In this dissertation, we study the density function of the time of ruin with exponential claims. More precisely, we present two different formulas, the one that Drekic and Willmot (2003) obtained for the classical model of ruin theory and the one that Borovkov and Dickson (2008) derived for a Sparre Andersen process with exponential claim sizes. After the presentation of the two formulas, we prove that they are equivalent for the special case of exponentially distributed times (classical model) and zero initial surplus. The theoretical presentation is followed by numerical examples, executed in the computational package Mathematica, where the behavior of the density function of the ruin time is studied for different values of its parameters. We also examine, if and how much, is there a good fit of the exponential distribution to the density of ruin time, empirically. Apart from this, we also study the failure rate of the distribution function and if there are cases that this is monotonic. Closing, various numerical examples are given, in order to come to a conclusion for the classiffcation of the distribution function of ruin time, in terms of its failure rate.