Pension accounts can't short or borrow: a reparameterized neural network cuts the duality gap to 2.87%
Synopsis
A defined-contribution pension member invests in cash, two nominal bonds, an inflation-linked bond and a stock index in a two-factor market, and may also buy term life insurance before retirement, with trading restricted to a non-empty closed convex constraint set. The approach builds a family of unconstrained artificial markets, computes dual upper bounds with a reparameterized neural network, and recovers feasible lower bounds via nested Monte Carlo and a Hamiltonian quadratic program. Relative duality gaps under two risk-aversion settings are below the SAMS and deep pricing kernel benchmarks, one of them at 2.87%.
Interpretation
Investment and life-insurance decisions for a defined-contribution pension can now be solved under realistic trading limits, without assuming the member can freely short and borrow. Earlier constrained pension studies mostly used simpler markets, specific constraint types, or did not model life insurance jointly; unconstrained solutions turn aggressive with leverage when maximizing retirement benefits. Under two risk-aversion settings the relative duality gaps are below the SAMS and deep pricing kernel benchmarks, one of them at 2.87%; the direct neural network's utility estimates fall outside the reported intervals.
Constraints enter through a family of artificial markets: drift terms of risky assets are adjusted so a rational investor chooses to trade in a constraint-compliant way. The reparameterized neural network parameterizes the transformed compensated drift terms directly and recovers the original quantity through a minimal nonnegative shift, avoiding an inverse transform on a near-singular volatility matrix. The calibrated market volatility matrix has a condition number indicating near singularity; the lower bound uses nested Monte Carlo for conditional moments and a Hamiltonian quadratic program to project candidate strategies back onto the original constraint set.
The constrained solution differs substantially from the unconstrained closed-form solution rather than being a small perturbation. The unconstrained benchmark shorts the short-term nominal bond heavily to finance long positions in other assets and insurance, violating the account rule against borrowing. Expected terminal wealth under the unconstrained benchmark exceeds that of the constrained policy by several orders of magnitude, with wealth, holdings and insurance positions enlarged accordingly.
Perspective
The demonstrated scope is a two-factor market calibrated to monthly U.S. data, with assets of cash, 3-year and 10-year nominal bonds, a 10-year inflation-linked bond and the stock index, under a portfolio-mix constraint banning short selling and borrowing; numerical comparisons use 10,000 paths and 210 time steps.
At lower risk aversion, the optimal life-insurance strategy turns negative after the early accumulation stage, interpreted as annuity demand; this sign reading depends on the setup where life insurance and annuity cash flows are opposite. The constrained-versus-unconstrained comparison rests on simulated paths, not actual account data.