Maximization of Nonconcave Utility Functions in Discrete-Time Financial Market Models

This paper investigates the problem of maximizing expected terminal utility in a (generically incomplete) discrete-time financial market model with finite time horizon. By contrast to the standard setting, a possibly nonconcave utility function U is considered, with domain of definition equal to the whole real line. Simple conditions are presented that guarantee the existence of an optimal strategy for the problem. In particular, the asymptotic elasticity of U plays a decisive role: Existence can be shown when it is strictly greater at -infinity than at +infinity

Nonconcave Robust Optimization with Discrete Strategies under Knightian Uncertainty

We study robust stochastic optimization problems in the quasi-sure setting in discrete-time. The strategies in the multi-period-case are restricted to those taking values in a discrete set. The optimization problems under consideration are not concave. We provide conditions under which a maximizer exists. The class of problems covered by our robust optimization problem includes optimal stopping and semi-static trading under Knightian uncertainty.Comment: arXiv admin note: text overlap with arXiv:1610.0923

Existence of solutions in non-convex dynamic programming and optimal investment

We establish the existence of minimizers in a rather general setting of dynamic stochastic optimization in finite discrete time without assuming either convexity or coercivity of the objective function. We apply this to prove the existence of optimal investment strategies for non-concave utility maximization problems in financial market models with frictions, a first result of its kind. The proofs are based on the dynamic programming principle whose validity is established under quite general assumptions. © 2016 Springer-Verlag Berlin Heidelber

Why should older people invest less in stock than younger people?

Financial planners typically advise people to shift investments away from stocks and toward bonds as they age. The planners commonly justify this advice in three ways. They argue that stocks are less risky over a young person’s long investment horizon, that stocks are often necessary for young people to meet large financial obligations (like college tuition for their children), and that younger people have more years of labor income ahead with which to recover from the potential losses associated with stock ownership. This article uses economic reasoning to evaluate these three different justifications. It finds that the first two arguments do not make economic sense. The last argument is valid—but only for people with labor income that is relatively uncorrelated with stock returns. If a person’s labor income is highly correlated with stock returns, then that investor is better off shifting investments toward stocks over time.Saving and investment ; Stock market

Multinomial Inverse Regression for Text Analysis

Text data, including speeches, stories, and other document forms, are often connected to sentiment variables that are of interest for research in marketing, economics, and elsewhere. It is also very high dimensional and difficult to incorporate into statistical analyses. This article introduces a straightforward framework of sentiment-preserving dimension reduction for text data. Multinomial inverse regression is introduced as a general tool for simplifying predictor sets that can be represented as draws from a multinomial distribution, and we show that logistic regression of phrase counts onto document annotations can be used to obtain low dimension document representations that are rich in sentiment information. To facilitate this modeling, a novel estimation technique is developed for multinomial logistic regression with very high-dimension response. In particular, independent Laplace priors with unknown variance are assigned to each regression coefficient, and we detail an efficient routine for maximization of the joint posterior over coefficients and their prior scale. This "gamma-lasso" scheme yields stable and effective estimation for general high-dimension logistic regression, and we argue that it will be superior to current methods in many settings. Guidelines for prior specification are provided, algorithm convergence is detailed, and estimator properties are outlined from the perspective of the literature on non-concave likelihood penalization. Related work on sentiment analysis from statistics, econometrics, and machine learning is surveyed and connected. Finally, the methods are applied in two detailed examples and we provide out-of-sample prediction studies to illustrate their effectiveness.Comment: Published in the Journal of the American Statistical Association 108, 2013, with discussion (rejoinder is here: http://arxiv.org/abs/1304.4200). Software is available in the textir package for

Skorohod's representation theorem and optimal strategies for markets with frictions

We prove the existence of optimal strategies for agents with cumulative prospect theory preferences who trade in a continuous-time illiquid market, transcending known results which previously pertained only to risk-averse utility maximizers. The arguments exploit an extension of Skorohod's representation theorem for tight sequences of probability measures. This method is applicable in a number of similar optimization problems. © 2017 Society for Industrial and Applied Mathematics