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?
remark to add
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}} $