stock-bond correlation

Return Stacked’s RSSB gives you a dollar of global equities and a dollar of Treasuries on the same dollar of capital. They just listed options on it this week. It will be a useful option market to watch if it gathers liquidity because its implied vol allows you to back out an implied stock-bond correlation since we know the vol of the legs.

Risk-parity funds and strategies are diversified, which gives them cover to use leverage but also means they are structurally short stock-bond correlation. In theory, they should be natural buyers of RSSB vol. If they actually did that, I’d expect the implied vol to trade at a healthy risk premium since it’s a one-way risk. There’s no real natural seller of that correlation.

I asked the Moontower Agent to work out the details and generate a deck (which is also a heat check on its abilities.) It produced a deck I could download, walking through the variance identity, a IEF-for-GOVT proxy swap, a scenario table for implied ρ across RSSB IV levels, and the noise associated with realized correlation.

The deck is short and educational:

Reading Implied Correlation Through RSSB Options

Confidence Intervals on Correlation

This made me look up a time series of the correlation between stocks and bonds using VTI and IEF as respective representatives of the asset classes. The 30-day came back +0.43, the 1-year at +0.28.

The agent volunteered the following statistical insight:

N=22 daily returns is a small sample. The 95% confidence interval on a correlation of +0.43 with N=22 is roughly [+0.02, +0.72] — wide. You cannot confidently distinguish +0.43 from +0.20 or +0.60 at that sample size. The 1Y number (N=251) is statistically much tighter — CI roughly [+0.16, +0.39].

Normally, when you compute a confidence interval, you effectively make a market by scaling the standard deviation to your desired confidence (so 1.96 or “2 sigma” for ~95% confidence)

But take notice of the first interval: +0.02 to +0.72 isn’t centered on +0.43. The upper tail runs 0.29 above, the lower tail 0.41 below. Weird. You wouldn’t see lopsided error bars if the math was something like “estimate ± 1.96 × standard error”.

Agent teach me what you did and why.

The problem is that the error bar’s width depends on the answer

Correlation lives on [−1, +1]. Bounded. If the true ρ is 0.9, your sample estimates can’t overshoot by much (the ceiling is 1.0) but they can undershoot plenty.

That boundedness shows up in the standard error of r itself, which runs about (1 − r²)/√(N − 1). Look at what that does. At r = 0.43 with N = 22 the SE is 0.177. At r = 0.9 with the same 22 observations it’s 0.042, four times tighter. Not because you learned more, but because you got squeezed against the wall.

The agent used something it called Fisher’s fix to move to a coordinate system where the standard error stops depending on r, do the easy symmetric thing there, and come back.

The recipe

1. Transform the point estimate: z = arctanh(r) = ½ · ln[(1 + r) / (1 − r)]

2. Take the standard error in z-space: SE = 1/√(N − 3). Note what’s missing. No r. Sample size is the only input.

3. Build the interval symmetrically, the boring way you already know: z ± 1.96 · SE

4. Recover each endpoint separately: tanh(z_low) and tanh(z_high)

Broadly educational bits to notice

Step 1 does almost nothing to the estimate.

0.4332 becomes 0.4635. The transform is only there to get the r out of the standard error in step 2.

Step 4 is where the lopsidedness comes from.

Both 30-day endpoints sit exactly 0.4497 away from the center in z-space. Perfectly symmetric. But tanh squashes hard once you’re out past 0.5 but does almost nothing near the origin, so the top end at 0.9131 gets crushed down to +0.72 while the bottom end at 0.0139 is nearly untouched at +0.014. What matters is each endpoint’s distance from zero, not its distance from your estimate.

Step 2 accounts for sample size

Ten times the sample and the standard error only comes down by a factor of 3.6. At large N the confidence grows by √N scaling, but at small N, subtracting 3 makes the confidence grow slower than square root scaling.

2 lessons

1. The object-level lesson: Correlations are volatile and because they are bounded from [-1,1] we need to recenter our point estimates before cuffing their ranges
2. The meta-lesson: LLMs are great math tutors so when they volunteer info of the “I don’t know what I don’t know” variety they can hold your hand. I’ve shared how I use LLMs as a tutor before in Socrates 2026: how to use highlights