There are many state-of-the-art techniques for loop bound analysis. Most of them target an upper bound for a given program, and others find a lower bound. \emph{Exact} bound analysis still remains largely unexplored, but it offers new applications. To compute an \emph{exact} bound for a program it is necessary to reason about the possible values the program's inputs can take and how they relate to each other. Since inputs can vary on any given execution of the program, it makes the problem of computing an exact bound challenging. In this work, we present a new approach to find an exact bound by way of \emph{precondition synthesis} which iteratively considers under-approximations of a program under which the bound can be precomputed over initial values of program variables. For each precondition, our approach synthesizes a function over program variables such that when the function is applied to the initial values of the program variables, its output is an exact bound for the program. We reduce the precondition synthesis problem to that of \emph{safety verification} which lends its correctness guarantees to the exact bounds we compute. {Our technique has been implemented in a tool called \textsc{Elba}, and we show that it is effective on a set of challenging single loop benchmarks under Linear Integer Arithmetic.}