Risk Depends On The Resolution

I am disappointed with investing section of my last post Plane With Zits. Let’s remediate the problem with it and see where we land.


a) We recalled how volatility, a first order quantity, “drags” down median returns in a non-linear fashion

The volatility term drag is a squared term. This is same intuition can be appreciated from another angle — if you lose X% you need to gain back X/(1-X) which you can plot in your trusty TI-82 to see it’s non-linear.

  • Lose 10% you need to make 11% to get back to even.

  • Lose 33%, need to gain 50%.

  • Lose 50%, need to gain 100%.

  • Lose 75%, need to gain 300%

b) I showed why the impact of large drawdowns have an outsize impact on CAGR

My toy example assumed compounded returns of 9% for 19 years then 45% drawdown.

c) In such an event you are roughly in the same place had you put 50% in stocks and 50% in bonds yielding 4%

As soon as I hit send I started to feel weird about it. I did something lazy. And the problem got worse because I got 3 messages from people saying it was one of the best things they’ve seen because it confirmed intuition but hadn’t seen it presented this way. But there’s a problem with it. In fact I told one of the readers to call me because I wanted to explain why this needed revision.

So as a mini-test, ask yourself what the problem is? (It’s not a tax thing either).


Ok, let’s just jump in to the thought process and the fix.

I originally picked 9% because I wanted a CAGR that our collective conscience would agree is a reasonable guess for what long-term equity index CAGR is.

The problem is I can’t use 9% for 19 out of 20 years because the 20 year CAGR needs to be about 9% inclusive of the drawdown! Our perception of what equities return includes all the terrible times already. I can’t just use that CAGR and then bolt on 45% drawdown.

Instead, I needed to:

  1. Pick a number for those 19 years that was higher than the CAGR

  2. Apply the 45% drawdown

  3. Make sure the resulting 20 year CAGR was 9%

Once I got to that point I just looked up what SP500 monthly returns were going back to 1926 via https://www.officialdata.org/us/stocks/s-p-500/1900 (The SP500 index didn’t exist then but since they base this on Robert Shiller’s work I’ll just assume the historical reconstitution is valid).

Using monthlies, the data set includes 1161 rolling 12-month returns. We find:

  • Annual Simple (arithmetic) Return 11.4%

  • Annual CAGR: 10.2%

  • Annualized volatility: 15.4%

  • .50% (ie 1 in 200) of these returns include a 12-month loss of 45% or greater

In the last post I made the disaster year occur 1 out of 20, but historically the odds were much small than that measured at monthly resolution.

I re-did the computation assuming that the typical year is an 11.4% return and allowed 2 variables to vary:

  1. the disaster year return (R)

  2. the probability of a disaster (p)

The formula in each cell is:

The table output:

(emphasis on cells with a roughly a 10.2% CAGR)

This is not a stock simulation so the 11.4% assumed return can just be thought of as a compounded return net of the volatility. This isolates the effect of a 12-month drawdown of R for probability p just to see how sensitive the total CAGR is.

It’s not until a 45% disaster occurs in 1 in 50 to 1 in 200 years does it threaten to knock a full 1% off the CAGR.

This might make readers now rush to the other side of the boat…”hey it’s a great idea to put 100% in stocks”

But remember, the history of the US stock market is a small sample size. The true sample size requires looking at non-overlapping returns as opposed to rolling 12-month returns. Which means you get as many data points as you do years.

Plus it’s only the US.

Jared appears again:

But let me add a mathematical point to the discussion…looking at monthly returns hides the emotional path as well as knowledge of the distribution.

Let me explain. Standard deviations are normalized measures. They are move sizes scaled to time.

The Socratic demonstration:

Is it more likely for a stock index to fall 10% in 1 year or 1 day?

That’s easy, in 1 year of course. But the return by itself is not normalized for time. It’s just a raw number…10%

Let’s ask this another way.

Is it more likely for the stock market to fall 3 standard deviations in 1 day or in 1 year?

You should now choose 1 day.

Think of it this way…in 1987 the stock market fell more than 20% in one day. I don’t know what SP500 volatility was leading up to the crash but I’d be surprised if the daily standard deviation was more than say 3%. That day would have been 7 standard deviations.

You have never seen a 1 year 7 standard deviation move.

Largest single day moves for the Dow:


Using the overlapping data from earlier we find 3 annual standard deviation moves occurring .50% of the time (fatter than normal distribution) but some of these daily moves would be considered impossible.

The shorter the sampling period, the fatter the tails.

Or said otherwise:

For a shorter time horizon, the 1% probability move will be more standard deviations than the longer time horizon. (You can see this implied in option surfaces as well)

So if you look at returns at low resolution, you miss the experience. Even if you look at 2020 monthlies, it doesn’t seem anywhere near as significant as the feelings you had as an investor through it.

Summing up:

  • Using monthly and annual resolution, I overstated the risk.

  • But risk depends on the resolution. If you are an investor and can avoid looking at your account, you actually witness less volatility (on a standard deviation basis)! This is an argument for ignoring path.

    Calm In The Distance, Turbulence Up Close
  • The problem is there’s nothing about past US returns that indicate what the future holds. Assuming real returns (after-inflation) of 6% is aggressive.

  • 100% stocks when your investing life is one draw from a 40-year series has more to do with faith than judgement.

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