I keep reading that linear returns aggregate across securities, but I'm having trouble proving it. I suspect there's some mistake in my approach; I'd appreciate some help in seeing it.
Suppose we have two securities, $A$ and $B$. They have prices at timestep $t$ of $P_A^t$ and $P_B^t$, respectively. The portfolio has weights $w_A$ and $w_B$, which sum to 1. The price of the portfolio is then
$$P_P^t = w_A P_A^t + w_B P_B^t.$$
We define linear returns via the formula
$$R^t = \frac{P^t}{P^{t-1}} - 1.$$
Thus, the linear returns of the portfolio are
$$R_P^t = \frac{P_P^t}{P_P^{t-1}} - 1 = \frac{w_A P_A^t + w_B P_B^t}{w_A P_A^{t-1} + w_B P_B^{t-1}} - 1$$
I see it claimed in various locations around the internet that these returns aggregate over securities, which is to say that
$$R_P^t = w_A R_A^t + w_B R_B^t.$$
However, this formula yields
$$ R_P^t = w_A \frac{P_A^t}{P_A^{t-1}} + w_B \frac{P_B^t}{P_B^{t-1}} - 1$$
which, as far as I can see, is not in general equal to the previous expression for returns. In particular,
$$ w_A \frac{P_A^t}{P_A^{t-1}} + w_B \frac{P_B^t}{P_B^{t-1}} \neq \frac{w_A P_A^t + w_B P_B^t}{w_A P_A^{t-1} + w_B P_B^{t-1}}.$$
What am I missing here? I need to be able to use linearity when justifying Markowitz-style optimization.