# What is my rate of return?

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.

- Page: https://www.acalculator.org/finance/rate-of-return-calculator
- JSON spec: https://www.acalculator.org/finance/rate-of-return-calculator.json
- Version: be66f6110471

## Default answer

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.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| solve | Find the | Which value to work out; type the others. |
| n | Number of periods | How long the money was invested, in periods: years when the periods a year is 1. |
| rate | Rate of return (yearly) | The yearly return in percent, compounded C/Y times a year. With 1 period a year it is the annualized return. |
| pv | Initial investment | The amount invested at the start. |
| pmt | Added each period | A level amount added every period. 0 when nothing was added. |
| fv | Final value | What the investment is worth at the end. |
| py | Periods a year | How many periods (and additions) there are in a year: 1 for yearly, 12 for monthly. |
| cy | Compounding a year | How many times a year the return compounds. Usually the same as the periods a year. |
| due | Additions at the | The end of each period or the start of each period. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| n | Number of periods | How long the money was invested, in periods: years when the periods a year is 1. |
| rate | Rate of return (yearly) | The yearly return in percent, compounded C/Y times a year. With 1 period a year it is the annualized return. |
| pv | Initial investment | The amount invested at the start. |
| pmt | Added each period | A level amount added every period. 0 when nothing was added. |
| fv | Final value | What the investment is worth at the end. |
| periodRate | Rate per period | The interest rate for one payment period, i = (1 + I/Y ÷ C/Y)^(C/Y ÷ P/Y) − 1. |
| ear | Effective yearly rate | The rate over a whole year once interest compounds: (1 + i)^(P/Y) − 1. |
| totalPmt | Total added | The number of periods × the amount added each period. |
| interest | Total gain | The final value minus the initial investment and all the additions. |
| totalReturn | Total return | The total gain as a percent of all the money put in (the initial investment plus the additions). |

## 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.

## Assumptions

- 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.

## Worked examples

1. solve = rate, n = 5, pv = $10,000.00, pmt = $0.00, fv = $15,000.00, py = 1, cy = 1, due = end gives rate = 8.447177%, interest = $5,000.00, totalReturn = 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. solve = rate, n = 12, pv = $1,000.00, pmt = $100.00, fv = $2,301.40, py = 12, cy = 12, due = begin gives rate = 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. solve = rate, n = 10, pv = $1,000.00, pmt = $100.00, fv = $3,000.00, py = 1, cy = 1, due = end gives rate = 5.519429%, totalPmt = $1,000.00, interest = $1,000.00, totalReturn = 50%. Source: The future value of a sum and an annuity, Broverman (2017), Mathematics of Investment and Credit, chapter 2, solved for the rate.

## FAQ

### 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.

## Sources

- U.S. Securities and Exchange Commission, Investor.gov compound interest calculator and formula. https://www.investor.gov/financial-tools-calculators/calculators/compound-interest-calculator
- Microsoft, Excel FV function, with worked examples. https://support.microsoft.com/en-us/office/fv-function-2eef9f44-a084-4c61-bdd8-4fe4bb1b71b3
- S. A. Broverman (2017), Mathematics of Investment and Credit, 7th edition, chapters 1 and 2: rates of return and annuities.