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I've found a few sources for instance this one, that makes the claim that over a portfolio, simple returns are additive. I can see why it doesn't make a difference if we use net or gross returns as long as the weights add to one, but I can't see why we should get this additive behaviour for a quantity that has a division in it. Perhaps I am using a bad definition of weight?

If someone could share a derivation for a portfolio of two assets that would be sufficient.

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We have a portfolio of two assets, with prices $P_1(0), P_2(0)$ at $t=1$. In the next period, their prices are $P_1(1), P_2(1)$.

The portfolio market value at $t=0$ is $V(0) = W_1 P_1(0) + W_2 P_2(0)$.

The portfolio weights $\omega_i$ (for $i =1,2$) are: $$ \omega_i = \frac{W_i P_i(0)}{W_1 P_1(0) + W_2 P_2(0)}. $$

(We can easily see that $\omega_1 + \omega_2 = 1$.)

The return for an asset $R_i$ is given by: $$ R_i = \frac{P_i(1)}{P_i(0)} - 1. $$

The market value of the portfolio $V(1)$ is equal to: $$ V(1) = W_1 P_1(0)(1+R_1) + W_2 P_2(0) (1+R_2). $$ Then, $$ V(1) = (W_1 P_1(0) + W_2P_2(0))(1 + \frac{W_1P_1(0)R_1}{W_1 P_1(0) + W_2P_2(0)} + \frac{W_2P_2(0)R_2}{W_1 P_1(0) + W_2P_2(0)}), $$ $$ V(1) = V(0) (1 + \omega_1 R_1 + \omega_2 R_2). $$ If we define the portfolio return $R$ as $\omega_1 R_1 + \omega_2 R_2$, then $$ V(1) = V(0) (1 + R). $$

This is easier to see if we normalise prices.

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It is not true.

$$w_t=p_{1,t}q_{1,t}+p_{2,t}q_{2,t}$$ Without loss of generality, to ignore splits, $q_{x,t}\equiv{q_{x,t+\Delta{t}}}.$

$$w_{t+\Delta{t}}=p_{1,t+\Delta{t}}q_{1,t+\Delta{t}}+p_{2,t+\Delta{t}}q_{2,t+\Delta{t}}.$$

$$\Delta{w}=p_{1,t+\Delta{t}}q_{1,t+\Delta{t}}+p_{2,t+\Delta{t}}q_{2,t+\Delta{t}}-p_{1,t}q_{1,t}-p_{2,t}q_{2,t}$$

$$\Delta{w}=\Delta{p_1}q_{1,t}+\Delta{p_2}q_{2,t}$$

$$r=\frac{\Delta{w}}{w_t}$$

$$r=\frac{\Delta{p_1}q_{1,t}}{p_{1,t}q_{1,t}+p_{2,t}q_{2,t}}+\frac{\Delta{p_2}q_{2,t}}{p_{1,t}q_{1,t}+p_{2,t}q_{2,t}}$$

Except for a couple of special cases, such as bankruptcy where the assets go to zero, this is not generally true. For example, if you made a 1000% return on a \$1 investment and a 1% return on a \$100,000,000 then clearly the returns are not additive.

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