# Optimal Rebalancing with the Markowitz-Van-Dijk Heuristic

Institutional investors usually employ mean–variance analysis to determine optimal portfolio weights. Almost immediately upon implementation, however, the portfolio’s weights become sub-optimal as changes in asset prices cause the portfolio to drift away from the optimal targets.

In an idealized world (without transaction costs) investors would rebalance continually to the optimal weights. In the presence of transaction costs, investors must trade-off the cost of suboptimality with the cost of restoring the optimal weights. Most investors employ heuristics to manage this trade off. Some implement calendar-based rebalancing policies, in which they rebalance each month, quarter, or year. Others impose tolerance bands in which they rebalance when the exposure to any asset class drifts more than two percentage points from its target, for example. While these approached are better than not rebalancing at all, they’re arbitrary.

*Is total cost minimized by tolerance bands of one-or-two percentage points? Should equity asset classes have wider bands than fixed-income asset classes? Should investors rebalance in periods when there has been little drift?*

We propose that investors implement a rebalancing policy that minimizes explicitly the sum of transaction costs and suboptimality costs, including the expected future costs associated with each decision, at each point in time. **Rebalancing is a multi-period problem: A decision to rebalance today, or not, has implications for expected suboptimality and transaction costs in the future.**

### Dynamic Programming and Rebalancing

In 2006, Sun, Fan, Chen, Schouwenaars, and Albota showed us how to use dynamic programming to develop a rebalancing road map. This approach gives the optimal rebalancing decision for every possible combination of portfolio weights that may arise at each decision point across the investment horizon. They showed that this approach outperforms calendar and tolerance band approaches by reducing both transaction costs and suboptimality costs. Unfortunately, this solution suffers from the *curse of dimensionality *— it is intractable for portfolios with more than a few asset classes.

### Optimal Reblaancing Using Markowitz-Van-Dijk

Markowitz and van Dijk (2003) introduced a quadratic heuristic to rebalance portfolios to capture changes in expected returns of assets through time. In 2009, Kritzman, Myrgren, and Page applied this approach, which they called the Markowitz-van Dijk (MvD) heuristic, to the rebalancing problem. Through this process, they show that the MvD solution is actually quite close to the dynamic programming solution for small numbers of assets– AND that it scales manageable to several hundred assets.

The 2009 article, “Optimal Rebalancing: A Scalable Solution,” explores the process by which the MvD heuristic applied to the rebalancing problem. These tests showed that the MvD heuristic performs almost as well as dynamic programming for up to four assets and *better* than dynamic programming for five assets. In theory, of course, dynamic programming always yields the best result. Unfortunately, however, we have no way of determining how the MvD heuristic would compare to the unobservable “correct” dynamic programming solution. However, based on these tests. the MvD heuristic is a better option for rebalancing portfolios with more than a few assets.

### Applications

The scalability of MvD opens the door to several new applications of portfolio rebalancing. Passive managers could use the MvD heuristic to optimize the trade-off between tracking error and transaction costs. Quantitative asset managers could use it to minimize alpha decay between rebalancing dates. Plan sponsors could benefit from the MvD heuristic in very meaningful ways, as they are continually confronted with asset mix rebalancing decisions. Furthermore, plan sponsors could customize the optimal rebalancing process to existing tolerance bands, tracking error targets, cash inflows, and benefit payments.

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