Building an Optimal Portfolio in Discrete Time in the Presence of Transaction Costs

Abstract

Portfolio theory covers different approaches to the construction of a portfolio offering maximum expected returns for a given level of risk tolerance where the goal is to find the optimal investment rule. Each investor has a certain utility for money which is reflected by the choice of a utility function. In this article, a risk averse power utility function is studied in discrete time for a large class of underlying probability distribution of the returns of the asset prices. Each investor chooses, at the beginning of an investment period, the feasible portfolio allocation which maximizes the expected value of the utility function for terminal wealth. Effects of both large and small proportional transaction costs on the choice of an optimal portfolio are taken into account. The transaction regions are approximated by using asymptotic methods when the proportional transaction costs are small and by using expansions about critical points for large transaction costs.     

Optimal dividend payout for classical risk model with risk constraint

In this paper we consider the problem of maximizing the total discounted utility of dividend payments for a Cramér-Lundberg risk model subject to both proportional and fixed transaction costs.We assume that dividend payments are prohibited unless the surplus of insurance company has reached a level b.Given fixed level b,we derive a integro-differential equation satisfied by the value function.By solving this equation we obtain the analytical solutions of the value function and the optimal dividend strategy when claims are exponentially distributed.Finally we show how the threshold b can be determined so that the expected ruin time is not less than some T.Also,numerical examples are presented to illustrate our results.     

Optimal impulse control of a portfolio with a fixed transaction cost

The aim of this work is to investigate a portfolio optimization problem in presence of fixed transaction costs. We consider an economy with two assets: one risky, modeled by a geometric Brownian motion, and one risk-free which grows at a certain fixed rate. The agent is fully described by his/her utility function and the objective is to maximize the expected utility from the liquidation of wealth at a terminal date. We deal with different forms of utility functions (power, logarithmic and exponential utility), describing in each case how the fixed transaction costs influence the agent’s behavior. We show when it is optimal to recalibrate his/her portfolio and which are the best adjusted portfolios. We also analyze how the optimal strategy is influenced by the risk-aversion, as well as other model parameters.     

An Algorithm for Portfolio Optimization with Variable Transaction Costs,Part 1: Theory

A portfolio optimization problem consists of maximizing an expected utility function of n assets. At the end of a typical time period, the portfolio will be modified by buying and selling assets in response to changing conditions. Associated with this buying and selling are variable transaction costs that depend on the size of the transaction. A straightforward way of incorporating these costs can be interpreted as the reduction of portfolios’ expected returns by transaction costs if the utility function is the mean-variance or the power utility function. This results in a substantially higher-dimensional problem than the original n-dimensional one, namely (2K+1)n-dimensional optimization problem with (4K+1)n additional constraints, where 2K is the number of different transaction costs functions. The higher-dimensional problem is computationally expensive to solve. This two-part paper presents a method for solving the (2K+1)n-dimensional problem by solving a sequence of n-dimensional optimization problems, which account for the transaction costs implicitly rather than explicitly. The key idea of the new method in Part 1 is to formulate the optimality conditions for the higher-dimensional problem and enforce them by solving a sequence of lower-dimensional problems under the nondegeneracy assumption. In Part 2, we propose a degeneracy resolving rule, address the efficiency of the new method and present the computational results comparing our method with the interior-point optimizer of Mosek. This research was supported by the National Science and Engineering Research Council of Canada and the Austrian National Bank. The authors acknowledge the valuable assistance of Rob Grauer and Associate Editor Franco Giannessi for thoughtful comments and suggestions.     

Optimal proportional reinsurance and investment with transaction costs, I: Maximizing the terminal wealth

We consider a problem of optimal reinsurance and investment with multiple risky assets for an insurance company whose surplus is governed by a linear diffusion. The insurance company’s risk can be reduced through reinsurance, while in addition the company invests its surplus in a financial market with one risk-free asset and n risky assets. In this paper, we consider the transaction costs when investing in the risky assets. Also, we use Conditional Value-at-Risk (CVaR) to control the whole risk. We consider the optimization problem of maximizing the expected exponential utility of terminal wealth and solve it by using the corresponding Hamilton-Jacobi-Bellman (HJB) equation. Explicit expression for the optimal value function and the corresponding optimal strategies are obtained.     

A sharp error estimate of the best approximation by algebraic polynomials in the weighted space L 2(−1, 1)

We obtain an exact error estimate of the best approximation by algebraic polynomials in the Lebesgue space L ₂(?1, 1) with the weight 1 ? x ² of degree λ > ?1.     

Asymptotic behavior of solutions to Schrödinger equations with a subcritical dissipative nonlinearity

We study the asymptotic behavior in time of solutions to the initial value problem of the nonlinear Schrödinger equation with a subcritical dissipative nonlinearity λ|u|^p−1u, where 1<p<1+2/n, n is the space dimension and λ is a complex constant satisfying Imλ<0. We show the time decay estimates and the large-time asymptotics of the solution, when the space dimension n?3, p is sufficiently close to 1+2/n and the initial data is sufficiently small.     

Economic optimization in a fixed sequence of unreliable inspections

Given a fixed sequence of unreliable inspection operations with known costs and inspection error probabilities of two types (classifying good items as defective and vice versa), we develop a model for selecting the set of inspections that should be activated in order to minimize expected total costs (inspection and penalties). We present an efficient branch and bound algorithm for finding the optimal solution, and two variations of a greedy heuristic that can be applied jointly to provide very good solutions at a O(n²) computational complexity. The conclusions are backed by a factorial experiment that included 1440 problem instances.