Money-Back Guarantees (MBGs) in Finance & Insurance

• MBGs are contractual provisions that return purchase premiums under defined contingencies such as early death, product returns, or cost inefficiencies.
• They play a crucial role in retirement annuities, tontine decumulation, and multiproduct bundling by embedding elements of life insurance, risk screening, and consumer learning.
• Analytical approaches include actuarial pricing, recursive fixed-point models, and dynamic programming to reveal how refund structures impact pricing and risk exposures.

Money-back guarantees (MBGs) are contractual provisions that ensure an initial premium or purchase price is returned, fully or partially, under specified contingencies such as early death, product return, or low realized value. In the cited literature, MBGs appear in life-contingent retirement products, pooled decumulation arrangements, consumer refund mechanisms, and bundling schemes with return rights. Although the contractual environments differ, the common design feature is a contingent transfer that changes participation incentives, ex post allocation, and the mapping between price and risk exposure (Milevsky et al., 2021, Orozco et al., 18 Feb 2026, Lyu, 2024, Alaei et al., 5 Jul 2025, Hinnosaar et al., 2018, Ma et al., 2015).

1. Conceptual scope and contractual forms

The annuity literature uses MBGs to describe refundability in immediate income annuities. A life-only income annuity pays a fixed income for life and stops at death, whereas a cash-refund income annuity pays for life and, if death occurs before the present value of payments received equals the premium $P$, pays a lump sum equal to $\max\{0, P - \text{value of income already paid}\}$. An installment-refund annuity instead continues post-death income payments until the premium is fully returned in discounted terms (Milevsky et al., 2021).

In pooled retirement decumulation, an MBG overlay guarantees that the member’s initial purchase price $L_0$ is “paid back” either through lifetime withdrawals or, if death occurs earlier, through a death benefit to the estate equal to the nominal shortfall. In the formulation studied for individual tontine accounts, the guarantee settles only upon death and therefore does not alter the retiree’s account dynamics while alive; it is priced ex post under the induced optimal decumulation policy (Orozco et al., 18 Feb 2026).

In product-market mechanism design, MBGs take the form of return policies. A deterministic refund contract is an option $(P,R)$ in which the buyer pays price $P$ up front and, after learning realized value $v$, returns the item if and only if $v \le R$. Other models allow stochastic return policies, partial refunds, or free returns. In a different but related application, Pure Bundling with Disposal for Cost allows a buyer who purchases a bundle to return any subset of items for their production costs, which the paper interprets as an MBG-like mechanism calibrated to costs rather than dissatisfaction (Alaei et al., 5 Jul 2025, Lyu, 2024, Ma et al., 2015).

This suggests a useful unifying distinction between MBGs that insure against mortality or longevity states, MBGs that insure against ex post valuation risk, and MBGs that insure against overinclusion in multiproduct sales. The guarantee is not a single economic object; its role depends on whether the underlying friction is bequest demand, retirement spending risk, private learning, informational robustness, or production cost.

2. Refundable annuities and the actuarial economics of MBGs

Milevsky and Salisbury study refundable income annuities as a quantitatively important but analytically neglected class of retirement products. They document that refundable income annuities represented 59.4% of U.S. income-annuity quotes in Q1 2021, up from 18.5% in 2011, while life-only contracts fell to 10.6% from 25.3%. Their central point is that refundability makes valuation recursive because the premium appears inside the benefit formula itself, especially for cash-refund designs (Milevsky et al., 2021).

With age at issue $x$, future lifetime $T_x$, hazard $\mu_x(t)$, survival

$\max\{0, P - \text{value of income already paid}\}$0

and constant valuation rate $\max\{0, P - \text{value of income already paid}\}$1, the standard life-only actuarial present value for continuous payout rate $\max\{0, P - \text{value of income already paid}\}$2 is

$\max\{0, P - \text{value of income already paid}\}$3

For a cash-refund annuity, discounted cumulative income paid by time $\max\{0, P - \text{value of income already paid}\}$4 is

$\max\{0, P - \text{value of income already paid}\}$5

and the refund paid at death is

$\max\{0, P - \text{value of income already paid}\}$6

Pricing with insurer loading $\max\{0, P - \text{value of income already paid}\}$7 therefore becomes a fixed-point problem: $\max\{0, P - \text{value of income already paid}\}$8

The unloaded cash-refund price per unit payout, $\max\{0, P - \text{value of income already paid}\}$9, satisfies the recursive identity

$L_0$0

The paper also gives an equivalent two-account form,

$L_0$1

with a phase-one account funding early payments and generating interest that finances phase-two liabilities. For $L_0$2 and standard increasing hazard $L_0$3, the unloaded cash-refund price is unique, decreases in both $L_0$4 and age $L_0$5, and exceeds the life-only value. A monotone bisection algorithm on $L_0$6 is sufficient because the associated integral equation has a unique crossing.

Installment-refund annuities are structurally simpler. Their unloaded price per unit payout, $L_0$7, satisfies

$L_0$8

and with loading $L_0$9,

$(P,R)$0

For $(P,R)$1, a unique installment-refund price always exists, it decreases in $(P,R)$2 and $(P,R)$3, and it is strictly below the corresponding cash-refund price.

The most counterintuitive result concerns loaded cash-refund annuities. Whereas a loaded life-only price is simply $(P,R)$4 and decreases with age, the loaded cash-refund fixed point may imply that older buyers pay more than younger ones. Writing $(P,R)$5 and $(P,R)$6, the loaded cash-refund equation becomes

$(P,R)$7

where

$(P,R)$8

The additional deterministic term $(P,R)$9 can shift the solution rightward as survival declines, yielding $P$0 at older ages. Viability also requires a threshold valuation rate: $P$1 If this fails, no finite loaded cash-refund price exists. The paper links this directly to the disappearance of inflation-adjusted cash-refund annuities when real rates are too low.

The same framework alters duration and money’s-worth diagnostics. For life-only annuities, duration $P$2 coincides with Macaulay duration, but for refundable annuities recursivity breaks that equivalence. The paper reports that unloaded cash-refund and installment-refund durations exceed life-only duration by 8–10 years around retirement ages, while loaded cash-refund duration can “blow up” near the age at which viability fails. Reported empirical money’s-worth ratios using Fidelity quotes and a Gompertz calibration with $P$3, $P$4 for males and $P$5 for females, include approximately 0.996 for life-only and 1.031 for cash-refund at male age 65, and approximately 1.002 versus 1.017 at age 80. The paper interprets the higher cash-refund ratio as reflecting the embedded life-insurance component and possible selection differences.

3. MBG overlays in tontine-based retirement decumulation

The tontine literature extends MBGs from insurer-issued annuities to pooled retirement income products. In the individual money-back tontine analyzed in 2026, the retiree contributes $P$6 at inception and receives withdrawals at annual decision dates $P$7, while the estate receives a death benefit equal to the remaining nominal shortfall if death occurs before nominal withdrawals sum to $P$8. Because settlement occurs only at death, the MBG does not enter the wealth recursion while the retiree is alive; instead, it is priced ex post under the optimal policy induced by the decumulation problem (Orozco et al., 18 Feb 2026).

The account dynamics combine mortality credits, fees, withdrawals, and dynamic asset allocation. For a large approximately homogeneous pool, the per-member mortality credit at time $P$9 is

$v$0

where $v$1 is the one-year conditional death probability and $v$2 is pre-withdrawal wealth after credits and fees. The retiree then withdraws $v$3 subject to bounds and rebalances across domestic stock and bonds and foreign stock and bonds, all expressed in real domestic-currency terms. The optimization is solved under a “plan-to-live” convention, with stochastic mortality affecting outcomes only through mortality credits at the pool level.

The control objective is an expected-withdrawal/terminal-risk trade-off. Expected total real withdrawals are

$v$4

and risk is measured by $v$5, using the Rockafellar–Uryasev representation inside a scalarized pre-commitment objective. The paper solves the resulting high-dimensional constrained problem with two feedforward fully connected neural networks: a withdrawal network with sigmoid output scaled to the feasible range, and a rebalancing network with softmax output to enforce nonnegativity and sum-to-one constraints. Training uses Adam on simulated paths generated from either a Kou double-exponential jump–diffusion benchmark or a stationary block bootstrap of long-horizon historical monthly real AUD returns.

The ex post MBG payout is the real-dollar value of the nominal shortfall at death, converted through CPI. If death is recorded in $v$6, the guarantee cost reported at $v$7 is

$v$8

with zero real discounting. The equivalent up-front MBG load factor is determined by

$v$9

so the total load decomposes into expected guarantee cost and a prudential tail buffer.

Quantitatively, the paper finds that international diversification and tontine pooling jointly generate the largest improvements in the $v \le R$0–$v \le R$1 frontier, while stochastic mortality modeled through Lee–Carter or Cairns–Blake–Dowd shifts the frontier only modestly. At $v \le R$2 thousand real AUD, the four-asset tontine supports expected annual withdrawals around 70, compared with 40 for the “4% rule” benchmark. Optimal controls use domestic bonds as the main stabilizer in moderate and high wealth states and shift aggressively into U.S. equities in low-wealth states as a state-dependent catch-up instrument. For the base case with four assets, Lee–Carter mortality, $v \le R$3, $v \le R$4, $v \le R$5, and $v \le R$6, the paper reports $v \le R$7, $v \le R$8, and $v \le R$9. Under actuarially fair pricing $x$0, the load is approximately 5–7%, implying that tail capital rather than mean payouts is the dominant driver of MBG cost.

4. Consumer learning, free returns, and robust MBG pricing

A distinct MBG literature studies experience goods when buyers learn their valuations only gradually or only after purchase. In the dynamic-learning model of refund design, the buyer’s value is binary, $x$1, the common prior of high value is $x$2, and learning follows a good-news Poisson process with flow cost $x$3. The seller may offer a stochastic return policy, but the main result is that the optimal refund mechanism is deterministic: either a sufficiently low non-refundable price that deters learning, or a sufficiently high price with free returns that induces maximal learning. The optimal regime is non-monotone in the buyer’s prior, with no-return optimal at very low and very high priors and MBG/free-return optimal at intermediate priors (Lyu, 2024).

The buyer’s pre-purchase option value satisfies a Bellman equation, and the stopping region is characterized by the quitting belief

$x$4

and the trial belief $x$5. For priors in the interior learning region, stochastic policies can make demand increase with price because a higher price can be paired with a return policy that prolongs learning and raises the chance of sale. Yet, once price is optimized jointly with refunds, revenue becomes quasi-convex in price, so the optimum lies at boundary prices where the return policy is deterministic. In the MBG regime, seller profit under full learning at price $x$6 is

$x$7

This result directly rejects the idea that richer stochastic refund menus are necessarily superior when price is endogenous.

A related but more robust framework studies a seller who knows only the buyer’s mean prior $x$8, not the information structure $x$9 generating posterior beliefs $T_x$0. With price-refund pair $T_x$1, a buyer purchases if

$T_x$2

and under a refundable offer returns if $T_x$3. The paper defines normalized restocking cost

$T_x$4

shows that the fully refundable full-price offer $T_x$5 is a “generous refund,” and derives its expected profit as

$T_x$6

This MBG performs best when buyers are relatively informed.

When buyers may instead be relatively uninformed, the robust mechanism mixes the generous refund with non-refundable random discounts. The random-discount component draws a non-refundable price from a log-uniform distribution on $T_x$7, and the mixing weight is

$T_x$8

The main theorem is

$T_x$9

so the robust refund policy achieves the best guaranteed profit against all feasible information structures. For $\mu_x(t)$0, the robust policy is the generous refund alone; for higher return costs it becomes a mixture of MBG and non-refundable random discounting (Hinnosaar et al., 2018).

Taken together, these two papers imply that free-return MBGs are optimal only under specific informational objectives. They can be optimal as a learning-inducing boundary solution, and they can be part of a max–min robust policy, but neither result implies that full refunds are generically optimal once heterogeneity, incentive constraints, or tail capital are made explicit.

5. Deterministic refund mechanisms and virtual-value screening

The 2025 paper on deterministic refund mechanisms provides a general revenue-maximizing framework for MBGs when a buyer knows her ex ante type $\mu_x(t)$1 but learns realized value $\mu_x(t)$2 only after receiving the item. Types are ordered by first-order stochastic dominance: $\mu_x(t)$3 FOSD-dominates $\mu_x(t)$4 whenever $\mu_x(t)$5. A deterministic MBG is a contract $\mu_x(t)$6, and under quasi-linearity the buyer returns if and only if

$\mu_x(t)$7

This induces a second-stage threshold allocation $\mu_x(t)$8, while expected seller revenue from type $\mu_x(t)$9 is

$\max\{0, P - \text{value of income already paid}\}$00

in the zero-salvage benchmark (Alaei et al., 5 Jul 2025).

The paper’s central characterization is incentive-theoretic. For deterministic mechanisms, first-stage IC is equivalent to the allocation being weakly increasing in both $\max\{0, P - \text{value of income already paid}\}$01 and $\max\{0, P - \text{value of income already paid}\}$02, which in turn is equivalent to refunds $\max\{0, P - \text{value of income already paid}\}$03 being weakly decreasing in type. Prices are pinned down by payment identities based on

$\max\{0, P - \text{value of income already paid}\}$04

with, for discrete types $\max\{0, P - \text{value of income already paid}\}$05,

$\max\{0, P - \text{value of income already paid}\}$06

Expected revenue can then be written as expected virtual surplus. In the continuous setting,

$\max\{0, P - \text{value of income already paid}\}$07

and the optimal deterministic MBG maximizes $\max\{0, P - \text{value of income already paid}\}$08 subject to monotonicity. Because $\max\{0, P - \text{value of income already paid}\}$09 under FOSD, $\max\{0, P - \text{value of income already paid}\}$10. Returns are therefore not merely insurance; they are a screening device that switches off allocation in low-virtual-value regions. The paper states that MBGs strictly dominate no-refund whenever $\max\{0, P - \text{value of income already paid}\}$11 is positive on a set of positive measure.

The contribution is also computational. If values take $\max\{0, P - \text{value of income already paid}\}$12 discrete levels, an optimal deterministic mechanism requires at most $\max\{0, P - \text{value of income already paid}\}$13 menu options, with refunds chosen from the value support; the corresponding dynamic program runs in $\max\{0, P - \text{value of income already paid}\}$14. For ordered item qualities $\max\{0, P - \text{value of income already paid}\}$15, the menu size bound is $\max\{0, P - \text{value of income already paid}\}$16 and the runtime is $\max\{0, P - \text{value of income already paid}\}$17. If the designer restricts attention to at most $\max\{0, P - \text{value of income already paid}\}$18 options, the paper gives dynamic programs with runtimes $\max\{0, P - \text{value of income already paid}\}$19 or $\max\{0, P - \text{value of income already paid}\}$20, depending on the environment. A notable implication is that deterministic MBGs need not generate large menu complexity to approximate optimal screening.

The paper also states that, within its risk-neutral baseline and absent explicit salvage or restocking costs, it does not identify general conditions under which full refunds are optimal. This sharply contrasts with settings in which free returns arise as boundary solutions. In this framework, partial refunds or restocking fees are typically the revenue-relevant instrument because they implement threshold screening across both realized values and ex ante types.

6. Cost-calibrated MBGs in bundling and broader interpretation

The bundling literature applies MBG logic to multiproduct pricing with production costs. Pure Bundling with Disposal for Cost (PBDC) sets a grand-bundle price $\max\{0, P - \text{value of income already paid}\}$21, allows the buyer to purchase the bundle, and then permits return of any subset $\max\{0, P - \text{value of income already paid}\}$22 for a refund equal to the production costs of returned items. The induced menu over retained subsets is

$\max\{0, P - \text{value of income already paid}\}$23

with $\max\{0, P - \text{value of income already paid}\}$24. The buyer returns item $\max\{0, P - \text{value of income already paid}\}$25 if and only if $\max\{0, P - \text{value of income already paid}\}$26, so PBDC eliminates overinclusion losses while preserving the concentration benefits of bundling (Ma et al., 2015).

The core welfare variable is

$\max\{0, P - \text{value of income already paid}\}$27

with mean $\max\{0, P - \text{value of income already paid}\}$28, standard deviation $\max\{0, P - \text{value of income already paid}\}$29, and coefficient of variation

$\max\{0, P - \text{value of income already paid}\}$30

Under independent additive valuations, the paper proves a distribution-dependent guarantee: $\max\{0, P - \text{value of income already paid}\}$31 It also proves a distribution-free guarantee that either PBDC or individual sales earns at least $\max\{0, P - \text{value of income already paid}\}$32 of optimal IC–IR revenue, improving the earlier constant of $\max\{0, P - \text{value of income already paid}\}$33. Conversely, no partitioning mechanism can in general exceed approximately $\max\{0, P - \text{value of income already paid}\}$34 of optimal revenue. These results reinterpret MBG-style refunds as a way to restore bundling profitability when costs are non-negligible.

The paper’s pricing guidance sets the net-of-cost bundle price at

$\max\{0, P - \text{value of income already paid}\}$35

with the optimized choice

$\max\{0, P - \text{value of income already paid}\}$36

Its simulations, conducted over 15,000 randomly generated instances and compared with individual sales, pure bundling, and bundle-size pricing, show that when costs vary, PBDC achieves at least 97.5% of the best profit among those simple schemes in worst-case instances, compared with 79.9% for individual sales, 16.8% for pure bundling, and 59.5% for bundle-size pricing. The paper also reports zero overinclusion loss under PBDC and decreasing deadweight loss as the number of items increases.

Across the cited literatures, MBGs therefore admit several distinct interpretations. In annuities, they embed life insurance into longevity insurance and generate recursive pricing, non-monotone age effects, and feasibility thresholds. In tontines, they add bequest protection to pooled decumulation but shift cost determination toward tail-sensitive ex post pricing. In refund design, they affect learning and screening, sometimes making free returns optimal only as a boundary solution. In deterministic mechanism design, they implement virtual-value screening via type-dependent refund thresholds. In bundling, they remove overinclusion by tying refunds to costs rather than to dissatisfaction. A plausible implication is that “money back” is not economically primitive; its meaning depends on which hidden state—mortality, fit, information precision, or cost inefficiency—the guarantee is designed to insure.