Implications

Clearly, the magnitude of rebalancing is greater, the higher the portfolio risk or any of its components, where, again, each of these variables are period specific, for example, . It is lower, on the other hand, the higher the marginal costs of rebalancing or the higher the discount rate. Moreover, the rebalancing rate is lower for assets with more inherent volatility (captured by the Wiener process) but higher for portfolios with greater allowable drift between and . Since can be either positive or negative, then optimal rebalancing can move in either direction (as in the Leland case). There is a no-trade region as well, where and is defined by setting to zero and solving to get:

As an example, let and assume for monthly monitoring. Moreover, let and assume that c is proportional to the value of the portfolio so that . Finally, keep the target weights constants so that . Then the allowable portfolio drift is contained in:

Thus, the no-trade zone is proportional to the Wiener process. As long as portfolio drift does not exceed the amount on the right hand side, then no rebalancing is required.

Theoretically, any portfolio with drift—and hence, tracking error—will need rebalancing. The question is when. In the context of this optimal control problem, when is not as important as how much. Sometimes, the amount will be zero. Other times, not. Nevertheless, the problem itself is analyzed continuously. Rebalancing, however, is undertaken periodically. We focused in our analysis on the interval and implicitly set this interval to length one (that is, one year, or ). We then solved for the optimal rebalancing rule that minimizes the net risk in terms of value at risk over that interval. Since choosing to rebalance in any period affects the value and therefore the risk position of the portfolio from that point forward (just as fishing affects the stock of fish and its value from that point forward), then we face what is essentially a control problem through time (hence, the optimal control framework).

Nevertheless, the interval could easily be subdivided. For example, year can be divided into 4 or 12 intervals. Rebalancing is then performed at each of these subintervals in time as indicated by . In the limit, rebalancing could be continuous. Theoretically, it is continuous rebalancing that is optimal. If we choose to rebalance, say, quarterly, then all of the adjustment is performed four times per period. This obviously entails higher absolute costs (and benefits), as these have accumulated through the intervening periods. Waiting too long may increase risk to unacceptable levels. The empirical simulations in the next section are designed to test this implication.