A Closed-Form Solution of the Multi-Period Portfolio Choice Problem for a Quadratic Utility Function

In the present paper, we derive a closed-form solution of the multi-period portfolio choice problem for a quadratic utility function with and without a riskless asset. All results are derived under weak conditions on the asset returns. No assumption on the correlation structure between different time points is needed and no assumption on the distribution is imposed. All expressions are presented in terms of the conditional mean vectors and the conditional covariance matrices. If the multivariate process of the asset returns is independent it is shown that in the case without a riskless asset the solution is presented as a sequence of optimal portfolio weights obtained by solving the single-period Markowitz optimization problem. The process dynamics are included only in the shape parameter of the utility function. If a riskless asset is present then the multi-period optimal portfolio weights are proportional to the single-period solutions multiplied by time-varying constants which are depending on the process dynamics. Remarkably, in the case of a portfolio selection with the tangency portfolio the multi-period solution coincides with the sequence of the simple-period solutions. Finally, we compare the suggested strategies with existing multi-period portfolio allocation methods for real data.Comment: 38 pages, 9 figures, 3 tables, changes: VAR(1)-CCC-GARCH(1,1) process dynamics and the analysis of increasing horizon are included in the simulation study, under revision in Annals of Operations Researc

Learning in Financial Markets

We survey the recent literature on learning in financial markets. Our main theme is that many financial market phenomena that appear puzzling at first sight are easier to understand once we recognize that parameters in financial models are uncertain and subject to learning. We discuss phenomena related to the volatility and predictability of asset returns, stock price bubbles, portfolio choice, mutual fund flows, trading volume, and firm profitability, among others.Bayesian, uncertainty, bubble, volatility, predictability

Author Correction: CHD3 helicase domain mutations cause a neurodevelopmental syndrome with macrocephaly and impaired speech and language

Correction to: Nature Communications https://doi.org/10.1038/s41467-018-06014-6, published online 05 November 2018. The HTML and PDF versions of this Article were updated after publication to remove images of one individual from Fig. 1