Market-cap weighted portfolios are self-rebalancing

This is a very simple note about a proof that I regularly forget. It is often said that a great advantage of market-cap weighted portfolios is that they are self rebalancing. That is, when you choose the weights of each security in a portfolio according to the proportion of their market-cap, the weights remain consistent with the original definition (under some rather mild assumptions).

Return of a portfolio

First, an even simpler result that we need in the rest of that note: The return of a portfolio is simply the weighted average of the returns of the securities in the portfolio. Let’s introduce some notation: let’s call $X_t$ the \$-value of the portfolio at time $t$, and $X_t = \sum_i X_{i,t}$ where $X_{i,t} = w_{i,t} X_t$ is the \$-value of the $i^\text{th}$ security in the portfolio at time $t$. The sum is taken over all securities in the portfolio, and we assume the weights sum to 1. Next the return of the portfolio (similarly for any security $i$) is defined by $r_t = X_t/X_{t-1}-1$. Then, \(\begin{align} X_t & = \sum_i X_{i,t} = \sum_i (1+r_{i,t}) X_{i,t-1} = \sum_i (1+r_{i,t}) w_{i,t-1} X_{t-1} \notag \\ & = X_{t-1} \left( 1 + \sum_i w_{i,t-1} r_{i,t} \right), \end{align}\) since by definition $\sum_i w_{i,t} = 1$ at any time $t$. Note that the same conclusion applies to all type of returns, including continuously compounded returns \(\bar{r}_{i,t}\), since $1+r_{i,t} = e^{\bar{r}_{i,t}}$.

Self-rebalancing

Let’s define $M_{i,t}$ as the market-cap of security $i$, and let $M_t = \sum_i M_{i,t}$. Now let’s assume that in the period $t-1$, the weights are calculated as a proportion of their relative market-cap, i.e., $w_{i,t-1} = M_{i,t-1} / M_{t-1}$. And let’s assume that from period $t-1$ to $t$, the market-cap of all securities $i$ only vary through its price (number of shares remain constant, no M$\&$A,…). Then, the ratios of market-cap will vary from $t-1$ to $t$ as \[ \frac{M_{i,t}/M_t}{M_{i,t-1}/M_{t-1}} = \frac{M_{i,t}}{M_{i,t-1}} \frac{M_{t-1}}{M_t} = \frac{1+r_{i,t}}{1+r_t} . \] On the other hand, the weights of the portfolios will vary as \[ \frac{w_{i,t}}{w_{i,t-1}} = \frac{X_{i,t}/X_t}{X_{i,t-1}/X_{t-1}} = \frac{X_{i,t}}{X_{i,t-1}} \frac{X_{t-1}}{X_t} = \frac{1+r_{i,t}}{1+r_t} . \]