How the profitability index works

The profitability index asks how many dollars of present value you get per dollar outlaid:

$$PI = \frac{\text{PV of inflows}}{\text{Outlay}}$$

A machine that costs $10,000 and pays $3,000 at the end of each of five years has inflows worth $11,978.13 today at 8 percent. PI is $11978.13 / 10000 = 1.1978$. NPV is the surplus, $1,978.13, which is $1.1978 - 1$ times the $10,000.

For a conventional project, PI above 1 is NPV above 0. They cannot disagree on accept or reject. The NPV calculator on this page is the present values. This page is the division.

Payback against NPV is a different clock. Profitability index against NPV is the ratio against the surplus.

At 12 percent the same five receipts are worth $10,814.33 today. PI is $10814.33 / 10000 = 1.0814$. NPV is $814.33. Four points on the rate cut the surplus by more than half and pulled the index toward 1, because every future receipt shrank and the outlay did not.

Push the rate to the IRR of 15.24 percent and NPV is 0, so PI is 1. Past that, both flip. The crossing is one fact.

A second $10,000 project pays $1,000, then $2,000, $3,000, $4,000, $5,000. Same total cash, later. At 8 percent the inflows are worth $11,365.14. PI is 1.1365. NPV is $1,365.14, below the level machine's $1,978.13.

The index ranks them the same way NPV does when the outlay is the same size. That is the usual classroom case. When outlays differ, PI ranks value per dollar outlaid and NPV ranks dollars. A smaller project can win on PI and lose on NPV.

Scale every cash flow on the machine by two. NPV doubles. PI does not move, because both the inflows and the outlay doubled. A rate is silent on size. So is this ratio. When projects compete for a limited outlay, PI is the ranking some capital-rationing rules use. When they do not, NPV in money is the ranking NPV against IRR already argued for.

This page will not pick the rule. It will only divide the present values it is given.

The $10,000 is not discounted. Discounting it would shrink the denominator, inflate PI, and describe a different series. Time value of money is why the receipts are discounted and the outlay is not: the outlay is already today.

It is not a capital-rationing policy, not IRR, and not a claim that 1.1978 is a hurdle. The three sheets are PI 1.1978 at 8 percent (NPV $1,978.13, inflows $11,978.13), PI 1.0814 at 12 percent (NPV $814.33, inflows $10,814.33), and PI 1.1365 on the back-loaded series at 8 percent (NPV $1,365.14, inflows $11,365.14). This is educational material, not financial advice.