  Let $(\xi_1, \eta_1)$, $(\xi_2, \eta_2),\ldots$ be independent identically
distributed $\mathbb{R}^2$-valued random vectors. We prove a strong law of
large numbers, a functional central limit theorem and a law of the iterated
logarithm for convergent perpetuities $\sum_{k\geq
0}b^{\xi_1+\ldots+\xi_k}\eta_{k+1}$ as $b\to 1-$. Under the standard actuarial
interpretation, these results correspond to the situation when the actuarial
market is close to the customer-friendly scenario of no risk.
