Multiperiod Performance Attribution

Most attribution models are very simple when applied to a single period.  However, applying them over multiple periods can be difficult.  This is the problem of multiperiod attribution.

Damien Laker has written several papers advocating an approach known variously as the "notional portfolio method" or the "exact method".  The strengths and weaknesses of this method are generally made plain by comparing the method with other leading methods.  The following material should give you a good understanding of the notional portfolio method, and a good idea of which other methods are worth knowing about.

The "Toward Consensus" Paper

In Spring 2005, the Journal of Performance Measurement published my paper Toward Consensus on Multiple-period Arithmetic Attribution.  The purpose of this paper was to clarify some areas of agreement (and disagreement) that have emerged through debates in the professional literature over the past few years.  In this paper, I accept some criticisms of my earlier work, and reject other criticisms.  Above all, this paper seeks to clarify the areas of agreement and disagreement in the hope of building consensus.

This paper may be a good introduction to the topic, if you wonder why so many learned people fail to agree on this problem.

There is a companion spreadsheet, Toward consensus on multiperiod attribution.xls, which works through some of the examples discussed in the "Toward Consensus..." paper.  The worksheet "Example 1" shows the calculations for the numbers used on page 3 of the paper.  The second worksheet (for those who may be interested) shows how the same problem would be treated by David Cariňo's method.  The third worksheet ("Example 2: Interaction") provides all the detail for the problem that is discussed on pp. 5 - 7 of the paper.

Menchero's 2004 Paper from Financial Analysts Journal

In the middle of 2004, Financial Analysts Journal published Jose Menchero's paper 'Multiperiod Arithmetic Attribution'.  Damien Laker has written a paper in response: Return Compounding and Multiperiod Arithmetic Attribution.pdf 

If you wish to check the calculations used in that document, here is a spreadsheet that contains Menchero's two examples, plus the the logic for calculating attributes using the notional portfolio method, and also using Menchero's optimized method: Menchero Examples.xls

If you would like to make comments or questions about Damien's new paper, please have a look at the Discussion thread at investment-performance.com whose topic is Damien's new paper.

Worked Example of the Calculation

The easiest way to understand the notional portfolio calculation is by setting aside ideas of performance attribution for one moment, to focus simply on what is involved in comparing the performance of different portfolios.

We can do that by reproducing some tables from the paper Return Compounding and Multiperiod Arithmetic Attribution.pdf, retaining the original numbering for the tables.

Table 1 shows the weights and returns over three periods for a portfolio which makes no active Asset Allocation decisions.  Consistently with the principle of "no active Asset Allocation", the portfolio sector weights exactly match the benchmark sector weights in each period.  Hence, the only source of value added in this portfolio is Stock Selection (also known as "Issue Selection").

Within each period, the total-level returns are simply weighted sums of the sector-level returns.  The single-period value added is simply the difference between the portfolio return and the benchmark return.  Over all three periods, the value added is simply the difference between the compound portfolio return and the compound benchmark return.

These calculations are simply conventional performance measurement calculations.  They don't involve any particular theory of performance attribution.

Next, we can consider a portfolio which makes no active Issue Selection (i.e. no Stock Selection).  Table 4 shows the numbers for this portfolio.  As you can see, each portfolio sector return is held equal with the benchmark sector return.  This means that the only source of active performance in this portfolio can be active Sector Allocation.

As in the previous table, this table simply uses universally-accepted performance measurement calculations.  The value added over three periods is simply the difference between the compound portfolio performance and the compound benchmark performance.

So far, we have discovered that the active performance in Table 1 was 0.19%, while the active performance in Table 4 was 1.15%.  Do these numbers have any implications for performance attribution?  Indeed they do, as you will find as we analyze Table 9 and Table 7.