CIMA - C3 Fundamentals Of Business Mathematics

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240 9: Discounting and basic investment appraisal ⏐ Part D Financial mathematics

2.4 Example: Expected net present value

An organisation with a cost of capital of 5% is contemplating investing $340,000 in a project which has a 25%

chance of being a big success and producing cash inflows of $210,000 after one and two years. There is, however,

a 75% chance of the project not being quite so successful, in which case the cash inflows will be $162,000 after

one year and $174,000 after two years.

Required

Calculate an NPV and hence advise the organisation.

Solution

Discount

Success

Failure

Year

factor

Cash flow

$'000

1.000

(340)

(340.00)

(340)

(340.000)

0.952

210

199.92

162

154.224

0.907

210

190.47

174

157.818

50.39

(27.958)

NPV = (25% × 50.39) + (75% × –27.958) = –8.371

The NPV is –$8,371 and hence the organisation should not invest in the project.

2.5 Limitations of using the NPV method

There are a number of problems associated with using the NPV method in practice.

(a) The future discount factors (or interest rates) which are used in calculating NPVs can only be

estimated and are not known with certainty. Discount rates that are estimated for time periods far

into the future are therefore less likely to be accurate, thereby leading to less accurate NPV values.

(b) Similarly, NPV calculations make use of estimated

future cash flows. As with future discount

factors, cash flows which are estimated for cash flows several years into the future cannot really be

predicted with any real certainty.

at the end of the

year

. However, this assumption is also likely to give rise to less accurate NPV values.

There are a number of computer programs available these days which enable a range of NPVs to be calculated for a

number of different circumstances (best-case and worst-case situations and so on). Such programs allow some of the

limitations mentioned above to be alleviated. We will look at how Excel can be used to calculate NPVs at the end of this

chapter.

Part D Financial mathematics ⏐ 9: Discounting and basic investment appraisal 241

3 The Internal Rate of Return (IRR) method

3.1 IRR method

The IRR method determines the rate of interest (the IRR) at which the NPV is 0. Interpolation, using the following

formula, is often necessary. The project is viable if the IRR exceeds the minimum acceptable return.

IRR = a% + %)ab(

NPVNPV

NPV

⎥

⎦

⎤

⎢

⎣

⎡

−

⎟

⎠

⎞

⎜

⎝

⎛

−

The internal rate of return (IRR) method of evaluating investments is an alternative to the NPV method. The NPV

method of discounted cash flow determines whether an investment earns a

positive or a negative NPV when

discounted at a given rate of interest

. If the NPV is zero (that is, the present values of costs and benefits are

equal) the return from the project would be exactly the rate used for discounting.

The IRR method will indicate that a project is viable

if the IRR exceeds the minimum acceptable rate of return.

Thus if the company expects a minimum return of, say, 15%, a project would be viable if its IRR is more than 15%.

3.2 Example: The IRR method over one year

If $500 is invested today and generates $600 in one year's time, the internal rate of return (r) can be calculated as

follows.

PV of cost = PV of benefits

500 =

)r1(

600

500 (1 + r) = 600

1 + r =

500

600

= 1.2

r = 0.2 = 20%

3.3 Interpolation method

The arithmetic for calculating the IRR is more complicated for investments and cash flows extending over a period

of time longer than one year. A technique known as the

interpolation method can be used to calculate an

approximate IRR.

3.4 Example: Interpolation

A project costing $800 in year 0 is expected to earn $400 in year 1, $300 in year 2 and $200 in year 3.

Required

Calculate the internal rate of return.

T F

RWAR

242 9: Discounting and basic investment appraisal ⏐ Part D Financial mathematics

Solution

The IRR is calculated by first of all finding the NPV at each of two interest rates. Ideally, one interest rate should

give a small positive NPV and the other a small negative NPV. The IRR would then be somewhere between these

two interest rates: above the rate where the NPV is positive, but below the rate where the NPV is negative.

A very rough guideline for estimating at what interest rate the NPV might be close to zero, is to take

⎟

⎠

⎞

⎜

⎝

⎛

rojectt of the pcos

profit

In our example, the total profit over three years is $(400 + 300 + 200 – 800) = $100. An approximate IRR is

therefore calculated as:

800

100

× = 0.08 approx.

A starting point is to try 8%.

(a) Try 8%

Year

Cash flow

Discount factor

Present value

(800)

1.000

(800.0)

400

0.926

370.4

300

0.857

257.1

200

0.794

158.8

NPV

(13.7)

The NPV is negative, therefore the project fails to earn 8% and the IRR must be less than 8%.

(b) Try 6%

Year

Cash flow

Discount factor

Present value

(800)

1.000

(800.0)

400

0.943

377.2

300

0.890

267.0

200

0.840

168.0

NPV

12.2

The NPV is positive, therefore the project earns more than 6% and less than 8%.

The

IRR is now calculated by interpolation. The result will not be exact, but it will be a close

approximation. Interpolation assumes that the NPV falls in a straight line from +12.2 at 6% to –13.7 at 8%.

Part D Financial mathematics ⏐ 9: Discounting and basic investment appraisal 243

The IRR, where the NPV is zero, can be calculated as follows.

IRR =

NPV

a(ba)%

NPV NPV

⎡⎤

⎛⎞

+−

⎢⎥

⎜⎟

−

⎝⎠

⎣⎦

Where a is one interest rate

b is the other interest rate

NPV

is the NPV at rate a

NPV

is the NPV at rate b

()

%)68(

7.132.12

2.12

⎥

⎦

⎤

⎢

⎣

⎡

−×

= 6% + 0.942%

= 6.942% approx

(d) The answer is only an

approximation because the NPV falls in a slightly curved line and not a straight line

between +12.2 and –13.7. Provided that NPVs close to zero are used, the linear assumption used in the

interpolation method is nevertheless fairly accurate.

(e) Note that the formula will still work if A and B are both positive, or both negative, and even if a and b are a

long way from the true IRR, but the results will be less accurate.

Question

Internal rate of return

The net present value of an investment at 15% is $50,000 and at 20% is – $10,000. The internal rate of return of

this investment (to the nearest whole number) is:

A 16%

B 17%

C 18%

D 19%

Formula to

learn

244 9: Discounting and basic investment appraisal ⏐ Part D Financial mathematics

Answe

IRR = a% +

%)ab(

NPVNPV

NPV

⎥

⎦

⎤

⎢

⎣

⎡

−

⎟

⎠

⎞

⎜

⎝

⎛

−

Where a = one interest rate = 15%

b = other interest rate = 20%

NPV

= NPV at rate a = $50,000

NPV

= NPV at rate b = –$10,000

IRR = 15% +

⎥

⎦

⎤

⎢

⎣

⎡

−×

−−

)1520(

)000,10£(000,50£

000,50£

= 15% + 4.17%

= 19.17%

= 19%

The correct answer is therefore D.

4 Annuities and perpetuities

4.1 Annuities

An annuity is a constant sum of money received or paid each year for a given number of years.

Many individuals nowadays may invest in annuities which can be purchased either through a single payment or a

number of payments. For example, individuals planning for their retirement might make regular payments into a

pension fund over a number of years. Over the years, the pension fund should (hopefully) grow and the final value

of the fund can be used to buy an annuity.

annuity might run until the recipient's death, or it might run for a guaranteed term of n years.

4.2 The annuity formula

The syllabus for Business Mathematics states that you need to be able to calculate the present value of an annuity

using both a formula and CIMA Tables. Let's have a look at the formula you need to be able to use when

calculating the PV of an annuity.

The

present value of an annuity of $1 per annum receivable or payable for n years commencing in one year,

discounted at r% per annum, can be calculated using the following formula.

PV =

(1 r)

⎛⎞

−

⎜⎟

⎝⎠

Note that it is the PV of an annuity of $1 and so you need to multiply it by the actual value of the annuity.

Assessment

formula

T F

RWAR

Part D Financial mathematics ⏐ 9: Discounting and basic investment appraisal 245

4.3 Example: The annuity formula

What is the present value of $4,000 per annum for years 1 to 4, at a discount rate of 10% per annum?

Solution

Using the annuity formula with r = 0.1 and n = 4.

PV = $4,000

⎟

⎠

⎞

⎜

⎝

⎛

⎟

⎠

⎞

⎜

⎝

⎛

−

)

1.01

(

1.0

= $4,000 × 3.170 = $12,680

4.4 Calculating a required annuity

If PV of $1 =

⎟

⎠

⎞

⎜

⎝

⎛

−

r)1(

, then PV of $a = a

⎟

⎠

⎞

⎜

⎝

⎛

⎟

⎠

⎞

⎜

⎝

⎛

−

)

(

∴ a =

⎟

⎠

⎞

⎜

⎝

⎛

⎟

⎠

⎞

⎜

⎝

⎛

−

)r1(

$a of PV

This enables us to calculate the annuity required to yield a given rate of return (r) on a given investment (P).

4.5 Example: required annuity

The present value of a ten-year receivable annuity which begins in one year's time at 7% per annum compound is

$3,000. What is the annual amount of the annuity?

Solution

PV of $a = $3,000

r = 0.07

t = 10

a =

⎟

⎠

⎞

⎜

⎝

⎛

⎟

⎠

⎞

⎜

⎝

⎛

−

)07.1(

07.0

000,3$

024.7

000,3$

= 427.11

Question

Annuity formula (1)

(a) It is important to practise using the annuity factor formula. Calculate annuity factors in the following cases.

(i) n = 4, r = 10%

(ii) n = 3, r = 9.5%

(iii) For twenty years at a rate of 25%

246 9: Discounting and basic investment appraisal ⏐ Part D Financial mathematics

(b) What is the present value of $4,000 per annum for four years, years 2 to 5, at a discount rate of 10% per

annum? Use the annuity formula.

Answe

(a) (i)

⎟

⎠

⎞

⎜

⎝

⎛

−

)1.01(

1.0

= 3.170

(ii)

⎟

⎠

⎞

⎜

⎝

⎛

−

)095.01(

095.0

= 2.509

(iii)

⎟

⎠

⎞

⎜

⎝

⎛

−

)25.01(

25.0

= 3.954

(b) The formula will give the value of $4,000 at 10% per annum, not as a year 0 present value, but as a value at

the year preceding the first annuity cash flow, that is, at year (2 – 1) = year 1. We must therefore discount

our solution in paragraph 4.3 further, from a year 1 to a year 0 value.

PV = $12,680 ×

10.1

= $11,527.27

Question

Annuity formula (2)

In the formula

PV =

−

⎛⎞

⎜⎟

⎝⎠

(1 r)

r = 0.04

n = 10

What is the PV?

A 6.41

B 7.32

C 8.11

D 9.22

Answe

PV =

⎟

⎠

⎞

⎜

⎝

⎛

−

)04.01(

04.0

= 8.11

The correct answer is therefore C.

Part D Financial mathematics ⏐ 9: Discounting and basic investment appraisal 247

4.6 Annuity tables

To calculate the present value of a constant annual cash flow, or annuity, we can multiply the annual cash flows by

the sum of the discount factors for the relevant years. These total factors are known as

cumulative present value

factors

or annuity factors. As with 'present value factors of $1 in year n', there are tables for annuity factors, which

are shown at the end of this text. (For example, the cumulative present value factor of $1 per annum for five years

at 11% per annum is in the column for 11% and the year 5 row, and is 3.696).

4.7 The use of annuity tables to calculate a required annuity

The present value of an annuity can also be calculated using the annuity factors found in annuity tables.

Annuity (a) =

factor Annuity

annuityanofvaluePresent

4.8 Example: Annuity tables

A bank grants a loan of $3,000 at 7% per annum. The borrower is to repay the loan in ten annual instalments. How

much must she pay each year?

Solution

Since the bank pays out the loan money now, the present value (PV) of the loan is $3,000. The annual repayment

on the loan can be thought of as an annuity. We can therefore use the annuity formula

Annuity =

factor annuity

in order to calculate the loan repayments. The annuity factor is found by looking in the cumulative present value

tables under n = 10 and r = 7%. The corresponding factor = 7.024.

Therefore, annuity =

024.7

000,3$

= $427.11

The loan repayments are therefore $427.11 per annum.

4.9 Perpetuities

A perpetuity is an annuity which lasts for ever, instead of stopping after n years. The present value of a perpetuity

is PV = a/r where r is the cost of capital as a proportion.

The present value of $1 per annum, payable or receivable in perpetuity, commencing in one year, discounted at r%

per annum

PV =

Assessment

formula

T F

RWAR

T F

RWAR

248 9: Discounting and basic investment appraisal ⏐ Part D Financial mathematics

4.10 Example: A perpetuity

How much should be invested now (to the nearest $) to receive $35,000 per annum in perpetuity if the annual rate

of interest is 9%?

Solution

PV =

Where a = $35,000

r = 9%

∴

PV =

09.0

000,35$

= $388,889

4.11 Example: A perpetuity again

Mostly Co is considering a project which would cost $50,000 now and yield $9,000 per annum every year in

perpetuity, starting a year from now. The cost of capital is 15%.

Required

Assess whether the project is viable.

Solution

Year

Cash flow

Discount factor

Present value

15%

(50,000)

1.0

(50,000)

1– ∞

9,000

1/0.15

60,000

NPV

10,000

The project is viable because it has a positive net present value when discounted at 15%.

4.12 The timing of cash flows

Note that both annuity tables and the formulae assume that the first payment or receipt is a year from now. Always

check assessment questions for when the first payment falls.

For example, if there are five equal payments starting now, and the interest rate is 8%, we should use a factor of 1

(for today's payment) + 3.312 (for the other four payments) = 4.312.

Question

Present value of a lease

Hilarious Jokes Co has arranged a fifteen year lease, at an annual rent of $9,000. The first rental payment is to be

paid immediately, and the others are to be paid at the end of each year.

What is the present value of the lease at 9%?

A $79,074 C $81,549

B $72,549 D $70,074

Part D Financial mathematics ⏐ 9: Discounting and basic investment appraisal 249

Answe

The correct way to answer this question is to use the cumulative present value tables for r = 9% and n = 14

because the first payment is to be paid immediately (and not in one year's time). A common trap in a question like

this would be to look up r = 9% and n = 15 in the tables. If you did this, get out of the habit now, before you take

your assessment!

From the cumulative present value tables, when r = 9% and n = 14, the annuity factor is 7.786.

The first payment is made now, and so has a PV of $9,000 ($9,000

× 1.00). Payments 2-15 have a PV of $9,000 ×

7.786 = $70,074.

∴ The total PV = $9,000 (1st payment) + $70,074 (Payments 2-15)

= $79,074.

The correct answer is A.

(Alternatively, the annuity factor can be increased by 1 to take account of the fact that the first payment is

now.

∴ annuity factor = 7.786 + 1 = 8.786

∴ PV = annuity × annuity factor

= $9,000

× 8.786 = $79,074)

Question

Perpetuities

How much should be invested now (to the nearest $) to receive $20,000 per annum in perpetuity if the annual rate

of interest is 20%?

A $4,000

B $24,000

C $93,500

D $100,000

Answe

PV =

Where a = annuity = $20,000

r = cost of capital as a proportion = 0.2

PV =

2.0

000,20$

= $100,000

The correct answer is therefore D.