Hi, yumbrad-ga:
Suppose a "portfolio" is defined by initial nonnegative allocations
a_1,...,a_n to n asset classes, some or all of which can be zero.
Each asset class has a corresponding "relative participation" target
r_i > 0. The goal is to increment these allocations using funds that
become available in some piecemeal fashion in an "optimal" way.
The construction below is optimal in two senses. First, it minimizes
the maximum relative "deficiency" of allocations with respect to their
respective participation targets at each step along the way to
reaching the exact targets. Second, it minimizes the overall amount
of additional allocations used to reach the exact targets.
* * * * * * * * * * * *
For each asset class i, define the normalized allocation z_i =
a_i/r_i. [Note: We can arrange without loss of generality that SUM
r_i = 1, but this would not simplify the following account enough to
be worth insisting upon. It would make a step in the proof of
optimality easier, so we may return to this point at a later time.]
The set {z_i : i = 1,..,n } contains s distinct values:
t_1 < t_2 < ... < t_s
where s is at least 1 and at most n.
Let I_1 U I_2 U ... U I_s be the natural partition of asset classes
{1,..,n} such that I_k = { i : z_i = t_k }, ie. indexes that
correspond to the same normalized allocation.
Although the additional allocations are expected to become available
in unpredictable fashion, their scheduling can be conceptually divided
into s phases. In all but the last of these phases, there is a limit
to the overall amount of additional allocation possible before one
advance to the next phase.
In the first phase we only add to allocations for asset classes in
I_1, and these allocations are made proportionally to their relative
participation targets r_i. At the end of the first phase all of the
normalized allocations z_i from I_1 will have risen to equal those
from I_2, ie. from t_1 to t_2.
In the second phase allocations are added across I_1 U I_2, and again
these increments are assigned proportionally to the respective r_i
targets. At the end of the second phase the normalized allocations
will haver risen from t_2 to t_3, ie. so as to match those already
attained in I_3.
Continuing in this fashion, phase s begins when all the normalized
allocations have been raised to a common value t_s. The targets are
exactly attained at this point, and any further incremental
allocations are made proportionally to all asset classes according to
relative participation targets r_i. As a result the relative
participation targets are preserved.
* * * * * * * * * * * *
The earlier example can now be made more complicated. Suppose that
for the three initial allocations a_1 = 1, a_2 = 2, a_3 = 3 we set
unequal targets:
r_1 = 1/2, r_2 = 1/3, r_3 = 1/6
The normalized allocations are then:
z_1 = 2, z_2 = 6, z_3 = 18
In this case the normalized allocations are all distinct, and thus we
have three phases. In the first phase allocations are added to a_1
until the revised normalized allocation reaches 6:
a'_1/r_1 = 6 ==> a'_1 = 3
We must increase a_1 = 1 to a'_1 = 3, an incremental allocation of 2.
In the next phase both of the first two asset classes will receive
extra allocations, and for every r_1 = 1/2 that is added to a'_1 = 3,
we'll add r_2 = 1/3 to a'_2 = a_2 = 2. This continues until the
normalized allocations for both (which rise together, remaining equal)
become z_3 = 18. Therefore:
a"_1/r_1 = 18 ==> a"_1 = 9
and because we've incremented a'_1 by 9 - 3 = 6 = 12*(1/2), we must
also increment a'_2 = a_2 = 2 by 12*(1/3) = 4. Thus a"_2 = 2 + 4 = 6.
We have reached the beginning of the third phase, and the
participation targets have been attained:
a"_ 1 = 9, a"_2 = 6, a"_3 = 3
The total amount added is 12, which turns out to be the least amount
necessary to reach the prescribed participation targets.
* * * * * * * * * * * *
We now give some additional formulas that arise in connection with
this schedule of allocation increases. The overall amount added in
phase k, k < s, is:
D_k = (z_{k+1} - z_k) * SUM r_i
where the sum is taken over S_k = I_1 U ... U I_k.
The specific amount added to asset class i during phase k is:
(z_{k+1} - z_k) * r_i
if i belongs to S_k, and zero otherwise.
It can be shown that SUM D_k for k = 1,..,s-1 is the minimum overall
increment which allows all allocations to reach their prescribed
targets. Details available on request.
regards, mathtalk-ga |