How to calculate breadth: An evolution of the fundamental law of active portfolio management
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David Buckle is a senior fund manager with Merrill Lynch Investment Managers. He has ten years’ experience as a practising asset manager with JP Morgan Investment Management, Putnam Investments and Lee Overlay Partners, specialising in quantitative approaches to asset management. Prior to this, he earned a PhD in portfolio construction from Imperial College, London. Merrill Lynch Investment Managers, 33 King William Street, London EC4R 9AS, UK Tel: ⫹44 (0)207 743 4399; Fax: ⫹44 (0)207 743 1000; e-mail: [email protected]
Abstract Grinold and Kahn’s highly acclaimed fundamental law of active management has as a weak point the lack of a measurable definition of breadth. Buckle (2003) developed a more general model of active portfolio management that uses fewer assumptions than Grinold and Kahn, resulting in the generalised fundamental law of active management. This law unfortunately has quite a complex mathematical representation. By applying some of Grinold and Kahn’s assumptions to this generalised law, however, one derives a semi-generalised law, which is identical in form to Grinold and Kahn’s original law, but with a measurable definition of breadth. The simplicity of application of this semi-generalised law is illustrated in several worked examples. Keywords: active portfolio management, fundamental law of active management, information ratio, information coefficient, breadth
Introduction The seminal work of Grinold and Kahn (eg 1999), culminating in their fundamental law of active management, has arguably been the most high profile contribution in the development of active portfolio management theory. In particular, their law says that the active manager should maximise performance by maximising both the quality of the forecasting and the number of ‘bets’. Specifically, the law is expressed as IR ⬇ 兹BR ⫻ IC
(1)
where IR denotes the information ratio defined as active portfolio return divided
䉷 Henry Stewart Publications 1479-179X (2004)
by active portfolio risk and is proven by the mainstream financial theory to be the measure that should be maximised by the active manager, IC denotes the information coefficient defined as the correlation between the forecast of asset abnormal return and the return itself, thereby measuring the skill of the active manager, and BR denotes the breadth, defined as the number of independent signals behind these forecasts. The shortcoming of Grinold and Kahn’s law has always been that breadth is not easy to specify. For example, although the number of forecasts that are used in a particular situation may be known, it can be far from clear how
Vol. 4, 6, 393-405
Journal of Asset Management
393
Buckle
many independent signals this constitutes. As just one example, the difficulty in measuring breadth in a tactical asset allocation setting is discussed by Lundin (2004). A less frequently discussed shortcoming of Grinold and Kahn’s work is that it assumes a single investment position whereas, in practice, active portfolio managers make frequent rebalances to their portfolio
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