I just finished reading "Fooled by Randomness" by Nassim Taleb. He, inter alia, gives the following example to prove one of his points:
A 15% return with 10% volatility per annum translates into a 93% probability of making money in any given year. However, when looking at a quarter, this probability decreases to 77%. The respective probability in a given month/day/hour/minute/second is 67%, 54%, 51.3%, 50.17% and 50.02%, respectively.
Although probably trivial, this was an astonishing realizition for me. Given one is equally happy/sad by making gains/losses, a person that uses a smaller timeframe to assess his performance will be much sadder, than one that uses a larger timeframe (this was the point he was arguing for). He does not show the maths behind the aforementioned probability decrease conditional on narrowing down the timeframe, though. Could someone clarify this intuition mathematically? Thanks in advance.