Explain present Value in mathematics

Present value is the value today of a future amount of money discounted at the rate of return expected.

Present value = Future value / (1 + r) ⁿ.

Explanation:

What would you prefer to receive $1,000 today or the same amount in a year?

You should prefer to receive the $1,000 today than in a year from now.

What would you prefer to receive $1,000 today or $1,200 in a year?

If you do not have an urgent need today, most likely you would be willing to receive $1,200 in a year because $1,200 is much more money than $1,200.

What if I ask to choose between $,1000 today and $1,050 in a year?

How would you tell what is, financially, better?

The present value is the amount of money today that is financially equal to an amount of in the future.

How can you mathematically calculate the present value?

Assume you are offered $1,000 in a year, how much money does it represent today?

That depends on the opportunities that you have to do with the money. Let's say that you can invest your money today in a business that gives you a 10% return in a year. This is, r = 10%.

To calculate the value in a year of an investment today, you would multiply your investment for 1 + 10% = 1 + 0.01 = 1.01. This is:

Value today Value in a year

$1,000 $1,000 * 1.10 = $1,100

That means that $1,100 in a year are equal to $1,000 in the present. Hence, the present value of $1,100 in a year is $1,000.

$1,000 = $1,100 / (1 + 0.10) = $1,100 / (1.10)

In general, the present value is the value of a future amount of money discounted at the rate of return expected.

Present value = Value in one year / (1 + r).

But it also depends on the number of years elapsed. If it is 2 years you have to discount it two times, this is divide the value in two years by (1+r) two times:

Present value = Value in two year / (1 + r) ².

For a number n of years, you obtain:

Present value = Value in the futre / (1 + r) ⁿ.