Optimal portfolio strategies under a shortfall constraint

One of the major challenges faced by most pension fund managers in the defined pension (DC) scheme is how best member’s contributions can be invested to yield maximum returns. To achieve this, there is need to model and developed a robust investment plan which takes into consideration the volatility of the stock market price, tax on investment on risky assets and the mortality risk of its members. Based on this, the optimal portfolio distribution of a DC pension scheme with return of premium clause is studied where the mortality force function is characterized by the Weibull model and the investment in risky asset is subject to a certain proportion of tax. A portfolio with a risk-free asset and a risky asset modeled by the geometric Brownian motion such that the remaining accumulations are equally distributed between the remaining members is considered. Furthermore, the game theoretic approach is used to establish an optimization problem from the extended Hamilton Jacobi Bellman (HJB) equation which is a non-linear partial differential equation (PDE). Using variable separation method, closed form solutions of the optimal portfolio distribution and the efficient frontier are obtained. Lastly, some numerical simulations are used to study the impact of some the parameters on the optimal portfolio distribution with observations that the optimal portfolio distribution developed by the fund manager is inversely proportional to the tax imposed on the risky asset, risk averse coefficient, initial fund size, and risk free interest rate but directly proportional to time.

A behavioral mean-risk portfolio selection problem in continuous time is formulated and studied in this article. Based on the standard mean-variance portfolio selection problem, the cumulative distribution function of the cash flow is distorted by a probability distortion function. This probability distortion function represents the risk preference in a different way. Then the problem is no longer a convex optimization problem. This feature distinguishes it from the conventional linear-quadratic (LQ) problems. The stochastic optimal LQ control theory no longer applies. We take the quantile function of the terminal cash flow as the decision variable. The corresponding optimal terminal cash flow can be recovered by the optimal quantile function. Then the efficient strategy is the hedging strategy of the optimal terminal cash flow.

Dynamic revenue maximization: A continuous time approach

This paper is dedicated to studying the optimal investment proportions of three types of assets with symmetry, namely, risky assets, risk-free assets, and wealth management products, when the stochastic expenditure process follows a jump-diffusion model. The stochastic expenditure process is treated as an exogenous cash flow and is assumed to follow a stochastic differential process with jumps. Under the Cox–Ingersoll–Ross interest rate term structure, it is presumed that the prices of multiple risky assets evolve according to a multi-dimensional geometric Brownian motion. By employing stochastic control theory, the Hamilton–Jacobi–Bellman (HJB) equation for the household portfolio problem is formulated. Considering various risk-preference functions, particularly the Hyperbolic Absolute Risk Aversion (HARA) function, and given the algebraic form of the objective function through the terminal-value maximization condition, an explicit solution for the optimal investment strategy is derived. The findings indicate that when household investment behavior is characterized by random expenditures and symmetry, as the risk-free interest rate rises, the optimal proportion of investment in wealth-management products also increases, whereas the proportion of investment in risky assets continually declines. As the expected future expenditure increases, households will decrease their acquisition of risky assets, and the proportion of risky-asset purchases is sensitive to changes in the expectation of unexpected expenditures.

Although household savings in Japan are among the highest in the world, investment in risky assets is still very low. This study examines whether financial literacy explains the lack of investment in risky assets in Japan. We use data from the Preference Parameter Study, a nationwide survey in Japan that has been conducted by Osaka University. We use investment in stocks, investment trusts, futures/options, Japanese government bonds, government bonds of foreign countries, and foreign currency deposits as a proxy for investment in risky assets. Our results show that investment in risky assets is higher among financially literate people. Moreover, financial literacy has a significantly positive association with investment in risky assets even after controlling the demographic, socio-economic, and psychological factors. We check the robustness of the association between financial literacy and investment in risky assets by segregating investment in risky assets into investment in equity securities and investment in bonds and foreign currencies. Financial literacy is found to be associated with both investment in equity securities and investment in bonds and foreign currencies. Our results are also robust in terms of the endogeneity issue. The results imply that investment in risky assets in financial markets could be increased by introducing financial literacy programs at a mass level.

<p style='text-indent:20px;'>In this paper, we investigate a non-zero-sum stochastic differential reinsurance-investment game problem between two insurers. Both insurers can purchase proportional reinsurance and invest in a financial market that contains a risk-free asset and a risky asset. We consider the insurers' wealth processes with delay to characterize the bounded memory feature. For considering the effect of asymmetric information, we assume the insurers have access to different levels of information in the financial market. Each insurer's objective is to maximize the expected utility of its performance relative to its competitor. We derive the Hamilton-Jacobi-Bellman (HJB) equations and the general Nash equilibrium strategies associated with the control problem by applying the dynamic programming principle. For constant absolute risk aversion (CARA) insurers, the explicit Nash equilibrium strategies and the value functions are obtained. Finally, we present some numerical studies to draw economic interpretations and find the following interesting results: (1) the insurer with less information completely ignores its own risk aversion factor, but imitates the investment strategy of its competitor who has more information on the financial market, which is a manifestation of the herd effect in economics; (2) the difference between the effects of different delay weights on the strategies is related to the length of the delay time in the framework of the non-zero-sum stochastic differential game, which illustrates that insurers should rationally estimate the correlation between historical performance and future performance based on their own risk tolerance, especially when decision makers consider historical performance over a long period of time.

By this study, we want to provide tools for an economic agent who wants to maintain her maximized intertemporal consumption rate over time period regularly by trading activity in capital market. She would use the diversification principle to compose portfolio that is consisted of risky and risk free assets, with initial wealth by using principle of dynamic programming in stochastic control problem. While our target function is to maximize the discounted utility function over consumption rate, the agent may use this portion of wealth for hedonistic lifestyle without thinking the bequest aim. Motivated by observation that seminal work of Samuelson (1969)’s optimal investment on risky asset was independent of time and wealth process, we were replacing the source of uncertainty in the risky asset by binomial process in order to improve Samuelson's solution. The risky asset price evolution is determined to move in two possible directions of up and down in the next time instantly with certain probability, following binomial trees scheme. Working within this setting, we manage to obtain trading strategies as explicit solution for the optimal investment and consumption decisions which were being expressed as time-dependent functions as well as wealth process feedback form. This solution goes along with inter-temporal concept that our consumption and investment strategies tomorrow will be affected with our wealth today. We exemplify the results by numerical examples and perform Monte Carlo simulation.

The accumulated wealth from investment in risky assets is a random variable. If investment strategies are to be ordered, so that one is preferred to another, then the ordering of random variables is required. In this paper the levels of stochastic dominance for random variables are used to define bi-criteria problems for determining an efficient investment strategy. The criteria are characterized as growth and security, respectively, and produce an ordering of strategies consistent with stochastic dominance. In the case where the dynamics of asset returns follow geometric Brownian motion in continuous time, the efficient strategies are shown to be proportional to the growth optimum or Kelly strategy. The analogous problem in discrete time requires solving a stochastic program. An example is provided which compares the continuous and discrete time solutions.KeywordsAsset PricePlanning HorizonInvestment StrategyRisky AssetStochastic DominanceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Worst-Case Portfolio Optimization with Proportional Transaction Costs

This paper investigates the optimal reinsurance-investment strategy for an insurer whose premium is subject to extrapolative bias. In other words, the insurance premium is dynamically updated by a weighted average of prior claims and the initial estimation of claims. The insurer's surplus follows a diffusion approximation process. He purchases proportional reinsurance or acquires new business to manage insurance risk, and invests his surplus in the financial market, containing a risk-free asset and a risky asset (stock). The price of the risky asset is described by a constant elasticity of variance (CEV) model. The insurer is uncertain about the models of claims and risky asset. In order to derive robust optimal reinsurance-investment strategies, we establish an optimal control problem by maximizing the insurer's expected exponential utility of terminal wealth and solve the optimization problem explicitly. Finally, we present several numerical examples to illustrate our theoretical results.

The lagrange method of optimization with applications to portfolio and investment decisions

Time-consistent reinsurance and investment strategies for mean–variance insurer under partial information

Jump-Diffusions with Controlled Jumps: Existence and Numerical Methods

We extend the numerical methods of [Kushner, H.J. and Dupuis, P., 1992, Numerical Methods for Stochastic Control Problems in Continuous Time, 2nd ed., 2001 (Berlin and New York: Springer Verlag], known as the Markov chain approximation methods, to controlled general nonlinear delayed reflected diffusion models. Both the path and the control can be delayed. For the no-delay case, the method covers virtually all models of current interest. The method is robust, the approximations have physical interpretations as control problems closely related to the original one, and there are many effective methods for getting the approximations, and for solving the Bellman equation for low-dimensional problems. These advantages carry over to the delay problem. It is shown how to adapt the methods for getting the approximations, and the convergence proofs are outlined for the discounted cost function. Extensions to all of the cost functions of current interest as well as to models with Poisson jump terms are possible. The paper is particularly concerned with representations of the state and algorithms that minimize the memory requirements.

Optimal Hedging in Discrete and Continuous Time