Why is "Using accounting profit instead of net cash flow in the payback or NPV calculation." a common mistake in Investment appraisal?

Why it happens: Students confuse the three methods and assume 'return' always means profit, or mistakenly carry a profit figure (which already deducts depreciation) into a cash-based method. How to avoid it: Remember: payback and NPV use NET CASH FLOW; only ARR works from profit. Read the data label carefully before you start. Source: IB DP Business Management Guide (first assessment 2024)

Why is "Forgetting to subtract the initial investment when calculating NPV (stopping at the sum of present values)." a common mistake in Investment appraisal?

Why it happens: After discounting five years of cash flows, students treat the total present value as the answer and overlook the final subtraction. How to avoid it: The word is NET present value — always finish with 'sum of PVs − initial investment'. Add the cost as a Year-0 line (−$200,000) so you cannot forget it. Source: IB DP Business Management Guide (first assessment 2024)

Why is "Comparing payback or ARR with NPV without recognising that only NPV accounts for the time value of money." a common mistake in Investment appraisal?

Why it happens: Payback and ARR are simpler and calculated first, so students trust them equally and miss why NPV can give a different, better-founded verdict. How to avoid it: State explicitly that payback and ARR ignore the timing of cash flows, whereas NPV discounts them. Weight NPV most heavily when methods conflict. Source: IB DP Business Management Guide (first assessment 2024)

Why is "Recommending whether to invest on the basis of the numbers alone, with no qualitative factors." a common mistake in Investment appraisal?

Why it happens: The calculation feels like the whole task, so students stop once they have a positive NPV or a short payback. How to avoid it: For any AO3 recommendation, add at least two qualitative factors (finance available, objectives, risk, staff, ethics, the economy) and end with a substantiated, conditional judgement. Source: IB DP Business Management Guide (first assessment 2024)

Why is "Converting the part-year of the payback period incorrectly — e.g. writing 0.875 of a year as '8.75 months'." a common mistake in Investment appraisal?

Why it happens: Students treat the decimal fraction of a year as if it were already months, or forget to multiply by 12. How to avoid it: Multiply the fraction by 12: (amount still owed ÷ that year's cash flow) × 12. So 0.875 × 12 = 10.5 months, giving 2 years 10.5 months. Source: IB DP Business Management Guide (first assessment 2024)

Why is "Calculating ARR from the average annual cash flow instead of the average annual PROFIT (forgetting to subtract the initial cost first)." a common mistake in Investment appraisal?

Why it happens: Students average the cash inflows directly and divide by the investment, skipping the step that turns total return into total profit. How to avoid it: Follow the order: total net cash flow − initial investment = total profit; ÷ years = average annual profit; ÷ initial investment × 100 = ARR. Source: IB DP Business Management Guide (first assessment 2024)

IB DP Business Management (147712)

Investment appraisal

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Detailed Study Notes

Detailed notes on Finance and accounts for IB DP Business Management, covering key concepts, explanations, examples, and exam-focused revision points.

Investment appraisal — payback period, average rate of return (ARR) and net present value (NPV)

Investment appraisal is the set of quantitative techniques a business uses to judge whether a capital investment (a machine, a new outlet, a project) is financially worthwhile. This subtopic teaches you to calculate and interpret the three examined methods — payback period, average rate of return (ARR) and net present value (NPV) using given discount factors — to know the advantages and drawbacks of each, and — the top-band skill — to recommend WHETHER to invest by weighing the numbers alongside qualitative factors.

At a glance

Investment appraisal evaluates the profitability/desirability of a capital investment using forecast NET CASH FLOWS (not profit).
Payback period = how long the investment takes to repay its initial cost (shorter = better; measures speed/liquidity, not profit).
Average rate of return (ARR) = average annual profit as a % of the initial investment (higher = better; ignores timing).
Net present value (NPV) = today's value of all future net cash flows (discounted) minus the initial cost (positive NPV = accept).
NPV is the only method that accounts for the TIME VALUE of money via discount factors given in the exam.
Qualitative factors (finance available, objectives, risk, staff, ethics, economy) must sit alongside the numbers in any recommendation.
Command terms: Define/State (AO1, 2m), Calculate/Explain (AO2, 4–6m), Recommend/Evaluate/Discuss (AO3, ~10m). Paper 2 is heavily quantitative — a discount-factor table is provided.

What you’ll learn

Mapped to the IB DP Business Management subject guide (2024 onwards (first assessment May 2024)).

Explain the purpose of investment appraisal and distinguish quantitative from qualitative appraisal.
Calculate and interpret the payback period from even and uneven (cumulative) cash flows.
Calculate and interpret the average rate of return (ARR).
Calculate and interpret the net present value (NPV) using given discount factors, applying the time value of money.
Compare the advantages and limitations of payback, ARR and NPV.
Recommend, with justified judgement, whether a business should undertake an investment, integrating the numerical results with qualitative factors (HL: across multiple methods and multi-year scenarios).

What investment appraisal is — and why cash flow, not profit

Investment appraisal is the quantitative assessment of whether a capital investment is financially worthwhile, based on forecast net cash flows. The three examined methods are payback, ARR and NPV. Qualitative factors sit alongside the numbers.

Investment appraisal is the process of evaluating the attractiveness of an investment proposal — typically capital expenditure on a fixed (non-current) asset such as machinery, a vehicle fleet, a new outlet or a whole project — using quantitative techniques.

Managers appraise investments to answer two questions: (1) Is this project worth doing at all? and (2) If we can only afford one of several projects, which is best? Appraisal reduces the risk of committing large, often irreversible sums to a poor project.

Cash flow, not profit. All three methods work from forecast net cash flow = cash inflows − cash outflows in each year — not accounting profit. Cash flow reflects the actual money the investment generates and consumes. Using profit (which includes non-cash items like depreciation) is a classic error.

The three examined methods:

Payback period — how fast the project repays its cost (a measure of speed/liquidity).
Average rate of return (ARR) — the average annual profitability as a % of the cost.
Net present value (NPV) — the value created today once the time value of money is taken into account.

Quantitative vs qualitative appraisal. The three methods are quantitative (numbers-based). A full decision also weighs qualitative factors — the finance available, corporate objectives, staff, risk appetite, ethics/sustainability and the state of the economy. The numbers inform the decision; they do not make it on their own.

Investment appraisal = quantitative evaluation of a capital investment.
All methods use forecast NET CASH FLOW (inflows − outflows), not accounting profit.
Three methods: payback (speed), ARR (average % return), NPV (today's value, time-adjusted).
A recommendation combines quantitative results with qualitative factors.

Payback period — how fast the investment repays itself

The payback period is the time taken for the cumulative net cash flow to repay the initial investment. With even cash flows, divide cost by annual flow; with uneven flows, track cumulative cash flow and interpolate the part-year in months.

The payback period is the length of time it takes for an investment's net cash flows to repay (pay back) its initial cost. Shorter is better — a quick payback means capital is tied up for less time and the project is less exposed to risk.

Even (constant) cash flows — divide the initial cost by the annual net cash flow:

Payback = Initial investment ÷ Net annual cash flow

Example: a $48,000 machine generating$ 12,000 net cash flow a year pays back in 48,000 ÷ 12,000 = 4 years exactly.

Uneven cash flows — build a running cumulative cash flow and find the year the total turns from negative to positive; then convert the remaining fraction of that year into months.

Worked example (Project A — $200,000 machine):

Year	Net cash flow ($)	Cumulative cash flow ($)
0	(200,000)	(200,000)
1	60,000	(140,000)
2	70,000	(70,000)
3	80,000	10,000
4	60,000	70,000
5	40,000	110,000

By the end of Year 2 the firm is still $70,000 short. Year 3 brings in$ 80,000, so payback happens during Year 3:

Months into Year 3 = (70,000 ÷ 80,000) × 12 = 0.875 × 12 = 10.5 months

So the payback period is 2 years and 10.5 months (≈ 2.88 years).

Advantages: simple and quick to calculate; useful for firms with cash-flow/liquidity concerns; good where technology dates fast (recover cash before the asset is obsolete); easy for non-financial managers to understand.

Disadvantages: ignores all cash flows after payback (so it ignores total profitability); ignores the time value of money; a fast payback can hide a low-return project.

Even flows: Payback = initial investment ÷ net annual cash flow.
Uneven flows: track cumulative cash flow; convert the part-year to months = (amount still owed ÷ that year's flow) × 12.
Project A pays back in 2 years 10.5 months.
Strength: simple, liquidity-focused. Weakness: ignores later cash flows and the time value of money.

See the full worked example for investment appraisal →

Average rate of return (ARR) — average yearly profitability

ARR expresses the average annual profit of an investment as a percentage of its initial cost. Total profit = total net cash flow − initial investment; divide by the number of years for the average, then over the initial cost.

The average rate of return (ARR), also called the accounting rate of return, expresses the average annual profit an investment earns as a percentage of its initial cost. A higher ARR is better, and firms usually compare the ARR against a target/criterion rate (e.g. the interest they could earn elsewhere, or a company hurdle rate).

Step-by-step:

Add up all the net cash flows over the project's life → total return.
Subtract the initial investment → total profit over the whole life.
Divide by the number of years → average annual profit.
Express as a % of the initial investment → ARR.

Example: a $200,000 investment returns a total of$ 290,000 over 5 years.

Total profit = 290,000 − 200,000 = $90,000
Average annual profit = 90,000 ÷ 5 = $18,000
ARR = (18,000 ÷ 200,000) × 100 = 9%

If the firm could earn, say, 5% in a bank deposit, a 9% ARR looks attractive; if its hurdle rate is 12%, the project fails the test.

Advantages: uses all the cash flows across the whole life (unlike payback); shows profitability as a clear %; easy to compare with interest rates or a target return.

Disadvantages: ignores the timing of cash flows and therefore the time value of money (a dollar in Year 5 counts the same as one in Year 1); based on forecast figures that may be inaccurate; can be confused with other 'return' measures.

ARR = (average annual profit ÷ initial investment) × 100.
Average annual profit = (total net cash flow − initial investment) ÷ number of years.
Compare the ARR against a target/criterion rate to judge it.
Strength: uses all cash flows and gives a % return. Weakness: ignores timing / time value of money.

See the full worked example for investment appraisal →

Net present value (NPV) — the time value of money

Money received in the future is worth less than money today. NPV discounts each future net cash flow to its present value using given discount factors, sums them, and subtracts the initial investment. A positive NPV means accept.

The time value of money. $100 today is worth more than$ 100 in three years — you could invest today's $100 and earn interest, and inflation erodes future money's buying power. So future cash flows must be discounted back to their value in today's money before we compare them with a cost paid today.

Discount factors and present value. The exam gives you a table of discount factors (they depend on the chosen discount rate / cost of capital). The present value (PV) of a future cash flow is simply:

Present value = future net cash flow × discount factor

For example, $12,000 received in one year at a 10% discount rate has a present value of 12,000 × 0.909 = **$ 10,908** — that future $12,000 is worth only$ 10,908 in today's money.

Net present value (NPV) sums the present values of all the project's future net cash flows and subtracts the initial investment:

NPV = (sum of discounted cash flows) − initial investment

Positive NPV → accept (the project adds value in today's money).
Negative NPV → reject (it destroys value).
When choosing between projects, the higher positive NPV is usually preferred.

Worked example (Project A, discount rate 10%):

Year	Net cash flow ($)	Discount factor (10%)	Present value ($)
1	60,000	0.909	54,540
2	70,000	0.826	57,820
3	80,000	0.751	60,080
4	60,000	0.683	40,980
5	40,000	0.621	24,840
		Sum of PV	238,260

NPV = 238,260 − 200,000 = +$38,260. The NPV is positive, so on this method the machine is worthwhile.

Advantages: the only method that accounts for the time value of money; uses all the cash flows over the whole life; gives a clear accept/reject signal in today's dollars.

Disadvantages: the most complex to calculate and explain; the result depends heavily on the discount rate chosen — pick the wrong rate and the decision can flip; still relies on uncertain forecast cash flows.

Time value of money: future cash is worth less than cash today, so discount it.
Present value = future net cash flow × discount factor (factors given in the exam).
NPV = sum of discounted cash flows − initial investment.
Positive NPV → accept; negative → reject; Project A's NPV at 10% is +$38,260.

See the full worked example for investment appraisal →

Comparing the methods and the qualitative factors

No single method is 'best' in the abstract. Payback measures speed, ARR measures average return, NPV measures value with timing. A sound recommendation weighs the three results against qualitative factors and the firm's objectives.

The three methods answer different questions, so they can point in different directions. A fast-paying project may have a modest ARR; a high-NPV project may pay back slowly. Choose the emphasis to fit the firm's priorities.

Method	Measures	Main strength	Main weakness
Payback	Time to recover the cost	Simple; focuses on liquidity/risk	Ignores cash after payback & time value of money
ARR	Average % profit on cost	Uses all cash flows; easy % to compare	Ignores timing / time value of money
NPV	Value added in today's money	Accounts for the time value of money; uses all cash flows	Complex; very sensitive to the discount rate

Which to prioritise? A cash-strapped or high-risk business (e.g. fast-changing tech) leans on payback. A firm comparing a project against an interest rate leans on ARR. A firm making a large, long-term decision and able to estimate a sensible discount rate leans on NPV, the most theoretically complete method. Best practice is to use them together.

Qualitative factors that must sit alongside the numbers in any recommendation:

Finance available — even a positive-NPV project is off the table if the firm cannot raise the cash.
Corporate objectives & strategy — does the project fit the mission (e.g. sustainability, market leadership)?
Risk & reliability of forecasts — cash flows and the discount rate are estimates; how confident are they?
Human & operational factors — staff skills, morale, capacity, and the opportunity cost of managers' time.
Ethics & the environment — reputational and CSR implications beyond the numbers.
The external environment — interest rates, the state of the economy, competitor moves.

Recommending (AO3): state the numerical verdict from each method, note where they agree or conflict, then commit to a decision and justify it — usually conditionally (‘invest, provided the finance can be raised and the 10% discount rate is realistic’). A recommendation on numbers alone, with no qualitative factors, cannot reach the top band.

Payback = speed; ARR = average % return; NPV = time-adjusted value — use them together.
Priorities differ: liquidity → payback; comparison vs interest → ARR; large long-term decision → NPV.
Qualitative factors: finance available, objectives, risk, staff, ethics, the economy.
AO3: give the numerical verdict, note conflicts, then recommend and justify — never numbers alone.

See the full worked example for investment appraisal →

HL only — multi-year scenarios and integrated evaluation

HL exam material demands more: fuller multi-year NPV tables, comparing projects across all three methods when they conflict, testing sensitivity to the discount rate, and deeper qualitative evaluation woven into a justified recommendation.

This section applies to Higher Level candidates. SL and HL share the same three methods and calculation techniques, but HL Paper 2 pushes further in three ways.

1. Fuller multi-year NPV. HL data can span five or more years with uneven cash flows, requiring a complete discount-factor table (Year × cash flow × factor × present value) worked cleanly and totalled. Accuracy and clear presentation of the table earn method marks even if an arithmetic slip occurs.

2. Comparing projects when methods conflict. HL frequently gives two competing projects whose rankings disagree. Consider two $100,000 projects at a 10% discount rate:

Method	Project X	Project Y
Payback	2 years 4 months	3 years
ARR	10%	15%
NPV (10%)	+$14,680	+$21,490

Project X pays back faster (better for liquidity/risk), but Project Y has the higher ARR and the higher NPV (better for long-term value). The HL skill is to resolve the conflict with reference to the firm's objectives and circumstances — not to duck it.

3. Sensitivity to the discount rate. Because NPV depends on the discount rate, HL candidates should recognise that a higher rate (reflecting greater risk or a higher cost of capital) lowers the NPV. Project A's NPV falls from + $38,260 at 10% to +$ 11,960 at 15% — still positive, so the decision is robust, but a project whose NPV flips negative under a modest rate rise is far riskier.

4. Deeper qualitative evaluation. HL 10-mark answers must integrate the quantitative results with a developed discussion of qualitative factors and end with a substantiated, context-specific judgement — typically an integrated recommendation using all three methods together rather than one in isolation.

HL = same three methods, greater depth: full multi-year NPV tables and conflicting-project comparisons.
When payback, ARR and NPV disagree, resolve the conflict using the firm's objectives — don't ignore it.
A higher discount rate lowers NPV (Project A: + $38,260 at 10% → +$ 11,960 at 15%); test robustness.
HL recommendations integrate all methods with developed qualitative evaluation and a justified judgement.

See the full worked example for investment appraisal →

Quick recap

Investment appraisal evaluates a capital investment using forecast NET CASH FLOW, not profit.
Payback = time to recover the cost (even: cost ÷ annual flow; uneven: cumulative flow + months). Shorter is better.
ARR = (average annual profit ÷ initial investment) × 100; average annual profit = (total return − cost) ÷ years.
NPV = sum of discounted cash flows − initial investment, using given discount factors; positive → accept.
Only NPV accounts for the time value of money; payback and ARR ignore the timing of cash flows.
A top recommendation combines the numerical verdicts with qualitative factors (HL: across multiple methods and multi-year scenarios).

Memorise this

Verbatim phrases, formulae and definitions IB DP mark schemes credit (key for AO1 knowledge marks on Paper 1).

Payback (even) = initial investment ÷ net annual cash flow
Payback (uneven): build cumulative cash flow; part-year months = (amount owed ÷ that year's flow) × 12
ARR = (average annual profit ÷ initial investment) × 100
average annual profit = (total net cash flow − initial investment) ÷ number of years
Present value = future net cash flow × discount factor
NPV = sum of discounted (present) values − initial investment
Positive NPV = accept; negative NPV = reject
Only NPV allows for the time value of money; Year 0 cost is NOT discounted
Always add qualitative factors to a recommendation

How it’s examined

Assessed on Paper 1 and (heavily) Paper 2, where a discount-factor table is provided and the data are quantitative. AO1 (Define/State, 2 marks) asks you to define payback period, ARR or net present value, or a discount factor. AO2 (Calculate/Explain, 4–6 marks) asks you to calculate the payback period, ARR or NPV from a table of forecast net cash flows and interpret the result (e.g. is the NPV positive?). AO3 (Recommend/Evaluate/Discuss, ~10 marks) asks you to recommend WHETHER a business should undertake an investment, or to evaluate which appraisal method is most appropriate — requiring the numbers to be combined with qualitative factors (finance available, objectives, risk, the economy) and ending with a substantiated judgement. Examiners reward correct method (shown working earns marks even after a slip) and penalise using profit instead of cash flow, forgetting to subtract the initial investment for NPV, ignoring the time value of money, and recommending on numbers alone.

Sources: IB Diploma Programme Business Management Guide (first teaching 2022, first assessment 2024). Last reviewed 2026-07-24.

Take this whole topic with you

Step-by-step worked examples — Investment appraisal

Step-by-step solutions to past-paper-style questions on investment appraisal, written exactly the way a tutor would explain them at the board.

1Payback with even (constant) cash flows
Getting started• payback, AO2
Question
A firm buys a packaging machine for $48,000. It generates a constant net cash flow of$ 12,000 per year. Calculate the payback period. [2]
Step-by-step solution
1. Step 1
  With EQUAL annual cash flows, divide the initial cost by the annual net cash flow.
  $Payback = Initial investment ÷ Net annual cash flow$
2. Step 2
  Substitute the figures.
  $= 48,000 ÷ 12,000 = 4$
Answer
The payback period is 4 years exactly.
Examiner tip
Always state the unit (years). The even-cash-flow formula only works when every year's flow is the same — if they differ, you must use cumulative cash flow instead.
2Present value of a single future sum (discount factor)
Getting started• NPV, present value, AO2
Question
Using a discount rate of 10% (Year 1 discount factor = 0.909), calculate the present value of $12,000 to be received in one year, and explain what it tells us. [2]
Step-by-step solution
1. Step 1
  Present value = future net cash flow × discount factor. The factor is given in the exam's discount table.
  $PV = 12,000 × 0.909$
2. Step 2
  Multiply out. The result is worth less than the $12,000 because future money is worth less than money today (the time value of money).
  $= 10,908$
Answer
The present value is $10,908 — i.e.$ 12,000 received in a year is worth only $10,908 in today's money at a 10% discount rate.
Examiner tip
This single-sum PV is the building block of NPV: an NPV is just the sum of many such present values, minus the initial cost.
3Average rate of return (ARR)
Getting started• ARR, AO2
Question
An investment costs $200,000 and returns a total net cash flow of$ 290,000 over 5 years. Calculate the average rate of return (ARR). [2]
Step-by-step solution
1. Step 1
  Find the total profit over the life = total net cash flow − initial investment.
  $Total profit = 290,000 − 200,000 = 90,000$
2. Step 2
  Find the average annual profit by dividing by the number of years.
  $Average annual profit = 90,000 ÷ 5 = 18,000$
3. Step 3
  Express the average annual profit as a % of the initial investment.
  $ARR = (18,000 ÷ 200,000) × 100 = 9%$
Answer
The ARR is 9%.
Examiner tip
Subtract the initial cost BEFORE averaging — the numerator is average annual PROFIT, not average annual cash flow. A 9% ARR beats a 5% bank rate but fails a 12% hurdle rate.
4Payback with uneven cash flows (cumulative method)
Building confidence• payback, cumulative, AO2
Question
Project A costs $200,000. Forecast net cash flows are: Year 1$ 60,000; Year 2 $70,000; Year 3$ 80,000; Year 4 $60,000; Year 5$ 40,000. Calculate the payback period. [4]
Step-by-step solution
1. Step 1
  Build the cumulative cash flow year by year, starting at −200,000.
  $End Y1: -140,000 | End Y2: -70,000 | End Y3: +10,000$
2. Step 2
  The cumulative flow turns positive during Year 3. At the end of Year 2 the firm is still $70,000 short, and Year 3 brings in$ 80,000.
  $Amount still owed entering Y3 = 70,000; Y3 flow = 80,000$
3. Step 3
  Convert the fraction of Year 3 needed into months (× 12).
  $Months = (70,000 ÷ 80,000) × 12 = 0.875 × 12 = 10.5$
Answer
Payback = 2 years and 10.5 months (≈ 2.88 years).
Examiner tip
Show the cumulative column — it earns method marks. Convert the final fraction to months by multiplying by 12; do not write 0.875 as '8.75 months'.
5NPV of a project with even cash flows
Building confidence• NPV, AO2
Question
A project costs $100,000 and returns$ 40,000 net cash flow each year for 4 years. Using a 10% discount rate (factors: Y1 0.909, Y2 0.826, Y3 0.751, Y4 0.683), calculate the NPV and state whether the project should go ahead. [6]
Step-by-step solution
1. Step 1
  Discount each year's cash flow: present value = cash flow × discount factor.
  $Y1: 40,000 × 0.909 = 36,360 | Y2: 40,000 × 0.826 = 33,040$
2. Step 2
  Continue for Years 3 and 4.
  $Y3: 40,000 × 0.751 = 30,040 | Y4: 40,000 × 0.683 = 27,320$
3. Step 3
  Sum the present values.
  $Sum of PV = 36,360 + 33,040 + 30,040 + 27,320 = 126,760$
4. Step 4
  Subtract the initial investment to get the NET present value.
  $NPV = 126,760 − 100,000 = +26,760$
Answer
NPV = + $26,760. As the NPV is positive, on this method the project should go ahead — it adds$ 26,760 of value in today's money.
Examiner tip
Do not forget the final subtraction of the initial cost — the sum of PVs alone ($126,760) is NOT the NPV. Year 0 is not discounted (it is already today's money).
6Calculate and interpret ARR against a target return
Building confidence• ARR, interpretation, AO2
Question
A $150,000 investment yields net cash flows of$ 50,000, $55,000,$ 45,000 and $30,000 over four years. The firm's target (hurdle) rate of return is 12%. Calculate the ARR and comment on whether the investment meets the target. [6]
Step-by-step solution
1. Step 1
  Add the net cash flows for the total return.
  $Total return = 50,000 + 55,000 + 45,000 + 30,000 = 180,000$
2. Step 2
  Total profit = total return − initial investment.
  $Total profit = 180,000 − 150,000 = 30,000$
3. Step 3
  Average annual profit = total profit ÷ number of years.
  $Average annual profit = 30,000 ÷ 4 = 7,500$
4. Step 4
  ARR = average annual profit as a % of the initial investment.
  $ARR = (7,500 ÷ 150,000) × 100 = 5%$
Answer
ARR = 5%. Because 5% is well below the 12% target rate, the investment fails the firm's hurdle-rate test on this method and looks unattractive.
Examiner tip
Interpretation carries marks at AO2 — compare the calculated ARR with the stated target rate and state the verdict, don't just leave the number.
7Full multi-year NPV with a discount-factor table
Stretch• NPV, multi-year, AO2
Question
Project A costs $200,000, with net cash flows of$ 60,000, $70,000,$ 80,000, $60,000 and$ 40,000 over five years. Using a 10% discount rate (factors: 0.909, 0.826, 0.751, 0.683, 0.621), construct the NPV table and calculate the NPV. [6]
Step-by-step solution
1. Step 1
  Discount Years 1–3: present value = cash flow × discount factor.
  $Y1: 60,000 × 0.909 = 54,540 | Y2: 70,000 × 0.826 = 57,820 | Y3: 80,000 × 0.751 = 60,080$
2. Step 2
  Discount Years 4 and 5.
  $Y4: 60,000 × 0.683 = 40,980 | Y5: 40,000 × 0.621 = 24,840$
3. Step 3
  Sum all five present values.
  $Sum of PV = 54,540 + 57,820 + 60,080 + 40,980 + 24,840 = 238,260$
4. Step 4
  Subtract the initial investment.
  $NPV = 238,260 − 200,000 = +38,260$
Answer
NPV = +$38,260. The positive NPV means the machine creates value in today's money and, on this method, should be accepted.
Examiner tip
Lay the table out as Year | cash flow | discount factor | present value — clear presentation earns method marks and protects you if a single multiplication slips.
8Choosing between two projects across all three methods
Stretch• comparison, recommendation, AO3
Question
Two mutually exclusive projects each cost $100,000. Project X: NPV (10%) +$ 14,680, payback 2 years 4 months, ARR 10%. Project Y: NPV (10%) +$21,490, payback 3 years, ARR 15%. Which project should the firm choose, and why? [6]
Step-by-step solution
1. Step 1
  Compare on each method. Payback favours X (faster); ARR favours Y (15% vs 10%); NPV favours Y (higher value added).
  $Payback: X < Y | ARR: Y > X | NPV: Y > X$
2. Step 2
  Note that two of three methods — including NPV, the only time-adjusted one — favour Project Y. X only wins on payback (speed/liquidity).
3. Step 3
  Resolve using the firm's priorities: if it can bear a slightly slower payback, Y's higher return and higher value make it the stronger choice; if liquidity/risk is critical, X's faster payback may win.
Answer
On the numbers, Project Y is generally preferable — it has the higher ARR (15% vs 10%) and the higher NPV (+ $21,490 vs +$ 14,680). Project X's only advantage is a faster payback (2 yrs 4 mths vs 3 yrs), so X would be chosen only if the firm's liquidity or risk position makes speed of recovery the priority.
Examiner tip
When methods conflict, do not pick arbitrarily — weight NPV most heavily (it accounts for the time value of money) but justify the choice against the firm's objectives and cash position.
9HL — NPV sensitivity to the discount rate
Stretch• NPV, sensitivity, HL, AO3
Question
HL. Project A (cost $200,000; flows$ 60,000, $70,000,$ 80,000, $60,000,$ 40,000) has an NPV of +$38,260 at a 10% discount rate. Recalculate the NPV at a 15% discount rate (factors: 0.870, 0.756, 0.658, 0.572, 0.497) and comment on what this reveals. [6]
Step-by-step solution
1. Step 1
  Discount Years 1–3 at 15%.
  $Y1: 60,000 × 0.870 = 52,200 | Y2: 70,000 × 0.756 = 52,920 | Y3: 80,000 × 0.658 = 52,640$
2. Step 2
  Discount Years 4 and 5 at 15%.
  $Y4: 60,000 × 0.572 = 34,320 | Y5: 40,000 × 0.497 = 19,880$
3. Step 3
  Sum the present values and subtract the initial cost.
  $Sum of PV = 52,200 + 52,920 + 52,640 + 34,320 + 19,880 = 211,960; NPV = 211,960 − 200,000 = +11,960$
4. Step 4
  Compare: raising the discount rate from 10% to 15% cuts the NPV from + $38,260 to +$ 11,960 — a large fall, but the NPV stays positive.
Answer
At 15% the NPV falls to + $11,960 (from +$ 38,260 at 10%). NPV is highly sensitive to the discount rate, but because the project remains positive even at the higher rate, the decision to invest is robust to a plausible rise in the cost of capital.
Examiner tip
HL sensitivity analysis: a higher discount rate (more risk / higher cost of capital) always lowers NPV. Testing whether the verdict survives a pessimistic rate is a hallmark of a top-band evaluation.

Model Answers — Investment appraisal

High-scoring sample answers for investment appraisal on the Cambridge IGCSE paper, with examiner-style notes mapping each response to the mark scheme and assessment objectives.

Question 1
IB DP Business Management Paper 1/2 style2 marks
Define the term payback period. [2]
Model answer
The payback period is the length of time it takes for an investment's net cash flows to repay (recover) its initial cost. For example, a $48,000 machine earning$ 12,000 net cash flow per year has a payback period of 4 years.
Why this scores
2 marks: 1 for the core idea (time to recover the initial investment), 1 for accuracy/example. Note it uses net cash flow, not profit.
Question 2
IB DP Business Management Paper 1/2 style2 marks
Define the term net present value (NPV). [2]
Model answer
Net present value is the sum of the discounted (present) values of all an investment's future net cash flows minus the initial investment. It converts future cash flows into today's money using discount factors, so it accounts for the time value of money; a positive NPV means the project adds value.
Why this scores
2 marks: 1 for 'discounted future cash flows minus initial cost', 1 for the time-value-of-money idea or the accept-if-positive rule.
Question 3
IB DP Business Management Paper 1/2 style2 marks
State what the average rate of return (ARR) measures. [2]
Model answer
The ARR measures the average annual profit generated by an investment expressed as a percentage of its initial cost. It shows the average yearly profitability of the investment, allowing it to be compared with a target rate or an interest rate.
Why this scores
2 marks: 1 for 'average annual profit as a % of initial cost', 1 for the purpose (comparison with a target/interest rate).
Question 4
IB DP Business Management Paper 2 style4 marks
A firm is appraising a machine costing $200,000 with forecast net cash flows of$ 60,000 (Y1), $70,000 (Y2),$ 80,000 (Y3), $60,000 (Y4) and$ 40,000 (Y5). Calculate the payback period and interpret your result. [4]
Model answer
Cumulative cash flow: end of Y1 = − $140,000; end of Y2 = −$ 70,000; end of Y3 = + $10,000. The investment is repaid during Year 3. Entering Year 3 the firm still needs$ 70,000, and Year 3 generates $80,000, so the fraction of the year = (70,000 ÷ 80,000) × 12 = 10.5 months. Payback = 2 years 10.5 months.

Interpretation: the machine returns its $200,000 cost in just under three years of its five-year life, leaving over two years of cash flow as gain. Whether this is 'good' depends on the firm's target payback and how quickly the technology dates.
Why this scores
AO2: marks for the cumulative working, the correct months conversion (× 12), the stated answer, and a contextual interpretation rather than a bare number.
Question 5
IB DP Business Management Paper 2 style6 marks
Using a 10% discount rate (factors: Y1 0.909, Y2 0.826, Y3 0.751, Y4 0.683, Y5 0.621), calculate the NPV of the $200,000 machine above and advise whether, on this method alone, it should be purchased. [6]
Model answer
Present values: Y1 60,000 × 0.909 = $54,540; Y2 70,000 × 0.826 =$ 57,820; Y3 80,000 × 0.751 = $60,080; Y4 60,000 × 0.683 =$ 40,980; Y5 40,000 × 0.621 = $24,840. Sum of present values = **$ 238,260**. NPV = 238,260 − 200,000 = +$38,260.

Advice: the NPV is positive, so — accounting for the time value of money — the machine adds $38,260 of value in today's money and, on this method alone, should be purchased.
Why this scores
AO2: full method marks for the discounting table, the correct sum, subtracting the initial cost, and a clear accept decision tied to the positive NPV.
Question 6
IB DP Business Management Paper 2 style4 marks
For the same $200,000 machine (total net cash flow over 5 years =$ 310,000), calculate the ARR and comment on it. [4]
Model answer
Total profit = total net cash flow − initial cost = 310,000 − 200,000 = $110,000. Average annual profit = 110,000 ÷ 5 =$ 22,000. ARR = (22,000 ÷ 200,000) × 100 = 11%.

Comment: an 11% average annual return comfortably exceeds a typical bank deposit rate, so the investment looks profitable; however, the ARR ignores the timing of the cash flows, so it should be read alongside the NPV.
Why this scores
AO2: marks for total profit, average annual profit, the % calculation, and a comment that both judges the figure and notes ARR's timing weakness.
Question 7
IB DP Business Management Paper 2 style10 marks
Evaluate the usefulness of the payback period as a method of investment appraisal for a small technology start-up. [10]
Model answer
Payback measures how quickly an investment recovers its cost, and for a small technology start-up this is often highly relevant. Start-ups typically face tight cash flow and limited access to finance, so recovering capital quickly (e.g. a $48,000 machine paying back in 4 years, or Project A in 2 years 10.5 months) reduces the time cash is at risk and frees funds for reinvestment. In fast-moving technology, where products and equipment can become obsolete within a couple of years, a short payback also lowers the danger of committing to an asset that is superseded before it earns its keep. Payback is quick to calculate and easy for non-financial founders to understand and communicate to lenders.

However, payback has serious limitations. It ignores all cash flows after the payback point, so it says nothing about a project's total profitability — a project that pays back in two years but then stops earning is ranked above one that pays back in three years and earns strongly for a decade. It also ignores the time value of money, treating a dollar in Year 3 as equal to a dollar today. For a start-up choosing between projects, this can lead to short-termism — favouring quick, low-value projects over more valuable long-term ones that NPV would identify.

Overall, payback is a useful first filter for a cash-constrained, fast-changing start-up because liquidity and obsolescence risk genuinely matter to it, but it should not be used alone. Its blindness to later cash flows and to the time value of money means it must be combined with ARR and, ideally, NPV, plus qualitative factors such as the reliability of the cash-flow forecasts and the firm's strategic objectives. Its usefulness therefore depends on context: valuable for screening and liquidity management, inadequate as the sole basis for a major decision.
Why this scores
AO3 top band: balanced (strengths AND weaknesses), applied to the start-up context throughout, and a substantiated final judgement ('depends on context'), not a list.
Question 8
IB DP Business Management Paper 2 (HL) style10 marks
HL. Using the appraisal of the $200,000 machine (payback 2 years 10.5 months; ARR 11%; NPV at 10% = +$ 38,260), recommend whether the firm should purchase it. [10]
Model answer
All three quantitative methods point the same way. The NPV is clearly positive at +$38,260, meaning that after allowing for the time value of money the machine adds substantial value in today's money — the strongest single signal, because NPV is the only method that discounts future cash flows. The ARR of 11% exceeds a typical deposit or borrowing rate, indicating healthy average profitability, and the payback of 2 years 10.5 months recovers the cost in under three years of a five-year life, limiting the firm's exposure. When payback, ARR and NPV agree, the numerical case for investing is strong and consistent.

That said, the recommendation should not rest on numbers alone. First, the results depend on forecast cash flows and a chosen 10% discount rate; if demand is weaker than projected, or the cost of capital rises, the NPV shrinks (at 15% it would fall to roughly + $11,960 — still positive, so the decision is fairly robust). Second, the firm must be able to *finance* the$ 200,000 without straining liquidity or over-borrowing. Third, qualitative factors matter: does the machine fit the firm's strategic objectives, will staff need retraining, and are there ethical or environmental implications? Finally, the opportunity cost — what else the $200,000 could achieve — should be considered.

On balance, I recommend that the firm purchase the machine, because all three methods indicate a profitable, value-adding investment and the positive NPV survives a pessimistic discount rate. This recommendation is conditional: the firm should confirm it can raise the finance affordably, sense-check the cash-flow forecasts, and ensure the investment aligns with its wider objectives. Provided these conditions hold, the quantitative and qualitative case both support going ahead.
Why this scores
AO3 (HL) top band: integrates ALL THREE methods, weights NPV correctly, tests robustness to the discount rate, brings in qualitative factors and finance, and ends with a justified, conditional recommendation.
Question 9
IB DP Business Management Paper 2 (HL) style10 marks
HL. A firm must choose between two mutually exclusive projects each costing $100,000. Project X: payback 2 years 4 months, ARR 10%, NPV (10%) +$ 14,680. Project Y: payback 3 years, ARR 15%, NPV (10%) +$21,490. Discuss which project the firm should undertake. [10]
Model answer
The two projects send conflicting signals, so the choice depends on which criteria the firm weights most heavily. Project X is superior on payback, recovering its cost in 2 years 4 months versus 3 years for Y — attractive if the firm's priority is liquidity, minimising risk, or coping with fast-changing markets where distant cash flows are uncertain. However, Project Y is superior on both ARR (15% vs 10%) and NPV (+ $21,490 vs +$ 14,680). Because NPV is the only method that accounts for the time value of money and measures the actual value added, its verdict — favouring Y — should carry the greatest weight, and it is reinforced by the higher ARR.

Evaluating the trade-off: X wins on only one metric (speed), while Y wins on the two that measure profitability and value. Unless the firm has an urgent liquidity constraint or unusually high risk (e.g. rapid obsolescence, unreliable long-term forecasts), sacrificing $6,810 of extra NPV and five percentage points of ARR simply to recover the cash eight months sooner is hard to justify. Qualitative factors could tip the balance: if Project Y's later cash flows are far less certain, or if the firm is cash-strapped and cannot wait the extra time, X becomes defensible; the firm's objectives (growth vs survival) and the reliability of each project's forecasts are decisive.

Overall, I recommend Project Y, because it delivers the higher NPV and ARR and the modest difference in payback is unlikely to outweigh the greater value created — provided the firm can bear the slightly longer wait and Y's forecasts are as reliable as X's. If liquidity or forecast risk were severe, Project X's faster payback would justify the safer, if less profitable, choice. The decision therefore hinges on the firm's risk appetite and cash position rather than the numbers in isolation.
Why this scores
AO3 (HL) top band: resolves the conflict between methods rather than restating it, prioritises NPV with reasoning, weaves in qualitative factors, and lands a justified, conditional recommendation.

Key Formulae — Investment appraisal

The formulae you need to memorise for investment appraisal on the Cambridge IGCSE paper, with every variable defined in plain English and a note on when to use it.

Payback period (even cash flows)
$\text{Payback} = \dfrac{\text{Initial investment}}{\text{Net annual cash flow}}$
$Initial investment$
the up-front capital cost of the project ($)
$Net annual cash flow$
constant yearly cash inflow − outflow ($). For UNEVEN flows, use cumulative cash flow and interpolate the part-year in months instead
When to use
To find how long an investment takes to repay its cost when the annual net cash flow is the same each year. For uneven flows, build a cumulative cash flow column and find where it turns positive: months = (amount still owed ÷ that year's flow) × 12.
Example
48,000 / 12,000 = 4 years (even); Project A (uneven) = 2 years 10.5 months
Average rate of return (ARR)
$\text{ARR} = \dfrac{\text{Average annual profit}}{\text{Initial investment}} \times 100$
$Average annual profit$
(total net cash flow − initial investment) ÷ number of years
$Initial investment$
the up-front capital cost of the project ($)
When to use
To express the average yearly profit of an investment as a percentage of its initial cost, then compare it with a target/hurdle rate or an interest rate.
Example
(90,000 / 5) / 200,000 × 100 = 9%
Net present value (NPV)
$\text{NPV} = \sum \left( \text{Cash flow} \times \text{Discount factor} \right) - \text{Initial investment}$
$Discount factor$
the multiplier (given in the exam) that converts a future cash flow into its present value; smaller for later years and higher discount rates
$Cash flow × Discount factor$
the present value (PV) of each year's net cash flow ($)
$Initial investment$
the up-front cost, paid in Year 0 and NOT discounted ($)
When to use
To value an investment in today's money, accounting for the time value of money. Discount every future net cash flow, sum the present values, then subtract the initial cost. Positive NPV → accept; negative → reject.
Example
PV of 12,000 in Y1 = 12,000 × 0.909 = 10,908; Project A: 238,260 − 200,000 = +38,260

Key Definitions and Keywords — Investment appraisal

Definitions to memorise and the exact keywords mark schemes credit for investment appraisal answers — sharpened from recent examiner reports for the 2026 Cambridge IGCSE sitting.

Investment appraisal
Examiner keyword
The quantitative techniques used to evaluate whether a capital investment is financially worthwhile, based on forecast net cash flows.
Related: Capital expenditure, Payback period, Net present value (NPV)
Capital expenditure
Spending on fixed (non-current) assets such as machinery, vehicles or premises — the kind of long-term outlay that investment appraisal evaluates.
Example
Buying a $200,000 packaging machine.
Net cash flow
Examiner keyword
Cash inflows minus cash outflows in a period. All appraisal methods use net cash flow — not accounting profit.
Related: Investment appraisal, Initial investment
Initial investment
The up-front capital cost of a project, paid in Year 0. It is subtracted in the NPV calculation and is not discounted (it is already in today's money).
Related: Net present value (NPV), Capital expenditure
Payback period
Examiner keyword
The length of time it takes for an investment's net cash flows to repay its initial cost. A shorter payback is preferred.
Example
A $48,000 machine earning$ 12,000 a year pays back in 4 years.
Related: Net cash flow, Cumulative cash flow
Cumulative cash flow
The running total of net cash flows year by year, starting from the negative initial investment. Used to find the payback period when cash flows are uneven.
Related: Payback period
Average rate of return (ARR)
Examiner keyword
The average annual profit of an investment expressed as a percentage of its initial cost. Also called the accounting rate of return.
Example
Average annual profit $18,000 on a$ 200,000 investment gives an ARR of 9%.
Related: Net cash flow, Target rate of return
Target (hurdle / criterion) rate of return
The minimum acceptable rate of return a firm sets for an investment. A project is attractive if its ARR exceeds this rate.
Related: Average rate of return (ARR)
Net present value (NPV)
Examiner keyword
The sum of the present values of all an investment's future net cash flows minus the initial investment. Positive NPV means the project adds value.
Example
Project A: sum of PVs $238,260 − cost$ 200,000 = NPV +$38,260.
Related: Present value, Discount factor, Time value of money
Time value of money
Examiner keyword
The principle that money available now is worth more than the same amount in the future, because it can earn interest and because inflation erodes future purchasing power.
Related: Discount factor, Present value, Net present value (NPV)
Discount factor
Examiner keyword
A multiplier (given in the exam and based on the chosen discount rate) that converts a future cash flow into its present value. It is smaller for later years and higher discount rates.
Example
At 10%, the Year 1 factor is 0.909 and the Year 5 factor is 0.621.
Related: Present value, Discount rate
Present value (PV)
The value today of a future cash flow, calculated as future net cash flow × discount factor.
Example
$12,000 in one year at 10% = 12,000 × 0.909 =$ 10,908.
Related: Discount factor, Net present value (NPV)
Discount rate (cost of capital)
The percentage rate used to discount future cash flows, reflecting the cost of finance and the riskiness of the project. A higher discount rate lowers the NPV.
Related: Discount factor, Net present value (NPV)
Qualitative investment appraisal
The non-numerical factors weighed alongside the calculations — finance available, corporate objectives, risk, staff, ethics/sustainability and the state of the economy — when deciding whether to invest.
Related: Investment appraisal

Common Mistakes and Misconceptions — Investment appraisal

The traps other students keep falling into on investment appraisal questions — taken from recent Cambridge IGCSE examiner reports and mark schemes — and how to avoid them.

✕Using accounting profit instead of net cash flow in the payback or NPV calculation.
IB DP Business Management Guide (first assessment 2024)
Why it happens
Students confuse the three methods and assume 'return' always means profit, or mistakenly carry a profit figure (which already deducts depreciation) into a cash-based method.
How to avoid it
Remember: payback and NPV use NET CASH FLOW; only ARR works from profit. Read the data label carefully before you start.
✕Forgetting to subtract the initial investment when calculating NPV (stopping at the sum of present values).
IB DP Business Management Guide (first assessment 2024)
Why it happens
After discounting five years of cash flows, students treat the total present value as the answer and overlook the final subtraction.
How to avoid it
The word is NET present value — always finish with 'sum of PVs − initial investment'. Add the cost as a Year-0 line (−$200,000) so you cannot forget it.
✕Comparing payback or ARR with NPV without recognising that only NPV accounts for the time value of money.
IB DP Business Management Guide (first assessment 2024)
Why it happens
Payback and ARR are simpler and calculated first, so students trust them equally and miss why NPV can give a different, better-founded verdict.
How to avoid it
State explicitly that payback and ARR ignore the timing of cash flows, whereas NPV discounts them. Weight NPV most heavily when methods conflict.
✕Recommending whether to invest on the basis of the numbers alone, with no qualitative factors.
IB DP Business Management Guide (first assessment 2024)
Why it happens
The calculation feels like the whole task, so students stop once they have a positive NPV or a short payback.
How to avoid it
For any AO3 recommendation, add at least two qualitative factors (finance available, objectives, risk, staff, ethics, the economy) and end with a substantiated, conditional judgement.
✕Converting the part-year of the payback period incorrectly — e.g. writing 0.875 of a year as '8.75 months'.
IB DP Business Management Guide (first assessment 2024)
Why it happens
Students treat the decimal fraction of a year as if it were already months, or forget to multiply by 12.
How to avoid it
Multiply the fraction by 12: (amount still owed ÷ that year's cash flow) × 12. So 0.875 × 12 = 10.5 months, giving 2 years 10.5 months.
✕Calculating ARR from the average annual cash flow instead of the average annual PROFIT (forgetting to subtract the initial cost first).
IB DP Business Management Guide (first assessment 2024)
Why it happens
Students average the cash inflows directly and divide by the investment, skipping the step that turns total return into total profit.
How to avoid it
Follow the order: total net cash flow − initial investment = total profit; ÷ years = average annual profit; ÷ initial investment × 100 = ARR.

Investment appraisal — payback period, average rate of return (ARR) and net present value (NPV)

Year

Net cash flow ($)

Cumulative cash flow ($)

(200,000)

60,000

(140,000)

70,000

(70,000)

80,000

10,000

60,000

70,000

40,000

110,000

Year

Net cash flow ($)

Discount factor (10%)

Present value ($)

60,000

0.909

54,540

70,000

0.826

57,820

80,000

0.751

60,080

60,000

0.683

40,980

40,000

0.621

24,840

Sum of PV

238,260

Method

Measures

Main strength

Main weakness

Payback

Time to recover the cost

Simple; focuses on liquidity/risk

Ignores cash after payback & time value of money

ARR

Average % profit on cost

Uses all cash flows; easy % to compare

Ignores timing / time value of money

NPV

Value added in today's money

Accounts for the time value of money; uses all cash flows

Complex; very sensitive to the discount rate

Method

Project X

Project Y

Payback

2 years 4 months

3 years

ARR

10%

15%

NPV (10%)

+$14,680

+$21,490

Investment appraisal

Detailed Study Notes

At a glance

What you’ll learn

What investment appraisal is — and why cash flow, not profit

Payback period — how fast the investment repays itself

Average rate of return (ARR) — average yearly profitability

Net present value (NPV) — the time value of money

Comparing the methods and the qualitative factors

HL only — multi-year scenarios and integrated evaluation

Quick recap

Memorise this

How it’s examined

Take this whole topic with you

1Payback with even (constant) cash flows

2Present value of a single future sum (discount factor)

3Average rate of return (ARR)

4Payback with uneven cash flows (cumulative method)

5NPV of a project with even cash flows

6Calculate and interpret ARR against a target return

7Full multi-year NPV with a discount-factor table

8Choosing between two projects across all three methods

9HL — NPV sensitivity to the discount rate

Payback period (even cash flows)

Average rate of return (ARR)

Net present value (NPV)

Investment appraisal

Capital expenditure

Net cash flow

Initial investment

Payback period

Cumulative cash flow

Average rate of return (ARR)

Target (hurdle / criterion) rate of return

Net present value (NPV)

Time value of money

Discount factor

Present value (PV)

Discount rate (cost of capital)

Qualitative investment appraisal

✕Using accounting profit instead of net cash flow in the payback or NPV calculation.

✕Forgetting to subtract the initial investment when calculating NPV (stopping at the sum of present values).

✕Comparing payback or ARR with NPV without recognising that only NPV accounts for the time value of money.

✕Recommending whether to invest on the basis of the numbers alone, with no qualitative factors.

✕Converting the part-year of the payback period incorrectly — e.g. writing 0.875 of a year as '8.75 months'.

✕Calculating ARR from the average annual cash flow instead of the average annual PROFIT (forgetting to subtract the initial cost first).

Investment appraisal

Detailed Study Notes

At a glance

What you’ll learn

What investment appraisal is — and why cash flow, not profit

Payback period — how fast the investment repays itself

Average rate of return (ARR) — average yearly profitability

Net present value (NPV) — the time value of money

Comparing the methods and the qualitative factors

HL only — multi-year scenarios and integrated evaluation

Quick recap

Memorise this

How it’s examined

Take this whole topic with you

1Payback with even (constant) cash flows

2Present value of a single future sum (discount factor)

3Average rate of return (ARR)

4Payback with uneven cash flows (cumulative method)

5NPV of a project with even cash flows

6Calculate and interpret ARR against a target return

7Full multi-year NPV with a discount-factor table

8Choosing between two projects across all three methods

9HL — NPV sensitivity to the discount rate

Payback period (even cash flows)