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.
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 !