Mortality improvement changes the timing of insurance, while the asset mix barely reacts. The paper normalises initial wealth to zero and the initial contribution rate to one, so every position is measured as a multiple of that first contribution. Replacing static 2017 mortality with the SOA projected table shifts the bond position by between -0.75 and +1.00. The stock position moves by between 0 and +0.5, against baseline bond positions ranging from about -150 to +150. Insurance premiums and insurance proportions change materially under the same switch. Bond and stock proportions scarcely change (Figures 4(c), 4(f), 4(d), 4(e)). The paper establishes an insurance result.
We could not test any of it.
The central object is a jointly optimised insurance premium, and we cannot trade or price life-insurance coverage. Substituting bond and equity ETFs would leave us with the two legs whose allocations barely respond to the mortality-table switch (Figures 4(d) and 4(e)). Our data also contain no mortality-improvement curves.
Feng, Li, Xu and Wei acknowledge this in two places. Their conclusion describes the insurance strategy as "particularly sensitive to this mortality change compared with bonds and stocks". The abstract shifts insurance "toward early adulthood to protect the high value of future income" before reducing it "significantly at later ages". Its other half argues that "longevity raises expectations of future contributions, allowing pension members to adopt a less risky investment strategy". Yet the abstract leads with investment guidance. Section 4.3 gives the opposite direction for equity, while Figures 4(d) and 4(e) show proportions that barely budge.
Closed form, three controls
A member enters a defined-contribution plan at age 22 with zero wealth and retires at 67. The objective seeks a target expected payout K while minimising the variance of that payout. Survival to 67 produces terminal wealth. Earlier death produces the account balance plus a life-insurance face value at the date of death.
The model has three controls: dollars invested in a rolling zero-coupon bond with a fixed 10-year time-to-maturity, dollars in an equity index, and a continuous stream of insurance premiums. A single-factor Vasicek process governs interest rates. Contributions follow a geometric Brownian motion with deterministic age-linear drift. Each premium buys face value equal to the premium divided by the force of mortality, making coverage actuarially fair and unloaded.
The martingale approach supplies the technical shortcut. The authors solve for the terminal payoff first, then recover the controls. Construction makes the market complete, reducing the dynamic control problem to a static Lagrangian over terminal wealth and bequest value. Both are affine in the pricing kernel. Closed-form wealth, portfolio and insurance paths follow, along with a verification theorem establishing global optimality.
On the efficient branch, K is at or above W_0^eff/A(0,r0), and the efficient frontier is exactly linear in the standard deviation of the mortality-contingent payout. That linearity remains under stochastic rates, stochastic contributions and a random death date. Across the range considered, the authors also find a non-monotonic effect from the interest-rate risk price on the frontier.
Munk and Sørensen (2010) supply the market parameters. The authors say these "are not separately estimated in this paper; they are adopted from standard values in the literature". Annualised mean reversion is 0.50. The long-run and initial rates are both 0.02, with rate vol at 0.02. Equity vol is 0.20 and contribution vol is 0.20. The equity risk price is 0.20; the interest-rate risk price is 0.
Mortality is estimated separately. The specification is a generalized Gompertz-Makeham GM(2,3), selected from a 3x4 grid of polynomial orders using coefficient significance and BIC (258,111.64). It uses U.S. mortality rates for ages 22 to 67 from the SOA Mortality Improvement Model MIM-2021-v4. For the projected table, the paper reads diagonally from age 22 in 2017 to age 67 in 2062. Euler-Maruyama simulation uses 400 steps and 10,000 paths.
A target of K = 200 corresponds to roughly a 49.3% replacement ratio, assuming a 10% contribution rate and 20 years of level retirement income. The comparison is the OECD gross replacement rate for full-career average earners, about 52%. Targets K = 100 and K = 300 produce 24.7% and 74.0%. This is a closed-form model rather than an empirical strategy study. It has no asset-price dataset, return, Sharpe or backtest, which suits the question being asked.
Completeness carries the result
The model imposes completeness by setting the squared bond loading plus the squared adjusted stock loading of contribution risk equal to one. At baseline, the bond loading is zero and the stock loading is one. Contribution volatility is 20%, matching equity volatility. The member's wage stream therefore has equity volatility, is perfectly correlated with the stock, and can be priced through it. Its present value enters wealth as a tradable asset. Contribution risk loads 1.00 on the equity shock and 0.00 on the rate shock. Every hedging demand consequently addresses a replicable risk.
The optimizer uses that freedom. Reported bond proportions range from about -2.0 to +1.0, while the stock proportion reaches about 2.5. The wealth dynamics impose no constraints on short sales, borrowing or gross exposure. They also include no transaction cost or contribution cap. Mortality remains strictly independent of the market filtration, excluding longevity-equity correlation and any systematic mortality risk premium.
The static Lagrangian yields a pre-commitment solution in the standard mean-variance sense. The plan is chosen at inception and never re-optimised. Variance penalises upside.
Does longevity raise equity risk?
The abstract says longevity "raises expectations of future contributions, allowing pension members to adopt a less risky investment strategy". Section 4.3 says mortality improvements "overall make pension members more risk-seeking in their investments" and reports more stock at every age under the projected table. Those statements send equity in opposite directions.
The authors immediately narrow the claim: "this investment change is relatively small and doesn't change the proportions allocated between bond and stock." Taking that qualification seriously leaves little of either the contradiction or the investment conclusion.
The insurance finding survives. At younger ages, projected mortality remains close to the static table. It falls substantially through the 50s and 60s, corresponding to the 2040s and 2050s for this cohort. Across ages 22 to 67, the log force of mortality runs roughly -6.5 to -4.0. Greater expected human capital draws protection into early adulthood and reduces cover during mid-to-late life.
Even at baseline, the plotted premium path remains below 0.7 and the insurance proportion below 0.008 of wealth. Giving the optimizer an insurance control changes the timing of protection much more than the balance sheet. The sensitivity runs tell the same story. Increasing income drift from 0 to 0.0001 to 0.0002 reduces the stock position and increases insurance demand. Raising contribution volatility from 0.15 through 0.20 to 0.25 reduces bond, stock and insurance allocations. The paper identifies one exception: at early ages the stock amount rises slightly, although the stock proportion falls overall. Each run varies one parameter at a time.
For a plan designer
A group-cover calibration based on a single base-year table places the error in the age profile of coverage. The asset mix barely notices (Figures 4(d) and 4(e)). For trustees, the result is small in size and precise in shape. The plotted insurance proportion stays below 0.008 of wealth, peaks in early adulthood, then declines through mid-to-late life. The authors explicitly note that Australian superannuation deducts life premiums directly from member balances. Under their model, face value equals the premium divided by the force of mortality, without loading.
An unspanned-income version would change my mind. Restore the idiosyncratic component of the contribution vector, then show that the insurance-timing shift survives once human capital ceases to be replicable. The paper solves an easier problem.
The paper carries a memorial note for Pengyu Wei, one of its authors.