Compounding, Time Horizons, and Expected Return in a Two-Outcome Bet

Summary

The document compares several ways to annualize returns from a bet costing one unit today, with an equal chance of receiving either two units after one year or half a unit after two years. It contrasts averaging the two annualized outcomes, using an average payoff and average duration, calculating an internal rate of return across a pair of outcomes, and repeatedly reinvesting the bet. The author argues that repeated compounding is the relevant view for a long sequence and illustrates that equal numbers of wins and losses leave the capital unchanged despite the unequal holding periods.

The example raises a useful distinction between arithmetic averages, cash-flow returns, and compounded growth, but it does not establish one universally correct expected-return measure. In particular, the zero-growth illustration assumes a sequence with equal counts of each outcome; random sequences need not have exactly that composition. The appropriate measure depends on whether the question concerns one bet, an expected cash flow, or long-run compounded wealth, and on how repeated bets and durations are modeled.

Key ideas

• The document compares arithmetic averaging, average payoff and duration, internal rate of return, and repeated compounding.
• The author argues that an equal count of doubling and halving outcomes leaves capital unchanged.
• Holding periods differ between outcomes, so annualizing the bet requires care about time as well as payoff.
• The illustration does not resolve every definition of expected return and assumes equal counts in its repeated sequence.

Tags

# What is the expected rate of return from paying 1 today for a 50/50 bet receiving either 2 in year 1 or 0.5 in year 2?


# What is the expected rate of return from paying 1 today for a 50/50 bet receiving either 2 in year 1 or 0.5 in year 2?












What do you think is the correct way to calculate expected return for this example?

I think Method4 below is correct.

Method1 35.4% = AVERAGE(1.0, 0.5^(1/2)-1) Incorrect, but some will argue that you have a 50% chance of a 100% annualised rate of return (if you received 2 in year 1), and a 50% chance of a -29.3% annualised rate of return (if you received 0.5 in year 2)

Method2 16.0% = 1.25^(1/1.5)-1 Better - you expect to receive an average of 1.25 after an average of 1.5 years

Method3 20.7% = IRR({-2,2,0.5}) in Excel Maybe - you repeat the experiment and find the IRR of the resulting accumulated cashflows (the numbers in the IRR formula should be scaled up for the number of experiments...assumed to be just 2 here for clarity).

Method4 0% Correct, I think. You repeatedly reinvest in the bet so that you are forever doubling and halving your money. So for a sequence of say 1000 such bets your cumulative money is expected to be 100 = 100*(2.0^500 * 0.5^500) after 1500 = (1*500 + 2*500) years, so your annualised expected return is 0%

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