Math

Weight drift. If the portfolio starts at the target weights $w_e$ and $w_b = 1 - w_e$ and the two blocks develop over a period by the factors $R_e$ and $R_b$, the equity share at the end is:

$$w_e(t) = \frac{w_e R_e}{w_e R_e + w_b R_b}$$

The portfolio therefore drifts by itself towards whichever block ran better. Without rebalancing, a mixed strategy turns over the years into a bet on the winner of the past.

Band condition. The band check compares the actual share with the target weight, counted in percentage points:

$$\bigl| w_i(t) - w_i^{*} \bigr| \ge b_i \quad \Rightarrow \quad \text{action is due}$$

With two blocks the two deviations are equal in magnitude; a band on one side therefore always acts on the other as well.

Rebalancing premium. The contribution of rebalancing can be written as a diversification return. With the volatilities $\sigma_i$ of the blocks and the volatility $\sigma_p$ of the rebalanced portfolio, approximately:

$$\Delta g \approx \frac{1}{2}\left( \sum_i w_i \sigma_i^2 - \sigma_p^2 \right)$$

The expression is positive as long as the blocks do not move perfectly together. That is the arithmetic core of the claim that regular rebalancing is more than cosmetics. Against holding, the contribution can be negative once one side runs away for years; the rule then keeps selling it.

Diversification under leverage. The portfolio volatility of two blocks with correlation $\rho$ is:

$$\sigma_p^2 = w_e^2 \sigma_e^2 + w_b^2 \sigma_b^2 + 2 w_e w_b \rho\, \sigma_e \sigma_b$$

A negative $\rho$ lowers $\sigma_p$ substantially. That is what the strategy bets on, and that is why it is sensitive: leverage acts on volatility linearly but on the volatility drag quadratically. If the counter-movement disappears, $\sigma_p$ rises and the damage grows more than proportionally.

Rate regime. $\rho$ is not a constant. The pronounced negative correlation between stocks and long-dated government bonds is a phenomenon of particular decades, above all those with falling inflation. In phases of rising inflation and rising rates both sides moved down together.

The strategy is therefore not only an allocation but implicitly an assumption about the interest rate environment. That assumption appears in no formula, but in every backtest.

Day by day

The standard version rebalances by date. On the first common day, on which both products have a price, it buys with the value 100 the unit counts for the target weight $w_e$:

$$\begin{gathered} n_e = \frac{100\, w_e}{P_e(0)} \\ n_b = \frac{100\,(1 - w_e)}{P_b(0)} \end{gathered}$$

Between two dates the unit counts stay fixed, the value is units times price, and the equity share drifts as in the weight drift above:

$$W(t) = n_e\, P_e(t) + n_b\, P_b(t)$$

The deadlines count from the first common day. By date they are fixed calendar days: the 1st of every month whose distance from January is a multiple of the period, so with three months the 1st of January, April, July and October. The first deadline is the first date after the first common day, every further one the next date after the one carried out.

With quarterly rebalancing, the rule uses the first trading day of each quarter (in general: the first trading day of each scheduled month). This is a convention, not a recommendation.

If the deadline falls on a day without a price, the rebalancing happens on the next trading day, at the close and at that day's value:

$$\begin{gathered} n_e = \frac{w_e\, W(t)}{P_e(t)} \\ n_b = \frac{(1 - w_e)\, W(t)}{P_b(t)} \end{gathered}$$

With date and band, the first deadline is the first common day plus the period, and every further one still counts from the actual rebalancing day plus the period; if the day is missing in the target month, its last day applies. After the deadline has passed the rebalancing happens only on the day on which the equity share before rebalancing reaches the band $b$:

$$\lvert w_e(t) - w_e \rvert \ge b$$

The site's backtest does not compute the band condition above today; the package shows it in the second run.