A company can spend ₦800m on a new plant today and receive cash over the next four years. Is it a good deal? Adding up the cash is not enough, because ₦100 received in three years is worth less than ₦100 today. A bank pays interest, so money in hand can earn.
Discounted cash flow (DCF) puts every cash flow on the same date, today, so that costs and benefits can be compared fairly.
This topic covers capital expenditure and investment appraisal, the time value of money, discounting and discount tables, the net present value (NPV) method, working capital in a project, annuities and perpetuities (including growing ones), and uses of annuity arithmetic: equivalent annual costs, loan repayments and sinking funds. The previous area covered governance. The next topic covers the internal rate of return, the modified internal rate of return and relevant costs.
The May 2014 and November 2014 Pathfinder papers tested NPV inside larger project questions.
In 30 seconds
A sum today is worth more than the same sum later, so future cash flows are discounted at the cost of capital: discount factor = 1 ÷ (1 + r)n. NPV is the present value of inflows less the present value of outflows. Accept if NPV is positive, because the project adds value. Cash flows are placed at year ends, with “time 0” as now.
Include working capital as an outflow at the start and an inflow at the end. An annuity is a constant cash flow for a set number of years, with annuity factor = [1 − (1 + r)−n] ÷ r. A perpetuity is a constant flow forever: PV = cash flow ÷ r. With constant growth g, PV = first cash flow ÷ (r − g).
Key terms in plain English
| Term | What it means |
|---|---|
| Capital expenditure | Spending on non-current assets or a new business that lasts several years |
| Investment appraisal | Judging whether a capital project is worth doing |
| Discounting | Turning a future amount into its equivalent today |
| Cost of capital | The return investors require, used as the discount rate |
| Net present value (NPV) | PV of inflows minus PV of outflows |
| Annuity | A constant cash flow each year for a fixed number of years |
| Perpetuity | A constant cash flow each year forever |
| Sinking fund | Equal yearly savings that build up to a target amount |
Syllabus check
This topic is in syllabus area B (advanced investment appraisal, 30% of the marks, the largest area). It covers:
- the time value of money and discounted cash flow;
- net present value and its decision rule;
- annuities, perpetuities and their uses in appraisal.
CAMA 2020 and investment appraisal
The Companies and Allied Matters Act 2020 (CAMA 2020) is the company law of Nigeria. The sections below apply to this topic.
The Companies and Allied Matters Act 2020 (CAMA 2020) is the company law of Nigeria.
| Section | What it says | Link to this topic |
|---|---|---|
| s. 305 | A director must act in good faith in the best interests of the company | Projects and decisions should add value to the company |
| s. 308 | A director must use the care, diligence and skill of a reasonably prudent director | Directors should test the assumptions behind a decision |
| s. 377(1) and s. 378 | Directors prepare statements that follow IFRS and give a true and fair view | Decisions later affect reported assets, profit and cash |
| s. 385 | The directors’ report covers performance and risk | Major plans and risks are reported to shareholders |
Capital expenditure and appraisal
Capital expenditure is spending on non-current assets such as buildings and equipment, or on a new business, that appears on the statement of financial position. Revenue expenditure is written off as an expense or creates a short-term asset such as inventory. Capital projects may be essential (replacing worn-out assets) or discretionary (new business that should earn a return). Examination questions mostly concern discretionary spending.
Investment appraisal evaluates proposals. A project should be undertaken only if it gives a suitable financial return and the risk is acceptable. Appraisal sits inside a capital budget, often a five-year plan reviewed each year, with approved projects inside spending limits. The steps are: generate proposals that match strategy; forecast relevant cash flows; evaluate; implement those that meet the criteria; and monitor performance against expectations.
Typical projects involve a large payment at the start, returns over several years, a possible residual value at the end and an investment in working capital. A project to buy a business has an initial outlay and possibly returns over a very long period.
The time value of money
A sum today is worth more than the same sum later, because it can be invested to earn a return. Two techniques deal with this:
- Compounding finds the future value of a sum invested at an interest rate.
- Discounting finds the present value (PV) of a future sum. It is the reverse of compounding.
Discount factor = 1 ÷ (1 + r)n, where r is the interest or cost of capital per period and n the number of periods. PV = future amount × discount factor. The present value is the sum that, invested today, would grow to the future amount, so an investor is indifferent between them. Discount tables list factors by rate and year. For example, at 10%: year 1, 0.909; year 2, 0.826; year 3, 0.751; year 4, 0.683.
Net present value
DCF has two forms, NPV and the internal rate of return (IRR). For NPV:
- List all cash flows: the initial investment, future inflows and outflows.
- Discount them at the company’s cost of capital.
- NPV = PV of benefits − PV of costs. Accept if NPV is positive. Reject if negative.
Cash flows are assumed to arise at the end of each year. “Year 0” (time 0) is now, year 1 is the end of the first year, and so on. Using NPV is consistent with the aim of maximising shareholders’ wealth, because a positive NPV is the amount by which the project should add to the value of the company.
Layouts
Format 1 lists each item with its factor. Format 2 sets out years across and nets the cash flows for each year, which saves discounting repeatedly when there are many items. Changes in working capital are cash flows. An increase, usually at the start, is an outflow. The investment is released (an inflow) at the end of the project, so working capital ends at zero.
Layout example. Machine ₦40,000, working capital ₦5,000, cash profits ₦20,000 a year for 3 years, sale of the machine ₦6,000 and recovery of working capital in year 3. Cost of capital 10%.
| Year | Item | Cash flow ₦ | Factor at 10% | PV ₦ |
|---|---|---|---|---|
| 0 | Machine | (40,000) | 1.000 | (40,000) |
| 0 | Working capital | (5,000) | 1.000 | (5,000) |
| 1 to 3 | Cash profits | 20,000 | 2.487 | 49,740 |
| 3 | Sale of machine | 6,000 | 0.751 | 4,506 |
| 3 | Recovery of working capital | 5,000 | 0.751 | 3,755 |
| NPV | 13,001 |
The NPV is positive, so the project is accepted.
Annuities and perpetuities
An annuity is a constant cash flow for a set number of years, for example ₦30,000 a year for years 1 to 5. Its present value is the cash flow times the annuity factor, which is the sum of the individual discount factors. The formula is [1 − (1 + r)−n] ÷ r. A table gives the same result. At 9%, years 1 to 3: 0.917 + 0.842 + 0.772 = 2.531. So ₦50,000 a year for three years has a PV of 50,000 × 2.531 = ₦126,550.
A perpetuity is a constant flow for ever. Perpetuity factor = 1 ÷ r. A cash flow of ₦2,000 a year from year 1 at 8% has a PV of 2,000 ÷ 0.08 = ₦25,000.
Growth
If the flow grows at a constant rate g, the growing annuity has PV = [CF1 ÷ (r − g)] × [1 − ((1 + g) ÷ (1 + r))n], where CF1 is the first cash flow. The growing perpetuity has PV = CF1 ÷ (r − g), which needs r greater than g.
Using annuity arithmetic in reverse
- Equivalent annual cash flow or cost: divide a present value by the annuity factor.
The minimum yearly return needed from a ₦70,000 project over 5 years at 9% is 70,000 ÷ 3.890 = ₦17,995. - Loan repayments: divide the amount borrowed by the annuity factor.
₦10,000,000 repaid in 5 equal yearly payments at 8% is 10,000,000 ÷ 3.993 = ₦2,504,383. - Sinking fund: equal yearly savings that reach a target. The sinking fund factor is [(1 + r)n − 1] ÷ r. To save ₦5,000,000 over 10 years at 6%, the factor is about 13.181, so the yearly saving is about ₦379,340.
The rules in more depth
Why the cost of capital is the discount rate
It is the return investors could get elsewhere at the same risk. A project must beat it to add value.
Why year-end timing is a simplification
Real cash flows arise throughout the year. The convention keeps the arithmetic simple. If a question puts a cash flow early in a year, it is usually treated as the end of the previous year.
Why NPV is an absolute measure
It gives the money value added, so a large project with a large NPV can be ranked above a small project with a higher percentage return.
Why financing costs are left out
Interest and loan repayments are in the discount rate. Counting them in the cash flows would count them twice.
Effect on analysis
List the cash flows by year and net them. Discount at the right rate. State the decision and any risk or non-financial factor that might change it.
Worked examples
Example 1: comparing cash flows over time. A borrower owes ₦120,000 in 3 years. He offers ₦140,000 if he can repay in 5 years. The lender can earn 7%.
- PV of the existing contract.
120,000 ÷ 1.073 = ₦97,955. - PV of the offer.
140,000 ÷ 1.075 = ₦99,816. - The offer has the higher PV, so it is acceptable.
Example 2: a simple NPV. A project costs ₦10,000 now and returns ₦6,000 in year 1 and ₦8,000 in year 2. Cost of capital 10%.
- PV of year 1 = 6,000 × 0.909 = ₦5,454.
- PV of year 2 = 8,000 × 0.826 = ₦6,608.
- NPV = 5,454 + 6,608 − 10,000 = ₦2,062.
Accept.
Example 3: a growing annuity. A receipt of ₦50,000 in today’s money grows at 10% a year for 5 years. Cost of capital 15%.
- First cash flow.
50,000 × 1.10 = ₦55,000. - PV = [55,000 ÷ (0.15 − 0.10)] × [1 − (1.10 ÷ 1.15)5] = 1,100,000 × (1 − 0.8006).
- PV is about ₦219,300. A year-by-year table with rounded factors gives ₦219,221, the difference being rounding.
Example 4: a loan repayment. A company borrows ₦20m, repayable in 4 equal yearly instalments at 10%.
- Annuity factor, 10%, 4 years = 3.170.
- Instalment = 20,000,000 ÷ 3.170 = ₦6,309,148.
- Total repaid.
4 × 6,309,148 = ₦25,236,592, so interest = about ₦5.24m.
How to score full marks
- Set out the cash flows by year in a table.
- Use the right factor and show the rate.
- Include working capital and its recovery.
- Leave out financing costs, depreciation and sunk costs.
- State the decision and the reason.
- Check that the signs are right and that annuities start in the right year.
Common mistakes in the exam
- Discounting depreciation, which is not a cash flow.
- Using the annuity factor for the wrong start year.
- Forgetting to release working capital at the end.
- Including interest in the cash flows as well as in the rate.
- Using the perpetuity formula when growth is higher than the rate.
- Giving a number without a recommendation.
Quick revision
- Discount factor = 1 ÷ (1 + r)n. NPV positive: accept.
- Annuity factor = [1 − (1 + r)−n] ÷ r.
Perpetuity = CF ÷ r.
Growing perpetuity = CF1 ÷ (r − g). - Working capital: out at the start, back at the end.
- Equivalent annual cash flow, loan repayment and sinking fund use annuity factors in reverse.
- CAMA 2020: ss. 305, 308, 377(1), 378, 385.
Practice questions
These are original questions written in the style of the Institute of Chartered Accountants of Nigeria (ICAN) exam. Try each one before you open the answer. Full model answers follow the written questions.
Recall check: multiple choice
Recent ICAN papers are written, although older papers such as May 2014 included multiple-choice questions. These questions are for quick recall. The written questions that follow are closer to the exam.
Question 1. A sum received in the future is worth:
- More than the same sum today
- Less than the same sum today
- The same as today
- Nothing
Show the answer
B. Because money today can earn a return.
Question 2. The discount factor for year n is:
- 1 ÷ (1 + r)^n
- (1 + r)^n
- r ÷ n
- 1 − r
Show the answer
A. The inverse of the compound factor.
Question 3. A project should be accepted when its NPV is:
- Negative
- Zero only
- Positive
- Below the IRR
Show the answer
C. It adds value.
Question 4. Which item is NOT included in NPV cash flows?
- Working capital
- Depreciation
- Scrap value
- Cash profits
Show the answer
B. It is not a cash flow.
Question 5. Working capital in a project is usually:
- An inflow at the start
- An outflow at the start and an inflow at the end
- Ignored
- Depreciated
Show the answer
B. It is released at the end.
Question 6. The PV of a perpetuity of ₦2,000 at 8% is:
- ₦16,000
- ₦25,000
- ₦2,160
- ₦250
Show the answer
B. 2,000 ÷ 0.08.
Question 7. A growing perpetuity has PV equal to:
- CF1 ÷ (r − g)
- CF1 ÷ r
- CF1 × r
- CF1 × (1 + g)
Show the answer
A. Provided r is greater than g.
Question 8. To find the equivalent annual cost, divide the PV by:
- The discount factor
- The annuity factor
- The perpetuity factor
- The tax rate
Show the answer
B. For the same period and rate.
Question 9. Interest on a project loan should be:
- Included in the cash flows
- Left out of the cash flows, because it is in the discount rate
- Discounted twice
- Deducted from NPV
Show the answer
B. Otherwise it is counted twice.
Question 10. Which section of CAMA 2020 sets the director’s duty of care and skill?
- Section 308
- Section 400
- Section 405
- Section 409
Show the answer
A. Directors should test the assumptions in an appraisal.
Written questions
Question 11 (10 marks). (a) Explain why future cash flows are discounted. (3 marks) (b) State the steps in an NPV appraisal and the decision rule. (4 marks) (c) State the treatment of working capital in an NPV calculation. (3 marks)
Question 12 (20 marks). Biso Plc is considering a plant costing ₦800m. It needs working capital of ₦100m at the start, which is recovered at the end of year 4. Cash profits are ₦300m a year for years 1 to 4, and the plant can be sold for ₦50m at the end of year 4. The cost of capital is 12% (annuity factor for 4 years 3.037, discount factor for year 4 0.636).
(a) Calculate the NPV and advise. (8 marks) (b) The company wants to borrow ₦50m, repaid in 5 equal yearly instalments at 10% (annuity factor 3.791). Calculate the instalment. (4 marks) (c) A property will pay ₦24m a year for ever. Calculate its value at 8%. (3 marks) (d) Explain why interest is not included in the project cash flows. (5 marks)
Question 13 (10 marks). (a) A company expects dividends of ₦20m next year, growing at 5% a year for ever. Calculate the present value at 13%. (4 marks) (b) Explain the difference between an annuity and a perpetuity and give a use of each in appraisal. (6 marks)
Model answers
Show the model answer to Question 11
(a) Why discount (3 marks). A sum today can be invested to earn a return, so it is worth more than the same sum later. (1.5) Discounting puts all cash flows on the same date so that they can be compared. (1.5)
(b) Steps and rule (4 marks). List all cash flows. (1) Discount them at the cost of capital. (1) Subtract the PV of costs from the PV of benefits. (1) Accept if NPV is positive and reject if negative. (1)
(c) Working capital (3 marks). The investment is an outflow, usually at the start. (1.5) It is released as an inflow at the end, so the balance returns to nil. (1.5)
Show the model answer to Question 12
(a) NPV (8 marks).
- PV of cash profits.
300 × 3.037 = ₦911.1m. (2) - PV of sale.
50 × 0.636 = ₦31.8m. (1.5) - PV of working capital recovered.
100 × 0.636 = ₦63.6m. (1.5) - Outflows = 800 + 100 = ₦900m. (1)
- NPV = 911.1 + 31.8 + 63.6 − 900 = ₦106.5m. (1)
- It is positive, so accept. (1)
(b) Instalment (4 marks). 50m ÷ 3.791 = ₦13.19m a year. (4)
(c) Perpetuity (3 marks). 24 ÷ 0.08 = ₦300m. (3)
(d) Interest (5 marks). The cost of capital already reflects the return that lenders and shareholders require. (2.5) Counting interest in the cash flows as well would count the cost of finance twice and understate NPV. (2.5)
Show the model answer to Question 13
(a) Growing perpetuity (4 marks). PV = 20 ÷ (0.13 − 0.05) = 20 ÷ 0.08 = ₦250m. (4)
(b) Annuity and perpetuity (6 marks). An annuity is a constant cash flow for a fixed number of years, and a perpetuity is a constant flow for ever. (2) Annuity use: yearly cash profits over a project life, or a loan repayment. (2) Perpetuity use: valuing a long-lived income stream or a share with constant dividends. (2)
Pathfinder on NPV, annuities and perpetuities
These are the ICAN Pathfinder questions that test this topic. ICAN owns the questions and the suggested solutions, so we point you to them and do not copy them. Open your Pathfinder, attempt each question under exam conditions, then check your answer against the notes above.
The compulsory 30-mark question asked for a project NPV for a cement plant at a project-specific cost of capital of 13%. The model answer lists the cash flows by year, takes tax on operating profit, adds the tax saving on tax depreciation, includes working capital at 10% of revenue as an outflow at the start of each year and a recovery at the end, and leaves out loan repayments, interest, apportioned overheads and sunk survey costs. Worked example 2 and the layout table show the same method on smaller numbers.
Revise first: Net present value
Questions use annuity factors for equivalent annual costs, loan repayments and sinking funds. See the section on annuities and perpetuities.
Revise first: Annuities and perpetuities
Source: ICAN Pathfinder, Strategic Financial Management. Diets covered: November 2014 question 1 and May 2014 question 1 (NPV inside larger questions).
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