When are Past Returns Indicative of Future Returns?
A Brief Exploration Through the Lens of Coin Flipping1


You are presented with two coins, one is fair, and the other has a 60% chance of coming up heads. Unfortunately, you don't know which is which.2

What number of flips would you want to see performed in parallel on the two coins to give yourself a 95% chance of identifying the biased one? Please just give it a quick guess, without working it out on a pad of paper. You need to hit 'answer' in order to read the rest of the note.


How many flips?





1 A big thank you to James White, Jeff Rosenbluth, Andy Morton, Evi, Costas and Christos Kaplanis, PJ Beaghton, Ian Hall, Simon Bowden and Vlad Ragulin for much valuable input.

2 Example inspired by: ‘Good and bad properties of the Kelly criterion,’ MacLean, Thorp, Ziemba, 2010, page 8.

6 Merton, Robert, C., On Estimating the Expected Return on the Market: An Exploratory Investigation, Journal of Financial Economics 8 (1980) 323-361. See appendix A, pages 355-357. To see why you can’t get around this inconvenient truth, note that in continuous time, the number of years of observation, using the annual Sharpe Ratio (SR) as an input is 2*(1.645/SR)^2, where 1.645 is the 95% cumulative probability level in a Normal distribution. If we sample more frequently, f times per year, the required number of periods goes to 2*(1.645/(SR/sqrt(f)))^2 = 2f*(1.645/SR)^2. So, we'll need to observe f times as many periods as when we look annually, which is exactly how many more periods we get to observe by breaking the year into f intervals.

7 A comprehensive list of these would be long, and would include publication bias, base-rate bias, pervasive data 'snooping,' survivorship bias, non-normal distributions, out-of-the-money option selling, mean-reversion and benchmark mis-specification, to name a few.

8 Paraphrased from Harvey, Liu, Zhu, '……and the Cross-Section of Expected Returns,' (2015), on SSRN, 'Hundreds of papers and hundreds of factors attempt to explain the cross-section of expected returns. Given this extensive data mining, it does not make any economic or statistical sense to use the usual significance criteria for a newly discovered factor, e.g., a t-ratio greater than 2.0. [95% confidence]. …A newly discovered factor needs to clear a much higher hurdle, with a t-ratio greater than 3.0 [99.7% confidence]. Echoing a recent disturbing conclusion in the medical literature, we argue that most claimed research findings in financial economics are likely false.'

9 See our Elm paper on return-chasing here. Return Chasing results in the well-documented phenomenon of investor (aka dollar-weighted) returns being significantly lower than fund (aka time-weighted) returns. See Morningstar's 'Mind the Gap' notes and Dalbar's annual investor behaviour reports.

10 See 'The Most Important Number You Won't Find in the WSJ' for more discussion about driving your portfolio with eyes open and forward looking.