Cross-sectional Predictability


Kevin Crotty
BUSI 448: Investments

Where are we?

Last time:

  • CAPM and the Market Model Regression
  • CAPM: Theory
  • CAPM: Practice

Today:

  • The cross-section of expected stock returns
  • Portfolio sorts
  • Cross-sectional regression

Expected returns

Estimating a stock’s expected return

  • How should we think about estimating \(E[r_i]\)?
  • Core finance: use CAPM \[ E[r_i]=r_f + \beta_i E[r_{\text{mkt}}-r_f] \]
  • But we saw last time that this does poorly empirically
    • as does using historical returns
  • Today, we’ll discuss some firm characteristics and how they relate to expected returns

Portfolio sorts

Sorting stocks

  • Consider a characteristic of a firm, like its beta.
  • How can we test if high beta firms have high expected returns?
  • One method:
    • sort stocks on betas
    • form portfolios (b/c stock returns are noisy)
    • test if high beta portfolios have higher ex post returns than low beta portfolios

A litle history on sorting stocks

  • Market beta (Black, Jensen, Scholes 1972)
    • If anything, high beta firms underperform low beta
  • Size (Banz 1981)
    • Small firms outperform large firms
  • Book-to-market ratio (Fama-French 1992)
    • High B/M (value) outperform low B/M (growth)

Is the CAPM dead?

Source: Fama-French 1992

More history on sorting stocks

  • Liquidity (Amihud and Mendelson 1986)
    • less liquid \(\rightarrow\) high \(r_{t+1}\)
  • Momentum (Jegadeesh-Titman 1993)
    • past winners beat past losers
  • Idiosyncratic volatility (Ang, Hodrick, Xing, Zhang 2006)
    • High idiovol \(\rightarrow\) low \(r_{t+1}\)

Visualizing anomalies

One-way sorts

Two-way sorts

Sorting in python

# Sorting function
def cut_quintiles(x):
    try:
        out = pd.qcut(x, 5, labels=["Lo 20", "Qnt 2", "Qnt 3", "Qnt 4", "Hi 20"])
    except:
        out = pd.Series(np.nan, index=x.index)
    return out
CHAR = 'beta'
df["quintile"] = df.groupby("date")[CHAR].apply(cut_quintiles)

Cross-sectional regressions

Cross-sectional regression

  • An alternative approach to sorting is a cross-sectional regression

Regress each stock’s average return (a time-series average) on its average characteristic:\[ \overline{r}_{i} = a + b \cdot \text{characteristic}_{i} + e_{i} \]

  • If the characteristic is associated with higher returns, \(\hat{b}\) should be different from zero!

Fama-MacBeth

  • Characteristics are often time-varying, so it is preferable to use a series of cross-sectional regressions.

For each time period, run a cross-sectional regression:\[ r_{i,t} = a_t + b_t \cdot \text{characteristic}_{i,t-1} + e_{i,t} \]

  • This produces a time-series of coefficients \(b_t\). If the characteristic is associated with higher returns, the time-series average of \(\hat{b}_t\) should be different from zero!

Multivariate Fama-MacBeth

  • We can also characterize returns as a linear function of multiple characteristics:

For each time period, run a cross-sectional regression:\[ r_{i,t} = a_t + b_{1,t} \cdot \text{X1}_{i,t-1} + b_{2,t} \cdot \text{X2}_{i,t-1}+ e_{i,t} \]

Regressions or sorts?

  • Regression restricts the relation between returns and the characteristic to be linear
  • Sorting allows for a nonlinear relation across stocks
    • It is more flexible, but it is reassuring if the relation is monotonically increasing or decreasing.
  • Recent literature has applied various machine learning techniques to better characterize the cross-section of expected returns.

Persistence of anomalies

  • What should happen as market participants become aware of these results?
  • They should trade until the differences disappear.
  • Anomalies that do not go away may be compensation for risk and should be accounted for as part of the asset’s risk premium \[ E[r] = r_f + \text{risk premium} \]
  • More on this next time…

For next time: Multi-factor Models





BUSI 448