CFA Level I Exam · Capital Investments and Capital Allocation

NPV and IRR Investment Decision Rules Explained

Updated 7 October 2026 · Fact-checked

NPV is the present value of a project's cash flows, discounted at the required return, minus the initial outlay. IRR is the discount rate that makes NPV zero. For a standalone, conventional project, accept if NPV > 0 or IRR > required return. Both give the same decision. Use a calculator's cash flow worksheet.

Understand NPV and IRR Investment Decision Rules

A company invests money today to get cash flows later. A rupee, dollar or euro received later is worth less than one received today, so you must discount future cash flows before comparing them with the outlay. The discount rate is the project's required rate of return, usually the cost of capital adjusted for the project's risk.

Net present value (NPV) adds up the present values of all cash flows, including the initial outlay, which is negative. A positive NPV means the project earns more than the required return. The NPV is the amount by which the project is expected to increase the firm's value. This is why NPV links directly to shareholder wealth: accepting a project with NPV of $2 million should raise the value of the firm's equity by about $2 million, if the cash flows and discount rate are right.

The internal rate of return (IRR) is the discount rate at which NPV equals zero. It is the project's own break-even return. You cannot solve for it with algebra when there are more than two cash flows, so you use a calculator or trial and error. If IRR is higher than the required return, the project earns more than it needs to, and the NPV at the required return is positive.

The NPV profile is a graph of NPV against the discount rate. It slopes down for a conventional project. It cuts the horizontal axis at the IRR. When you compare two mutually exclusive projects, their profiles may cross. The rate where they cross is the crossover rate, which is the discount rate at which both projects have the same NPV. Below it, one project has the higher NPV. Above it, the other does. In such cases, NPV and IRR can rank the projects differently. When they conflict, NPV is the preferred criterion because it measures the value added and assumes reinvestment at the required return, which is more realistic than reinvestment at the IRR.

Key formulas to remember

Net present value: NPV = CF₀ + Σ [CFₜ ÷ (1 + r)ᵗ], t = 1 to n; CF₀ is the initial outlay and is negative. r is the required rate of return.
Internal rate of return: 0 = CF₀ + Σ [CFₜ ÷ (1 + IRR)ᵗ]; IRR is the rate that sets NPV to zero. Solve it on the calculator.
NPV decision rule: Accept if NPV > 0; reject if NPV < 0; For independent projects, accept every project with positive NPV. For mutually exclusive projects, choose the highest positive NPV.
IRR decision rule: Accept if IRR > required return; reject if IRR < required return; Gives the same answer as NPV for a standalone project with conventional cash flows.
Crossover rate: Crossover rate = IRR of the difference in cash flows (Project A − Project B); At this rate both projects have equal NPV. Subtract so the first cash flow is negative.
Conventional cash flow pattern: One outflow first, then only inflows; Non-conventional patterns, with more than one sign change, can give multiple IRRs or no IRR.

How to solve NPV and IRR Investment Decision Rules questions

Use the same routine for any NPV or IRR question. It keeps you from mixing up signs and timing.

1Draw a timeline. Mark time 0 and each later period, and write the cash flow below each point.
2Check the sign of each cash flow. The initial outlay is negative, and later inflows are positive unless the question says otherwise.
3Identify what is asked: NPV, IRR, a decision, or a comparison of two projects.
4For NPV, discount each cash flow at the required return and add them, including CF₀. For IRR, enter the cash flows in the calculator and compute IRR.
5Apply the rule. NPV > 0 or IRR > required return means accept. If projects are mutually exclusive, pick the higher NPV.
6For a ranking conflict, find the crossover rate from the difference in cash flows. Compare it with the required return to see which project has the higher NPV.
7Check the answer: a positive NPV should go with an IRR above the required return, and a negative NPV with an IRR below it.

Quickest way: Calculator cash flow worksheet (TI BA II Plus)

When to use it: Use this whenever there are uneven cash flows over three or more periods. It is faster and safer than discounting each flow by hand.

Press CF, then 2nd CLR WORK to clear old data.
Enter CF0 as a negative number, press ENTER, then press the down arrow.
For each later cash flow, key the amount into C01, C02 and so on, press ENTER, then the down arrow. If the same amount repeats, type the count into F01, F02 and so on, then ENTER.
For NPV, press NPV, key the required return as a percent into I, press ENTER, press the down arrow, then CPT.
For IRR, press IRR then CPT.
On the HP 12C, use g CF₀ for the outlay and g CFⱼ for the later flows. For NPV, store the required return as a percent with the i key (for example, 10 then i), then press f NPV. For IRR, press f IRR.

Common mistakes in NPV and IRR Investment Decision Rules

Forgetting to subtract the initial outlay and reporting the present value of inflows as the NPV.
You discount the later cash flows and stop, because the outlay at time 0 needs no discounting.
Fix: Always include CF₀ with its negative sign. NPV = PV of inflows − outlay. Check that your answer is plausible against the size of the outlay.
Entering the outlay as a positive number in the calculator.
You type the amount as stated in the question, and it is usually written without a sign.
Fix: Press the +/− key when you enter CF0. If IRR returns an error or a strange number, check your signs first.
Choosing the project with the higher IRR when two mutually exclusive projects conflict.
IRR is a percentage and looks like a clean ranking. It ignores project scale and timing.
Fix: For mutually exclusive projects, pick the higher NPV at the required return. NPV measures the value added to shareholders.
Entering the discount rate as a decimal such as 0.10 in the NPV function.
Formulas use decimals, but the BA II Plus expects a percent.
Fix: Type 10, not 0.10, for 10%. Check that the result is in a sensible range.
Computing the crossover rate from the individual IRRs.
You assume the crossover is the average, or one of the IRRs.
Fix: Subtract one project's cash flows from the other's period by period, then find the IRR of the difference. Put the first non-zero cash flow as negative.
Assuming a project with a positive NPV always has a single IRR.
Conventional projects do have one IRR, so you carry that idea to every case.
Fix: Count the sign changes in the cash flows. More than one sign change can give multiple IRRs or none, and then NPV is the reliable guide.

Worked examples

Example 1

A project requires an initial outlay of $1,000,000 and is expected to produce cash flows of $400,000, $450,000 and $500,000 at the end of years 1, 2 and 3. The required return is 10%. The NPV is closest to: A) $88,450, B) $111,195, C) $1,111,195.

Show the solution

Year 1: 400,000 ÷ 1.10 = 363,636.36.
Year 2: 450,000 ÷ 1.10² = 450,000 ÷ 1.21 = 371,900.83.
Year 3: 500,000 ÷ 1.10³ = 500,000 ÷ 1.331 = 375,657.40.
Sum of present values = 363,636.36 + 371,900.83 + 375,657.40 = 1,111,194.59.
NPV = 1,111,194.59 − 1,000,000 = 111,194.59, which rounds to $111,195.
Option C is the present value of inflows only, so it forgets the outlay. The NPV is positive, so the project should be accepted, and its IRR is above 10%.
Calculator check: CF0 = −1,000,000; C01 = 400,000; C02 = 450,000; C03 = 500,000; NPV with I = 10, then CPT.

Answer: B) $111,195. The NPV is positive, so accept the project.

Example 2

A project costs 100 today and returns 60 at the end of year 1 and 60 at the end of year 2. The required return is 12%. The IRR is closest to: A) 9.8%, B) 13.1%, C) 20.0%. What is the decision?

Show the solution

Set NPV to zero: 100 = 60 ÷ (1 + r) + 60 ÷ (1 + r)².
Let x = 1 ÷ (1 + r). Then 60x² + 60x − 100 = 0, which simplifies to 3x² + 3x − 5 = 0.
x = (−3 + √(9 + 60)) ÷ 6 = (−3 + 8.3066) ÷ 6 = 0.8844.
1 + r = 1 ÷ 0.8844 = 1.1307, so r = 13.07%, which is closest to 13.1%.
Check: 60 × 0.8844 = 53.07 and 60 × 0.7822 = 46.93. The total is 100.00.
Option C, 20%, is total profit ÷ outlay (20 ÷ 100), which ignores timing.
IRR of 13.07% is above the required return of 12%, so the NPV at 12% is positive and the project is accepted.
Calculator: CF0 = −100; C01 = 60 with F01 = 2; IRR then CPT.

Answer: B) 13.1%. IRR is above the 12% required return, so accept.

Exam tips

Questions often give a decision and ask which statement is correct. Think about the sign of NPV and the IRR against the required return first, and you can usually eliminate two options without calculating.
If two options are numerical and close, check for the usual traps: the outlay left out, a wrong sign, or the IRR confused with the required return.
When NPV and IRR rankings conflict for mutually exclusive projects, the answer is the higher NPV. Do not be tempted by a higher IRR.
Link NPV to shareholder wealth in theory questions: a positive NPV is the expected increase in firm value, and the NPV rule assumes reinvestment at the required return.
Practise the calculator worksheet until it is automatic. With about 90 seconds per question, a clean keystroke routine saves time and avoids errors.

Practice questions from Capital Investments and Capital Allocation

NPV and IRR Investment Decision Rules in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

NPV and IRR Investment Decision Rules: frequently asked questions

What is the difference between NPV and IRR?

NPV gives the value added in currency terms at a chosen discount rate. IRR gives the discount rate at which NPV is zero, as a percentage. For a standalone conventional project they give the same decision. For mutually exclusive projects they can disagree, and NPV is preferred.

How do I calculate IRR step by step on a financial calculator?

Clear the cash flow worksheet, enter CF0 as a negative number, then enter each later cash flow and its frequency. Press IRR and then CPT on the BA II Plus. On the HP 12C, enter the flows with g CF₀ and g CFⱼ, then press f IRR.

What is the crossover rate?

It is the discount rate at which two projects have the same NPV, which is where their NPV profiles cross. Find it by taking the difference in the projects' cash flows and computing the IRR of that difference. It matters when the projects are mutually exclusive and the rankings conflict.

Why is NPV preferred over IRR?

NPV measures the value created in currency terms and assumes cash flows are reinvested at the required return. IRR assumes reinvestment at the IRR itself, which can be unrealistic. IRR can also give multiple or no answers for non-conventional cash flows.