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*This article is a viewpoint of modified fundamental law for active portfolio management and how it can be used to improve investment performance.*

One of the positive outcomes of the global financial crisis has been a refocusing on the need for proper portfolio diversification and risk management. Our research at Pyramis Global Advisors indicates that portfolio managers can, in fact, achieve higher information ratios or IRs (the active return over active risk, or a portfolio manager’s value added) using newly developed diversification techniques.

**The law and its assumptions**

We began by revisiting the fundamental law of active management, a prominent philosophy especially favoured by quantitative portfolio managers. This law essentially states that the IR of a given investment strategy equals the information coefficient or IC of the underlying alpha process times the square‑root of the number of investment opportunities available to a portfolio manager (ÖN). IC, in other words, measures a manager’s skill or stock-picking ability, where N represents the number of independent assets or securities employed in a particular strategy. The law makes four key assumptions: 1) the underlying assets are uncorrelated; 2) asset return distribution and statistics are stationary through time, 3) there are no constraints on budgets and turnover; and 4) ICs are constant through time.

**Analysis and application**

To find out how the law works in the real world, we analyzed it using a mathematical technique known as principal components analysis. PCA uses observed returns of stocks, and the correlations of returns of individual stocks to one another, to arrive at a comprehensive view of the risk structure in the marketplace. It’s possible to build a PCA model that will rank risk factors in descending order of importance, as well as their orientation within the market. By applying an entropy‑inspired technique (see “Managing Diversification” by Atillio Meucci, 2009) to this PCA exercise, we were able to define the “effective dimensionality” of a portfolio, which captures the effective diversification within an investment strategy. This dimensionality number was then used to modify the law so that ÖN became the number of *effective* dimensions, or the number of effective portfolio decision bets.

The original law and its modified version were then applied to real world market data, in this case 20 years worth of daily total returns for a universe of U.S. stocks. To accomplish the comparison, we used a well‑known theoretical alpha generating engine: one‑month price reversal. In this strategy, a manager would bet on stocks that had gone down the most in the previous month while shorting stocks that had risen the most in the same period.

**More effective diversification**

The results made this complex analytical exercise worthwhile. By applying the basic law, a “naïve calculation,” we found that there was a dramatic *overstatement* of strategy diversification benefits, and an *overestimation* of the expected IR of investment strategies (roughly by a factor of four). But using the modified law, we found that the IR was very close to what had actually happened over the 20 year history. These results were shown to also hold for a number of different alpha strategies. Using these findings as a backdrop, we then demonstrate that techniques that increase strategy diversification can lead to significant improvement in the IR of an investment strategy.

In general, the renewed concern with diversification and risk management has inspired the investment community to improve portfolio construction techniques. Our research shows that diversification is about more than the number of investment opportunities as indicated in the fundamental law of active management. By modifying the law with an improved measure of diversification and applying it to investment strategies, portfolio managers can not only reduce risk, but also improve the performance of their strategies.

*Peter Millington is director of quantitative research and Tim Choe is a quantitative analyst with Pyramis Global Advisors.*