HMMT 2013 C10

Rosencrantz and Guildenstern each start with $\$ 2013$ and are flipping a fair coin. When the coin comes up heads Rosencrantz pays Guildenstern $\$ 1$ and when the coin comes up tails Guildenstern pays Rosencrantz $\$ 1$. Let $f(n)$ be the number of dollars Rosencrantz is ahead of his starting amount after $n$ flips. Compute the expected value of $\max \{f(0), f(1), f(2), \ldots, f(2013)\}$.