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Given total index return for a single period can be characterized as :

$$TR_{1}=\sum_{i=0}^Nw_i \frac {(p_{1i}-p_{0i}+inc_i)}{p_{0i}} $$

Is there a way to rewrite or derive a multi-period form of the above where inputs are returns (in percentage form) of the price and income components of the total return?

Ex:

If a security market index has a price return of 10% and an income return of 2% for period 1, and a price return of 15% and an income return of 3% for period 2, what is the approach to finding total return based only off of these kind of percentage inputs?

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Assuming $N$ is the number of securities in the index and $w_i$ is the weight of security $i$, one can rewrite $TR_{1,0}$, the total return of the index from time $0$ through $1$, as

$$ \begin{align*} TR_{1,0} &= \sum_{i=1}^Nw_i \frac {(p_{1,i}-p_{0,i}+c_{1,i})}{p_{0,i}} &&<\text{definition}> \\ &= \sum_{i=1}^Nw_i (p_{1,i}/p_{0,i}-p_{0,i}/p_{0,i}+c_{1,i}/p_{0,i}) &&<\text{algebra}> \\ &= \sum_{i=1}^Nw_i (p_{1,i}/p_{0,i}-1+c_{1,i}/p_{0,i}) &&<\text{algebra}> \\ &= \sum_{i=1}^Nw_i (pr_{1,i}+cr_{1,i}) &&<\text{substitute }pr_{1,i}=p_{1,i}/p_{0,i}-1, cr_{1,i}=c_{1,i}/p_{0,i}> \tag 1 \end{align*} $$

where $p_{j,i}$ is the price of security $i$ at time $j$, $c_{j,i}$ is the income of security $i$ from time $j-1$ through time $j$, $pr_{j,i}$ is the price return of security $i$ from time $j-1$ through $j$, $cr_{j,i}$ is the income return of security $i$ from time $j-1$ through $j$.

Equation (1) can be generalized as

$$ TR_{j,j-1} = \sum_{i=1}^Nw_i (pr_{j,i}+cr_{j,i}) \tag 2 $$

where $TR_{j,j-1}$ is the total return of the index from time $j-1$ through time $j$.

The total return of the index from time 0 through time $K$ (the multi-period return) can be calculated by compounding single period returns. This can be expressed as

$$ \begin{align*} TR_{K,0} &= \left( \prod\limits_{j=1}^{K} (TR_{j,j-1}+1) \right) - 1 &&<\text{definition}> \\ &= \left( \prod\limits_{j=1}^{K} \left( \left( \sum_{i=1}^N w_i (pr_{j,i}+cr_{j,i}) \right) +1 \right) \right) - 1. &&<\text{substitute equation (2)}> \tag 3 \end{align*} $$

It is possible to generalize equations (2) and (3) further by assuming weights can change dynamically. Dynamic weights can be expressed as $w_{j-1,i}$ which would be the weight of security $i$ at time $j-1$. Then equation (3) can be rewritten as

$$ TR_{K,0} = \left( \prod\limits_{j=1}^{K} \left( \left( \sum_{i=1}^N w_{j-1,i} (pr_{j,i}+cr_{j,i}) \right) +1 \right) \right) - 1. \tag 4 $$

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