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### Performance Measurement Theory 1. Introduction

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1.1          The main purpose of investment performance measurement and attribution is to determine in a quantitative sense how well a portfolio has performed and where that performance has come from. Mathematically, performance measurement is relatively straightforward compared with risk measurement, although careful attention to accounting detail is required. Different audiences may want to see attribution subdivisions in different ways. Because results are often highly sensitive to the accuracy of input data, it can also provide a useful check of the accuracy of the underlying accounting process. Performance attribution involves calculating the total returns for both fund and benchmark (for the relevant period), creating suitably accurate models of how these total returns can be built up from the various constituent parts, and then decomposing the differences in ways that are illuminating to the various audiences. For a hedge fund or a trading account, there might be no explicit benchmark as such, so performance attribution might instead concentrate on a cash benchmark.

1.2          The modelling process will subdivide time into various periods. Returns do not compound additively over time, but geometrically. The root time period can be as short as a single day, although such a short period can create extra work without necessarily offering any material improvement in accuracy. Even over very short periods it may be necessary to make assumptions or approximations, or equivalently you may have to accept that there will be residuals that need explaining or quantifying.

1.3          Ideally any performance attribution should start with the contributions to performance arising from each individual line of stock for both the fund and the benchmark. These would then be grouped together in some suitable fashion, e.g. a country/sector classification/portfolio design structure (for equity and managed funds) and/or using factor exposures such as duration (for bond funds). This may involve a hierarchical structure, drilling down potentially several levels. Sometimes cash is kept separate, and sometimes aggregated with the rest of the portfolio. Security classifications need to be maintained (including relevant factor exposures). The classification of a given security and its factor exposures may change over time. If the portfolio contains derivatives or similar instruments their values may need to be divided between two or more characteristics/factors simultaneously, often positive to one characteristic/factor and negative to another, see e.g. Kemp (1997), LIFFE (1992a) or LIFFE (1992b). Carrying out the same calculations for large numbers of funds simultaneously is facilitated by giving careful consideration to how to store all of this data in a suitable fashion, and how to process it efficiently. Many of the same data management issues also arise in practical risk management systems.