HW #6: Portfolio Sorts and Long-Horizon Return Inference

In anomalies papers, the core empirical workflow is:

Choose a signal (a number you can compute for each stock today that might predict future returns). Examples of signals might include past returns (momentum), valuation ratios (price to earnings P/E), firm characteristics (market size; dividend yield), trading activity (recent volume or volatilty), etc.
Sort stocks into groups based on that signal.
Form portfolios and track their returns.
Compute a winner-minus-loser (“long-short”) strategy.
Summarize the strategy’s performance with an average and a t-stat.

This assignment practices that workflow with small, friendly calculations.

1) A signal and a sort

A signal is a number we compute for each stock at the start of a period. A classic example is past 12-month return (“momentum” style), but signals could be many things.

You have 10 stocks with the following signal values:

library(tidyverse)

data <- tibble(
  stock = letters[1:10],
  signal = c(-.08, 0.04, 0.01, 0.12, -0.02, 0.20, 0.06, -0.10, 0.15, 0.09)
  ) # %>%
  # arrange(___) %>%
  # mutate(grp = if_else(___, ___, ___))

data

# A tibble: 10 × 2
   stock signal
   <chr>  <dbl>
 1 a      -0.08
 2 b       0.04
 3 c       0.01
 4 d       0.12
 5 e      -0.02
 6 f       0.2 
 7 g       0.06
 8 h      -0.1 
 9 i       0.15
10 j       0.09

Sort into 2 groups:

Losers = bottom 5 signal values
Winners = top 5 signal values

Use mutate to edit the variable definition of data to add a new variable grp which takes on “winner” or “loser”. You may want to arrange first.

2) Equal-weight portfolio return

An equal-weight portfolio gives each stock the same weight. So if there are 5 stocks, each gets weight (1/5). A portfolio return is just the weighted average of its component returns.

Suppose next month the returns of the Winner group stocks are given by returns. Compute the equal-weight Winner portfolio return and the equal-weight Loser portfolio return.

data %>%
  mutate(returns = c(.02, .01, .04, -.01, .03,
                     -.02, .00, .01, -.03, .02)) # %>%

# A tibble: 10 × 3
   stock signal returns
   <chr>  <dbl>   <dbl>
 1 a      -0.08    0.02
 2 b       0.04    0.01
 3 c       0.01    0.04
 4 d       0.12   -0.01
 5 e      -0.02    0.03
 6 f       0.2    -0.02
 7 g       0.06    0   
 8 h      -0.1     0.01
 9 i       0.15   -0.03
10 j       0.09    0.02

  # ___

Anomalies papers often report a long-short portfolio, which means: \(r_{LS} = r_{\text{Winners}} - r_{\text{Losers}}\) This helps cancel out market-wide effects that hit all stocks: if the market as a whole is up 5%, winners might be up 7% and losers might be 4%, which yields a long-short return of 3%. The effect of the market being up is largely cancelled out.

3) Holding periods

A holding period is how long you keep a portfolio after formation. If you hold for multiple months, your total return compounds. This is different from re-forming the portfolio every month.

Suppose you form a portfolio and then hold it for 3 months. The monthly returns are:

Month 1: (0.01)
Month 2: (-0.02)
Month 3: (0.03)

Compute the 3-month buy-and-hold total return: \((1+r_1)(1+r_2)(1+r_3)-1\).

4) Event-time vs calendar-time

In event-time portfolios, you track performance relative to when the strategy was formed (like “Month 1 after formation,” “Month 2 after formation,” etc.). In calendar-time portfolios, you create a strategy return series month-by-month (January return, February return, March return, …). Long-horizon strategies often create overlapping portfolios, where multiple formation cohorts are active at the same time.

You form a 2-month holding strategy every month: - You form Cohort 1 in January (holds Jan and Feb) - You form Cohort 2 in February (holds Feb and Mar) - You form Cohort 3 in March (holds Mar and Apr)

In February, how many cohorts are active at the same time?

5) Why overlapping creates dependence

Overlapping portfolios reuse some of the same months of returns. For example, both Cohort 1 and Cohort 2 include February. That means the strategy returns you compute across months are not perfectly independent over time. This matters for standard errors and t-stats.

True or False: If a strategy uses overlapping holding periods, then the strategy’s monthly returns are more likely to be correlated over time than if it used non-overlapping periods.

6) Measuring abnormal performance

Before we do CAPM, we can still benchmark performance in simple ways.

Excess return: \(r - r_f\)
Market-adjusted return: \(r - r_m\)
Long-short spread: winners minus losers (already benchmark-like)

This month: - Long-short strategy return: \(r_{LS} = 0.015\) - Risk-free rate: \(r_f = 0.002\)
- Market return: \(r_m = 0.010\)

Compute: 1) Excess long-short return: (r_{LS} - r_f)
2) Market-adjusted long-short return: (r_{LS} - r_m)

7) Mean return and a simple t-stat

A strategy’s average return is its typical performance. The t-stat compares the average return to its standard error:

\[t = \frac{\bar{r}}{SE(\bar{r})}\]

A large magnitude t-stat means the average return is far from zero relative to uncertainty.

Suppose a strategy has:

Average monthly return: \(\bar{r} = 0.004\)
Standard error: \(SE(\bar{r}) = 0.0016\)

Compute the t-stat.

8) Time-series dependence and Newey-West intuition

The usual t-stat formula assumes observations are independent over time. But many finance strategies (especially long-horizon and overlapping ones) have time-series dependence, so the usual standard errors can be too small. Newey-West is a common tool that adjusts standard errors to be more realistic when returns are autocorrelated, without changing the average return.

If you ignore time-series dependence in an overlapping strategy, your t-stat is most likely to look: