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 t = [ R - R _ - A ∑ k = 1 T ( R k - R _ k ) ∑ k = 1 T ( R k - R _ k ) 2 ] ( R t - R _ t ) . 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. 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 , Vestek ) ( 1 + S it G , Vestek ) , where 1 + I it G , Vestek = ( 1 + w it r it 1 + w it r _ it ) Γ t is the geometric issue selection and 1 + S it G , Vestek = ( 1 + w it r _ it 1 + w _ it r _ it ) ( 1 + w _ it R _ t 1 + w it R _ t ) Γ t is the geometric sector selection, with Γ t = [ ( 1 + R t 1 + R _ t ) ∏ j = 1 N ( 1 + w _ jt r _ jt ) ( 1 + w jt R _ t ) ( 1 + w jt r jt ) ( 1 + w _ jt R _ t ) ] 1 2 N . The attribution effects employed in the invention 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. A computer system, comprising:
a processor which performs an arithmetic performance attribution computation to determine portfolio performance, relative to a benchmark, over multiple time periods t, where t varies from 1 to T, by determining coefficients (A+α t ), where A is a coefficient, 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 R t is a portfolio return for period t, R t is a benchmark return for period t, R k is a portfolio return for period k, R k is a benchmark return for period k, k varies from 1 to T, R k =R t for k=t, R k = R t for k=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 determining the portfolio relative 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, S it A is a sector selection for sector i and period t, and i ranges from 1 to N; and
a display device coupled to the processor for displaying a result of the arithmetic performance attribution computation.
2. A computer system, comprising:
a processor which performs a geometric performance attribution computation to determine portfolio performance, relative to a benchmark, over multiple time periods t, where t varies from 1 to T, by determining attribution effects for issue selection (1+I it G,Vestek ) given by
1
+
I
it
G
,
Vestek
=
(
1
+
w
it
r
it
1
+
w
it
r
_
it
)
Γ
t
and determining attribution effects for sector selection (1+S it G,Vestek ) given by
1
+
S
it
G
,
Vestek
=
(
1
+
w
it
r
_
it
1
+
w
_
it
r
_
it
)
(
1
+
w
_
it
R
_
t
1
+
w
it
R
_
t
)
Γ
t
,
where the values of Γ t are,
Γ
t
=
[
(
1
+
R
t
1
+
R
_
t
)
∏
j
=
1
N
(
1
+
w
_
jt
r
_
jt
)
(
1
+
w
jt
R
_
t
)
(
1
+
w
jt
r
jt
)
(
1
+
w
_
jt
R
_
t
)
]
1
2
N
,
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 it is a portfolio return for sector i for period t, r it is a benchmark return for sector i for period t, w it is a weight for r it , w it is a weight for r it , i ranges from 1 to N, r it =r jt for i=j, r it = r jt for i=j, w it =w jt for i=j, w it = w jt for i=j, 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 determining the portfolio performance as
1
+
R
1
+
R
_
=
∏
t
=
1
T
∏
i
=
1
N
(
1
+
I
it
G
,
Vestek
)
(
1
+
S
it
G
,
Vestek
)
;
and
a display device coupled to the processor for displaying a result of the geometric performance attribution computation.
3. A computer readable medium containing instructions for programming a processor to perform an arithmetic performance attribution computation to determine portfolio performance, relative to a benchmark, over multiple time periods t, where t varies from 1 to T, by determining coefficients (A+α t ), where A is a coefficient, 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 R t is a portfolio return for period t, R t is a benchmark return for period t, R k is a portfolio return for period k, R k is a benchmark return for period k, k varies from 1 to T, R k =R t for k=t, R k = R t for k=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 determining the portfolio relative 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, S it A is a sector selection for sector i and period t, and i ranges from 1 to N.
4. A computer readable medium containing instructions for programming a processor to perform a geometric performance attribution computation to determine portfolio performance, relative to a benchmark, over multiple time periods t, where t varies from 1 to T, by determining attribution effects for issue selection (1+I it G,Vestek ) given by
1
+
I
it
G
,
Vestek
=
(
1
+
w
it
r
it
1
+
w
it
r
_
it
)
Γ
t
and determining attribution effects for sector selection (1+S it G,Vestek ) given by
1
+
S
it
G
,
Vestek
=
(
1
+
w
it
r
_
it
1
+
w
_
it
r
_
it
)
(
1
+
w
_
it
R
_
t
1
+
w
it
R
_
t
)
Γ
t
,
where the values of Γ t are,
Γ
t
=
[
(
1
+
R
t
1
+
R
_
t
)
∏
j
=
1
N
(
1
+
w
_
jt
r
_
jt
)
(
1
+
w
jt
R
_
t
)
(
1
+
w
jt
r
jt
)
(
1
+
w
_
jt
R
_
t
)
]
1
2
N
,
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 it is a portfolio return for sector i for period t, r it is a benchmark return for sector i for period t, w it is a weight for r it , w it is a weight for r it , i ranges from 1 to N, r it =r jt for i=j, r it = r jt for i=j, w it =w jt for i=j, w it = w jt for i=j, 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 determining the portfolio performance as
1
+
R
1
+
R
_
=
∏
t
=
1
T
∏
i
=
1
N
(
1
+
I
it
G
,
Vestek
)
(
1
+
S
it
G
,
Vestek
)
.
5. 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:
generating data indicative of 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 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
processing the data indicative of coefficients (A+α t ) to generate data indicative of (A+α t )(I it A +S it A ), for each value of i in the range 1≦i≦T and each value of t in the range 1≦i≦N, where I it A is an issue selection for sector i and period t, S it A is a sector selection for sector i and period t, and
R
-
R
_
=
∑
t
=
1
T
∑
i
=
1
N
(
A
+
α
t
)
(
I
it
A
+
S
it
A
)
.
6. The method of claim 5 , 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 .
7. The method of claim 5 , wherein A=1.
8. A geometric 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:
generating data indicative of attribution effects for issue selection (1+I it G,Vestek ) defined as
1
+
I
it
G
,
Vestek
=
(
1
+
w
it
r
it
1
+
w
it
r
_
it
)
Γ
t
and generating data indicative of attribution effects for sector selection (1+S it G,Vestek ) defined as
1
+
S
it
G
,
Vestek
=
(
1
+
w
it
r
_
it
1
+
w
_
it
r
_
it
)
(
1
+
w
_
it
R
_
t
1
+
w
it
R
_
t
)
Γ
t
,
where R t is a portfolio return for period t, R t is a benchmark return for period t, and the values of Γ t are
Γ
t
=
[
(
1
+
R
t
1
+
R
_
t
)
∏
j
=
1
N
(
1
+
w
_
jt
r
_
jt
)
(
1
+
w
jt
R
_
t
)
(
1
+
w
jt
r
jt
)
(
1
+
w
_
jt
R
_
t
)
]
1
2
N
,
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
processing the data indicative of attribution effects for issue selection (1+I it G,Vestek ) and processing the data indicative of attribution effects for sector selection (1+S it G,Vestek ) to generate data indicative of (1+I it G,Vestek )(1+S it G,Vestek ), for each value of i in the range of 1≦i≦T and each value of t in the range 1≦i≦N, where I it G,Vestek is an issue selection for sector i and period t, S it G,Vestek is a sector selection for sector i and period t, and
1
+
R
1
+
R
_
=
∏
t
=
1
T
∏
i
=
1
N
(
1
+
I
it
G
,
Vestek
)
(
1
+
S
it
G
,
Vestek
)
,
where
1
+
R
1
+
R
_
is the portfolio performance.