Capital injections delay annuitization and preserve risk at zero wealth

At zero liquid wealth, the pension manager should still hold a positive risky position, with the sponsor absorbing its losses. This is the strongest proved result in Huang, Liang, Xiong and Yu. Their model charges β per unit for the minimum capital injection needed to guarantee solvency. The claim closing the abstract, that cheaper annuities can cost the sponsor more, has a narrower basis: a Monte Carlo plot under a single calibration. The authors themselves say it "may" happen.

Who pays at zero wealth?

Liquid wealth and existing annuity income a are the model's two state variables. The manager chooses a dollar position in one risky asset, a non-negative consumption rate and irreversible purchases of life annuity income.

An annuity income unit costs θ = 1/(r+λo). With benchmark values r = 0.04 and λo = 0.04, the premium is 12.5 per unit. Preferences are CARA, with γ = 0.02.

The sponsor supplies just enough cash to keep wealth at zero whenever it would otherwise turn negative. That cash remains in the account, earning r. Each injected unit incurs a penalty of β in the objective, discounted at δ = ρ + λs. The benchmark sets the discount rate to 0.10 and β = 1.20. The paper specifies that injections "are available solely to fulfill the solvency guarantee and cannot be used as a discretionary borrowing facility."

To solve the policy, the authors use a Legendre transform. It converts the free-boundary problem into an Euler ODE in the dual variable; a verification theorem completes the solution. An income threshold, a_β, separates the states that require injections from those that do not. For β ≥ 1, the threshold equals κ/(γβ^{1/κ}), with κ a constant between 1/(1-ν-) and 1. Below a_β, the marginal-value cap v_x(0,a) = β binds and the sponsor injects cash whenever wealth reaches zero. Above the threshold, the optimal policy maintains non-negative wealth without support. Combining those two injection states with zero, mixed or positive consumption yields the six regimes.

The trading implication is in the risky position at the floor: π(0,a) = ((μ-r)/σ²)(κ/m)(a_β - a). This is strictly positive when a is below a_β. Liang and Young (2018), the unfunded benchmark, instead forces the position to 0. With sponsor support, risk at the floor declines linearly as income approaches a_β. The sponsor absorbs every loss on that position through reflection.

Consumption at zero wealth equals (-log β)^+/γ. It is zero at β = 1.20. Below 1, β makes consumption positive, and injections "may then finance consumption as well as offset investment losses." The model defines β as a utility-equivalent cost. It never translates that penalty into the sponsor's dollar funding cost.

The guarantee is priced in utils.

A straight boundary for funded annuitization

With injections active, the annuitization boundary takes the form b(a) = b^N(a_β) + θ(a_β - a). The slope is -θ, or -12.5 at the benchmark. Thus b(a) + θa stays constant along the boundary. In the authors' words, "annuity income and liquid wealth act as substitutes along this part of the boundary."

For a < a_β, Figure 5 places the funded boundary above the unfunded boundary; they meet at a_β. The authors interpret this ordering as the guarantee increasing "the option value of remaining in the continuation region," which "delays annuity purchases." The ordering holds beyond this calibration. Combining (3.54) with Lemma 3.3's (b^N)'(a) > -θ gives b^I(a) > b^N(a) for every a < a_β. Figure 5(a) supplies the paper's illustration; the general argument is ours.

Purchases work differently on this segment too. Buying income moves the state in direction (1, -θ), tangent to the active boundary. Continuous purchases therefore cannot keep the state below it. The optimal action is a discrete jump to (b(a_β), a_β), followed by ordinary reflection. Because the value function is constant along the active segment, that jump carries no loss of value.

The authors allow second derivatives to differ across this segment, a weaker condition than classical C² smooth fit. In Theorem 3.6, second derivatives at the boundaries are one-sided limits taken from the adjacent smooth regions. This part is proved. We accept it.

How far does the cheaper-annuity result go?

The abstract states: "Numerical results further reveal that cheaper annuities may increase the expected capital contributions required from the pension sponsor." Figure 7(b) is the evidence. Starting wealth is 10, initial income takes values 2, 4, 6 and 8, and λo ranges from 0.02 to 0.10. According to the text, injections "generally increase with λo under the benchmark calibration."

The proposed explanation runs through a_β: a higher λo raises the threshold, expanding the income range in which support is used. The design cleanly isolates the price effect. The insurer prices mortality using λo, whereas λs governs the retiree's horizon. Increasing λo therefore reduces annuity prices without altering δ.

But the explanation depends on the sign of ∂a_β/∂λo. The expression for a_β is explicit in κ; κ in turn depends on θ through z_0b, the root of (3.21). The authors had the ingredients to establish the sign or a bound. We did not find a proof. Rising a_β appears only in the plotted line in Figure 6(c). Nor did we find the simulated path count, time step or horizon truncation used for Figure 7.

The subjective-mortality finding has the same evidential status. At a = 2, the funded boundary falls and then rises as λs increases from 0.02 to 0.12. The unfunded boundary rises monotonically. Figure 6(d) evaluates only a = 2 and keeps all other parameters at their benchmark values.

No backtest, and an unresolved model calculation

We could not backtest this model. Our data has neither immediate life annuity contracts nor historical premiums. Substituting bond ETFs would strip out the mortality-contingent income that determines every boundary. Daily bars also cannot reproduce continuous reflection at zero wealth. A discretized run would therefore test an approximation of the guarantee.

Within the model, Figure 7 still leaves path counts, standard errors, the time step and horizon truncation unspecified. An analytic sign for ∂a_β/∂λo would make the cheaper-annuity finding a property of the model. So would evidence that the Figure 7(b) pattern persists across the β grid of about 0.6 to 1.8 already plotted in Figure 7(a), including values below 1.