Repeated 50% Gains and Losses: Expected Wealth Versus Typical Growth

Summary

The document asks how wealth changes when each repeated bet either increases the current balance by 50% or reduces it by 50%, with equal probability. It highlights the difference between calculating expected wealth and describing the path followed by a typical sequence of wins and losses. Because each round multiplies wealth by a factor whose arithmetic mean is one, the expected balance remains unchanged over repeated independent rounds. A sequence with roughly equal numbers of wins and losses instead has a multiplicative factor based on the geometric mean of those outcomes, which is below one.

The answer points to the asymmetry of percentage gains and losses: recovering from a 50% decline requires a 100% gain. Its stated expression for growth assumes an equal number of wins and losses, so it describes that kind of path rather than the expected value across all possible paths. The distinction matters when interpreting compounding, but the discussion omits transaction costs, varying bet sizes, and other return distributions.

Key ideas

• Expected wealth can remain constant even when typical compounded paths decline.
• A 50% loss requires a 100% gain to recover the original balance.
• The arithmetic average of returns differs from the geometric average that drives compounded wealth.
• The answer's equal-win-and-loss calculation describes a particular path pattern, not expected wealth across all outcomes.

Tags

# What's the expected value of a repeated game with 50% chance to win 0.5 and 50% to lose 0.5?


# What's the expected value of a repeated game with 50% chance to win 0.5 and 50% to lose 0.5?












Assume we start with 1.

In the first bet the expected value of remained balance is 1.5 * 0.5 + 0.5 * 0.5 = 1 For N times, is it still 1 according to E(XYZ)=E(X)E(Y)E(Z)? But 1.5^50 * 0.5^50 is not 1.

If the game is repeated N times, what's the expected value of remained balance in the end?

Thanks in advance!

## Answer by roz (score 3)

https://quant.stackexchange.com/a/50530

On average half the time you will win 0.5 times your current bankroll and half the time you will lose 0.5 times your current bankroll. Over N plays your expected growth will be (0.5)^(N/2)(1.5)^(N/2) and you will tend to lose money over time and in the limit since 0.5*1.5 = 0.75 < 1. This happens because gaining and losing 50% are not equivalent. Think about starting with 1 dollar and then losing 50%. In order to get back up to a dollar you have to gain 100%, not 50%.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.