# Why the portfolio return is defined as a weighted return?

After reading the modern portfolio theory, I am wondering why the portfolio return is defined that way. Suppose there are $$n$$ assets in a portfolio, the simple return of an individual asset $$i$$ at time $$t$$ is defined as $$r^i_{t} = (P^i_{t} - P^i_{t-1})/(P^i_{t-1})$$. The portfolio return is then defined as the weighted sum of individual returns,

$$R_t = \sum_{i=1}^n w_i r^i_t,$$

where $$w_i$$ is the weight for the individual asset $$i$$. However, this is not the "natural definition of return" I am thinking in my head. At time $$(t-1)$$, the cost we spend to form the portfolio is given by

$$C_{t-1} = \sum_{i=1}^n w_i P^i_{t-1}.$$

At time $$t$$, suppose the price of the assets $$i$$ rises to $$P^i_{t}$$, then the "value of the portfolio at $$t$$" is

$$V_t = \sum_{i=1}^n w_i P^i_{t}$$

So, the return of the portfolio should be

$$R'_t = (V_t - C_{t-1}) / C_{t-1} = \frac{\sum_{i=1}^n w_i P^i_{t}}{\sum_{i=1}^n w_i P^i_{t-1}} - 1$$

which is different from the textbook definition $$R_t$$.

Can someone explain why we use $$R_t$$ rather than $$R'_t$$ in defining the portfolio return?

When googling for the weighted return used in modern portfolio theory, I found people saying that log return is additive across time but not across assets, while simple return is additive across assets but not across time. This is why people use simple return in modern portfolio theory.

However, isn't the discrepancy between $$R_t$$ and $$R'_t$$ a proof that simple return is NOT additive across assets either? I wish I can resolve this so that I can proceed..

Proof of the equivalence

Thanks @Dave Harris for the helpful solution. First of all, the formulas regarding $$R'_t$$ should be corrected as

$$C_{t-1} = \sum_{i=1}^n n_i P^i_{t-1},$$

$$V_t = \sum_{i=1}^n n_i P^i_{t},$$

where $$n_i$$ is the quantity of the asset $$i$$, and so

$$R'_t = (V_t - C_{t-1}) / C_{t-1} = \frac{\sum_{i=1}^n n_i P^i_{t}}{\sum_{i=1}^n n_i P^i_{t-1}} - 1.$$

Then, it is actually quite trivial to rewrite $$R_t$$ in the form of $$R'_t$$:

$$R_t = \sum_{i=1}^n w_i r^i_t = \sum_{i=1}^n \frac{n_iP_{t-1}^i}{\sum_{j=1}^n n_j P^j_{t-1}} \frac{P_t^i - P^i_{t-1}}{P_{t-1}^i} = \frac{\sum_{i=1}^n n_i P^i_{t}}{\sum_{j=1}^n n_j P^j_{t-1}} - 1 = (V_t - C_{t-1}) / C_{t-1} = R'_t,$$

where the weights are the proportion of wealth

$$w_i := \sum_{i=1}^n\frac{n_iP_{t-1}^i}{\sum_{j=1}^n n_j P^j_{t-1}}$$

• You stopped too soon. I believe $R_t$ and $R'_t$ are the same if you take it 1 step further.... Jul 23, 2022 at 10:21
• After realizing the exact definitions of the weights and quantities, the proof is indeed one step away. Jul 23, 2022 at 13:53

There is an error in your assumtions. Once you have invested weights $$w_i$$ of your wealth at time $$t-1$$, these weights have no reason to remain constant when the prices of your assets change. Imagine that you have an equally weighted portfolio of assets at time $$t-1$$, then $$w_i = \frac{1}{n}$$ if you have $$n$$ assets. If all your assets except the first one become worthless between $$t-1$$ and $$t$$, then your new weights will be $$w_1^\prime =1$$, $$w_i^\prime=0\forall i\neq1$$. You are confusing weights, which refer to amounts of wealth, and quantities, which refers to units of assets. They are not the same things.
• Thanks, the "weights" are indeed dynamical. With this key remark, it can be shown that $R_t = R'_t$ indeed. Thanks! Jul 23, 2022 at 13:02