Optimal Annuitization, Consumption, and Investment with Costly Capital Injections

We study a joint optimal decision of portfolio, consumption, and annuitization for a pension-plan manager who balances retirement consumption benefits against the cost of maintaining the solvency of pension account. The pension sponsor provides costly capital injection to prevent negative wealth. Annuity purchases are irreversible, and injected funds remain in the liquid account and earn interest. Using the Legendre transform, we derive a semi-explicit solution to the associated free-boundary problem and verify the optimality of the resulting strategies. We identify an endogenous annuity-income threshold separating injection-active and injection-inactive regions and characterize six policy regimes arising from the interplay between capital injections and consumption constraints. Compared with the no-injection benchmark, capital injections delay annuitization, alter the structure of the annuitization boundary, and permit positive holdings in risky assets even at zero liquid wealth. Numerical results further reveal that cheaper annuities may increase the expected capital contributions required from the pension sponsor.