I think you’re confusing NET Present Value with regular ol’ Present Value or future value. If your IRR is positive, then your PV of future cash flows will also be positive. Your FV will also be positive.
However, NET PV compares an initial cash outflow with future cash inflows. In other words, it NETS the difference. As such, with NPV, you take your PV of all future cash flows and SUBTRACT out today’s cost. This value can be positive or negative.
To illustrate, let’s keep it easy: Let’s say you have a project that costs $100 today, and our time horizon is only one year. In other words, we have a one-time cash outflow today of $100, and a one time cash inflow in one year, amount unknown.
IRR = Required Rate, NPV = zero
If our IRR is 10% and our required rate is also 10%, then our cash inflow (in one year) will be $110. ($100 x 1.1) Working the problem in reverse, if you take the PV of $110 at 10%, you get $100. ($110 / 1.1.) So as you can see, $110 in one year is worth $100 today. We paid in $100, and the discounted $110 is worth $100. The discounted $100 inflow NETTED AGAINST the $100 outflow equals zero. As such, our NET PV is zero.
IRR < Required Rate, NPV is negative
If, however, the cash inflow (in one year) is only $106, then your IRR is only 6%, which is less than the required return of 10%. If you discount $106 back for one year at 10%, you get $96.36. ($106 / 1.1 = 96.36.) You paid out $100, and got $106 back, so you did make some money. However, it’s less than the $110 that is required by the 10% discount rate. Our discounted inflow of $96.36 is NETTED AGAINST our outflow of $100, so our NET PV is -3.64. (+96.36 - 100.00 = -3.64)
IRR > Required, NPV is positive.
If our cash inflow in one year is $115, then our IRR is 15%, which is greater than the required return of 10%. $115 discounted back at 10% = $104.55. (115 / 1.1 = $104.55.) Our cash outflow was $100, our discounted cash inflow is $104.55. These two are NETTED AGAINST EACH OTHER, so our NET PV is +4.55. ($104.55 - 100 = 4.55)