---
title: "Anatomy of a Predictor"
book: "Research Craft: Predictors, Backtests, Measurement, Portfolios"
subject: quant
language: en
chapter: 6
exercises: 8
source: https://one-course.com/books/quant/7/en/chapter/6-anatomy-of-a-predictor
---

# Chapter 6 — Anatomy of a Predictor

Two researchers report the [information coefficient](#def-rs-anatomy-of-a-predictor-ic) of the same reversal signal on the same stocks: 0.038 and 0.007. Both computed a rank correlation between minus a past return and a future return, and both computed it correctly. The first correlated yesterday’s return with tomorrow’s; the second the last five days’ return with the next five days’. A [predictor](#def-rs-anatomy-of-a-predictor-predictor) is not a formula: it is a formula, a target, a horizon, a normalisation and a [neutralisation](#def-rs-anatomy-of-a-predictor-neutralise), and a number describing it means nothing without all five. This chapter takes a [predictor](#def-rs-anatomy-of-a-predictor-predictor) apart into those pieces, measures it with the [information coefficient](#def-rs-anatomy-of-a-predictor-ic) and the standard errors that the measurement needs, and ends with the [predictor card](#def-rs-anatomy-of-a-predictor-card), the one-page record that the rest of Part II fills for every [predictor](#def-rs-anatomy-of-a-predictor-predictor) it studies. Its data are the book’s synthetic market (chapter 5), whose one-day reversal is known to be real.

## 6.1 The target

**Definition 6.1 (Predictor, prediction target, forecast horizon).**

A *predictor* is a number computed for each security at a decision time from data known at that time, intended to be correlated across securities with a later quantity, its *prediction target* (usually a return). The *forecast horizon* is the span of the target: the return over the next day, the next five days, the next month.

![The timing of a predictor. Everything it uses is known at the close of the decision day t; its target begins the next day. A predictor whose window reaches past t, or a target that starts at t, has look-ahead (chapter 3).](https://one-course.com/images/onecourse/chapters/quant-7/rs-anatomy-of-a-predictor/fig-8a6e489c721d.svg)

***Figure 6.1.** The timing of a [predictor](#def-rs-anatomy-of-a-predictor-predictor). Everything it uses is known at the close of the decision day $t$; its target begins the next day. A [predictor](#def-rs-anatomy-of-a-predictor-predictor) whose window reaches past $t$, or a target that starts at $t$, has look-ahead (chapter 3).*

The horizon is part of the [predictor](#def-rs-anatomy-of-a-predictor-predictor). On the synthetic market, reversal measured four ways gives four different numbers: minus the last day’s return has a mean rank IC of 0.0385 with the next day’s return ($t = 17.8$), and 0.0162 with the next five days’ return; minus the last five days’ return has 0.0187 with the next day and 0.0074 with the next five ($t = 2.4$). The simulator plants a reversal of one day, and every measurement that dilutes that day, on either side, sees less of it. A real [predictor](#def-rs-anatomy-of-a-predictor-predictor)’s horizon is unknown and must be found (chapter 13), but the lesson carries: the pair ([predictor](#def-rs-anatomy-of-a-predictor-predictor) window, target horizon) is what is measured.

**Definition 6.2 (Residual return).**

The *residual return* of a security over a period is its return minus the part explained by chosen factor exposures, estimated on each date by a cross-sectional regression of the returns on the exposures: what is left after the market, the industries or the styles.

Three targets are common: the raw return, the return in excess of a market (or benchmark), and a [residual return](#def-rs-anatomy-of-a-predictor-residual). For a rank IC across securities on one date the first two are the same, since subtracting one number from every return leaves their ranks unchanged: on the synthetic market both give 0.0074 for the five-day reversal. The residual is different, and the next section says when it matters.

## 6.2 Normalisation

**Definition 6.3 (Cross-sectional z-score, rank transform).**

The *cross-sectional z-score* of a [predictor](#def-rs-anatomy-of-a-predictor-predictor) on a date is its value minus that date’s mean across securities, divided by that date’s standard deviation, usually clipped at a few units. Its *rank transform* replaces each value by its rank on the date, scaled to lie in $(-\tfrac12, \tfrac12)$.

Raw [predictors](#def-rs-anatomy-of-a-predictor-predictor) are not comparable across dates (a five-day return of 3% is large in a calm week and small in a crash) or across [predictors](#def-rs-anatomy-of-a-predictor-predictor) (a return and a ratio of book value to price have different units). Normalising each date’s cross-section removes both problems and turns a [predictor](#def-rs-anatomy-of-a-predictor-predictor) into a set of relative bets, which is what a market-neutral book holds. The z-score keeps the shape of the distribution and must be clipped (or the [predictor](#def-rs-anatomy-of-a-predictor-predictor) winsorised, Book 4, chapter 15), because one extreme name otherwise dominates the date. The [rank transform](#def-rs-anatomy-of-a-predictor-normalise) discards the shape entirely: it is robust, and the rank IC, Spearman’s correlation of Book 4, chapter 15, is the Pearson correlation of two rank-transformed variables. Nothing is free: the [rank transform](#def-rs-anatomy-of-a-predictor-normalise) treats the difference between the first and second names like that between the 500th and the 501st.

## 6.3 Neutralisation

**Definition 6.4 (Neutralisation).**

*Neutralisation* is the removal from a [predictor](#def-rs-anatomy-of-a-predictor-predictor), or from a target, of its component along chosen exposures, by taking on each date the residual of a cross-sectional regression on them: industry neutralisation removes industry means, beta neutralisation the component along market beta.

**Proposition 6.5 (What neutralising the target does to the IC).**

On one date, write the target as $y = f + u$, where $f$ is its projection on the exposures and $u$ the residual, uncorrelated with $f$. If the [predictor](#def-rs-anatomy-of-a-predictor-predictor) $s$ is uncorrelated with $f$, then

$$
\operatorname{corr}(s, u) = \operatorname{corr}(s, y)\,\sqrt{\frac{\Var(y)}{\Var(u)}}.
$$

**Proof.** $\Cov(s, y) = \Cov(s, u)$, since $s$ is uncorrelated with $f$. Divide by $\sqrt{\Var(s)\Var(y)}$ and by $\sqrt{\Var(s)\Var(u)}$ in turn. ∎

If the exposures explain a share $1 - \rho$ of the target’s cross-sectional variance, measuring against the residual raises the IC by $1/\sqrt\rho$: the [predictor](#def-rs-anatomy-of-a-predictor-predictor) was right about the part it could see, and the factor part was noise to it. In the default synthetic market the market beta and ten industries explain only 10% of the variance of five-day returns across stocks, and neutralising changes the five-day reversal’s IC from 0.0074 to 0.0064, within noise. In a variant whose industries move more (30% volatility a year instead of 8%) they explain 52%, and the proposition predicts a ratio of $1/\sqrt{0.476} = 1.45$: the IC goes from 0.0043 to 0.0067 against the residual target, and to 0.0086 when the [predictor](#def-rs-anatomy-of-a-predictor-predictor) is industry-neutralised too, with the $t$-statistic going from 0.8 to 3.5 ([Figure 6.2](#fig-rs-anatomy-of-a-predictor-neutral)). Neutralising the [predictor](#def-rs-anatomy-of-a-predictor-predictor) removes its own industry component, which in this market carries no information about the future.

![Five-day reversal against five-day returns in a synthetic market whose industries move strongly (30% a year): raw predictor and raw target, raw predictor and residual target, industry-neutral predictor and residual target. Labels are the HAC t-statistics. Data: firm.synthmkt with ind_vol=0.30, seed 1.](https://one-course.com/images/onecourse/chapters/quant-7/rs-anatomy-of-a-predictor/fig-51e40ee0b6db.svg)

***Figure 6.2.** Five-day reversal against five-day returns in a synthetic market whose industries move strongly (30% a year): raw [predictor](#def-rs-anatomy-of-a-predictor-predictor) and raw target, raw [predictor](#def-rs-anatomy-of-a-predictor-predictor) and residual target, industry-neutral [predictor](#def-rs-anatomy-of-a-predictor-predictor) and residual target. Labels are the HAC $t$-statistics. Data: `firm.synthmkt` with `ind_vol=0.30`, seed 1.*

## 6.4 The information coefficient and its standard error

**Definition 6.6 (Information coefficient, rank information coefficient).**

The *information coefficient* (IC) of a [predictor](#def-rs-anatomy-of-a-predictor-predictor) on a date is the correlation, across the securities of that date, between the [predictor](#def-rs-anatomy-of-a-predictor-predictor) and its target. The *rank information coefficient* is the same with ranks (Spearman’s correlation). A [predictor](#def-rs-anatomy-of-a-predictor-predictor) is summarised by the time series of its ICs: their mean, their standard deviation, and the ratio of the two.

**Definition 6.7 (IC information ratio, quantile spread).**

The *IC information ratio* (ICIR) is the mean IC divided by its standard deviation over dates, annualised by $\sqrt{252/h}$ for an $h$-day horizon sampled every $h$ days. The *quantile spread* is the mean target of the securities in the [predictor](#def-rs-anatomy-of-a-predictor-predictor)’s top quantile minus that of the bottom quantile, date by date.

Two facts about the IC’s noise decide how long a [predictor](#def-rs-anatomy-of-a-predictor-predictor) must be observed. On one date with $N$ names and no predictability, the IC has a standard deviation of about $1/\sqrt{N - 1}$ (0.032 for 1 000 names); a mean IC of 0.01 is therefore invisible on any single date and must be accumulated over hundreds. And when the target spans $h$ days but the IC is computed every day, the targets of consecutive dates share $h - 1$ days, the ICs are strongly autocorrelated, and the naive standard error of their mean is too small.

**Proposition 6.8 (Overlapping targets).**

If the daily IC series of an $h$-day target has autocorrelations $\rho_k$ that decline linearly to zero at lag $h$ (as for a persistent [predictor](#def-rs-anatomy-of-a-predictor-predictor) and non-overlapping daily returns), the variance of its mean over $n$ dates is approximately $h$ times the naive $\Var/n$: the naive $t$-statistic overstates by up to $\sqrt h$. The Newey–West estimator with $h - 1$ lags (Book 4, chapter 11) corrects it, and so does using every $h$-th date.

**Proof.** $\Var(\bar x) \approx \frac{\sigma^2}{n}\bigl(1 + 2\sum_{k=1}^{h-1}\rho_k\bigr)$ with $\rho_k = 1 - k/h$; the sum is $(h - 1)/2$, so the bracket is $h$. ∎

For the five-day reversal against five-day targets, the naive $t$-statistic is 3.85 and the corrected one 2.39: the ratio 1.61 is below $\sqrt5 = 2.24$ because the reversal is not persistent. Using only every fifth date gives 1.31, a valid but less efficient estimate from a fifth of the data. For the one-day reversal against one-day returns there is no overlap: 0.0385, $t = 17.8$, an annualised ICIR of 6.4 once the [predictor](#def-rs-anatomy-of-a-predictor-predictor) is neutralised, a number no real daily [predictor](#def-rs-anatomy-of-a-predictor-predictor) reaches after costs; the simulator plants it strongly so that later chapters’ tools have something to find.

![The one-day reversal on the synthetic market. Left: its rank IC with the return of each of the next ten days; the information is spent on the first day (0.0385), the second has 0.0008. Right: the cumulative sum of its daily IC over ten years, neutralised to beta and industries; a straight line is a stable predictor. Data: firm.synthmkt, seed 1.](https://one-course.com/images/onecourse/chapters/quant-7/rs-anatomy-of-a-predictor/fig-36624d9b47ca.svg)

![The one-day reversal on the synthetic market. Left: its rank IC with the return of each of the next ten days; the information is spent on the first day (0.0385), the second has 0.0008. Right: the cumulative sum of its daily IC over ten years, neutralised to beta and industries; a straight line is a stable predictor. Data: firm.synthmkt, seed 1.](https://one-course.com/images/onecourse/chapters/quant-7/rs-anatomy-of-a-predictor/fig-cf6e7f393337.svg)

***Figure 6.3.** The one-day reversal on the synthetic market. Left: its rank IC with the return of each of the next ten days; the information is spent on the first day (0.0385), the second has 0.0008. Right: the cumulative sum of its daily IC over ten years, neutralised to beta and industries; a straight line is a stable [predictor](#def-rs-anatomy-of-a-predictor-predictor). Data: `firm.synthmkt`, seed 1.*

## 6.5 The predictor card

**Definition 6.9 (Predictor card).**

A *predictor card* is the one-page record of a [predictor](#def-rs-anatomy-of-a-predictor-predictor): its definition as a formula; its inputs and the [knowledge time](https://one-course.com/books/quant/7/en/chapter/3-point-in-time-data-and-the-biases#def-rs-point-in-time-data-and-the-biases-bitemporal) of each; its rationale; its horizon and measured half-life; its normalisation and [neutralisation](#def-rs-anatomy-of-a-predictor-neutralise); its known failure modes; its sources. The measured statistics come from a tested script, never from memory.

The card is the unit in which Part II of this book, and a research group, accumulates knowledge: a [predictor](#def-rs-anatomy-of-a-predictor-predictor) without a card has not been studied, only tried. Its fields force the questions of this chapter to be answered before the [predictor](#def-rs-anatomy-of-a-predictor-predictor) is used. Here is the first.

**Predictor card 6.1 — One-day reversal.**

**Definition.** $s_{i,t} = -r_{i,t}$, the day’s return with the sign changed, ranked across the universe and neutralised to market beta and industries.

**Inputs and timestamps.** The closes of days $t-1$ and $t$, known at the close of $t$; betas and industries as of $t$.

**Rationale.** Liquidity providers are paid, through a reversal, for absorbing demand for immediacy (chapter 1).

**Horizon and half-life.** One day. On the synthetic market the rank IC is 0.0385 with day $t+1$ and 0.0008 with day $t+2$: the half-life is under a day.

**Normalisation.** Cross-sectional rank; residual on beta and industry dummies. Neutralised, its mean rank IC is 0.040 with an annualised ICIR of 6.4 ($t = 20.1$) on the synthetic market, which plants it.

**Failure modes.** Moves on news (earnings days) do not revert; the daily turnover costs more than the IC earns in liquid large stocks; crowding by many liquidity providers.

**Sources.** N. Jegadeesh (1990); B. N. Lehmann (1990); `rs_anatomy.card()` for the numbers.

## 6.6 Tutorial: measuring a predictor

**Goal.** Measure reversal on the synthetic market with every choice of this chapter made explicitly, and fill its card. **End state:** Figures [6.2](#fig-rs-anatomy-of-a-predictor-neutral) and [6.3](#fig-rs-anatomy-of-a-predictor-lags); the horizon table; the $t$-statistics 3.85 (naive) and 2.39 (corrected); [Predictor card 6.1](#pred-rs-anatomy-of-a-predictor-reversal).

1. **Neutralise.** A cross-sectional regression per date on a constant and the exposures; names with a missing value are skipped, not filled. `def residualise (y: np.ndarray, X: np.ndarray) -> np.ndarray: """Per date, the residual of y on a constant and the exposures (names with any missing value are skipped).""" y = np.asarray(y, float ) out = np.full(y.shape, np.nan) for t in range (y.shape[0 ]): Xt = X[t] if X.ndim == 3 else X Xt = np.column_stack([np.ones(y.shape[1 ]), Xt]) ok = ~np.isnan(y[t]) & ~np.isnan(Xt).any(axis=1 ) if ok.sum() <= Xt.shape[1 ] + 1 : continue b = np.linalg.lstsq(Xt[ok], y[t, ok], rcond=None )[0 ] out[t, ok] = y[t, ok] - Xt[ok] @ b return out` **Listing 6.1.** Per-date residuals on the exposures. code/firm/predictor/firm_predictor.py
2. **Summarise.** The IC series’ mean, its information ratio and two $t$-statistics, the second with $h - 1$ Newey–West lags for overlapping targets. `def ic_summary (ic, h: int = 1 , periods: int = 252 ) -> dict : """Mean IC and its t-statistics: naive (as if the dates were independent) and Newey-West with h - 1 lags, the right correction when h-day targets overlap on consecutive dates. The IC information ratio is the mean over the standard deviation; its annual version scales by sqrt(periods / h).""" x = np.asarray(ic, float ) x = x[~np.isnan(x)] n = len (x) mean, sd = float (x.mean()), float (x.std(ddof=1 )) se_naive = sd / math.sqrt(n) se_hac = math.sqrt(long_run_variance(x, max (0 , h - 1 )) / n) if h > 1 else se_naive return {" mean " : mean, " sd " : sd, " icir " : mean / sd, " icir_annual " : mean / sd * math.sqrt(periods / h), " t_naive " : mean / se_naive, " t_hac " : mean / se_hac, " n " : n}` **Listing 6.2.** The IC summary. code/firm/predictor/firm_predictor.py
3. **Run** `horizon_ics()` , `study()` and `card()` ; then `neutralisation` on a market simulated with `ind_vol=0.30` , and `fig_anatomy.py` .

**What to change next.** Replace the [rank transform](#def-rs-anatomy-of-a-predictor-normalise) by a z-score clipped at 3, then at 10, and watch the IC’s standard deviation; neutralise to the size exposure as well and check that the reversal survives.

## 6.7 Build: the predictor toolkit

**Purpose.** One implementation of targets, normalisers, [neutralisation](#def-rs-anatomy-of-a-predictor-neutralise) and the IC for every [predictor](#def-rs-anatomy-of-a-predictor-predictor) of the book, so that numbers from different chapters (and, later, Books 8 and 9) are comparable.

**Interface.** `forward_return(ret, h)`, `excess(target, market)`, `residualise(y, X)` and `neutralise`, `zscore(x, clip)`, `rank_transform(x)`, `ic_series(signal, target, method)`, `ic_summary(ic, h, periods)`, `quantile_spread(signal, target, q)`, `PredictorCard`.

**Rules.** Panels are (dates, names) with NaN for absent names; a [predictor](#def-rs-anatomy-of-a-predictor-predictor) at row $t$ is known at the close of $t$ and its target starts at $t+1$; missing values are skipped per date, never imputed; a date with fewer than 20 names has no IC.

**Acceptance tests.** `code/firm/predictor/tests/`: compounding and gaps in forward returns; clipping and ranks; residuals orthogonal to the exposures; an IC of 0.05 recovered with the $1/\sqrt N$ noise; a naive $t$ more than 2.5 times the HAC $t$ for a persistent [predictor](#def-rs-anatomy-of-a-predictor-predictor) and overlapping 20-day targets; a card’s fields.

**Stretch.** Weighted ICs (by liquidity); ICs by group (industry, size bucket); a bootstrap of the IC’s mean over dates.

Sources and further reading

- R. C. Grinold and R. N. Kahn, *Active Portfolio Management* , 2nd ed., McGraw-Hill, 2000.
- E. E. Qian, R. H. Hua and E. H. Sorensen, *Quantitative Equity Portfolio Management* , Chapman & Hall/CRC, 2007.
- W. K. Newey and K. D. West, “A simple, positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix”, *Econometrica* 55(3), 1987.
- N. Jegadeesh, “Evidence of predictable behavior of security returns”, *Journal of Finance* 45(3), 1990; B. N. Lehmann, “Fads, martingales, and market efficiency”, *Quarterly Journal of Economics* 105(1), 1990.

## 6.8 Exercises

**Exercise 6.1 ★.**

A universe has 800 names. What is the standard deviation of a single date’s IC for a useless [predictor](#def-rs-anatomy-of-a-predictor-predictor), and how many independent dates are needed for a mean IC of 0.01 to reach $t = 2$?

**Solution of Exercise 6.1.**

$1/\sqrt{799} = 0.0354$. A mean of 0.01 reaches $t = 2$ when $0.01\sqrt n/0.0354 = 2$, $n = 50$ independent dates, if the only noise were the cross-sectional sampling; common shocks make real IC series noisier, and several times as many dates are needed in practice.

**Exercise 6.2 ★.**

A [predictor](#def-rs-anatomy-of-a-predictor-predictor)’s daily rank IC against the next day’s raw return is 0.02. What is its rank IC against the next day’s return in excess of the equal-weighted market? And against the return in excess of each stock’s own beta times the market?

**Solution of Exercise 6.2.**

Still 0.02 against the market-excess return: subtracting one number from every return leaves the ranks unchanged. Against beta-adjusted returns it changes whenever betas differ, because each stock subtracts a different amount; how much depends on how the [predictor](#def-rs-anatomy-of-a-predictor-predictor) correlates with beta.

**Exercise 6.3 ★.**

A daily IC series has mean 0.012 and standard deviation 0.08. What is the annualised ICIR?

**Solution of Exercise 6.3.**

$0.012/0.08 \times \sqrt{252} = 2.38$.

**Exercise 6.4 ★★.**

Industry factors explain 40% of the cross-sectional variance of monthly returns, and a stock-selection [predictor](#def-rs-anatomy-of-a-predictor-predictor) with no industry content has an IC of 0.03 against raw returns. What IC should it show against industry-residual returns?

**Solution of Exercise 6.4.**

The residual keeps $\rho = 0.6$ of the variance: $0.03/\sqrt{0.6} = 0.039$.

**Exercise 6.5 ★★.**

A 20-day target is scored every day for a very persistent [predictor](#def-rs-anatomy-of-a-predictor-predictor). By the proposition, by how much can the naive $t$-statistic overstate? What $t$ does a naive 6.0 correspond to?

**Solution of Exercise 6.5.**

Up to $\sqrt{20} = 4.47$; a naive 6.0 may be as little as $6.0/4.47 = 1.34$.

**Exercise 6.6 ★★.**

Give one reason to prefer a rank IC to a Pearson IC and one reason to prefer the Pearson IC, in terms of what a portfolio built from the [predictor](#def-rs-anatomy-of-a-predictor-predictor) earns.

**Solution of Exercise 6.6.**

Rank IC: robust to outliers in the [predictor](#def-rs-anatomy-of-a-predictor-predictor) and in returns, so one extreme name does not decide a date. Pearson IC: a portfolio whose weights are proportional to the [predictor](#def-rs-anatomy-of-a-predictor-predictor) earns in proportion to the Pearson correlation, magnitudes included, so it measures what a linear book would make.

**Exercise 6.7 ★★★.**

*Coding.* With `ic_series` and `ic_summary`, measure the one-day reversal against the next day’s return separately on days when the market’s absolute return was in its top decile and on the other days. Which days carry the [predictor](#def-rs-anatomy-of-a-predictor-predictor)?

**Solution of Exercise 6.7.**

On the tenth of days with the largest absolute market return the rank IC is 0.037 ($t = 5.0$); on the others 0.039 ($t = 17.1$). The reversal is carried evenly: the simulator reverses a fixed share of each day’s specific shock, whatever the market does. On real data this is the kind of dependence to test, not to assume.

**Exercise 6.8 ★★★.**

*Find the flaw.* “Our 60-day [predictor](#def-rs-anatomy-of-a-predictor-predictor) has a mean IC of 0.04 against 60-day forward returns, computed daily over five years: 1 260 dates, so $t = 0.04/(0.10/\sqrt{1\,260}) = 14$.”

**Solution of Exercise 6.8.**

The 60-day targets of consecutive days overlap by 59 days, so the 1 260 ICs are nowhere near independent: there are about 21 non-overlapping periods. The proposition bounds the overstatement at $\sqrt{60}$: the honest $t$ may be as low as $14/7.75 = 1.8$. Use Newey–West with 59 lags, or every 60th date.

## 6.9 Problem: Two ICs for One Signal

**Problem 6.1.**

Weekend problem — what the choices of a measurement do to a predictor’s numbers

The synthetic market with its default configuration (1 000 names, ten years, seed 1), and a variant with industry volatility of 30% a year. The [predictor](#def-rs-anatomy-of-a-predictor-predictor) is reversal: minus a past return.

**Part I — Horizon.**

1. What is the rank IC of one-day reversal against the next day’s return, and its $t$ -statistic?
2. And of five-day reversal against the next five days’ return?
3. What are the two mixed pairs (one-day against five, five-day against one)?
4. Which of the four is the planted effect, and why are the others smaller?
5. What is the one-day reversal’s IC with the return of the second day after the decision?

**Part II — Targets and [neutralisation](#def-rs-anatomy-of-a-predictor-neutralise).**

6. Why do the raw and the market-excess targets give the same rank IC?
7. What share of five-day return variance do beta and industries explain in the default market?
8. In the strong-industry variant, what share, and what ratio does the proposition predict?
9. What are the three ICs of [Figure 6.2](#fig-rs-anatomy-of-a-predictor-neutral) and their $t$ -statistics?
10. Why does neutralising the [predictor](#def-rs-anatomy-of-a-predictor-predictor) add to neutralising the target?

**Part III — Standard errors.**

11. What are the naive and corrected $t$ -statistics of the five-day reversal against five-day targets?
12. Why is their ratio below $\sqrt5$ ?
13. What does using every fifth date give, and why is it lower still?
14. What is the standard deviation of one date’s IC with 1 000 names and no predictability?

**Part IV — The card.**

15. What are the card’s mean IC, annualised ICIR and $t$ -statistic?
16. What does the card say about the half-life, and from which numbers?
17. Which field of the card would change first if the market’s liquidity providers became more numerous?
18. State the *named result* : the five-day reversal’s naive and HAC $t$ -statistics for overlapping targets, and the IC gained by neutralising target and [predictor](#def-rs-anatomy-of-a-predictor-predictor) in the strong-industry market.
19. What should a research report say when it quotes an IC?
20. In one sentence: what is a [predictor](#def-rs-anatomy-of-a-predictor-predictor) ?

**Solution of Problem 6.1.**

**1.** 0.0385, $t = 17.8$. **2.** 0.0074, $t = 2.4$. **3.** 0.0162 (one-day against five) and 0.0187 (five-day against one). **4.** One-day against one day: the simulator reverses a share of each day’s specific shock the next day, and each other pair adds four days of unrelated returns to the [predictor](#def-rs-anatomy-of-a-predictor-predictor) or to the target. **5.** 0.0008: nothing. **6.** Ranks do not change when one number is subtracted from every return. **7.** 10% (the residual keeps 90%). **8.** 52%; $1/\sqrt{0.476} = 1.45$. **9.** 0.0043 ($t = 0.8$), 0.0067 ($t = 3.5$), 0.0086 ($t = 3.5$). **10.** The [predictor](#def-rs-anatomy-of-a-predictor-predictor)’s own industry component carries no information about the future in this market, so removing it removes noise from the [predictor](#def-rs-anatomy-of-a-predictor-predictor) as the target’s residual removes noise from the target. **11.** 3.85 and 2.39. **12.** The bound $\sqrt h$ holds for a persistent [predictor](#def-rs-anatomy-of-a-predictor-predictor); reversal changes daily, so the ICs of consecutive dates overlap less. **13.** $t = 1.31$: valid, but from a fifth of the dates. **14.** $1/\sqrt{999} =
0.032$. **15.** 0.040, 6.4, 20.1. **16.** Under a day: the IC is 0.0385 with day $t+1$ and 0.0008 with day $t+2$. **17.** The failure modes and then the IC itself: more liquidity providers compete the reversal away (crowding, chapter 28). **18.** *Named result:* for five-day reversal scored daily against overlapping five-day targets the naive $t$ is 3.85 and the HAC $t$ 2.39; in the strong-industry market, neutralising the target and the [predictor](#def-rs-anatomy-of-a-predictor-predictor) raises the IC from 0.0043 to 0.0086 and its $t$ from 0.8 to 3.5. **19.** The [predictor](#def-rs-anatomy-of-a-predictor-predictor)’s formula and window, the target and its horizon, the normalisation and [neutralisation](#def-rs-anatomy-of-a-predictor-neutralise), the universe and period, the IC type, and the standard error used. **20.** A formula with a target, a horizon, a normalisation and a [neutralisation](#def-rs-anatomy-of-a-predictor-neutralise), of which the formula is the least informative part.

## 6.10 Interview questions

**Interview question 6.1 ★ researcher.**

What is an [information coefficient](#def-rs-anatomy-of-a-predictor-ic), and what values are good?

**Solution of Interview question 6.1.**

The cross-sectional correlation between a [predictor](#def-rs-anatomy-of-a-predictor-predictor) and the return it predicts, computed per date and averaged. For stock returns a mean IC of 0.02 to 0.05 is good if it is stable; the information ratio of the IC series and its persistence matter as much as its mean.

*What the interviewer is looking for: a definition per date, realistic magnitudes, and attention to stability.*

**Interview question 6.2 ★★ researcher, mle.**

You compute a 20-day [predictor](#def-rs-anatomy-of-a-predictor-predictor)’s IC every day and get $t = 8$. Do you believe it?

**Solution of Interview question 6.2.**

Not as stated: consecutive 20-day targets overlap by 19 days, so the naive $t$ can overstate by up to $\sqrt{20} = 4.5$; $t = 8$ may be about 1.8. Recompute with Newey–West (19 lags) or every 20th date.

*What the interviewer is looking for: the overlap correction with a number.*

**Interview question 6.3 ★★ researcher.**

Why neutralise a [predictor](#def-rs-anatomy-of-a-predictor-predictor) to industries? When would you not?

**Solution of Interview question 6.3.**

To remove a component that carries no information but adds noise (industry moves), and to avoid a book that is really an industry bet. Not when the [predictor](#def-rs-anatomy-of-a-predictor-predictor)’s information is at the industry level (industry momentum), or when the book is meant to take industry risk.

*What the interviewer is looking for: [neutralisation](#def-rs-anatomy-of-a-predictor-neutralise) as a statement about where the information is.*

**Interview question 6.4 ★★ researcher, trader.**

A [predictor](#def-rs-anatomy-of-a-predictor-predictor) has an IC of 0.05 against raw returns and 0.02 against [residual returns](#def-rs-anatomy-of-a-predictor-residual). What is going on?

**Solution of Interview question 6.4.**

Most of its apparent power comes from its correlation with the factors that were removed (industry, beta, size): it is partly a factor bet. The 0.02 is its stock-specific skill; whether the rest is worth having depends on whether that factor exposure is rewarded and wanted.

*What the interviewer is looking for: the decomposition into factor and specific parts.*

**Interview question 6.5 ★★ mle, researcher.**

Rank IC or Pearson IC for a machine-learned return forecast? Why?

**Solution of Interview question 6.5.**

Pearson, if positions will be proportional to the forecast (the magnitudes matter), checked alongside the rank IC for robustness; rank if the forecast is used only to order names or if its tails are unreliable.

*What the interviewer is looking for: the link between the metric and how the forecast is used.*

**Interview question 6.6 ★★★ researcher.**

Derive the standard deviation of a single date’s IC under no predictability, and use it to size the history needed to detect a mean IC of 0.02.

**Solution of Interview question 6.6.**

Under no predictability and independent names, the sample correlation of $N$ pairs has variance about $1/(N - 1)$. The mean of $n$ independent dates has standard error $1/\sqrt{n(N - 1)}$; $t = 2$ for a mean of 0.02 with $N = 1\,000$ needs $n = (2/(0.02 \times
31.6))^2 = 10$ dates in theory, but common shocks make the IC’s true dispersion several times larger, and in practice hundreds of dates are needed.

*What the interviewer is looking for: the $1/\sqrt{N-1}$ result and awareness that it understates real noise.*