Agnostic Risk Parity: Taming Known and Unknown-Unknowns

Abstract

Markowitz' celebrated optimal portfolio theory generally fails to deliver out-of-sample diversification. In this note, we propose a new portfolio construction strategy based on symmetry arguments only, leading to "Eigenrisk Parity" portfolios that achieve equal realized risk on all the principal components of the covariance matrix. This holds true for any other definition of uncorrelated factors. We then specialize our general formula to the most agnostic case where the indicators of future returns are assumed to be uncorrelated and of equal variance. This "Agnostic Risk Parity" (AGP) portfolio minimizes unknown-unknown risks generated by over-optimistic hedging of the different bets. AGP is shown to fare quite well when applied to standard technical strategies such as trend following.

Optimal trend following portfolios

“…To target the same unconditional risk on any eigenvector of the correlation matrix, Benichou et al proposed the agnostic risk parity portfolio [17]. This specific asset allocation allows to balance the risk between all the principal components of the correlation matrix.…”

“…where we omitted the proportionality constant (u † 1 s) 2 −1/2 . This is the agnostic risk parity portfolio [17], which differs from the naive Markowitz portfolio by the power −1/2 of the correlation matrix C. We note that the goal here was to provide a simple sufficient condition under which the agnostic risk-parity portfolio would be optimal. To get Eq.…”

“…Our goal here is to propose a minimal theoretical setting that can produce explicit, easily interpretable and practically implementable solutions of the allocation problem. In particular, we show that, under certain assumptions specified below, the optimal solution can be seen as a linear combination of four basic portfolios: risk parity, naive Markowitz solution (when expectations of returns are based on trends), trend on risk parity, and agnostic risk portfolio [17]. The choice of involving risk parity in the research design is motivated by a more efficient way of allocating assets according to their risk contribution to the portfolio because weights are proportional to inverse volatility, which seeks more or less equal risk-exposure between all the asset classes within a portfolio.…”

Data-driven integration of norm-penalized mean-variance portfolios

“…Ledoit and Wolf [49,50], and Benichou et al [8], demonstrate that when the covariance matrix is poorly conditioned, the portfolio construction process can become highly unstable, resulting in concentrated 'error-maximizing' portfolio weights. Their methods focus on providing improved covariance estimates based on James-Stein shrinkage [65], L 2 -regularization [68] and rotationally invariant estimators [17].…”

The Case for Long-Only Agnostic Allocation Portfolios

“…As mentioned in section III C, a = 1/2 corresponds to the case where the risk is allocated proportionally to the natural projection of the predictor onto the eigenmodes. In order to justify this choice, Benichou et al proposed to think about portfolio diversification in terms of symmetries rather than in terms of optimization [1]. Let us consider returns r with a covariance matrix C and predictors p with a covariance matrix Q.…”