Imagine I write a contract which will give you exactly 1 dollar on 1st January of each year forever. Now I put up this contract for sale. What is the value of this contract?

A 100$, a 1000$, a million dollars?

It turns out there is a rational answer to this question. But first lets dive into the concept of **time value of money**. What it means is that a dollar today is worth more than a dollar tomorrow due its growth potential when invested, risk and inflation.

For example a investing 100$ today at 5% annual interest would yield 105$ one year from now. Therefore 105$ a year from now is same as 100$ today at a 5% interest rate.

We can verify this with.

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

where FV = Future value

PV = Present value

r = rate of interest

n = no of periods

Coming back to our problem:

Compute the value of a contract that gives 1$ on the 1st of January of each year forever.

Assume a discount rate of 5%. [the return that could be earned per unit of time on an investment with similar risk]

To compute the value of a our contract we need to discount each payout to its present value and sum them up.

PV = FV/(1+r)^n

Present value of 1$ **1** year from now = 1/(1.05) = .95$

Present value of 1$ **2** years from now = 1/(1.05)^2 = .90$

Present value of 1$ **3** years from now = 1/(1.05)^3 = .86$

………..

Present value of 1$ **10** years from now =1/(1.05)^10 = .61$

………..

Present value of 1$ **100** years from now = 1/(1.05)^100 = $.007

……..

Note that as the no of years increase the present value of a future 1$ tends to 0.

Using the NPV (net present value) formula found in excel or summing all individual present values that were computed gives us an approximate valuation of 19.85$. Note that changing the discount rate would change the valuation.

To conclude given a discount rate of 5% the value of a eternal contract paying out 1$ every year is only 19.85$ today !

References:

https://en.wikipedia.org/wiki/Time_value_of_money

https://en.wikipedia.org/wiki/Net_present_value