Active Management, the Fundamental Law and Choosing Managers

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.