Standard Deviation of Returns (Part 2)

I left off by illustrating how standard deviation (SD) of returns is greater for my new friends-and-family account than for the underlying index.

I think of naked puts (NP) as having a significantly lower SD of returns so this was puzzling. Is it a bona fide finding?

Applying the rolling-returns rationale, we can observe SD to be lower for NPs (red line) most of the time. When the market pulled back, SD moved higher for the NPs and has remained higher to date.

Max drawdown (MDD), also discussed in the previous post, would produce a similar result. MDD was roughly 4.8% and 6.6% for the NPs and underlying index, respectively. Despite the small sample size, this is a significant difference in favor of NPs.

I calculated risk-adjusted return (RAR) by dividing total return by SD of returns. This improved NP return from 6.41% to 6.66% and benchmark return from 0.12% to 0.17%. On a percentage basis this represents a much greater improvement for the benchmark but the magnitude of numbers is so disparate that the NP return is still far greater.

Weighing all the evidence, even if the finding is real it certainly may not be relevant.

But then I stumbled upon a solution.

The finding is real and a function of the leverage ratio. Leverage ratio is notional risk of the account divided by net liquidation value. It makes sense to say the more NP positions I hold, the wider account value swings I will see. The underlying index has no position size so its SD will remain constant. Studying SD of the daily account value changes is probably not very meaningful for this reason.

If I wanted to compare SD between NPs and the underlying index then looking at a large sample size of matched trades would probably be best. As one example, suppose I shorted a 700 put. The notional risk is 700 * $100/contract = $70,000. I would therefore use $70,000 as the initial account value and calculate the SD of daily % changes in account value until position close. For the underlying index, I would start out by purchasing a number of shares equal to $70,000 / index price. Daily index account value then equals number of shares multiplied by the index value on each day. From that I could calculate SD of the daily % changes. These two SDs could be compared.

So there you have it: leverage ratio is the culprit making my SD of returns larger than the benchmark! As long as the total returns are significantly better, though, I don’t expect much client pushback.