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Large tontine books diversify 97% of the spouse-cost tail

Capped payments average AUD 433k to 440k over 35 years; uncapping roughly quadruples the residual market tail.

2026-10-01 · 5 min read · Retirement portfolio allocation and decumulation · None; the required tontine exposure cannot be traded or replicated with the available instruments

Reviewing: Spouse-Protected Tontines: Household Decumulation via Neural-Network Optimization · Duy-Minh Dang and Yukan Perumal · Read it on arxiv

Dang and Perumal's 97% diversification result should change how a large tontine book is buffered. It leaves the expected spouse bill untouched. In the baseline capped runs, the authors find that sizing a buffer to one contract's excess tail (1027.5 to 1096.2 thousand) overstates the large-book requirement by about 40x. Holding no buffer understates it by 24.9 to 29.0.

The household keeps its account after a first death

An ordinary individual tontine forfeits the retiree's account at death and distributes its balance as mortality credits to surviving accounts. Here, a household contract passes through four states: both alive (11), retiree only (10), spouse only (01), and neither (00). Only 00 ends the contract. The first death changes the state; the account stays to fund the survivor, with no transfer.

Each active account earns a credit equal to its own balance multiplied by exit probability over survival probability. In state 11, both partners must die in the same year for the contract to exit. The rate therefore uses the product of their annual death probabilities, a small quantity. Spouse protection, the authors write, "delays forfeitures and attenuates mortality-credit gains" while both partners live.

The household chooses withdrawals and a four-asset allocation while the retiree lives. Calibrated to ASFA, the couple's spending band is 51.299 to 77.375 thousand real AUD a year; the single band is 35.503 to 54.840. If the retiree dies first, the spouse receives the last withdrawal, capped at 54.840, and the portfolio rebalances passively.

The objective maximizes expected cumulative real payments minus λ times CVaR at 95% of terminal shortfall. Reserve targets are 720.93 for a couple and 522.35 for a single survivor. Two small networks, each with two hidden layers of 10 nodes, learn the controls from 256,000 block-bootstrapped paths. Those paths resample monthly real AUD returns from 1935 to 2022, using a 24-month expected block. The starting household is a 65-year-old man and a 60-year-old woman with AUD 1 million; the horizon is 35 years.

How much does the spouse cost?

A great deal, across the policies studied. Under 2021 Australian period tables, 71% of households enter the spouse-only state and spend an average of 12.25 years there within the 35-year horizon. For Australia, Canada, the US, Japan and the Netherlands, entry ranges from 68.1% to 77.0%, with conditional duration of 12.25 to 13.34 years. This is a mortality-only comparison. The paper does not recalibrate returns, spending bands or product rules by country.

Moving from λ=0.3 to λ=1 reduces terminal-shortfall CVaR by 31.0% and ever-insolvency by 14.1 points. Expected payments fall 8.4%. Yet expected capped continuation payments, cumulative over 35 years, remain between 433.1 and 440.1 thousand across the three policies. Retiree-alive payments fall by 270.5, from 1422.2 at λ=0.3 to 1151.7 at λ=15. The spouse's share of expected total payments, the expected-cost load, consequently rises from 23.5% to 27.3%. The more conservative household still owes nearly the same spouse bill, and that bill takes a larger share.

What a large book sheds

The excess spouse-cost tail is CVaR at the 5% upper tail of continuation payments minus the mean. On one contract, it reaches 1027.5 to 1096.2, more than twice the mean. About 97% reflects household mortality: the timing of the retiree's death and the length of the spouse's survival. Those risks diversify across households.

Put J contracts on one market path, with deaths independent across households. At J=1000, the excess tail falls by 95% to 96%. In the conditional limit, which takes expected cost given the market path, 24.9 to 29.0 remains. This is the common-market component. D_c, the diversified share, reaches 97.35% to 97.62%. With a buffer coefficient of 1, the risk-loaded load falls from 51.9% to 55.9% to 24.7% to 28.5%, about one point above the expected-cost load.

The mean stays put.

Linearity of expectation gives the same mean at every book size, and the tower property gives it in the limit. Estimates differ by under 0.2%. Convergence appeared faster than the square-root heuristic: at J=1000, the normalized gap was 0.018 to 0.024 against 0.0316. The authors correctly say this does not establish an empirical rate.

The cap matters more. Removing it under the same learned policies raises expected continuation cost to 476.3 to 555.6. The residual market tail roughly quadruples, to 103.6 to 128.4, while D_c drops to 87.91% to 92.96%. The cap binds for 50.3% of households under the payment-seeking policy and 22.7% under the conservative one. More contracts cannot diversify that market-driven remainder. The authors keep policies fixed without reoptimization, so the comparison "does not establish an optimal cap." Its funding benefit, they add, "must be weighed against the restriction on the payment inherited by the spouse."

Before this becomes a product

The paper derives a finite-pool rule that rescales credits until payouts exactly match realized forfeitures. It then sets the scaling factor to 1 throughout. The authors warn that a large pool may still have material bias under spouse protection, since state 11 generates few forfeitures. Every reported credit stream therefore pays the fair rate with certainty.

Insolvency also has a forgiving treatment. Even the conservative policy becomes insolvent at some point on 12.0% of paths; the payment-seeking policy does so on 26.7%. Minimum payments continue afterward, funded by debt at the domestic bond return plus 2%, and count in the reward. CVaR penalizes that debt only if someone survives to year 35. Debt remaining when the household becomes extinct sits outside the model. Mortality comes from fixed 2021 tables and assumes independent spouses, which the authors identify as future work. Training and evaluation both resample the same 1935 to 2022 history. Monte Carlo errors of 0.6 to 0.7 on reward exclude retraining with other seeds.

We could not run this. Account returns include mortality credits transferred from dead households within a closed pool; no ETF or index position can produce those transfers.

I would want to see the finite-pool scaling switched on in a pool of realistic size. If state 11 credit shortfalls widen the 24.9 to 29.0 residual, that wider figure is what a product's buffer would have to cover.