On Entropy, Divergence and Portfolio Diversification

On Entropy, Divergence and Portfolio Diversification Hellinton H. Takada 1,2 Julio M. Stern 1 1 Department of Applied Mathematics, Institute of Mathematics and Statistics, University of São Paulo, Brazil 2 Quantitative Research, Itaú Asset Management, Banco Itaú-Unibanco S.A., Brazil 36th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering Takada & Stern (University of São Paulo) Portfolio Diversification MaxEnt2016 1 / 34

Outline 1 Introduction 2 Objective 3 Risk Diversification 4 Factor Risk Parity Portfolios 5 Factor Risk Budgeting Portfolios 6 Final Comments 7 References Takada & Stern (University of São Paulo) Portfolio Diversification MaxEnt2016 2 / 34

Introduction Asset Allocation Conceptually, asset allocation is an investment strategy and consists in determining the proportion of each asset class or investment factor in the portfolio. There are some different asset allocation methodologies using entropy measures and divergences to ensure diversification [1] [13]. Takada & Stern (University of São Paulo) Portfolio Diversification MaxEnt2016 3 / 34

Introduction Diversification Diversification is a risk management approach to smooth out unsystematic risks from the portfolio. Entropy is an accepted measure of diversity. Actually, the greater the level of entropy, the higher the degree of portfolio diversification. The concept of diversification has also slight variations depending on the context. Takada & Stern (University of São Paulo) Portfolio Diversification MaxEnt2016 4 / 34

Diversification Approaches Introduction There are two distinct contexts: the allocation weights-based approaches [1] [9][12][13] and the risk-based approaches [10][11]. Takada & Stern (University of São Paulo) Portfolio Diversification MaxEnt2016 5 / 34

Introduction Allocation Weights-Based Approaches The idea of diversification is related to the allocation weights over the available asset classes or investment factors. The unrestricted maximum diversification is achieved when the set of allocation weights is given by a uniform distribution. The related methodologies are derived from the Markowitz s [14] mean-variance. Mean? Variance? Bera and Park (2005) [2] introduced both the maximization of the Shannon entropy and the minimization of the Kullback-Leibler divergence to achieve such diversified allocation weights. Takada & Stern (University of São Paulo) Portfolio Diversification MaxEnt2016 6 / 34

Introduction Allocation Weights-Based Approaches Mean-Variance Asset Allocation Considering a portfolio of n risky assets, an investor needs to find the allocation weights w from the following optimization problem: max E U (w, R, λ) = max E [ w R ] (λ/2) Var [ w R ], (1) w w s.t. : w 1 n = 1, (2) where R represents the excess returns of the risky assets, λ 0 is the risk aversion parameter of the investor and U ( ) a utility function (the expected utility function adopted is exact for elliptical distributions of excess returns). Takada & Stern (University of São Paulo) Portfolio Diversification MaxEnt2016 7 / 34

Introduction Allocation Weights-Based Approaches Entropy Approach for Diversification Considering a portfolio of n risky assets, an investor needs to find the allocation weights w from the following optimization problem to achieve diversification: n max w i log (w i ), (3) w i=1 s.t. : w 0, w 1 n = 1, w Σw σ 0, w r r 0, (4) where r is the estimated mean and Σ is the estimated covariance of excess returns of the n risky assets. In addition, it is not allowed to short assets and to leverage the portfolio. Finally, σ 0 is the maximum desired risk and r 0 is the minimum desired excess return for the portfolio. Takada & Stern (University of São Paulo) Portfolio Diversification MaxEnt2016 8 / 34

Introduction Allocation Weights-Based Approaches Divergence Approach for Diversification The Kullback-Leibler divergence KL ( ) is used as the objective function: n min KL (w q) = min w i log (w i /q i ), (5) w w s.t. : w 0, w 1 n = 1, i=1 w Σw σ 0, w r r 0, (6) where q is a predefined portfolio. Usually, it is considered q i = 1/n to ensure portfolio diversification. In addition, it is not allowed to short assets and to leverage the portfolio. Again, σ 0 is the maximum desired risk and r 0 is the minimum desired excess return for the portfolio. Takada & Stern (University of São Paulo) Portfolio Diversification MaxEnt2016 9 / 34

Introduction Risk-Based Approaches The idea of diversification is related to the risk contribution of each available asset class or investment factor to the total portfolio risk. The maximum diversification or the risk parity allocation is achieved when the set of risk contributions is given by a uniform distribution. Meucci (2009) [10] introduced the maximization of the Rényi entropy as part of a leverage constrained optimization problem to achieve such diversified risk contributions. Takada & Stern (University of São Paulo) Portfolio Diversification MaxEnt2016 10 / 34

Introduction Risk-Based Approaches Risk Parity After the subprime crisis, the risk tolerance of investors decreased and risk-based allocation methodologies have arisen with the idea of combining both risk management and asset allocation. A risk-based allocation seeks for risk diversification and does not use performance forecasts of assets as inputs of the methodologies. Risk parity is the most widely used risk-based allocation methodology and has been used by several fund managers. Starting in 1996, one of the pioneers to use risk parity was the All Weather hedge fund from Bridgewater. The risk parity portfolio has been extensively studied in terms of its properties (Maillard et al., 2010) [18] and compared to other asset allocation heuristics (Chaves et al., 2011) [19]. Takada & Stern (University of São Paulo) Portfolio Diversification MaxEnt2016 11 / 34

Introduction Risk-Based Approaches Risk Budgeting Risk budgeting is a generalization of risk parity. A theoretical and applied study about risk budgeting techniques is presented by Bruder and Roncalli (2013) [22]. An example of risk budgeting was investigated by Bruder et al. (2011) [21] where they have studied a portfolio in which the risk contribution of each sovereign bond from a set of countries is done proportional to the gross domestic product (GDP) of the respective country. In the literature, there are some proposed optimization frameworks to obtain the risk budgeting and, consequently, risk parity portfolios [22][23]. Takada & Stern (University of São Paulo) Portfolio Diversification MaxEnt2016 12 / 34

Introduction Risk-Based Approaches Optimization Frameworks The existence and uniqueness (for some cases) of the solution for the optimization problems is proved for particular cases with the imposition of some restrictions such as long-only allocations [22][23], impossibility of leveraging the portfolio [22] and an upper bound for portfolio volatility [23]. In this work, these restrictions are relaxed. The objective functions used in the optimizations from [22][23] do not possess a strong theoretical basis related to the idea of maximizing risk diversification. However, Meucci (2009) [10] introduced a measure called effective number of bets ENB α based on Rényi entropy with parameter α. The maximization of the objective function ENB α brings the idea of maximizing risk diversification. Takada & Stern (University of São Paulo) Portfolio Diversification MaxEnt2016 13 / 34

Introduction Risk-Based Approaches Factors or Assets? In [11][26][27], they have applied principal component analysis (PCA) to extract uncorrelated factors and analyze the performance of the called factor risk parity (FRP) portfolios. Deguest et al. (2013) [11] have developed an optimization framework for FRP portfolios using the concept of ENB α. Their approach has a leverage restriction and does not include the most general risk budgeting case. Takada & Stern (University of São Paulo) Portfolio Diversification MaxEnt2016 14 / 34

Objective Objective A generalization of the risk parity is the risk budgeting when there is a prior for the distribution of the risk contributions. In this presentation, we generalize the existent optimization frameworks to be able to solve the risk budgeting problem. In addition, we do not impose any leverage constraint. Instead of working with assets, we are going to work with orthogonal factors. Takada & Stern (University of São Paulo) Portfolio Diversification MaxEnt2016 15 / 34

Risk Diversification Risk Diversification The risk-based allocations depend on some measure of each individual asset, asset class or investment factor risk contribution to the total portfolio risk. Definition 1. (relative marginal contribution for assets) Considering n available risky assets, the relative marginal contribution of each asset to the total portfolio risk is given by the following n 1 vector: p := diag(w)σw w Σw, (7) where Σ is the n n covariance matrix of risky assets excess returns, w is the vector of weights such that w R n 1 0 n 1 and diag(w) is a diagonal matrix with w as its diagonal. w 0 n 1 because w = 0 n 1 represents the absence of allocation in the risky assets and, then, it is not a desired solution. Finally, it is clear that n i=1 p i = 1. Takada & Stern (University of São Paulo) Portfolio Diversification MaxEnt2016 16 / 34

Risk Diversification Risk Diversification Meucci (2009) [10] proposed the use of the Rényi entropy with parameter α as a measure of diversification and the measure was called the effective number of bets ENB α. Instead of using assets or asset classes, ENB α uses the distribution of the relative marginal risk contribution of each uncorrelated factor as a measure of risk diversification. In practical terms, the uncorrelated factors can be obtained using principal component analysis (PCA). Considering T excess returns of the risky assets represented by the n T matrix r, the n T matrix of uncorrelated factors is given by r F = A r (8) and the corresponding covariance matrix is given by Σ F = A ΣA, (9) where Σ F is a positive definite n n diagonal matrix and A is an invertible n n matrix. Takada & Stern (University of São Paulo) Portfolio Diversification MaxEnt2016 17 / 34

Risk Diversification Risk Diversification We define the relative marginal contribution of each uncorrelated factor to the total portfolio risk analogously to (7). Definition 2. (relative marginal contribution for uncorrelated factor) Considering n available risky assets, the relative marginal contribution of each uncorrelated factor to the total portfolio risk is given by the following n 1 vector: p F := diag(w F )Σ F w F w F Σ F w F. (10) where Σ F is the n n diagonal covariance matrix (9), w F is the vector of weights such that w F R n 1 0 n 1 and diag(w F ) is a diagonal matrix with w F as its diagonal. It is important to notice that n i=1 p F,i = 1 and p F,i 0, i = 1,..., n. Takada & Stern (University of São Paulo) Portfolio Diversification MaxEnt2016 18 / 34

Risk Diversification Risk Diversification Using (10), the ENB α ( ) is defined in the following. Definition 3. (effective number of bets) The effective number of bets of portfolio w of n risky assets is given by where the α is the α-norm. α 1 α ENB α (w) := p F α, α 0, α 1, (11) Takada & Stern (University of São Paulo) Portfolio Diversification MaxEnt2016 19 / 34

Risk Diversification Risk Diversification By definition, Then, ENB α (w) = p F α = ( n i=1 ( n p α F,i i=1 p α F,i ) 1 1 α ) 1 α. (12), α 0, α 1. (13) It is important to notice that log(enb α (w)) is the Rényi entropy H α (p F ). In addition, we write ENB α as a function of w since p F is a function of w F (see (10)) and w F is a function of w (see the next property). Takada & Stern (University of São Paulo) Portfolio Diversification MaxEnt2016 20 / 34

Risk Diversification Risk Diversification Property 1. w F = A 1 w. Proof of Property 1. The excess return of the portfolio in terms of assets or uncorrelated factors is the same: w r = w F r F. Using (8), w r = w F A r w = Aw F. Finally, w F = A 1 w. The ENB α measure achieves its minimum equal to 1 when the portfolio is risk concentrated in only one factor. On the other hand, the ENB α measure achieves its maximum when the portfolio is totally risk diversified with p F,i = 1/n, i = 1,..., n. Takada & Stern (University of São Paulo) Portfolio Diversification MaxEnt2016 21 / 34

Risk Diversification Risk Diversification The following property concerning ENB α is presented because it will be useful in the following slides... Property 2. ENB α (λw) = ENB α (w), λ R 0. Proof of Property 2. It is straightforward to state that Using Property 1, ENB α (λw) = ENB α (p F (w F (λw))). (14) ENB α (λw) = ENB α (p F (λa 1 w)). (15) Takada & Stern (University of São Paulo) Portfolio Diversification MaxEnt2016 22 / 34

Risk Diversification Risk Diversification Additionally, using (10), it is trivial to see that p F (λw F ) = p F (w F ), λ R 0. (16) Consequently, ENB α (λw) = ENB α (p F (A 1 w)) = ENB α (p F (w F )) = ENB α (p F (w F (w))), λ R 0. ENB α (λw) = ENB α (w), λ R 0. (17) Takada & Stern (University of São Paulo) Portfolio Diversification MaxEnt2016 23 / 34

Factor Risk Parity Portfolios Factor Risk Parity Portfolios The factor risk parity (FRP) portfolios were defined by Deguest et al. (2013) [11] using the ENB α ( ) measure. Since the FRP portfolios from [11] were developed under a leverage restriction, we are going to refer to them as restricted FRP (RFRP) portfolios. Using the ENB α ( ) measure, the optimization problem from [11] to obtain the RFRP portfolios is max w ENB α(w), (18) s.t. 1 nw = 1, (19) where 1 n is a n 1 vector of ones. It is important to notice that the restriction in (18) is a non-leverage constraint or budget condition. Takada & Stern (University of São Paulo) Portfolio Diversification MaxEnt2016 24 / 34

Factor Risk Parity Portfolios Factor Risk Parity Portfolios Theorem 1. (RFRP portfolios) The family of RFRP portfolios is given by 1 w RF RP = AΣ 2 F ı n, (20) 1 naσ 1 2 F ı n where ı n := ( ±1 ±1 ) is a vector of size n 1 representing all the combinations of ±1. Proof of Theorem 1. It was shown in [11] that the closed-form expression for the solutions of (18) is given by (20). It is important to notice that (20) implies in 2 n 1 possible solutions (for details, see [11]). Additionally, the portfolios with only positive signs coincide with the solutions provided by [18]. Takada & Stern (University of São Paulo) Portfolio Diversification MaxEnt2016 25 / 34

Factor Risk Parity Portfolios Factor Risk Parity Portfolios As we have already mentioned, the non-leverage constraint in (18) is easily relaxed. Our optimization problem to obtain the generalized factor risk parity (GFRP) portfolios is given by max w ENB α(w). (21) Theorem 2. (GFRP portfolios) The family of GFRP portfolios is given by w GF RP = λ AΣ F 1 2 ı n 1 naσ 1 2 F ı n, λ R 0. (22) Takada & Stern (University of São Paulo) Portfolio Diversification MaxEnt2016 26 / 34

Factor Risk Parity Portfolios Factor Risk Parity Portfolios Proof of Theorem 2. It is straightforward to prove (22). Considering a modified version of the optimization problem (18): w λ = arg max w ENB α(w), (23) s.t. 1 nw = λ, λ R 0, (24) the solution using Property 2 is w λ = w RF RP λ. Consequently, it is necessary to solve the problem for each λ R 0 to obtain (22). Takada & Stern (University of São Paulo) Portfolio Diversification MaxEnt2016 27 / 34

Factor Risk Parity Portfolios Factor Risk Parity Portfolios Since the volatility of the portfolio is given by σ RF RP := w RF RP Σw RF RP, the volatility of w GF RP (λ) is σ GF RP (λ) = λ σ RF RP. It is important to notice that unconstraining the problem, it is possible to obtain risk parity portfolios for any desired volatility or, in other words, risk level. Consequently, our GFRP portfolios are important in terms of asset allocation because it will adapt better to the investor s risk preference than the RFRP portfolios. Takada & Stern (University of São Paulo) Portfolio Diversification MaxEnt2016 28 / 34

Factor Risk Budgeting Portfolios Factor Risk Budgeting Portfolios We generalize the FRP to the factor risk budgeting (FRB). Our optimization problem to obtain the FRB portfolios is given by min w D α (p F b), (25) where b is a n 1 vector containing the budgets or targets b i, i = 1,..., n such that n i=1 b i = 1 and b i 0, and D α (p F b) is the Rényi divergence or α-divergence given by ( n ) D α (p F b) = 1 α 1 log p α F,i b α 1, (26) i where 0 < α < and α 1. Considering α 1, the Rényi divergence becomes the Kullback-Leibler divergence. i=1 Takada & Stern (University of São Paulo) Portfolio Diversification MaxEnt2016 29 / 34

Factor Risk Budgeting Portfolios Factor Risk Budgeting Portfolios Obviously, b works like a probability mass distribution and represents a prior to the risk allocation process. Additionally, it is trivial to prove that when b is uniformly distributed the optimization problem (25) reduces to max w ENB α (w) (22). Theorem 3. (FRB portfolios) The family of FRB portfolios is given by 1 w F RB = λ AΣ 2 F ı n b 1 2, λ 1 naσ 1 R 0, (27) 2 F ı n b 1 2 where is the Hadamard product and is the Hadamard power. Takada & Stern (University of São Paulo) Portfolio Diversification MaxEnt2016 30 / 34

Factor Risk Budgeting Portfolios Factor Risk Budgeting Portfolios Proof of Theorem 3. Since D α (p F b) is a divergence, D α (p F b) is minimum and equal to zero when p F = b. Consequently, Then, p F,k = b k (σ F,kw F,k ) 2 w F Σ F w F = b k, k = 1,..., n. (28) w F,k = ± w F Σ F w F σ F,k bk, k = 1,..., n. (29) Using matrix notation, the factor weights are given by: w F = w F Σ F w F Σ 1 2 F ı n b 1 2. (30) Takada & Stern (University of São Paulo) Portfolio Diversification MaxEnt2016 31 / 34

Factor Risk Budgeting Portfolios Factor Risk Budgeting Portfolios Considering the relation w = Aw F (see Property 1) and the restriction 1 nw = 1, it is possible to obtain the following expression for leveragerestricted factor weights: w RF RB = 1 AΣ 2 F ı n b 1 2 1 naσ 1 2 F ı n b 1 2. (31) Using the same argument from Proof of Theorem 2, we obtain (27). Takada & Stern (University of São Paulo) Portfolio Diversification MaxEnt2016 32 / 34

Factor Risk Budgeting Portfolios Factor Risk Budgeting Portfolios Considering that σ RF RB := w RF RB Σw RF RB, the volatility of w F RB (λ) is given by σ F RB (λ) = λ σ F RB (1). It is important to point that the FRB portfolios achieve any desired volatility or, in other words, risk level. Consequently, our FRB portfolios are flexible because it will adapt to the investor s risk preference when varying λ. Takada & Stern (University of São Paulo) Portfolio Diversification MaxEnt2016 33 / 34

Final Comments Final Comments We organize the existent literature on portfolio diversification using entropies and divergences. In terms of theoretical contributions, we generalize the existent optimization framework to obtain risk parity portfolios based on Rényi entropy taking out the leverage constraint. In addition, we give the analytical solutions to the risk parity unconstrained optimization problem. It is important to point that unconstraining the problem, we obtain risk parity portfolios for any desired volatility or, in other words, risk level. Consequently, our risk parity portfolios adapt better to the investor s risk preference. Finally, we generalize the risk parity optimization framework based on Rényi entropy introducing the risk budgeting optimization framework based on Rényi divergence. We also give the analytical solutions to the risk budgeting optimization problem. Takada & Stern (University of São Paulo) Portfolio Diversification MaxEnt2016 34 / 34

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