  Given a spectrally negative L\'evy process, we predict, in a $L_1$ sense, the
last passage time of the process below zero before an independent exponential
time. This optimal prediction problem generalises Baurdoux and Pedraza (2020)
where the infinite horizon problem is solved. Using a similar argument as that
in Urusov (2005), we show that this optimal prediction problem is equivalent to
solving an optimal prediction problem in a finite horizon setting. Surprisingly
(unlike the infinite horizon problem) an optimal stopping time is based on a
curve that is killed at the moment the mean of the exponential time is reached.
That is, an optimal stopping time is the first time the process crosses above a
non-negative, continuous and non-increasing curve depending on time. This curve
and the value function are characterised as a solution of a system of
non-linear integral equations which can be understood as a generalisation of
the free boundary equations (see e.g. Peskir and Shiryaev (2006) Chapter
IV.14.1) in the presence of jumps. As an example, we calculate numerically such
curve in the Brownian motion case and a compound Poisson process with
exponential sized jumps perturbed by a Brownian motion.
