Annualized Return Calculator

Annualized return = ((1 + total return ÷ 100)^(1 ÷ years) − 1) × 100

Total return

The whole gain or loss over the holding period as a percentage — 60 means the money grew by sixty percent — including dividends that were reinvested

Years

The length of the holding period in whole years, which must be at least one

Annualized return

The constant yearly compounded rate that reproduces the total return over that many years; the main answer

Divided by years

The total return split evenly across the years with no compounding, which is the annual percentage rate convention and a different number

Multiple

The ending value divided by the starting value, which is one plus the total return as a decimal: 60 percent is a multiple of 1.6

Use it whenever you have a total return and a holding period and you want to compare that result against something quoted per year — another fund, a savings rate, a benchmark index. It is the form in which returns are actually published, because a fund's advertised average annual total return is the compounded figure and not the divided one, and the disclosure rule that governs fund advertising requires it over fixed one, five and ten year windows. It is the right measure for a single lump sum held for a period and then sold. It is not the right measure for a series of deposits, because money added along the way was not invested for the whole period; an account that grew through monthly contributions needs a money-weighted return instead, and this page will understate or overstate it depending on when the money arrived. It is also not a forecast: a past annualized return is an average of a path that went up and down, and the fact that a fund compounded at 9.856 percent over five years says nothing about the next five.

1. 60 percent over five years: 9.856 percent a year, not 12

   1. The money grew to 1.6 times its starting value
   2. Annualized: 1.6^(1 ÷ 5) − 1 = 1.09856 − 1 = 9.856%
   3. Divided by years: 60 ÷ 5 = 12.000%
   4. The two differ by 2.144 percentage points

   This is the page's default because it is the sentence people actually say out loud. Five years at 12 percent compounded would give 76.2 percent, not 60, and the reason it does not is that the 12 percent figure is a straight line while the money grew along a curve. The divided figure is not a mistake — it is the convention behind an annual percentage rate — but it is a different number, and on a page that prints both you can see which one a fund prospectus is quoting.

2. A loss: minus 50 percent over four years

   1. The money fell to 0.5 times its starting value
   2. Annualized: 0.5^(1 ÷ 4) − 1 = 0.840896 − 1 = −15.910%
   3. Divided by years: −50 ÷ 4 = −12.500%
   4. The two differ by 3.41 percentage points, in the opposite direction to the gain case

   The sign of the comparison flips here and it catches people out. With a gain the compounded figure is the smaller of the two; with a loss it is the larger in magnitude, so annualizing makes the loss look worse and dividing makes it look milder. Four years at 12.5 percent a year compounded would lose 41.4 percent rather than 50, which is the same asymmetry seen from the other side.

3. Twenty-five years at 200 percent, where the gap is widest

   1. The money tripled, to 3 times its starting value
   2. Annualized: 3^(1 ÷ 25) − 1 = 1.04492 − 1 = 4.492%
   3. Divided by years: 200 ÷ 25 = 8.000%
   4. The two differ by 3.508 percentage points — the widest gap in the table

   Tripling your money sounds like a triumph and 4.492 percent a year sounds like a disappointment, and they are the same event. This is why the holding period has to be attached to any total return before it means anything: 200 percent over twenty-five years is a mediocre rate, the same 200 percent over eight years is 14.7 percent a year, and a bare percentage cannot tell you which one you are looking at.

The figure is only as good as the percentage you typed, and that percentage has to be the total return on the money rather than the change in a published price. If a fund rose 40 percent and paid distributions that you reinvested, the total return is higher than 40 percent, and entering the price change understates the annualized figure — an error that produces a plausible smaller number and therefore never looks like an error. The holding period is in whole years, so a position held for eighteen months has to be expressed as 1.5 years only by accepting that the page will reject a fractional entry; use the total return and a whole number of years, or a money-weighted calculation. The calculation assumes a single lump sum invested at the start and left alone until the end, so contributions or withdrawals along the way break it. It assumes the same rate every year, which no investment delivers: an annualized return is an average over a path, and two funds with the same average can have very different paths. Nothing here is after tax or after inflation, and nothing here is a forecast — a past annualized return is a description of what happened, not a rate you can expect to repeat.

How do I calculate an annualized return?

Divide the ending value by the starting value, raise that to the power of one over the number of years, and subtract one. A total return of 60 percent over five years is a multiple of 1.6, and 1.6 to the power of one fifth is 1.09856, so the annualized return is 9.856 percent. Dividing 60 by 5 gives 12 percent, which is a different figure under a different convention.

Why is the annualized return lower than dividing by the years?

Because compounding works on a curve and division works in a straight line. Each year's gain is earned on the previous year's gain as well as on the original amount, so a constant rate that arrives at the same endpoint has to be smaller than the average of the total. On a gain the compounded figure is always the smaller of the two; the gap grows with both the size of the return and the length of the period, reaching 3.5 percentage points on 200 percent over twenty-five years.

Does the same thing happen with a loss?

Yes, but the comparison flips. Four years at minus 50 percent annualizes to minus 15.910 percent against minus 12.5 percent by division, so this time the compounded figure is the larger loss in magnitude and the divided figure flatters it. The reason is the same curve read from the other direction: recovering from a loss requires a larger proportional gain than the loss itself, and the constant annual rate that produces a halving in four years has to be steeper than a quarter of it.

What is the difference between this page and the CAGR calculator?

The input, not the arithmetic. Both solve the same equation and both return a compound annual growth rate. The CAGR page takes two amounts, which is what you have when you are reading an account statement, and this page takes a total return percentage and a holding period, which is what you have when you are reading a fund report. Neither is more correct, and the two pages link to each other so a reader can enter from whichever side matches what is on the paper in front of them.

Is the divided-by-years figure wrong?

No, it is a different convention. Dividing a periodic rate by the period and multiplying gives what the United States securities regulator calls the annual percentage rate, and it is used widely for quoting and comparing. It is simply not a compounded rate, so it does not describe what the money actually did year by year, and it should not be compared against a savings rate or a fund's advertised average annual total return, both of which are compounded figures.

What counts as the total return?

The change in the value of the money itself, which means price change plus any distributions that were reinvested. If a fund's price rose 40 percent and it also paid dividends that you reinvested, the total return is above 40 percent and using the price change alone understates the annualized figure. Distributions that were taken as cash rather than reinvested are a separate matter, because the money left the investment and was not there to compound.

• Annual Return — Investor.gov glossary (the many ways an annual rate can be computed, and which of them is called the A.P.R.) — U.S. Securities and Exchange Commission, Investor.gov (United States)
• 17 CFR 230.482 — Advertising by an investment company (the rule that fixes the one, five and ten year average annual total return disclosures) — U.S. Securities and Exchange Commission rule, via the Electronic Code of Federal Regulations (United States)
• H.15 Selected Interest Rates — the published benchmark rates a computed rate is compared against — Board of Governors of the Federal Reserve System (United States)