Maximize Head Ratio I

Say that you are flipping a fair coin where you can stop flipping whenever you want. Your goal is to maximize the ratio between the number of heads you get with the total number of flips. The ratios are in the form $a:1$. What is the expectation of “a” when your strategy is to target a ratio of $0.5:1$ in a long run of flips but if its greater than $0.5:1$, you immediately stop there.