Comparative statics of trading boundary in finite-horizon portfolio selection problem with proportional transaction costs

We consider Merton’s problem with proportional transaction costs. It is well known that the optimal investment strategy is characterized by two trading boundaries–the buy boundary and the sell boundary–between which lies the no-trading region. We investigate how these two trading boundaries vary with the transaction cost rates. We show that the cost-adjusted trading boundaries are monotone in the transaction costs. Our result implies the following: (i) the Merton line must lie between the two cost-adjusted trading boundaries; and (ii) when the Merton line is positive, both the buy and sell boundaries are monotone in the transaction cost rates, and consequently the Merton line lies in the no-trading region.