Compounding Multi-Period Index Returns from Price and Income Returns
Summary
The document derives an index’s single-period total return from security weights, price changes, and income. It rewrites each holding’s contribution as the sum of its price return and income return, then takes the weighted sum across holdings. This makes the approach usable when the inputs are expressed as percentage returns rather than raw prices and cash income.
For multiple periods, the single-period index returns are compounded by multiplying one plus each period’s return and subtracting one. The answer also extends the calculation to weights that change between periods, using each security’s weight at the start of that period. The method assumes the supplied returns and weights are defined consistently for each interval; it does not claim that simple addition of period returns gives the multi-period result. It also does not spell out index rebalancing, income reinvestment, or special treatment for unusual cash flows.
Key ideas
- A security’s single-period total return is the sum of its price return and income return.
- The index return for a period is the weighted sum of its constituents’ total returns.
- Multi-period index returns are found by compounding the period returns.
- When weights change, use the weights prevailing at the start of each period.
Tags
Full text
# Calculate total index return given only capital gains and income returns
# 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?
## Answer by Alper (score 1, accepted)
https://quant.stackexchange.com/a/68941
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
$$ TR_{K,0} = \left( \prod\limits_{j=1}^{K} \left( \left( \sum_{i=1}^N w_{j-1,i} (pr_{j,i}+cr_{j,i}) \right) +1 \right) \right) - 1. \tag 4 $$Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.