Three criteria complete the toolkit, and each is useful for a stated purpose and wrong for the obvious one.
Payback period is the time until cumulative undiscounted cash flow turns positive. It ignores the time value of money entirely and it ignores every cash flow after the cutoff. Both defects can be demonstrated in one pair of projects. Project one costs $100,000 and returns $50,000 in each of two years: payback is exactly 2.0000 years and the net present value at 10% is -$13,223.1405. Project two costs the same $100,000 and returns $20,000 a year for twelve years: payback is 5.0000 years and the net present value is +$36,273.8365.
A three-year payback rule accepts the first and rejects the second, which is both possible errors from one policy on one afternoon.
Discounted payback fixes the first defect and not the second. Discount each flow before accumulating, and project one never pays back at all, so the rule now rejects it correctly. Project two's discounted payback is 7.2821 years, so a three-year rule still rejects a project worth $36,274. Truncation is the deeper defect and discounting does not touch it.
Both measures do answer a real question, and it is a liquidity question rather than a value question. How long is capital at risk? How exposed is the decision to a forecast that has to hold for many years? Used as a constraint alongside net present value, payback is defensible.
Used as the criterion, it is a policy for rejecting long-lived assets.
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What is the profitability index?
The profitability index is the present value of inflows divided by the initial outlay, so PI = PV(inflows) / |CF0| and PI > 1 is exactly equivalent to NPV > 0. Its content is per-rupee-invested value, which makes it the natural ranking device under a hard budget, subject to a limit stated in a moment, and the wrong device for choosing between mutually exclusive projects of different size. A project with a profitability index of 1.36 on a ₹1 crore outlay creates ₹36 lakh; one with a profitability index of 1.20 on ₹10 crore creates ₹2 crore. If you can only do one, take the second.
The profitability index also gets miscomputed in a specific way often enough to be worth naming: some analysts divide the net present value by the outlay rather than the present value of inflows. That is the profitability index minus one, which ranks projects identically and fails the "greater than one" test everyone quotes. Say which definition you used.