Diversification Return, Portfolio Rebalancing, and the Commodity Return Puzzle

Abstract

Diversification return is an incremental return earned by a rebalanced portfolio of assets. The author argues that the underlying source of the diversification return is the rebalancing; in contrast, the incremental return of a buy-and-hold portfolio is driven by the fact that the best-performing assets become a greater fraction of the portfolio. On the basis of these findings, the author resolves two aspects of a puzzle associated with Gorton and Rouwenhorst’s index of commodity futures.

The term diversification return was coined in the context of a rebalanced portfolio—that is, a portfolio with a constant percentage invested in each asset. The contribution of each asset to the portfolio’s compound return, dubbed the return contribution, exceeds the asset’s compound return by an incremental amount called the diversification return. The portfolio’s diversification return is the weighted average of the assets’ diversification returns.

One is led to wonder whether the diversification return has two separate, perhaps related, aspects: diversification and rebalancing. Several authors have argued in favor of this point of view to various degrees. For example, some have regarded a portfolio’s diversification return as the difference between its geometric average return and the (weighted) geometric average returns of its individual assets, regardless of whether the weights are constant.

In this article, I revisited the issue of diversification return and portfolio rebalancing. I showed that diversification return can be precisely defined in the context of a rebalanced portfolio. I argued that the reduction in variance inherent in a diversified portfolio is a necessary, but not sufficient, condition to earning a diversification return.

I clarified the underlying source of diversification return. The diversification return is usually expressed in terms of the difference between the variance of an asset and its covariance with the portfolio. Although this approach is elegant and useful, it masks the fact that the diversification return stems from selling assets that have appreciated in relative value and buying assets that have declined in relative value, as measured by their weights in the portfolio.

I also analyzed a buy-and-hold portfolio. Although such a portfolio can have no diversification return, it can have an incremental return relative to the initially weighted average of the compound returns of the assets. This result stems from the fact that, over time, a buy-and-hold portfolio will increase the weights of the best-performing assets. This result, however, also changes the risk profile of the portfolio. In contrast, an investor earns a diversification return in a rebalanced portfolio while maintaining a constant risk profile.

Finally, I used these results to resolve two aspects of the commodity return puzzle. I argued that the excess return (above the risk-free rate) of a commodity futures index, which is rebalanced monthly, can be largely accounted for by the diversification return. If the index is not rebalanced, however, it generates a significant incremental excess return as a buy-and-hold portfolio because the compound returns of the underlying commodity futures in the index have a wide range of values. Thus, no contradiction exists; the commodity futures index generates an excess return in both its rebalanced and unrebalanced incarnations but for totally different and unrelated reasons.