Cumulative Returns as the Growth of an Invested Dollar
Summary
A cumulative return series tracks the compounded performance of an investment over successive periods. The document explains it by imagining that one dollar is invested in the asset whose periodic returns are being measured. After each period, the investment value changes according to that period’s return, and cumulative return is the gain or loss relative to the original dollar.
The example compounds monthly returns of 10%, 20%, and 5%: each period’s growth factor multiplies the value reached in the previous period. This distinguishes cumulative performance from simply adding the periodic percentages. The series can be expressed as cumulative returns, starting at zero return before the first period, or as the corresponding investment value, starting at one dollar. The answer does not discuss adjustments for fees, cash flows, or the precise convention used by the paper that prompted the question, so those details should be checked in the paper’s methodology.
Key ideas
- Cumulative return measures the compounded gain or loss on an initial investment.
- Each period’s return is applied to the value accumulated through prior periods.
- The return series begins at zero before the first period, while an investment-value series begins at one unit of currency.
- Adding periodic returns generally does not give the cumulative return when returns compound.
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Full text
# What is a cumulative return series?
# What is a cumulative return series?
I guess this is pretty easy but I cannot find a definition anywhere. I am trying to reproduce a paper and they say they use a cumulative return series at some point. Does anyone know exactly what this is?
I have two ideas, first: may it just be the cumulative (total) returns so that the series is:
```
total return cumulative return series
10% 10%
20% 32% (10%+10%*20%+20%)
5% 37.016% (32%+5%+5%*32%)
```
and so on.
Or is it like the price series but calculated with total return data? like this:
```
price total return cumulative return series
5 10% 5.5
5.4(ignored) 20% 6.6 (5.5*(1+0.2))
```
and so on.
If someone has the definition of what a cumulative return series is and especially what is its starting value, it would be great if you could kindly tell me! Thank you very much.
## Answer by mark leeds (score 3, accepted)
https://quant.stackexchange.com/a/66240
O7-30-2021 : POSTING COMMENT AS ANSWER BASED ON SUGGESTION OF RICHARD HARDY.
Hi: The cumulative return is defined as the return on 1 dollar if it had been invested in whatever asset the returns came from. So, suppose the three total return numbers were monthly and that one was compounding monthly. Then, the cumulative return would be, 10 percent ( i.e: (1+0.10) -1 ) at the end of the first month. At the end of the second month, it would be (1+ 0.1) *(1+0.2) - 1 = (1.10 * 1.2) - 1. For the third month, you'd take the previous number and multiply it by 1.05 and subtract 1. This last return multiplied by 1 dollar would be dollars gained or lost by the investor. –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.