We learned about compounding, and we learned about discounting. But each of our lessons only involved a one-year time period (T = 1). In this lesson we’ll take a look at compounding and discounting over multiple years.

The **future value** **(FV)** of an investment today is simply the value our investment will attain over a certain number of years, given an interest rate.

If I invest $1,000 today at a 5% interest rate. What will be the value next year?

$1,000 x 1.05 = $1,050.

Now what if I want to keep that money invested for another year? Simply take the value of the investment at the end of year 1, and multiply it again by (1 + r):

$1,050 x 1.05 = $1,102.50

In reality, we are multiplying our original investment by 1.05, twice:

$1,000 x 1.05 x 1.05

There is a cleaner way to write this, however:

$1,000 x (1.05)^2 … that is 1.05 raised to the second power.

This is the formula for future value: Present Value (PV) x (1 + r)^T, where T is the number of years we want our investment to compound.

*A fun exercise, pretend that grandma and grandpa deposited $10,000 in an account for you when you were born. They set it up so that you could take control of the funds after college, when you are 22 years old. How much would that $10,000 investment be worth if it were invested at 7%?*

The **present value** **(PV) **of an investment is today’s value of some future payout. We approach discounting in the same manner as compounding over multiple years in that we raise our (1 + r) to the power T (where T is the number of years we want to compound). To find our present value, we first need to know the future value and the number of years we’ll be invested. Remember to divide by (1 + r) to discount!

Example: You just had a baby boy named John. In 18 years you plan to make a lump sum payment to Stanford so that John can go to college. This lump sum will be $300,000. What is the present value of that $300,000, given a 7% interest rate?

We know that…

- T = 18
- r = 7%
- FV = $300,000

This is all we need! Remember our PV formula: FV / (1+r)^T

$300,000 / (1.07)^18 =

$300,000 / 3.379 = $88,759.17

This amount ($88,759.17) is what we would need to invest today, at a 7% interest rate, for 18 years, if we wanted to make a lump sum payment of $300,000 when John goes to college.

Remember our two formulae:

**FV = PV x (1+r)^T**

**PV = FV / (1+r)^T**