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Annualized Return Calculator

Turn any gain over any period into a comparable yearly rate, step by step
Annualized return from 12,000 to 19,500 over 4 years
Annualize a 1.8% quarterly return
A deferred balance grows from 50,000 to 86,000 in 9 years - what rate?
What rate turns 6,000 a year for 15 years into 150,000?

Annualizing a Holding-Period Return

An annualized return is the constant yearly rate that reproduces an observed gain. From a beginning and an ending value over nn years:

rann=(VendVbegin)1n−1r_{\text{ann}} = \left(\frac{V_{\text{end}}}{V_{\text{begin}}}\right)^{\frac{1}{n}} - 1

  • VbeginV_{\text{begin}}, VendV_{\text{end}} — the values at the two dates
  • nn — elapsed years, as a decimal if the period is not whole

To go the other way, from a return rpr_p earned once per period with mm periods a year:

rann=(1+rp)m−1r_{\text{ann}} = (1 + r_p)^m - 1

This is compounding, not multiplication: a 1.8% quarterly return is not 7.2% a year, because each quarter's gain earns in the quarters that follow. The same relation converts back down, rp=(1+rann)1/m−1r_p = (1 + r_{\text{ann}})^{1/m} - 1, which is how a stated annual rate becomes the periodic rate an annuity schedule actually uses.

The Rate Inside an Annuity

When money goes in gradually, no closed form exists for the rate. A stream of nn level deposits CC accumulating to FVFV satisfies

FVC=(1+r)n−1r\frac{FV}{C} = \frac{(1+r)^n - 1}{r}

The right-hand side is the future-value annuity factor. It rises steadily with rr, so you solve by trial and interpolation: evaluate the factor at two rates that bracket FV/CFV/C, then interpolate linearly between them and refine. Three iterations are usually enough for four significant figures.

Two related quantities:

interest earned=FV−nC,PV=C⋅1−(1+r)−nr\text{interest earned} = FV - nC, \qquad PV = C \cdot \frac{1 - (1+r)^{-n}}{r}

The second is what a stream of future payments is worth today at rate rr — the calculation behind pricing any fixed payment stream. Contract terms, crediting methods and fees differ by product and jurisdiction; this page computes the arithmetic on whatever rate and payments you enter.

Common Mistakes to Avoid

  • Dividing the total gain by the number of years: a 62.5% gain over 4 years is not 15.6% a year. Take the 4th root of the growth factor.
  • Multiplying a periodic return by the number of periods: use (1+rp)m−1(1+r_p)^m - 1.
  • Ignoring deposits and withdrawals: a start-and-end comparison silently credits contributions as if they were returns. With cash flows, you need the annuity or IRR calculation.
  • Annualizing a very short period: raising a one-week return to the 52nd power magnifies noise into an implausible figure.
  • Mismatching rr and nn in the annuity factor: monthly deposits need a monthly rate and a count of months.
  • Confusing a stated annual rate with an effective one: they differ whenever compounding happens more than once a year.

示例题目

Step 1: Growth factor: 19500/12000=1.62519500/12000 = 1.625
Step 2: r=1.6251/4−1r = 1.625^{1/4} - 1; ln⁡1.625≈0.4855078\ln 1.625 \approx 0.4855078, divided by 4 gives 0.12137700.1213770
Step 3: e0.1213770≈1.1290504e^{0.1213770} \approx 1.1290504
Step 4: r≈0.1290504r \approx 0.1290504
Step 5: Check: 12000×(1.1290504)4≈19,50012000 \times (1.1290504)^4 \approx 19{,}500
Answer: ≈12.905%\approx 12.905\% a year

Step 1: m=4m = 4 periods a year, rp=0.018r_p = 0.018
Step 2: (1.018)2=1.036324(1.018)^2 = 1.036324
Step 3: (1.018)4=(1.036324)2≈1.0739674(1.018)^4 = (1.036324)^2 \approx 1.0739674
Step 4: rann≈0.0739674r_{\text{ann}} \approx 0.0739674
Step 5: Simply multiplying would have given 4×1.8%=7.2%4 \times 1.8\% = 7.2\%, understating it by about 0.2 points
Answer: ≈7.397%\approx 7.397\% a year

Step 1: Required factor: FV/C=150000/6000=25FV/C = 150000/6000 = 25
Step 2: Try r=6.9%r = 6.9\%: (1.069)15≈2.7203632(1.069)^{15} \approx 2.7203632, so factor =1.7203632/0.069≈24.9363= 1.7203632/0.069 \approx 24.9363
Step 3: Try r=7.0%r = 7.0\%: (1.07)15≈2.7590315(1.07)^{15} \approx 2.7590315, so factor =1.7590315/0.07≈25.1290= 1.7590315/0.07 \approx 25.1290
Step 4: Interpolate: 6.9%+0.1%×(25−24.9363)/(25.1290−24.9363)≈6.9%+0.0331%6.9\% + 0.1\% \times (25 - 24.9363)/(25.1290 - 24.9363) \approx 6.9\% + 0.0331\%
Step 5: r≈6.933%r \approx 6.933\%; deposits total 15×6000=90,00015 \times 6000 = 90{,}000, so \60{,}000$ is interest
Answer: r≈6.93%r \approx 6.93\% a year

常见问题

Divide the ending value by the beginning value, raise the result to the power 1/n where n is the number of years, then subtract 1. For periods shorter than a year, express n as a decimal — 9 months is n = 0.75.

Because each month's gain earns a return in the months that follow. The correct conversion is (1 + monthly)^12 − 1. At 1% a month that is 12.68%, not 12%, and the gap grows with the size of the periodic return.

Set FV/C equal to the annuity factor ((1+r)^n − 1)/r and solve for r numerically — the equation has no closed-form solution. Evaluate the factor at two bracketing rates, interpolate, and repeat until the factor matches to the precision you need.

No. The simple start-and-end formula treats every dollar of growth as investment return, so contributions inflate it and withdrawals deflate it. When cash flows occur mid-period you need an internal rate of return, which discounts each flow on its own date.

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