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We find the minimum probability of lifetime ruin of an investor who caninvest in a market with a risky and a riskless asset and who can purchase areversible life annuity. The surrender charge of a life annuity is a proportionof its value. Ruin occurs when the total of the value of the risky and risklessassets and the surrender value of the life annuity reaches zero. We find theoptimal investment strategy and optimal annuity purchase and surrenderstrategies in two situations: (i) the value of the risky and riskless assets isallowed to be negative, with the imputed surrender value of the life annuitykeeping the total positive; or (ii) the value of the risky and riskless assetsis required to be non-negative. In the first case, although the individual hasthe flexiblity to buy or sell at any time, we find that the individual will notbuy a life annuity unless she can cover all her consumption via the annuity andshe will never sell her annuity. In the second case, the individual surrendersjust enough annuity income to keep her total assets positive. However, in thissecond case, the individual's annuity purchasing strategy depends on the sizeof the proportional surrender charge. When the charge is large enough, theindividual will not buy a life annuity unless she can cover all herconsumption, the so-called safe level. When the charge is small enough, theindividual will buy a life annuity at a wealth lower than this safe level.

Optimal Reversible Annuities to Minimize the Probability of Lifetime Ruin