Rebalancing and Returns.indd - Dimensional Fund Advisors

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8 Dimensional Fund Advisors exists when comparing average returns of different rebalancing strategies. Depending on how much drift is allowed by the strategy, some return difference is expected due to allocation differences. Portfolios with more asset class drift will tend to have greater allocations in higher-return assets. The evolution of portfolio weights due to drift and rebalancing will also affect the portfolio return. The average return can be broken down into the expected return based on the average allocation and the effects from the time-varying weights: 5 (2) E(∑wr) = ∑E(w)E(r) + ∑cov(w, r) The term on the left-hand side is the arithmetic average portfolio return, which is computed by taking the weighted sum of returns each period, and averaging them over all periods. The first term on the right-hand side is the return on a portfolio with weights fixed at the average allocation. It is computed by first averaging the weights and returns on each asset class, then taking the weighted sum of the averages. Notice that if the weights are independent of returns, the average of the products would equal the product of the averages. Because portfolio weights are not constant through time, it is possible for the weights to have nonzero covariance with returns. If this occurs, the portfolio can have higher or lower returns than what would be expected given its average allocation. Proponents of positive rebalancing returns argue that it is possible to formulate rebalancing strategies that exploit long-term price reversals, making this term positive on average. However, there is also the potential for momentum effects to make this term negative on average. In either case, unless asset class returns are strongly correlated across time, this covariance term should be small. In Table 3, I show how much of the variation in average returns can be expected due to differences in the average allocation. Panel A of Table 3 displays the annualized arithmetic average return for each rebalancing strategy, which is the term on the lefthand side of (2). In Panel B, the expected return given the average allocation, which is the first term on the right-hand side of (2), is computed by first averaging the weights and returns of the individual asset classes, then taking a weighted sum. Once again, each return in Panels A and B is an average of returns on 12 portfolios, each starting in a different month. The variation in expected returns in Panel B is due completely to differences in the amount of drift allowed by the rebalancing algorithm. There is zero drift in the (1, 0%) strategy, as the portfolio is rebalanced to target each day. All the other strategies allow the portfolio weights to stray from target to some extent, with greater drift occurring in strategies with wider rebalancing bands. Therefore, rebalancing with wide bands, such as 20%-25%, is expected to increase returns by a few basis points relative to rebalancing with narrow bands, because the additional drift implies a slightly larger allocation to high-return assets on average. 5. This is the definition of covariance. For any two variables X and Y, cov(X,Y) = E(XY) – E(X)E(Y).
Rebalancing and Returns 9 Table 3 Average Rebalanced Returns February 1996-December 2004 Panel A: Gross Annualized Arithmetic Average Returns Look Interval (market days) Rebalance Band 1 5 10 20 60 125 250 0% 9.35 9.36 9.32 9.34 9.33 9.30 9.39 5% 9.44 9.42 9.41 9.41 9.37 9.33 9.39 10% 9.51 9.54 9.53 9.52 9.48 9.34 9.40 15% 9.59 9.63 9.60 9.50 9.43 9.36 9.40 20% 9.62 9.63 9.64 9.52 9.41 9.35 9.36 25% 9.44 9.39 9.38 9.31 9.29 9.25 9.27 100% 8.96 8.96 8.96 8.96 8.96 8.96 8.96 Panel B: Expected Returns for Average Allocations Look Interval (market days) Rebalance Band 1 5 10 20 60 125 250 0% 9.34 9.35 9.35 9.35 9.35 9.36 9.38 5% 9.36 9.36 9.36 9.36 9.36 9.37 9.38 10% 9.39 9.38 9.38 9.38 9.38 9.38 9.39 15% 9.40 9.40 9.40 9.40 9.39 9.39 9.40 20% 9.40 9.39 9.39 9.40 9.38 9.38 9.40 25% 9.42 9.41 9.40 9.40 9.39 9.39 9.40 100% 9.56 9.56 9.56 9.56 9.56 9.56 9.56 Panel C: Return Difference from Covariance of Weights with Returns Look Interval (market days) Rebalance Band 1 5 10 20 60 125 250 0% 0.00 0.02 -0.03 -0.01 -0.02 -0.06 0.01 5% 0.08 0.06 0.05 0.05 0.01 -0.04 0.01 10% 0.13 0.15 0.15 0.14 0.10 -0.04 0.00 15% 0.19 0.23 0.20 0.11 0.04 -0.03 0.00 20% 0.21 0.23 0.25 0.12 0.03 -0.03 -0.03 25% 0.02 -0.02 -0.03 -0.09 -0.10 -0.14 -0.13 100% -0.59 -0.59 -0.59 -0.59 -0.59 -0.59 -0.59
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