# Are 'annualized return' and 'annualized total return' synonymous?

Here's what I wrote for a draft blog article. But now I don't know that annualized total return means the CAGR/geometric average whereas annualized return does not have anything to do with compounding and just means investment returns held for a period other than one year, scaled to one year.

"The geometric average (also termed the annualized total return or compounded annual growth rate [CAGR]) is the average return of investments over a certain period of time, which takes compounding into consideration (but gives no indication of the investment's volatility).

'Annualized returns,' not to be confused with 'annualized total returns,' are simply investment returns held for a period other than one year, scaled to one year."

The difference between "annualized return" and "annualized total return," if there is any, is that the total return includes the effect of the dividends and computes the return you would get if you reinvested these dividends as they came out. In this case "annualized return" would represent what you would earn if you threw the dividends away.

It's a little ambiguous what you mean by "compounding." Normally a return is a way of representing wealth in an investment over a single period. Appropriately aggregating these always takes into account compounding in the usual sense--earning money on appreciated capital. However, if you consider dividends to be the thing to be compounded, then only the "total return" takes them into account.

You can fix what you wrote and make it reasonably clear by replacing "which takes compounding into consideration" with "which assumes reinvestment of any dividends" and then removing the comment in parentheses.

Your last sentence seems to me that it should be cut. Annualized return is often used as a synonym for annualized total return and even when it is not, is also a CAGR, just as annualized total return is. The only difference between them mathematically is whether dividends are included in the computation of the periodic (monthly or daily) returns.

• In the case of stocks, if dividends aren't causing compounding (by using them to buy additional stocks), what is? In the case of some types of bonds coupons/interest can be compounded. If returns aren't being reinvested by adding to the principal (or called 'face value' in the case of bonds, I learned) from which future interest is calculated or by purchasing additional stocks, what would compounding mean otherwise? Sorry if I'm slow to understand, I'm new to investing. Thanks for your help! – Natalie Mar 30 '17 at 19:52
• Start with \$100. Price goes up to \$110, no dividends. Return is 10% and wealth increase is \$10. Price goes up again to \$121, no dividends. Return is again 10% and wealth increase is \$11. Some kind of no-compounding solution would see the 10% increase and say earnings were \$10 in the second period because they don't account for price appreciation. Or see the \$11 and say 11%. There could be other interpretations, though, as this math is wrong. You see people use it often enough, regardless. – farnsy Mar 30 '17 at 23:07

They are synonymous. A side note is that 'total return' can mean dividends are reinvested, as opposed to net return which more accurately traces the stock's volatility. (Dividends skew the volatility.)

As for annualisation and geometric averaging, here are a couple of simple scenarios:

four annual returns: `(a, b, c, d) = (1%, 2%, 3%, 4%)`

total return over four years: `(1 + a) (1 + b) (1 + c) (1 + d) - 1 =`

``````(1 + 0.01) (1 + 0.02) (1 + 0.03) (1 + 0.04) - 1 = 0.10355 = 10.355%
``````

annualised total return: `((1 + a) (1 + b) (1 + c) (1 + d))^(1/4) - 1 =`

``````((1 + 0.01) (1 + 0.02) (1 + 0.03) (1 + 0.04))^(1/4) - 1 = 0.024939 = 2.4939%
``````

check: `(1 + 0.024939)^4 - 1 = 10.355%` over 4 years.

Strictly speaking the geometric mean is `(a b c d)^(1/4)` but adding in the base gives the annualised total return: `((1 + a) (1 + b) (1 + c) (1 + d))^(1/4) - 1 = 2.4939%`

Second scenario

four quarterly returns: `(a, b, c, d) = (1%, 2%, 3%, 4%)`

total annual return: `(1 + a) (1 + b) (1 + c) (1 + d) - 1 = 10.355%`

mean quarterly return: `((1 + a) (1 + b) (1 + c) (1 + d))^(1/4) - 1 = 2.4939%`

annualising the mean quarterly return: `(1 + 0.024939)^4 - 1 = 0.10355 = 10.355%`

• Dividends skew the volatility? I think not. If you compute volatility sans dividends you get a regular negative spike whenever dividends are paid and the price drops by the same amount. This will artificially inflate the volatility relative to that of the underlying firm and cash flows. – farnsy Mar 30 '17 at 14:21
• @farnsy Hmm, that's interesting. I'll look into it. Thanks. – Chris Degnen Mar 30 '17 at 15:59
• Thanks! I read elsewhere that one of the problems with the CAGR is that it does not reflect volatility, but I didn't know dividends skew volatility. Confusing stuff... – Natalie Mar 30 '17 at 19:46
• I wonder how to interpret "it does not reflect volatility." in this context. Cumulative returns do not measure volatility, but that is not a shortcoming. "My car gets 30 miles to the gallon." Would you say it's a downside that this statement doesn't tell us what color the car is? I would not. – farnsy Apr 1 '17 at 16:55