Annualized Returns: Linear Averages Versus CAGR
Summary
The document explains two ways to express a return over multiple years as an annual figure. With annualized linear returns, the total return over the period is divided by the number of years. With compound annual growth rate (CAGR), the annual rate is the one that compounds to the observed total return across the full period. The distinction matters when interpreting a financial report’s one-year and longer-period return figures.
The answer notes that linear returns and CAGR are close when returns are small, because the compounding cross-term becomes relatively minor. It also points out that fractional-year periods require a fractional exponent or year count, and briefly mentions continuous compounding as another convention. The question does not provide the report’s methodology, so the applicable convention cannot be determined from the table alone; the response suggests assuming a linear annualization if compounding is not specified, while recognizing that this depends on reporting practice.
Key ideas
- Annualized linear return divides the total period return by the number of years.
- CAGR is the annual rate that compounds to the observed total return over the period.
- Linear annualization and CAGR become similar when returns are small.
- Fractional-year periods require using a fractional year count in the calculation.
- Continuous compounding is another return convention, though the answer says it is uncommon in financial reports.
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Full text
# What does 2 Year Annualized mean compared to 1 Year Annualized
# What does 2 Year Annualized mean compared to 1 Year Annualized
I am looking at a company's financial report and there is a table in it that lists returns over different annualized periods. It ranges from 1 year to 20 years. Would a 2 year return in this table be defined as the average return you would get in two years based off previous returns, or is it the average return based only on the last two years of data?
Also, how is this calculated? Annualized returns are raised to the 1/n power, where n is the number of years, so would a 2 year return just be 2/n power?
Thank you.
## Answer by pincopallino (score 1)
https://quant.stackexchange.com/a/12684
In your question you do not provide any reference. I believe that we are in front of two possibilities: annualized linear returns and Compound Annual Growth Rate (CAGR). If compounding is not mentioned, I would assume annualized linear returns.
$n$-years Annualized linear returns
$n$ = number of years
$ n * r_A = r_* $, where $r_*$ is the return over the $n$ years time span.
n-years CAGR
$ (1 + r_{CAGR})^n = (1 + r_*) $, where $r_*$ is the return over the $n$ years time span.
A few additional notes:
- With "small" returns, linear returns approximate CAGR. In $ (1 + r) * (1 + r) = (1 + 2*r + r^2) $, if $r$ is "small", $r^2$ may be negligible
- Fractions of years may be tricky: for example 1 year and six months leads to $n = 1.5$
- A third type of returns exists: continuous compounding. I list continuous compounding here for completeness sake but I do not think, based on my experience, that it is used in practice to present results in a financial report. It is interesting to look at their relationship with logarithmic returnsShown 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.