Is it easy to write down the large deviations rate for the maximizer of a random walk with negative drift?

Let $X_i$ be the (iid, mean $-\mu$, variance $\sigma$, arbitrarily nice tails) jumps of a random walk $S_i$. I am interested in the location of the maximum of $S_i$, i.e. in $\arg\max_k S_k$. (If $X_i$ are continuous, the maximizer is almost surely unique.)

There's a trivial exponential upper bound $$ \mathbb{P}(\arg\max_k(S_k) > n) \leq \mathbb{P}(S_{n+1} > 0) = \mathbb{P}(\sum_{i=1}^n X_i > 0) $$ and the trivial exponential lower bound $$ \mathbb{P}(\arg\max_k(S_k) > n) \geq \mathbb{P}(X_i > 0\ \forall i \leq n) $$ and they (of course) don't match. Is there a limit $$ \lim_{n \to \infty}n^{-1}\log\mathbb{P}(\arg\max_k(S_k) > n), $$ and is it possible to write it down in terms of the distribution of the jumps $X_i$?