The basic law assumes the manager builds the portfolio her forecasts imply. No manager does. Long-only is a constraint, so is a cap on any single position, so is a sector band, a turnover limit, a liquidity screen and an exclusion list. Every one of them moves the implemented portfolio away from the optimal one, and the full law puts a number on the gap:
E(R_A) = TC x IC x sqrt(BR) x sigma_A
What is the transfer coefficient and why do constraints lower it?
The new term is the transfer coefficient, TC, introduced by Clarke, de Silva and Thorley in Financial Analysts Journal in 2002. It is the risk-weighted correlation between the active weights the manager would hold unconstrained and the active weights she actually holds. A transfer coefficient of 1.0 means the constraints bind on nothing. A transfer coefficient of 0.50 means half the manager's expected value added never reaches the portfolio, and she is being paid a full fee for half a product.
The mechanism is easiest to see on the long-only constraint, which is the one that binds hardest and the one that gets discussed least. A manager's negative view on a stock can be expressed only down to a zero weight. If the benchmark weight is 8 percent, her maximum short is minus 8 percentage points of active weight, however strong the view. Small benchmark weights are where this bites: a stock at 0.2 percent of the index carries a maximum expressible negative view of 0.2 percentage points, so a long-only manager can be right about the worst company in the index and earn almost nothing for it. Worked Example 3 prices exactly this case, and the answer is a transfer coefficient of 0.9487 on a five-stock book with one binding constraint.
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What does a low transfer coefficient cost?
Constraints stack multiplicatively rather than additively, which is why real transfer coefficients are lower than intuition suggests. Clarke, de Silva and Thorley reported values in the range of roughly 0.3 to 0.8 for constrained institutional portfolios, with long-only alone typically costing a good deal more than managers expect. At the information coefficient of 0.106066 that Worked Example 3 implies, with a breadth of 5 and 4 percent active risk, the expected active return runs 0.2846 percent at a transfer coefficient of 0.30, 0.4743 percent at 0.50, 0.6641 percent at 0.70 and 0.9012 percent at 0.95. The same skill, the same breadth, the same risk budget, and a threefold range in what the client actually receives.
Two uses follow. When a client considers lifting a constraint, the full law prices the decision: the value of relaxing a short-sale prohibition is the rise in the transfer coefficient multiplied by everything else. And when a manager underperforms, the transfer coefficient separates two diagnoses. Good forecasts with a low transfer coefficient is an implementation problem, fixable by changing the mandate. A high transfer coefficient with a poor result is a forecasting problem, which is not.