and local returns. Note that for a given investment style, the sum of managed contributions across all groups is equal to the portfolio's managed return. 4.    Calculate the total and local return of each decile group. Asset-Level Contributions to Return There are four different types of asset level contributions that we define in addition to managed, benchmark, and active contribution. These are: 1.   Relative vs. group. For the ?zth asset at time t, this is defined as: Active weight X (Security return - Total return on the 2th group based on the benchmark). 2.   Relative vs. total. For the nth asset at time t, this is defined as: Active weight X (Security return - Benchmark total return). 3.   Absolute vs. group. For the nxh asset at time t, this is defined as: Managed weight X (Security return - Total return on the rth group based on the benchmark). 4.   Absolute vs. yotal. For the ?zth asset at time t, this is defined as: Managed weight X (Security return - Benchmark total return). IMPORTANT PRACTICAL MATTERS In this section we explain how to compute a portfolio's residual return that is the difference between the officially reported return and the estimated return. Under certain conditions where the residual return is small, an algorithm to minimize the residual, while simultaneously not impacting any single source of return in a substantial way, can be applied. Performance Measurement and Return Attribution As stated at the outset of the discussion on return attribution, for a given account and time period, the identified sources of return are not necessarily the sources of the officially reported return. Return attribution relies, instead, on an estimate of the portfolio's official return. This estimate is derived from time t - 1 portfolio weights and time t returns. When there are no intraday cash flows or trades, then the estimate and the official return should be identical if: II The prices used to compute the portfolio weights in return attribution are the same prices used to compute the officially reported return. II The holdings used to compute the portfolio weights in return attribution are the same holdings used to compute the officially reported return. 11 The asset (constituent) level returns used in return attribution are derived from the same prices and cash flows (e.g., dividends) as those used to compute the officially reported return. The difference between the officially reported portfolio return and the estimated portfolio return is called the residual. The sources of return become distorted whenever the residual is not zero. Naturally, the problem becomes bigger