  We investigate models of the life annuity insurance when the company invests
its reserve into a risky asset with price following a geometric Brownian
motion. Our main result is an exact asymptotic of the ruin probabilities for
the case of exponentially distributed benefits. As in the case of non-life
insurance with exponential claims, the ruin probabilities are either decreasing
with a rate given by a power function (the case of small volatility) or equal
to unit identically (the case of large volatility). The result allows us to
quantify the share of reserve to invest into such a risky asset to avoid a
catastrophic outcome: the ruin with probability one. We address also the
question of smoothness of the ruin probabilities as a function of the initial
reserve for generally distributed jumps.
