Information-Theoretic Approaches to Portfolio Selection

Kernel Minimum Divergence Portfolios

“…Portfolio optimization techniques based on finite many moments have recently been superseded by the so-called target distribution method [7,33]. The idea of the target distribution technique is to find a portfolio weight w ∈ W ⊆ R d by minimizing the discrepancy (measured in terms of a divergence D) between the associated portfolio returns P = w ⊤ R ∼ P w and the investor's target distribution of portfolio returns P T : w * = arg min w∈W D (P w , P T ) .…”

“…to determine the optimal portfolio weights with supp(•) denoting the support of its argument. [33] specialized the φ-divergence framework to Kullback-Leibler divergence when φ(x) = x log(x) − x + 1, and assumed P T to be a generalized normal (GN) distribution with pdf…”

“…With this specific choice, one can show [33,Proposition 5.1] that (1.1) is equivalent to w * = arg min w∈W 1 2β γ E P ∼Pw [|P − α| γ ] − H(P ), (1.4) where H(P ) denotes the Shannon differential entropy of P . By further assuming that P is a mixture of Gaussian distribution (MoG) E P ∼P P [|P − α| γ ] takes a closed form expressed via the Kummer's confluent hypergeometric function [33,Proposition 5.4].…”

“…Despite the pioneering nature of these works, they unfortunately suffer from serious computational and modelling bottlenecks: [7] requires ad-hoc heuristics for encoding the constraints in (1.2), [33] needs strong parametric assumptions (GN) in addition to the moderate compatibility of the GN and MoG assumptions. It is also worth mentioning the related approach of [22] who proposed an optimal-transport based technique to reach a target terminal wealth distribution in the continuous time setting.…”

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Portfolio optimization is a key challenge in finance with the aim of creating portfolios matching the investors' preference. The target distribution approach relying on the Kullback-Leibler [33] or the f -divergence [7] represents one of the most effective forms of achieving this goal. In this paper, we propose to use kernel and optimal transport (KOT) based divergences to tackle the task, which relax the assumptions and the optimization constraints of the previous approaches.

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