Sector-level Attribution Effects
with
Compounded Notional Portfolios
Why Would We Want Them
and
How Can We Get Them?
Mark R. David, CFA
1
The Setup – What is Arithmetic Time Period
Linking Trying to Accomplish?
Ø Additivity
• of sectors to the total portfolio
• of attribution effects to the total value add
• of time periods to the total attribution period
Ø As contrasted to geometric attribution
methods…
2
Single Period Sector Performance…
Is easy. For Portfolio P:
Period t
Return
Weight
R P , i ,t
WP,i ,t
Contribution
Sector i
Sector i
CP ,i ,t = WP ,i ,t * RP ,i ,t
Sector i
Total
RP ,t = ∑ C P ,i ,t
i
3
Multi-Period Sector Performance…
Is easy. For Portfolio P:
Period 1
R
Sector i
W
C
Period t
Adjusted Contribution
~
C P,i ,t = C P ,i ,t * (1 + RP ,t −1 )
R
W
C
Full Performance
Period 0 - t
Adjusted Contribution
Adjusted Contribution
~
CP ,i ,t = CP ,i ,t * (1 + R P ,t −1 )
~
~
C P ,i = ∑ C P ,i ,t
t
Sector i
Sector i
TOTAL
~
~
C P ,t = ∑ C P ,i ,t
i
~
~
CP, t = ∑ C P, i,t
i
~
~
RP = ∑ CP ,i = ∑ CP ,t
i
t
4
Multi-Period Sector Performance - 2
Ø “Adjusted” contributions are scaled to prior
cumulative Portfolio return:
RP , t
 t

= ∏ (1 + R P , s )  − 1
 s =1

Ø Consistent with intuition for dollar contributions,
which are additive: 10% return on $100 = $10 in
period 1 makes 10% return in period 2 “worth” $11,
or 11% in base-period terms.
5
Single Period Sector Attribution…
Is easy.
Period t
Portfolio P
R
Sector i
W
C
Benchmark B
R
W
C
Attribution Effects
Allocation
Selection
S i ,t
Ai, t
Value Added
Interaction
Vi ,t = C P ,i ,t − C B ,i ,t
I i, t
= Ai ,t + S i ,t + I i ,t
Sector i
Sector i
Total
At = ∑ Ai , t
i
St = ∑ Si ,t
i
I t = ∑ I i ,t
i
Vt = ∑Vi ,t
i
= At + S t + I t
6
Single Period Attribution - 2
Ø Using the familiar, “vanilla” Brinson method:
Ai ,t = (WP ,i ,t − WB,i ,t ) * RB,i ,t
S i ,t = WB,i ,t * ( RP,i ,t − RB ,i ,t )
I i,t = (WP ,i ,t − WB,i,t ) * ( RP ,i,t − RB ,i,t )
Ø Many use Brinson-Fachler, in which:
Ai ,t = (W P ,i ,t − W B ,i ,t ) * ( RB ,i ,t − RB ,t )
Ø but then
Vi ,t = C P ,i ,t − C B ,i ,t ≠ Ai ,t + S i ,t + I i ,t
7
Multi-Period Sector Attribution
Is hard!
Period 1
A
Sector i
S
I
Period t
V
A
S
I
Full Period Attribution 0 - t
V
Allocation
Selection
~
~
Ai = ??? S i = ???
Interaction
Value Added
~
~
~
Vi = C P ,i − C B,i
I i = ???
~ ~ ~
= Ai + Si + I i
Sector i
Sector i
Total
~
~
A = ∑ Ai S = ∑ Si
i
i
~
I = ∑ Ii
i
V = RP − RB
= A+ S + I
8
Multi-period Sector Attribution - 2
Ø It’s hard, because the standard Brinson
formulas include weight & return from two
entities, the Portfolio and the Benchmark
Ø What is the “adjustment” factor when these
two entities do not track?
9
Solutions: A Simple Attempt
Ø Just use the prior cumulative Portfolio return, like we did with
single period Portfolio performance:
~
Ai ,t = Ai ,t * (1 + RP ,t −1 )
~
S i ,t = S i , t * (1 + RP ,t −1 )
~
I i ,t = I i ,t * (1 + RP ,t −1 )
Ø Not exact
Ø The further Portfolio and Benchmark returns drift, the worse it
gets.
10
Something a Tad More Sophisticated?
Ø Scale the weights by their respective entity’s prior cumulative
performance:
~
Ai ,t = [(WP ,i ,t *(1 + RP ,t −1 )) − (WB ,i ,t (1 + RB ,t −1 ))]* RB ,i ,t
~
Si ,t = [(WB ,i ,t (1 + RB ,t −1 ))]* ( RP ,i ,t − RB ,i ,t )
~
I i ,t = [(WP ,i ,t *(1 + RP ,t −1 )) − (WB ,i ,t (1 + RB ,t −1 ))]* ( RP ,i ,t − RB ,i ,t )
Ø Still not exact
Ø There is an algebraic solution for the error, but it is hard to
explain, and can be larger than the effect itself.
11
The First Real Deal: Cariño
Ø Cariño, David, “Combining Attribution Effects over Time”, The Journal of
Performance Measurement, Summer 1999
Ø Attempts to solve by viewing continuously compounding effects
 ln(1 + RP ,t ) − ln(1 + RB ,t ) 
~ ~ ~
{ Ai ,t , S i ,t , I i ,t } = 
 *{ Ai ,t , S i , t , I i ,t }
R
−
R


P ,t
B ,t
Ø But the approach still leaves an “unexplained residual … it is fair to distribute
the residual proportionately”.
Ø Hence, a final re-adjustment occurs after summing up the adjusted effects:
[
]
 ln(1 + RP ) − ln(1 + RB ) 
~ ~ ~
~ ~ ~
{ Ai , S i , I i } =∑ { Ai ,t , S i , t , I i , t } / 

R
−
R
t
P
B


12
Menchero
Ø Menchero, Jose, “An Optimized Approach to Linking Attribution Effects over
Time,” The Journal of Performance Measurement, Fall 2000
Ø Based on geometric compounding, constructs a scaling factor, such that:
~ ~ ~
{ Ai ,t , S i ,t , I i ,t } = F *{ Ai, t , Si ,t , I i, t }
F=
1
T


RP − RB

1/T
1/T 
 (1 + RP ) − (1 + RB ) 
Ø But again, “still leaves a small residual … introduce a set of corrective terms at
that distribute this small residual among the different periods so that the following
equation exactly holds”
RP − RB = ∑ ( F * α t ) * ( RP,t − RB ,t )
t
Ø And proceeds by optimizing the residual to make at as small as possible
13
Frongello, Wilshire
Ø Frongello, Andrew, “Linking Single Period Attribution Results,” The
Journal of Performance Measurement, Spring 2002
Ø Bonafede, Julia K., Steven J. Foresti, and Peter Matheos, “A Multiperiod Linking Algorithm That Has Stood the Test of Time,” The Journal
of Performance Measurement, Fall 2002
 t −1

 t −1 ~ ~ ~ 
~ ~ ~
{ Ai ,t , Si ,t , I i ,t } = { Ai ,t , Si ,t , I i ,t } * ∏1 + RP,t  + RB,t * ∑ { Ai ,t , Si ,t , I i ,t }
 j =1

 j =1

14
Frongello, Wilshire - 2
Sources of this period value
added
Cumulative
Prior
Portfolio
Return =
This period portfolio return =
This period
Benchmark
Cumulative
Benchmark
Benchmark
Cumulative
Allocation
Allocation
Cumulative
Selection
Selection
Cumulative
Interaction
Interaction
This period
Allocation
This period
Selection
This period
Interaction
Allocation
Selection
Interaction
Ø Decomposes a periods attribution effect into:
§ This period’s effect * cumulative prior portfolio return
§ Plus cumulative prior periods’ effect * this period’s benchmark return
Ø Valtonnen later shows that this is a valid though arbitrary decomposition, and
is one of a continuum of exact solutions
15
Davies & Laker
Ø Davies, Owen and Laker, Damien, “Multiple-period Performance
Attribution Using the Brinson Model”, The Journal of Performance
Measurement, Fall 2001
Ø Goes back to the “first principles” of Brinson, Hood, Beebower (1986),
defining “Notional Portfolios”:
Returns of Portfolio
Returns of Benchmark
Weights of
Portfolio
Portfolio
Notional Allocation
Weights of
Benchmark
Notional Selection
Benchmark
Ø In period t, then,
At = RA,t − RB ,t
St = RS ,t − RB ,t
It = RP ,t + RB ,t − R A,t − RS ,t
16
Compounded Notional Portfolios
Ø Davies & Laker called it the “Exact Brinson Method”
Ø Currently referred to by this more neutral moniker
Ø Stated that any linking methodology, however it
works, should equal the results of CNP, or it isn’t
Brinson
Ø Has intuitive appeal based on its real-world
feasibility
17
CNP Doesn’t Do Sector-Level?
Ø But, as late as Summer of 2005, the primary
downside of CNP was that no one had put forth a
method of producing sector-level attribution effects
that summed to the total portfolio effects.
§ Actually, Laker himself showed an example using Cariño
under CNP, but it wasn’t exact
§ Valtonnen showed Frongello under CNP. Exact, but still a
hybrid – and the interaction effect was a monster.
18
The Solution
Ø You’ve probably seen, however, that we already solved this problem back on
page 4
Ø Since with CNP we are dealing with four individual portfolios (even if two of
them are notional), we can simply apply the multi-period single portfolio
method to each of them, and apply the “first principles” Brinson:
~
~
~
Ai,t = C A,i,t − CB ,i ,t
~
~
~
Si , t = C S , i , t − C B , i , t
~
~
~
~
~
I i ,t = C P ,i ,t + C B ,i ,t − C A,i ,t − C S ,i ,t
Ø And everything sums exactly every which way
19
CNP vs. Other Methods
Ø Robustness, Absence of Residuals:
§ Equivalent
Ø Intuitiveness:
§ Superior, IMHO
Ø Transparency:
§ Superior, by virtue of simplicity
Ø Commutativity:
§ “simply interchanging two of the periods should not change the
results”.
§ Only Frongello is not commutative, and he argues that that is a
desirable aspect, calling it “Order Dependence”
20
CNP vs. Other Methods - 2
Ø Metric Preservation
§ “Two periods that have identical relative performance should
contribute equally to relative performance when they are linked
together.”
§ This criteria, advanced by Menchero, is only evidenced in
Menchero’s method
Ø A-causality
§ “August’s stock selection contribution to this year’s excess return
does not become available until after the end of December”
§ Put another way, a report produced at the end of May will have
different numbers for May’s attribution effects than a report
produced at the end of June
§ IMHO, a big deal
§ Cariño and Menchero both exhibit a-causality
21
Biggest Remaining Issue with CNP:
Ø Spurious Interaction Effects
§ Interaction appears over multiple periods, even when no
single period exhibits Interaction at the Total Portfolio level.
§ Laker later addresses persuasively, by pointing out that
Interaction arises not only from simultaneous effect of
Allocation and Selection, but also from combined effects
over multiple periods.
§ Frongello has interesting example, where Interaction
effects in separate sectors exactly cancel each other out.
Can produce alarmingly large Interaction effects over
multiple periods.
22