I still my remember my first introduction to the “wisdom of crowds.” I was sitting in my first econometrics class at Yale, and my professor (Doug Mckee) placed a big bucket of tennis balls on one side of Davies Auditorium and an empty bucket on the other side of the room. He asked us all to guess how long it would take for a student to move all the balls from bucket to the other. All 150 of us guessed on our clickers; a student was picked at random (and properly incentivized to do well with a bar of chocolate); and he raced to move all the balls while the professor timed him. Upon completion, the professor tabulated the results and showed, to our shock, that the class’s average ex ante guess was closer to the true time than 95%(!) of the individual guesses.

In that vein, here’s a fascinating quote from a WSJ article this week by Jason Zweig:

On Jan. 16, 2010, I reported that “a nationwide survey last year found that investors expect the U.S. stock market to return an annual average of 13.7% over the next 10 years.”...Over the 10 years ended Dec. 31, 2019, the S&P 500 returned 13.6%—almost exactly what those investors I mocked had expected.

My first reaction — and likely your first reaction as well — is awe at the sheer accuracy of the forecast. A lot happens in a decade — new asset classes (crypto!), systemic risk events (covid!), investor composition (retail!) — and yet the average expectation still turns out to be correct ex post. Such accuracy testifies either to a large degree of luck or to the wisdom of crowds.

What makes this quote fascinating, is not the level of accuracy, but rather the recognition that this is different from the usual wisdom of crowds intuition we are familiar with. Normally when we speak of wisdom of crowds, we are implicitly making a statement about (1) “rational expectations” combined with (2) a central limit theorem / law of large numbers. Take for example, a poll about the final point differential between two Super Bowl teams. If on average, each individual has correct beliefs (rational expectations), and we survey a large enough sample, then the average guess will be quite close to the true mean.

The key difference between this point differential example and the stock return example is that beliefs and expectations have no effect on the outcome of the Super Bowl. That is, if everybody woke up one day irrationally more optimistic about the Ram’s chance of winning by a lot, the Rams would not play better or worse, and so the wisdom of crowds would no longer hold. Similarly, if everyone had guessed differently in the tennis ball example, those guesses would have no causal effect on the duration of the exercise.

But with asset prices, we have entirely different logic. Today’s prices and hence tomorrow’s returns are entirely a function of the aggregated beliefs of participants. If everyone wakes up tomorrow and believes something different about the future return of an asset, the prices will change as a result, and so will future returns. In other words, with asset prices, there is a causal channel between beliefs and outcomes.

To illustrate this formally is nearly trivial. Suppose a representative agent in a two period model speculates in an asset with some (random) terminal dividend $D_{t+1}$. He expects future gross returns $R_{t+1}$ to be equal to 10%. Then the basic definition of returns implies that:

$$ E_t[R_{t+1}]=1.1 \iff E_t\left[\frac{D_{t+1}}{P_t}\right]=1.1 \iff P_t=\frac{E_t[D_{t+1}]}{1.1} $$

In other words, taking expectations about the dividend distribution as given, today’s price adjusts such that the expected distribution of returns is whatever the representative agent thinks it is! If he expected returns to be higher, today’s price would drop by more, and so for any realized path of $D_{t+1}$, returns would also be higher. Even if our representative agent is very wrong about the true distribution of $D_{t+1}$, the higher the expected return ($E_t[R_{t+1}]$), the higher will be the realized return ($D_{t+1}/P_t$). By contrast, in a sports market, a higher expected point differential has no bearing on the actual point differential.

The broader intuition here is the Hayekian insight that prices aggregate all available information. A price is precisely that which the marginal buyer is willing to purchase a good for, and so reflects a type of average valuations across a group of market participants. Returns — which are mere ratios of prices — are not simply correlated with beliefs of market participants; they are the beliefs of market participants. Another way of saying this is there are many sets of equilibrium expectations that are self-fulfilling in traded asset markets.