Active Management, the Fundamental Law and Choosing Managers

Asset Allocation & Performance branch · Node AA1.07 · ~16 focused hours · Mastery gate >= 85%

Two managers walk into an investment committee. The first says her research process gets a stock right 52.5 percent of the time, which sounds like nothing, and she makes two hundred independent calls a year. The second says he is right three times in four on a much smaller number of very large decisions, two a year, so his skill on each call is ten times the first manager's. On the arithmetic that governs active management, they have exactly the same expected information ratio: 0.7071 for both. Skill and repetition are substitutes, and the exchange rate between them is a square root.

That arithmetic is the fundamental law of active management, and almost everything a committee argues about is downstream of it. Whether concentration is a virtue. Whether a long-only constraint is expensive. Whether a manager who holds forty names and turns them over once a year can possibly earn her fee. Whether a three-year record is evidence. All four have numerical answers, and the answers are frequently uncomfortable.

The law also explains the industry's favourite deception, which is not lying about returns. It is inflating breadth. A sector rotator with five hundred positions and eight genuinely independent views has a breadth of eight, not five hundred, and the difference between those two numbers is a factor of nearly eight in the information ratio he is entitled to claim. Correlation between forecasts destroys breadth violently: on a book of 240 forecasts a year, a pairwise correlation of just 0.05 between them cuts the effective breadth from 240 to 18.5 and the expected information ratio from 0.77 to 0.22.

Then there is the second half of the job, which is choosing the person. A performance record tells you what happened. It does not tell you whether the manager is a stock picker or a closet indexer, whether her style is the one you hired, what her fee structure will cost you in the three states of the world that matter, or whether her track record is long enough to mean anything. Four measurements answer those questions: active share, active risk, style analysis and a fee priced at three outcomes. None of them is difficult. All of them are routinely skipped.

The last question is the hardest and the least often asked out loud. Hiring a manager is a statistical decision under uncertainty, so it has a Type I error and a Type II error, and both are large. Take a plausible institutional setup: one manager in five in the search pool has genuine skill at an information ratio of 0.50, and you screen on three years of data at a 5 percent significance level. The screen has 21.8 percent power. A manager who passes it has a 52.2 percent chance of actually having skill, which is a coin flip dressed as a process. Stretch the record to ten years and the same arithmetic gets you to 70.4 percent, which is better and still not comfortable.

Learning objectives

You can:

1. State the value-added objective that active management optimises, find the level of active risk that maximises it for a given information ratio and risk aversion, and show what taking twice that much active risk does to the value added.
2. Derive and apply the basic fundamental law, IR ~= IC x sqrt(BR), state the assumptions it rests on, and compute any one of the three quantities from the other two.
3. Define breadth as the number of genuinely independent decisions per year, adjust a claimed breadth for the correlation between forecasts using BR_eff = N / (1 + (N - 1) x rho), and quantify how much of a claimed information ratio survives.
4. Distinguish the two live definitions of "information ratio" in circulation, the portfolio measure and the signal-evaluation measure, say which one the fundamental law uses, and convert a signal's information coefficient into a defensible portfolio expectation.
5. Compute a transfer coefficient as the risk-weighted correlation between the active weights a manager wants and the ones her constraints allow, and price the value added that a long-only constraint destroys.
6. Compute active share and active risk by hand, the first from a holdings list and the second from a stated covariance structure, decompose each into its two components, and place the book on the active-share-against-active-risk grid, including the case where a high active share carries a low tracking error.
7. Compute the Sortino ratio, maximum drawdown, drawdown duration and the upside and downside capture ratios from a return series, and read the capture spread as the single most direct statement of whether a manager's excess return was skill or leverage.
8. Run a returns-based style analysis as a constrained regression, detect style drift between two windows, and test whether the drift is larger than the estimation error.
9. Price an ad valorem fee, a symmetric performance fee and an asymmetric fee with a hurdle and a high-water mark at three return outcomes and over a two-year path, and say which structure transfers which risk to whom.
10. Frame a hire-or-fire decision as a hypothesis test, compute how much power a track-record screen of a given length has, and combine that power with a base rate to get the probability that a manager who passed your screen actually has skill.

The by-hand skill and the productivity payoff. The gate rewards the arithmetic: an active share off a holdings table, a tracking error off a covariance structure, a breadth adjustment, a capture ratio, a Bayes calculation on the back of a page. The payoff is running all of it over four hundred funds in a morning. The order matters and it is not negotiable, because every quantity here is one an assistant will produce fluently and wrongly. An active share computed against the wrong benchmark variant, a breadth taken from a position count, a Sortino ratio whose downside deviation divides by the number of shortfalls instead of the number of periods: each of those is a plausible number with a wrong meaning, and none of them looks wrong on a page.

Prerequisites & connections

Builds on. AA1.03 supplies the whole measurement layer used here without re-deriving it: time-weighted and money-weighted returns, the Brinson decomposition, benchmark validity and the five risk-adjusted measures (Sharpe, Treynor, Jensen's alpha, the information ratio and M²). The twelve-quarter data set in Worked Example 2 is AA1.03's own, deliberately, so the comparison table extends from five measures to eight without changing a single input. AA1.06 supplies the multifactor decomposition of active return and active risk into factor and idiosyncratic parts; the one-factor structure used here for tracking error is the simplest instance of it. QD1.02 supplies the empirical information coefficient, measured period by period on real cross-sections, which is the input the fundamental law consumes. QM1.02 supplies regression and its diagnostics, which is what a returns-based style analysis is. QM1.04 supplies significance, power and multiple testing. M9.02 supplies the concentrated-investor case and the active-versus-passive argument, and M9.04 supplies the power calculation on track records, quoted here rather than rebuilt. M3.02 supplies beta and the security market line.

Feeds forward. AA1.08 takes the implementation side: index construction, exchange-traded funds and the mechanics of running a passive or an enhanced-index book, which is what the active-share grid's bottom-left cell is actually buying. AL1.04 owns operational due diligence for alternative managers specifically, including the service-provider checklist, the valuation policy and the unsmoothing of appraisal-based returns; what follows owns the general investment-versus-operational split and hands the alternatives-specific version over. QD2.03 owns the coded implementation of Sortino, drawdown and attribution, including the deflated Sharpe ratio, which is the multiple-testing correction that the hiring section's base-rate argument makes in words. ES1.02 owns the standards and regulation around presenting all of this to a client.

Six neighbouring modules own material that touches this one, and none of it is rebuilt here. AA1.03 owns performance measurement and attribution: if the question is how a return was computed or how an excess return splits into allocation and selection, it belongs there. AA1.06 owns the factor models: a decomposition of active risk into named factor exposures is its work, and the single-factor structure used here is a teaching simplification of it. QD1.02 owns the construction and empirical evaluation of signals, including how an information coefficient is actually measured on a panel. M9.02 owns the argument for concentration and the efficient-markets debate; the arithmetic here prices that argument rather than restating it. M9.04 owns the power calculation on track records, and its numbers are cited here, not recomputed. AL1.04 owns operational due diligence for hedge funds and private vehicles.

Core teaching content

2. The Sharpe ratio, the information ratio and why only one of them is invariant

The information ratio is IR = R_A / sigma_A, active return per unit of active risk. The Sharpe ratio is (R_P - R_f) / sigma_P, total excess return per unit of total risk. They answer different questions and behave differently under a change of scale, and the difference is worth being precise about because a committee will ask you to reconcile them.

Scale the manager's active positions up by a factor of k. If her skill and her breadth are unchanged and she is simply taking bigger versions of the same bets, both the expected active return and the active risk multiply by k. The information ratio does not move. At 4 percent active risk with an information ratio of 0.40 the expected active return is 1.60 percent, and at 12 percent it is 4.80 percent; the ratio is 0.40 in both cases. The information ratio is a property of the process, not of how hard it is being pushed.

The Sharpe ratio behaves in the opposite way, and it has an optimum. Suppose the active position is a pure residual bet, uncorrelated with the benchmark, so the portfolio's variance is sigma_B^2 + omega^2 where omega is the residual risk. Take a benchmark with a Sharpe ratio of 0.55 and a volatility of 15 percent, and a manager with an information ratio of 0.40. The portfolio Sharpe ratio is 0.5980 at 2 percent residual risk, 0.6345 at 4 percent, 0.6801 at 10.9091 percent, and back down to 0.6500 at 20 percent.

The maximum is not an accident of these numbers. Maximising (SR_B x sigma_B + IR x omega) / sqrt(sigma_B^2 + omega^2) over omega gives omega* = IR x sigma_B / SR_B = 0.40 x 15 / 0.55 = 10.9091%, and at that point the portfolio Sharpe ratio equals sqrt(SR_B^2 + IR^2) = sqrt(0.3025 + 0.1600) = 0.6801. An information ratio of 0.40 bolted onto a benchmark Sharpe ratio of 0.55 buys 0.1301 of Sharpe ratio, a 23.65 percent improvement, and it buys it only if you are allowed to take almost 11 percent of tracking error.

Most institutional mandates allow nothing close to that. A typical active equity mandate caps tracking error at 3 to 5 percent, which at this manager's skill delivers about 0.63 rather than 0.68. That gap is the price of the risk budget, and the asset owner set it.

Two riders keep the identity honest. SR_P^2 = SR_B^2 + IR^2 holds only when the active position is genuinely residual, with no beta to the benchmark, so a manager whose active return is one third disguised beta breaks it. And it assumes the manager can be scaled without her information ratio decaying, which is false at some size, because market impact rises with position size.

6. Where breadth goes to die

Breadth is the number of independent decisions, and independence is where every inflated claim hides. Three failure modes account for nearly all of them.

The first is counting positions instead of decisions. A fund holding four hundred stocks does not have a breadth of four hundred if those positions come out of one model run quarterly on one factor; that is four decisions a year expressed four hundred ways. The test is not "how many holdings" but "how many times a year does a genuinely new and separate piece of information change what I hold". A manager who cannot answer that question has not thought about her own process.

The second is correlation between forecasts, and it is quantitative. If N raw forecasts share a common component with average pairwise correlation rho, the effective breadth is

BR_eff = N / (1 + (N - 1) x rho)

which is the same effective-sample-size arithmetic that governs a portfolio of correlated positions. The shrinkage is savage for large N. Take 240 raw forecasts a year at an information coefficient of 0.05. At zero correlation the effective breadth is 240 and the information ratio 0.7746. At a correlation of 0.02 the effective breadth is 41.52 and the information ratio 0.3222. At 0.05 it is 18.53 and 0.2152. At 0.10 it is 9.64 and 0.1552. At 0.30 it is 3.30 and 0.0908, so a correlation of 0.30 has cut the claimed information ratio by a factor of 8.53. Notice the limiting behaviour: as N grows, BR_eff tends to 1/rho and stops rising at all. A manager whose forecasts correlate at 0.10 can never have an effective breadth above 10 no matter how many stocks she covers.

The third failure mode is the annual forecaster who claims a high information ratio. A global macro manager who makes four independent calls a year and claims an information ratio of 0.50 is claiming an information coefficient of 0.25. A systematic manager with a thousand genuinely independent bets needs 0.01581 for the same result, which is 15.81 times less skill per decision. Both claims should be tested, but only one of them is testable against a measured benchmark: information coefficients on equity signals are routinely measured and routinely land far below 0.25.

Rebalancing frequency is where breadth is genuinely created, and it is not free. A signal whose predictive power decays over three months can be re-used four times a year; one that decays over three years cannot. Higher rebalancing raises breadth and therefore the information ratio, and it raises turnover and therefore transaction costs, which come straight out of the numerator. The trade-off is real and it is what separates a strategy that works on paper from one that works after costs.

11. Due diligence: the two halves, and which one actually fails

Manager selection has two independent workstreams, and treating them as one is the most common structural error in the process. Investment due diligence asks whether the strategy can work. Operational due diligence asks whether the firm can be trusted to run it. A manager can pass either and fail the other, and the failures look nothing alike.

Investment due diligence works best as a set of questions with falsifiable answers rather than a checklist of nouns. Start with the source of edge, stated in a sentence that could be wrong, and make the attribution show that process actually operating. Then the people: who decides, how long they have worked together, and what happens if the two who matter leave. Then the portfolio as it stands, which either matches the strategy in the pitch or does not. Then capacity, meaning the asset level at which the edge decays. And last the fee, priced at three outcomes rather than one and set against the expected value added it is charged on.

The active-share and breadth arithmetic slots directly into that. A manager who describes herself as a bottom-up stock picker and shows an across-sector active share of 43.75 percent of her total has a process-consistency question to answer. A manager claiming an information ratio of 0.60 with eight decisions a year is claiming an information coefficient of 0.2121, and the right response is to ask her to show the evidence for a skill level nobody has measured.

Operational due diligence covers the things that lose the whole investment rather than a few points of it. The service providers and whether they are independent and reputable: the auditor, the fund administrator, the custodian and the prime broker. The valuation policy, and specifically who marks the hard-to-value positions and whether the person marking them is paid on the marks. Cash controls and who can authorise a movement of money. Compliance, personal-account dealing rules and the trade-allocation policy across accounts. Business continuity and key-person cover. And the regulatory record, checked at the primary source rather than taken from the pitchbook.

The red flags are structural rather than statistical, and the classic set is short enough to memorise. An auditor or administrator nobody has heard of, or one affiliated with the manager. The manager pricing her own illiquid book. Returns implausibly smooth for the stated strategy. Reluctance to allow direct contact with the administrator, and refusal of position-level transparency to an investor asked to commit for years. The alternatives-specific version of this work, including unsmoothing an appraisal-based return series, belongs to the alternatives allocation material. What belongs here is the general principle: operational due diligence is a veto rather than a score. A manager who fails it does not proceed to a discussion of her information ratio.

14. Hiring and firing as a decision under uncertainty

Every hire is a hypothesis test whose null hypothesis is that the manager has no skill. Rejecting it wrongly is a Type I error: you hire a lucky manager. Failing to reject it when you should is a Type II error: you pass on, or fire, a genuinely skilled one. Both errors are expensive, and in manager selection both are large, which is unusual and worth sitting with.

Compute the power. With a true information ratio of IR over T years, the test statistic has a non-centrality of IR x sqrt(T), so at a one-sided 5 percent level the power is 1 - Phi(1.645 - IR x sqrt(T)). At a true information ratio of 0.50, three years of data gives 21.8 percent power, five years 29.9 percent, ten years 47.5 percent and twenty years 72.3 percent. At 0.30, ten years still gives only 24.3 percent. Even at an exceptional 1.00, three years gives 53.5 percent. The screen most institutions actually run, three years of data, is close to a coin flip against a genuinely good manager.

Now add the base rate, which is the step almost nobody takes. Suppose one manager in five in your search pool has real skill at an information ratio of 0.50, and your screen rejects the null 5 percent of the time when the manager has no skill. Then

P(skill | passes) = 0.20 x 0.218 / (0.20 x 0.218 + 0.80 x 0.05) = 0.5215

A manager who passed a three-year statistical screen has a 52.15 percent chance of having skill. Lower the base rate to one in ten and it falls to 32.63 percent, so most of the managers who pass your screen have no skill at all. Raise the record to ten years at the same one-in-five base rate and it rises to 70.35 percent. The base rate does more work than the test does, which is the general property of screening for rare things, and it is why the qualitative half of due diligence is not a formality: its whole job is to raise the base rate of the pool before the statistics are applied.

The same arithmetic runs on firing, and it is the more damaging error in practice. Goyal and Wahal, studying plan-sponsor hiring and firing decisions in the Journal of Finance in 2008, found that the managers sponsors terminated went on to perform about as well as, and over some horizons better than, the managers hired to replace them, and that the round trip cost the sponsors money in transition costs alone. Read the paper for the exact horizons and magnitudes before quoting numbers from it. The mechanism is not mysterious: three years of underperformance from a skilled manager is entirely ordinary, the committee experiences it as evidence, and the replacement is chosen from the top of a three-year ranking that is mostly noise.

What follows from all of this is a discipline rather than a formula. Write down, before you hire, what would make you fire: a change in the people, a change in the process, a style drift larger than its estimation error, a capacity breach, an operational finding. Underperformance over a horizon shorter than the power calculation supports is not on that list, and putting it on the list guarantees that you will buy high and sell low with a written rationale.

{
  "skill": "Turning a manager's disclosed track record into an honest statement of what it can and cannot support",
  "use": "Have an assistant build the power and base-rate calculation for a specific search: given the length of the records you actually have, the level of skill you are trying to detect and an assumed base rate for the pool, it produces the probability that a manager passing your screen has real skill, so the committee argues about the base rate rather than about the ranking.",
  "tools": ["Chat assistants with code execution", "Python: scipy.stats for the normal CDF, or math.erf if you want no dependencies", "The manager's own audited or GIPS-compliant presentation as the source of the return series"],
  "prompt": "Write Python that takes (a) a list of candidate managers with their realised annualised active return and tracking error and the number of years of record, (b) an assumed base rate for genuine skill in the pool, and (c) a target true information ratio to detect. For each manager compute the realised information ratio, its t-statistic, the standard error of the ratio using sqrt((1 + IR^2/2)/T), and the years that would be needed to reach t = 1.96. Then compute the power of a one-sided 5 percent screen at the target information ratio over the record length available, and the posterior probability of skill given the manager passed. Print the assumptions block at the top of the output and refuse to run if the base rate is not supplied.",
  "verify": "Recompute the power for one manager by hand from the normal table, because a one-sided and a two-sided test differ by enough to change a decision and the code will not tell you which one it used. Confirm the return series behind every realised information ratio is net of fees and against the benchmark variant the mandate actually names, since a ratio computed gross or against a price-return index is measuring the wrong manager. Trace each record to the primary presentation, Match its period-end and currency, and Date-check the record length you fed the model against the one the presentation actually covers. Cite the presentation and page. The posterior probability is only as good as the base rate you assumed, so state that assumption in the output rather than burying it.",
  "diy": "You must be able to compute the power at a given information ratio and horizon, and the Bayes step, on paper, which is what Practice Problem 12 and the Form A and Form B items drill. Without that, the model produces a number that sounds authoritative and the committee will not know that changing the base rate from 0.10 to 0.30 moves the answer from 0.33 to 0.65.",
  "market": "US"
}

Common mistakes & how experts think differently

1. Reading breadth off the position count. A four-hundred-stock book run on one quarterly model has a breadth of four, not four hundred, and the difference is a factor of ten in the information ratio it can claim. The expert's question is never how many holdings but how many genuinely independent pieces of information changed the book this year, and she asks the manager to answer it out loud, because a manager who has not thought about her own breadth has not thought about her own edge.

1. Ignoring the correlation between forecasts. Effective breadth is N / (1 + (N - 1) x rho), and it tends to 1 / rho as N grows, so a manager whose forecasts correlate at 0.10 can never have an effective breadth above 10 however many names she covers. A pairwise correlation of 0.039 on 720 forecasts destroys 81.5 percent of the naive information ratio. Experts treat any breadth claim above about fifty as requiring evidence of independence, not just evidence of activity.

1. Treating a long-only constraint as free. Every constraint lowers the transfer coefficient, and the full law says expected active return falls in exact proportion. A manager with excellent forecasts and a transfer coefficient of 0.40 delivers 40 percent of her value added, and the fix is a mandate change rather than a manager change. Experts price the constraint before they blame the person.

1. Using active share as a synonym for active risk. One is a sum of absolute deviations and the other is a square root of a quadratic form, so they can move independently. The same 64 percent active share carries an 8.89 percent tracking error over 14 names and a 1.66 percent tracking error over 400. Experts compute both and place the manager on the grid, because the grid is what distinguishes a diversified stock picker from a closet indexer with a marketing department.

1. Reading a positive alpha without looking at the capture spread. Beta lifts both capture ratios roughly equally and leaves a capture spread near zero; skill produces asymmetry. A fund at 130.67 upside and 157.95 downside has bought its excess return with leverage, and its beta of 1.392 and negative Jensen's alpha say the same thing. Experts read the three together and trust the pattern rather than any single measure.

1. Believing a three-year record. At a true information ratio of 0.50 a three-year screen has 21.8 percent power, and with a one-in-five base rate a manager who passes it has a 52.15 percent chance of having skill. Experts treat short records as weak evidence to be combined with process evidence, and they know the number of years their own conclusion would actually require.

1. Firing on performance and hiring on performance. Goyal and Wahal found that terminated managers subsequently did about as well as their replacements, and the round trip cost the sponsor transition costs on top. Experts write the firing criteria down before the hire, and underperformance over a horizon shorter than the power calculation supports is not one of them.

1. Pricing a fee at one outcome. The ranking of three fee structures changes between minus 10 percent, plus 5 percent and plus 20 percent, and it changes again over a two-year path because the fee alters the base that compounds. Experts price the schedule at three outcomes and over a path, and they ask whether the performance fee is measured against a benchmark or against zero.

1. Letting operational due diligence be a score. A weak auditor, a manager who marks her own illiquid book, or a refusal of administrator contact are vetoes, not points to be traded against a good information ratio. Experts run operational work as a gate before the investment discussion, because the failure mode it protects against is losing the whole allocation rather than a few points of it.

1. Skipping the benchmark question because the attribution work already asked it. Every quantity here, active share, active risk, capture, style weights, is defined against a benchmark, and an active share computed against the wrong index variant or a stale constituent file is wrong in a way that no downstream check catches. Experts confirm the benchmark and the as-of dates on both files before computing anything at all.

Worked examples

Worked example 1: A ten-name Indian book, measured four ways (India, ₹)

The Konkan Focused Equity Fund (synthetic) runs a ten-name Indian large-cap book against the Bharat Ten (synthetic), a ten-name benchmark. The manager describes the fund as bottom-up stock picking, claims an information coefficient of 0.06 from her firm's own back-test, and says the process makes 160 decisions a year because forty names are reviewed every quarter. Her realised active return over the last three years has averaged +1.90 percent a year.

The common factor is the broad Indian equity market with an annual volatility of 16.00 percent. Residual returns are uncorrelated across names. Weights are in percent, betas and residual volatilities as stated.

Solution. Step 0, the check before the work. Both weight columns sum to 100 and the active weights sum to zero. If either failed, nothing below would be valid.

Step 1, active share. Two routes, and they must agree. The overlap, sum of min(w_P, w_B), is 8 + 0 + 0 + 3 + 0 + 6 + 9 + 0 + 5 + 0 + 0 + 5 + 0 + 0 = 36, so active share is 100 − 36 = 64.0%. Half the sum of absolute active weights is 0.5 x 128 = 64.0%. The sum of the overweights alone is 64. All three agree, which is the identity working.

Step 2, the sector split. Portfolio sector weights are Financials 20, Technology 3, Consumer 34, Industrials 43 against benchmark weights of 25, 26, 21 and 28. The deviations are −5, −23, +13 and +15, summing to zero, so the across-sector active share is 0.5 x 56 = 28.0%.

Within each sector, compute the selection deviation w_P,i − (W_P,s / W_B,s) x w_B,i. The scaling ratios are 0.800000 for Financials, 0.115385 for Technology, 1.619048 for Consumer and 1.535714 for Industrials. Kaveri Bank's selection deviation is 8 − 0.800000 x 18 = −6.400000; Nilgiri Foods' is 6 − 1.619048 x 12 = −13.428571; every off-benchmark name contributes its full portfolio weight. Summing the absolute values gives 0.5 x 115.181319 = 57.5907% of within-sector active share.

Step 3, active risk. The active beta is sum of w_A,i x beta_i = +0.003500, so the systematic active variance is (0.003500 x 16.00)^2 = 0.003136 in squared percentage points. The stock-specific term, sum of w_A,i^2 x sigma_e,i^2, comes to 107.159900, dominated by Arcot Logistics at 23.04 (a 16-point active weight on a 30 percent residual volatility) and Bhima Chemicals at 18.40. Total active variance is 107.163036, so

active risk = sqrt(107.163036) = 10.3520%

and the stock-specific share of that variance is 99.997 percent.

Step 4, the classification and the law. Active share 64 percent, active risk 10.35 percent, active beta effectively zero: a concentrated stock picker, and the process description matches the risk profile, which is the first genuinely good news in the file.

The breadth claim does not match. At her stated information coefficient of 0.06 and her claimed breadth of 160, the law predicts an information ratio of 0.06 x sqrt(160) = 0.758947 and an expected active return of 0.758947 x 10.3520 = 7.857% a year. At an honest breadth of 12, the number of genuinely independent theses the book turns over in a year, it predicts an information ratio of 0.207846 and an expected active return of 2.152 percent.

Her realised active return of 1.90 percent implies a realised information ratio of 1.90 / 10.3520 = 0.183540. Invert the law: the breadth consistent with that realised ratio at her stated information coefficient is (0.183540 / 0.06)^2 = 9.36 decisions a year.

Read it. The record is consistent with a competent stock picker making about nine or ten independent decisions a year. It is not consistent with the 160 decisions the marketing claims, which would have entitled her to an information ratio of 0.7589 against the 0.1835 she achieved, a factor of 4.14. Neither number condemns her. What the arithmetic does is convert an unfalsifiable process claim into a testable one, and this claim failed.

The realised information ratio also says how little three years proves. Its t-statistic is 0.183540 x sqrt(3) = 0.3179, and the years needed to reach t = 1.96 at that ratio are (1.96 / 0.183540)^2 = 114.0. Nobody should be hired or fired on this evidence, and the only defensible use of it is as a consistency check against the process description, which is exactly what was just done.

Worked example 2: Three more measures on the same twelve quarters (US, $)

The Cascadia Core Equity Fund (synthetic) and its benchmark, twelve quarters of gross returns in percent, with a risk-free rate of 1.00 percent per quarter. This is the same series the five standard risk-adjusted measures were computed on, deliberately unchanged, so the comparison table extends rather than restarts.

Solution. Step 1, the figures carried forward. Annualised portfolio return 10.800%, benchmark 9.1333%, portfolio standard deviation 13.2838%, benchmark 9.3536%, beta 1.391956, tracking error 4.5149%. Sharpe 0.5119, Treynor 4.885%, Jensen's alpha −0.345%, information ratio 0.3691, M² 8.788%.

Step 2, the Sortino ratio at a stated minimum acceptable return. Take MAR = 1.00% per quarter, matching the risk-free rate. Five quarters fall short: −4.3, −8.9, −1.9, 0.6 and −5.6, with shortfalls of −5.3, −9.9, −2.9, −0.4 and −6.6. The sum of squared shortfalls is 178.2300. Dividing by n = 12,

downside deviation = sqrt(178.2300 / 12) = 3.853894% per quarter, or 7.707788% annualised

and the Sortino ratio is (10.800 − 4.000) / 7.707788 = 0.882225. The benchmark's downside deviation on the same basis is 2.748484 percent per quarter and its Sortino ratio is 0.933848, so the fund loses on Sortino too.

Two conventions, both stated. Dividing by n − 1 = 11 instead gives an annualised downside deviation of 8.050522 and a Sortino ratio of 0.844666. Changing the MAR to zero gives 6.650063 and 1.624045. The same twelve numbers support Sortino ratios of 0.84, 0.88 and 1.62 depending on two choices nobody writes on the factsheet.

Step 3, drawdown. Build the wealth index from 1.000000: 1.074000, 1.027818, 1.111071, 1.117738, 1.018259, 1.135359, 1.167149, 1.144973, 1.219396, 1.338897, 1.263919, 1.344810. The running peak reaches 1.117738 at the end of quarter 4 and the index falls to 1.018259 at the end of quarter 5.

maximum drawdown = 1 − 1.018259 / 1.117738 = 8.900%

Drawdown duration is one quarter and recovery takes one more, since quarter 6 closes at 1.135359 above the old peak. The benchmark's maximum drawdown over the same window is 6.200 percent, from the same peak to the same trough, so the fund fell 1.435 times as far.

Step 4, capture. The benchmark rose in eight quarters and fell in four. In the up quarters the portfolio returned 7.4, 8.1, 0.6, 11.5, 2.8, 6.5, 9.8 and 6.4 against benchmark returns of 5.0, 7.1, 1.6, 8.0, 3.9, 4.4, 7.6 and 2.9. Geometric means are 6.585638 percent and 5.040067 percent, so

upside capture = 6.585638 / 5.040067 = 130.67%

In the four down quarters the portfolio returned −4.3, −8.9, −1.9 and −5.6 against −2.4, −6.2, −0.5 and −4.0. Geometric means are −5.208842 and −3.297714 percent, so downside capture = 157.95% and the capture spread is −27.29 percentage points. Computing the same ratios on arithmetic means gives 131.11 and 158.02 for a spread of −26.90, close enough to confirm the arithmetic and different enough that the convention has to be stated.

Step 5, the table that now has eight rows.

Read it. The five original measures said this fund bought its 1.67 points of excess return with more systematic risk than it was worth, and pointed at a beta of 1.392 as the reason. The three new measures say the same thing in the language a client understands. It fell 44 percent further than the index in the worst quarter of the three years. It captured 130.67 percent of the up moves and 157.95 percent of the down moves, which is a levered index fund's signature, since leverage lifts both sides and skill does not. And its downside deviation is high enough that even the measure most sympathetic to a volatile winner, Sortino, puts it behind its benchmark.

One professional caution on the drawdown. The 8.90 percent figure is computed on quarterly data and is therefore a floor: any trough that opened and closed inside a quarter is invisible to it. Quote it as "8.90 percent, quarterly data" or do not quote it.

Worked example 3: What a long-only constraint costs (US, $)

Cascade Ridge Advisors (synthetic) runs a five-stock US mandate. Residual volatilities are 20 percent for every name and residuals are uncorrelated. The manager's forecast residual returns for the year, and the benchmark weights, are:

The risk budget is 4.00 percent of active risk. Find the unconstrained optimal active weights, the long-only ones, the transfer coefficient and the value added the constraint destroys.

Solution. Step 1, the unconstrained optimum. With equal, uncorrelated residual risks the optimal active weights are proportional to the alphas, and the alphas already sum to zero so the zero-sum constraint costs nothing. Scale to the risk budget: sigma_A^2 = sigma_e^2 x sum of w^2, so with w = c x alpha and sum of alpha^2 = 0.00225, 0.04 x c^2 x 0.00225 = 0.0016 gives c = 4.216370. The active weights are +12.6491, +6.3246, 0.0000, −6.3246 and −12.6491 percentage points.

The implied portfolio weights are 52.6491, 31.3246, 15.0000, 5.6754 and −4.6491 percent. The last one is a short position, so the long-only constraint binds on Summit Regional Bank and only on it.

Step 2, the constrained optimum. Pin Summit Regional Bank at a zero portfolio weight, so its active weight is fixed at −8.00 points, and re-optimise the other four subject to their active weights summing to +8.00 and the risk budget staying at 4.00 percent. Solving the first-order conditions gives active weights of +14.00, +6.00, −2.00, −10.00 and −8.00 percentage points, and portfolio weights of 54, 31, 13, 2 and 0 percent. Check the risk: 0.04 x (0.0196 + 0.0036 + 0.0004 + 0.0100 + 0.0064) = 0.0016, so active risk is 4.00 percent exactly, the same budget as the unconstrained book.

Step 3, the transfer coefficient. It is the risk-weighted correlation between the two active-weight vectors, which with equal residual risks reduces to the cosine between them:

TC = (w_c . w_u) / (|w_c| x |w_u|) = 0.03794733 / (0.2 x 0.2) = 0.948683

Step 4, the value added, both ways. Expected active return is alpha . w. Unconstrained: 0.03 x 0.126491 + 0.015 x 0.063246 + 0 + 0.015 x 0.063246 + 0.03 x 0.126491 = 0.948683%. Constrained: 0.03 x 0.14 + 0.015 x 0.06 + 0 + 0.015 x 0.10 + 0.03 x 0.08 = 0.900000%. The ratio is 0.948683, identical to the transfer coefficient, which is the full law's claim demonstrated rather than asserted.

Step 5, tie the law together. The scores implied by these alphas are z = alpha / (IC x sigma_e), and they have unit variance when the information coefficient is 0.106066, which gives z = +1.4142, +0.7071, 0, −0.7071, −1.4142. With five independent forecasts the unconstrained information ratio is IC x sqrt(5) = 0.237171, matching 0.948683 / 4. The constrained one is TC x IC x sqrt(5) = 0.948683 x 0.106066 x 2.236068 = 0.225000, matching 0.900 / 4. Every route agrees.

Read it. One binding constraint on the smallest position in a five-name book cost 0.048683 percentage points a year, which is 5.13 percent of the manager's value added. That sounds small, and it is small, because the constraint bound on exactly one name with an 8 percent benchmark weight. Scale the same logic to a five-hundred-name index where the median weight is 0.1 percent and the manager has negative views on two hundred of them, and the transfer coefficient falls into the 0.3 to 0.6 region that Clarke, de Silva and Thorley measured. The transfer coefficient is not a small correction on a real portfolio; it is frequently the largest single term in the full law after skill itself.

Worked example 4: One fee schedule, three outcomes, and a path (India, ₹)

A ₹100 crore Indian mandate is offered under three fee structures. All three charge on the average of the opening value and the closing value before fees.

• Structure A, ad valorem: 1.25 percent a year.
• Structure B, symmetric: 0.75 percent base, plus or minus 20 percent of the gross return relative to a benchmark return of 8.00 percent, with the total floored at zero.
• Structure C, asymmetric: 1.00 percent management fee, plus 15 percent of the closing value above a high-water mark grown at a 6.00 percent hurdle. The hurdle is applied one year at a time to whatever the high-water mark currently is, so a year that starts with the mark at 100 tests against 106 whether it is the first year or the fifth. Written the other way, as a hurdle compounding from inception, year two of the path below would test against 100 x 1.06^2 = 112.36 and charge no performance fee at all. Which of the two a contract means is worth reading before signing it.

Price all three at gross returns of −10 percent, +5 percent and +20 percent, then over a two-year path of −10 percent followed by +20 percent.

Solution. Step 1, at minus 10 percent gross. Closing value before fees is ₹90.0000 crore and average assets ₹95.0000 crore. Structure A charges 0.0125 x 95 = ₹1.18750 crore, leaving a net return of −11.1875 percent. Structure B's base is 0.0075 x 95 = ₹0.71250 crore and its performance component is 0.20 x (−0.10 − 0.08) x 95 = −₹3.42000 crore, so the total is negative and floors at zero, leaving the client's net return at exactly −10.00 percent. Structure C charges 0.0100 x 95 = ₹0.95000 crore of management fee, and no performance fee, because the closing value of 90 is below the hurdle value of 100 x 1.06 = ₹106.0000 crore.

Step 2, at plus 5 percent gross. Closing 105.0000, average 102.5000. A charges ₹1.28125 crore. B charges 0.76875 − 0.61500 = ₹0.15375 crore, because the return fell 3 points short of the benchmark. C charges ₹1.02500 crore, again with no performance fee, since 105 is still below the 106 hurdle value.

Step 3, at plus 20 percent gross. Closing 120.0000, average 110.0000. A charges ₹1.37500 crore. B charges 0.82500 + 2.64000 = ₹3.46500 crore. C charges 1.10000 + 0.15 x (120 − 106) = 1.10000 + 2.10000 = ₹3.20000 crore.

Step 4, the two-year path. Year one at −10 percent gross, year two at +20 percent gross on whatever survived.

Under A the client ends year one at ₹88.81250 crore, and year two runs 88.81250 x 1.20 = 106.57500 before a fee of 0.0125 x 97.69375 = ₹1.22117 crore, closing at ₹105.35383 crore. Under B the client ends year one at ₹90.00000 crore because the fee was zero, and year two runs to 108.00000 before a fee of 0.74250 + 2.37600 = ₹3.11850 crore, closing at ₹104.88150 crore. Under C the client ends year one at ₹89.05000 crore with the high-water mark still at 100, and year two runs to 106.86000 before a management fee of ₹0.97955 crore plus a performance fee of 0.15 x (106.86000 − 106.00000) = ₹0.12900 crore, closing at ₹105.75145 crore.

Read it. The ranking flips twice. Structure B is much the cheapest at both loss and flat outcomes and much the most expensive in a good year, which is what symmetry means and why managers dislike it. Structure A is the most expensive when things go badly and the cheapest when they go well, which is the opposite of what most clients believe they are buying. Over the two-year path the asymmetric structure C is the best of the three for the client, at ₹105.75 crore against ₹105.35 and ₹104.88, entirely because its high-water mark suppressed almost all of the performance fee in the recovery year while structure B paid a full performance fee for a year that merely got back to roughly where it started.

That last sentence is the finding. A symmetric fee measured against an absolute benchmark rather than against the client's own high-water mark can charge a full performance fee for recovering a loss it already refunded nothing for. Ask which reference the performance fee uses before admiring its symmetry.

Worked example 5: Detecting style drift in a returns-based analysis (US, $)

Northgate Equity Partners (synthetic) was hired as a US large-cap value manager six years ago. You have quarterly returns for the fund and for two style indices, a large-cap value index and a large-cap growth index, and no holdings. Run the analysis on each three-year window.

Window 1, quarters 1 to 12, all figures in percent:

Window 2, quarters 13 to 24:

Solution. Step 1, reduce the constrained problem to one regression. With two style indices, weights w and 1 − w that are non-negative and sum to one, the fitted style return is w x r_V + (1 − w) x r_G = r_G + w x (r_V − r_G). Minimising the residual variance is therefore an ordinary regression of y = r_F − r_G on x = r_V − r_G, with the slope clipped to the interval from zero to one.

Step 2, window 1. The means are x̄ = −0.225000 and ȳ = −0.116667. The cross-product S_xy is 49.295000 and S_xx is 63.542500, so

w = 49.295000 / 63.542500 = 0.775780

The value weight is 77.58 percent and the growth weight 22.42 percent. The intercept is ȳ − w x̄ = 0.057884 percent a quarter, or 0.2315 percent a year of selection return. The residual standard deviation is 0.394283 percent a quarter, the standard error of the weight is 0.049463, and the fit explains 96.09 percent of the variation in y.

Step 3, window 2. Now x̄ = −0.100000, ȳ = 0.000000, S_xy = 28.660000 and S_xx = 83.020000, so

w = 28.660000 / 83.020000 = 0.345218

The value weight has fallen to 34.52 percent and the growth weight risen to 65.48 percent. The intercept is 0.034522 percent a quarter, 0.1381 percent a year, the residual standard deviation is 0.388079 and the standard error of the weight is 0.042592, on a fit explaining 86.79 percent of the variation.

Step 4, is the drift real? The change in the value weight is 43.0562 percentage points. The standard error of the difference is sqrt(0.049463^2 + 0.042592^2) = 0.065273, so the t-statistic is 0.430562 / 0.065273 = 6.596. A drift of this size in these windows is roughly six and a half standard errors from zero.

Read it. The fund hired as a value manager now behaves like a two-thirds growth fund, and the change is far too large to be estimation noise. Three things follow and they are separable. First, the client's own asset allocation is wrong, because it was built assuming a value exposure that no longer exists, and that is the client's problem to fix regardless of what the manager says. Second, the benchmark in the contract is now the wrong benchmark, so every risk-adjusted number computed against it is measuring a mismatch rather than a manager. Third, and only third, there is a question for the manager, and it has a defensible answer available to her: the value opportunity set may genuinely have narrowed. What is not defensible is that the client had to discover the change from a regression.

One limit worth stating. The R^2 fell from 96.09 percent to 86.79 percent between the windows, which means the style pair explains the fund less well in the second window than the first. Before concluding growth, check whether a third index, small-cap or quality, would fit better, because a two-index model forced onto a fund that has moved outside both will load onto whichever of the two is closer rather than reporting that neither is right.

Practice set

Work each problem fully before reading its solution. A calculator is assumed. Numeric answers within ±2% of the stated figure score as correct, except where an identity is demanded, in which case it must tie exactly.

P1 (guided). The basic law, both directions. (a) A manager's information coefficient is 0.04 and she makes 150 independent decisions a year. What information ratio does the law predict? (b) A client wants an information ratio of 0.50 from a manager who makes 25 independent decisions a year. What information coefficient is she being asked to have?

Solution. (a) IR = 0.04 x sqrt(150) = 0.04 x 12.247449 = **0.489898**. (b) IC = 0.50 / sqrt(25) = 0.50 / 5 = **0.100000**. Read (b) before believing any pitch: an information coefficient of 0.10 is at the top of the range that has ever been measured on equity signals, so a 25-decision manager promising an information ratio of 0.50 is promising best-in-the-world skill on every one of those 25 calls.

P2 (guided). What doubling costs. At an information coefficient of 0.05, how many independent decisions a year does an information ratio of 0.25 require, and how many does 0.50 require?

Solution. BR = (IR / IC)^2, so 0.25 needs (0.25 / 0.05)^2 = **25** and 0.50 needs (0.50 / 0.05)^2 = **100**. The extra 75 decisions a year buy the second 0.25 of information ratio that the first 25 decisions bought. Breadth has quadratic cost in the ratio, which is why doubling an information ratio by working harder is nearly always the wrong plan and improving the information coefficient is the right one.

P3 (guided). Active share by hand. A benchmark holds four names at 30, 25, 20 and 15 percent, with 10 percent in a fifth. The portfolio holds them at 42, 25, 0 and 8 percent, with 25 percent in the fifth. Compute the active share two ways.

Solution. Active weights are +12, 0, −20, −7 and +15, summing to zero. The sum of absolute values is 54, so active share is 0.5 x 54 = **27.0%**. By the overlap route, sum of min(w_P, w_B) = 30 + 25 + 0 + 8 + 10 = 73, so active share is 100 − 73 = **27.0%**. Both routes agree, which is the check. At 27 percent this is a low-conviction book against a five-name benchmark, and no threshold calibrated on a broad index applies to it.

P4. Active risk from a one-factor structure. Three names carry active weights of +8, −5 and −3 percentage points, with betas of 1.30, 0.90 and 1.10 and residual volatilities of 30, 24 and 28 percent. The factor's volatility is 18 percent. Compute the active risk and the stock-specific share of the active variance.

Solution. Active beta is 0.08 x 1.30 − 0.05 x 0.90 − 0.03 x 1.10 = 0.104 − 0.045 − 0.033 = **0.026000**. The systematic term is (0.026 x 0.18)^2 = 0.00002190. The stock-specific term is 0.0064 x 0.09 + 0.0025 x 0.0576 + 0.0009 x 0.0784 = 0.000576 + 0.000144 + 0.00007056 = 0.00079056. Total variance 0.00081246, so active risk = sqrt(0.00081246) = **2.8504%** and the stock-specific share is 0.00079056 / 0.00081246 = **97.30%**. Almost all of the risk is name-specific, which is what a stock picker's profile looks like.

P5. The full law with a transfer coefficient. A manager has an information coefficient of 0.045, a breadth of 90 and a risk budget of 5 percent. Her mandate's constraints give her a transfer coefficient of 0.55. What active return should the client expect, and what is the constraint costing?

Solution. E(R_A) = TC x IC x sqrt(BR) x sigma_A = 0.55 x 0.045 x 9.486833 x 5 = **1.1740%**. Unconstrained, at a transfer coefficient of 1.0, it would be 0.045 x 9.486833 x 5 = **2.1345%**. The constraints destroy **0.9605 percentage points** a year, or 45 percent of the value added. Before renegotiating the fee, find out which constraint is binding: a short-sale prohibition, a sector band and a position cap are three different conversations.

P6. Correlated forecasts. A manager makes 100 forecasts a year with an information coefficient of 0.045. Their average pairwise correlation is 0.10. Compute the effective breadth, the honest information ratio and the factor by which the naive figure overstates it.

Solution. BR_eff = 100 / (1 + 99 x 0.10) = 100 / 10.9 = **9.1743**. Honest information ratio = 0.045 x sqrt(9.1743) = **0.136301**. Naive = 0.045 x 10 = **0.450000**. The overstatement factor is 0.450000 / 0.136301 = **3.3015**, which equals sqrt(1 + 99 x 0.10) = sqrt(10.9). Correlation of 0.10 among a hundred forecasts leaves the manager with the breadth of a nine-decision process.

P7. Sortino on a short series. Eight quarterly returns are +3.0, −2.0, +5.0, −6.0, +4.0, +1.0, −1.0 and +7.0 percent. The minimum acceptable return is 0.5 percent a quarter. Compute the downside deviation dividing by the number of periods, the quarterly Sortino ratio, and the annualised figure.

Solution. The mean is 11.0 / 8 = **1.3750%**. Three quarters fall short, with shortfalls of −2.5, −6.5 and −1.5, so the sum of squares is 6.25 + 42.25 + 2.25 = **50.7500**. Downside deviation = sqrt(50.75 / 8) = **2.5187%** a quarter. Quarterly Sortino = (1.3750 − 0.5) / 2.5187 = **0.3474**, and annualising the numerator by 4 and the deviation by 2 gives **0.6948**. For contrast, the total standard deviation is 4.2405 percent and the corresponding total-risk ratio is 0.2063: the Sortino ratio is 68 percent higher because five of the eight quarters were above the target and the total-risk measure charges for their volatility too.

P8. Drawdown, duration and recovery. Monthly returns are +5, −3, −4, −2, +6, +3, −8, +4, +5 and +2 percent. Compute the maximum drawdown, its duration and its recovery time from the trough.

Solution. The wealth index runs 1.000000, 1.050000, 1.018500, 0.977760, 0.958205, 1.015697, 1.046168, 0.962475, 1.000974, 1.051022, 1.072043. The running peak of 1.050000 is set at month 1, and the trough is 0.958205 at month 4, so maximum drawdown = 1 − 0.958205 / 1.050000 = **8.7424%**. Duration is 4 − 1 = **3 months**. Recovery arrives at month 9, when the index reaches 1.051022 above the old peak, so recovery time is 9 − 4 = **5 months** and the peak-to-recovery span is 8 months. The later fall is worth seeing too: month 7's index of 0.962475 is an 8.00 percent drop from the month-6 peak of 1.046168, deep but not the deepest, and a report quoting only the largest number hides how much of this sample was spent under water.

P9. Capture ratios and the spread. Six quarters of portfolio returns are +6.0, −5.0, +4.0, −7.0, +9.0 and +2.0 percent against benchmark returns of +4.0, −4.0, +3.0, −5.0, +6.0 and −1.0 percent. Compute the upside and downside capture on geometric means and the capture spread.

Solution. The benchmark rose in quarters 1, 3 and 5 and fell in 2, 4 and 6. Up-market geometric means: portfolio (1.06 x 1.04 x 1.09)^(1/3) − 1 = **6.3135%**; benchmark (1.04 x 1.03 x 1.06)^(1/3) − 1 = **4.3259%**; so upside capture = **145.95%**. Down-market geometric means: portfolio (0.95 x 0.93 x 1.02)^(1/3) − 1 = **−3.4092%**; benchmark (0.96 x 0.95 x 0.99)^(1/3) − 1 = **−3.3482%**; so downside capture = **101.82%**. The capture spread is **+44.12 percentage points**. This is the opposite pattern to Worked Example 2's fund: it kept up in falling markets and beat badly in rising ones, which leverage cannot produce, and the sixth quarter is the reason, since the portfolio made +2.0 percent while the benchmark lost 1.0 percent.

P10. Two fee structures at one outcome. A ₹50 crore mandate returns 15 percent gross. Structure one charges 1.00 percent of the average of the opening and closing values. Structure two charges 0.50 percent of the same base plus 20 percent of the gain above an 8 percent hurdle applied to the opening value. Compute both fees and both net returns.

Solution. The closing value before fees is ₹57.50 crore and the average is ₹53.75 crore. Structure one charges 0.0100 x 53.75 = **₹0.5375 crore**, leaving 57.50 − 0.5375 = ₹56.9625 crore and a net return of **+13.925%**. Structure two's hurdle value is 50 x 1.08 = ₹54.00 crore, so the performance fee is 0.20 x (57.50 − 54.00) = ₹0.70 crore, and the base is 0.005 x 53.75 = ₹0.26875 crore, for a total of **₹0.96875 crore** and a net return of **+13.0625%**. The performance structure is the more expensive one at this outcome, which is the general result whenever the return comfortably clears the hurdle.

P11. Style weights and drift. A returns-based analysis on two style indices gives S_xy = 41.6 and S_xx = 52.0 in the first window and S_xy = 22.5, S_xx = 60.0 in the second, where x is the value index minus the growth index and y is the fund minus the growth index. Compute both value weights and the drift.

Solution. w_1 = 41.6 / 52.0 = **0.800000**, so 80.0 percent value and 20.0 percent growth. w_2 = 22.5 / 60.0 = **0.375000**, so 37.5 percent value and 62.5 percent growth. The drift is **42.50 percentage points** toward growth. Whether that is real needs the standard errors, which need the residual variances; without them the point estimate alone is a hypothesis, not a finding.

P12. Type I, Type II and the base rate. You screen managers on a five-year record at a one-sided 5 percent level, and you want to detect a true information ratio of 0.75. Fifteen percent of the pool has genuine skill at that level. How much power does the screen have, and what is the probability that a manager who passes actually has skill?

Solution. The non-centrality is 0.75 x sqrt(5) = 1.677051, so the power is 1 − Phi(1.645 − 1.677051) = 1 − Phi(−0.032051) = **0.512784**. By Bayes, P(skill | passes) = 0.15 x 0.512784 / (0.15 x 0.512784 + 0.85 x 0.05) = 0.076918 / 0.119418 = **0.644106**. So even a screen with better-than-even power, hunting for an exceptional level of skill, over five years, leaves you with a 64 percent chance of being right about a manager who passed. Run the other side too: P(skill | fails) = 0.15 x 0.487216 / (0.15 x 0.487216 + 0.85 x 0.95) = **0.082993**, so the screen is much better at ruling managers out than at ruling them in, which is how a low-power test with a low false-positive rate always behaves.

Verify. Every figure printed above, restated at the precision it was computed to, so a disagreement in your own working can be located rather than argued about.

Applied mini-project

Run a full manager appraisal and selection memo on two real funds, in about 4 hours. You will take two publicly available funds, run the full appraisal and selection process on both, and write a two-page memo with a hire-or-pass recommendation. Everything you compute must reconcile, and everything you cite must be traceable to a named primary source with a pull date.

Step 1, choose and source (about 30 minutes). Pick one India fund and one US fund: for India, an equity scheme whose monthly portfolio disclosure and scheme information document you can download from the asset manager's own site; for the US, a fund or exchange-traded fund whose holdings and annual report you can pull from the securities regulator's filing system. Record for each: the benchmark's exact name including the variant, the expense ratio, the inception date, the holdings as-of date, and the URL and pull date of every document.

Step 2, benchmark and return validity (about 25 minutes). Confirm the fund's reported return and its benchmark's return are the same kind of return, and that the holdings file and the constituent file share an as-of date. Compute the fund's time-weighted return over three years from published net asset values and confirm it ties to the published figure. Then build a hypothetical investor cash-flow series and compute the money-weighted return alongside it, and state the gap.

Step 3, the eight measures (about 45 minutes). From your quarterly or monthly series and the benchmark's over the same dates, with a documented risk-free rate, compute Sharpe, Treynor, Jensen's alpha, the information ratio, M², the Sortino ratio, maximum drawdown with its duration and recovery, and the upside and downside capture with the spread. State your MAR, your downside-deviation denominator, your annualisation convention and your degrees of freedom. Then write the pattern reading: which measures agree, which disagree, and what the capture spread says that the others do not.

Step 4, active share and active risk (about 40 minutes). Where holdings are disclosed on both sides, compute active share by both routes and confirm they agree. Decompose it into the across-sector and within-sector components and report the cancellation gap. Compute realised tracking error from the return series, place the fund on the active-share-against-active-risk grid, and say which of the four cells it occupies and why. Where holdings are not disclosed for one side, say so explicitly rather than approximating, and note what that costs your conclusion.

Step 5, breadth, the law and the fee (about 40 minutes). From the fund's stated process, estimate an honest breadth and defend the number in two sentences. Invert the law on the realised record to get the implied breadth at a stated information coefficient, and compare. Then price the fee schedule at three gross-return outcomes and over a two-year path, and compute the information ratio the fund must achieve at its actual tracking error before the client sees anything at all.

Step 6, style and the decision (about 40 minutes). Run a two-index returns-based style analysis over two adjacent windows, report both weight sets with standard errors, and test the drift. Then write the memo: two pages, at most 900 words. What the fund is, what it charges, what it would have to be true for the fee to be worth paying, what the evidence supports and what it cannot support, and a hire-or-pass recommendation with the Type I and Type II errors named and the base rate you assumed stated out loud.

Rubric (36 points across 12 rows scored 0 to 3; pass at 29 with no zero on any row). A criterion scores full marks only if it is complete, correct, dated and sourced. Any artefact whose stated test fails scores zero for its row regardless of how well the memo reads, because a decision memo built on a number that does not tie is worse than no memo.

Reading & resources

Core, do these.

• Grinold, R. and Kahn, R., Active Portfolio Management, second edition. The primary source for everything in sections 1 through 6: the value-added objective, the fundamental law, the information coefficient and breadth, and the alpha-scaling rule that makes the law's derivation work. Chapters 5 and 6 are the ones to read first. [Paid] [Advanced]
• Grinold, R. (1989), "The Fundamental Law of Active Management," Journal of Portfolio Management. The original paper, short, and clearer about its own assumptions than most of what has been written about it since. [Paid] [Intermediate]
• Clarke, R., de Silva, H. and Thorley, S. (2002), "Portfolio Constraints and the Fundamental Law of Active Management," Financial Analysts Journal. The transfer coefficient, with measured values for constrained institutional portfolios. This is the paper that turns "constraints are costly" into a number. [Free/Paid] [Advanced]
• Cremers, M. and Petajisto, A. (2009), "How Active Is Your Fund Manager? A New Measure That Predicts Performance," Review of Financial Studies. The active-share measure, its construction and the empirical case for it. [Free/Paid] [Intermediate]
• Petajisto, A. (2013), "Active Share and Mutual Fund Performance," Financial Analysts Journal. The active-share-against-tracking-error grid and the five fund categories it produces, including the closet-indexer diagnosis. [Free/Paid] [Intermediate]
• Sharpe, W. (1992), "Asset Allocation: Management Style and Performance Measurement," Journal of Portfolio Management. Returns-based style analysis as originally formulated, constraints and all. [Paid] [Intermediate]

Going deeper.

• Goyal, A. and Wahal, S. (2008), "The Selection and Termination of Investment Management Firms by Plan Sponsors," Journal of Finance. The evidence on what institutional hiring and firing decisions actually earn. Read the tables rather than the summaries of it. [Paid] [Intermediate]
• Sortino, F. and van der Meer, R. (1991), "Downside Risk," Journal of Portfolio Management. The primary source for the downside-deviation construction and for why the minimum acceptable return has to be chosen rather than defaulted. [Paid] [Intermediate]
• Berk, J. and Green, R. (2004), "Mutual Fund Flows and Performance in Rational Markets," Journal of Political Economy. The capacity argument: why a skilled manager's information ratio decays as assets arrive, and why persistent outperformance would be surprising even in a world full of skill. [Paid] [Advanced]
• CFA Institute, Standards of Practice Handbook, and the CFA Institute manager due-diligence material. The professional framing of the selection process, including what a fiduciary owes a client when recommending a manager. [Free] [Beginner]
• SEBI's mutual fund regulations and circulars on portfolio disclosure and expense ratios, at sebi.gov.in. The Indian disclosure regime that determines what holdings data you can actually get and at what lag. Verify the current disclosure frequency and expense-ratio slabs at the source before relying on either; both have been revised more than once. [Free] [Beginner]
• US SEC fund holdings filings and prospectuses, at sec.gov/edgar. The US equivalent, and the source for the holdings file that any active-share computation on a US fund has to start from. [Free] [Beginner]

Data and tools, free.

• Index providers' own constituent and factsheet pages (niftyindices.com for the Indian index family, and the equivalent for whichever US index a fund names). The only defensible source for benchmark weights, and the place where the total-return and price-return variants are distinguished properly. [Free]
• Python: pandas and NumPy for the holdings join and the return statistics, SciPy for the constrained optimisation a three-or-more-index style analysis requires. [Free]

One thing to do rather than read. Download one fund's holdings file and its benchmark's constituent file for the same month-end and compute the active share by both routes. The two routes will disagree the first time, and the reason will be one of four things: a name held in a different share class, a cash position nobody classified, a corporate action that changed an identifier between the two files, or two different as-of dates. Finding out which teaches more about active share than any paper on it.

Teach it back & journal

Feynman prompt. Explain to a trustee who has just been handed two manager pitches why "we make 400 decisions a year" is a claim about arithmetic rather than a claim about effort, and why you intend to test it. Walk through the law itself, IR ~= IC x sqrt(BR), without writing anything the trustee cannot follow: skill per decision, multiplied by the square root of how many genuinely different decisions there are. Then show what correlation does to it, using the concrete case where 240 forecasts a year that correlate at 0.05 have the breadth of an eighteen-decision process, and the manager's claimed information ratio of 0.77 becomes 0.22. Then do the second half of the job, which is the part trustees find harder: explain why the three-year track record in front of them, whatever it says, has a 21.8 percent chance of detecting a genuinely skilled manager, and why that makes the qualitative work the load-bearing half of the process rather than the soft half. You have taught it properly if the trustee's next question is about the base rate of skill in the search pool rather than about the ranking.

Journal prompt. Take one fund you actually hold, or one your employer recommends, and run four measurements on it: the active share against its stated benchmark, the realised tracking error, the upside and downside capture with the spread, and the fee expressed as the information ratio it demands at that tracking error. Write down what each one told you and, more usefully, where you could not get the data and what you did instead. Then write a paragraph on what all of this is really for. Every quantity here converts a claim that cannot be argued with into one that can: "we are bottom-up stock pickers" becomes an across-sector active share; "our process is highly repeatable" becomes a breadth you can invert out of the record; "the fee is competitive" becomes an information ratio the manager must clear before the client sees a rupee. Which of your own investment beliefs is currently stated in the unfalsifiable form, and what is the number that would test it? And what will you do this month, one concrete thing, to move it from the first form to the second, because a belief you cannot test is a belief you cannot improve.

This module's flashcards and mastery quiz are wired into the app: see the node's Quiz and Reviews.