Present Value (PV)


Money now is more valuable than money later on.

Why? Because you can use money to make more money!

You could run a business, or buy something now and sell it later for more, or simply put the money in the bank to earn interest.

Example: Let us say you can get 10% interest on your money.

So $1,000 now could earn $1,000 x 10% = $100 in a year.

Your $1,000 now would become $1,100 in a year's time.

Present Value


So $1,000 now is the same as $1,100 next year (at 10% interest).

We say the Present Value of $1,100 next year is $1,000

Because we could turn $1,000 into $1,100 (if we could earn 10% interest).

Now let us extend this idea further into the future ...

How to Calculate Future Payments

Let us stay with 10% Interest. That means that money grows by 10% every year, like this:

So:

  • $1,100 next year is the same as $1,000 now.
  • And $1,210 in 2 years is the same as $1,000 now.
  • etc

In fact all those amounts are the same (considering when they occur and the 10% interest).

Easier Calculation

But instead of "adding 10%" to each year it is easier to multiply by 1.10 (explained at Compound Interest):

So we get this (same result as above):

Future Back to Now

And to see what money in the future is worth now, go backwards (dividing by 1.10 each year instead of multiplying):

Example: Sam promises you $500 next year, what is the Present Value?

To take a future payment backwards one year divide by 1.10

So $500 next year is $500 ÷ 1.10 = $454.55 now (to nearest cent).

The Present Value is $454.55

 

Example: Alex promises you $900 in 3 years, what is the Present Value?

To take a future payment backwards three years divide by 1.10 three times

So $900 in 3 years is:

$900 ÷ 1.10 ÷ 1.10 ÷ 1.10
$900 ÷ (1.10 × 1.10 × 1.10)
$900 ÷ 1.331
$676.18 now (to nearest cent).

Better With Exponents

But instead of $900 ÷ (1.10 × 1.10 × 1.10) it is better to use exponents (the exponent says how many times to use the number in a multiplication).

Example: (continued)

The Present Value of $900 in 3 years (in one go):

$900 ÷ 1.103 = $676.18 now (to nearest cent).

As a formula it is:

PV = FV / (1+r)n

  • PV is Present Value
  • FV is Future Value
  • r is the interest rate (as a decimal, so 0.10, not 10%)
  • n is the number of years

Example: (continued)

Use the formula to calculate Present Value of $900 in 3 years:

PV = FV / (1+r)n
PV = $900 / (1 + 0.10)3 = $900 / 1.103 = $676.18 (to nearest cent).

 

 

Exponents are easier to use, particularly with a calculator.

For example 1.106 is quicker than 1.10 × 1.10 × 1.10 × 1.10 × 1.10 × 1.10

Let us use the formula a little more:

Example: What is $570 next year worth now, at an interest Rate of 10% ?

PV = $570 / (1+0.10)1 = $570 / 1.10 = $518.18 (to nearest cent)

But your choice of interest rate can change things!

Example: What is $570 next year worth now, at an interest Rate of 15% ?

PV = $570 / (1+0.15)1 = $570 / 1.15 = $495.65 (to nearest cent)

Or what if you don't get the money for 3 years

Example: What is $570 in 3 years time worth now, at an interest Rate of 10% ?

PV = $570 / (1+0.10)3 = $570 / 1.331 = $428.25 (to nearest cent)

One last example:

Example: You are promised $800 in 10 years time. What is its Present Value at an interest rate of 6% ?

PV = $800 / (1+0.06)10 = $800 / 1.7908... = $446.72 (to nearest cent)

 

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