## A beginnner's question on optimal leverage ratio

Discussions about Money Management and Risk Control.
ezbentley
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### A beginnner's question on optimal leverage ratio

In the book Quantitative Trading Strategies, Lars Kestner mentioned that the optimal leverage for a trading strategy is equal to the mean of return divided by the variance of return. This seems to be a result of Harry Markowitz's modern portfolio theory.
Can anyone help me understand the concept and derivation behind this equation? And how is the optimal leverage ratio related to Kelly's criteria and Vince's optimal f?

Thanks

sluggo
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Maybe you could put together a little spreadsheet with some return numbers and then experiment with different amounts of leverage.

Perhaps you will find that the leverage value which looks best to you, happens to be the same as Kestner's formula result. Or perhaps you will find that Kestner's formula does a rotten job of predicting what you find optimum.

As a practical matter, I'm betting on the latter. I mean, how could Kestner possibly guess what you will like the most? He hasn't even met you.

I looked in Kestner's book and I think you're referring to this passage on page 317. Notice that the mean "mu" and the variance "sigma squared" refer to unleveraged investing. Which is a standard practice in stock trading but wildly atypical in futures or forex trading.

sluggo
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sluggo wrote: ... Or perhaps you will find that Kestner's formula does a rotten job of predicting what you find optimum.

Good old Markowitz himself, said the same thing. You may not like the "optimum" calculated value of leverage.

ezbentley
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Joined: Sat Mar 15, 2008 1:12 am
Thanks a lot for both replies. I am fully aware that the optimal leverage is only optimal when the only objective is to maximize return without regard to the potential risk, and therefore can be pretty useless in actual trading.

I am however curious in the theoretical derivation of this mu/variance formula. Can anyone refer me to some book/article that explains this concept? Also I still don't understand the comparison that Kestner make between this optimal leverage ratio and Kelly's criteria. It seems to me that they are different concepts.

Thanks,

sluggo
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Consider the textbook positive expectaction coinflipping game: You bet B dollars and flip a fair coin. If it comes up heads, you win 2B dollars. If it comes up tails, you lose B dollars. Your expectation per play is (+B/2). Too bad the Bellagio doesn't offer this game!

Kestner calculates the mean (mu = B * (+1/2)) and the variance (sigma squared = B * (9/4)) and thus his optimum betsize ("leverage") is 2/9 = 0.22222.

Kelly calculates the optimum betsize in the textbook way: numerator = -1 + (Pwin * (1 + (W/L))), denominator = (W/L). Thus Kelly's optimal betsize ("leverage") is 1/4 = 0.25000.

We now possess one example for which the Kestner optimum (0.22222) is different than the Kelly optimum (0.25000). But, what does this mean, if anything?