Digital Signal Processing Reference
In-Depth Information
⎛
⎝
π(
i
,
j
)
⎞
4
a
e
(
Θ
)
t
i
⎠
ς
≤
A
t
j
⎛
⎝
π(
⎞
4
d
a
e
(
Θ
)
t
i
a
k
⎠
≤
a
k
|
a
k
||
t
j
|
i
,
j
)
k
=
1
⎛
⎝
π(
⎞
4
d
a
e
(
Θ
)
⎠
≤
a
k
||
a
k
||
q
||
a
k
||
r
,
i
,
j
)
k
=
1
by virtue of Hölder inequality (
q
,
r
≤∞
), using the fact that
T
is orthonormal.
Taking
q
=
r
=
2 and simplifying yields the statement of the Theorem.
and
A
. It says that as
the “gap” in the eigenvalues of the market natural allocation increases compared
to the eigenvalues of the investor's allocation, the magnitude of the interaction term
decreases. Thus, the risk premium tends to depend mainly on the discrepancies (mar-
ket vs investor) between “spectral” allocations for each asset, which is the separable
term in (
15.52
).
Notice that (
15.72
) depends only on the eigenvalues of
Θ
15.7 Conclusion
In this paper, we have first proposed a generalization of Markowitz' mean-variance
model, in the case where returns are not supposed anymore to be Gaussian, but
are rather distributed according to exponential families of distributions with matrix
arguments. Information geometry suggests that this step should be tried [
2
]. Indeed,
because the duality collapses in this case [
2
], the Gaussian assumption makes that
the expectation and natural parameter spaces are
identical
, which, in financial terms,
represents the identity between the space of returns and the space of allocations.
This, in general, can work at best only when returns are non-negative (unless short
sales are allowed). Experiments suggest that the generalized model may be more
accurate to spot peaks of premia, and alert investors on important market events.
Our model generalizes one that we recently published, which basically uses plain
Bregman divergences on vectors, which we used to learn portfolio based on their
certainty equivalent [
20
]. The matrix extension of the model reveals interesting and
non trivial roles for the two parts of the diagonalization of allocations matrices in the
risk premium: the premium can indeed be split into a
separable
part which computes a
premium over the spectral allocation, thus being a plain (vector) Bregman divergence
part like in our former model ([
20
]),
plus
a non separable part which computes an
interaction between stocks due to the transition matrices. We have also proposed in
this paper an analysis of the magnitude of this interaction term.
