NPV & IRR: practice problems with step-by-step solutions

Also called: net present value · internal rate of return

Free NPV & IRR practice problems with diagrams, hints and full worked solutions. New values every time, at three levels: equal cash flows, uneven cash flows, internal rate of return.

Free practice problemEasy
years0123−20 0009 0009 0009 000

A project costs €20 000 today and returns €9 000 at the end of each year for 3 years. The required rate of return is 12%. Find the net present value NPV.

r = 12%CF0 = −20 000CF1 = 9 000CF2 = 9 000CF3 = 9 000
€
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What you practise

  1. Easyequal cash flows
  2. Mediumuneven cash flows
  3. Hardinternal rate of return

How to solve NPV & IRR problems

  1. List all cash flows with their year; the investment at year 0 is negative.
  2. Discount each cash flow: CFt/(1 + r)t.
  3. Add them up to get the NPV. Positive means the project earns more than r.
  4. For the IRR, find the rate that makes NPV zero by trial or interpolation.

Common mistakes

  • Discounting the year-0 cash flow.
  • Forgetting the initial investment.
  • Adding cash flows without discounting.

Key formulas

Net present value
NPV = −C0 + Σ CFt/(1 + r)t
Equal cash flows
NPV = −C0 + CF·[1 − (1 + r)−n]/r
Internal rate of return
the r at which NPV = 0
Decision
accept when NPV > 0 (or IRR > required return)

Learn NPV & IRR from scratch

Before this: Time value of money

1Why money today is worth more

Would you rather have €100 today or €100 in a year? Today: you could put it in the bank and have more than €100 in a year.

So a euro in the future is worth less than a euro now. Businesses use this idea to decide whether an investment is worth making.

The net present value (NPV) answers one question: what are all the money flows of a project worth today, after paying for it?

  • NPV above zero: the project earns more than the required return. Accept it.
  • NPV below zero: the money would do better elsewhere. Reject it.
Self-check A project has an NPV of +€500. What does that mean?
  1. It earns more than the required rate of return
  2. It makes a loss of €500
  3. It returns exactly the required rate
  4. Its cash flows add up to €500

Answer: It earns more than the required rate of return. A positive NPV means the project adds value on top of the required return. An NPV of exactly zero would mean it earns just the required rate.

2Discounting one amount

Bringing a future amount back to today is called discounting. With a rate r per year, an amount received after t years is worth:

PV = CFt / (1 + r)t

Example: €1100 in one year at 10% is worth 1100/1.10 = €1000 today. Check: €1000 in the bank at 10% grows to €1100 in a year.

The further away the money, the bigger the power and the smaller its value today.

Always write the rate as a decimal: 8% is 0.08, so you divide by 1.08.

Self-check What is €121, received in 2 years, worth today at 10% per year?
  1. €100
  2. €110
  3. €146.41
  4. €96.80

Answer: €100. 121/1.10² = 121/1.21 = €100. Dividing by 1.10 only once gives €110, which forgets the second year.

3NPV step by step

The NPV is the sum of all discounted cash flows. The investment at year 0 is negative and is not discounted, because it is already today.

NPV = −I0 + CF1/(1 + r) + CF2/(1 + r)² + … + CFn/(1 + r)n
  1. Draw a timeline with every cash flow at its year. Money out is negative, money in is positive.
  2. Discount every future cash flow with its own power of (1 + r).
  3. Add all present values, including the −I0 at year 0.
  4. Decide: accept if the NPV is positive.

When the cash flow is the same every year, an annuity factor does all the discounting in one go:

PV = C · [1 − (1 + r)−n] / r

4A worked example

Follow every step. Try to predict the next one before you open it.

Worked example

A machine costs €9000 today and saves €3600 at the end of each year for 3 years. The required rate of return is 8%. Find the NPV. Should the company buy it?

1

Draw the timeline

0123−9000+3600+3600+3600t

One payment out now, three equal amounts in at the end of years 1, 2 and 3.

2

Discount each cash flow

3600 / 1.08 = 3333.33
3600 / 1.08² = 3600 / 1.1664 = 3086.42
3600 / 1.08³ = 3600 / 1.259712 = 2857.80
3

Add them up

The savings are worth 3333.33 + 3086.42 + 2857.80 = €9277.55 today.

NPV = −9000 + 9277.55 = €277.55
4

Check with the annuity factor

[1 − 1.08−3] / 0.08 = 2.5771
3600 · 2.5771 = 9277.55

Same result, in one line. Use this shortcut whenever the cash flows are equal.

5

Decide

The NPV is positive, so the machine earns more than 8% a year. The company should buy it.

Answer

NPV ≈ €277.55, which is positive, so accept the investment.

5The internal rate of return

The internal rate of return (IRR) is the rate at which the NPV is exactly zero. It is the project’s own yearly return.

There is no simple formula for it. You try rates until the NPV changes sign, then interpolate between the two.

In the example, the NPV is +277.55 at 8%. At 10% it is 3600 · 2.4869 − 9000 = −47.33. So the IRR lies between 8% and 10%.

IRR ≈ 8% + 2% · 277.55 / (277.55 + 47.33) ≈ 9.7%

Decision rule: accept when the IRR is higher than the required rate. Here 9.7% is above 8%, the same answer as the NPV gave.

Self-check A project’s NPV is positive at 8% and negative at 12%. Where is its IRR?
  1. Above 12%
  2. Below 8%
  3. Between 8% and 12%
  4. Exactly 10%

Answer: Between 8% and 12%. The IRR is where the NPV crosses zero, so it lies between the two rates where the sign changes. Interpolation gives a good estimate, but it is not automatically the midpoint.

6Common mistakes

  • Discounting the investment at year 0. It is already in today’s money.
  • Forgetting the investment altogether, and reporting only the present value of the inflows.
  • Adding cash flows without discounting them first.
  • Using the percentage instead of the decimal: dividing by 1.8 instead of 1.08.
  • Using the wrong power: a cash flow in year 3 is divided by (1 + r)³, not (1 + r).
Self-check A project costs €5000 and returns €5500 after one year. The required rate is 12%. What is the NPV?
  1. +€4910.71
  2. +€500
  3. −€500
  4. −€89.29

Answer: −€89.29. 5500/1.12 = 4910.71, and 4910.71 − 5000 = −€89.29. The €500 profit is not enough to reach a 12% return, so reject.

7How exam questions look

  • Easy: an investment and equal yearly cash flows. Use the annuity factor.
  • Medium: uneven cash flows. Discount each year on its own and add them.
  • Hard: find the IRR. Try two rates, find where the NPV changes sign, and interpolate.

Always state the decision in words: accept or reject, and why.

Learn it step by step in the appLearn it step by step in the app

Worked examples

Example 1Easy
years012345−80 00030 00020 00025 50031 00030 500

A project costs €80 000 today and returns €30 000, €20 000, €25 500, €31 000, €30 500 at the end of years 1 to 5. Assume cash flows arrive evenly through each year. Find the payback period PB in years.

CF0 = −80 000CF1 = 30 000CF2 = 20 000CF3 = 25 500CF4 = 31 000CF5 = 30 500

Find PB (years)

Show hints
  1. Add the cash flows year by year until the total reaches the initial cost.
  2. In the last year, only part of the cash flow is needed: take the fraction still missing.
  3. after year 3: €75 500.
Show the answer and the step-by-step solution

Payback period: the time until the cumulative cash flows repay the initial investment. Cash is assumed to come in evenly during each year, so the last part is a fraction of a year. The simple payback ignores the time value of money; the discounted payback period uses the discounted cash flows instead.

Cumulative cash flows

year 1: €30 000
year 2: €50 000
year 3: €75 500
year 4: €106 500

Fraction of the last year

still needed after year 3: 80 000 − 75 500 = 4 500
PB = 3 + 4 500/31 000 = 3.145

Answer

PB = 3.145 years
Example 2Medium
years01234−45 00011 00010 00013 00014 000

An investment of €45 000 gives these cash flows at the end of years 1 to 4: €11 000, €10 000, €13 000, €14 000. The cost of capital is 12%. Find the net present value NPV.

r = 12%CF0 = −45 000CF1 = 11 000CF2 = 10 000CF3 = 13 000CF4 = 14 000

Find NPV (€)

Show hints
  1. Bring each cash flow back to year 0 by dividing by (1 + r)ᵗ.
  2. The initial cost is already at year 0 and counts as negative.
  3. CF1/(1 + r) = 9 821.43.
Show the answer and the step-by-step solution

Net present value: discount every cash flow back to today with the required rate of return r, then add them up: NPV = Σ CFₜ/(1 + r)ᵗ. A project with NPV > 0 earns more than r and adds value. The internal rate of return (IRR) is the rate at which NPV = 0.

Present value of each cash flow

CF0 = −45 000
CF1/(1 + 0.12)1 = 11 000/1.12 = 9 821.43
CF2/(1 + 0.12)2 = 10 000/1.254 = 7 971.94
CF3/(1 + 0.12)3 = 13 000/1.405 = 9 253.14
CF4/(1 + 0.12)4 = 14 000/1.574 = 8 897.25

Add them up

NPV = €−9 056.24

Answer

NPV = €−9 056.24

NPV is negative, so the project earns less than the required return.

Example 3Hard
years0123−40 00015 50016 00027 500

A project costs €40 000 and returns €15 500, €16 000, €27 500 at the end of years 1 to 3. Find its internal rate of return IRR in percent.

CF0 = −40 000CF1 = 15 500CF2 = 16 000CF3 = 27 500

Find IRR (%)

Show hints
  1. IRR is the discount rate that makes NPV exactly zero.
  2. Try two rates, one giving NPV > 0 and one giving NPV < 0, then narrow down (or interpolate between them).
  3. NPV(15%) = 3 658.26, NPV(20%) = −57.87.
Show the answer and the step-by-step solution

Net present value: discount every cash flow back to today with the required rate of return r, then add them up: NPV = Σ CFₜ/(1 + r)ᵗ. A project with NPV > 0 earns more than r and adds value. The internal rate of return (IRR) is the rate at which NPV = 0.

Bracket the IRR

NPV falls as the rate rises, so the IRR lies between a rate with positive NPV and one with negative NPV.

NPV(15%) = 3 658.26
NPV(20%) = −57.87

Narrow down

Linear interpolation gives a first estimate; repeating it (or a calculator's IRR function) gives the exact rate.

IRR ≈ 15 + 5·3 658.26/(3 658.26 − (−57.87)) = 19.92%
exact: NPV(19.92%) = 0 → IRR = 19.92%

Answer

IRR = 19.92%

Questions and answers

What is NPV & IRR?

Would you rather have €100 today or €100 in a year? Today: you could put it in the bank and have more than €100 in a year.

So a euro in the future is worth less than a euro now. Businesses use this idea to decide whether an investment is worth making.

How do I solve NPV & IRR problems step by step?
  1. List all cash flows with their year; the investment at year 0 is negative.
  2. Discount each cash flow: CFt/(1 + r)t.
  3. Add them up to get the NPV. Positive means the project earns more than r.
  4. For the IRR, find the rate that makes NPV zero by trial or interpolation.
What are common mistakes in NPV & IRR?
  • Discounting the year-0 cash flow.
  • Forgetting the initial investment.
  • Adding cash flows without discounting.
What do the NPV & IRR exercises cover?
  • Easy: equal cash flows
  • Medium: uneven cash flows
  • Hard: internal rate of return

Every problem has new values, so you learn the method instead of an answer. Stuck? Open a small hint first, and the full explanation only when you need it.

How practising works

  • Every problem has new values, so you learn the method instead of an answer.
  • Stuck? Open a small hint first, and the full explanation only when you need it.
  • Your progress, mistakes and saved problems are kept, so you can review before the exam.

Topics before and after this one

Learn first

Time value of money

Calculators

NPV and IRR calculatorROI calculator (return on investment)Payback period calculator (simple and discounted)

Courses with this topic

Corporate FinanceInvestments and ValuationFE Civil exam practiceCFA Level I quantitative methods and fixed income practice

More topics in Business & economics

Time value of moneyBond pricing & yieldCAPM & WACCPrice elasticity of demandEconomics conceptsBreak-even analysisFinancial accountingDepreciationMarketing principles

Printable worksheet

NPV & IRR worksheet with answers
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