the 2 vectors of volatility scaling

Friends,

I fired up Corey Hoffstein’s goated Flirting With Models podcast to hear Scott Phillips discuss “ugly” edges in crypto. This episode came highly recommended in my corner of twitter. It does not disappoint.

But I want to zoom in on one part. Scott says:

If you did what I do perfectly, you'd be up well over Sharpe two, and probably with size. But the way that I do it, our four-year Sharpe is 1.7 with retail-level costs. So cross-sectional momentum is a little bit better than that, but you don't have the really nice positive skew of trend. And cross-sectional carry is about a Sharpe 1.7 as well, and slightly orthogonal. So you blend the three of them together, and then you're at Sharpe two easily — and without even good execution…The math holds. Returns scale with the square root of independent bets.

Scott misspeaks here (easy to do in a conversation) but what Scott means is volatility scales with the square root of independent bets (returns scale linearly). This concept underpins one of my most important posts — Understanding Edge. It is the basis of all trading businesses without exception. It’s Day 1 learning at a prop shop. Munger has that “Take a simple idea and take it seriously” advice…This is THE idea Jeff Yass took seriously.

I’m repetitive on log and compounding math for 2 reasons that extend beyond the shock factor of the “lilypads in a pond” puzzle:

a) Investing is a serially repeated game so compound returns are our primary concern

b) Option theory sits atop logreturn math which is just continuous compounding (in fact e, the Euler constant of 2.718, is your growth of $1 if you continuously compound at 100% rate for 1 unit of time. See Using Log Returns And Volatility To Normalize Strike Distances)

We’ve confronted this math many times from different angles:

• The Volatility Drain

• Path: How Compounding Alters Return Distributions

• Examples Of Comparing Interest Rates With Different Compounding Intervals

• Well What Did You “Expect”?

• Geometric vs Arithmetic Mean In The Wild

• Understanding Log Returns

• I Felt Bad For Picking My 3rd Grader Off

• Getting Comfortable With Log Charts

• you can ONLY eat risk-adjusted returns

• Growth rate = 70% * (doublings/years)

Compounding is a multiplicative process that describes how returns scale across time. “Volatility drain” reminds us that volatility’s interaction with growth is embedded in that process.

We already understand how compounding scales across time as a function of volatility.

But volatility itself has scaling properties.

That’s what Scott means when he says (or meant to say) it grows by the square root of independent bets. Regular readers have seen me scale a daily return to an annual return by √251 or ~16 many times. But I don’t always remember to say that this assumes no correlation between daily vols (mapping to the “independent bets” language).

In multiple posts, we have seen that volatility is understated in the presence of autocorrelation:

• Thinking in N not T

• Volatility Depends On The Resolution

• the option market's point spread (part 2)

Autocorrelation affects how we scale vol through time per asset.

This is only half the volatility scaling story. It’s one vector.

The second vector is how we scale vol across our portfolio.

This depends not on autocorrelation, but on pair-wise correlation.

Independent bets are simply bets that have no (ie zero) correlation with one another.

It is why you can blend several strategies and end up with a composite Sharpe greater than any of the components.

Let’s look closer.