Active share is an L1 measure: it adds up absolute deviations without caring whether they offset. Active risk is an L2 measure: it squares them and accounts for how they covary. The two therefore answer different questions, and a manager can score high on one and low on the other.
How do you split active risk into factor and stock-specific parts?
Compute active risk from the active weights and a covariance structure. With a single common factor, security i returns r_i = beta_i x f + e_i, the residuals uncorrelated with each other and with the factor, and the active variance splits cleanly:
sigma_A^2 = (sum of w_A,i x beta_i)^2 x sigma_f^2 + sum of w_A,i^2 x sigma_e,i^2
The first term is the factor contribution, driven by the book's active beta. The second is the stock-specific contribution. The split is the diagnosis: a book whose active risk is 80 percent factor is making a macro bet whatever its marketing says, and a book whose active risk is 95 percent stock-specific is doing what a stock picker is paid for. Worked Example 1's Indian book runs an active beta of +0.0035, so 99.997 percent of its active variance is stock-specific, which is about as clean a stock-picking profile as a real book produces.
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What does the active share and tracking error grid show?
Now put the two measures on a grid, following Petajisto's Financial Analysts Journal study of 2013. High active share with high active risk is a concentrated stock picker: few names, big deviations, a tracking error that will make a trustee uncomfortable in the third bad quarter. High active share with low active risk is a diversified stock picker: many small deviations that diversify against one another. Low active share with high active risk is a factor bet: the manager holds most of the index but tilts it hard toward one exposure. Low active share with low active risk is a closet indexer, and the fee is the whole story there.
The counter-intuitive cell is the second one, so here is the arithmetic that makes it work. Hold active share fixed at 64 percent and spread it over N names with equal absolute active weights, all residuals uncorrelated with a common residual volatility of 26 percent. Each name carries 2 x 64 / N percentage points of active weight, and the tracking error is 26 x (1.28 / N) x sqrt(N) = 33.28 / sqrt(N) percent. At 14 names that is 8.894 percent. At 40 names, 5.262 percent. At 100 names, 3.328 percent. At 400 names, 1.664 percent. Identical active share, and a 5.35-fold range in tracking error, purely because summing absolute values ignores diversification and summing squares does not.
Which is better, a concentrated or a diversified book?
That arithmetic also settles an argument about which of the two books is better, and the answer is that they are better at different things. Take the concentrated case as Worked Example 1 actually builds it rather than as the equal-weight idealisation above, so its tracking error is 10.352 percent rather than 8.894: at an information coefficient of 0.06 and a breadth of 12 that book has an expected information ratio of 0.2078 and an expected active return of 0.2078 x 10.352 = 2.152 percent. The diversified book at 400 names, an information coefficient of 0.03 and a breadth of 400 has an information ratio of 0.60 and an expected active return of 0.60 x 1.664 = 0.998 percent. That breadth of 400 is an assumption about independence rather than a position count, and it is the assumption the breadth section exists to test. The concentrated manager is expected to deliver more than twice the active return; the diversified manager delivers hers with one sixth of the tracking error and three times the reliability. Which one a client should hire depends on the risk budget and on how many bad quarters the committee can survive, and it is a policy question rather than a skill question.
How do fees change the answer?
The pairing also drives the fee arithmetic, which is where most active management dies. Charge the concentrated book 1.00 percent and its expected active return net of fee is 1.152 percent a year. Charge the diversified book 0.35 percent and its net expected active return is 0.648 percent. Both survive. Charge either of them 1.50 percent and the diversified one is finished before it starts, because an information ratio of 0.60 at 1.664 percent tracking error simply cannot generate 1.50 percent. The general test is the one to carry: a fee of f at active risk sigma_A demands an information ratio of at least f / sigma_A before the client sees anything. A 0.75 percent fee at 4 percent tracking error needs 0.1875, which at an information coefficient of 0.05 needs 14.06 independent decisions a year. The same fee at 10 percent tracking error needs only 0.075 and 2.25 decisions. Low-tracking-error products need high information ratios to justify any fee at all, which is precisely the cell where closet indexers live.