# Calculate total index return given only capital gains and income returns

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?

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

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