back-of-the-envelope hedging cost

a little exercise in proportion

Here’s a way to think about options that has nothing to do with vol.

Consider a strategy that:

1. Buys a 1-year 15% OTM put on SPX for 3% on Jan 1.

2. Puts the remaining capital in SPX

3. Every Jan 1, rebalances back into that position. So if your put expires in-the-money, you reinvest the proceeds back into SPX, if it doesn’t you sell enough SPX to buy the next year’s protection.

Ask Claude to backtest it for, say, 25 years. It only needs the SPX prices on every Jan 1 because we are not using market-based pricing for the put. We’re just saying it costs 3%

Then compare the performance to just owning SPY unhedged.

The exercise provides a confidence interval for: what premium would you be happy to pay for the put so that you could skip diversifying and just hold 100% SPY plus the insurance policy?

If you run this exercise, you’ll find for anyone with a typical risk preference*:

At 1% it’s a no-brainer.

At 5% it’s meh.

*It’s a bit chaotic to throw the phrase “risk preference” into that sentence so freely, but you can, in the spirit of upgrading your context, learn a lot from Elm Wealth’s whitepaper How We Invest

I posted that thought exercise on X and received a reply I really liked:

It’s a great example of the type of on-the-fly reasoning one develops when they don’t have the luxury of time.

I’ll rephrase the tweet’s approach in my own way, showing my logic at every step.

1. Which years had a point-to-point return of -18% (the put’s breakeven) or more?

• 2001 or 2002. One of those years, I can’t remember, but if the put paid, it was very small.

• 2008. I’d guess the market was down ~40%, so the put paid (40-18)/3 or about 7x

• 2020. Oh wait, the market didn’t actually finish lower iirc. The put didn’t pay out.

• 2022, it might have paid a small amount

2. Estimating net losses on the put

Let’s be conservative and say the put broke even in the wake of the dot-com era and 2022. So the put loses its full value in 22 years, pushes in 2 years, and makes 7x one-time.

Net net, you lost 15 out of 25 bets or 60% of your premium. If the put costs 3%, the hedge pulls down your arithmetic return by 1.8% per year.

3. What does this mean for the strategy?

Back of the envelope, I’d guess your annual arithmetic return on a naked SPX position for the past generation has been close to 10% and lop off about 1.3% for vol drag* to get a CAGR ~8.7%

This put cost you another 1.8% per year, leaving you with about 8.2% arithmetic but a touch less vol drag because it lowered your volatility. It’s a bit crunchy to import the savings from a 2008 tail event into volatility, which is more descriptive of a distribution’s meat, but a 40% move in variance space is nearly 10x bigger than a typical 15% move, so if you think of 2008 as 10 years, then the GFC represents about ⅓ of the sample in variance time.

Notice, the put doesn’t erase 2008, but it makes the loss much smaller. Without the put, 2008 was down 40%, but with the put, your loss caps at -18%.

Now measure each version as a miss from an average year:

• Unhedged, the average year is about +10%, so −40% is a 50-point miss.

• Hedged, the average year is about +8%, so −18% is a 26-point miss.

Variance cares about miss squared.

50² = 2500.

26² ≈ 676.

676 is about ¼ of 2500.

So the hedged 2008 carries about ¼ of the variance the unhedged 2008 did.

In other words, the put removed ¾ of 2008’s variance!

Since 2008, a single year, carried about ⅓ of the whole sample’s variance, the put removed about ¼ (¾ of ⅓) of the total variance.

I’m going to repeat this for tedious emphasis:

Removing ¾ of the variance for a single giant year out of 25 years removed ¼ of the total variance

¾ of the total variance remains, but since vol is the square root of variance, we need to compute √0.75 ≈ 0.87.

87% of the volatility remains, so vol goes from 16% to about 14%.

Vol drag is half of vol squared: ½ × 0.14² ≈ 1%.

So we get 8.2% arithmetic return with the put minus ~1% of vol drag and we end up with a CAGR of about 7.2%

The put cost 1.8% per year, but because it reduced drag as well, it only lops ~1.5% from CAGR. This represents somewhere between a third and a half of historical equity risk premium. Whether this is worth it or not is in the eye of the beholder, but it doesn’t sound like puts, which we expect to have a premium, are egregiously high.

4. What if the put only cost 1%?

We’ll stay conservative and say you push in 2002 and 2022. But in 2008 you make (40-16)/1 or 24x on the puts. You basically break even on owning the puts and cut off your left tail!

You get better at this the more you practice, but moving in between risk, return, and payoff scenarios like this should not only be fun but inoculate you from the viral disease of fuzzy thinking