arXiv subjects

Peter A. Forsyth

Publications and source records attributed to Peter A. Forsyth.

13 recordsLinked to original sources

arXiv2026

Numerical methods for optimal decumulation of a defined contribution pension plan

The decumulation of a defined contribution (DC) pension plan is well known to be one of the hardest problems in finance. We model this decumulation challenge as an optimal stochastic control problem. The control problem is solved, at each rebalancing date, by alternatively solving a linear partial-integro differential equation (PIDE) followed by an optimization step. We solve the PIDE by using a $δ$-monotone Fourier method, which ensures that monotonicity holds to $O(δ)$. We allow for the use of leverage (i.e. borrowing to invest in stocks), as well as minimum constraints on bond holdings. We pay particular attention to minimizing wrap-around error, an issue which is endemic for Fourier methods and central to the effective use of these methods for optimal control problems. Rather unexpectedly, we find that restricting the portfolio equity fraction to a maximum of 50\% does not reduce portfolio efficiency noticeably. This may be a useful strategy for risk-averse retirees.

cs.CE↗

arXiv2026

Money-Back Tontines for Retirement Decumulation: Neural-Network Optimization under Systematic Longevity Risk

Money-back guarantees (MBGs) are features of pooled retirement income products that address bequest concerns by ensuring the initial premium is returned through lifetime payments or, upon early death, as a death benefit to the estate. This paper studies optimal retirement decumulation in an individual tontine account with an MBG overlay under international diversification and systematic longevity risk. The retiree chooses withdrawals and asset allocation dynamically to trade off expected total withdrawals (EW) against the Conditional Value-at-Risk (CVaR) of terminal wealth, subject to realistic investment constraints. The optimization is solved under a plan-to-live convention, while stochastic mortality affects outcomes through its impact on mortality credits at the pool level. We develop a neural-network based computational approach for the resulting high-dimensional, constrained control problem. The MBG is priced ex post under the induced EW--CVaR optimal policy via a simulation-based actuarial rule that combines expected guarantee costs with a prudential tail buffer. Using long-horizon historical return data expressed in real domestic-currency terms, we find that international diversification and longevity pooling jointly deliver the largest improvements in the EW--CVaR trade-off, while stochastic mortality shifts the frontier modestly in the expected direction. The optimal controls use foreign equity primarily as a state-dependent catch-up instrument, and implied MBG loads are driven mainly by tail outcomes (and the chosen prudential buffer) rather than by mean payouts.

q-fin.PM↗

arXiv2025

Risk Measures for DC Pension Plan Decumulation

As the developed world replaces Defined Benefit (DB) pension plans with Defined Contribution (DC) plans, there is a need to develop decumulation strategies for DC plan holders. Optimal decumulation can be viewed as a problem in optimal stochastic control. Formulation as a control problem requires specification of an objective function, which in turn requires a definition of reward and risk. An intuitive specification of reward is the total withdrawals over the retirement period. Most retirees view risk as the possibility of running out of savings. This paper investigates several possible left tail risk measures, in conjunction with DC plan decumulation. The risk measures studied include (i) expected shortfall (ii) linear shortfall and (iii) probability of shortfall. We establish that, under certain assumptions, the set of optimal controls associated with all expected reward and expected shortfall Pareto efficient frontier curves is identical to the set of optimal controls for all expected reward and linear shortfall Pareto efficient frontier curves. Optimal efficient frontiers are determined computationally for each risk measure, based on a parametric market model. Robustness of these strategies is determined by testing the strategies out-of-sample using block bootstrapping of historical data.

math.OC↗

arXiv2023

Machine Learning and Hamilton-Jacobi-Bellman Equation for Optimal Decumulation: a Comparison Study

We propose a novel data-driven neural network (NN) optimization framework for solving an optimal stochastic control problem under stochastic constraints. Customized activation functions for the output layers of the NN are applied, which permits training via standard unconstrained optimization. The optimal solution yields a multi-period asset allocation and decumulation strategy for a holder of a defined contribution (DC) pension plan. The objective function of the optimal control problem is based on expected wealth withdrawn (EW) and expected shortfall (ES) that directly targets left-tail risk. The stochastic bound constraints enforce a guaranteed minimum withdrawal each year. We demonstrate that the data-driven approach is capable of learning a near-optimal solution by benchmarking it against the numerical results from a Hamilton-Jacobi-Bellman (HJB) Partial Differential Equation (PDE) computational framework.

math.OC↗

arXiv2023

Neural Network Approach to Portfolio Optimization with Leverage Constraints:a Case Study on High Inflation Investment

Motivated by the current global high inflation scenario, we aim to discover a dynamic multi-period allocation strategy to optimally outperform a passive benchmark while adhering to a bounded leverage limit. To this end, we formulate an optimal control problem to outperform a benchmark portfolio throughout the investment horizon. Assuming the asset prices follow the jump-diffusion model during high inflation periods, we first establish a closed-form solution for the optimal strategy that outperforms a passive strategy under the cumulative quadratic tracking difference (CD) objective, assuming continuous trading and no bankruptcy. To obtain strategies under the bounded leverage constraint among other realistic constraints, we then propose a novel leverage-feasible neural network (LFNN) to represent control, which converts the original constrained optimization problem into an unconstrained optimization problem that is computationally feasible with standard optimization methods. We establish mathematically that the LFNN approximation can yield a solution that is arbitrarily close to the solution of the original optimal control problem with bounded leverage. We further apply the LFNN approach to a four-asset investment scenario with bootstrap resampled asset returns from the filtered high inflation regime data. The LFNN strategy is shown to consistently outperform the passive benchmark strategy by about 200 bps (median annualized return), with a greater than 90% probability of outperforming the benchmark at the end of the investment horizon.

q-fin.PM↗

arXiv2023

A parsimonious neural network approach to solve portfolio optimization problems without using dynamic programming

We present a parsimonious neural network approach, which does not rely on dynamic programming techniques, to solve dynamic portfolio optimization problems subject to multiple investment constraints. The number of parameters of the (potentially deep) neural network remains independent of the number of portfolio rebalancing events, and in contrast to, for example, reinforcement learning, the approach avoids the computation of high-dimensional conditional expectations. As a result, the approach remains practical even when considering large numbers of underlying assets, long investment time horizons or very frequent rebalancing events. We prove convergence of the numerical solution to the theoretical optimal solution of a large class of problems under fairly general conditions, and present ground truth analyses for a number of popular formulations, including mean-variance and mean-conditional value-at-risk problems. We also show that it is feasible to solve Sortino ratio-inspired objectives (penalizing only the variance of wealth outcomes below the mean) in dynamic trading settings with the proposed approach. Using numerical experiments, we demonstrate that if the investment objective functional is separable in the sense of dynamic programming, the correct time-consistent optimal investment strategy is recovered, otherwise we obtain the correct pre-commitment (time-inconsistent) investment strategy. The proposed approach remains agnostic as to the underlying data generating assumptions, and results are illustrated using (i) parametric models for underlying asset returns, (ii) stationary block bootstrap resampling of empirical returns, and (iii) generative adversarial network (GAN)-generated synthetic asset returns.

q-fin.CP↗

arXiv2022

Optimal performance of a tontine overlay subject to withdrawal constraints

We consider the holder of an individual tontine retirement account, with maximum and minimum withdrawal amounts (per year) specified. The tontine account holder initiates the account at age 65, and earns mortality credits while alive, but forfeits all wealth in the account upon death. The holder desires to maximize total withdrawals, and minimize the expected shortfall, assuming the holder survives to age 95. The investor controls the amount withdrawn each year and the fraction of the investments in stocks and bonds. The optimal controls are determined based on a parametric model fitted to almost a century of market data. The optimal control algorithm is based on dynamic programming and solution of a partial integro differential equation (PIDE) using Fourier methods. The optimal strategy (based on the parametric model) is tested out of sample using stationary block bootstrap resampling of the historical data. In terms of an expected total withdrawal, expected shortfall (EW-ES) efficient frontier, the tontine overlay greatly outperforms an optimal strategy (without the tontine overlay), which in turn outperforms a constant weight strategy with withdrawals based on the ubiquitous four per cent rule.

q-fin.CP↗

arXiv2021

Optimal control of the decumulation of a retirement portfolio with variable spending and dynamic asset allocation

We extend the Annually Recalculated Virtual Annuity (ARVA) spending rule for retirement savings decumulation to include a cap and a floor on withdrawals. With a minimum withdrawal constraint, the ARVA strategy runs the risk of depleting the investment portfolio. We determine the dynamic asset allocation strategy which maximizes a weighted combination of expected total withdrawals (EW) and expected shortfall (ES), defined as the average of the worst five per cent of the outcomes of real terminal wealth. We compare the performance of our dynamic strategy to simpler alternatives which maintain constant asset allocation weights over time accompanied by either our same modified ARVA spending rule or withdrawals that are constant over time in real terms. Tests are carried out using both a parametric model of historical asset returns as well as bootstrap resampling of historical data. Consistent with previous literature that has used different measures of reward and risk than EW and ES, we find that allowing some variability in withdrawals leads to large improvements in efficiency. However, unlike the prior literature, we also demonstrate that further significant enhancements are possible through incorporating a dynamic asset allocation strategy rather than simply keeping asset allocation weights constant throughout retirement.

q-fin.CP↗

arXiv2020

A Stochastic Control Approach to Defined Contribution Plan Decumulation: "The Nastiest, Hardest Problem in Finance"

We pose the decumulation strategy for a Defined Contribution (DC) pension plan as a problem in optimal stochastic control. The controls are the withdrawal amounts and the asset allocation strategy. We impose maximum and minimum constraints on the withdrawal amounts, and impose no-shorting no-leverage constraints on the asset allocation strategy. Our objective function measures reward as the expected total withdrawals over the decumulation horizon, and risk is measured by Expected Shortfall (ES) at the end of the decumulation period. We solve the stochastic control problem numerically, based on a parametric model of market stochastic processes. We find that, compared to a fixed constant withdrawal strategy, with minimum withdrawal set to the constant withdrawal amount, the optimal strategy has a significantly higher expected average withdrawal, at the cost of a very small increase in ES risk. Tests on bootstrapped resampled historical market data indicate that this strategy is robust to parametric model misspecification.

q-fin.CP↗

arXiv2018

$ε$-Monotone Fourier Methods for Optimal Stochastic Control in Finance

Stochastic control problems in finance often involve complex controls at discrete times. As a result numerically solving such problems, for example using methods based on partial differential or integro-differential equations, inevitably give rise to low order accuracy, usually at most second order. In many cases one can make use of Fourier methods to efficiently advance solutions between control monitoring dates and then apply numerical optimization methods across decision times. However Fourier methods are not monotone and as a result give rise to possible violations of arbitrage inequalities. This is problematic in the context of control problems, where the control is determined by comparing value functions. In this paper we give a preprocessing step for Fourier methods which involves projecting the Green's function onto the set of linear basis functions. The resulting algorithm is guaranteed to be monotone (to within a tolerance), $\ell_\infty$-stable and satisfies an $ε$-discrete comparison principle. In addition the algorithm has the same complexity per step as a standard Fourier method while at the same time having second order accuracy for smooth problems.

q-fin.CP↗

arXiv2017

Are target date funds dinosaurs? Failure to adapt can lead to extinction

Investors in Target Date Funds are automatically switched from high risk to low risk assets as their retirements approach. Such funds have become very popular, but our analysis brings into question the rationale for them. Based on both a model with parameters fitted to historical returns and on bootstrap resampling, we find that adaptive investment strategies significantly outperform typical Target Date Fund strategies. This suggests that the vast majority of Target Date Funds are serving investors poorly.

q-fin.PM↗

arXiv2015

Weakly chained matrices, policy iteration, and impulse control

This work is motivated by numerical solutions to Hamilton-Jacobi-Bellman quasi-variational inequalities (HJBQVIs) associated with combined stochastic and impulse control problems. In particular, we consider (i) direct control, (ii) penalized, and (iii) semi-Lagrangian discretization schemes applied to the HJBQVI problem. Scheme (i) takes the form of a Bellman problem involving an operator which is not necessarily contractive. We consider the well-posedness of the Bellman problem and give sufficient conditions for convergence of the corresponding policy iteration. To do so, we use weakly chained diagonally dominant matrices, which give a graph-theoretic characterization of nonsingular weakly diagonally dominant M-matrices. We compare schemes (i)--(iii) under the following examples: (a) optimal control of the exchange rate, (b) optimal consumption with fixed and proportional transaction costs, and (c) pricing guaranteed minimum withdrawal benefits in variable annuities. We find that one should abstain from using scheme (i).

math.NA↗

arXiv2015

The existence of optimal bang-bang controls for GMxB contracts

A large collection of financial contracts offering guaranteed minimum benefits are often posed as control problems, in which at any point in the solution domain, a control is able to take any one of an uncountable number of values from the admissible set. Often, such contracts specify that the holder exert control at a finite number of deterministic times. The existence of an optimal bang-bang control, an optimal control taking on only a finite subset of values from the admissible set, is a common assumption in the literature. In this case, the numerical complexity of searching for an optimal control is considerably reduced. However, no rigorous treatment as to when an optimal bang-bang control exists is present in the literature. We provide the reader with a bang-bang principle from which the existence of such a control can be established for contracts satisfying some simple conditions. The bang-bang principle relies on the convexity and monotonicity of the solution and is developed using basic results in convex analysis and parabolic partial differential equations. We show that a guaranteed lifelong withdrawal benefit (GLWB) contract admits an optimal bang-bang control. In particular, we find that the holder of a GLWB can maximize a writer's losses by only ever performing nonwithdrawal, withdrawal at exactly the contract rate, or full surrender. We demonstrate that the related guaranteed minimum withdrawal benefit contract is not convexity preserving, and hence does not satisfy the bang-bang principle other than in certain degenerate cases.

q-fin.PR↗