolio resampling (Michaud, 1989) • Robust asset allocation (T¨ ut¨ unc¨ u and Koenig, 2004) ⇒ Market practice: many investors prefer more heuristic solutions, which are computationally simple to implement and appear robust as they are not dependent on expected returns. • The minimum variance (mv) portfolio It is obtained for φ = 0 in the mean-variance problem and does not depend on the expected returns. • The equally-weighted (ew or 1/n) portfolio Another simple way is to attribute the same weight to all the assets of the portfolio (Bernartzi and Thaler, 2001). Equally-weighted risk contributions portfolios The asset allocation problem 1-6
The minimum variance portfolio In the previous example, the mv portfolio is x 1 = 11 . 7%, x 2 = 49 . 1%, x 3 = 27 . 2%, x 4 = 0 . 0% and x 5 = 11 . 7%. ⇒ This portfolio suffers from large concentration in the second asset whose volatility is the lowest. The 1/n portfolio In the previous example, the ew portfolio is x 1 = x 2 = x 3 = x 4 = x 5 = 20%. ⇒ This portfolio does not take into account volatilities and cross-correlations! How to build a 1/n portfolio which takes into account risk? Equally-weighted risk contributions portfolios The asset allocation problem 1-7
2 Theoretical aspects of the ERC portfolio • Definition • Mathematical link between the ew, mv and erc portfolios • Some analytics • σ mv ≤ σ erc ≤ σ 1 / n • Numerical algorithm to compute the ERC portfolio • The existence of the ERC portfolio Equally-weighted risk contributions portfolios Theoretical aspects of the ERC portfolio 2-1
2.1 Definition � x ⊤ Σ x be the risk of the portfolio x . The Euler Let σ ( x ) = decomposition gives us: n n x i × ∂ σ ( x ) � � σ ( x ) = σ i ( x ) = . ∂ x i i =1 i =1 with ∂ x i σ ( x ) the marginal risk contribution and σ i ( x ) = x i × ∂ x i σ ( x ) the risk contribution of the i th asset. Starting from the definition of the risk contribution σ i ( x ), the idea of the ERC strategy is to find a risk-balanced portfolio such that the risk contribution is the same for all assets of the portfolio: σ i ( x ) = σ j ( x ) Remark: We restrict ourseleves to cases without short selling. Equally-weighted risk contributions portfolios Theoretical aspects of the ERC portfolio 2-2
An example Let us consider the previous case (0). We have: σ ( x ) = 9 . 5% x i ∂ x i σ ( x ) x i × ∂ x i σ ( x ) c i ( x ) 1 19.2% 0.099 0.019 20% 2 23.0% 0.082 0.019 20% 3 20.8% 0.091 0.019 20% 4 17.7% 0.101 0.019 20% 5 19.2% 0.099 0.019 20% c i ( x ) is the relative contribution of the asset i to the risk of the portfolio: c i ( x ) = σ i ( x ) /σ ( x ) x i × ∂ x i σ ( x ) = σ ( x ) Equally-weighted risk contributions portfolios Theoretical aspects of the ERC portfolio 2-3
2.2 Mathematical comparison of the ew, mv and erc portfolios • ew x i = x j • mv ∂ σ ( x ) = ∂ σ ( x ) ∂ x i ∂ x j • erc ∂ σ ( x ) ∂ σ ( x ) x i = x j ∂ x i ∂ x j ⇒ The ERC portfolio may be viewed as a portfolio “between” the 1/n portfolio and the minimum variance portfolio. Equally-weighted risk contributions portfolios Theoretical aspects of the ERC portfolio 2-4
2.3 Some analytics The correlations are the same The solution is: σ − 1 i x i = j =1 σ − 1 � n j The weight allocated to each component i is given by the ratio of the inverse of its volatility with the harmonic average of the volatilities. The volatilities are the same We have: − 1 n � x i ∝ x k ρ ik k =1 The weight of the asset i is proportional to the inverse of the weighted average of correlations of component i with other components. Equally-weighted risk contributions portfolios Theoretical aspects of the ERC portfolio 2-5
The general case We have: x i ∝ β − 1 i The weight of the asset i is proportional to the inverse of its beta. Remark 1 The solution for the two previous cases is endogeneous since x i is a function of itself directly and through the constraint that � x i = 1 . Equally-weighted risk contributions portfolios Theoretical aspects of the ERC portfolio 2-6
2.4 The main theorem ( σ mv ≤ σ erc ≤ σ 1 / n ) Consider the following optimization problem: � x ⋆ ( c ) x ⊤ Σ x = arg min � n i =1 ln x i ≥ c 1 ⊤ x = 1 u.c. 0 ≤ x ≤ 1 Notice that if c 1 ≤ c 2 , we have σ ( x ⋆ ( c 1 )) ≤ σ ( x ⋆ ( c 2 )). Moreover, we have x ⋆ ( −∞ ) = x mv and x ⋆ ( − n ln n ) = x 1 / n . The ERC portfolio corresponds to a particular value of c such that −∞ ≤ c ≤ − n ln n . We also obtain the following inequality: σ mv ≤ σ erc ≤ σ 1 / n ⇒ The ERC portfolio may be viewed as a form of variance-minimizing portfolio subject to a constraint of sufficient diversification in terms of component weights. Equally-weighted risk contributions portfolios Theoretical aspects of the ERC portfolio 2-7
2.5 Another result Let us consider the minimum variance portfolio with a constant correlation matrix C n ( ρ ). The solution is: � − 1 − (( n − 1) ρ + 1) σ − 2 � + ρ � n σ i σ j i j =1 x i = � − 1 � � − (( n − 1) ρ + 1) σ − 2 � � n + ρ � n σ k σ j k =1 j =1 k The lower bound of C n ( ρ ) is achieved for ρ = − ( n − 1) − 1 . and we have: � − 1 � � n σ − 1 σ i σ j j =1 i x i = � − 1 = → erc k =1 σ − 1 � n � � n � n σ k σ j k j =1 k =1 Equally-weighted risk contributions portfolios Theoretical aspects of the ERC portfolio 2-8
2.6 Numerical algorithm to compute the ERC portfolio The ERC portfolio is the solution of the following problem: x ⋆ = x ∈ [0 , 1] n : x ⊤ 1 = 1 , σ i ( x ) = σ j ( x ) for all i, j � � If the ERC portfolio exists, there exists a constant c > 0 such that: x ⊙ Σ x = c 1 u.c. 1 ⊤ x = 1 and 0 ≤ x ≤ 1 We may show that solving this problem is equivalent to solving the following non-linear optimisation problem: x ⋆ = arg min f ( x ) u.c. 1 ⊤ x = 1 and 0 ≤ x ≤ 1 with: n n x 2 i (Σ x ) 2 � � f ( x ) = n x i x j (Σ x ) i (Σ x ) j i − i =1 i,j =1 ⇒ numerical solution may be obtained with a SQP algorithm. Equally-weighted risk contributions portfolios Theoretical aspects of the ERC portfolio 2-9
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