Investment Appraisal Practice: Payback, NPV and IRR

The original eleven exercises contained useful cash-flow practice, but several answers depended on assumptions that were missing or applied inconsistently. This edition keeps the source tables and rebuilds the calculations from them. It does not preserve a result merely because it appeared on the old page.

The cases are hypothetical. Project A, B and C are treated as one-off, mutually exclusive alternatives. Under that restricted assumption, the project with the highest positive net present value is preferred. ACCA describes NPV as the principal investment-appraisal method, while its IRR guidance explains why IRR can produce the wrong ranking for mutually exclusive projects.

Calculation review date: 14 August 2026.

Calculation rules

1. Payback is the time required for undiscounted cumulative inflows to recover the period-0 outlay. Where recovery occurs within a year, the calculation assumes that year’s cash flow accrues evenly. The displayed month is rounded to the nearest month. ACCA’s payback guidance also distinguishes this within-year assumption from an end-of-year convention.
2. NPV is Σ CFₜ ÷ (1 + r)ᵗ for a constant annual discount rate. Full-precision factors are used; only the final answer is rounded to two decimal places.
3. IRR is the constant annual rate at which NPV equals zero. Each table has one initial outflow followed by non-negative inflows, producing one positive IRR. The legacy pages generally used linear interpolation between two trial rates. That is an estimate, not the exact root reported here.
4. ARR is omitted. Cash flow is not accounting profit. The source records do not consistently specify depreciation, residual value or average capital employed, so a comparable ARR cannot be reconstructed safely.
5. A stated residual or scrap value is not added to the base calculation unless it already appears in the cash-flow table. The sources do not specify disposal timing. The separate sensitivity therefore uses one explicit scenario for comparison: add the stated value to every alternative at the end of year 7.

OpenStax’s guidance on choosing between projects explains the highest-NPV rule and the complications introduced by capital rationing and unequal project lives. HM Treasury’s Green Book addresses public-sector social appraisal rather than private capital budgeting, but it supports the general discipline of expressing future values in present-value terms and testing material assumptions.

Exercise 01

Source ID: 2854. Discount rate: 10%. Amounts: £000.

Year	Project A	Project B	Project C
0	−50,000	−50,000	−50,000
1	11,000	8,000	20,000
2	11,000	9,000	20,000
3	11,000	12,000	20,000
4	11,000	14,000	1,000
5	11,000	16,000	1,000
6	11,000	13,000	1,000
7	11,000	11,000	1,000

Measure	Project A	Project B	Project C
Payback	4.55 years (about 4 years 7 months)	4.44 years (about 4 years 5 months)	2.50 years (2 years 6 months)
NPV (£000)	3,552.61	6,206.35	2,118.61
IRR	12.13%	13.37%	12.21%

Interpretation: Project C returns the initial outlay fastest, but Project B has the highest NPV. Project B is preferred under the stated mutually exclusive, one-off assumption.

Exercise 02

Source ID: 2885. Discount rates: 10% in years 1–3, then 12% in years 4–7. Amounts: £000.

Year	Project A	Project B	Project C
0	−150,000	−250,000	−550,000
1	25,000	48,000	120,000
2	25,000	39,000	120,000
3	37,000	52,000	120,000
4	49,000	64,000	230,000
5	61,000	66,000	130,000
6	51,000	63,000	30,000
7	31,000	61,000	30,000

For years after year 3, the corrected discount factor is 1 ÷ [(1.10)³ × (1.12)^(t−3)]. This carries the first three years forward before applying the new annual rate.

Measure	Project A	Project B	Project C
Payback	4.23 years (about 4 years 3 months)	4.71 years (about 4 years 9 months)	3.83 years (about 3 years 10 months)
NPV (£000)	32,667.68	10,215.19	10,940.12
IRR	16.37%	11.84%	11.15%

Correction: The legacy calculation used 10% for years 1–3, then applied 1 ÷ (1.12)ᵗ to years 4–7 as though 12% had applied from period 0. At full precision, that different spot-rate model gives NPVs (£000) of 26,801.51, 2,570.53 and −2,873.70. It does not match the statement that the rate changes after year 3. Under cumulative discounting, Project A remains the preferred alternative.

Exercise 03

Source ID: 2892. Discount rate: 12%. Amounts: source units.

The introductory sentence gives a 10% discount rate, but the later calculation instruction specifies 12%. This edition uses 12% because it is the explicit instruction attached to the calculation; the source does not explain the conflict.

Year	Project A	Project B	Project C
0	−75,000	−80,000	−200,000
1	20,000	25,000	80,000
2	20,000	25,000	80,000
3	20,000	25,000	80,000
4	20,000	15,000	40,000
5	20,000	15,000	20,000
6	20,000	15,000	0
7	20,000	0	0

Measure	Project A	Project B	Project C
Payback	3.75 years (3 years 9 months)	3.33 years (3 years 4 months)	2.50 years (2 years 6 months)
NPV	16,275.13	5,689.42	28,915.76
IRR	18.58%	14.79%	18.87%

The source does not state a residual value. Its ARR answers can be reverse-engineered only by assuming an unreported value of 10,000. That inference is not adopted here. Project C has the highest NPV under the cash flows actually shown.

Exercise 04

Source ID: 2905. Discount rate: 10%. Stated residual value: 0. Amounts: source units.

Year	Project A	Project B	Project C
0	−20,000	−40,000	−80,000
1	2,500	18,000	20,000
2	2,500	18,000	20,000
3	5,000	10,000	20,000
4	5,000	10,000	10,000
5	2,500	4,000	15,000
6	15,000	0	20,000
7	0	0	8,000

Measure	Project A	Project B	Project C
Payback	5.17 years (about 5 years 2 months)	2.40 years (about 2 years 5 months)	4.67 years (about 4 years 8 months)
NPV	1,529.90	8,066.64	1,275.74
IRR	11.97%	19.60%	10.53%

The legacy ARR results match Exercise 05’s figures and implicitly use a residual value of 10,000 despite this exercise stating zero. ARR is therefore not reproduced. Project B is preferred by NPV.

Exercise 05

Source ID: 2901. Discount rate: 12%. Stated residual value: 10,000, with no cash-flow treatment specified. Amounts: source units.

The introductory sentence gives a 10% discount rate, but the later calculation instruction specifies 12%. This edition uses 12% because it is the explicit instruction attached to the calculation; the source does not explain the conflict.

Year	Project A	Project B	Project C
0	−20,000	−40,000	−80,000
1	2,500	18,000	20,000
2	2,500	18,000	20,000
3	5,000	10,000	20,000
4	5,000	10,000	10,000
5	2,500	4,000	15,000
6	15,000	0	20,000
7	0	0	8,000

Base measure: displayed cash flows only	Project A	Project B	Project C
Payback	5.17 years (about 5 years 2 months)	2.40 years (about 2 years 5 months)	4.67 years (about 4 years 8 months)
NPV	−20.35	6,163.61	−3,345.37
IRR	11.97%	19.60%	10.53%

If the 10,000 residual value is an additional terminal cash inflow, the uniform end-of-year-7 scenario gives:

Terminal-residual sensitivity	Project A (year 7)	Project B (year 7)	Project C (year 7)
NPV	4,503.15	10,687.10	1,178.12
IRR	17.10%	23.06%	12.48%

Project B is preferred under either treatment. The sensitivity does not resolve what the source author intended; it shows why the timing and meaning of a residual value must be stated.

Exercise 06

Source ID: 2915. Discount rate: 12%. Stated residual value: 75,000, with no cash-flow treatment specified. Amounts: source units.

Year	Project A	Project B	Project C
0	−250,000	−250,000	−250,000
1	100,000	80,000	100,000
2	100,000	80,000	50,000
3	80,000	80,000	50,000
4	80,000	80,000	100,000
5	30,000	40,000	50,000
6	0	40,000	0
7	0	10,000	0

Base measure: displayed cash flows only	Project A	Project B	Project C
Payback	2.63 years (about 2 years 8 months)	3.13 years (about 3 years 2 months)	3.50 years (3 years 6 months)
NPV	43,811.77	40,473.76	6,657.57
IRR	19.94%	18.11%	13.13%

If 75,000 is added uniformly at the end of year 7:

Terminal-residual sensitivity	Project A (year 7)	Project B (year 7)	Project C (year 7)
NPV	77,737.96	74,399.95	40,583.76
IRR	23.68%	21.60%	17.69%

Project A has the highest NPV under both treatments. The source does not say whether the same residual value applies to every alternative; the second table is therefore a sensitivity, not a recovered source fact.

Exercise 07

Source ID: 2916. Discount rate: 10%. Stated residual value: 0. Amounts: source units.

Year	Project A	Project B	Project C
0	−20,000	−40,000	−100,000
1	5,000	8,000	25,000
2	5,000	8,000	25,000
3	8,000	8,000	25,000
4	8,000	8,000	15,000
5	1,000	10,000	25,000
6	0	10,000	25,000
7	0	10,000	0

Measure	Project A	Project B	Project C
Payback	3.25 years (3 years 3 months)	4.80 years (about 4 years 10 months)	4.40 years (about 4 years 5 months)
NPV	773.23	2,344.46	2,051.38
IRR	11.57%	11.67%	10.71%

The margins are narrow, but Project B has the highest NPV under the stated inputs.

Exercise 08

Source ID: 2917. Discount rate: 12%. Stated residual value: 0. Amounts: source units.

Year	Project A	Project B	Project C
0	−11,000	−22,000	−33,000
1	3,000	6,000	11,000
2	3,000	6,000	10,000
3	3,000	6,000	10,000
4	3,000	6,000	10,000
5	3,000	6,000	5,000
6	3,000	6,000	1,000
7	0	6,000	0

Measure	Project A	Project B	Project C
Payback	3.67 years (3 years 8 months)	3.67 years (3 years 8 months)	3.20 years (about 3 years 2 months)
NPV	1,334.22	5,382.54	1,610.12
IRR	16.19%	19.38%	14.09%

Project C pays back first. Project B has the highest NPV and is preferred under the decision rule used here.

Exercise 09

Source ID: 2955. Discount rate: 12%. Stated residual value: 0. Amounts: source units.

Year	Project A	Project B	Project C
0	−100,000	−100,000	−100,000
1	30,000	25,000	20,000
2	30,000	25,000	40,000
3	30,000	20,000	40,000
4	30,000	50,000	20,000
5	30,000	25,000	20,000
6	30,000	35,000	20,000
7	0	0	20,000

Measure	Project A	Project B	Project C
Payback	3.33 years (3 years 4 months)	3.60 years (about 3 years 7 months)	3.00 years
NPV	23,342.22	20,180.55	21,454.61
IRR	19.91%	18.36%	19.04%

Project C has the shortest payback, while Project A has the highest NPV and IRR. Project A is preferred under the stated rule.

Exercise 10

Source ID: 2967. Discount rate: 10%. Stated residual value: 25,000, with no cash-flow treatment specified. Amounts: source units.

Year	Project A	Project B	Project C
0	−100,000	−250,000	−550,000
1	20,000	50,000	100,000
2	20,000	50,000	100,000
3	50,000	100,000	100,000
4	50,000	100,000	200,000
5	20,000	100,000	100,000
6	20,000	35,000	100,000
7	0	0	100,000

Base measure: displayed cash flows only	Project A	Project B	Project C
Payback	3.20 years (about 3 years 2 months)	3.50 years (3 years 6 months)	4.50 years (4 years 6 months)
NPV	30,135.06	62,058.40	5,143.23
IRR	19.36%	17.62%	10.28%

If 25,000 is added uniformly at the end of year 7:

Terminal-residual sensitivity	Project A (year 7)	Project B (year 7)	Project C (year 7)
NPV	42,964.01	74,887.36	17,972.18
IRR	21.86%	18.74%	10.95%

Project A has the highest IRR, but Project B has the highest NPV under both residual treatments. For these mutually exclusive alternatives, Project B is preferred under the stated NPV rule.

Exercise 11

Source ID: 3064. Discount rate: 12%. Stated residual value: 10,000, with no cash-flow treatment specified. Amounts: source units.

Year	Project A	Project B	Project C
0	−75,000	−135,000	−265,000
1	17,000	25,000	50,000
2	17,000	25,000	100,000
3	17,000	50,000	100,000
4	17,000	50,000	50,000
5	17,000	25,000	50,000
6	17,000	25,000	50,000
7	17,000	25,000	50,000

Base measure: displayed cash flows only	Project A	Project B	Project C
Payback	4.41 years (about 4 years 5 months)	3.70 years (about 3 years 8 months)	3.30 years (about 3 years 4 months)
NPV	2,583.86	12,776.37	38,636.53
IRR	13.08%	14.94%	16.84%

If 10,000 is added uniformly at the end of year 7:

Terminal-residual sensitivity	Project A	Project B	Project C
NPV	7,107.35	17,299.86	43,160.03
IRR	14.78%	15.84%	17.30%

Project C has the shortest payback, highest NPV and highest IRR under both treatments.

What these exercises can and cannot decide

The calculations answer a narrow question: how three stipulated cash-flow patterns compare under a given discount rate. They do not establish whether the estimates are credible, whether risks differ, whether capital is rationed or whether an organisation should proceed with any real project.

Three checks should precede a real appraisal:

• confirm that every cash flow is future, incremental and assigned to the correct period;
• state whether residual value is already included and test changes in the assumptions that drive the result;
• examine tax, inflation, working capital, project life, risk and material effects that cannot be reduced to the supplied cash-flow table.

Payback can describe liquidity exposure, and IRR can express a break-even discount rate. Neither replaces the underlying evidence. NPV is only as reliable as the cash flows, timing and discount rate entered into it.