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The role of diversification return in factor portfolios

The Diversification Return has long been supposed to represent the incremental return associated with portfolios that are regularly rebalanced compared to those that are not. In this paper, we test the hypothesis that it can be correctly
associated with a “rebalancing premium” for various factor portfolios. We then go on to determine how much of the
excess return of these factor portfolios may be attributed to the diversification return, and therefore whether or not, a
traditional factor-based explanation of their performance is more appropriate.

The notion of “diversification return” has been widely discussed by academics and practitioners over the years [1, 2, 3, 4,5, 6]. The consensus is that it represents the incremental return associated with portfolios that are regularly rebalanced, compared to those that are not, and consequently, it is often referred to as the “rebalancing premium.”
The idea first surfaced with Fernholz and Shay [1], who used the methods of Stochastic Portfolio Theory to derive an
expression for the log-return of a rebalanced portfolio, in terms of the weighted average of individual stock log-returns and an “excess growth rate.” Later, Booth and Fama [2] provided an independent derivation of this relationship and
furthermore coined the term “diversification return” for the contribution they attributed to the benefits of portfolio
diversification. Both sets of researchers express the diversification return as half the difference between the weighted
average stock volatility of the portfolio’s constituents and the portfolio’s volatility. Hence its contribution is positive and can be readily interpreted as arising from the risk reduction benefit of holding a portfolio rather than individual stocks.

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The derived relationship is elegant, but not universal, as it requires that a portfolio is rebalanced back to a constant
(unchanging) set of portfolio weights. This is quite a restrictive condition as, in most cases, portfolio weights will vary from rebalance to rebalance (for example, to counter decaying factor exposures, or to reduce portfolio risk under changing market conditions). This, however, has not prevented subsequent attempts to widen the scope of the relationship and to draw conclusions about how rebalancing and diversification relates to the return of more general portfolios [5]. Indeed, the concept has even been applied to several smart beta portfolios, leading to the conclusion that the diversification return is the sole source of their excess return [3, 4].

To evaluate the true nature of the diversification return and its role in the performance of factor portfolios, we will construct portfolios that strictly adhere to the conditions under which the diversification return has previously been specified. That is, we will construct portfolios that are regularly rebalanced back to the same set of weights and then examine whether their performance properties are in line with expectations. It is instructive to compare these rebalanced portfolios to the equivalent non-rebalanced portfolios; this allows us to tease out the effect of rebalancing and address the question whether rebalancing is always advantageous. In order to fully realize this comparison, we extend the definition of diversification return to one that is applicable to the non-rebalanced portfolios. Furthermore, we will examine how significant a contribution of the diversification return is to our portfolios’ performance, and compare it to other more traditional sources of excess return.

The structure of this note is as follows. In Section 2, we present a definition of the “diversification return” that may be
applied to both rebalanced and non-rebalanced portfolios. Using this as a starting point, we also derive an approximate result, which we believe is the basis for the widespread belief that “rebalancing is best.” In Section 3, we construct several simple factor portfolios that rebalance back to fixed weights, and compare their performance to equivalent non-rebalanced portfolios. In Section 4, we decompose the geometric return of our rebalanced portfolios into the diversification return and the weighted average stock geometric return (or “strategic return”). We apply this decomposition to determine the dominant contributor to each portfolio’s absolute and relative returns and specify how this relates back to the factor exposure of our portfolios. In Section 5, we draw our conclusions.

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