Proof that linear returns aggregate across securities

Question

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.

Answer 1

I think you are simply confusing percentage weights and number of assets.

In your definition the initial percentage weight of the $m$ assets in the portfolio are given by $w_i^{t - 1}$ and they sum to one, i.e.

\begin{equation} \sum_{i = 1}^m w_i^{t - 1} = 1. \end{equation}

Now define the absolute number of assets as $n_i$. They are linked to the percentage weights through

\begin{equation} w_i^{t - 1} = \frac{n_i P_i^{t - 1}}{\sum_{j = 1}^m n_j P_j^{t - 1}}. \end{equation}

As you don't rebalance your portfolio, the number of assets stays the same in $t - 1$ and $t$. The value of the portfolio at any time $t$ is

\begin{equation} P_P^t = \sum_{i = 1}^m n_i P_i^t. \end{equation}

Then

\begin{eqnarray} R_P^t & = & \frac{P_P^t}{P_P^{t - 1}} - 1\\ & = & \frac{\sum_{i = 1}^m n_i P_i^t}{\sum_{j = 1}^m n_j P_j^{t - 1}} - 1\\ & = & \sum_{i = 1}^m \left( \frac{n_i P_i^{t - 1}}{\sum_{j = 1}^m n_j P_j^{t - 1}} \right)\frac{P_i^t}{P_i^{t - 1}} - 1\\ & = & \sum_{i = 1}^m w_i^{t - 1} \frac{P_i^t}{P_i^{t - 1}} - 1\\ & = & \sum_{i = 1}^m w_i^{t - 1} R_i^t \end{eqnarray}

Note that generally $w_i^t \neq w_i^{t - 1}$.