Optimal constrained investment in the Cramer-Lundberg model

We consider an insurance company whose surplus is represented by the classical Cramer-Lundberg process. The company can invest its surplus in a risk-free asset and in a risky asset, governed by the Black-Scholes equation. There is a constraint that the insurance company can only invest in the risky asset at a limited leveraging level; more precisely, when purchasing, the ratio of the investment amount in the risky asset to the surplus level is no more than a; and when short-selling, the proportion of the proceeds from the short-selling to the surplus level is no more than b. The objective is to find an optimal investment policy that minimizes the probability of ruin. The minimal ruin probability as a function of the initial surplus is characterized by a classical solution to the corresponding Hamilton-Jacobi-Bellman (HJB) equation. We study the optimal control policy and its properties. The interrelation between the parameters of the model plays a crucial role in the qualitative behavior of the optimal policy. For example, for some ratios between a and b, quite unusual and at first ostensibly counterintuitive policies may appear, like short-selling a stock with a higher rate of return to earn lower interest, or borrowing at a higher rate to invest in a stock with lower rate of return. This is in sharp contrast with the unrestricted case, first studied in Hipp and Plum, or with the case of no short-selling and no borrowing studied in Azcue and Muler.     

Dynamic Optimization of Long-Term Growth Rate for a Portfolio with Transaction Costs and Logarithmic Utility

We study the optimal investment policy for an investor who has available one bank account and n risky assets modeled by log-normal diffusions. The objective is to maximize the long-run average growth of wealth for a logarithmic utility function in the presence of proportional transaction costs. This problem is formulated as an ergodic singular stochastic control problem and interpreted as the limit of a discounted control problem for vanishing discount factor. The variational inequalities for the discounted control problem and the limiting ergodic problem are established in the viscosity sense. The ergodic variational inequality is solved by using a numerical algorithm based on policy iterations and multigrid methods. A numerical example is displayed for two risky assets.     

Optimal proportional reinsurance policies for diffusion models

Abstract

When applying a proportional reinsurance policy π the reserve of the insurance company is governed by a SDE =(a_π (t)u dt + a_π (t)σ dW_t where {W_t } is a standard Brownian motion, µ, π, > 0 are constants and 0 ? a_π (t) ? 1 is the control process, where a_π (t) denotes the fraction, that is reinsured at time t. The aim of this paper is to find a policy that maximizes the return function V_π (x) = where c > 0, τ_π is the time of ruin and x refers to the initial reserve.     

Asset Pricing and Hedging in Financial Markets with Transaction Costs: An Approach Based on the Von Neumann–Gale Model

The paper develops a general discrete-time framework for asset pricing and hedging in financial markets with proportional transaction costs and trading constraints. The framework is suggested by analogies between dynamic models of financial markets and (stochastic versions of) the von Neumann–Gale model of economic growth. The main results are hedging criteria stated in terms of “dual variables” – consistent prices and consistent discount factors. It is shown how these results can be applied to specialized models involving transaction costs and portfolio restrictions.     

Controlling Risk Exposure and Dividends Payout Schemes:Insurance Company Example

The paper represents a model for financial valuation of a firm which has control of the dividend payment stream and its risk as well as potential profit by choosing different business activities among those available to it. This model extends the classical Miller–Modigliani theory of firm valuation to the situation of controllable business activities in a stochastic environment. We associate the value of the company with the expected present value of the net dividend distributions (under the optimal policy). The example we consider is a large corporation, such as an insurance company, whose liquid assets in the absence of control fluctuate as a Brownian motion with a constant positive drift and a constant diffusion coefficient. We interpret the diffusion coefficient as risk exposure, and drift is understood as potential profit. At each moment of time there is an option to reduce risk exposure while simultaneously reducing the potential profit—for example, by using proportional reinsurance with another carrier for an insurance company. Management of a company controls the dividends paid out to the shareholders, and the objective is to find a policy that maximizes the expected total discounted dividends paid out until the time of bankruptcy. Two cases are considered: one in which the rate of dividend payout is bounded by some positive constant M, and one in which there is no restriction on the rate of dividend payout. We use recently developed techniques of mathematical finance to obtain an easy understandable closed form solution. We show that there are two levels u₀ and u₁ with u₀≤u₁. As a function of currently available reserve, the risk exposure monotonically increases on (0,u₀) from 0 to the maximum possible. When the reserve exceeds u₁ the dividends are paid at the maximal rate in the first case and in the second case every excess above u₁ is distributed as dividend. We also show that for M small enough u₀=u₁ and the optimal risk exposure is always less than the maximal.     

CLASSICAL AND IMPULSE STOCHASTIC CONTROL FOR THE OPTIMIZATION OF THE DIVIDEND AND RISK POLICIES OF AN INSURANCE FIRM

This paper deals with the dividend optimization problem for a financial or an insurance entity which can control its business activities, simultaneously reducing the risk and potential profits. It also controls the timing and the amount of dividends paid out to the shareholders. The objective of the corporation is to maximize the expected total discounted dividends paid out until the time of bankruptcy. Due to the presence of a fixed transaction cost, the resulting mathematical problem becomes a mixed classical-impulse stochastic control problem. The analytical part of the solution to this problem is reduced to quasivariational inequalities for a second-order nonlinear differential equation. We solve this problem explicitly and construct the value function together with the optimal policy. We also compute the expected time between dividend payments under the optimal policy.     

Explicit solution of a general consumption/portfolio problem with subsistence consumption and bankruptcy

This paper solves a general continuous-time single-agent consumption and portfolio decision problem with subsistence consumption in closed form. The analysis allows for general continuously differentiable concave utility functions. The model takes into consideration that consumption must be no smaller than a given subsistence rate and that bankruptcy can occur. Thus the paper generalizes the results of Karatzas, Lehoczky, Sethi, and Shreve (1986).     

Optimal Financing of a Corporation Subject To Random Returns

We consider the problem of finding an optimal financing mix of retained earnings and external equity for maximizing the value of a corporation in a stochastic environment. We formulate the problem as a singular stochastic control for a diffusion process. We show that the value function satisfies a free-boundary problem. We characterize the value function and show that the optimal policy can be characterized in terms of two threshold parameters. With asset level below the lower threshold, optimal policy is to finance the firm's growth by retaining all earnings and raising the required external equity financing. With asset level above the higher threshold, optimal policy is to pay all retained earnings as dividends and to bring in no new equity. Between the two thresholds, the optimal policy is to retain all earnings but not raise any external equity. We obtain an explicit solution for the value function when there is no brokerage commission in floating external equity. We provide economic interpretations of the results obtained in the paper.