Alternatively, we can say that the $100 is a present value of the $105.

We can ask the investment question a little differently.

If you want $100 in your bank account after 1 year and

the bank is paying an interest of 5% per year, how much should you invest today?

For now, I'll give you the answer.

It is $95.24.

We'll get to the actual calculation later.

If you invest $95.24 today and the bank pays you 5% interest,

you will earn 95.24 times 0.05 which is equal $4.76 in interest.

That combine with their initial investment of $95.24,

will give you 95.24 plus 4.76 which equals $100.

Here again, we say that the present value of $100 is $95.24.

Or conversely, the future value of $95.24 after one year is $100.

Time value of money is one of the most important concepts of finance.

Almost all calculations in finance are based on it.

It is also referred to as the discounted cash flow methodology, in short DCF,

because we are discounting a future cash flow to the present using a discount rate.

In our example, we discounted the $100 at 5% a year back to

today which yielded a present value of $95.24.

What if the bank offers you a higher interest rate of 10% a year?

How much must you invest today to have $100 after a year?

The answer is $90.91, an interest rate of 10% will

give you $90.91 times 0.01, which equals $90.91 in interest.

Which when added to your initial investment of $90.91

will give you 90.91 plus 9.09 equals $100,

as you can see the present value is lower when the interest rate is higher.

This makes sense because you're earning more through interest and hence,

you can invest a smaller amount today.

Next, we need a formula to calculate the present value given a future value or

vice versa.

Notice that I said that the present value of $100 when

interest rate is 10% is 90.91.

But I didn't tell you how I arrived that $90.91.

Let's be with the equation we wrote earlier,

90.91 plus 9.09 equals $100 $90.91 is the present value.

$9.09 is the interest earned, and $100 is the future value.

Let's denote present value as PV, future value as FV, and interest rate as r.

Interest earned, 9.09, is equal to

90.91 times 0.10, which is equal to PV times r.

So we can rewrite a first equation as as follows,

90.91 + 9.09 = 100 in other words PV + PV*r= FV,

and we can simplify that to PV ( l + r ) = FV which is borrow one investment.

What if you make a two year investment?

You invest $90.91 for two years at 10% a year, after one year you will have $100.

The $100 will continue to be in the bank account and

own an additional 10% in the second year.

The interest on in the second year is 100 times 0.10 which equals $10.

Add that to the investment of $100 at the start of the second year, and

your bank account will have $110 after two years.