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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    $\begingroup$ I'm voting to close this question as off-topic because it doesn't ask a question. $\endgroup$ – AfterWorkGuinness Oct 30 '15 at 15:12

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