The Fundamental Law

There are three factors that determine the information ratio:

• The information coefficient (IC) is a measure of a manager’s skill. IC is the ex ante (i.e., expected), risk weighted correlation between active returns and forecasted active returns. The ex post information coefficient, ICR measures actual correlation between active returns and expected active returns.
• The transfer coefficient (TC) can be thought of as the correlation between actual active weights and optimal active weights. The optimal active weight for a security is positively related to its expected active return and negatively related to its expected active risk. For an unconstrained active portfolio, the active weights will be equal to the optimal weights and TC = 1.
• For a constrained portfolio (e.g., constraints on short positions or active risk), TC may be less than 1. More precisely, transfer coefficient is the cross sectional correlation between the forecasted active returns and the actual weights adjusted for risk:
• Breadth (BR) is the number of independent active bets taken per year. For example, if a manager takes active positions in 10 securities each month, then BR = 10 × 12 = 120.

The Grinold rule allows us to compute the expected active return based on the information coefficient, active risk, and a standardized score:

The expected value added by active management is:

For an unconstrained portfolio, TC = 1 and optimal values are denoted by asterisks (*):

For constrained portfolios, the actual active weights (∆wi) will differ from the optimal

active weights (∆wi

*) and the transfer coefficient will be less than 1. In this case, we have:

Because transfer coefficients are always less than one (TC<1), the information ratio must be less than the optimal information ratio (IR<IR*), and the expected active return must be less than the optimal expected active return E(RA)<E(RA)*.

Recall that the optimal level of active risk (in an unconstrained portfolio) is a function of the information ratio, the Sharpe ratio of the benchmark, and the standard deviation of the benchmark return:

For a constrained portfolio, the optimal level of active risk (σCA

*) is calculated as:

This implies that the optimal active risk of a constrained portfolio will be less than the optimal active risk of an unconstrained portfolio. Similarly, the Sharpe ratio of a constrained portfolio is lower than the Sharpe ratio of an unconstrained portfolio and is given by:

Ex Post Performance Measurement

Realized value added from active management is the ex post active return that the manager achieves. Using the ex post information coefficient ICR, the fundamental law can be written as:

The actual return on the active portfolio can be expressed as its conditional expected return and a noise term:

where E(RA | ICR) represents the expected value added, given the realized skill of the investor that period, and the noise term is the constraint induced noise

The proportion of realized active return variance attributed to variation in the realized information coefficient is TC2. The remaining (1 − TC2) is attributed to constraint- induced noise.

Portfolio theory concludes that investors will choose some combination of the risk free asset and an optimal risky portfolio, with the weights determined by their preferences (risk tolerance). The optimal risky portfolio is the portfolio with the highest Sharpe ratio. The Sharpe ratio of an actively managed portfolio is higher than the Sharpe ratio of the benchmark based on the information ratio of the actively managed portfolio.

The portfolio with the highest information ratio will also be the portfolio with the highest Sharpe ratio, so investors will choose the active manager with the highest information ratio; the actively managed portfolio with the highest information ratio is the optimal (active) portfolio for all investors regardless of their risk tolerance.

The information ratio can be used to determine the expected active return for a given target level of active risk:

E(RA) = IR × σA