  Refundable income annuities (IA), such as cash-refund and instalment-refund,
differ in material ways from the life-only version beloved by economists. In
addition to lifetime income they guarantee the annuitant or beneficiary will
receive their money back albeit slowly over time. We document that refundable
IAs now represent the majority of sales in the U.S., yet they are mostly
ignored by insurance and pension economists. And, although their pricing,
duration, and money's-worth-ratio is complicated by recursivity which will be
explained, we offer a path forward to make refundable IAs tractable.
  A key result concerns the market price of cash-refund IAs, when the actuarial
present value is grossed-up by an insurance loading. We prove that price is
counterintuitively no longer a declining function of age and older buyers might
pay more than younger ones. Moreover, there exists a threshold valuation rate
below which no price is viable. This may also explain why inflation-adjusted
IAs have all but disappeared.
