Your earnings are equivalent to $S_N=X_1+\cdots+X_N$, where each $X_i$ is iid, satisfies $P(X_i=\pm 1)=1/2$, has mean $\mu=0$ and standard deviation $\sigma=1$. By the CLT, $S_N/\sigma\sqrt{N}$ is approximately normally distributed. So the coefficient is just 1.

If you're interested in the maximum earnings (for any time between 1 and $N$), the law of the iterated logarithm says that $max_{n\leq N}S_n/\sqrt{2n\log\log(n)}\approx 1$, which gives a "coefficient" of $\sqrt{2\log\log(n_m)}$, where $n_m$ is the time of the maximum.

Your earnings are equivalent to $S_N=X_1+\cdots+X_N$, where each $X_i$ is iid, satisfies $P(X_i=\pm 1)=1/2$, has mean $\mu=0$ and standard deviation $\sigma=1$. By the CLT, $\frac{S_N}{\sigma\sqrt{N}}=\frac{S_N}{\sqrt{N}}$ is approximately normally distributed. So the coefficient will fluctuate roughly between -3 and 3. as that encapsulates 99.7% of all cases.

If you're interested in the absolute deviations from 0, then this is a half-normal distribution, with which gives a mean coefficient of $\sqrt{\frac{2}{\pi}}.$