If I had an investment that with 50% likelihood quadruples your investment on a given day and you lose it all also with 50% likelihood, what percent of your money should you invest each day to maximize your median return? I believe from memory the answer is 25%, but I'd like to understand the math behind this and believe the answer may be related to the Sharpe ratio.

I've asked a related question over on the money stack exchange and I got back the nonsensical (to my thinking) answer of infinite leverage, so I'd like to offer some math to better address this question more objectively:



closed as off topic by Raskolnikov, Thomas, tomasz, Norbert, J. M. is a poor mathematician Oct 7 '12 at 13:28

Questions on Mathematics Stack Exchange are expected to relate to math within the scope defined by the community. Consider editing the question or leaving comments for improvement if you believe the question can be reworded to fit within the scope. Read more about reopening questions here. If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ I think you should try here. $\endgroup$ – Raskolnikov Sep 27 '12 at 13:37
  • $\begingroup$ I think this is more a basic math question than a quantitative finance question. Let's try here first and if no takers, I can repost over there. $\endgroup$ – WilliamKF Sep 27 '12 at 13:38
  • 1
    $\begingroup$ I beg to differ, but let's do as you suggest. Wait and see. $\endgroup$ – Raskolnikov Sep 27 '12 at 13:43

For your first question, if you invest a fraction $f$ of your bankroll you have $\frac 12$ chance of ending with $1-f$ and $\frac 12$ of ending with $1+3f$. The median result over a span of days will have you win and lose an equal number of times, so you will have some power of $(1-f)(1+3f)=1+2f-3f^2$. By the usual take the derivative and set to zero, this is maximized at $f=\frac 13$ with a gain of $\frac 1{3}$ of your bankroll every two days. Where can I get this deal?


Not the answer you're looking for? Browse other questions tagged or ask your own question.