We have a market portfolio with weight \(\mu_{i,t}\) on stock \(i\) at date \(t\) (cap weights)
We have an index, which I assume has constant weights (simplification) \(\pi_{i}\); this portfolio is rebalanced discretely
\(\pi\) rebalances while \(\mu\) is buy-and-hold
How do we define the rebalancing gain? Not by a direct comparison of the performance of the two portfolios!
Here are two methods
One defines the buy-and-hold portfolio initialized with weights \(\pi\), which I call \(\pi^{bh}\)
The difference between \(\pi^{bh}\) and \(\mu\) isolates the contribution of different initial active weights
The difference between \(\pi\) and \(\pi^{bh}\) identifies the contribution of contrarian trading
Instead we create a fictitious portfolio \(\pi^{c}\) which rebalances continuously (high frequency)
We compute its performance assuming price trajectories are smooth in continuous time
The performance of this portfolio can be computed analytically
The relative performance of \(\pi\) against \(\mu\) can then be decomposed into the relative performance of \(\pi^{c}\) against \(\mu\) and a residual usually called the excess growth-rate which I write \(\gamma\).
\(\gamma\) is not a pure rebalancing contribution as it compares the discretely rebalanced portfolio \(\pi\) and the continuously rebalanced portfolio \(\pi^{c}\)
The relative performance of \(\pi^{c}\) against \(\mu\) includes a rebalancing component.
However:
From this standpoint, \(\gamma\) is sometimes described as encapsulating rebalancing gains. This is somewhat abusive.
The residual \(\gamma\) is positive and depends positively on micro volatility.
Method 1 is quite pure but has drawbacks. It cannot be run period by period. One has to look at it across periods, and then, the outcome depends heavily on the starting and end points.
Method 2 relies on a dubious interpretation of \(\gamma\) as a rebalancing contribution. However, it has some advantages. It can be carried-out period by period.
Constant weight portfolios beat the market portfolio when all \(\mu_{i,t}\) remain uniformly bounded away from zero at all times and the stocks are sufficiently volatile so that \(\gamma\) is always above a fixed strictly positive threshold \(\gamma\).
This is easy to prove using Method 2. It is also consistent with method 1. In method 1, under the above assumptions, the two buy and hold portfolios \(\mu\) and \(\pi^{bh}\) achieve the same growth rates and the trading component \(\pi-\pi^{bh}\) grows at the rate \(\gamma\).
From this standpoint, \(\gamma\) indeed measures the long-term benefit of rebalancing. It measures the excess growth in the rebalanced portfolio versus the cap-weighted portfolio.
Remark: the assumption that cap weights remain bounded away from zero is highly debatable!
It implies, I think, that all stocks share the same stochastic trend (all relative prices are stationnary). This needs to be further investigated.
Assuming the market portfolio (actually the buy-and-hold portfolio we have decided to rely on) is the numéraire.
All values and all prices are expressed in units of the market portfolio. Cap weights \(\mu_{i,t}\) are thus prices.
The value of the market portfolio is one at all times.
Portfolio Values initialized at one. Cap weights can't be.
Method 2 leads to:
\[\log(V_{\pi,T})=\log(\Phi(\mu_{T})/\Phi(\mu_{0}))+\sum_{t=1}^{T}\gamma_{\pi}(\mu_{t}/\mu_{t-1}),\]
with:
\[\log(\Phi(\mu_{t}))=\sum_{i=1}^{N}\pi_{i}\log(\mu_{i}),\]
and:
\[\gamma_{\pi}(x)=\log(\sum_{i=1}^{N}\pi_{i} x_{i})-\sum_{i=1}^{N}\pi_{i} \log(x_{i}).\]
Method 1 gives for the rebalancing gain (\(\pi-\pi^{bh}\), see my paper):
\[\sum_{t=1}^{T}\gamma_{\pi}(\mu_{t}/\mu_{t-1})-\gamma_{\pi}(\mu_{T}/\mu_{0}),\]
and for the overall decomposition:
\[\log(V_{\pi,T})=\sum_{t=1}^{T}\gamma_{\pi}(\mu_{t}/\mu_{t-1})-\gamma_{\pi}(\mu_{T}/\mu_{0})+\log(\sum_{i=1}^{N}\pi_{i}\frac{\mu_{i,T}}{\mu_{i,0}}).\]
This is consistent since we have:
\[\log(\Phi(\mu_{T})/\Phi(\mu_{0}))=-\gamma_{\pi}(\mu_{T}/\mu_{0})+\log(\sum_{i=1}^{N}\pi_{i}\frac{\mu_{i,T}}{\mu_{i,0}})\]
\[=-\log(\sum_{i=1}^{N}\pi_{i}\frac{\mu_{i,T}}{\mu_{i,0}})+\sum_{i=1}^{N}\pi_{i}\log(\frac{\mu_{i,T}}{\mu_{i,0}})+\log(\sum_{i=1}^{N}\pi_{i}\frac{\mu_{i,T}}{\mu_{i,0}}).\]