  For a given Markov process $X$ and survival function $\overline{H}$ on
$\mathbb{R}^+$, the inverse first-passage time problem (IFPT) is to find a
barrier function $b:\mathbb{R}^+\to[-\infty,+\infty]$ such that the survival
function of the first-passage time $\tau_b=\inf \{t\ge0:X(t)<b(t)\}$ is given
by $\overline{H}$. In this paper, we consider a version of the IFPT problem
where the barrier is fixed at zero and the problem is to find an initial
distribution $\mu$ and a time-change $I$ such that for the time-changed process
$X\circ I$ the IFPT problem is solved by a constant barrier at the level zero.
For any L\'{e}vy process $X$ satisfying an exponential moment condition, we
derive the solution of this problem in terms of $\lambda$-invariant
distributions of the process $X$ killed at the epoch of first entrance into the
negative half-axis. We provide an explicit characterization of such
distributions, which is a result of independent interest. For a given
multi-variate survival function $\overline{H}$ of generalized frailty type, we
construct subsequently an explicit solution to the corresponding IFPT with the
barrier level fixed at zero. We apply these results to the valuation of
financial contracts that are subject to counterparty credit risk.
