The Comparative Statics of Transaction Costs

Treating the determinant Δ as a constant, then can be derived solely from Δ₁, Δ₂, and Δ₃ by differentiating each with respect to k,j for . (Recall, that transaction costs are captured by the aggregate term ϕ_i, and so differentiation is with respect to this term.) For the three-asset case, these comparative statics are:

where k, j, and i index the assets. The first result says that an increase in the own t-cost will increase the drift. The second result examines the cross-asset weight sensitivity, whose sign depends on the relative return spreads. More specifically, the asset i weight change generated by a change in the asset j transactions cost depends on a comparison of the returns to these assets relative to the return on the third asset. In general, if they both exceed (both are less than) the return to the remaining asset k, the sign is negative. To make sense from this, consider a specific case using Δ₁ and differentiate this determinant with respect to ϕ₂ (representing asset two's transactions cost). When k₂ rises, w₂ rises unequivocally. But the effect on w₁ depends on and on . If , then w₁ wants to rise. But, if , then the net effect on w₁ is negative. Why? Two reasons. The first is that the sum of the new weights must satisfy the adding-up constraint. So, if w₂ rises, then the sum must decrease by an equal amount. Since , to keep risk minimized, then w₁ must fall and w₃ will rise. If, on the other hand, , but , then the following adjustment takes place: An increase in k₂ raises w₂, and as before, puts positive pressure on w₁. However, now, since , then this upward pressure is reinforced and w₁ rises (and w₃ falls to satisfy the adding-up constraint). The following tables summarize all the possibilities (each cell is the sign of the partial derivative with respect to k_j).

Optimization Formulas

Optimization routine for targeted tracking error

Subject to:

Optimization routine for rebalancing to a percentage boundary

Subject to:

Example and Definitions

w_t = N × 1 vector containing the optimized active portfolio weights (where _t represents new weights and _t–1 represents weights before optimization).

V = N × N matrix containing the VRS AA covariance matrix conditioned to the subperiod.

k = N × 1 vector containing the expected t-costs for each asset in percentage terms.

te = scalar representing the targeted expected tracking error.

t = scalar representing the number of subperiods in each year.

wmin = N × 1 vector of minimum underweights for each asset.

wmin1 = N × 1 vector of underweights that will be targeted if a lower boundary is violated.

wmax = N × 1 vector of maximum overweights for each asset.

wmax1 = N × 1 vector of overweights that will be targeted if a upper boundary is violated.