Method and system for multi-period performance attribution with metric-preserving coefficients
View Patent ↗A method for arithmetic performance attribution which accurately links single-period attribution effects over multiple periods. In preferred embodiments, the method determines portfolio relative performance over multiple time periods (t=1, 2, . . . , T) as a sum of terms of form R - R _ = ∑ i t ⌊ c 1 a i t + c 2 a i t 2 ⌋ , where a it is a component of active return for period t, the summation over index i is a summation over all components a it for period t, R is R = [ ∏ t = 1 T ( 1 + R t ) ] - 1 , R _ is R _ = [ ∏ t = 1 T ( 1 + R _ t ) ] - 1 , R t is a portfolio return for period t, R t is a benchmark return for period t, and the coefficients c 1 and c 2 are c 1 =A, and c 2 = [ R - R _ - A ∑ j t a j t ∑ j t a j t 2 ] . More generally, the invention is an arithmetic method for determining portfolio relative performance over multiple time periods (t=1, 2, . . . , T) as a sum of terms of form: R - R _ = ∑ i t ∑ k = 1 ∞ c k a i t k , where a it is a component of active return for period t. In preferred quadratic implementations (in which the only nonzero coefficients c k are those for which k=1 or k=2), the coefficients c 1 and c 2 are defined as in the above-mentioned preferred embodiments. In all embodiments, the method of the invention is metric preserving at the component portfolio level. 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 c 1 =A, and
c
2
=
[
R
-
R
_
-
A
∑
j
t
a
j
t
∑
j
t
a
j
t
2
]
,
where A has any predetermined value, a jt is a component of active return, the summation over index j is a summation over all components a jt for period t,
R
=
[
∏
t
=
1
T
(
1
+
R
t
)
]
-
1
,
R
_
=
[
∏
t
=
1
T
(
1
+
R
_
t
)
]
-
1
,
is a portfolio return for period t, R t is a benchmark return for period t, and the components a jt for each period t satisfy
∑
j
a
j
t
=
R
t
-
R
_
t
;
and
(b) determining the portfolio performance as
R
-
R
_
=
∑
it
⌊
c
1
a
it
+
c
2
a
it
2
⌋
,
where the summation over index i is a summation over all the terms (c 1 a it +c 2 a it 2 ) for period t.
2. The method of claim 1 , wherein A is
A
=
1
T
[
(
R
-
R
_
)
(
1
+
R
)
1
/
T
-
(
1
+
R
_
)
1
/
T
]
,
or for the special case R= R :
A = (1+ R ) (T−1 )/T .
3. The method of claim 1 , wherein A=1.
4. 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 a set of coefficients c k , including a coefficient c k for each positive integer k; and
(b) determining the portfolio performance as
R
-
R
_
=
∑
i
t
∑
k
=
1
∞
c
k
a
i
t
k
,
where a it is a component of active return for period t, the summation over index i is a summation over all components a it for period t,
R
=
[
∏
t
=
1
T
(
1
+
R
t
)
]
-
1
,
R
_
=
[
∏
t
=
1
T
(
1
+
R
_
t
)
]
-
1
,
R 1 is a portfolio return for period t, R t is a benchmark return for period t, and the components a it for each period t satisfy
∑
i
a
i
t
=
R
t
-
R
_
t
,
where the summation over index i is a summation over all components α it for said each period t.
5. The method of claim 4 , 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 .
6. The method of claim 4 , wherein c k =0 for each integer k greater than two,
c
1
=
A
,
c
2
=
[
R
-
R
_
-
A
∑
j
t
a
j
t
∑
j
t
a
j
t
2
]
,
A has any predetermined value, the summation over index j is a summation over all components a jt for period t,
R
=
[
∏
t
=
1
T
(
1
+
R
t
)
]
-
1
,
R
_
=
[
∏
t
=
1
T
(
1
+
R
_
t
)
]
-
1
,
R t is a portfolio return for period t, R t is a benchmark return for period t, and the components a jt for each period t satisfy
∑
j
a
j
t
=
R
t
-
R
_
t
.