The Riccati tontine: how to satisfy regulators on average

This paper introduces a novel accumulation-based tontine, which we have called the Riccati tontine, named after two Italians: mathematician Jacobo Riccati (b. 1676, d. 1754) and financier Lorenzo di Tonti (b. 1602, d. 1684). The Riccati tontine is another way of pooling and sharing longevity risk, but is different from competing designs and historical tontines in two key ways: First, in the Riccati tontine, the representative investor is expected—although not guaranteed—to receive their money back, if they die or lapse. This design feature, in which an investor exiting early will receive their initial deposit back on average, is also strongly encouraged by regulators. The second innovation in a Riccati tontine is that the underlying funds within the pool are deliberately not indexed to the market. Instead, the underlying investments are selected so that portfolio return shocks are negatively correlated with stochastic mortality. This negative correlation to mortality and positive correlation to longevity reduces the volatility of outcomes and typically will improve utility. In addition to explaining the rationale for non-indexed investments, or effectively active management, the paper provides a mathematical proof that the required recovery schedule satisfies a first-order ODE that is quadratic in the unknown function, which (yes) is known as a Riccati equation.