The three retirement guarantees look cheap at a 4.9% riskless rate. At a 3.3% net bill rate, none is feasible. That gap matters more to a trader than the elegance of the solution.
Ren, Wang, Wong and Yam prove that optimal wealth cannot cross its floor in their model (Theorem 4.11). Across 10,000 simulated paths, the 5th and 20th percentiles of wealth and benefits remain at or above each floor. Prices and rebalancing are continuous in the model, so wealth cannot jump through the boundary. The control theory is careful. The practical claim about the price of a guarantee needs closer scrutiny.
The policy and its floor
A retiree gives a manager a lump sum x0. The manager continuously chooses the risky-asset weight π and the benefit rate b. Wealth must remain above αx0, while benefits must remain above b0. Funding b0 indefinitely from interest alone takes at least b0/r in wealth, with r denoting the riskless rate after fees. The effective wealth floor is the larger of αx0 and b0/r. The objective combines the retiree's utility from benefits and bequest with the manager's utility from fees. Death arrives at the constant rate λ = 0.042, corresponding to 23.6 remaining years at age 60.
The authors establish existence and uniqueness of a classical solution to the HJB equation for general utilities. They invert marginal value to turn the fully nonlinear equation into a semilinear ODE. For αx0 strictly above b0/r, a shooting argument determines the unknown boundary slope. For αx0 ≤ b0/r, they apply the Schauder fixed point theorem. Near the floor, the value function's curvature diverges and the optimal risky weight tends to zero. The optimal π also stays nonnegative without an imposed short-selling constraint. As the authors put it, "the no short-selling feature emerges endogenously from the very nature of the problem."
In a daily SPY-and-cash test, this rule becomes a lookup table. Solve the ODE, then tabulate π(x) and b(x) = max(b0, (U')⁻¹(v'(x))) across a wealth grid. Today's wealth determines the equity weight and withdrawal rate.
What do the reported percentages measure?
With x0 = 1 (in units of USD 100,000), the constrained policy loses 1.12%, 0.66% and 0.89% of total value against the unconstrained policy. Those figures correspond, respectively, to (b0, α) = (0.035, 0.95), (0.035, 0.8) and (0.04, 0.8). The retiree's losses are 2.22%, 1.24% and 1.32%. The manager's are −0.81%, −0.38% and 0.10%; the first two mean the manager fares better with the constraint.
The paper says the cost "is at most 2.22% for either the retiree or the fund manager across the three baseline scenarios". That describes those scenarios accurately. The authors acknowledge that most of the cost falls on the retiree, then call the price acceptable because the manager's utility changes "only marginally". Under joint variation, the manager's cost reaches 6.42%. The phrase "only marginally" belongs to the three baseline scenarios.
There is also a problem with reading these percentages as economic prices. The authors note that a constant can be added to the utilities without changing the optimal strategies; they assume positive utilities without loss of generality. Yet such an addition changes (V_un − V)/V_un. A 1.12% loss measured against a freely shiftable utility level is no certainty-equivalent figure. The baseline calculations use pure CRRA x^γ/γ without an added constant, so this concerns interpretation rather than the arithmetic. The authors acknowledge a related dependence: with heterogeneous exponents, the USD 100,000 unit sets the relative cardinal scaling of retiree and manager utility, and therefore their implicit weights. Joint variation of α, b0, γ and θ takes the largest costs to 9.01% for the retiree and 8.21% combined. The authors call those costs moderate. For the retiree, 9.01% strains that description.
The simulated paths show why a floor appeals. Unconstrained 5th-percentile wealth slips below 0.8 after about 0.71 years; the 20th percentile does so after about 3.4 years. Unconstrained 5th-percentile benefits fall below 0.035 after about 1.25 years. Under the constraint, the 5th and 20th percentiles remain at or above their floors. These are model paths, driven by μ = 8% and σ = 20% from PGIM's forward-looking ten-year assumptions. A policy respecting the constraint in the model that defines it confirms the solver. We did not find any test against historical returns.
The rate behind the guarantee
The paper uses r1 = 5.3%, the 30-year zero-coupon Treasury yield, as the fund's riskless rate. After its 0.4% fee, r = 4.9%. The authors argue that the fund earns a liquidity premium unavailable to individuals; they also say the theory requires only r > 0.
The numerical guarantees face a tighter condition. The paper requires b0 ≤ r x0 and says no admissible strategy exists otherwise. Use the 3.7% four-week bill rate for the cash sleeve, subtract the same 0.4% fee, and r becomes 3.3%. Each b0, 0.035 or 0.04, exceeds 0.033. All three guarantees are infeasible at that rate.
The baseline guarantees thus depend on treating a 30-year zero-coupon yield as riskless. The floor calculation assumes away that bond's price risk.
Daily bars, discrete trades
We are building a discretized construction of the policy, with a narrow scope. It approximates the continuous-time solution on SPY and a modeled cash sleeve. Daily closes cannot establish whether a pathwise floor held between observations. A large one-day fall could carry wealth through the floor before rebalancing; the paper's guarantee says nothing about that case.
Our construction rebalances once a day at the close using π(x) from the tabulated grid. It accrues benefits daily at b(x)/252 and records b0 separately from any excess. After a breach, it moves fully into cash and continues paying b0. Below b0/r, the shortfall grows because interest rX no longer funds b0, the reasoning behind the paper's Lemma 2.1.
Breach frequency and depth are the first outputs to judge, followed by cumulative benefits split between guaranteed and discretionary payments. Account value comes last. The benefit-to-wealth ratio falls and then rises as wealth increases. Above about 1.02, it exceeds the 3.7% bill rate in all three constrained scenarios; a retiree at x0 = 1 starts just below that level.
The theory holds up. Its 0.66% to 1.12% figures come from continuous prices and a 4.9% riskless rate, while a 3.3% net bill rate makes the three baseline guarantees infeasible. Our daily construction can measure how often a discretized version breaches its floor at a cash rate a retiree can earn. It cannot adjudicate the 0.66% to 1.12% figures inside the paper's model, and one automated pass would not be a verdict on the authors' work.