A Brinson Model Alternative: an Equity Attribution …

1 Fall 2007 The Journal of Performance Measurement-59-A Brinson Model Alternative: an Equity AttributionModel with orthogonal Risk ContributionsThis paper presents an algorithm for Equity Attribution that circumvents some of the less intuitive features of thestandard Brinson Model . The proposed alternative framework provides an unambiguous division of returns bysource of risk, in a manner consistent with first-principles interpretation. Andrew Colin, Director of Fixed Income Research for the StatPro Group plc. He holds a in Mathematics from theUniversity of St. Andrews and has worked or consulted for a wide range of banks and other companies. He alsois adjunct professor at the Faculty of Business, Queensland University of Technology, where his team researchesrisk in financial and engineering asset Brinson (or sector) Attribution Model and its suc-cessors ( Brinson , Hood, Beebower, 1986; Brinson ,Fachler, 1985) are widely used in portfolio manage-ment as tools to assess the skills of the manager in run-ning an Equity investment portfolio.

2 These modelsdecompose a portfolio s excess return above the bench-mark into returns arising from stock selection and assetallocation decisions. In na ve terms, stock selection is deciding whichstocksto hold, while asset allocation is deciding how muchofeach stock to hold. Stock selection is a binary, yes/notype of decision, while asset allocation forms a contin-uous, constrained set of us put this another way. Consider an investment tipping newsletter, where a group of stocks are sin-gled out by an expert as being a good buy. These rec-ommendations form pure stock selection advice. Howmuch of each stock to buy is never suggested, since thisdecision cannot be made in isolation but must take intoaccount the other holdings of the investor. If one based an investment strategy on the use of suchrecommendations, the ideal Attribution analysis wouldbe one that measures the return generated by the use ofthe newsletter (stock selection) and the return generat-ed by decisions on amounts bought (asset allocation).

3 Sectoral Attribution cannot provide such an analysis,and this is one of the motivations for the algorithm pre-sented in the addition, the sector Attribution Model does not clear-ly discriminate between the returns generated by thesetwo types of decisions. Specifically, there are two waysin which the Model falls short:1. The returns attributed to stock selection in theBrinson Model also include the effects of asset allo-cation decisions at the subsector level. This extraasset allocation return is not split out from the stockselection return, even though it arises from a quitedifferent source of risk. This is illustrated in the following example. Supposethat a manager decides that a $100 million portfolioshould contain 15% by market weight of IT stocks. Thebenchmark weight is 10%, so the portfolio is over-weight IT stocks by 5 percent.

4 This is an asset alloca-tion decision, and the return generated by this decisionis classed under asset allocation return. Asset allocationdoes not form a decision on which specific IT stocks tohold; it just indicates how much exposure there shouldbe to a particular class of manager now decides to invest the 15% of hisfunds allocated to IT into Microsoft and Borland purchases $10 million of Microsoft and $5 millionof Borland stock. The returns from this decision areattributed to stock selection, even though the return hasbeen generated by a mixture of stock selection and assetallocation decisions, ,to buy Microsoft and Borland,and not to buy IBM or Dell and to purchase stock in theabove amounts rather than $2 million of Microsoft and$13 million of Borland. To put it another way, the deci-sion of how much exposure there should be to individ-ual stocks is a continuation of the sector-level assetallocation decision and should be included in the assetThe Journal of Performance MeasurementFall 2007-60-allocation returns.

5 This is not the case in the conven-tional sector Attribution Despite the qualitatively different nature of thestock selection and the asset allocation decisions,the Brinson framework does not clearly discrimi-nate between the two due to the appearance of aninteractionterm. To quote Bacon (2004), A flawof both Brinson models is the inclusion of the inter-action or other term. Interaction is not part of theinvestment process; you are unlikely to identify inany asset management firm individuals responsiblefor adding value through interaction. To put itanother way, the returns are not orthogonal , sincethe returns from both decision types can Brinson Attribution ModelIf one is managing a portfolio from the stock levelupwards, one cannot decide upon the quantityof eachstock to buy without first deciding whichstocks to this case, the stock selection decision precedes theasset allocation decision.

6 If the portfolio is managed inthis manner, it uses a bottom-up investment commonly used alternative approach is top-downinvestment. The investor first partitionsthe portfolioand benchmark into sectors or buckets based on indus-try sector, large or small capitalization, or some similarclassification. The use of such a partition has, in fact,nothing to do with the portfolio s risk or overall return;it is an externally imposed means of allowing invest-ment decisions to be taken in a hierarchical imposing this partition on the universe of investiblestocks, it is possible to reverse the order in which theprevious decisions were taken. A broad-brush assetallocation decision is made by assigning desired hold-ing weights to each sector, and using this as a constrainton further investment. Typically, such weights aremeasured relative to the corresponding weights in thebenchmark, so we talk of being over-weight or under-weight with respect to a given sector.

7 In the Brinson framework, the excess generated by thisover- or under-weighting at the sector level can bemeasured and is referred to as asset allocation return; itis measured by()AAiiiiSrwWb = where wiand Wiare the weight of the portfolio andbenchmark for sectori, respectively, and biis the returnof the benchmark for the sector weights have been chosen, the compo-sition of each sector is decided. This process involvesboth stock selection choices and further asset allocationdecisions within the sector. The returns by both thesedecisions at this level are aggregated together as stockselection returns. In the Brinson framework, thesereturns are measured bywhere riis the return of the portfolio for sector i, and theother terms are as defined for Equation sum of these two terms does not sum to the activereturn, defined as the return of the portfolio above thatof the benchmark.

8 To fix this, one can either introducean additional interaction term where the terms are as defined above, or use the portfo-lio sector weight wiin Equation 2 so that The Brinson framework allows a hierarchical decompo-sition of sectors into subsectors, so that transport stockscan be decomposed into railways, aircraft, etc. In prin-ciple one could continue to break the portfolio down sothat eventually each sector contains exactly one securi-ty. However, this is not particularly useful, since thenthe return of the security and the benchmark sector willbe the same, and the stock selection return will be zero. An Alternative to the Brinson Attribution ModelSuppose that the set of benchmark stocks Brepresentsthe investible universe (we cover the case of stocks thatlie outside the benchmark below). The benchmark hasnon-zero holdings in every stock within B, with return , (1)()SSiiiiSrWrb = , (2)()()IiiiiiSrwWrb = , (3)()SSiiiiSrwrb =.

9 (4)Fall 2007 The Journal of Performance Measurement-61-riand weight ai. The weights {ai} satisfy , and1iia= the return of the benchmark RBis given by BiiiRar= .We now select the stocks for our portfolio P, where . The first step is to build an intermediate port-folio composed of only these stocks, but with thesame relative holdings that they have in the is similar to the construction of a semi-notionalportfolio in the Brinson framework, but at the securitylevel instead of the sector level. Here, we are simplyconsidering the effect of modifying the benchmarkholdings by including some stocks and excluding oth-ers; we are expressly not considering asset allocationeffects. The weights in this portfolio are given byPB P iiiiPWwW = for , andfor .The return due to the stock selection decision is the dif-ference in return between this new portfolio and thebenchmark, given byiP 0iw =iP P , (5), (6)()SSiiiiBRwWR = , (7)The return due to the asset allocation decision is the dif-ference in return between and the portfolio P, givenbyIn both (7) and (8), the sum is over all securities in instance, suppose the portfolio and benchmark arecomposed of five stocks with the following weights andP ()AAiiiiBRwwR =.

10 (8) Stock 1 2 3 4 5 Portfolio weight Benchmark weight Return 2% -2% 1% -2% 0% Table 1: Sample Portfolio Structurereturns. Note that returns are the same for portfolio andbenchmark, since we are working at the security a security is traded at a non-end of day level, then thereturn of that security in the portfolio may differ to thereturn of the same security in the benchmark. In thiscase, we may attribute the difference in return to trad-ing effects, and proceed as before using the decide to build a portfolio from stocks 1 and on their relative holdings within the benchmark,their weights within will be ( + ) = ( + ) = , difference in return between the benchmark Bandthis intermediate portfolio is the return due to thestock selection decision. The benchmark return is thesum of the performance contributions, which is ( *2%) + ( * -2%) + ( * 1%) + ( * -2%) + ( *0%) = The return of the intermediate portfolio is( * 2%) + ( * 1%) = Therefore, thereturn due to stock selection is - ( ) = next step is to include the effect of the asset alloca-tion decision.