  In this paper, we discuss the Cram\'er-Lundberg model with investments, where
the price of the invested risk asset follows a geometric Brownian motion with
drift $a$ and volatility $\sigma> 0.$ By assuming there is a cap on the claim
sizes, we prove that the probability of ruin has at least an algebraic decay
rate if $2a/\sigma^2 > 1$. More importantly, without this assumption, we show
that the probability of ruin is certain for all initial capital $u$, if
$2a/\sigma^2 \le 1$.
