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# Ideal leverage ratio based on volatility drag equation

Discussion in 'Risk Management' started by wintermute0, Nov 6, 2019.

1. ### wintermute0

Hi, this is my first post. I'm an aspiring quant and finished my first blog post (part 1): https://smabie.github.io/posts/2019/10/04/vol.html

I was motivated to write this post by what I felt was unsound advice cautioning individuals from investing in leveraged index ETFs. I'm currently working on part 2 and am looking for some feedback. The general idea is that we can take modify the volatility drag equation (r_final = r - var/2) to incorporate leverage. After that we can take the derivative with respect to the leverage ratio to find the maximum return given a forecasted return and variance. In short, the ideal leverage ratio assuming a normal distribution is returns/variance.

I've looked for papers talking about this but can't find anything. I'm an amateur (I work as a quantitative developer, not a real quant), so I'm sure this is a known result. Anyone who could review the post and give me feedback and maybe links to papers covering this topic would be greatly appreciated.

Thanks!

2. ### Magic

I skimmed your post, are you aware you can just cut to the chase and put mean return / sigma^2 to get a leverage figure for optimal leverage for maximizing logarithmic growth?

Optimal sizing for maximizing logarithmic growth is technically going to be a function of worst singular loss.. size via fraction of capital at risk for a given trade / period. Using expected long-term volatility or any sort of forecast for the sizing input is a little dangerous if you're flying so close to the sun at true max growth, because of unexpected jumps and the like. And an inefficiency at the wrong time can be very damaging to your overall return series and end result.

Given a long enough time horizon, optimal growth is going to put your eventual worst DD into the 90%'s. Also ironically the better systems will score higher on optimal risk measures so the swings will actually be bigger and the drawdowns steeper for them than for worst systems. Many recommend backing down significantly from the optimal growth figure in order to make your band of possible paths through time a lot narrower and more predictable. Or in slightly more sophisticated fashion you can target the inflection point on the growth curve where the incremental increase in "risk" becomes too large for the additional boost to the geo return.

Discussions on this have been going on at least since Bernoulli wrote about the St. Petersburg paradox. Markowitz and Kelly are foundation to modern thought. Thorpe is a big contributor for bringing these ideas into practice. Ralph Vince develops the ideas a little further and incorporates the effect of multiple simultaneous systems, and is a big proponent of fractional risk over volatility as an input. There's at least a handful of people on the board that are more knowledgeable in this area than I am but hope that helps give some food for thought.

Hi Sturm
Nice post. I think you can just plug your adjusted sharpe ratio and expected vol into the standard kelly equation so you don't need to do any more calculus if you have a closed form equation for expected SR / vol given some expected unleveraged return / vol / leverage. Of course since we don't know the true Sharpe Ratio, what you will actually find is a multiplying factor to apply to a Sharpe Ratio in the presence of leverage / vol-drag.

GAT

4. ### wintermute0

Thanks guys! I was aware of the Kelly criterion but not the fractional one. Seems very similar to mine except it's trading off between risky and risk-free instead of adjusting leverage which comes out to: (r - rf) / var

Wish I knew this before I wrote it but maybe I can slip it into part 2 so I don't look stupid

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