Investment Science: Sum of a Geometric series and applications in computing Net Present Value

Thursday, September 2, 2010

In this post, we will see two very simple, yet powerful shortcuts for calculating the Net Present Value (NPV) of a cash flow, though you may have learnt these formulas already, but they are very useful when finding the Net Present Value of a cash flow stream, especially for long cash flow streams spanning multiple periods.

1. Sum of a geometric series from 0 to n (n +1 terms):

Given a constant 'a' and a constant ratio 'r'


Example:

Consider a cash flow stream (40, 40, 40, 40) at an yearly interest rate 10%, the present value of this cash flow stream is given by


Through the above geometric series formula (with a = 40, r = 1/1.1 and n = 4), it's given by


2. Sum of a geometric series from 1 to n: 

Given a constant 'a' and a constant ratio 'r'


Example:

Consider a cash flow stream (-10, 40, 40, 40) at an yearly interest rate 10%, the present value of this cash flow stream is given by


Through the above geometric series formula, we get (add -10 to the sum of geometric series from 1 to n with a = 40, r = 1/1.1 and n = 3)


Thus, the sum of a geometric progression has a practical application when computing the Net Present Value of a cash flow stream.

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