Maxing Out: Stocks as Lotteries and the Cross-Section of Expected Returns

Source: Bali, T. G., Cakici, N. & Whitelaw, R. F. (2011). Journal of Financial Economics

99(2), 427–446.

TL;DR

Stocks with high maximum daily returns over the previous month (MAX) — a lottery-like feature —

earn low subsequent returns. A strategy long low-MAX and short high-MAX stocks is profitable, and

MAX largely explains the otherwise puzzling negative relation between idiosyncratic volatility and

returns documented by Ang et al. (2006).

What anomaly it documents

Investors appear to overpay for stocks that offer a small chance of a large payoff (positive

skewness / lottery characteristics). Those stocks become overpriced and subsequently underperform —

a cross-sectional anomaly driven by a behavioural preference for skewness.

How to construct it

Sorting variable: MAX = the average of the five highest daily returns of the stock in the prior

month (MAX(1) uses just the single highest day; MAX(5) is the common version).

Universe: US common stocks.

Portfolio: long the bottom decile (low MAX), short the top decile (high MAX).

Weighting / rebalancing: value-weighted legs, rebalanced monthly.

Evidence and replication

Period	Result	Source
IS (1962–2005)	Raw decile spread ~1%/month; significant after Fama-French and momentum controls	this paper
Interaction	MAX absorbs much of the negative idiosyncratic-volatility effect (Ang et al. 2006)	this paper

Like most anomalies, it should be expected to decay out of sample, and its short leg sits in volatile

small-caps where costs bite.

Why it might work

Lottery preference / skewness demand: investors overweight low-probability, high-payoff

outcomes (Barberis & Huang, 2008).

Under-diversified retail investors (Kumar, 2009) tilt toward lottery stocks.

Limits to arbitrage keep these stocks overpriced.

Limitations and risks

Transaction costs and shorting constraints on the high-MAX short leg.

Overlap with idiosyncratic volatility and illiquidity effects.

Out-of-sample decay and sensitivity to the MAX definition.

Key references

Bali, T., Cakici, N. & Whitelaw, R. (2011) — Maxing Out: Stocks as Lotteries — Journal of Financial Economics

Ang, A., Hodrick, R., Xing, Y. & Zhang, X. (2006) — The Cross-Section of Volatility and Expected Returns — Journal of Finance

Barberis, N. & Huang, M. (2008) — Stocks as Lotteries — American Economic Review

Kumar, A. (2009) — Who Gambles in the Stock Market? — Journal of Finance

Reference replication on ConvexPi

An open, verified replication of this strategy is maintained at convexpi/replications. It recomputes the strategy from underlying building blocks and scores it out of sample (the McLean & Pontiff test):

Period	Annualized Sharpe
In-sample (pre-2011)	-0.11
Out-of-sample (≥ 2011)	-0.29
Last 10 years	-0.41

Verdict: dormant. Run it on live data in Colab · view the code