Project Appraisal — Payback, ARR, Time Value of Money, NPV & IRR
What project appraisal is and its key issues, non-discounting methods (payback, ARR), the time value of money and discounting, NPV with the four-project example, IRR by interpolation (23.38%), and which method is best.
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Project Appraisal: Part 1
Project appraisal, its key issues, and the non-discounting methods: payback period and accounting rate of return.
00:04Introduction and lecture outcomes01:58What project appraisal is (the Hambantota Port example)03:41Goal: maximise the firm's wealth04:41Discounting vs non-discounting techniques06:27Payback period: definition and acceptance criteria08:07Example: project A ($900k) vs project B ($1m)13:31Payback advantages14:38Payback disadvantages18:56Discounted payback period at 20%25:19Project A 3.3 years vs project B 2.96 years27:39Accounting rate of return (ARR)30:06ARR acceptance criteria32:28ARR example: $400k machine → 31.875%36:10ARR advantages and disadvantagesProject Appraisal: Part 2
Time value of money, discounting, NPV (four-project example) and IRR (interpolation example).
00:48Lecture outcomes: NPV and IRR02:25Why time matters: project A ($110 in year 1) vs B ($121 in year 2)04:48Three reasons for the time value of money05:29Opportunity cost07:48Inflation09:47Uncertainty and risk (risk premium)12:59Discounting: PV = FV ÷ (1 + r)ⁿ13:37Example: 3,000 today vs 1,100 + 1,210 + 1,33115:36The NPV method and its steps17:45NPV decision rules19:41Example: four projects at 12%23:31Limited resources: choose the highest NPV24:56Advantages of NPV26:22IRR: definition27:44Calculating IRR by trial and interpolation30:26Example: $400k, $200k a year for 3 years → IRR 23.38%33:22Comparing the four techniques: NPV is best1What is project appraisal?
Project appraisal is the process of assessing and questioning proposals before resources are committed. It evaluates whether an investment project is worthwhile.
Key issues of project appraisal:
- environmental factors
- options (alternatives)
- inputs
- value for money
- implementation
- risk and uncertainty
| Non-discounting methods (ignore the time value of money) | Discounting methods (consider the time value of money) |
|---|---|
| Payback period | Net present value (NPV) |
| Accounting rate of return (ARR) | Internal rate of return (IRR) |
2Non-discounting methods
Payback period
Payback is the length of time taken to repay the initial capital cost from the project's cash inflows 06:27.
- Equal cash flows: payback = initial investment ÷ annual cash inflow.
- Unequal cash flows: accumulate the cash flows year by year until the cost is recovered. Then add the fraction of the year: (amount still needed ÷ that year's cash flow).
- Decision: accept if payback ≤ the benchmark period. Among mutually exclusive projects, choose the shortest payback.
| Year | Project A | Cumulative A | Project B | Cumulative B |
|---|---|---|---|---|
| 0 | (900,000) | (900,000) | (1,000,000) | (1,000,000) |
| 1 | 700,000 | (200,000) | 600,000 | (400,000) |
| 2 | 100,000 | (100,000) | 400,000 | 0 |
| 3 | 100,000 | 0 | 400,000 | 400,000 |
| 4 | 1,300,000 | 1,300,000 | 400,000 | 800,000 |
Payback: A = 3 years, B = 2 years, so choose B 12:51.
Notice: payback ignores A's huge 1.3m in year 4. That's its main weakness 15:08.
| Advantages 13:31 | Disadvantages 14:38 |
|---|---|
| Easy to calculate and interpret | Ignores cash flows after payback |
| Favours quick liquidity; less risk | The benchmark period is arbitrary 16:05 |
| Facilitates further evaluation: a first screen before NPV and IRR | Ignores the time value of money 17:59 |
Discounted payback period
This fixes the time-value problem: discount each cash flow first, then do payback on the discounted figures 18:56.
| Year | DF = 1/1.2ⁿ | A: PV | A: cumulative | B: PV | B: cumulative |
|---|---|---|---|---|---|
| 1 | 0.8333 | 583,333 | 583,333 | 500,000 | 500,000 |
| 2 | 0.6944 | 69,444 | 652,778 | 277,778 | 777,778 |
| 3 | 0.5787 | 57,870 | 710,648 | 231,481 | 1,009,259 |
| 4 | 0.4823 | 626,929 | 1,337,577 | 192,901 |
- A = 3 + (900,000 − 710,648) ÷ 626,929 = 3.30 years
- B = 2 + (1,000,000 − 777,778) ÷ 231,481 = 2.96 years
Choose B 27:14.
Accounting rate of return (ARR)
PDF: ÷ initial cost. Video: ÷ average investment = (cost + scrap) ÷ 2
- ARR compares accounting profit (after depreciation and tax), not cash flows, with the investment 27:43.
- Decision: accept if ARR ≥ the predetermined minimum ARR (benchmark); choose the highest ARR 30:06.
| Year | 1 | 2 | 3 | 4 | Total |
|---|---|---|---|---|---|
| Profit after 15% tax | 51,000 | 68,000 | 59,500 | 76,500 | 255,000 |
- Average profit = 255,000 ÷ 4 = 63,750
- Average investment = (400,000 + 0) ÷ 2 = 200,000
- ARR = 63,750 ÷ 200,000 = 31.875%. That's above 20%, so invest 35:41.
On initial cost (the PDF formula), ARR = 63,750 ÷ 400,000 = 15.9%. State which base you use.
| Advantages | Disadvantages |
|---|---|
| Easy to compute and understand | Ignores the time value of money |
| Uses averages, so projects with different lives can be compared 36:31 | Ignores cash flows: profit can be manipulated 37:14 |
3Time value of money
Money available today is worth more than the same amount in the future, because of its earning potential: money can earn interest and grow.
Three reasons 04:48:
- Opportunity cost: money in hand could earn a return elsewhere, such as a fixed deposit. Waiting means giving up the best alternative 05:29.
- Inflation: prices rise, so the purchasing power of money falls. Rs 1,000 buys fewer loaves next year 07:48.
- Uncertainty and risk: the future is uncertain (nobody predicted COVID in 2019), so investors demand a risk premium 09:47.
Discount factor = 1 ÷ (1 + i)ⁿ
At 10%: receive 3,000 today, or receive 1,100 (year 1) + 1,210 (year 2) + 1,331 (year 3)?
| Year | Cash | ÷ (1.1)ⁿ | PV |
|---|---|---|---|
| 1 | 1,100 | 1.100 | 1,000 |
| 2 | 1,210 | 1.210 | 1,000 |
| 3 | 1,331 | 1.331 | 1,000 |
| Total | 3,000 |
The two options have the same present value. Likewise, $110 in year 1 and $121 in year 2 are both worth $100 today 02:25.
4Net present value (NPV)
Steps 16:25:
- Identify the discount rate (cost of capital).
- Identify the project's cash flows.
- Discount each cash flow to present value.
- Add them and deduct the initial investment.
| NPV | Decision 17:45 |
|---|---|
| > 0 | Accept: the return beats an equal-risk alternative |
| < 0 | Reject |
| = 0 | Indifferent: accept if the project is necessary |
| Several projects with limited funds | Choose the highest NPV |
| Project | Initial investment | NPV | Decision |
|---|---|---|---|
| A | 900,000 (cash flows 120k, 250k, 400k, 1.3m) | +517,327 | Accept |
| B | 1,000,000 | +214,920 | Accept |
| C | 303,730 | 0 | Indifferent |
| D | 1,500,000 | negative | Reject |
Resources for only one project: choose A, the highest NPV 24:33.
Check for A:
| Year | Working | PV |
|---|---|---|
| 1 | 120,000 ÷ 1.12 | 107,143 |
| 2 | 250,000 ÷ 1.2544 | 199,298 |
| 3 | 400,000 ÷ 1.4049 | 284,712 |
| 4 | 1,300,000 ÷ 1.5735 | 826,174 |
| Total PV | 1,417,327 | |
| Less investment | (900,000) | |
| NPV | 517,327 ✓ |
Advantages of NPV 24:56:
- it uses the time value of money
- it considers all cash flows over the project's life
- it uses cash flows, not accounting profit (which can be manipulated)
- it's consistent with maximising shareholder wealth
5Internal rate of return (IRR)
IRR is the discount rate that makes NPV = 0: discounted inflows exactly equal discounted outflows 26:22.
Accept if IRR ≥ the cost of capital (benchmark); reject if it's lower.
- IRR helps measure the worth of an investment against internal standards of return.
- It allows comparison of projects with different initial outlays.
Calculating IRR (trial and interpolation) 27:44:
- Find the NPV at a rate that gives a positive NPV (P).
- Find it at a higher rate that gives a negative NPV (N).
- Interpolate:
L = lower rate, H = higher rate, P = NPV at L (positive), N = NPV at H (negative)
Cost $400,000; inflows $200,000 a year for 3 years; benchmark 15%.
| Rate | Annuity factor (3 years) | PV of inflows | NPV |
|---|---|---|---|
| 22% | 2.0422 | 408,440 | +8.4k (P) |
| 24% | 1.9813 | 396,260 | −3.8k (N) |
IRR = 22 + [8.4 ÷ (8.4 − (−3.8))] × (24 − 22) = 22 + (8.4 ÷ 12.2) × 2 = 23.38% 32:24
Since 23.38% > 15%, accept.
Watch the signs: P − N = 8.4 + 3.8.
IRR disadvantages:
- complex
- time-consuming
- needs expertise to calculate and interpret 32:45
6Which method is best?
| Method | Time value of money? | All cash flows? | Basis | Verdict |
|---|---|---|---|---|
| Payback | ✗ | ✗ (ignores later flows) | Cash | Simple, liquidity-focused |
| ARR | ✗ | ✓ | Accounting profit | Simple; profit can be manipulated |
| NPV | ✓ | ✓ | Cash | Best: wealth in absolute Rs 33:38 |
| IRR | ✓ | ✓ | Cash | Good, but complex; gives a % |
Calculate:
- payback, especially with unequal flows
- ARR
- NPV using discount factors
- IRR by interpolation (watch the minus sign)
Then:
- state the accept or reject decision and choose between projects
- explain the time value of money (opportunity cost, inflation, risk)
- give the pros and cons of each method, and why NPV is best
7Quick revision sheet
| Concept | Remember it as |
|---|---|
| Project appraisal | Assess proposals before committing resources |
| Key issues | Environment · options · inputs · value for money · implementation · risk |
| Non-discounting | Payback, ARR (ignore the time value of money) |
| Discounting | NPV, IRR |
| Time value of money | Rs today > Rs tomorrow: opportunity cost, inflation, risk |
| PV | FV ÷ (1 + i)ⁿ |
| Payback | Time to recover the cost; shorter is better (A 3 years vs B 2 years) |
| Discounted payback | Payback on PVs (A 3.30 vs B 2.96 years at 20%) |
| ARR | Average profit ÷ investment (63,750 ÷ 200,000 = 31.875%) |
| NPV | Σ PV of inflows − investment; > 0 accept, highest wins |
| Video NPV | A 517,327 · B 214,920 · C 0 · D negative → choose A |
| IRR | Rate where NPV = 0; L + P ÷ (P − N) × (H − L) |
| Video IRR | 22 + 8.4 ÷ 12.2 × 2 = 23.38% > 15% → accept |
| Best method | NPV |
?Self-check quiz
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