An active portfolio is a benchmark plus a set of deliberate deviations from it. The deviations are the product. Everything else, the benchmark exposure, is available for a handful of basis points from an index fund, and a client who pays an active fee for it has been sold something twice.
Write the deviations down as a vector of active weights, one per security, w_A,i = w_P,i - w_B,i. Because both weight vectors sum to one, the active weights sum to zero: every overweight is funded by an underweight somewhere. Two quantities follow immediately, and the rest of the module is about them.
What are active return and active risk?
Active return is what the deviations earned: R_A = R_P - R_B = sum of w_A,i x r_i. Active risk, or tracking error, is the standard deviation of that active return series, and it is the risk the manager chose to take on the client's behalf over and above the risk the benchmark was already carrying.
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A manager who only wanted to maximise active return would take unbounded active risk, so the objective has to charge for it. The standard form is the value-added objective:
VA = R_A - lambda_A x sigma_A^2
where sigma_A is active risk and lambda_A is the client's aversion to it. Both R_A and sigma_A are in percent, so lambda_A is in units of percent-per-percent-squared; institutional values sit between roughly 0.05 and 0.15.
Take a manager whose information ratio is 0.60 against a client with lambda_A = 0.075. Expected active return rises linearly with active risk, R_A = IR x sigma_A, while the charge rises with its square, so there is an interior optimum. Differentiate and set to zero: sigma_A* = IR / (2 lambda_A) = 0.60 / 0.15 = 4.00%. At that level the expected active return is 0.60 x 4.00 = 2.40% and the value added is 2.40 - 0.075 x 16 = 1.20%, which is also IR^2 / (4 lambda_A).
How much active risk is too much?
Now run the same manager at other risk levels and read the shape. At 2 percent active risk the value added is 0.90 percent; at 4 percent it is 1.20 percent; at 8 percent it is exactly zero. Doubling the active risk past the optimum does not halve the benefit. It destroys all of it, because the charge grows as the square while the reward grows linearly. A manager who is twice as aggressive as her skill justifies has added nothing at all, and she has done it while looking busy.
The objective also settles a question that gets asked badly. "Should this manager be more concentrated?" is unanswerable without her information ratio, because concentration is a decision about sigma_A and the optimum scales with IR. At an information ratio of 0.30 and the same risk aversion the answer is 2 percent of active risk, not 4. Concentration is a conclusion rather than a philosophy.