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As discussed in the previous section, the fundamental law of active management can be employed in sectors such as security selection, sector rotation, and market timing.

The information coefficient (IC) is the correlation between investors’ forecasts and the actual outcomes of a portfolio. It is a critical element of the fundamental law. Besides investors have the tendency to overestimate their skills in the assumed IC. We should, therefore, acknowledge the challenge of accurately assessing an investor’s skills. Besides, ability forecasts are not constant across the asset segments. In fact, ability forecasts vary over time.

Since the information coefficient varies over time, Qian and Hua (2004) added uncertainty about the level of skill to the basic form of the fundamental law. They showed that:

$$ \sigma_A=\sigma_{IC}\sqrt N\sigma_{RM} $$

Where:

- \(\sigma_A\) is the realized active portfolio risk.
- \(\sigma_{RM}\) is the benchmark tracking risk predicted by the risk model.
- \(\sigma_{IC}\) is the additional risk induced by the uncertainty of the information coefficient.

The number of individual assets does not adequately measure the strategy breadth in cases where there is a correlation of the active returns between individual assets, and forecasts are dependent from period to period. For instance, applying the fundamental law concepts to hedging strategies using any form of arbitrage increases the breadth beyond the number of securities.

Clarke, de Silva, and Thorley (2006) proposed a more practical measure of breadth given by:

$$ BR=\frac{N}{1+\left(N-1\right)\rho} $$

Where:

- \(N\) is the number of decisions.
- \(\rho\) is the same correlation between the decisions.

## Question

Two active investment managers, A and B, construct a portfolio every quarter by selecting individual stocks of the S&P 500. Both managers calculate their information ratios based on the fundamental law of active management. Manager A has a substantially higher information ratio relative to manager B. In a bid to establish why manager A’s information coefficient is higher, the two managers come up with the following possible reasons:

- Correlation of the active returns on the stocks of the S&P 500 stocks.
- Correlation of the stocks from quarter to quarter.
- Different information coefficients for different stocks.
Based on the limitations of the fundamental law, which of the following statements gives the most accurate reason for manager A’s higher information ratio relative to manager B?

- I and II
- II and III
- I, II and III
## Solution

The correct answer is C.

Statement I is correct. The active returns on the individual stocks in the S&P 500 are probably correlated. For this reason, the number of independent monthly decisions is lower than 500. This is referred to as cross-sectional dependence.

Statement II is correct. A stock that is forecasted to outperform in one quarter is likely to retain the outperformance forecast for several consecutive quarters. This is referred to as time-series dependence.

Statement III is correct. The fundamental law of active management assumes an identical information coefficient for all forecasts. However, there is a possibility of information coefficient changing with time. Indeed, it could vary between different sets of stocks.

Reading 45: Analysis of Active Portfolio Management

*LOS 45 (f) Describe the practical strengths and limitations of the fundamental law of active management.*