Replica approach to mean-variance portfolio optimization

Abstract

We consider the problem of mean-variance portfolio optimization for a generic covariance matrix subject to the budget constraint and the constraint for the expected return, with the application of the replica method borrowed from the statistical physics of disordered systems. We find that the replica symmetry of the solution does not need to be assumed, but emerges as the unique solution of the optimization problem. We also check the stability of this solution and find that the eigenvalues of the Hessian are positive for r = N/T < 1, where N is the dimension of the portfolio and T the length of the time series used to estimate the covariance matrix. At the critical point r = 1 a phase transition is taking place. The out of sample estimation error blows up at this point as 1/(1 − r), independently of the covariance matrix or the expected return, displaying the universality not only of the critical exponent, but also the critical point. As a conspicuous illustration of the dangers of in-sample estimates, the optimal in-sample variance is found to vanish at the critical point inversely proportional to the divergent estimation error.

Replica Approach for Minimal Investment Risk with Cost

“…First, the portfolio of asset i(= 1, 2, · · · , N) is w i ∈ R, and the portfolio of all N assets is w = (w 1 , w 2 , · · · , w N ) T ∈ R N . The notation T indicates the transpose of a vector or matrix and, using the same setting as in previous works [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23] , the budget constraint of the portfolio w is defined as…”

“…Recently, however, such analysis of the quenched disordered system desired by rational investors in the context of diversified investment has started to be investigated using the analytical approaches developed in statistical mechanical informatics and econophysics. [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23] For instance, Ciliberti et al examined the investment risk of the absolute deviation model and the expected shortfall model in the portfolio * shinzato@eng.tamagawa.ac.jp optimization problem with a budget constraint by using replica analysis. Specifically, they analyzed the typical behavior of the ground state in the limit of absolute zero temperature (the optimal solution of the portfolio optimization problem).…”

“…They also derived the faster solver algorithm for the optimal solution using the trial distribution. 11) As described above, various investment models have been examined using replica analysis and a belief propagation method in these previous studies, [12][13][14][15][16][17][18][19][20][21][22][23] but in recent years, attention has been given to the mathematical similarity between the Hopfield model and the most representative investment models, that is mean-variance model.…”

“…Therefore, in order to provide a seamless connection between the analytical approach discussed in previous works [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23] and the analysis of utility functions, that is, as a first step of an analysis of utility functions, we examine the minimization problem of a loss function defined by two objective criteria under a budget constraint by using replica analysis. In particular, we assume the utility function of the rational investors whose hope is to reduce two negative aspects of investment, the investment risk (fluctuation risk of the held asset occurring during the investment period) and purchasing (or selling) cost (cost incurred in investing).…”

Financial Time Series Forecasting Using Empirical Mode Decomposition and Support Vector Regression

“…Future prices are difficult to predict because market imperfections are quickly discovered, exploited and corrected by market participants. Nonetheless, forecasting financial time series is a very active research area with applications spanning from trading strategies to risk management (Alexander 2001;Aymanns et al 2016;Caccioli et al 2016;Clements et al 2004;Kim 2003;Varga-Haszonits et al 2016).…”