# What is the present value?

Computes the present value of a future amount, of level payments (an annuity), or of both, at a yearly discount rate, with payments at the end or the start of each period.

- Page: https://www.acalculator.org/finance/present-value-calculator
- JSON spec: https://www.acalculator.org/finance/present-value-calculator.json
- Version: 5cd076338c98

## Default answer

Example with the default inputs (Find the Present, Number of periods 10, Discount rate (yearly) 5%, Payment each period $0.00, Future value $10,000.00, Periods a year 1, Compounding a year 1, Payments at the End (END)): At 5% a year, $10,000.00 after 10 periods plus $0.00 each period is worth $6,139.13 today.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| solve | Find the | Which value to work out; type the others. |
| n | Number of periods | How many periods until the future amount, and how many payments. Years when P/Y is 1. |
| rate | Discount rate (yearly) | The yearly interest rate used to discount, in percent, compounded C/Y times a year. |
| pv | Present value | What the future amount and the payments are worth today. |
| pmt | Payment each period | A level amount received every period (an annuity). 0 for a single future amount. |
| fv | Future value | A single amount received after the last period. 0 for payments only. |
| py | Periods a year | How many periods (and payments) there are in a year: 1 for yearly, 12 for monthly. |
| cy | Compounding a year | How many times a year the rate compounds. Usually the same as the periods a year. |
| due | Payments at the | The end of each period (an ordinary annuity) or the start of each period (an annuity due). |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| n | Number of periods | How many periods until the future amount, and how many payments. Years when P/Y is 1. |
| rate | Discount rate (yearly) | The yearly interest rate used to discount, in percent, compounded C/Y times a year. |
| pv | Present value | What the future amount and the payments are worth today. |
| pmt | Payment each period | A level amount received every period (an annuity). 0 for a single future amount. |
| fv | Future value | A single amount received after the last period. 0 for payments only. |
| 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 of payments | N × the payment each period. |
| interest | Interest (the discount) | Everything received (N × payment + future value) minus the present value. |

## Method

Present value = (future value + payment × (1 + i × t) × ((1 + i)^N − 1) ÷ i) ÷ (1 + i)^N, with i the rate per period and t = 1 for payments at the start of each period, 0 at the end.

## 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 payments and the future amount are money received, and the present value is what they are worth today.
- No fees or taxes are included. This is an estimate for planning, not financial advice.

## Worked examples

1. solve = pv, n = 10, rate = 5%, pmt = $0.00, fv = $10,000.00, py = 1, cy = 1, due = end gives pv = $6,139.13, interest = $3,860.87. Source: The compound interest formula A = P(1 + r)^n solved for P (SEC Investor.gov, https://www.investor.gov/financial-tools-calculators/calculators/compound-interest-calculator).
2. solve = pv, n = 240, rate = 8%, pmt = $500.00, fv = $0.00, py = 12, cy = 12, due = end gives pv = $59,777.15, totalPmt = $120,000.00. Source: Microsoft Excel PV function example: ($59,777.15) (https://support.microsoft.com/en-us/office/pv-function-23879d31-0e02-4321-be01-da16e8168cbd).
3. solve = pv, n = 5, rate = 6%, pmt = $1,000.00, fv = $0.00, py = 1, cy = 1, due = begin gives pv = $4,465.11. Source: Present value of an annuity due, Broverman (2017), Mathematics of Investment and Credit, section 2.1.
4. solve = pv, n = 20, rate = 4%, pmt = $50.00, fv = $1,000.00, py = 2, cy = 2, due = end gives pv = $1,490.54. Source: Present value of an annuity plus a lump sum, Broverman (2017), Mathematics of Investment and Credit, chapter 2.

## FAQ

### What is present value?

Present value (PV) is what money you will receive later is worth today, given a rate of interest you could earn in the meantime. $10,000 received in 10 years is worth $6,139.13 today at 5% a year, because $6,139.13 invested at 5% grows to $10,000 in 10 years.

### What is the present value formula?

For a single future amount: PV = FV ÷ (1 + i)^N, where i is the rate per period and N the number of periods. For a level payment PMT at the end of each period: PV = PMT × (1 − (1 + i)^−N) ÷ i. For payments at the start of each period, multiply the payment part by (1 + i). This page adds the two parts.

### What discount rate should I use?

Use the return you could earn elsewhere on money of the same risk: a savings or CD rate for money that is sure to arrive, a higher rate for payments that might not. A higher discount rate gives a lower present value. This page does not look up any rate; you type it.

### What is the present value of an annuity?

It is today’s value of a series of equal payments. $500 a month for 20 years at 8% a year is worth $59,777.15 today, even though the payments add up to $120,000. Set the future value to 0 for payments only.

### What is the difference between an ordinary annuity and an annuity due?

In an ordinary annuity each payment comes at the end of its period; in an annuity due it comes at the start. Each payment of an annuity due arrives one period sooner, so its present value is (1 + i) times larger: $1,000 a year for 5 years at 6% is worth $4,212.36 at the end of each year and $4,465.11 at the start.

### How is present value used to price a bond?

A bond’s price is the present value of its coupons and its face value. A 10-year bond paying $50 twice a year plus $1,000 at the end, at 4% a year compounded twice a year, is worth $1,490.54 on a coupon date. For prices between coupon dates, with accrued interest and a day count, use the bond calculator.

## 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 PV function, with a worked example. https://support.microsoft.com/en-us/office/pv-function-23879d31-0e02-4321-be01-da16e8168cbd
- S. A. Broverman (2017), Mathematics of Investment and Credit, 7th edition, chapters 1 and 2: present value, annuities immediate and annuities due.