---
title: "From Signal to Forecast"
book: "Research Craft: Predictors, Backtests, Measurement, Portfolios"
subject: quant
language: en
chapter: 15
exercises: 8
source: https://one-course.com/books/quant/7/en/chapter/15-from-signal-to-forecast
---

# Chapter 15 — From Signal to Forecast

A stock’s composite score is $+2$. Is its expected return over the next month two basis points or two hundred? The composite of chapter 14 ranks stocks; a portfolio optimiser, a risk budget and a trader need expected returns in units of return, per stock and per horizon, and a manager needs to know what information ratio the forecasts can deliver. This chapter turns scores into expected returns (calibration), states the law that links a forecast’s quality to a portfolio’s information ratio, and measures on a planted forecast where the law’s promise is lost. The answer to the opening question, for a score with an IC of 0.05 on a stock whose monthly residual volatility is 7.7%, is 77 basis points.

## 15.1 Calibrating a score

**Definition 15.1 (Forecast calibration, alpha scaling rule).**

*Forecast calibration* is the mapping of a score into an expected return such that, among the names given any forecast, the average realised return equals the forecast. The *alpha scaling rule* sets the expected [residual return](https://one-course.com/books/quant/7/en/chapter/6-anatomy-of-a-predictor#def-rs-anatomy-of-a-predictor-residual) of a name to $\alpha_i = \mathrm{IC}\cdot\omega_i\cdot z_i$, with $z_i$ the standardised score and $\omega_i$ the name’s residual volatility over the horizon.

**Proposition 15.2 (The scaling rule is a regression).**

If the standardised score $z$ and the volatility-scaled [residual return](https://one-course.com/books/quant/7/en/chapter/6-anatomy-of-a-predictor#def-rs-anatomy-of-a-predictor-residual) $r/\omega$ are jointly normal with correlation $\mathrm{IC}$ and unit variances, the expected return given the score is $\mathrm{E}[r \mid z] = \mathrm{IC}\cdot\omega\cdot z$.

**Proof.** For jointly normal standardised variables, $\mathrm{E}[r/\omega \mid z] = \mathrm{IC}\cdot z$; multiply by $\omega$. ∎

The rule makes three assumptions visible: that the score is standardised, that the IC is the same for all names (so that riskier names get proportionally larger forecasts), and that the relation is linear. Each can be checked. The chapter’s planted forecast (`rs_forecast`) adds to the monthly market-adjusted returns of the 704 names that `firm.synthmkt` lists for ten years a true expected return $0.05\,\omega_i s_i$, with $s_i$ a standard normal score redrawn each month and $\omega_i$ the name’s residual volatility (median 7.7% a month). Estimated on the first five years, the IC is 0.055. On the last five, the Mincer–Zarnowitz regression of realised returns on the scaled forecasts (Book 4, chapter 18) has a slope of 0.92, close to the 1 of a calibrated forecast, and an $R^2$ of 0.24%: calibrated forecasts explain almost none of the variance of returns, and that is what an IC of 0.05 means. The same regression on the raw score has a slope of 0.0039, the IC times a typical volatility: the score was never in units of return.

**Definition 15.3 (Calibration curve, isotonic regression).**

A *calibration curve* plots the mean realised return of names grouped by forecast (in bins) against the mean forecast. *Isotonic regression* fits the best non-decreasing function of the score to realised returns by least squares, by pooling adjacent groups that violate the order (Ayer, Brunk, Ewing, Reid and Silverman, 1955).

Binned by score decile, the realised volatility-scaled returns follow the scaling rule’s line, with the noise of 4 224 observations a decile ([Figure 15.1](#fig-rs-from-signal-to-forecast-calibration)). [Isotonic regression](#def-rs-from-signal-to-forecast-curve) gives a calibration without the linear assumption and pays for it in steps: flat stretches where the data could not order neighbouring scores, and larger values in the tails, which are fitted on few names. Between the two, a linear calibration with an estimated slope is the default; the curve is the check.

![Calibration of the planted score on the last five years: mean realised return over residual volatility by score decile against the scaling rule with the IC estimated on the first five (0.055), and the isotonic fit. Data: rs_forecast on firm.synthmkt.](https://one-course.com/images/onecourse/chapters/quant-7/rs-from-signal-to-forecast/fig-837b25efceab.svg)

***Figure 15.1.** Calibration of the planted score on the last five years: mean realised return over residual volatility by score decile against the scaling rule with the IC estimated on the first five (0.055), and the isotonic fit. Data: `rs_forecast` on `firm.synthmkt`.*

## 15.2 Scaling across assets and horizons

The rule scales the forecast by each name’s volatility, so a forecast is comparable across assets only through its IC. A signal that is equally informative about a utility and a biotechnology stock forecasts a larger return for the biotechnology stock, because its returns are larger; a forecast that gave both the same expected return would imply a much higher IC for the utility. Across horizons the scaling follows the [IC decay curve](https://one-course.com/books/quant/7/en/chapter/13-half-life-decay-and-stability#def-rs-half-life-decay-and-stability-decay) of chapter 13: the expected return over $h$ periods is $\omega\,z$ times the sum of the single-period ICs up to $h$ (for volatility per period $\omega$), not the one-period forecast times $h$. A signal whose information lasts a day forecasts the same expected return over a month as over a day. Across asset classes, the standardisation must be within each class, and the IC estimated per class; a blend of an equity and a bond forecast adds expected returns, never scores.

## 15.3 The fundamental law of active management

**Definition 15.4 (Breadth, fundamental law of active management).**

The *breadth* of a strategy is the number of independent forecasts it acts on per year. The *fundamental law of active management* (Grinold, 1989) states that the information ratio of the optimal portfolio built on forecasts with [information coefficient](https://one-course.com/books/quant/7/en/chapter/6-anatomy-of-a-predictor#def-rs-anatomy-of-a-predictor-ic) $\mathrm{IC}$ and breadth $\mathrm{BR}$ is $\mathrm{IR} \approx \mathrm{IC}\sqrt{\mathrm{BR}}$.

With 704 names forecast each month and an IC of 0.05, the law promises $0.05\sqrt{12 \times 704} = 4.60$. The book that acts on the planted forecast holds each name in proportion to $\alpha_i/\omega_i^2$ (the mean-variance weights for uncorrelated residuals), rebalanced monthly. When the residuals are replaced by Gaussian noise of the same volatilities, independent across names, the book’s realised information ratio over ten years is 4.92, within its sampling error of the law (about 0.45 for ten years of monthly returns). The law holds where its assumptions do.

## 15.4 Where the law breaks

**Definition 15.5 (Effective breadth, transfer coefficient).**

The *effective breadth* of a set of forecasts is the number of independent forecasts that would give the same information ratio; $n$ bets with average pairwise correlation $\rho$ are worth about $n/(1 + (n - 1)\rho)$. The *transfer coefficient* of a portfolio is the cross-sectional correlation between its risk-adjusted active weights $w_i\omega_i$ and the risk-adjusted forecasts $\alpha_i/\omega_i$.

**Proposition 15.6 (The generalised law).**

Under the assumptions of the law, a portfolio whose risk-adjusted active weights have correlation $\mathrm{TC}$ with the risk-adjusted forecasts has an expected information ratio of $\mathrm{TC}\cdot\mathrm{IC}\sqrt{\mathrm{BR}}$.

**Proof.** Write $x_i = w_i\omega_i$ and $u_i = r_i/\omega_i$, so that the active return is $\sum_i x_iu_i$ with $\mathrm{E}[u_i] = \mathrm{IC}\,z_i$ and independent unit-variance noise. For a fixed $x$ with $\sum x_i^2 = 1$, the expected active return is $\mathrm{IC}\sum x_iz_i = \mathrm{IC}\sqrt n\,\mathrm{TC}$ (the correlation of $x$ with $z$, both with norm fixed) and the risk is 1; over $\mathrm{BR}/n$ independent periods a year the ratio scales by $\sqrt{\mathrm{BR}/n}$, giving $\mathrm{TC}\cdot\mathrm{IC}\sqrt{\mathrm{BR}}$. ∎

Clarke, de Silva and Thorley (2002) derived this generalised version for constrained portfolios, with an ex post decomposition of performance into the success of the forecasts and the noise of the constraints. On the chapter’s market the book loses its promise in three steps ([Figure 15.2](#fig-rs-from-signal-to-forecast-shortfall)):

**The IC is noisy.** Qian and Hua reformulated the law with a random IC: the information ratio is the IC’s mean over its standard deviation (as summarised by Zhang, Wang and Cao, 2021). With the market’s own residuals (industry and style factors, volatility clustering) the monthly IC has a mean of 0.0505 and a standard deviation of 0.0424, against 0.0331 with Gaussian residuals; the ratio, annualised, is 4.13, and the book’s realised information ratio is 4.26. The factors make neighbouring bets move together, and the 704 names are worth $(\mu_{\mathrm{IC}}/\sigma_{\mathrm{IC}})^2/\mathrm{IC}^2 = 569$ independent forecasts a month.

**The portfolio cannot follow the forecasts.** A long-only book around an equal-weight benchmark cannot hold a name below zero. Asked for a monthly tracking error of 4%, it clips 47% of its intended holdings at zero, and its [transfer coefficient](#def-rs-from-signal-to-forecast-breadth) falls to 0.86. The generalised law then predicts $0.857 \times 4.13 = 3.54$.

**The constraint adds noise.** The book’s realised information ratio is 3.00: the part of the clipped book that is not proportional to the forecasts is uncorrelated with them and adds risk without return. From 4.60 to 3.00, two thirds of the promise survives.

![The information ratio of the planted forecast (IC 0.05, 704 names, monthly): the law’s promise, the book’s with Gaussian residuals, the IC’s mean over its standard deviation and the book’s with the market’s residuals, the generalised law with the long-only transfer coefficient, and the long-only book. Data: rs_forecast.law.](https://one-course.com/images/onecourse/chapters/quant-7/rs-from-signal-to-forecast/fig-2c967a3fbf03.svg)

***Figure 15.2.** The information ratio of the planted forecast (IC 0.05, 704 names, monthly): the law’s promise, the book’s with Gaussian residuals, the IC’s mean over its standard deviation and the book’s with the market’s residuals, the generalised law with the long-only [transfer coefficient](#def-rs-from-signal-to-forecast-breadth), and the long-only book. Data: `rs_forecast.law`.*

The tracking error is the lever: the [transfer coefficient](#def-rs-from-signal-to-forecast-breadth) falls from 0.98 at a requested 0.25% a month to 0.86 at 4%, and the information ratio from 3.91 to 3.00, while the share of clipped holdings rises from 13% to 47%.

| requested tracking error (a month) | 0.25% | 0.5% | 1% | 2% | 4% |
| --- | --- | --- | --- | --- | --- |
| holdings clipped at zero | 13% | 28% | 38% | 44% | 47% |
| [transfer coefficient](#def-rs-from-signal-to-forecast-breadth) | 0.98 | 0.93 | 0.89 | 0.87 | 0.86 |
| information ratio | 3.91 | 3.54 | 3.25 | 3.09 | 3.00 |

## 15.5 Evaluating forecasts

A forecast is evaluated as a forecast before it is evaluated as a strategy: its IC (mean, dispersion, and their ratio), its calibration (the Mincer–Zarnowitz slope near 1, the [calibration curve](#def-rs-from-signal-to-forecast-curve)), its decay (chapter 13), its stability (chapter 13), and its incremental value over the firm’s other forecasts (chapter 12). The law then says what portfolio to expect, and the gap between the law and the backtest is not a mystery: it is the IC’s noise, the constraints and the costs (chapter 27), each measurable. Michaud, Esch and Michaud (2020) argued that applications of Grinold’s law to portfolio design, adding securities or factors or trading more often, are often unreliable because they ignore estimation error and the constraints of practice; the decomposition above is the check that keeps a manager honest.

## 15.6 Predictor cards

**Predictor card 15.1 — Calibrated composite.**

**Definition.** $\alpha_i = \widehat{\mathrm{IC}}\,\omega_i z_i$, $z$ the composite of chapter 14 standardised by date, $\widehat{\mathrm{IC}}$ its trailing mean IC, $\omega_i$ the risk model’s residual volatility over the horizon.

**Inputs and timestamps.** The composite and the risk model as of the decision date; the IC estimated on earlier dates only.

**Rationale.** [Proposition 15.2](#prop-rs-from-signal-to-forecast-scaling).

**Horizon and half-life.** The composite’s; over $h$ periods, the sum of its single-period ICs.

**Normalisation.** Already in return units; checked each month by the Mincer–Zarnowitz slope.

**Failure modes.** An IC estimated in sample; volatilities from a stale risk model; tails extrapolated from a linear fit.

**Sources.** Grinold (1989); `firm.forecast`; `rs_forecast.calibration`.

## 15.7 Tutorial: why the IR is not what the law promised

**Goal.** Calibrate a planted score into expected returns three ways, check the law with Gaussian residuals, then measure the shortfall on the market’s residuals and under a long-only constraint. **End state:** Figures [15.1](#fig-rs-from-signal-to-forecast-calibration) and [15.2](#fig-rs-from-signal-to-forecast-shortfall); the tracking-error table.

1. **[Isotonic regression](#def-rs-from-signal-to-forecast-curve)** by pooling adjacent violators. `def isotonic (x, y, w=None ) -> np.ndarray: """Pool adjacent violators: sort by x, merge neighbouring blocks whose means decrease, return the fit at each original point (Ayer, Brunk, Ewing, Reid and Silverman, 1955).""" x, y = np.asarray(x, float ), np.asarray(y, float ) w = np.ones(len (y)) if w is None else np.asarray(w, float ) order = np.argsort(x, kind=" stable " ) means, weights, sizes = [], [], [] for yi, wi in zip (y[order], w[order], strict=True ): means.append(yi) weights.append(wi) sizes.append(1 ) while len (means) > 1 and means[-2 ] > means[-1 ]: m2, w2, n2 = means.pop(), weights.pop(), sizes.pop() m1, w1, n1 = means.pop(), weights.pop(), sizes.pop() means.append((m1 * w1 + m2 * w2) / (w1 + w2)) weights.append(w1 + w2) sizes.append(n1 + n2) fit = np.repeat(means, sizes) out = np.empty(len (y)) out[order] = fit return out` **Listing 15.1.** Pool adjacent violators. code/firm/forecast/firm_forecast.py
2. **The law and its corrections.** `def effective_breadth (n: float , rho: float ) -> float : return float (n / (1.0 + (n - 1.0 ) * rho)) def transfer_coefficient (w, alpha, vol) -> float : w, alpha, vol = (np.asarray(a, float ) for a in (w, alpha, vol)) return float (np.corrcoef(w * vol, alpha / vol)[0 , 1 ]) def law_ir (ic: float , breadth: float , tc: float = 1.0 ) -> float : return float (tc * ic * math.sqrt(breadth)) def qian_hua_ir (mu_ic: float , sd_ic: float ) -> float : return float (mu_ic / sd_ic) def ding_martin_ir (mu_ic: float , sd_ic: float , n: int ) -> float : return float (mu_ic / math.sqrt(sd_ic ** 2 + (1.0 - mu_ic ** 2 - sd_ic ** 2 ) / n))` **Listing 15.2.** Effective breadth, transfer coefficient, and the laws. code/firm/forecast/firm_forecast.py
3. **Run** `calibration()` , `law()` , `tracking_error_grid()` and `fig_forecast.py` .

**What to change next.** Make the IC depend on the name’s volatility (higher for small, volatile names) and see the scaling rule miscalibrate; add a sector-neutrality constraint and measure its [transfer coefficient](#def-rs-from-signal-to-forecast-breadth).

## 15.8 Build: the forecast toolkit

**Purpose.** Every composite that reaches a portfolio arrives as an expected return, with its calibration checked and its expected information ratio decomposed.

**Interface.** `scale_rule(z, ic, vol)`, `binned`, `isotonic`, `mincer_zarnowitz`, `effective_breadth`, `transfer_coefficient(w, alpha, vol)`, `law_ir(ic, breadth, tc)`, `qian_hua_ir`, `ding_martin_ir`, `information_ratio`.

**Rules.** ICs used for scaling are estimated before the dates they scale; forecasts are in return units per horizon; every backtest reports the law’s promise beside its realised information ratio.

**Acceptance tests.** `code/firm/forecast/tests/`: the scaling rule and bins by hand; [isotonic regression](#def-rs-from-signal-to-forecast-curve) on a hand sequence, on unsorted inputs and on a noisy line; Mincer–Zarnowitz on a planted slope; [effective breadth](#def-rs-from-signal-to-forecast-breadth) at the extremes; the laws by hand; a [transfer coefficient](#def-rs-from-signal-to-forecast-breadth) of 1 for the unconstrained book and below 1 when shorts are removed.

**Stretch.** Calibration by regime; IC by volatility bucket; the Ding and Martin form with an estimated conditional IC.

Sources and further reading

- R. C. Grinold, “The fundamental law of active management”, *Journal of Portfolio Management* 15(3), 1989.
- R. Clarke, H. de Silva and S. Thorley, “Portfolio constraints and the fundamental law of active management”, *Financial Analysts Journal* 58(5), 2002.
- F. Zhang, X. Wang and H. Cao, “Turnover-adjusted information ratio”, arXiv:2105.10306, 2021 (the law and its Qian–Hua and Ding–Martin forms).
- R. O. Michaud, D. N. Esch and R. O. Michaud, “Estimation error and the fundamental law of active management”, *Journal of Investing* 29(4), 2020.
- M. Ayer, H. D. Brunk, G. M. Ewing, W. T. Reid and E. Silverman, “An empirical distribution function for sampling with incomplete information”, *Annals of Mathematical Statistics* 26(4), 1955.

## 15.9 Exercises

**Exercise 15.1 ★.**

A score of $-1.5$, an IC of 0.04 and a monthly residual volatility of 10%: what is the expected [residual return](https://one-course.com/books/quant/7/en/chapter/6-anatomy-of-a-predictor#def-rs-anatomy-of-a-predictor-residual) for the month? And with a score of $+2$ and a volatility of 5%?

**Solution of Exercise 15.1.**

$0.04 \times 10\% \times (-1.5) = -0.6\%$ for the month; $0.04 \times 5\% \times 2 = 0.4\%$.

**Exercise 15.2 ★.**

What information ratio does the law promise for an IC of 0.03 on 500 names rebalanced monthly? On 50 names rebalanced weekly (52 times a year)?

**Solution of Exercise 15.2.**

$0.03\sqrt{12 \times 500} = 2.32$; $0.03\sqrt{52 \times 50} = 1.53$, if the weekly forecasts are independent, which a slow signal’s are not.

**Exercise 15.3 ★.**

Fit [isotonic regression](#def-rs-from-signal-to-forecast-curve) by hand to the values 2, 1, 3, 3, 2, 4 at increasing scores.

**Solution of Exercise 15.3.**

2 and 1 violate the order: pool to 1.5, 1.5. 3, 3 are in order; the next 2 violates: pool the last 3 with it (2.5), which still violates the first 3: pool the three to $8/3 = 2.667$. 4 is in order. Fit: 1.5, 1.5, 2.667, 2.667, 2.667, 4.

**Exercise 15.4 ★★.**

One hundred bets with an average pairwise correlation of 0.05: what is their [effective breadth](#def-rs-from-signal-to-forecast-breadth)? What correlation would halve it?

**Solution of Exercise 15.4.**

$100/(1 + 99 \times 0.05) = 16.8$. To halve it to 8.4: $1 + 99\rho = 100/8.4$, $\rho = 0.110$.

**Exercise 15.5 ★★.**

A monthly IC has mean 0.04 and standard deviation 0.05. What annual information ratio do Qian and Hua’s form and the law with 704 names predict, and why do they differ?

**Solution of Exercise 15.5.**

Qian and Hua: $0.04/0.05 \times \sqrt{12} = 2.77$. The law: $0.04\sqrt{12 \times 704} = 3.68$. The law assumes the IC varies only by sampling across 704 independent names, a standard deviation of about $1/\sqrt{704} = 0.038$; the measured 0.05 includes the factors’ common moves, which the law ignores.

**Exercise 15.6 ★★.**

A signal’s single-day ICs are 0.02, 0.01 and 0 after the second day, with a daily residual volatility of 2%. What expected return does a score of $+1$ imply over one day and over a month?

**Solution of Exercise 15.6.**

Over one day, $0.02 \times 2\% = 0.04\%$. Over a month, the sum of the single-day ICs, $0.02 + 0.01 = 0.03$, times the daily volatility: $0.06\%$; not 21 times the daily forecast.

**Exercise 15.7 ★★★.**

*Coding.* With `rs_forecast.portfolio`, measure the long-only book’s [transfer coefficient](#def-rs-from-signal-to-forecast-breadth) and information ratio at a requested tracking error of 1% a month when the benchmark is concentrated (weights proportional to a lognormal size). Why does it change?

**Solution of Exercise 15.7.**

`rs_forecast.concentrated`: the [transfer coefficient](#def-rs-from-signal-to-forecast-breadth) falls to 0.59 and the information ratio to 2.18, against 0.89 and 3.25 around the equal-weight benchmark. Most names now have tiny benchmark weights, so the book can underweight them only by their tiny weight, while the few large names absorb the negative bets: the risk-adjusted weights lose their resemblance to the forecasts.

**Exercise 15.8 ★★★.**

*Find the flaw.* “Our IC is 0.03 on 3 000 stocks rebalanced daily, so our information ratio should be $0.03\sqrt{252 \times 3000} = 26$.”

**Solution of Exercise 15.8.**

[Breadth](#def-rs-from-signal-to-forecast-law) counts independent forecasts: a signal whose ranks change little from day to day (chapter 13) gives far fewer than 252 new forecasts a year, and 3 000 names that share industries and factors are fewer than 3 000 independent bets. The daily IC of a slow signal is also smaller than its monthly IC, the portfolio’s constraints lower the [transfer coefficient](#def-rs-from-signal-to-forecast-breadth), and trading daily costs money. The formula’s 26 is a ceiling nobody reaches.

## 15.10 Problem: Why Is Our Information Ratio Two-Thirds of the Law’s?

**Problem 15.1.**

Weekend problem — a shortfall, decomposed

The chapter’s planted forecast: IC 0.05, 704 names, ten years of months, on `firm.synthmkt`.

**Part I — Calibration.**

1. What expected return does a score of $+2$ imply for a name with the median volatility?
2. What IC is estimated on the first five years?
3. Give the Mincer–Zarnowitz slope and $R^2$ of the scaled forecast, and the slope on the raw score.
4. Why is the $R^2$ so small, and why does that not matter?
5. When would you prefer the isotonic calibration to the linear one?

**Part II — The law.**

6. What does the law promise?
7. What does the book realise with Gaussian residuals, and is it consistent with the law?
8. What are the IC’s mean and standard deviation with the market’s residuals and with Gaussian ones?
9. What [effective breadth](#def-rs-from-signal-to-forecast-breadth) do they imply?

**Part III — The constraint.**

10. What [transfer coefficient](#def-rs-from-signal-to-forecast-breadth) does the long-only book reach at a requested tracking error of 4% a month, and what share of holdings is clipped?
11. What does the generalised law predict, and what is realised?
12. How do the [transfer coefficient](#def-rs-from-signal-to-forecast-breadth) and information ratio change with the requested tracking error?
13. State the *named result* : the decomposition of the shortfall into [effective breadth](#def-rs-from-signal-to-forecast-breadth) , [transfer coefficient](#def-rs-from-signal-to-forecast-breadth) and the noise of the constraint.

**Part IV — Beyond.**

14. What would costs do to the last [bar](https://one-course.com/books/quant/7/en/chapter/2-market-data-for-research#def-rs-market-data-for-research-bar) ?
15. How would a cap-weighted benchmark change the [transfer coefficient](#def-rs-from-signal-to-forecast-breadth) ?
16. Why can adding names reduce the information ratio in practice?
17. What should the firm report for every strategy?
18. Which of the three steps can research improve, and which only portfolio construction?
19. What is the danger of using the law to size a new strategy?
20. In one sentence: what does the law measure?

**Solution of Problem 15.1.**

1. $0.05 \times 7.7\% \times 2 = 0.77\%$ for the month.
2. 0.055.
3. Slope 0.92 and $R^2$ 0.24% for the scaled forecast; slope 0.0039 on the raw score.
4. An IC of 0.05 explains about $0.05^2$ of the variance of each name’s return; the forecast is still valuable because a portfolio sums hundreds of such small edges.
5. When the relation is visibly non-linear in the tails or saturates; with enough data in the tails to fit steps reliably.
6. 4.60.
7. 4.92; the sampling error of an information ratio over ten years of monthly returns is about 0.45, so yes.
8. Market residuals: mean 0.0505, standard deviation 0.0424. Gaussian residuals: standard deviation 0.0331.
9. 569 independent forecasts a month, against 704 names.
10. 0.86, with 47% of holdings clipped at zero.
11. 3.54 predicted; 3.00 realised.
12. From 0.98 and 3.91 at 0.25% a month to 0.86 and 3.00 at 4% (table).
13. **Named result.** Of the law’s 4.60, the IC’s noise on the market’s residuals (569 effective names of 704) leaves 4.13, the long-only [transfer coefficient](#def-rs-from-signal-to-forecast-breadth) of 0.86 leaves 3.54, and the constraint’s noise leaves the realised 3.00: two thirds of the promise.
14. Lower it again, in proportion to turnover: the long-only book at a high requested tracking error trades the most.
15. Lower it (0.59 in exercise 7): small benchmark weights leave little room for negative bets.
16. More names add correlated bets (little [effective breadth](#def-rs-from-signal-to-forecast-breadth) ), smaller and less liquid positions, and more estimation error in the risk model and the IC.
17. The law’s promise, the IC’s mean over its standard deviation, the [transfer coefficient](#def-rs-from-signal-to-forecast-breadth) and the realised information ratio, side by side.
18. Research improves the IC and its stability, and the [effective breadth](#def-rs-from-signal-to-forecast-breadth) by finding independent signals; portfolio construction improves the [transfer coefficient](#def-rs-from-signal-to-forecast-breadth) and the constraints’ noise.
19. It promises an information ratio that ignores the IC’s noise, the constraints and costs, and invites adding names or trading more often.
20. The information ratio that forecasts of a given quality could deliver if they were independent, unconstrained and free to trade.

## 15.11 Interview questions

**Interview question 15.1 ★ researcher.**

State the [fundamental law of active management](#def-rs-from-signal-to-forecast-law) and its assumptions.

**Solution of Interview question 15.1.**

$\mathrm{IR} \approx \mathrm{IC}\sqrt{\mathrm{BR}}$ (Grinold, 1989): the information ratio of the optimal portfolio grows with the forecasts’ correlation with realised returns and with the square root of the number of independent forecasts a year. Assumptions: independent bets, a constant IC, an unconstrained mean-variance portfolio, no costs.

**Interview question 15.2 ★★ researcher.**

How do you turn an alpha score into an expected return?

**Solution of Interview question 15.2.**

Standardise the score by date, estimate its IC on past data, and set $\alpha_i = \mathrm{IC}\cdot\omega_i z_i$ with the risk model’s residual volatility over the horizon; check calibration out of sample (Mincer–Zarnowitz slope near 1, [calibration curve](#def-rs-from-signal-to-forecast-curve) by bins).

**Interview question 15.3 ★★ researcher, trader.**

Your backtest’s information ratio is half what the law predicts. Where do you look?

**Solution of Interview question 15.3.**

The IC’s noise (its standard deviation against $1/\sqrt n$: correlated bets), the [transfer coefficient](#def-rs-from-signal-to-forecast-breadth) of the constraints, the constraints’ noise, costs, and whether the IC used in the law was estimated in sample. Measure each: the chapter’s planted forecast lost 10%, 14% and 15% at the first three.

**Interview question 15.4 ★★ researcher.**

What is the [transfer coefficient](#def-rs-from-signal-to-forecast-breadth), and what lowers it?

**Solution of Interview question 15.4.**

The correlation between the portfolio’s risk-adjusted active weights and the risk-adjusted forecasts. Long-only constraints (especially against concentrated benchmarks), turnover limits, sector and factor neutrality, position limits and liquidity caps lower it.

**Interview question 15.5 ★★ researcher, mle.**

How would you check that a model’s return forecasts are calibrated?

**Solution of Interview question 15.5.**

Out of sample: regress realised on forecast returns (intercept near 0, slope near 1), plot mean realised against mean forecast by forecast decile, compare by volatility bucket and by period, and refit the calibration only on data before each date.

**Interview question 15.6 ★★★ researcher.**

Derive the generalised law $\mathrm{IR} = \mathrm{TC}\cdot\mathrm{IC}\sqrt{\mathrm{BR}}$ and say which assumption fails first in practice.

**Solution of Interview question 15.6.**

With $x = w\omega$ normalised and $u = r/\omega$ with mean $\mathrm{IC}\,z$ and unit independent noise, the active return has mean $\mathrm{IC}\,x^\top z =
\mathrm{IC}\sqrt n\,\mathrm{TC}$ and unit risk; annualising over independent periods gives $\mathrm{TC}\cdot\mathrm{IC}\sqrt{\mathrm{BR}}$. The independence of the noise fails first: common factors make bets move together, which is why the chapter’s IC had a larger standard deviation on the market’s residuals than on Gaussian ones.