# The value of a dollar.

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