Stable linear-time optimisation in arbitrage pricing theory models
Gordon Ritter presents an explicit formula for mean-variance optimisation in the context of arbitrage pricing theory models (also called multifactor models), and related generalisations with trading costs. Its solutions are well defined and numerically stable in the presence of approximate or exact collinearity in the design matrix and the computational complexity is (manifestly) linear with respect to the number of assets
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1 Markowitz optimisation and arbitrage pricing theory 1.1 Arbitrage pricing theory.Many models for asset returns in empirical finance, following Ross (1976), assume a linear functional form:
Rt+1=Xtft+ϵt,?[ϵ]=0,?[ϵ]=D (1)where Rt+1 is an n-dimensional random vector containing the cross-section of returns in excess of the risk-free rate over some time interval [t,t+1], Xt is a (non-random) n×p matrix that is known before time t, and ϵt is assumed
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