LO 66.7: Determine the optimal no-trade region for rebalancing with transaction

LO 66.7: Determine the optimal no-trade region for rebalancing with transaction costs.
If transaction costs are zero, a manager should revise a portfolio every time new information arrives. However, as a practical matter, a manager should make trading decisions based on expected active return, active risk, and transaction costs. The manager may wish to be conservative because these measures are uncertain. Underestimating transaction costs, for example, will lead to trading too frequently. In addition, the frequent trading and short time horizons would cause alpha estimates to exhibit a great deal of uncertainty. Therefore, the manager must choose an optimal time horizon where the certainty of the alpha is sufficient to justify a trade given the transaction costs.
The rebalancing decision depends on the tradeoff between transaction costs and the value added from changing the position. Portfolio managers must be aware of the existence of a no-trade region where the benefits of rebalancing are less than the costs. The benefit of adjusting the number of shares of a given portfolio asset is given by the following expression:
marginal contribution to value added = (alpha of asset) [2 x (risk aversion) x (active risk)
x (marginal contribution to active risk of asset)] 2018 Kaplan, Inc.
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Topic 66 Cross Reference to GARP Assigned Reading – Grinold and Kahn, Chapter 14
If this value is between the negative cost of selling and the cost of purchase, the manager would not trade that particular asset. In other words, the no-trade region is as follows:
(cost of selling) < (marginal contribution to value added) < (cost of purchase)
Rearranging this relationship with respect to alpha gives a no-trade region for alpha:
[2 x (risk aversion) x (active risk) x (marginal contribution to active risk)] (cost of selling) < alpha of asset < [2 x (risk aversion) x (active risk) x (marginal contribution to active risk)] + (cost of purchase)
The size of the no-trade region is determined by transaction costs, risk aversion, alpha, and the riskiness of the assets.
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