If this does not hold, then an inaccurate measure of attribution may result.

Additional terms and definitions that appear on variance analysis reports relate to asset-specific contributions. These terms include: relative versus group, relative versus total, absolute versus group, and absolute verus total.

For the nth asset at time t, these terms are defined as follows: Relative versus group: Active weight X (Security return - Total return on the z'th group based on benchmark)

t^{t-\{\{t)-r^{t)\ (19.20)

Relative versus total: Active weight X (Security return - Benchmark total return)

u^{t-\\\{t)-rh{t)\ (19.21)

Absolute versus group: Managed weight X (Security return - Total return on the 2th group based on benchmark)

wlit-l)^)-^)] (19.22)

Absolute versus total: Managed weight X (Security return - Benchmark total return)

wUt-l)[K(t)-rb(t)] (19.23)

In the preceding two sections, we presented methods for return attribution. The first method is based on a linear factor model and decomposes return into factor and specific components. In this section, an asset grouping methodology was introduced. According to this approach, no model is assumed. All that is required is a set of mappings that tell us how to classify assets. An example of a mapping would be an industry classification scheme.

Also, in the previous two parts we defined and explained one-period return attribution procedures. Various issues arise when we need to compute attribution over multiple periods. For example, one-period attribution may be one-day attribution. When we compute attribution over, say, a quarter, we need to "link"5 the daily sources of return so that the compounded quarterly portfolio return is consistent with the compounded sources of return.

Finally, we note an important difference between the asset grouping and factor model-based methodologies. In the factor model approach, at each point in time the returns to factors are estimated simultaneously. These estimates are the result of cross-sectional regressions6 using equation (19.4).

This process captures

^Linking is the process by which individual stocks, groups, or factors are compounded over time in such a way that the sum of the individual linked contributions is equal to the compounded total return on the portfolio. £See Chapter 20 for details on how factor returns are estimated via cross-sectional regression.