Method and system for multi-period performance attribution
View Patent ↗A method for arithmetic and geometric performance attribution which accurately links single-period attribution effects over multiple periods. The arithmetic attribution method optimally distributes the residual to yield a minimum-distortion residual-free arithmetic attribution system. The geometric attribution method defines the attribution effects in terms of ratios. The arithmetic method determines portfolio performance over multiple time periods as a sum of terms of form (R t − R t )(A+α t ), where the coefficients α t are defined as α 1 = [ R - R _ - A ∑ k = 1 T ( R k - R _ k ) ∑ k = 1 T ( R k - R _ k ) 2 ] ( R t - R _ t ) . In preferred embodiments, the value of A is A = 1 T [ ( R - R _ ) ( 1 + R ) 1 / T - ( 1 + R _ ) 1 / T ] , where R≠ R , or, for the special case of R= R , A=( 1+ R) (T−1)/T , where T is the number of time periods. In all embodiments, the inventive coefficient (A+α t ) have smaller (typically much smaller) standard deviation than conventional logarithmic coefficients, which reduces variation in the weights assigned to each time period relative to the other time periods in the attribution calculation. In preferred embodiments, the geometric method determines portfolio relative performance over multiple time periods as 1 + R 1 + R _ = ∏ t = 1 T ∏ i = 1 N ( 1 + I it G ) ( 1 + S it G ) , where 1 + I it G = ( 1 + w it r it 1 + w it r _ it ) Γ t I is the geometric issue selection and 1 + S it G = ( 1 + w it r _ it 1 + w _ it r _ it ) ( 1 + w _ it R _ t 1 + w it R _ t ) Γ t S is the geometric sector selection. The attribution effects employed in preferred embodiments of the inventive geometric attribution method have more natural form than the conventional counterparts, and allow geometric attribution to be performed more accurately. Other aspects of the invention are a computer system programmed to perform any embodiment of the inventive method, and a computer readable medium which stores code for implementing any embodiment of the inventive method.
1. An arithmetic performance attribution method for determining portfolio performance, relative to a benchmark, over multiple time periods t, where t varies from 1 to T, comprising the steps of:
(a) determining coefficients (A+α t ), where the values α t are defined as
α
t
=
[
R
-
R
_
-
A
∑
k
=
1
T
(
R
k
-
R
_
k
)
∑
k
=
1
T
(
R
k
-
R
_
k
)
2
]
(
R
t
-
R
_
t
)
,
where A has any predetermined value, R t is a portfolio return for period t, R t is a benchmark return for period t, R is determined by
R
=
[
∏
t
=
1
T
(
1
+
R
t
)
]
-
1
,
and R is determined by
R
_
=
[
∏
t
=
1
T
(
1
+
R
_
t
)
]
-
1
;
and
(b) determining the portfolio performance as
R
-
R
_
=
∏
t
=
1
T
(
A
+
α
t
)
(
R
t
-
R
_
t
)
.
2. The method of claim 1 , wherein A is
A
=
1
T
[
(
R
-
R
_
)
(
1
+
R
)
1
/
T
-
(
1
+
R
_
)
1
/
T
]
,
where R≠ R , or for the special case R= R :
A= (1 +R ) (T−1)/T.
3. The method of claim 1 , wherein A=1.
4. The method of claim 1 , wherein step (b) is performed by determining the portfolio performance as
R
-
R
_
=
∑
t
=
1
T
∑
i
=
1
N
(
A
+
α
t
)
(
I
it
A
+
S
it
A
)
,
where I it A is an issue selection for sector i and period t, and S it A is a sector selection for sector i and period t.
5. A geometric performance attribution method for determining portfolio performance, relative to a benchmark, over multiple time periods t, where t values from 1 to T, comprising the steps of:
determining attribution effects for issue selection (1+I it G ) given by
1
+
I
it
G
=
1
+
w
it
r
it
1
+
w
it
r
_
it
Γ
t
I
,
and determining attribution effects for sector selection (1+S it G ) given by
1
+
S
it
G
=
(
1
+
w
it
r
_
it
1
+
w
_
it
r
_
it
)
(
1
+
w
_
it
R
_
t
1
+
w
it
R
_
t
)
Γ
t
S
,
where r jt is a portfolio return for sector j for period t, r jt is a benchmark return for sector j for period t, w jt is a weight for r jt , w jt is a weight for r jt , R is determined by
R
=
[
∏
t
=
1
T
(
1
+
R
t
)
]
-
1
and R is determined by
R
_
=
[
∏
t
=
1
T
(
1
+
R
_
t
)
]
-
1
;
and determining the portfolio performance as
1
+
R
1
+
R
_
=
∏
t
=
1
T
∏
i
=
1
N
(
1
+
I
it
G
)
(
1
+
S
it
G
)
.