# 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?

## 2 Answers

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


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.

• Thanks for the answer HK47. So that means I can simply get the EAR by using the regular formula (1 + dividing that rate by 252)^(252) -1 correct? And this would be the effective annual rate for a 10 Year Treasury Rate Yield for year X1? – Dmitriy Dec 13 '17 at 20:04
• @Dmitriy Yes, precisely. There will be those with different opinions but generally speaking, getting the geometric mean and then using it to calculate EAR is effective. Be careful of using this strategy to price securities that have continuous compounding, though. – HK47 Dec 13 '17 at 20:37
• one more question. So what I've effectively done is calculate an effective annual rate for a 10 Year Treasury Yield using the geometric mean derived from the daily 10 Year Treasury Yield. This means that I can invest on any day in year X1 in a 10 Year Treasury security and expect a return of the EAR that I compounded on average. Is this a correct conclusion? – Dmitriy Dec 13 '17 at 21:47
• @Dmitriy Geometric mean is better for forecasting into the future or measuring returns on different time scales (quarterly returns, monthly, etc). EAR calculates based on annual compounding. Treasuries pay annual coupons, so the EAR would be a good approximation of your expected return. There's many other ways to bootstrap an expected return too, but your method is effective. – HK47 Dec 14 '17 at 15:32