MeanVariance optimization (MV) has been
the standard asset allocation model since the 1950s. PORTAX extends the basic MV model to
account for market value, cost basis, turnover, management fees, income, capital
appreciation, differential tax rates, and various types of weight constraints
among other issues.
The basic MV model is
min
wΣw
s.t.
wr
= r_{p}
wl
= 1
…
w =
vector of asset weights
Σ= beforetax
variancecovariance matrix
r = vector of
beforetax asset returns_{
}
r_{p}= portfolio
beforetax return target_{
}l = vector of 1’s
…
= other constraints
The aftertax extension for PORTAX is
min
(1 – α)wΣw + αwL
s.t.
wr
= r_{p}
wl
= 1
…
α =
optimization control index between 0
and 1
w =
vector of asset weights
Σ = aftertax
variancecovariance matrix= σρσ
σ
= aftertax standard deviations
ρ = correlations
L = iterative linear
tax optimization
r = vector of
marginal aftertax asset returns_{
}
r_{p}= portfolio
marginal aftertax return target_{
}
I =
vector of 1’s
…
= other constraints
While the standard MV optimization is a
quadratic programming problem, the PORTAX optimization is an iterative
quadraticlinear programming problem.
As discussed in the Details, actual aftertax returns are not independent
of asset allocations if market values and cost basis differ. This creates a simultaneity problem
where aftertax returns cannot be used in the quadratic or quadratic linear
programming. PORTAX instead uses the
marginal aftertax returns which are invariant to asset allocations. PORTAX then uses an iterative linear
optimization to minimize taxes based on market values and cost basis. An
iterative heuristic is necessary in that decreasing an asset allocation has
different linear tax impacts then increasing an asset allocation.
This means the quadratic component
minimizes aftertax risk while the iterative linear component minimizes taxes. The
α
controls the focus of the optimization.
For multiperiod analysis, the optimizer runs each period in conjunction
with the PORTAX Cash Projections that update market value and cost basis.


Single Period, BeforeTax Optimization
PORTAX Multperiod, AfterTax Optimization
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