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I want to compute the ideal investment leverage ratio based upon an investment's volatility.

For example, 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 is 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:

http://money.stackexchange.com/q/16990/1516

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closed as off topic by Raskolnikov, Thomas, tomasz, userNaN, J. M. Oct 7 '12 at 13:28

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I think you should try here. –  Raskolnikov Sep 27 '12 at 13:37
    
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. –  WilliamKF Sep 27 '12 at 13:38
1  
I beg to differ, but let's do as you suggest. Wait and see. –  Raskolnikov Sep 27 '12 at 13:43

1 Answer 1

up vote 3 down vote accepted

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 $1+4f$. 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 $(1-f)(1+4f)=1+3f-4f^2$. By the usual take the derivative and set to zero, this is maximized at $f=\frac 38$ with a gain of $\frac 9{16}$ of your bankroll every two days. Where can I get this deal?

Your second question about assessing volatility has nothing to do with mathematics.

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