Quadratic Transaction Costs (Return Penalty)
Quadratic transaction cost constraints (return penalties) impose quadratic penalties to the return in the case of larger transactions in addition to any linear transaction cost constraints. Both types of transaction cost constraints can be saved and loaded in transaction cost files (*.nftc). (To access the Quadratic Transaction Cost column, activate the transaction cost option in the Constraints Menu.)
Use quadratic transaction cost constraints to consider the effect of liquidity factors. Enter quadratic transaction cost constraints in the Quadratic Column in the Transaction Costs Section of the Constraints Worksheet. Quadratic transaction cost constraints require an initial portfolio and a quadratic reference point.
Define what makes a large transaction in the Quadratic Reference field below the Quadratic Transaction Cost Column. The quadratic reference point indicates the change in an asset's weight between the initial and optimal portfolios where a quadratic transaction cost penalty and a regular, linear transaction cost constraint of the same percentage are equal. In all the examples below, the asset's weight changed by 10% from the initial to the optimal portfolio. In the third row, the reference point is also 10%, so there is no difference between a 2% quadratic or a 2% linear transaction cost. Before the reference point, the quadratic transaction cost constraint penalizes the return of the optimal portfolio less than a linear transaction cost constraint does; after the reference point, the quadratic penalty is greater than a linear transaction cost constraint. A lower reference point forces a higher level of quadratic restraint as the quadratic return penalty surpasses the linear transaction cost sooner. See the chart below for a demonstration in how changing the reference point changes the penalty.
|
Initial Weight |
Optimal Weight |
Reference Point |
Quadratic Constraint |
Return Penalty |
|
2% |
12% |
2% |
2% |
1.00% |
|
2% |
12% |
6% |
2% |
0.33% |
|
2% |
12% |
10% |
2% |
0.20% |
|
2% |
12% |
14% |
2% |
0.14% |
The Optimizer calculates the quadratic transaction cost penalty that will be subtracted from the expected return along with any linear transaction costs as follows:
(Initial Portfolio Weight – Optimal Portfolio Weight)2 x (Quadratic Transaction Cost Constraint/Reference Point) = Penalty
So, changing an asset's allocation from an initial weight of 0% to an optimal weight of 15%, with a 2% quadratic transaction cost constraint and a 10% Quadratic Reference Point, results in a 0.45% transaction penalty: [(0.00-0.15)2 X 0.02/0.1=.0045]. A 2% linear transaction cost constraint in the same situation calculates to 0.3%. If you impose both a quadratic and a linear transaction cost constraint, both penalties are exacted from the return.