Discounted Investment Appraisal Techniques (Net Present Value and IRR)

CIMA-P2 · Chapter 11

Discounted Investment Appraisal Techniques (Net Present Value and IRR)

1 Introduction

Chapter 10 covered two appraisal methods that ignore the time value of money. This chapter covers the two that do not: net present value (NPV) and the internal rate of return (IRR). NPV is the technically correct method and the one the rest of this section of the syllabus is built on.

The chapter also covers two shortcuts that appear constantly in exam questions — annuities and perpetuities — which are simply faster ways of discounting a cash flow that repeats unchanged.

2 Discounted cash flow — net present value

This recording works Examples 1, 2 and 3 and covers the whole of sections 2 and 3. It is sound throughout; the notices are about what it points you at, not about what it teaches.

One. The tutor says the present value and annuity tables are "at the beginning of the notes … on page three". They are in the front matter of the printed 2023 PDF and are not in this digital chapter set. Use the formulae in sections 2.2, 4.1 and 4.2, which give the same numbers, and use the tables supplied in the examination itself.

Two. He describes the tables as running "up to 20%, for any number of years, up to 15%". He means up to 15 years; the 15% is a slip.

Three. The recording is in pounds throughout — "£80,000", "£11,500". The notes are in dollars and the figures are otherwise identical.

Four. Example 4 is not on the tape, although the closing minutes of this recording — on how far the cost of capital could be wrong before the decision changes — are the substance of its answer.

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The discounted cash flow approach takes the net cash flows arising from an investment decision and adjusts them for the time value of money.

2.1 The time value of money

Time value of money

$1 received or paid today is worth more than $1 received or paid at some future date, and the difference grows the further into the future the flow lies.

The reason is the cost of money itself. If the business has borrowed to fund the project it pays interest; if it used its own cash it has given up the interest that cash could have earned. Either way, money tied up in a project has a cost, and that cost is what discounting measures.

The time value of money is brought into an appraisal by discounting future cash flows at the company's required rate of return — variously called the cost of capital, the required rate, or the weighted average cost of capital. Each future cash flow is converted into its present value: the amount it is worth today.

2.2 Discount factors

The present value of a single cash flow is found by multiplying it by a discount factor:

Discount factor = 1 ÷ (1 + r)ⁿ

where r is the cost of capital as a decimal and n is the number of years away that the cash flow arises.

Present value tables give these factors, rounded to three decimal places, for rates up to 20% and for up to 15 years. They are provided in the P2 objective test examination. Where the rate you are given is not in the tables — 4.5%, say — use the formula above.

The sum of the discounted cash flows is netted against the initial cost of the project. If the result is positive the project is financially acceptable; if it is negative the project should be rejected.

The net present value is expressed as an absolute amount of money, not as a percentage or a ratio. That is one of its strengths — it states how much better off the shareholders are — and it is why NPV cannot be used on its own to rank projects of very different sizes when capital is rationed (chapter 13).

2.3 Interpreting the net present value

Result	What it means	Decision
NPV > 0 (positive)	The project earns more than the required rate of return.	Financially viable — accept.
NPV = 0	The project earns exactly the required rate of return; it breaks even in present value terms.	Marginal — normally accepted, since the required return has been met in full.
NPV < 0 (negative)	The project earns less than the required rate of return.	Not financially viable — reject.

The decision rule is NPV greater than zero, not greater than one. A rule written as "NPV > 1 (positive value)" would reject a project with an NPV of $0.50 — a positive value — under a rule that calls itself positive. There is no threshold: any NPV above zero means the project earns more than the cost of capital.

Net present value

A machine will cost $80,000. It has an expected life of 4 years with an anticipated scrap value of $10,000.

Expected net operating cash inflows for each year are as follows:

Year	$
1	20,000
2	30,000
3	40,000
4	10,000

The cost of capital is 10% p.a.

Required

Calculate the net present value of the investment and determine whether the project should be accepted.

Net present value

Time	Cash flow $	Discount factor @ 10%	Present value $
0	(80,000)	1.000	(80,000)
1	20,000	0.909	18,180
2	30,000	0.826	24,780
3	40,000	0.751	30,040
4	20,000	0.683	13,660
		Net present value	6,660

The year 4 flow is $20,000, not $10,000: the $10,000 operating inflow and the $10,000 scrap proceeds both arise at the end of year 4 and are discounted together.

The net present value is positive, $6,660, so the project earns more than the 10% required and should be accepted.

Beyond the net present value

Consider the project appraised in Example 1.

Required

Suggest other factors that should be considered before the investment is accepted.
State the advantages of the net present value method.
State the disadvantages of the net present value method.

Beyond the net present value

(a) Other factors

Are the cash flow forecasts reasonable, and were they produced by a reliable method? Every figure after time 0 is an estimate.
Is the cost of capital accurate? The whole appraisal turns on it, and it is difficult to measure (which is the reason for section 3).
Can the initial outlay be funded, and can the business afford to have the money tied up for four years?
Does the project fit the strategy — is it compatible with existing operations, will staff need training, will the new machine work with current systems?
Qualitative factors: environmental and ethical consequences, the effect on employee morale, and the effect on the quality of the product or service.
How sensitive is the conclusion to each estimate? An NPV of $6,660 on an $80,000 outlay is not a large margin (chapter 15).

(b) Advantages of NPV

It uses cash flows, which are objective, rather than profits, which depend on accounting policy.
It accounts for the time value of money, so a dollar in year 4 is not treated as equal to a dollar in year 1.
It uses all the cash flows over the whole life of the project — nothing is ignored after a cut-off date, as it is under payback.
It gives an absolute figure in money terms, which is directly the increase in shareholder wealth, so it is consistent with the objective of the business.
The results of separate projects are additive: two projects' NPVs can simply be added.

(c) Disadvantages of NPV

It depends on an accurate cost of capital, which in practice is very hard to establish and is assumed — usually wrongly — to stay constant over the project's life.
It depends on cash flow estimates that reach years into the future.
It is not intuitive to non-financial managers in the way payback is.
It ignores everything that cannot be expressed as a cash flow, which is most of what part (a) lists.
Shareholders are given a profit statement each year, and a project that is right on NPV can still depress reported profit and divisional return in its early years.

Any other reasonable factor is acceptable in part (a); the examiner is looking for the range of considerations, not a fixed list.

3 The internal rate of return (IRR)

The practical difficulty with discounted cash flow is that it is very hard to establish a company's true cost of capital. In Example 1 the project was accepted on the basis that the cost of capital is 10%. Suppose it is not 10% but 15%: a higher rate discounts the cash flows more heavily, so the NPV will be lower, and it may become negative.

Because the cost of capital is uncertain, it is useful to know the break-even rate of return for the project — the discount rate at which its NPV is exactly zero. That rate is the internal rate of return.

Internal rate of return

The discount rate at which the net present value of a project is zero.

The IRR is found by interpolation: the cash flows are discounted at two different rates, one giving a positive NPV and one a negative NPV, and the answer is apportioned between them.

IRR ≈ L + [NPV_L ÷ (NPV_L − NPV_H)] × (H − L)

where L is the lower discount rate, H the higher discount rate, NPV_L the NPV at the lower rate and NPV_H the NPV at the higher rate.

The result is only an approximation, because the relationship between the discount rate and the NPV is a curve rather than a straight line and interpolation treats it as straight. That is why the symbol is ≈, "approximately equal to". The approximation is good enough: the answer is used to judge how much room there is for error in the cost of capital, not as a precise figure.

Which two rates you choose does not matter much. If the question tells you which to use, use them. If not, any two sensible guesses give an answer within a fraction of a percent of each other — the technique is robust, and it is far quicker than plotting the curve.

Internal rate of return

Using the project cash flows and the result obtained in Example 1:

Required

Calculate the net present value of the project at a discount rate of 15%.
Estimate the internal rate of return of the project, using your result from Example 1 and your answer to part (a).
Explain the significance of the internal rate of return you have calculated.

Internal rate of return

(a) Net present value at 15%

Time	Cash flow $	Discount factor @ 15%	Present value $
0	(80,000)	1.000	(80,000)
1	20,000	0.870	17,400
2	30,000	0.756	22,680
3	40,000	0.658	26,320
4	20,000	0.572	11,440
		Net present value	(2,160)

Negative, so at a 15% cost of capital the project would be rejected.

(b) Internal rate of return

IRR ≈ 10% + [6,660 ÷ (6,660 − (−2,160))] × (15% − 10%)

= 10% + (6,660 ÷ 8,820) × 5% = 10% + 3.78% = 13.78%

Note the denominator: the two NPVs have opposite signs, so subtracting the negative one adds it. It is the total distance between them — 8,820 — that is equivalent to the 5 percentage points between the two rates.

(c) Significance

Part (b) tells us that the internal rate of return of the project is approximately 13.78%. Discounting the project's cash flows at that rate would give an NPV of approximately zero.

It is the break-even cost of capital. The project is worthwhile provided the true cost of capital is below 13.78%; above that, it is not.
It therefore measures how sensitive the decision is to the cost of capital estimate. We believe the cost of capital is 10%. There is 3.78 percentage points of room before the decision reverses — enough to be comfortable, but not a wide margin. Had the IRR come out at 11%, a small error in the estimate would flip the decision and the project would be a much riskier proposition.
It can be read as the return the project earns on the money invested in it, which makes it easier to explain to non-financial managers than an absolute NPV.

Advantages and disadvantages of IRR

Required

State the advantages of using the internal rate of return.
State the disadvantages of using the internal rate of return.

Advantages and disadvantages of IRR

(a) Advantages

It uses cash flows and recognises the time value of money, as NPV does.
It does not require the cost of capital to be known in order to be calculated — which is precisely the difficulty that motivates it.
It is a percentage, so it is easily understood and easily compared with a target return or with the returns available elsewhere.
It shows the margin of safety in the cost of capital estimate, which an NPV on its own cannot.

(b) Disadvantages

It is a relative measure and ignores the size of the project. A $10,000 project with an IRR of 40% adds less to shareholder wealth than a $10 million project with an IRR of 15%.
It can therefore rank mutually exclusive projects in a different order from NPV, and where the two conflict NPV is correct.
A project whose cash flows change sign more than once — an outflow, then inflows, then a large outflow for decommissioning — can have more than one IRR, and none of them is meaningful.
It assumes that interim cash flows can be reinvested at the IRR itself, which for a high-IRR project is unlikely to be true.
The interpolation method gives only an approximation, because the NPV curve is not linear.

4 Annuities and perpetuities

This recording works Examples 5 to 8 and is correct throughout. Three notices.

One. It is in pounds; the notes are in dollars.

Two. The tutor misreads Example 5's opening line as "a machine will cost £45,000 a year" and corrects himself immediately — the cost is a one-off $45,000.

Three. As in the previous recording, the annuity tables he refers to are in the printed front matter and not in this digital set. Section 4.1 gives the formula.

The perpetuity section is illustrated with a divorce-settlement analogy. The arithmetic it carries — $12,000 a year forever is worth $120,000 today at 10%, because $120,000 in the bank generates $12,000 a year — is the clearest explanation of 1 ÷ r anywhere in the paper.

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4.1 Annuities

Most appraisal questions produce a different cash flow in each year, so each year has to be discounted separately. Occasionally, though, a payment or receipt is the same amount each year for a fixed number of years — a fixed annual rent, for instance. A constant annual cash flow for a fixed number of years is an annuity.

Because the amount is the same each year, it can be multiplied by the total of the individual discount factors rather than by each of them in turn. That total is the annuity factor, and it is given by:

Annuity factor = (1 − (1 + r)⁻ⁿ) ÷ r

where r is the discount rate as a decimal and n the number of years. Cumulative present value tables give these factors and are provided in the P2 objective test examination; be able to use both the tables and the formula.

The annuity factor assumes the first cash flow arrives at the end of year 1. Where the flow starts later, the factor has to be adjusted — see Example 6. Reading "years 4 to 13" as a ten-year annuity factor, or as a 14-year one, are the two errors this topic generates.

Annuity

A machine will cost $45,000 and is expected to generate $8,000 p.a. for the following 8 years.

The cost of capital is 15% p.a.

Required

Calculate the net present value of the investment.

Annuity

Time	Cash flow $	Discount factor @ 15%	Present value $
0	(45,000)	1.000	(45,000)
1 – 8	8,000	4.487	35,896
		Net present value	(9,104)

4.487 is the 8-year annuity factor at 15% — the total of the eight individual discount factors. Discounting the $8,000 eight separate times gives the same answer and takes eight times as long.

The net present value is negative, $(9,104), so the project should be rejected.

Deferred annuity

The cost of capital is 12% p.a.

Required

What is the present value of $20,000 that is first receivable in 4 years' time and thereafter each year for a total of 10 receipts?

Deferred annuity

Ten receipts, the first at time 4, so the last is at time 13 — count them: 4, 5, 6, 7, 8, 9, 10, 11, 12, 13. The annuity runs from year 4 to year 13.

Annuity tables start at year 1, so the factor for years 4 to 13 is found by subtraction:

Years	Annuity factor @ 12%
1 – 13	6.424
Less: 1 – 3	(2.402)
4 – 13	4.022

PV = 20,000 × 4.022 = $80,440

4.2 Perpetuities

Where an identical cash flow is expected to continue indefinitely it is a perpetuity. Nothing genuinely lasts forever, but a flow expected to continue for so long that its end cannot sensibly be forecast is treated as one.

The present value factor for a perpetuity is:

Perpetuity factor = 1 ÷ r

where r is the discount rate as a decimal. The logic is worth holding on to: at 10%, $120,000 placed on deposit generates $12,000 a year forever, so the right to $12,000 a year forever is worth $120,000 today. That is 12,000 ÷ 0.10.

Perpetuity factors are not provided in the objective test examination, so 1 ÷ r must be remembered. And, exactly as with annuities, 1 ÷ r assumes the first flow arrives at the end of year 1 — a deferred perpetuity needs the same subtraction (Example 8).

Perpetuity

A machine costs $100,000 and is expected to generate $12,000 p.a. in perpetuity.

The cost of capital is 10% p.a.

Required

What is the net present value of the project?

Perpetuity

Time	Cash flow $	Factor @ 10%	Present value $
0	(100,000)	1.000	(100,000)
1 – ∞	12,000	10.000	120,000
		Net present value	20,000

The factor is 1 ÷ 0.10 = 10. The net present value is positive, $20,000, so the project should be accepted.

Deferred perpetuity

The rate of interest is 5% p.a.

Required

What is the present value of $18,000 first receivable in 5 years' time and thereafter annually in perpetuity?

Deferred perpetuity

The flow runs from year 5 to infinity. Take the factor for years 1 to infinity and subtract the factor for years 1 to 4:

Years	Factor @ 5%
1 – ∞ (= 1 ÷ 0.05)	20.000
Less: 1 – 4 (annuity factor)	(3.546)
5 – ∞	16.454

PV = 18,000 × 16.454 = $296,172

5 Choosing between NPV and IRR

Both methods use cash flows and both recognise the time value of money, so on an ordinary accept-or-reject decision they agree: a project with a positive NPV at the cost of capital has an IRR above the cost of capital, and vice versa.

They can disagree when ranking mutually exclusive projects, because IRR is a percentage and takes no account of scale. Where they conflict, NPV is correct, because NPV measures the absolute increase in shareholder wealth and that is the objective. IRR's proper role is to say how much room there is for the cost of capital to be wrong.

Discount each future cash flow by 1 ÷ (1+r)ⁿ and add them up. NPV > 0, accept — not NPV > 1. IRR is the rate at which NPV is zero, found by interpolating between one positive and one negative NPV, and it measures the margin of error in the cost of capital. An annuity factor is (1 − (1+r)⁻ⁿ) ÷ r; a perpetuity factor is 1 ÷ r; both assume the first flow is at the end of year 1, so subtract for a deferred start.

6 Test your knowledge

Two quick checks before you move on: work through the flashcards to fix this chapter’s key terms and definitions, then sit the objective questions for exam-style practice. Both mark themselves and explain the answers as you go.

Answer the questions one at a time. Your progress is saved so you can leave and come back.

Open chapter practice