## Taxable Investing > Complexities > Optimizing with Taxes

 Mean-Variance 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 = rp       wl = 1       … w = vector of asset weights Σ= before-tax variance-covariance matrix r = vector of before-tax asset returns rp= portfolio before-tax return target l = vector of 1’s … = other constraints    The after-tax extension for PORTAX is min (1 – α)wΣw + αwL s.t.  wr = rp       wl = 1       … α = optimization control index between 0 and 1 w = vector of asset weights Σ = after-tax variance-covariance matrix= σρσ σ = after-tax standard deviations ρ = correlations L = iterative linear tax optimization r = vector of marginal after-tax asset returns rp= portfolio marginal after-tax 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 quadratic-linear programming problem.  As discussed in the Details, actual after-tax returns are not independent of asset allocations if market values and cost basis differ.  This creates a simultaneity problem where after-tax returns cannot be used in the quadratic or quadratic linear programming.  PORTAX instead uses the marginal after-tax 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 after-tax risk while the iterative linear component minimizes taxes.  The α controls the focus of the optimization.  For multi-period analysis, the optimizer runs each period in conjunction with the PORTAX Cash Projections that update market value and cost basis. Single Period, Before-Tax Optimization PORTAX Mult-period, After-Tax Optimization Click on images to enlarge ...