# Present Value (PV) Calculator

### What is a Present Value (PV)?

Present value (PV) is a financial calculation used when determining the time value of money to determine the “present day value” of a series of financial inputs.
It takes into consideration a value at a time in the future as well as the interest rate and payments occurring each compounding period.
As this calculator is structured to parallel the results of a financial calculator, inputs and outputs will be similar – for example, a negative present value (or payment) means an outflow as opposed to a directly negative number.

### How is the Present Value calculated?

Present value is calculated through a financial formula used to determine the time value of money.
There are two separate calculations involved: The base sum as well as the payment schedule.
For the base sum, the formula is:
$text{Present Value} = frac{text{Future Value}}{(1 + text{Rate of Return})^text{Periods}}$
For a payment annuity that occurs at the end of each period, the formula is:
$text{Present Value} = text{Payment} times (frac{1 - (1 + text{Rate of Return})^(-text{Periods)}}{text{Rate of Return}})$
For a payment annuity that occurs at the beginning of each period, the formula is:
$text{Present Value} = text{Payment} times (frac{(1 + text{Rate of Return})^text{Periods} - 1}{text{Rate of Return}}) times (1 + text{Rate of Return})$

### Example

We are looking to save \$2,000 over 12 years in an account that earns 2.2% interest per year. At the end of each year, we will pay \$100 into the account. How much would we have to put into the account at the start of the saving period to reach the \$2,000 goal?
Step 1: Calculate the present value of the fixed portion
$text{Present Value} = frac{2000}{(1 + 0.022)^12}$
$text{Present Value} = frac{2000}{(1.022)^12}$
$text{Present Value} = frac{2000}{1.29840670}$
$text{Present Value} = 1540.35$
Step 2: Calculate the present value of the payment annuity
$text{Present Value} = -100 times (frac{1 - (1 + 0.022)^-12}{0.022})$
$text{Present Value} = -100 times (frac{1 - (1.022)^-12}{0.022})$
$text{Present Value} = -100 times (frac{1 - 0.77017470}{0.022})$
$text{Present Value} = -100 times (frac{0.2298253}{0.022})$
$text{Present Value} = -100 times 10.44660455$
$text{Present Value} = -100 times 10.44660455$
$text{Present Value} = -1044.66$
Step 3: Add the two together
$text{Present Value} = 1540.35 + (-1044.66)$
$text{Present Value} = 495.69$
The present value of the account would be 495.69. The result would be converted to negative (-495.69) as this would be the amount that would need to be paid into (cash flow out) the account to reach the balance of 495.69.

### What is Future Value?

Future Value (FV) is the total value at the end of the time period.

### What are periods?

Periods are the number of times that compounding (and payments) take place.

### What is the rate?

The rate is the amount of interest earned per compounding period.

### What is Payment (PMT)?

A payment is an amount either deposited or withdrawn at each compounding period. A negative number designates an amount that is deposited, while a positive one withdrawn. For example, if \$100 is deposited each compounding period, it would be entered as '-100', while if \$75 was payed out each compounding period, it would be entered as '75'.

### What is Payments at start or end of period?

A payment at the beginning of a period would mean that the payment (or deposit) occurs at the beginning of each period. A payment at the end of a period would mean that the payment (or deposit) occurs at the end of each period.