Using the Geometric Mean to Measure Compound Returns
Summary
The document explains why the geometric mean is useful for describing returns compounded over time. It contrasts the arithmetic average with the constant periodic return that would take an investment from its starting value to its ending value. A two-period example with a gain followed by an equal-sized loss shows that the arithmetic mean can be zero even when the account ends below its starting balance; the geometric mean reflects the compounded decline.
For a series of daily returns, the formula given computes the equivalent daily compound rate. The same approach can describe returns over other periods by adjusting the exponent to the number of observations. The discussion focuses on investment returns and time-series interpretation. It does not establish that a geometric mean is appropriate for averaging Treasury yields or calculating a market risk premium, which was part of the original question.
Key ideas
- The geometric mean gives the constant periodic return that matches the observed compounded outcome.
- Arithmetic averages can obscure losses caused by compounding.
- For daily returns, the formula yields an equivalent daily compound rate.
- The geometric mean can be adapted to other observation periods by changing the exponent.
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Full text
# Intuitive explanation of geometric mean
# Intuitive explanation of geometric mean
Suppose that the 10 Year Treasury Yield Rate varies every trading day during the year X1 (which in practice is accurate) what is the intuitive explanation behind calculating the geometric mean using this equation $[(1+R_1)(1+R_2)...(1+R_{252})]^{(1/252)}-1$? Is this simply the daily compound rate instead of the annualized compound rate? I realize that to calculate say the market risk premium for the year I can simply take the average of the rate for the whole year and use that but would I also be able to use the geometric mean as calculated above to do this and is it meaningful?
## Answer by gdlamp (score 4)
https://quant.stackexchange.com/a/37350
For a simple example, say you start with \$100 in an account.
In the first year, it makes 50% gain (+50% interest) => \$150
In the second year, it makes 50% loss (-50% interest) => \$75
The arithmetic mean is
```
(50% - 50%)/2 = 0%
```
The geometric mean is
```
(150% * 50%)^0.5 - 1 = 86.6% - 1 = -13.4% pear year
```
You know that you go from \$100 to \$75 over 2 years, so you have definitely lost money. The geometric mean will capture the reality better if you apply a -13.4% return on each year, yielding:
```
$100 -> $86.6 -> $75
```
## Answer by HK47 (score 1)
https://quant.stackexchange.com/a/37348
The arithmetic mean (simple mean) is not as useful for measuring rates of return over time because of compounding. When you are plotting a time series or forecasting into the future, it is more appropriate to use the geometric mean because it tells you what % return you would need per day/month/year (depends what time scale you are measuring).
Since the example you provided is based on days, you answered your own question. It is the daily compound rate of return.
Conversely, if you were to measure a fund's returns over 10 years it would be the same math except to ^1/10th power.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.