Optimal Asset Allocation Problem Research Articles

Risk management under weighted limited expected loss

We present and solve an optimal asset allocation problem under a weighted limited expected loss (WLEL) constraint. This formulation encompasses the risk management problem with a limited expected loss (LEL) constraint as a specialized instance and offers a pertinent internal risk management instrument for firms. We observe that a WLEL constraint makes the optimizing investor pursue less volatile payoffs than the unconstrained Merton solution. Compared to the LEL-constrained problem with the same weighted default threshold, the WLEL optimal terminal wealth displays a less dispersed distribution with a smaller variance, suggesting a more secure risk management framework. Conducting a comprehensive equilibrium analysis in the presence of a WLEL risk manager, we validate the relatively conservative investment approach undertaken by the WLEL manager. Subsequently, we expand our findings to encompass broader incomplete market settings, wherein the uniqueness of the equivalent local martingale measure is not assured.

Optimal investment and benefit payment adjustment strategies for the target benefit plan under partial information

This article investigates the optimal asset allocation and benefit adjustment problem for the target benefit plan (TBP) with predictable returns in an environment of partial information. We assume that the return rate of the stock depends on an observable and an unobservable state variables, and the pension manager estimates the unobservable component from known information through Bayesian learning. Under the criterion of expected exponential utility maximization, we obtain the optimal strategy in closed-form through the dynamic programming principle approach. Furthermore, numerical simulations are conducted by the Monte Carlo method to discuss the impacts of some parameters on the derived optimal strategy and to compare the optimal strategy with the suboptimal strategy when ignoring learning. It turns out that neglecting learning affects the optimal strategy seriously, thereby leading to significant utility losses.

Optimal Defined Contribution Pension Management with Jump Diffusions and Common Shock Dependence

This work deals with an optimal asset allocation problem for a defined contribution (DC) pension plan during its accumulation phase. The contribution rate is assumed to be proportional to the individual’s salary. The salary follows a Heston stochastic volatility model with jumps, and there exists common shock dependence between the salary and the volatility. Since the time horizon of pension management is quite long, the influence of inflation is considered in the given context. The aim of the pension plan described in this paper is to reduce fluctuations in terminal wealth by investing in the bond and the stock. Through the dynamic programming principle, the Hamilton–Jacobi–Bellman equation is shown. The explicit expression of the investment decision is derived by solving the Hamilton–Jacobi–Bellman equation. In the last part, a numerical analysis is shown to illustrate the impacts of different parameters on the optimal investment policy.

Portfolio performance under benchmarking relative loss and portfolio insurance: From omega ratio to loss aversion

AbstractWe study an optimal investment problem under a joint limited expected relative loss and portfolio insurance constraint with a general random benchmark. By making use of a static Lagrangian method in a complete market setting, the optimal wealth and investment strategy can be fully determined along with the existence and uniqueness of the Lagrangian multipliers. Our numerical demonstration for various commonly used random benchmarks shows a trade-off between the portfolio outperformance and underperformance relative to the benchmark, which may not be captured by the widely used Omega ratio and its utility-transformed version, reflecting the impact of the benchmarking loss constraint. Furthermore, we develop a new portfolio performance measurement indicator that incorporates the agent’s utility loss aversion relative to the benchmark via solving an equivalent optimal asset allocation problem with a benchmark-reference-based preference. We show that the expected utility performance is well depicted by looking at this new portfolio performance ratio, suggesting a more suitable portfolio performance measurement under a limited loss constraint relative to a possibly random benchmark.

Robust optimal investment strategy for a DC pension plan in the market with mispricing and constant elasticity of variance

This paper considers an optimal asset allocation problem for a defined contribution pension fund in a continuous time setting, and the manager is concerned about potential model misspecification. We suppose that the financial market consists of one risk-free asset, one market index fund whose price satisfies the constant elasticity of variance model, and a pair of risky assets with mispricing. Under the objective of maximizing the expected utility of the pension fund wealth at the retirement, the closed form expressions of the robust optimal investment strategy and the corresponding value function are obtained by using the stochastic control theory. Finally, some numerical examples are given to compare the difference between the ambiguity averse and the ambiguity neutral strategies, and to investigate the effect of relevant parameters on the strategy. We find that the optimal investment strategy is not a long-short symmetric strategy when the mispricing assets have different mispricing correct capability, and it will lead to significant utility loss for the manager if the model uncertainty risk is ignored.

Optimal Investment with Time-Varying Stochastic Endowments

This paper considers a utility maximization and optimal asset allocation problem in the presence of a stochastic endowment that cannot be fully hedged through trading in the financial market. We rely on the dynamic programming approach to solve the optimization problem. The properties of the value function, particularly the homogeneity, are used to reduce the HJB equation by one dimension. Furthermore, the optimal strategy is derived, and its asymptotic behavior is discussed.

Implicit incentives for fund managers with partial information

We study the optimal asset allocation problem for a fund manager whose compensation depends on the performance of her portfolio with respect to a benchmark. The objective of the manager is to maximise the expected utility of her final wealth. The manager observes the prices but not the values of the market price of risk that drives the expected returns. Estimates of the market price of risk get more precise as more observations are available. We formulate the problem as an optimization under partial information. The particular structure of the incentives makes the objective function not concave. Therefore, we solve the problem by combining the martingale method and a concavification procedure and we obtain the optimal wealth and the investment strategy. A numerical example shows the effect of learning on the optimal strategy.

Optimal asset allocation under search frictions and stochastic interest rate

In this paper, we investigate an optimal asset allocation problem in a financial market consisting of one risk-free asset, one liquid risky asset and one illiquid risky asset. The liquidity risk stems from the asset that cannot be traded continuously, and the trading opportunities are captured by a Poisson process with constant intensity. Also, it is assumed that the interest rate is stochastically varying and follows the Cox–Ingersoll–Ross model. The performance functional of the decision maker is selected as the expected logarithmic utility of the total wealth at terminal time. The dynamic programming principle coupled with the Hamilton–Jacobi–Bellman equation has been adopted to solve this stochastic optimal control problem. In order to reduce the dimension of the problem, we introduce the proportion of the wealth invested in the illiquid risky asset and derive the semi-analytical form of the value function using a separation principle. A finite difference method is employed to solve the controlled partial differential equation satisfied by a function which is an important component of the value function. The numerical examples and their economic interpretations are then discussed.

On the Equivalence Between Value-at-Risk and Expected Shortfall in Non-Concave Optimization

This paper studies an optimal asset allocation problem for a surplus-driven financial institution facing a Value-at-Risk (VaR) or an Expected Shortfall (ES) constraint corresponding to a non-concave optimization problem under constraints. We obtain the closed-form optimal wealth with the ES constraint as well as with the VaR constraint respectively, and explicitly calculate the optimal trading strategy for constant relative risk aversion (CRRA) utility functions. We find that both VaR and ES-based regulation can effectively reduce the probability of default for a surplus-driven financial institution. However, the liability holders' benefits cannot be fully protected under either VaR- or ES-based regulation. In addition, we show that the VaR and ES-based regulation can induce the same optimal portfolio choice for a surplus-driven financial institution. This differs from the conclusion drawn in Basak and Shapiro 2001 where the financial institution aims at maximizing the expected utility of the total assets, and ES provides better loss protection.

Linking Risk Management Under Expected Shortfall to Loss-Averse Behavior

We introduce and solve an optimal asset allocation problem under a weighted expected shortfall (WES) constraint, which contains the risk management problem under an expected shortfall constraint of Basak and Shapiro (2001) as a special case. Furthermore, we link our risk management problem under the WES constraint with an optimal asset allocation with a multiple-reference-based preference (MRBP) and find that the optimal wealth with MRBP owns the same form as the optimal solution under the WES constraint. For the degenerate case with a fixed reference level, we are able to determine the critical maximal allowed expected shortfall constraint as a function of the loss aversion parameters to achieve equivalence. It is interesting to observe that, while no equivalence can be in general obtained between the WES and the MRBP solution, the optimal terminal wealth of the WES can be made to coincide with the MRBP terminal wealth in the most favorable and in the worst market states. In addition, we carry out a general equilibrium analysis in the presence of a WES/MRBP risk manager.

Implicit Incentives for Fund Managers with Partial Information

We study the optimal asset allocation problem for a fund manager whose compensation depends on the performance of her portfolio with respect to a benchmark. The objective of the manager is to maximise the expected utility of her final wealth. The manager observes the prices but not the values of the market price of risk that drives the expected returns. The estimates of the market price of risk get more precise as more observations are available. We formulate the problem as an optimization under partial information. The particular structure of the incentives makes the objective function not concave. We solve the problem via the martingale method and, with a concavification procedure, we obtain the optimal wealth and the investment strategy. A numerical example shows the effect of learning on the optimal strategy.

Optimal asset allocation with heterogeneous discounting and stochastic income under CEV model

This article focuses on an optimal asset allocation problem with heterogeneous discounting and stochastic income under the constant elasticity of variance (CEV) model. Heterogeneous discounting is an important non-constant discounting model, which can describe the fact that a decision maker discounts in different ways the utility derived from consumption and that of the bequest or final function. It is consistent with the fact that the concern of a decision maker about the bequest left to her descendants when she is young is not the same as that when she is old. In our model, a decision maker with stochastic income can enjoy the consumption, purchase life insurance, and invest her wealth in a risk-free asset and a risky asset whose price process satisfies the CEV model. Meanwhile, the volatility of the stochastic income arises from the risky asset. Since the problem is time-inconsistent, the Bellman’s principle of optimality does not hold. To obtain the time-consistent solution, an equilibrium strategy is calculated. By applying the game theoretic framework and solving an extended Hamilton-Jacobi-Bellman system, we derive the time-consistent consumption, investment, and life insurance strategies for both exponential and logarithmic utility functions. Finally, we provide numerical simulations to illustrate our results.

Optimal Longevity Risk Management in the Retirement Stage of the Life Cycle

The uncertainty in life expectancy plays a critical role in individual financial planning. Its impact is magnified during the retirement years (the wealth distribution stage of the life cycle), as new sources of income typically are not available to retired persons. Utilizing a multi-stage stochastic program, the authors model and solve the optimal asset allocation problem of a retired couple with uncertain life expectancy in the presence of a term life insurance policy. In the base case, they find optimal policies assuming no longevity risk (i.e., lifetime scenarios are uncertain although life expectancy is fixed on the retirement date). Next, they introduce longevity risk in the scenario generation stage through either a shift in the expected lifetimes or an unexpected cut in periodic retirement income. The authors find that the optimal asset allocation policy depends on the presence and the type of these risks, as well as the relative price of insurance and the size of any cut in pension benefits.

Portfolio optimization in a defined benefit pension plan where the risky assets are processes with constant elasticity of variance

The paper studies the optimal asset allocation problem of a defined benefit pension plan that operates in a financial market composed of risky assets whose prices are constant elasticity variance processes. The benefits paid to the participants are deterministic. The contributions to the fund are designed by a spread amortization method, which takes into account the size of the unfunded actuarial liability, defined as the difference between the actuarial liability and the fund assets. We address the case where the fund manager wishes to minimize the solvency risk at the final date of the plan when the fund is underfunded, as well as the case where the fund manager wishes to maximize an increasing, constant elasticity utility function of the fund surplus, when the fund is overfunded. The optimal portfolio and contributions are obtained in both scenarios, with the help of the Hamilton–Jacobi–Bellman equation. A numerical illustration shows the evolution of the plan for several values of the elasticity parameter of the CEV price processes and the risk aversion of the manager, yielding some tips on the main properties of the optimal portfolio.

A Risk-Based Approach for Asset Allocation with A Defaultable Share

This paper presents a novel risk-based approach for an optimal asset allocation problem with default risk, where a money market account, an ordinary share and a defaultable security are investment opportunities in a general non-Markovian economy incorporating random market parameters. The objective of an investor is to select an optimal mix of these securities such that a risk metric of an investment portfolio is minimized. By adopting a sub-additive convex risk measure, which takes into account interest rate risk, as a measure for risk, the investment problem is discussed mathematically in a form of a two-player, zero-sum, stochastic differential game between the investor and the market. A backward stochastic differential equation approach is used to provide a flexible and theoretically sound way to solve the game problem. Closed-form expressions for the optimal strategies of the investor and the market are obtained when the penalty function is a quadratic function and when the risk measure is a sub-additive coherent risk measure. An important case of the general non-Markovian model, namely the self-exciting threshold diffusion model with time delay, is considered. Numerical examples based on simulations for the self-exciting threshold diffusion model with and without time delay are provided to illustrate how the proposed model can be applied in this important case. The proposed model can be implemented using Excel spreadsheets.

Optimal Longevity Risk Management in the Retirement Stage of the Life Cycle

The uncertainty in life expectancy plays a critical role in individual financial planning. Its impact is magnified during the retirement years (the wealth distribution stage of the life cycle), as new sources of income typically are not available to retired persons. Utilizing a multi-stage stochastic program, the authors model and solve the optimal asset allocation problem of a retired couple with uncertain life expectancy in the presence of a term life insurance policy. In the base case, they find optimal policies assuming no longevity risk (i.e., lifetime scenarios are uncertain although life expectancy is fixed on the retirement date). Next, they introduce longevity risk in the scenario generation stage through either a shift in the expected lifetimes or an unexpected cut in periodic retirement income. The authors find that the optimal asset allocation policy depends on the presence and the type of these risks as well as the relative price of insurance and the size of any cut in pension benefits.

Optimal Longevity Risk Management in the Retirement Stage of the Life Cycle

The uncertainty in life expectancy plays a critical role in individual financial planning. Its impact is magnified during the retirement years, also called the wealth distribution stage of the life-cycle, as new sources of income are typically not available to individuals. Utilizing a multi-stage stochastic program, we model and solve the optimal asset allocation problem of a retired couple with uncertain life expectancy in the presence of a term life insurance policy. In the base case, we find optimal policies assuming no longevity risk (i.e., lifetime scenarios are uncertain although life expectancy is fixed on the retirement date). Next, we introduce longevity risk in the scenario generation stage through either a shift in the expected lifetimes or an unexpected cut in periodic retirement income. We find that optimal asset allocation policy depends on the presence and the type of these risks as well as the relative price of insurance and the percentage cut in pension benefits.

Building the Markowitz Efficient Frontier

In this paper we combine some algorithms in the literature in order to effectively build the Markowitz efficient frontier, as a solution of the optimal asset allocation problem. The proposed method works for any size of the investment portfolio, and for any reasonable equality and inequality constraints.

An adaptive portfolio trading system: A risk-return portfolio optimization using recurrent reinforcement learning with expected maximum drawdown

A reinforcement learning trading algorithm with expected drawdown risk is proposed.The expected maximum drawdown is shown to improve portfolio signal generation.The effectiveness of the method is validated using different transaction costs.An adaptive portfolio rebalancing system with automated retraining is recommended. Dynamic control theory has long been used in solving optimal asset allocation problems, and a number of trading decision systems based on reinforcement learning methods have been applied in asset allocation and portfolio rebalancing. In this paper, we extend the existing work in recurrent reinforcement learning (RRL) and build an optimal variable weight portfolio allocation under a coherent downside risk measure, the expected maximum drawdown, E(MDD). In particular, we propose a recurrent reinforcement learning method, with a coherent risk adjusted performance objective function, the Calmar ratio, to obtain both buy and sell signals and asset allocation weights. Using a portfolio consisting of the most frequently traded exchange-traded funds, we show that the expected maximum drawdown risk based objective function yields superior return performance compared to previously proposed RRL objective functions (i.e. the Sharpe ratio and the Sterling ratio), and that variable weight RRL long/short portfolios outperform equal weight RRL long/short portfolios under different transaction cost scenarios. We further propose an adaptive E(MDD) risk based RRL portfolio rebalancing decision system with a transaction cost and market condition stop-loss retraining mechanism, and we show that the proposed portfolio trading system responds to transaction cost effects better and outperforms hedge fund benchmarks consistently.

Dual control Monte-Carlo method for tight bounds of value function in regime switching utility maximization

In this paper, we study the dual control approach for the optimal asset allocation problem in a continuous-time regime-switching market. We find the lower and upper bounds of the value function that is a solution to a system of fully coupled nonlinear partial differential equations. These bounds can be tightened with additional controls to the dual process. We suggest a Monte-Carlo algorithm for computing the tight lower and upper bounds and show the method is effective with a variety of utility functions, including power, non-HARA and Yaari utilities. The latter two utilities are beyond the scope of any current methods available in finding the value function.