  In this paper, we propose a new type of reversible modern tontine with
transaction costs. The wealth of the retiree is divided into a bequest account
and a tontine account. And consumption can only be withdrawn from the bequest
account. Each transaction between the two accounts incurs fixed and
proportional transaction costs depending on the transaction volume. The retiree
dynamically controls the allocation policy between the two accounts and the
consumption policy to maximize the consumption and bequest utilities. We
formulate the optimization problem as a combined stochastic and impulse control
problem with infinite time horizon, and characterize the value function as the
unique viscosity solution of a HJBQVI (Hamilton-Jacobi-Bellmen
quasi-variational inequality). The numerical results exhibit the V-shaped
transaction region which consists of two stages. In the former stage, since
longevity credits increase gradually, the retiree decreases the proportion of
wealth in the tontine account to smooth the wealth and reduce the volatility.
However, in the latter stage, the retiree increases the proportion of wealth in
the tontine account to gamble for the considerable longevity credits.
