Variance Allocation and Shapley Value

TL;DR

This paper introduces a simple, computationally efficient way to allocate variance and standard deviation among dependent variables using the Shapley value, inspired by portfolio optimization.

Contribution

It derives a simplified form of the Shapley value for variance allocation in dependent variables, making it easier to compute in portfolio contexts.

Findings

Derived a simple formula for the Shapley value in variance allocation

Extended the approach to standard deviation allocation

Formulated a conjecture relating the two allocation methods

Abstract

Motivated by the problem of utility allocation in a portfolio under a Markowitz mean-variance choice paradigm, we propose an allocation criterion for the variance of the sum of $n$ possibly dependent random variables. This criterion, the Shapley value, requires to translate the problem into a cooperative game. The Shapley value has nice properties, but, in general, is computationally demanding. The main result of this paper shows that in our particular case the Shapley value has a very simple form that can be easily computed. The same criterion is used also to allocate the standard deviation of the sum of $n$ random variables and a conjecture about the relation of the values in the two games is formulated.