By Victor Haghani and James White¹
ESTIMATED READING TIME: 6 min.

What sayeth our readers?

Thank you to the 2,792 readers (and counting) who have submitted their estimates for the probability of a 30% stock market drop over the coming year. If you haven’t read the article yet – or put in your personal estimate – it’s not too late: How Likely is a Stock Market Crash?

Summary of estimates

The average estimate was a 30.5% probability of a stock market crash in the coming year, while the median probability (half of responses higher and half lower) was 25%. Only 31% of respondents put the probability between 4% and 16%, in the range of 0.5x – 2x the 8% estimate we suggested in our article.

It seems our article, and our 8% estimate, did persuade many respondents to lower their estimates. For the roughly 820 who also submitted a revision after reading the article, their average estimate dropped from 34% to 20%. We are happy that only 5% of people read our article and then submitted a follow-up response further from 8% than their initial estimate.

Perhaps we shouldn’t be surprised by what seem like high estimates of the probability of a crash, given that the Shiller/Yale crash survey finds about 65% of respondents who estimate the probability of a crash (albeit defined differently than our crash) at greater than 20% over the next year.² We found 63% of our respondents estimating a stock market crash at over 20%, almost identical to the Shiller/Yale survey. This coincidence, despite our crash definition being quite different than theirs, suggests that perhaps the exact specification of “stock market crash” doesn’t matter that much to how people respond to the question. Perhaps to most, a crash is a crash?

Interpretations

What should we make of the 30% average estimated probability of a stock market crash?

There is an extensive academic literature that tries to explain this type of result, with one of the foremost contributions being the Goetzmann, Kim and Shiller 2024 article, “Emotions and Subjective Crash Beliefs.”

Their research tries to explain why investors’ survey‑based probabilities of a catastrophic stock market crash are far higher than what historical data or option‑implied crash risk would suggest. They decompose those beliefs and find that the extra “subjective” crash probability is strongly linked to negative, high‑arousal emotions and emotional reactions to salient local events, supporting a “risk‑as‑feelings” explanation for why people systematically overestimate crash risk.

To those we didn’t convince

First, we should recognize that the estimates we received were generally made without much analysis. As we saw, those who re-submitted after reading our note dropped their estimates dramatically. But what to make of the roughly one-quarter of those who read our article, weren’t terribly convinced by our analysis and submitted an updated estimate of a 30% or higher crash probability – about 200 respondents in total?

If you’re one of those readers, have we got a trade for you! You should want to buy out-of-the-money put options, and lots of them, as explained in the table below. If you believe there’s a 30% chance of the stock market being down 30% at some point over the next year, the 30% out-of-the-money put options will look like a great (though risky) investment. We show the optimal fraction of wealth to invest in these options, under the conservative assumption that it is uncorrelated with any other things you’d like to invest in. If we recognize the fact that the put option provides a hedge against other risky investments you may be attracted to, you’d want to own even more of it; of course, if you think the probability of a 30% crash is even higher than 30%, that should make you want to buy more of it too.

We’ve done the calculation for three different degrees of personal risk-aversion. For an investor with a typical degree of risk-aversion, \(\gamma = 2\), the optimal allocation is to spend about 3% of your wealth on option premium, which gives you a short position (conditional on exercise) equal to three-times the size of your wealth. We’d love to know if there are readers who have actually bought such options? If you see these numbers but aren’t willing to buy much or any of these options, perhaps it will persuade you that you may not really have such a high estimate of a crash after all.

Only 2% of respondents estimated the probability of a crash at less than 5%³ so we won’t spend time on assessing whether those people should be selling put options (our short answer is it’s probably just not worth it).

A few interesting comments from readers

We received some interesting and useful feedback on our article, much of it directing us to a trove of research on the topic of our note. In addition to the Goetzmann et al. paper already mentioned, you may find an article by Ian Martin, “What is the expected return on the market” (2017) interesting, as it too tries to infer the probability of a crash from options prices.⁴

We heard from a few readers that they agreed with the quote we included from Mark Spitznagel, who “expect[s] an 80% crash…but only after a massive, euphoric, historic blow-off rally.” This view reminds us of the cognitive quirk illustrated by Kahneman and Tversky in the “Linda problem.”

We recognize that there’s not an exact correspondence between the Linda problem and the possibility that some of our readers view a huge rally followed by a crash as more likely than just a crash, but we suspect this line of thinking may explain some of the high probability estimates submitted.

Connecting the dots

We recognize that the options market is not the ultimate gospel in predicting the future, and that there was a degree of subjectivity in the adjustments we made to the “raw” prediction of options prices to account for the equity market risk premium in both the return of the stock market and the options market implied volatility. And yes, we’re living in times when investors will pay $16.7 million for 696 pennies – on average 2.4 million times the value of each penny – so perhaps some skepticism about market rationality and the wisdom of crowds is warranted.

In the spirit of Robert Aumann’s Agreement theorem, we are updating our estimate of the probability of a crash from 8% to 11% to incorporate the weight of our readers’ estimates.⁵ If you’ve read this second note all the way to this point, we believe you also agree that we should not agree to disagree on this estimate of a stock market crash. We trust that this note has brought us closer together, and even if we don’t all agree on the same exact number, we hope we’ve further narrowed the range of our disparate views.

Appendix: Using Brier scores to grade stock market crash predictions

We also were asked how to evaluate whether the prediction of the options market has been accurate historically. The most popular measure for assessing binary predictions is called the “Brier score,” widely used to measure the accuracy of probabilistic predictions, especially for binary outcomes like weather forecasts, by comparing predicted probabilities to actual outcomes.⁶ The lower the Brier score, the better the predictive power of the forecaster. For example, if the options market was generally predicting an 8% probability of a crash over the next year, and it happened in 8% of the past 50 years, then the Brier score would be 0.07.⁷ In Phil Tetlock’s Superforecaster trials, the best forecasters achieved Brier scores of around 0.2, which was considered excellent.⁸

1. This is not an offer or solicitation to invest, nor are we tax experts and nothing herein should be construed as tax advice. Past returns are not indicative of future performance.
   
2. We doubled the probability of 10% that the question poses for the next six months.
   
3. Excluding the roughly 3% who answered 0%, which we think was either not intended, or shouldn’t be taken too seriously.
   
4. We believe that Martin calculates the probability of a crash using a risk-neutral estimate of volatility, whereas we favor using an estimate which recognizes that there are material limits to arbitrage in this domain, the systemic nature of which merits a volatility risk premium.
   
5. Recall from our previous note that, without any adjustments from risk-neutral to real-world measure, the options market was “predicting” a 14% probability of crash.
   
6. \(\text{Brier score} = \frac{1}{N} \sum_{(i = 1)}^N ( f_i – o_i )^2\)
   where \(f_i = \text{forecasted probability for event i}\) and \(o_i = \text{outcome (1 if event occurs, 0 if not).}\)
   
7. If someone predicted it would happen 30% of the time, but it only happened 8% of the time, their Brier score would be 0.12.
   
8. Or, for a more relatable example, meteorologists in northern Europe achieved a Brier score of 0.14 in their 24-hour rain forecasts.