Optimal Portfolio Selection Problem Research Articles

Optimal payout strategies when Bruno de Finetti meets model uncertainty

Model uncertainty is ubiquitous and plays an important role in insurance and financial modeling. While a substantial effort has been given to studying optimal consumption, portfolio selection and investment problems in the presence of model uncertainty, relatively little attention is given to investigating optimal payout policies taking account of the impacts of model uncertainty. As one of the early attempts, this paper studies the optimal payout control problem under the classical risk model taking into account of model uncertainty about the claims arrival intensity. We aim to provide insights into understanding optimal decisions incorporating model uncertainty and to examine key impact of model uncertainty. We find that the optimal strategy robust to model uncertainty is of a band type. However, the presence of the model uncertainty alters the qualitative behavior of the optimal strategy in the sense that the optimal robust policy is no longer a barrier strategy for some particular cases. We provide numerical examples to illustrate the theoretical results and examine the impact of model uncertainty on optimal payout policies. We also provide examples that use real insurance data for calibration. We discover that the decision maker takes more conservative strategies under model uncertainty, which is consistent with the findings in the economic field and has not been addressed in the existing optimal payout problems without model uncertainty.

Necessary and Sufficient Conditions for Pareto Optimal Solution of Backward Stochastic System With Application

This paper is interested in studying a new kind of Pareto cooperative differential game of backward stochastic differential equation. Based on characterizations of Pareto optimal solution, the game problem is transformed into a set of single objective optimal control problems with constraints of backward stochastic differential equations. In the first place, a necessary condition for Pareto optimal strategy is established by virtue of Ekeland's variational principle, and then it is proved that the necessary condition is also sufficient under certain convex assumption. To shed light on the application of the above theoretical results, a linear-quadratic game and a kind of optimal portfolio and consumption selection problem are also solved explicitly.

Optimal portfolio selection with VaR and portfolio insurance constraints under rank-dependent expected utility theory

This paper investigates two optimal portfolio selection problems for a rank-dependent utility investor who needs to manage his risk exposure: one with a single Value-at-Risk (VaR) constraint and the other with joint VaR and portfolio insurance constraints. The two models generalize existing models under expected utility theory and behavioral theory. The martingale method, quantile formulation, and relaxation method are used to obtain explicit optimal solutions. We have specifically identified an equivalent condition under which the VaR constraint is effective. A numerical analysis is carried out to demonstrate theoretical results, and additional financial insights are presented. We find that, in bad market states, the risk of the optimal investment outcome is reduced when compared to existing models without or with one constraint.

Portfolio selection and job switching with CARA utility

We investigate an optimal consumption and portfolio selection problem with a job switching option. The agent with CARA utility can choose between two types of jobs: one with high labor income but high disutility from labor, and the other with low labor income and low disutility from labor. We use the dual/martingale approach to obtain closed-form solutions to the agent’s lifetime utility optimization problem. We also obtain some numerical implications.

Constructing portfolios using stable distributions: The case of S&P 500 sectors exchange-traded funds

Portfolio construction is an important practical problem in finance. In the traditional approach, introduced by Markowitz, one assumes normally distributed returns and constructs a portfolio with a minimum risk (measured by the standard deviation of portfolio returns) for a specified (and minimally acceptable) return.In practice, returns are not normally distributed and have heavy tails. As a result, the normality assumption severely underestimates risk. It has been long suggested that a more appropriate way is to model returns by using alpha-stable distributions.In this paper, we use elliptical stable distributions for optimal stable portfolio construction. We illustrate this by considering portfolios from S&P 500 sector exchange-traded funds (ETFs).Our main results indicate that stable portfolios are in general comparable with standard Markowitz portfolios. But we discovered a few properties of stable distributions that are promising for optimal portfolio selection problem. Stable risk estimation for next year is much closer to the actual portfolio risk, stable portfolios are more balanced, so they better handle maximum restriction on component weights, and stable portfolios are more resilient to extreme market conditions.Both stable and normal optimal portfolios outperform S&P 500 index and equally-weighted ETF portfolio in most years.We propose an investment strategy that uses both stable and normal distributions for building optimal portfolios. The return of this strategy exceeds the return of the strategies which use only stable or only normal distributions.

The optimal solution of ESG portfolio selection models that are based on the average ESG score

Recently, various models have been proposed to engage portfolio selection or ESG investments. In this brief report, we solve the problem of optimal portfolio selection of arbitrary ESG utility functions where the ESG preference function is based on the average ESG score. The proposed optimal solution shows that the impact of the ESG score and the expected return vectors on the optimal weights are equal, up to a scalar, regardless of the utility function of the investors.

Sampling distributions of optimal portfolio weights and characteristics in small and large dimensions

Optimal portfolio selection problems are determined by the (unknown) parameters of the data generating process. If an investor wants to realize the position suggested by the optimal portfolios, he/she needs to estimate the unknown parameters and to account for the parameter uncertainty in the decision process. Most often, the parameters of interest are the population mean vector and the population covariance matrix of the asset return distribution. In this paper, we characterize the exact sampling distribution of the estimated optimal portfolio weights and their characteristics. This is done by deriving their sampling distribution by its stochastic representation. This approach possesses several advantages, e.g. (i) it determines the sampling distribution of the estimated optimal portfolio weights by expressions, which could be used to draw samples from this distribution efficiently; (ii) the application of the derived stochastic representation provides an easy way to obtain the asymptotic approximation of the sampling distribution. The later property is used to show that the high-dimensional asymptotic distribution of optimal portfolio weights is a multivariate normal and to determine its parameters. Moreover, a consistent estimator of optimal portfolio weights and their characteristics is derived under the high-dimensional settings. Via an extensive simulation study, we investigate the finite-sample performance of the derived asymptotic approximation and study its robustness to the violation of the model assumptions used in the derivation of the theoretical results.

Application of maximal monotone operator method for solving Hamilton–Jacobi–Bellman equation arising from optimal portfolio selection problem

In this paper, we investigate a fully nonlinear evolutionary Hamilton–Jacobi–Bellman (HJB) parabolic equation utilizing the monotone operator technique. We consider the HJB equation arising from portfolio optimization selection, where the goal is to maximize the conditional expected value of the terminal utility of the portfolio. The fully nonlinear HJB equation is transformed into a quasilinear parabolic equation using the so-called Riccati transformation method. The transformed parabolic equation can be viewed as the porous media type of equation with source term. Under some assumptions, we obtain that the diffusion function to the quasilinear parabolic equation is globally Lipschitz continuous, which is a crucial requirement for solving the Cauchy problem. We employ Banach’s fixed point theorem to obtain the existence and uniqueness of a solution to the general form of the transformed parabolic equation in a suitable Sobolev space in an abstract setting. Some financial applications of the proposed result are presented in one-dimensional space.

Consumption and Portfolio Selection with Recursive Utility, Stochastic Income, and Liquidity Constraints

This paper studies a continuous-time optimal consumption and portfolio selection problem when an economic agent with recursive utility has stochastic income and liquidity constraints. To tackle this problem, we introduce a transform of the Hamilton-Jacobi-Bellman equation into a free boundary value ODE (ordinary differential equation). This transform is designed to simplify the procedure required for performing a duality method. Then, we characterize the optimal policies by deriving an integral equation with a fixed interval from the free boundary ODE. The integral equation is used to prove the verification theorem and to generate a stable numerical scheme. It is notable that the optimal portfolio depends on the elasticity of intertemporal substitution (EIS) due to liquidity constraints even if the investment opportunity is constant. Furthermore, we perform various analyses to investigate the impact of the borrowing constraints on the optimal policies and the marginal propensity to consume.

Optimal Portfolio Selection With VaR and Portfolio Insurance Constraints Under Rank-Dependent Expected Utility Theory

This paper investigates two optimal portfolio selection problems for a rank-dependent utility investor who needs to manage his risk exposure: one with a single Value-at-Risk (VaR) constraint and the other with joint VaR and portfolio insurance constraints. The two models generalize existing models under expected utility theory and behavioral theory. To obtain explicit optimal solutions, the martingale method, quantile formulation, and relaxation method are used. We have specifically identified an equivalent condition under which the VaR constraint is effective. A numerical analysis is carried out to demonstrate theoretical results, and additional financial insights are presented. We find that, in bad market states, the risk of the optimal investment outcome is reduced when compared to existing models without or with one constraint.

Portfolio optimization for jump-diffusion risky assets with regime switching: A time-consistent approach

<p style='text-indent:20px;'>In this paper, an optimal portfolio selection problem with mean-variance utility is considered for a financial market consisting of one risk-free asset and two risky assets, whose price processes are modulated by jump-diffusion model, the two jump number processes are correlated through a common shock, and the Brownian motions are supposed to be dependent. Moreover, it is assumed that not only the risk aversion coefficient but also the market parameters such as the appreciation and volatility rates as well as the jump amplitude depend on a Markov chain with finite states. In addition, short selling is supposed to be prohibited. Using the technique of stochastic control theory and the corresponding extended Hamilton-Jacobi-Bellman equation, the explicit expressions of the optimal strategies and value function are obtained within a game theoretic framework, and the existence and uniqueness of the solutions are proved as well. In the end, some numerical examples are presented to show the impact of the parameters on the optimal strategies, and some further discussions on the case of <inline-formula><tex-math id="M1">\begin{document}$ n\geq 3 $\end{document}</tex-math></inline-formula> risky assets are given to demonstrate the important effect of the correlation coefficient of the Brownian motions on the optimal results.</p>

Construction of a class of forward performance processes in stochastic factor models, and an extension of Widder’s theorem

We consider the problem of optimal portfolio selection under forward investment performance criteria in an incomplete market. Given multiple traded assets, the prices of which depend on multiple observable stochastic factors, we construct a large class of forward performance processes, as well as the corresponding optimal portfolios, with power-utility initial data and for stock–factor correlation matrices with eigenvalue equality (EVE) structure, which we introduce here. This is done by solving the associated nonlinear parabolic partial differential equations (PDEs) posed in the “wrong” time direction. Along the way, we establish on domains an explicit form of the generalised Widder theorem of Nadtochiy and Tehranchi (Math. Finance 27:438–470, 2015, Theorem 3.12) and rely for that on the Laplace inversion in time of the solutions to suitable linear parabolic PDEs posed in the “right” time direction.

Finite time-horizon optimal investment and consumption with time-varying subsistence consumption constraints

In this paper we consider a general optimal consumption and portfolio selection problem of a finitely-lived agent whose consumption rate process is subject to time-varying subsistence consumption constraints. That is, her consumption rate should be greater than or equal to some convex, non-decreasing and continuous function of time t. Using martingale duality approach and Feynman–Kac formula, we derive the partial differential equation of the Cauchy problem satisfied by the dual value function. We use the integral transform method for solving this Cauchy problem to obtain the general optimal policies in an explicit form. With constant relative risk aversion and constant absolute risk aversion utility functions we illustrate some numerical results of the optimal policies.

OPTIMAL PORTFOLIO SELECTION BY CVAR BASED SHARPE RATIO — GENETIC ALGORITHM APPROACH —

We address an optimal portfolio selection problem of maximizing so we call CVaR (Conditional Value-at-Risk) based Sharpe ratio of return rate of portfolio, which is defined as the ratio of the expected excess return to CVaR. The Sharpe ratio defined as the ratio of expected excess return to standard deviation, the most common traditional performance measure, takes standard deviation as a risk measure, however, its has been received a lot of criticisms. In our CVaR based Sharpe ratio, the standard deviation is replaced by CVaR, which is a remarkable coherent risk measure which overcomes essential defects of standard deviation. Although our new performance measure is expected to enlarge the applicable area of practical investment problems for which the original Sharpe ratio is not suitable, however, we should device effective computational methods to solve optimal portfolio selection problems with very large number of investment opportunities. In order to deal with rather complicated non-concave objective function, which comes from the introduction of CVaR, in this paper, we propose a Genetic Algorithm (GA) approach. The paper briefly reviews the literature in the area of application of Genetic Algorithms to financial problems, and then details the development of Genetic Algorithm for portfolio selection. In order to evaluate CVaR for each portfolio, by utilizing the results of Rockafellar and Uryasev (2000), we introduce an auxiliary decision variable to obtain a tractable concave maximization problem. Furthermore, if we estimate or approximate required expected values by sampling methods or historical data, we can reduce this concave maximization problem to an LP (Linear Programming) problem. Therefore, our problem could be solved by GA which incorporates LP for evaluating values of CVaR. Numerical experiments from real Japanese financial data are conducted to test our approach to conclude that, by suitable choice of probability parameter of three evolution operation, we could effectively solve optimal selection problems with practical sizes.

Multi-period mean-variance portfolio selection in a regime-switching market with a bankruptcy state and a state-dependent uncertain exit-time

In this paper, we study optimal multi-period portfolio selection problem with uncertain exit-time under mean-variance criterion in a Markovian regime-switching market. The market state space contains an absorbing state which represents the bankruptcy state. It is assumed that all random key parameters, i.e., asset returns, the recovery rate, and the exit-time depend on the current market state. Three common mean-variance formulations are considered, i.e., minimum variance formulation, maximum expected return formulation and the trade-off formulation. First, the problem with an uncertain exit-time is reformulated as a problem with a certain exit-time. Then, by applying the Lagrange duality method and the dynamic programming approach, the optimal multi-period portfolio strategies and the efficient frontier are derived in a closed form. Moreover, the conditions under which the aforementioned three problems are (mutually) equivalent are given. A numerical example is provided to illustrate the results.

A Numerical Solution of Optimal Portfolio Selection Problem with General Utility Functions

In this paper, we adopt a monotone numerical scheme to solve the Hamilton–Jacobi–Bellman equation arising from the optimal portfolio selection problem with general utility functions. To explore the relationship between the relative risk aversion and optimal investment strategy, three different examples are presented with a constant relative risk aversion (CRRA) utility, an increasing relative risk aversion (IRRA) utility and a decreasing relative risk aversion (DRRA) utility, respectively. Our numerical results suggest that non-CRRA investors should adjust the portfolio according to both wealth and time positions. More specifically, both IRRA and DRRA investors should reduce their allocation to the risky asset as time approaches the investment horizon, which is consistent with the life-cycle investment advice. On the other hand, as the wealth increases, IRRA investors should decrease their allocation to the risky asset, while DRRA investors should behave the other way around.

Analytic solution to the portfolio optimization problem in a mean-variance-skewness model

ABSTRACTIn portfolio theory, it is well-known that the distributions of stock returns are often unimodal asymmetric distributions. Therefore, many researches have suggested considering the skew-normal distribution as an adequate model in quantitative finance. Such asymmetry explains why the celebrated mean-variance theory, which does not account to the skewness of distribution of returns, frequently fails to provide an optimal portfolio selection rule. In this paper, we provide a novel approach for solving the problem of optimal portfolio selection for asymmetric distributions of the stock returns, by putting it into a framework of a mean-variance-skewness measure. Moreover, our optimal solutions are explicit and are closed-form. In particular, we provide an analytical portfolio optimization solution to the exponential utility of the well-known skew-normal distribution. Our analytical solution can be investigated in comparison to other portfolio selection rules, such as the standard mean-variance model. The new methodology is illustrated numerically.

Dynamic portfolio choice with return predictability and transaction costs

We derive a closed-form solution to a continuous-time optimal portfolio selection problem with return predictability and transaction costs. Specifically, we assume that asset returns are predicted by stochastic signals, and that transaction costs are of quadratic form. The agent chooses a trading strategy to maximize the expected exponential utility of his terminal wealth. Our feedback trading strategy indicates that the agent should trade gradually toward a dynamic aim portfolio, which is a weighted sum of the expected future Merton portfolios. The agent’s aim portfolio converges to the Merton portfolio as time approaches the terminal date. Our analysis offers new insights to the existing literature. First, our optimal trading strategy is affected by the volatility of return-predicting factors, while such an effect is absent in Gârleanu and Pedersen (2016). Secondly, the agent invests more into the assets with more persistent signals and with less transaction costs.

Expected shortfall and portfolio management in contagious markets

Abstract We study the impact of market contagion on portfolio management. To model possible recurrence in the arrival of extreme events, we equip classic Poisson jumps with long memory via past-weighted randomization of the likelihood of their occurrences (Hawkes processes). Within this framework, we tackle the problem of optimal portfolio selection in terms of Expected Shortfall (ES). We use the generalized method of moments to estimate the model on three US stock indexes, representing three major sectors of the US economy. The moment conditions of the model are computed efficiently in closed form applying a novel technique. Given parameter estimates we maximize (at a monthly frequency in the period 2001–2016) the expected return subject to a constraint on the ES of a portfolio consisting of the three US sector indexes. We find that the weights of the optimal portfolio are significantly adjusted when the level of contagion is high. Finally, we perform an extensive out-of-sample back-test of the model’s ability to measure ES and find that the Hawkes jump-diffusion model outperforms two traditional models that are commonly implemented.

Portfolio selection problem with nonlinear wealth equations under non-extensive statistical mechanics for time-varying SDE

This paper is dedicated to the study of continuous-time mean–variance optimal portfolio selection problem with non-linear wealth equations under non-extensive statistical mechanics for the time-varying stochastic differential equation model. Firstly, we allow the returns and variance of risky assets are time-varying functions, which can fit the financial data better. Secondly, we consider an investor with the non-linear wealth equation. In fact, the wealth equations are not linear in many cases. The investor has to pay some taxes, which leads to a non-linear wealth equation. Moreover, since the return of the stocks price may be affected by a large investors portfolio selection, the wealth equation is non-linear in this case. Thirdly, the non-linear wealth equation driven by Tsallis distribution is constructed under non-extensive statistical mechanics, which can capture the characteristics of fat tails and aiguilles of the risky asset’s return. The viscosity solution of the HJB equation for the portfolio problem is proposed by the optimal stochastic control theory and Lagrange multiplier method. Finally, the efficient portfolio strategy and efficient frontier under non-extensive statistical mechanics are obtained. Furthermore, numerical analysis and real data study are discussed to show our results.