Linking of Attributes Results - · PDF fileLINKING OF ATTRIBUTES RESULTS 3 although foreign to some, offers some benefits. As outlined by Bacon,1 these include the following. Bacon’s

1
INTRODUCTIONThis chapter deals with the linking of single-period attributionresults. Although the compounding of single-period total returns israther straightforward, the linking of attribution effects introducesadditional challenges. We will begin by looking at the two approachesfor defining excess return, geometric and arithmetic. We will thenproceed with a popular single-period attribution scheme called theBrinsonFachler (BF, 1985) method. Next we will see how these single-period results are presented in the following geometric methods: the Burnie, Knowles and Teder (BKT, 1998), the BKT exponential, thepure geometric, the Cario (1999) adjusted geometric and theMenchero (2005) adjusted geometric. We will see that the strength ofgeometric attribution lies in the ability to easily present multiple-period results. After the geometric presentations, we will compareand contrast the following arithmetic linking algorithms: Cario,Menchero and Frongello.
RETURNSBefore we can address the challenge of linking attribution effects, itis necessary to define the excess return that the attributes will even-tually explain. Intuitively, excess return is simply the performanceof the portfolio relative to some benchmark performance. Moneymangers aim for positive excess performance over the respectivebenchmark return. Mechanically, there are two approaches for
?
Linking of Attributes ResultsAndrew Scott Bay Frongello
MIT Sloan School of Business
1
Frongello.qxd 11/25/05 5:24 PM Page 1

PORTFOLIO ANALYSIS
2
calculating and presenting excess return. These include the arith-metic and geometric approaches.
Arithmetic excess returnThe arithmetic excess return in a single period is the portfolioreturn minus the return of the benchmark
Geometric excess returnThe geometric excess return is the ratio of one plus the portfolioreturn divided by one plus the benchmark return minus one
Single-period total return comparisonDue to the simple and immediate intuitive appeal of the arithmeticmethod, this is usually the preferred approach. The arithmeticapproach answers the question: What is the excess return relativeto a base of zero? On the other hand, the geometric approachanswers the question: What is the excess return relative to thelevel of the benchmark?. In other words, the arithmetic approachexplains how much better you did than the benchmark as a per-centage of the initial investment amount and the geometricapproach explains how much better you did than the benchmark asa percentage of the final value of the initial amount invested in thebenchmark. We will show the comparison of the presentation of thetwo methods with an example, see Table 1. Consider the arithmeticversus geometric example shown here. Which manager did better?It really depends on your definition of better. Remember that thetwo approaches answer different questions; the benefit of the arith-metic presentation is that it is very intuitive to the audience. Itis instinctive to simply subtract one return from the other to arriveat the relative performance. However, the geometric approach,
E
ERR
G
G
geometric excess return( )11
1( )
E
R
R
A
arithmetic excess returnportfolio return
benchmark reeturn
E R RA

LINKING OF ATTRIBUTES RESULTS
3
although foreign to some, offers some benefits. As outlined byBacon,1 these include the following.
Bacons arguments for the geometric method:
1. Proportional As explained earlier, the geometric approach meas-ures excess return relative to the performance of the benchmark.It is much more impressive to have a positive excess return whenthe market performs poorly than during a time when the marketperforms well. In the example above, manager A outperformedB arithmetically. However, manager A was operating in a bullishinvestment space, while manager B outperformed by roughlythe same arithmetic difference in a flat market. The geometricpresentation takes this into account and portrays manager B asthe outperforming manager.
2. Convertible Let us say our managers are domestic managers intwo countries A and B. Assume we measure manager performancein terms of currency B. What if market A was flat and all the bench-mark performance is due to currency effects? After conversion tocurrency B, manager As benchmark returned zero and the port-folio returned 9.09%. This is exactly the conclusion arrived at withthe geometric approach. The relative ranking of the managersremains the same regardless of the currency quoted.
3. Compoundable Geometric excess returns are easily compoundedto arrive at multiple-period excess returns.2 Arithmetic excessreturns alone cannot. This leads to relatively easy geometricmultiple-period analysis, more on this later.
Arithmetic versus geometric attributionAttribution effects explain the total excess return relative to abenchmark. In arithmetic attribution, these effects add to the total
Table 1
Manager A Manager B(%) (%)
Portfolio 20.00 9.50Benchmark 10.00 0.00Arithmetic excess 10.00 9.50Geometric excess 9.09 9.50

PORTFOLIO ANALYSIS
4
excess return. In geometric attribution, these effects compound tothe total excess return.
Single-period returnsThe single-period returns given in Table 2 will be used in the exam-ples to follow.
BF attributionThe arithmetic excess return is commonly decomposed by way ofthe BF (1985) method. To begin we need to introduce the followingvariables and formulas of the additive BF scheme3,4
w i t
wit
it
portfolio weight in sector in period benchmark wweight in sector in period portfolio return in sec
i t
rit ttor in period benchmark return in sector in peri
i t
r iit ood portfolio return in period
benchmark return in
t
R t
Rt
t
period
allocation effect in sector in period
t
A i t
AitBF
ttBF
itBF
t
S
total allocation effect in period
selection efffect in sector in period
total selection effect i
i t
StBF nn period t
R w r
R w r
A w w r R
A
t it iti
t it iti
itBF
it it it t
t
( ) ( )
BBFit it it t
i
w w r R ( ) ( )
Table 2
Portfolio Benchmark Arith. diff. Geo. diff.(%) (%) (%) (%)
Period 1 3.93 1.88 2.05 2.01

5
The BF (1985) example is shown in Table 3. Here we see possibleportfolio and benchmark compositions that agree with our totalreturns in Table 2. Note that the total allocation and selection effectssum to the total arithmetic excess return. Work it out! We will staywith these same figures throughout the examples to follow. In thefollowing sections we will take these additive effects and fit theminto a geometric context. If successful, these geometric equivalentswill compound to the geometric excess return.
BKT geometricInterestingly, the geometric methods also rely heavily on the arith-metic BF (1985) principles in one way or another. Below we illustratethe single-period geometric presentation attributed to BKT (1998)
AA
w r
A A
SS
w r
itBKT it
BF
it iti
tBKT
itBKT
i
itBKT it
BF
it iti
1
1
SS S
E A S
tBKT
itBKT
i
tG
tBKT
tBKT
( ) ( )1 1 1
itBF
it it it
tBF
it it
S w r r
S w r r
( ) ( )( ) iit
i
tA
tBF
tBFE A S
( )
Table 3
Period 1 Portfolio Benchmark Attribution
Weight Return Weight Return Allocation Selection(%) (%) (%) (%) (%) (%)
Equity 15.00 12.00 25.00 9.00 0.71 0.45Bond 85.00 2.50 75.00 0.50 0.24 2.55Total 100.00 3.93 100.00 1.88 0.95 3.00

PORTFOLIO ANALYSIS
6
The BKT (1998) geometric example is shown in Table 4. In the BKTgeometric model, the additive BF (1985) attributes are divided byone plus the return of the notional portfolio. Here, the notionalportfolio is defined as a hypothetical portfolio composed of port-folio weights and benchmark returns. After the transformation, thetotal allocation and selection effects are found by summing therespective sector level attributes. These totals then compound tothe total geometric excess return for the period.
BKT exponentialIn the original BKT approach, we note that the effects across sec-tors sum to their respective attribute totals. For those that prefer amultiplicative approach across all levels, Cario (1999) proposedthe following exponential adjustment to the BF (1985) arithmeticattributes
The BKT exponential example is given in Table 5. This approach isvery similar to the original BKT model. However, in the exponential
AR R
R RA
A
itEXP t t
t titBF
tEXP
exp
ln ln1 11
( ) ( )
1 1
1 1
A
SR R
R R
itEXP
i
itEXP t t
t t
( )
( ) ( )
expln ln
( )
( )
S
S S
E A
itBF
tEXP
itEXP
i
tG
tEXP
1
1 1
1 1 StEXP( ) 1
Table 4
Period 1 Allocation Selection(%) (%)
Equity 0.6994 0.4459Bond 0.2331 2.5266Total 0.9325 2.9725
Table 5
Period 1 Allocation Selection(%) (%)
Equity 0.6901 0.4383Bond 0.2305 2.5092Total 0.9190 2.9585

7

Documents

Linking of Attributes Results - · PDF fileLINKING OF ATTRIBUTES RESULTS 3 although foreign to some, offers some benefits. As outlined by Bacon,1 these include the following. Bacon’s