Approximations for the distribution of perpetuities with small discount rates

Abstract

Perpetuities (i.e., random variables of the form D=∫0∞e−Γ(t−)dΛ(t)$$ D={\int}_0^{\infty }{e}^{-\Gamma \left(t-\right)}d\Lambda (t) $$ play an important role in many application settings. We develop approximations for the distribution of D$$ D $$ when the “accumulated short rate process”, Γ$$ \Gamma $$ , is small. We provide: (1) characterizations for the distribution of D$$ D $$ when Γ$$ \Gamma $$ and Λ$$ \Lambda $$ are driven by Markov processes; (2) general sufficient conditions under which weak convergence results can be derived for D$$ D $$ , and (3) Edgeworth expansions for the distribution of D$$ D $$ in the iid case and the case in which Λ$$ \Lambda $$ is a Levy process and the interest rate is a function of an ergodic Markov process.