<p>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 $P_1(1), P_2(1)$. </p>

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

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

<p>(We can easily see that $\omega_1 + \omega_2 = 1$.)</p>

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

<p>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
$$
V(1) = V(0) (1 + R).
$$</p>

<p>This is easier to see if we normalise prices.</p>

CC BY-SA 3.0
2018-01-29T14:37:42.777Z
2018-01-29T14:37:42.777Z
Brian Romanchuk
<p>It is not true.</p>

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

<p>$$w_{t+\Delta{t}}=p_{1,t+\Delta{t}}q_{1,t+\Delta{t}}+p_{2,t+\Delta{t}}q_{2,t+\Delta{t}}.$$</p>

<p>$$\Delta{w}=p_{1,t+\Delta{t}}q_{1,t+\Delta{t}}+p_{2,t+\Delta{t}}q_{2,t+\Delta{t}}-p_{1,t}q_{1,t}-p_{2,t}q_{2,t}$$</p>

<p>$$\Delta{w}=\Delta{p_1}q_{1,t}+\Delta{p_2}q_{2,t}$$</p>

<p>$$r=\frac{\Delta{w}}{w_t}$$</p>

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

<p>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.</p>

CC BY-SA 4.0
2021-07-05T19:32:48.967Z
öhm, but yes, they are even in your example. You have to weight them accordingly. The total portfolio value is \$100,000,001 and the return is clearly \$1,000,010, no matter if you calculate the average return of the portfolio or the average returns of the assets and sum then up.
Peter Scholz
CC BY-SA 3.0
2018-01-27T05:50:16.883Z
2018-01-27T05:50:16.883Z
Dave Harris
<p>I've found a few sources <a href="https://assylias.wordpress.com/2011/10/27/linear-vs-logarithmic-returns/" rel="nofollow noreferrer">for instance this one</a>, 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?</p>

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

<econometrics><definition>
Why are simple returns additive over securities?
user16045
# Why are simple returns additive over securities?

(Asked by: username_0 on 2018-01-26)

I've found a few sources [for instance this one](https://assylias.wordpress.com/2011/10/27/linear-vs-logarithmic-returns/), 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.

**Tags:** econometrics, definition

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## 2 answers

### Answer by username_1 on 2018-01-29. Score: 3

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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### Answer by username_2 on 2018-01-27. Score: 1

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.

> **Comments:**
> **username_3**: öhm, but yes, they are even in your example. You have to weight them accordingly. The total portfolio value is \$100,000,001 and the return is clearly \$1,000,010, no matter if you calculate the average return of the portfolio or the average returns of the assets and sum then up.  

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