Stocks as Lottery Tickets: the MAX Effect

People buy lottery tickets. This is, from the point of view of standard economics, an embarrassing fact. A lottery ticket has a negative expected value by design. Buying one is a guaranteed way to lose money on average, and everybody knows it.

Yet millions of people buy them cheerfully, every week. The explanation is that human beings do not evaluate a small chance of an enormous payoff the way arithmetic says they should. We overweight tiny probabilities of huge wins. We are willing to pay more than fair value for the dream.

Turan Bali, Nusret Cakici and Robert Whitelaw asked the obvious follow-on question: do people do this in the stock market too? And they found a startlingly simple way to test it.

The problem: how do you find the lottery tickets?

If some stocks are the market's lottery tickets, and investors overpay for them, those stocks should earn poor returns. But which stocks are they?

You could try to measure skewness, or estimate the probability of a big jump, or build a complicated model of the return distribution. All of these are noisy and require choices.

The authors' measure is almost comically simple. Look at the stock's single largest daily return over the past month. That is it. Call it MAX.

Why does that work? Because a stock that jumped 15% on one day last month has demonstrated, in the most vivid possible way, that it is capable of doing so. It has advertised itself as a stock that can produce a spectacular day. It looks like a lottery ticket, and more importantly, it looks like one to the sort of investor who is shopping for lottery tickets. The single best day is a salient, memorable, attention-grabbing event, and salience is exactly what draws in the buyers who chase jackpots.

The key idea via analogy: paying for the dream

Sort every stock by its maximum daily return over the previous month, form portfolios, and look at what happens next.

The result: a strong negative relationship between MAX and future returns. The stocks with the biggest single-day pops went on to do badly. The gap between the lowest-MAX and highest-MAX groups exceeded one percent per month in both raw and risk-adjusted terms, which is a very large effect.

The interpretation follows the lottery logic exactly. A stock that recently had a spectacular day attracts a crowd of investors who want a chance at another one. That crowd bids the price up. An overbid price today is a poor return tomorrow. You are not being paid for taking risk; you are on the wrong side of somebody else's dream.

The buyers are not being irrational in the sense of being stupid. They are getting something they value: excitement, the possibility of a life-changing gain, a story. They are simply paying for it, and the payment comes out of their long-run returns. Exactly as with a real lottery ticket.

The kicker: it explains the volatility puzzle

Here is what makes the paper more than a curiosity.

There was already a famous, awkward finding, from Ang, Hodrick, Xing and Zhang, that stocks with high idiosyncratic volatility earn low returns. That is a genuine puzzle, because diversifiable risk is not supposed to be priced at all, let alone negatively.

Bali, Cakici and Whitelaw showed that once you control for MAX, the idiosyncratic volatility puzzle largely goes away. The two effects are not independent. High-volatility stocks are, mostly, stocks that have big up days. The reason high-volatility stocks earn low returns is that they are the lottery tickets.

That is a genuine unification. A well-known anomaly that nobody could explain turns out to be a symptom of a simpler, more human phenomenon that anybody who has ever watched somebody buy a scratch card can understand.

Why it mattered

• It gave the lottery-preference idea a hard, testable measure. Behavioural theories about a taste for skewness had existed for years. MAX made them tradeable and falsifiable.
• It resolved a standing puzzle. Subsuming the idiosyncratic volatility anomaly is a serious achievement, and it is the reason this paper is cited so heavily.
• It supports the whole low-volatility family. Lottery preferences give an economic reason why the exciting end of the market is overpriced, which complements the leverage-constraint story of Frazzini and Pedersen and the benchmarking story of Baker, Bradley and Wurgler. Several mechanisms, all pointing the same way.
• It is a practical screen. "Avoid stocks that just had a huge day" is advice a retail investor can act on, and it points in the opposite direction from most people's instincts, which is usually where the money is.

The honest limitations

• The profit is in the short leg. The return comes overwhelmingly from not owning, or shorting, the high-MAX stocks. But those are precisely the volatile, small, expensive-to-borrow names where shorting is costly and dangerous. Capturing the effect in practice is far harder than the paper's numbers suggest.
• One month is a short, noisy window. MAX is computed over a single month of daily returns, and it inherits all the noise of that window. It also requires frequent rebalancing, which means turnover, which means costs.
• The causal story is inferred, not observed. The paper does not watch investors buying lottery stocks. It infers the preference from the price pattern. Alternative explanations, including plain short-term reversal after a price spike, are hard to fully rule out.
• It overlaps with everything nearby. MAX, idiosyncratic volatility, skewness, beta and short-term reversal are all tangled together. Which one is the true underlying variable is genuinely unclear, and different papers give different answers.

The one-line takeaway

Bali, Cakici and Whitelaw showed that a stock's single biggest day in the past month predicts poor returns going forward, because such stocks look like lottery tickets, investors overpay for the chance of a jackpot exactly as they do with real lotteries, and this simple fact explains away the puzzle of why volatile stocks earn so little.