Imagine you’re a quantitative long-short equities trader. If you can predict which stocks will have above-average returns next period and which will have below-average returns, then you can profit by buying the winners and selling short the losers. Return predictability and trading profits are two

Imagine you’re an asset-pricing researcher. You’ve just thought up a new variable, X, that might predict the cross-section of returns. And you’ve regressed returns on X in a market environment e of your choosing (i.e., using data on some specific time period, country, asset class, set of test

An asset-pricing model is a claim about which optimization problem people are solving when they choose their investment portfolios. One way to make such a claim testable is to derive a condition that should hold if people were actually solving this optimization problem. And the standard approach to

Asset-pricing research consistently produces many different explanations for the same empirical facts. As a rule of thumb, you should expect asset-pricing researchers to wildly overachieve. Behavioral researchers can typically point to several psychological biases which might explain the same

Economists use machine learning (ML) to study asset prices in two different ways. Approach #1: use these techniques to predict the cross-section of expected returns—i.e., to predict which stocks are most likely to have high or low future returns. e.g., see here, here, or here. Approach #2: use

In standard cross-sectional asset-pricing models, expected returns are governed by exposure to aggregate risk factors in a market populated by fully rational investors. Here’s how these models work. Because investors are fully rational, they correctly anticipate which assets are most likely to

Behavioral-finance models are often written down so that, although each individual trader holds incorrect beliefs, market events nevertheless unfold in such a way that traders can rationalize their own errors. e.g., consider the model in Scheinkman and Xiong (2003). In this model, each individual

Journalists love to talk about bubbles. The Wall Street Journal has hinted at bubbles in both the Chinese stock market and the market for Bitcoin during the past month alone. But, financial economists are much more reluctant to call something a bubble. There’s debate about whether bubbles even

Imagine you’re looking for variables that predict the cross-section of expected returns. No search process is perfect. So, as you work, you will inevitably uncover both tradable anomalies as well as spurious correlations. To figure out which are which, you regress returns on each variables that

Motivation Trading has gotten a lot faster over the last two decades. The term “short-run trader” used to refer to people who traded multiple times a day. Now, it refers to algorithms that trade multiple times a second. Some people are worried about this new breed of short-run trader making

Imagine you’re a financial economist who thinks that some risk factor,{\color{white}i}f_t, explains the cross-section of expected returns. And, you decide to test your hunch. First, you regress the realized returns of N different assets on{\color{white}i}f_t to estimate each asset’s exposure to

Imagine we’re traders in a market where the cross-section of returns is related to V \geq 1 variables: In the equation above, \alpha^\star denotes the mean return, and each \beta_v^\star captures the relationship between returns and the vth right-hand-side variable. Some notation: I’m going to

Traders are constantly looking for variables that predict returns. If x is the only candidate variable traders are considering, then it’s easy to use the Bayesian information criterion to check whether x predicts returns. Previously, I showed that using the univariate version of the Bayesian

Imagine that we’re trying to predict the cross-section of expected returns, and we’ve got a sneaking suspicion that x might be a good predictor. So, we regress today’s returns on x to see if our hunch is right, \begin{align*} r_{n,t} = \hat{\mu}_{\text{OLS}} + \hat{\beta}_{\text{OLS}} \cdot

Imagine you’ve just seen Apple’s most recent return, r, which is Apple’s long-run expected return, \mu^\star, plus some random noise, \epsilon \overset{\scriptscriptstyle \mathrm{iid}}{\sim} \mathrm{N}(0, \, 1): (1) \begin{align*} r &= \mu^\star + \epsilon. \end{align*} You want to use

1. Motivating Example This post shows that, if traders face convex transaction costs (i.e., it costs them more per share to buy 2 shares of stock than to buy 1 share of stock), then it is possible to infer traders’ investment horizons from trading-volume data. To see why, imagine you are a trader

I have been trying to use dimensional analysis to understand asset-pricing problems. In many hard physical problems, it is possible to gain some insight about the functional form of the solution by examining the dimensions of the relevant input variables. In the canonical example of this brand of

This post looks at the consequences of ETF rebalancing. These funds follow pre-announced rules that involve discrete thresholds. The well-known SPDR tracks the S&P 500, but there are over 1400 different ETFs tracking a wide variety of different underlying indexes. When any of these underlying

1. Motivating Example A Popular Goal. Financial economists have been looking for variables that predict stock returns for as long as there have been financial economists. For some recent examples, think about Jegadeesh and Titman (1993), which shows that a stock’s current returns are predicted by

1. Motivating Example Jegadeesh and Titman (1993) show that, if you rank stocks according to their returns over the previous 12 months, then the past winners will outperform the past losers by 1.5{\scriptstyle \%} per month over the next 3 months. But, the authors don’t just test this particular

1. Motivation Evolutionarily Slow. In modern financial markets, people simultaneously trade the exact same assets on vastly different timescales. For example, a Jegadeesh and Titman (1993)-style momentum portfolio turns over half its holdings once every 6 months. By contrast, Kirilenko, Kyle,

Kyle (1985) introduces an information-based asset-pricing model where informed traders keep trading until the marginal benefit of holding one additional share of the asset is exactly offset by the marginal cost of this last trade’s price impact. This model has really nice intuition, but it also

1. Motivation New information-based asset-pricing models are often extensions of either Kyle (1985) or Grossman-Stiglitz (1980). At first glance, these two canonical models look quite similar. Both price an asset with an unknown payout, like a stock or bond, and both analyze the strategic behavior