  We find the minimum probability of lifetime ruin of an investor who can
invest in a market with a risky and a riskless asset and who can purchase a
reversible life annuity. The surrender charge of a life annuity is a proportion
of its value. Ruin occurs when the total of the value of the risky and riskless
assets and the surrender value of the life annuity reaches zero. We find the
optimal investment strategy and optimal annuity purchase and surrender
strategies in two situations: (i) the value of the risky and riskless assets is
allowed to be negative, with the imputed surrender value of the life annuity
keeping the total positive; or (ii) the value of the risky and riskless assets
is required to be non-negative. In the first case, although the individual has
the flexiblity to buy or sell at any time, we find that the individual will not
buy a life annuity unless she can cover all her consumption via the annuity and
she will never sell her annuity. In the second case, the individual surrenders
just enough annuity income to keep her total assets positive. However, in this
second case, the individual's annuity purchasing strategy depends on the size
of the proportional surrender charge. When the charge is large enough, the
individual will not buy a life annuity unless she can cover all her
consumption, the so-called safe level. When the charge is small enough, the
individual will buy a life annuity at a wealth lower than this safe level.
