IRR vs NPV: Verified Examples of Scale, Timing and Multiple Roots

Compare IRR and NPV using complete cash-flow schedules. Reproduce scale and timing conflicts, two valid IRRs, and the conditions behind the usual decision rule.

NPV gives a monetary value at a specified discount rate. IRR finds a rate that makes the same schedule's NPV zero. An IRR percentage does not show investment size, and a schedule can have more than one IRR.

The IRR & XIRR calculator reports NPV and detected rates for the same annual or dated inputs. This guide supplies complete hypothetical schedules so you can reproduce where the metrics agree and where rankings differ. Amounts below are in dollars and periods are years unless stated otherwise.

Start with the same amounts, dates and valuation point#

NPV(r) = CF0 + CF1/(1+r) + CF2/(1+r)^2 + ... + CFn/(1+r)^n
IRR is a rate r for which NPV(r) = 0.

An initial payment belongs at time zero. In Excel, if that payment is in B2 and the later annual flows are in B3:B7, use =NPV(rate,B3:B7)+B2. Putting B2 inside the NPV range discounts it one extra period. Microsoft's NPV documentation describes this timing convention.

A forecast IRR describes forecast cash flows; it does not establish what the investment will actually deliver. For cash paid on irregular dates, use XIRR and XNPV with the same dates rather than treating each row as a year.

When the usual accept-or-reject rule works#

For an initial negative investment followed only by nonnegative future receipts, with at least one positive receipt, NPV decreases as the discount rate increases within the domain above -100%. There is one IRR in that mathematical domain. Therefore:

For example, [-1000,1150] has a 15% annual IRR. At a 10% discount rate:

NPV = -1000 + 1150/1.10 = 45.454545...

At 20%, NPV is -$41.67. The agreement is conditional on this investment-shaped schedule; a loan-shaped schedule starting with an inflow reverses the direction, and later negative cash flows can make the relationship more complicated. A finite software search can also fail to find a mathematically existing rate outside its search range.

Scale: a higher IRR can accompany a lower NPV#

These are two one-year investments with no interim payments:

The dollar profits are not the NPVs. Discounting the final receipts makes the distinction visible:

A: -50000 + 62500/1.10 = 6818.181818...
B: -500000 + 570000/1.10 = 18181.818182...

A has the higher rate; B has the higher NPV at the stated required return. If these are mutually exclusive, both feasible, and the unused capital can earn the required return, B adds more present value under this model.

Capacity changes the decision. If A can be repeated ten times with identical economics, the combined A schedule uses $500,000 and has $68,181.82 NPV. You must compare feasible uses of the available capital, including repeatability and remaining funds, rather than mechanically selecting either the larger NPV or the larger percentage from a mismatched opportunity set.

Timing: even equal initial investments can rank differently#

Both projects below require $100 immediately. Project A distributes $130 after one year. Project B distributes $160 after two years. Year two is the common comparison horizon.

A IRR = 130/100 - 1 = 30%
B IRR = (160/100)^(1/2) - 1 = 26.49110641%

A NPV at 10% = -100 + 130/1.10 = 18.181818...
B NPV at 10% = -100 + 160/1.10^2 = 32.231405...

B has the higher NPV at 10% despite its lower IRR. At a different required return the ranking can change: both NPVs are equal when 130/(1+r) = 160/(1+r)^2, giving a crossover rate of approximately 23.0769%. That is a statement about these two schedules, not a general discount rate to use.

The zero in A's year-two row preserves the common horizon. Its IRR remains 30%; adding an empty period after the final receipt does not turn the receipt into money reinvested for another year.

Multiple roots: the highest reported IRR is not a decision rule#

The schedule [-100,230,-132] includes a later obligation after an early receipt. Multiplying the zero-NPV equation by (1+r)^2 gives:

-100*(1+r)^2 + 230*(1+r) - 132 = 0
-100*((1+r)-1.1)*((1+r)-1.2) = 0

Both 10% and 20% are valid IRRs. At a 15% required return, NPV is approximately +$0.1890. Comparing that required return only with the 10% root would suggest a different verdict from comparing it only with the 20% root.

The workspace includes this example, reports the detected roots, and states that its numerical search is finite. It may miss other roots in more complicated schedules. Multiple sign changes permit multiple roots without proving how many exist. Evaluate NPV at a justified rate and review the timing and later funding obligation instead of choosing the most favorable percentage.

Reinvestment and final wealth need their own assumptions#

Neither the zero-NPV equation nor the NPV sum explicitly sends distributions into a new investment account. The discount rate expresses how the model values money at different times. Interpreting an IRR as the compound growth of all proceeds through the final date adds an assumption about how interim distributions are used.

For Project A in the timing example, the final wealth at year two is $130 if the year-one receipt is held without interest. It is $143 if that receipt earns 10% for the remaining year. Those are distinct reinvestment scenarios applied to the same original project, whose IRR is still 30%.

MIRR explicitly combines the discounted negative flows with positive flows compounded to a stated final period. Its finance and reinvestment rates must match that period. It is a different measure under specified assumptions, and can be higher or lower than IRR. For a full numerical check, see the Excel IRR and MIRR example.

Keep the comparison auditable#

A useful side-by-side investment comparison includes the complete schedule, timing convention, required return, NPV, detected IRR or IRRs, and total contributions and receipts. Distinguish forecasts from realized amounts, and keep financing and tax treatment consistent across alternatives.

For a conventional investment, IRR is a compact rate description. For selecting among feasible alternatives, NPV helps compare present value at a consistent required return. Both depend on the cash-flow assumptions; neither validates those assumptions or accounts for every business constraint.

The verified IRR examples include CSV schedules you can use to practice checking both outputs together.