How to Calculate Present Value

If you were the head of your state’s department of transportation, how would you choose between the following expenditures?

Repaving a road: A large, one time expenditure which eliminates smaller expenditures for a long time.
Filling potholes: A small, one time expenditure which eliminates the need for re-paving today, but the big expenditure is still required relatively soon.

Present value calculations are an essential part of many industries. Any project which represents a stream of cash flow that has a negative (spend) component and a positive (income) component where either or both of these component vary over time, a present value analysis should be used to justify the project.

From a program management perspective (i.e. management of multiple projects), present value should be determined for every project before a decision is made to pursue it. This ensures a comparison of apples to apples because of the sporadic nature of the cash flows.

From a project management perspective (i.e. within projects), the present value analysis is used to compare alternatives, for example the different types of equipment that could be purchased, what they would initially cost versus how much ore they will move.

A single payment

The present value, in today’s dollars, of one single future payment is:

P = (1 + i)^-n

Where:

P = Value in today’s dollars
i = interest rate (%)
n = number of compounding periods

The common notation to express this calculation is P = (P/F, i%, n). For example, if I was trying to communicate that a project will receive a $1,000 income in 2 years time, with a discount o 8%, I would write (P/F, 8%, 2) = $961.17.

Multiple payments

If there is more than one payment, as inevitably all projects have, it’s the same calculation but many times over. There are two variations on this, the uniform series and the uniform gradient.

Uniform Series

This is a series of payments of the same size and spacing. The notation is:

P = (P/A, i%, n)

The present value calculation is:

$P = \frac{(1 + i)^n - 1}{i(1 + i)^n}$

Uniform Gradient

This is a series of payments that are changing at a uniform rate. The notation is:

P = (P/G, i%, n)

The present value calculation is:

$P = \frac{(1 + i)^n - 1}{i^2(1 + i)^n} - \frac{n}{i(1 + i)^n}$

Example

Let’s say a project represents a series of cash flows that looks like this:

Year	Cash Inflows	Present Value (at 10% discount rate)
1	$1,000	$909
2	$2,000	$1,653
3	$2,000	$1,503
4	$5,000	$3,415
5	$2,000	$1,242
Present Value of cash inflows (Total)		$8,722
Less investment		$10,000
Net Present Value		($1,278)

In this hypothetical example, each of the cash flows discounted to the present will not recover the initial investment, and this investment should not be made.