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."
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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%
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.
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