Geometric Brownian Motion and Volatility Drag

A game starts with $100. Each round a fair coin is flipped. Heads, your money goes up 50%. Tails, it goes down 40%. The average move is

so every flip adds 5% in expectation, and after $n$ flips your expected wealth is $1.05^n$ times the start.

Now play one heads and one tails. Heads takes $100 to $150. Tails takes 40% of $150, which is $60, and leaves $90. The order makes no difference: $1.5 \times 0.6 = 0.6 \times 1.5 = 0.9$. Each heads-tails pair costs you 10%, even though the average flip gains 5%.

Play ten flips that split five and five and you hold $0.9^5 = 0.59$ of your start, while the expectation says $1.05^{10} = 1.63$. At 100 flips the gap is enormous. Expected wealth is $1.05^{100} \approx 131$ times the start. The median player gets 50 heads and holds $0.9^{50} \approx 0.005$ of the start, about 52 cents. You finish ahead only with 56 or more heads, which happens with probability 13.6%.

Both numbers are correct. The expectation is an average over all players, and a handful of lucky players with long runs of heads end up so rich that they drag the average up. A typical player, with around 50 heads out of 100, is left with about half a percent of what they started with.