WHAT THIS TELLS US
There are several lessons we can take away from these equations.
1) Without a positive expectation, risk of ruin is 100%. When the mean return is negative, these equations no longer apply and they give answers in excess of 100% risk of ruin. Play long enough with a negative expectation and you will lose all your money.
2) As the standard deviation increases, risk of ruin rises sharply. The second equation has the standard deviation squared in the denominator of the exponent. Doubling the standard deviation in that example from 13% to 26% changes risk of ruin from 2.9% to 41.2%. If a market becomes more volatile, a wise trader will reduce position size accordingly. Leveraged traders who were long stock in high volatility crashes like October 2008 felt pain.
3) It is difficult to know the mean and standard deviation accurately. Over-optimizing on past data may lead to inflated estimates of mean return. The standard deviations of markets are not stable. Volatility measures like VIX may increase and remain high for months. Risk of ruin is probably not less than that calculated by these equations. The output of these equations will not be accurate if the input variables change.
4) There is no limit to the maximum possible drawdown. Large losses may be unlikely, but they are possible. Increasing the average return and/or decreasing the standard deviation of returns will make large losses less likely.
David E. Chamness is a professional trader with a background in engineering and statistics. Contact him at firstname.lastname@example.org .