Existence of solutions to participant actuarial fairness equations

Establish the existence, under explicit regularity conditions on survival indicators and share allocations, of at least one investment vector π = (π1, …, πn, πn+1) that satisfies the n actuarial fairness conditions for participants in the one-period tontine fund (I, π, f), namely E[Wi] = (1 + R)πi for i = 1, …, n, where each participant’s payout is defined by Wi = (1 + R) × (∑j=1^{n+1} πj) × (fi × Ii) / (∑j=1^{n} fj × Ij) when at least one participant survives and Wi = 0 otherwise.

Background

Section 4 introduces actuarial fairness for a one-period tontine fund, requiring each participant’s expected payout to equal the time-1 value of their initial investment. This yields a system of n equations relating the investment vector π to the payouts determined by the share allocation vector f and survival indicators I.

While Theorem 1 shows scale invariance (so uniqueness cannot be expected), the existence of at least one solution for π is not established. The authors explicitly defer a formal proof of existence under suitable conditions, making this a concrete open problem central to the construction of actuarially fair tontine schemes.