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Grinold and Kahn's ( Active Portfolio Management: A Quantitative Approach for Providing Superior Returns and Controlling Risk, 1999) fundamental law of active portfolio management states that active managers should maximize an information ratio (IR) that incorporates an information coefficient (IC) and breadth (BR). The author explains the law's limitations, generalizes the law, and derives a semigeneralized law. The semigeneralized law applies to multiperiod portfolios and provides a measurable BR that varies with correlations in forecasts and forecast errors. The fundamental law states that IR, the ratio of active portfolio return to active portfolio risk, equals the IC times the square root of the BR. IC, the correlation between the manager's forecasted and realized excess returns, measures the active manager's skill. BR is the number of signals behind the manager's forecast, or the number of “bets” in the actively managed portfolio. The original law applies only to a single-period investment, and BR is not measurable. The author's generalized law applies the capital asset pricing model to isolate the investment manager's residual return, which equals forecast return plus forecast error. He makes four assumptions regarding residual returns, forecasts, and forecast errors: (1) There will be no long-term excess returns, (2) there will be independence between forecasts and forecast errors, (3) there will be conversion of forecasts into portfolio positions according to modern portfolio theory, and (4) there will be a normal distribution of forecasts. These assumptions are explicitly or implicitly present in the original formulation of the fundamental law. The mathematical presentation of the generalized fundamental law is complex, but adding three assumptions in the original enables the author to show a much simpler semigeneralized fundamental law.

These assumptions are: (1) The portfolio is constructed using the forecast error covariance matrix, (2) the ICs are all equal, and (3) the typical IC is small, perhaps 0.1, and seldom above 0.3. The semigeneralized law has the same form as the original, IR ≈ √BR × IC. BR depends on the correlation between forecasts of asset pairs and the corresponding elements in (the inverse of) the forecast error correlation matrix.

The semigeneralized law provides a measure for BR and allows multiperiod investing with fewer restrictive assumptions than the original law, and it does not restrict how forecasts are correlated across assets or over time. The author shows the ease of calculating BR through five examples. Anwb Klein Vaarbewijs Download. First, if the forecast errors are mutually independent, BR equals the number of assets in the investment universe ( n). Second, if a single forecast is applied over numerous assets, as in applying an industry forecast to individual company stocks, BR depends on the correlations in the forecast error matrix.

If the correlations are all equal and mutually independent, BR again equals n. If the forecast errors are perfectly correlated, BR ≈ 1, the equivalent of having a single asset. Anne Friedberg The Virtual Window Pdf. BR declines rapidly toward unity as the correlation increases, and declines relatively faster for larger asset universes. Thus, the IR of a manager who invests in a small universe of uncorrelated assets may be greater than that of one who invests in a large universe of correlated assets. Third, if forecasts are independent and errors are dependent, BR increases with correlation and approaches infinity as error correlation approaches unity.

This follows because highly correlated securities present arbitrage opportunities. This result again depends on the assumption that error correlations are used in portfolio construction.

If zero correlations are used in portfolio construction, BR = n. Fourth, if forecasts and errors both are dependent, BR = n if the forecast and error correlations are equal. If zero correlations are used in portfolio construction, BR declines with forecast correlation. And fifth, the author identifies strange results that occur if errors are negatively correlated.

For example, combining negatively correlated assets may be riskier than is generally perceived.