Rebalancing is the act of restoring portfolio weights or numbers of shares to prespecified values. The need to rebalance is usually created by the natural tendency of portfolio weights to drift in the absence of any action. Seen at the level of the number of shares held, the need to rebalance is the result of a need to change the number of shares.

As explained in the previous post, rebalancing is easily understood when the target portfolio weights are constant. In the absence of rebalancing, portfolio weights drift away from their targets unless all asset returns turn out to be equal. But the term rebalancing is often used even when targeted portfolio weights are time varying. In the equity context, smart beta indices typically have time varying portfolio weights. A conjecture in Bouchey[2015] is that smart beta indices beat cap weighted portfolios because they rebalance. Technically, they rebalance since they are not buy-and-hold portfolios, but why should that contribute to performance? What assumptions are needed to rationalize the idea that rebalancing towards time varying weights adds value?

Rebalancing is just trading and the question can then be recast as: which characteristic of the trading process induced by smart beta indices would ensure that it adds value? No explicit answer is given to that question by defenders of the rebalancing edge, as far as I am aware. So here is my proposition.

One should distinguish contrarian trading from momentum trading. Let's cast this in a set up where there are two stocks. The first stock is taken as the numeraire, so its price is constant. The second stock has a time varying price \(p\) quoted in terms of the numeraire, which I initialize at \(1\) at inception. I assume that the portfolio is initialized with \(n_{0}\) shares of the second stock. Subsequently, trading ensures that I hold \(n(p)\) shares of the second stock if the price is \(p\). I'll fix \(n(1)=n_{0}\) as a simplification.

First, buy-and-hold is the rule \(n(p)=n_{0}\) whatever the price \(p\). Contrarian trading is the situation where \(n(\cdot)\) is a decreasing function of \(p\). I sell shares as the price rises, or I buy shares as the price falls. Momentum trading is the situation where \(n(\cdot)\) is an increasing function of \(p\). I buy shares as the price rises, or I sell shares as the price decreases.

Intuitively, contrarian trading might beat buy-and-hold if the price has a tendency to move away from its initial value and then reverts back. Momentum trading might beat buy-and-hold if the price moves away and never comes back. I say might because we need other assumptions to make sense of these intuitions (discrete trading for instance). If these intuitions are robust, then we might hope that the following situation holds for smart beta indices versus cap weighted indices (which buy and hold):

- smart beta indices do induce contrarian trading on average,
- prices do exhibit a tendency to cycle.

Now the case is not won. Market prices are not cyclical in any obvious way. Smart beta indices have time varying weights and it is therefore not obvious at all that smart beta indices induce contrarian trading. One needs other assumptions to make sense of these claims and subsequently, a structured empirical investigation is needed to check whether these assumptions hold in the data. But it looks like we now have a plan.

Bouchey[2015]: P. Bouchey, V Nemtchinov and T.L. Wong, *Volatility Harvesting in Theory and in Practice*, The Journal of Wealth Management, Vol 18,pp 89-100.

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