  This paper presents a new type of modern accumulation-based tontine, 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 yet another way of pooling and sharing longevity risk, but is
different from competing designs in two key ways. The first is that in the
Riccati tontine, the representative investor is expected -- although not
guaranteed -- to receive their money back if they die, or when the tontine
lapses. The second is that the underlying funds within the tontine are
deliberately {\em not} indexed to the stock market. Instead, the risky assets
or underlying investments are selected so that return shocks are negatively
correlated with stochastic mortality, which will maximize the expected payout
to survivors. This means that during a pandemic, for example, the Riccati
tontine fund's performance will be impaired relative to the market index, but
will not be expected to lose money for participants. In addition to describing
and explaining the rationale for this non-traditional asset allocation, the
paper provides a mathematical proof that the recovery schedule that generates
this financial outcome satisfies a first-order ODE that is quadratic in the
unknown function, which (yes) is known as a Riccati equation.
