acalculator

What is my rate of return?

Enter what you invested, anything you added along the way, the final value and how long it took. See the yearly rate of return and the total return.

Your numbers

Find the
Optional: 0 when nothing was added.
Periods, compounding and timing of additions
Additions at the
Rate of return (yearly)
8.4472%

Growing $10,000.00 into $15,000.00 over 5 periods is a rate of return of 8.4472% a year.

Rate per period
8.447177%
Effective yearly rate
8.4472%
Total added
$0.00
Total gain
$5,000.00
Total return
50%

Rate of return (yearly): 8.4472%. Growing $10,000.00 into $15,000.00 over 5 periods is a rate of return of 8.4472% a year.

The results are estimates for information only. They are not financial, tax, or legal advice. Check the numbers with your lender or a qualified professional before you decide. Terms of use

How to calculate

Computes the yearly rate of return (the annualized return) that grows an initial investment, plus optional regular additions, into a final value, and the total return.

Example with the default inputs (Find the Return, Number of periods 5, Initial investment $10,000.00, Added each period $0.00, Final value $15,000.00, Periods a year 1, Compounding a year 1, Additions at the End (END)): Growing $10,000.00 into $15,000.00 over 5 periods is a rate of return of 8.4472% a year.

Method: Final value = investment × (1 + i)^N + addition × (1 + i × t) × ((1 + i)^N − 1) ÷ i, solved for the return per period i by search; with 1 period a year and no additions, rate = (final ÷ investment)^(1 ÷ N) − 1.

  • The interest rate is the same in every period, and every payment is the same size.
  • The yearly rate is nominal, compounded as many times a year as set; payments are made as many times a year as set.
  • The initial investment and the additions are money paid in; the final value is money received.
  • No fees or taxes are included. This is an estimate for planning, not financial advice.

Machine-readable copies: Markdown, JSON.

Worked examples

Each example is checked against the calculator on every build.

  1. Find the Return, Number of periods 5, Initial investment $10,000.00, Added each period $0.00, Final value $15,000.00, Periods a year 1, Compounding a year 1, Additions at the End (END) gives Rate of return (yearly) 8.447177%, Total gain $5,000.00, Total return 50%.Source: The compound interest formula A = P(1 + r)^n solved for r (SEC Investor.gov, https://www.investor.gov/financial-tools-calculators/calculators/compound-interest-calculator)
  2. Find the Return, Number of periods 12, Initial investment $1,000.00, Added each period $100.00, Final value $2,301.40, Periods a year 12, Compounding a year 12, Additions at the Start (BGN) gives Rate of return (yearly) 6%.Source: Microsoft Excel FV function, example 4: 6% a year gives $2,301.40 (https://support.microsoft.com/en-us/office/fv-function-2eef9f44-a084-4c61-bdd8-4fe4bb1b71b3), solved back for the rate
  3. Find the Return, Number of periods 10, Initial investment $1,000.00, Added each period $100.00, Final value $3,000.00, Periods a year 1, Compounding a year 1, Additions at the End (END) gives Rate of return (yearly) 5.519429%, Total added $1,000.00, Total gain $1,000.00, Total return 50%.Source: The future value of a sum and an annuity, Broverman (2017), Mathematics of Investment and Credit, chapter 2, solved for the rate

How the rate of return is worked out

Inputs. The initial investment I, an amount A added each period (0 when nothing was added), the final value V, the number of periods N, the periods a year P/Y and the compounding times a year C/Y (whole numbers from 1 to 365; both 1 by default, so N is in years and the rate is the annualized return), and whether additions come at the end or the start of each period. Amounts are positive; an empty addition or final value counts as 0. Find the picks the value to work out: the rate by default, or the final value, the investment, the addition or the number of periods.

The growth equation. With i the return per period (a decimal), g = (1 + i)^N, t = 0 for additions at the end and 1 at the start, and a = (1 + i × t) × (g − 1) ÷ i (a = N when i = 0):

V = I × g + A × a

This is the time value of money equation of a financial calculator, with the investment and the additions as money paid and the final value as money received.

Yearly rate. The rate is a nominal yearly rate compounded C/Y times a year: i = (1 + rate ÷ (100 × C/Y))^(C/Y ÷ P/Y) − 1. With P/Y = C/Y = 1, rate = 100 × i and, with no additions, rate = 100 × ((V ÷ I)^(1 ÷ N) − 1).

Finding the rate. The calculator checks every yearly rate above −100% and at most 1,000% where I × g + A × a − V changes sign: every 0.01% from −20% to 50%, every 0.1% from −99.9% to −20% and from 50% to 1,000%, and at −99.95%, −99.99%, −99.999%, −99.9999%, −99.99999% and −99.999999%, each refined by bisection. Money paid in and then received once changes sign once, so there is one rate.

The other values. V = I × g + A × a. I = (V − A × a) ÷ g. A = (V − I × g) ÷ a. N = ln(g) ÷ ln(1 + i) with g = (A × (1 + i × t) + V × i) ÷ (A × (1 + i × t) + I × i); at 0%, N = (V − I) ÷ A.

Other results.

  • Total added = N × A.
  • Total gain = V − I − N × A.
  • Total return = total gain ÷ (I + N × A) × 100, shown only when I + N × A is above 0.
  • Rate per period = 100 × i; effective yearly rate = 100 × ((1 + i)^(P/Y) − 1).

Precision and overflow. The rate search reads the sign of the equation from the form above when (1 + i)^N is at most 1, and from the same equation divided by (1 + i)^N when it is above 1, so no step overflows; a rate at which (1 + i)^N is not a finite number above 0 in 64-bit floating point is dropped. With the investment, the addition and the final value all 0 (or empty) the equation holds at every rate, so there is no rate. The other computed values use the same divided equation when (1 + i)^N is above 1, so an answer inside the limits never fails on a larger number on the way. The total added, the total gain and the total return are exact decimal arithmetic on the typed amounts and the computed value, each read as the decimal it prints as, rounded once at the end. The computed value, the rates and the powers are 64-bit floating point, so a value that is exactly half a cent in decimals can land just below it and show rounded down.

Limits and no answer. N is above 0 and at most 12,000; the rate is above −100% and at most 1,000%; amounts are from $0 to $1 trillion. A computed amount below $0 or over $1 trillion, or a computed N of 0 or less or over 12,000, is no answer. So is a (1 + i)^N that is not a finite number above 0 in 64-bit floating point, N at 0% with no addition, N when g ≤ 0 or g = 1, and a rate when none fits.

Display. Money shows to the cent, rounded half up from the value worked out; the rate and the effective rate at most 4 decimals, the rate per period 6, the total return 2; periods at most 2 decimals.

Assumptions

  • The return is the same every period; real investments go up and down, so this is the steady rate that gives the same result.
  • Every addition is the same size. No taxes or fees are included.
  • This is an estimate for planning, not financial advice.

Worked examples by hand

$10,000 to $15,000 in 5 years. Rate = 1.5^(1/5) − 1 = 8.4472% a year. The gain is $5,000, a total return of 5,000 ÷ 10,000 = 50%.

Excel's FV example 4, read backwards. $1,000 plus $100 at the start of each month for 12 months grew to $2,301.40. At 6% a year (0.5% a month): g = 1.005^12 = 1.061678, a = 1.005 × 0.061678 ÷ 0.005 = 12.397240, and 1,000 × 1.061678 + 100 × 12.397240 = 2,301.4018, which Excel shows rounded down to the cent as $2,301.40. Typed as 2,301.40, that final value needs a little less than 6%: the rate is 5.9999% a year (6% when the final value is typed in full, 2,301.4018303409).

Yearly additions. $1,000 plus $100 at the end of each year for 10 years, worth $3,000. At i = 5.519429%: g = 1.711293, a = 12.887072, and 1,000 × 1.711293 + 100 × 12.887072 = 3,000.00. So the rate is 5.5194% a year. The additions total $1,000, the gain is 3,000 − 1,000 − 1,000 = $1,000, and the total return is 1,000 ÷ 2,000 = 50%.

Other questions people ask

What is a rate of return?

A rate of return is the gain on an investment as a percent. The total return compares the gain with the money put in: $10,000 that became $15,000 is a 50% total return. The yearly (annualized) rate of return spreads that gain over the years with compounding: 50% over 5 years is 8.4472% a year, because 1.084472^5 = 1.5.

How do I calculate the annualized rate of return?

With no additions: (final value ÷ initial investment)^(1 ÷ years) − 1. For $10,000 to $15,000 in 5 years: 1.5^(1/5) − 1 = 8.4472%. With regular additions there is no formula: the calculator finds the rate at which the investment and every addition, each grown at that rate, add up to the final value.

Why is the yearly rate lower than the total return divided by the years?

Because returns compound: each year’s gain also earns a return in later years. 50% over 5 years is 10% a year without compounding, but only 8.4472% a year with it, since 8.4472% compounded five times makes 50%.

How do regular additions change the rate of return?

Money added later has less time to grow, so the calculator counts each addition only from the day it went in. $1,000 plus $100 a year for 10 years, worth $3,000 at the end, is a 50% total return on $2,000 but a 5.5194% yearly rate, higher than the 4.1% a year that 50% over 10 years would suggest, because most of the money was invested for less than 10 years.

Is this the same as CAGR or IRR?

With no additions, the rate of return here is the CAGR (compound annual growth rate). With equal additions each period it is the internal rate of return (IRR) of those cash flows. For uneven additions or withdrawals, use the IRR calculator.

Can the rate of return be negative?

Yes. If the final value is less than the money put in, the rate is negative: $10,000 that fell to $8,000 in 3 years is −7.1682% a year. The calculator looks for rates above −100% and at most 1,000% a year.