Trigger Strategies and No-Trade Regions

Trigger strategies are attractive because they are outcomes to an optimization problem, meaning that they minimize the possibly unnecessary (and redundant) costs of rebalancing on a purely calendar basis. But they are also static, meaning that they assume costs and benefits are constant through time. To understand this better, consider a setup for which costs are constant but benefits are measured as a Value-at-Risk (VaR) dollar amount portfolio risk exposure. VaR is a function of the correlations in the portfolio returns, the returns themselves, and the value of the portfolio, at each point in time. Clearly, this quantity is not constant, and in fact, may be highly volatile. In such a setting, the optimal solution to the problem of choosing a time to rebalance, as well as the degree of rebalancing, is one that maximizes the net benefits of doing so over all points in time. It is a dynamic strategy. As such, it is not so much a problem of knowing when to rebalance; technically, maximizing the net benefits occurs at each point, so rebalancing is really a continuous process—it is a random variable. Rather, it is knowing how much to rebalance in each period. In this type of problem—an optimal control problem—the objective is to maximize the function (say, net benefits) over time, that is, to solve for the optimal rebalancing path that maximizes the objective function continuously. What is attractive about this approach is that it combines the aspects of a calendar strategy (defining each point in time, for example, quarterly, monthly, daily) within an optimization problem (at each point in time, how much rebalancing is indeed optimal).

Two points need to be made. With respect to a trigger strategy, the time to rebalancing is a random variable, but the amount is not. On the other hand, the optimal control problem chooses the amount as a random variable but the timing is continuous (or practically speaking, it is periodic, say, monthly). The timing choice in a sense becomes one of the control variables—should rebalancing occur monthly, quarterly, or annually? The answer lies in the VaR. An optimal path (continuous, that is, defined in terms of dt) for rebalancing can be solved explicitly. Choosing to exercise rebalancing periodically is acceptable and does not change the nature of the problem or its solution. The classic control problem involves extracting fish from a lake. The fish reproduce at a rate and fishing depletes this renewable resource. The objective is to choose the optimal rate of fishing (analogous to our rebalancing argument) that keep the fish population within acceptable levels while maximizing the market value of the fish sold. Fishing is technically continuous, but practically speaking, periodic, for example, daily or weekly. But, choosing to wait too long within the interval increases risk, perhaps to unacceptable levels. Thus, the timing decision should be a function of the volatility in VaR. More volatile environments require more frequent rebalancing.

Leland (1999) suggests that the optimal policy involves a no-trade region around the target weights. Asset volatility and trading costs produce a region, or interval, around the targeted weights for which no rebalancing is optimal. Choosing not to rebalance will, in turn, reduce turnover. The no-trade region takes the form of a convex region, which, in the n-asset case, lies in Rⁿ. When actual portfolio weights lie to the right (left) of this region, then rebalancing occurs back to the boundary points, but not to the original target levels. This is a trigger strategy, but it is a continuously updated procedure for determining that strategy because the no-trade region continuously changes as a function of the movements in the underlying asset prices. Basically, the objective is to maximize, again, net benefits. Drift from the target produces utility losses, which are reduced through rebalancing. Drift, itself, is a function of the underlying asset dynamics (Wiener process), and since portfolio weights are functions of the asset price movements, then Ito's lemma produces a functional whose arguments include the weights (to be rebalanced) and the costs. The solution is a partial differential equation(s) that maximizes the discounted present value of the optimal path for net benefits. Underlying volatility in asset prices is the source of the no-trade region. Thus, we have a trader who wishes to hold risky assets in target proportions but for whom divergence between actual and target levels create expected losses (tracking error). Losses can be reduced by trading more frequently, but that leads to higher transaction costs. The optimal strategy minimizes the sum of the tracking error and transaction costs.

Application of this principle is costly because of the difficulty in locating the no-trade region—it is the outcome to the solution of a set of partial differential equations (assuming a solution exists—in general, it does not and quasi-optimal solutions need be identified). At present, this strategy is not readily implementable, certainly not without significant costs.