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For a given Markov process $X$ and survival function $\overline{H}$ on$\mathbb{R}^+$, the inverse first-passage time problem (IFPT) is to find abarrier function $b:\mathbb{R}^+\to[-\infty,+\infty]$ such that the survivalfunction of the first-passage time $\tau_b=\inf \{t\ge0:X(t)<b(t)\}$ is givenby $\overline{H}$. In this paper, we consider a version of the IFPT problemwhere the barrier is fixed at zero and the problem is to find an initialdistribution $\mu$ 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, wederive the solution of this problem in terms of $\lambda$-invariantdistributions of the process $X$ killed at the epoch of first entrance into thenegative half-axis. We provide an explicit characterization of suchdistributions, which is a result of independent interest. For a givenmulti-variate survival function $\overline{H}$ of generalized frailty type, weconstruct subsequently an explicit solution to the corresponding IFPT with thebarrier level fixed at zero. We apply these results to the valuation offinancial contracts that are subject to counterparty credit risk.

Explicit solution of an inverse first-passage time problem for L\'{e}vy processes and counterparty credit risk