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Second Order Conic Programming Approach for Equal Risk Contribution Portfolios

IP.com Disclosure Number: IPCOM000239094D
Publication Date: 2014-Oct-10
Document File: 6 page(s) / 192K

Publishing Venue

The IP.com Prior Art Database

Abstract

Disclosed is a new formulation of the Equal Risk Contribution (ERC) problem, related to portolio management,which can be solved with a computationally robust Second Order Conic Programming (SOCP) algorithm.

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Second Order Conic Programming Approach for Equal Risk Contribution Portfolios

Consider a portfolio x comprising positions in a set of N assets, where xi is the weight of asset i . Let Q be the covariance matrix of the asset returns and let denote the (convex) set of all feasible portfolio weights. We assume that x in satisfies the following linear constraints:

Let R (x ) be some measure of the risk of portfolio x and define Ci (x ) to be the risk contribution of asset i . By definition, it holds that

. If the portfolio's risk is measured by the variance of its return then and

, where

. Similarly, using the standard deviation of the return as the risk measure results in

and

.

if and only if

         , it follows that the ERC portfolios for the variance and standard deviation risk measures are the same . Hereafter, we consider only the variance risk measure, recognizing that all results apply equally to standard deviation.

If M = 0 then a unique ERC portfolio exists, and it can be found by solving

1

An Equal Risk Contribution(ERC) portfolio x* satisfies for i = 1, ..., N . Since


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for any constant c to obtain y* and then normalizing , i = 1, ..., N . The Karush-Kuhn-Tucker (KKT) conditions imply that

the optimal solution to Problem 1satisfies for i = 1, ..., N , where is the Lagrange multiplier of the

constraint.

In practice, investment managers typically face constraints on the composition of their portfolios (beyond simply the exclusion of short positions as in (ii)). Since the ERC portfolio may not be feasible in this case, when M > 0 the goal is to find x' in that is "as close as possible" to x*. In light of this, one can let D (x ) measure the discrepancy between x and x* , and solve

Two forms of D (x ) have been proposed in the literature:


1. The sum of the squared differences between the contributions of all pairs of assets , i.e.,

2


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2. The sum of the squared differences between each asset's risk contribution and some (variable) target contribution τ, i.e.,

Observe that for the ERC portfolio D (x* ) = 0 in both Equations 3and 4, where in the latter case.

Current practice is to solve either Problem 1or Problem 2using general non-linear programming (NLP) algorithms. Given the numerical complexity of NLP, it can be difficult to obtain solutions, particularly when N is large (Maillard et al. (2010), Hallerbach (2013)). Simplified methods have been proposed for some special cases, such as when the portfolio weights are unconstrained (Chaves et al. (2012)) or when the asset returns derive from a linear factor model (Clarke et al. (2013)).

In fact, certain classes of convex non-linear optimization problems can be solved effectively by specialized convex optimization algorithms. In order of increasing solution difficulty, these classes include quadratic programs (QP), second order conic programs (SOCP) and semi-definite programs (SDP).

It is possible to formulate Problem 1and a modified version of Problem 2...