Victor Haghani suggested a coin flipping model to illustrate the challenge of properly selecting a manager with skill, based on past performance.
We can select a manager from several managers, but only one of them has skill, and the remaining are without skill.
The skilled manager has a slightly greater probability of positive returns than negative ones, while the unskilled managers have a slightly greater probability of negative returns, so that the average performance of all the managers is zero.
If the probability of positive returns is equal to p>0.5, then the annual Sharpe ratio is equal to 250−−−√∗(2p−1).
If the excess annual Sharpe ratio is equal to 0.4, then the probability of positive returns is equal to (0.4/250−−−√+1)/2=51.2%.