CIMA - C3 Fundamentals Of Business Mathematics

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230 8: Compounding ⏐ Part D Financial mathematics

∴ $1,724,616 =

11.09

1)(1.09A

−

∴ A = $20,361.25 per annum

If annual repayment = $20,361.25

Monthly repayment =

$20,361.25

= $1,696.77

You will probably find it useful to draw a time line to identify the time periods and interest rates involved when

answering questions on financial mathematics. Don't be afraid to include a quick sketch of a time line in an

assessment – it should help to clarify exactly when investments are made in saving funds or repayments are made

on a loan.

Chapter Roundup

• Simple interest is interest which is earned in equal amounts every year (or month) and which is a given proportion

of the original investment (the principal). The simple interest formula is S = X + nrX.

• Compounding means that, as interest is earned, it is added to the original investment and starts to earn interest

itself. The basic formula for compound interest is S = X(1 + r)

• The basic compound interest formula can be used to calculate the net book value of an asset depreciated using the

reducing balance method of depreciation by using a negative rate of 'interest' (reverse compounding).

• If the rate of interest changes during the period of an investment, the compounding formula must be amended

slightly to S = X(1+r

)

(1 + r

)

n–y

• An effective annual rate of interest is the corresponding annual rate when interest is compounded at intervals

shorter than a year.

• A nominal rate of interest is an interest rate expressed as a per annum figure although the interest is compounded

over a period of less than one year. The corresponding effective rate of interest shortened to one decimal place is

the annual percentage rate (APR) (sometimes called the compound annual rate, CAR).

• A geometric progression is a sequence of numbers in which there is a common or constant ratio between adjacent

terms.

• The final value (or terminal value), S, of an investment to which equal annual amounts will be added is found

using the following formula.

S =

(

)

()

−

AR 1

(the formula for a geometric progression).

• A sinking fund is an investment into which equal annual instalments are paid in order to earn interest, so that by

the end of a given number of years, the investment is large enough to pay off a known commitment at that time.

Commitments include the replacement of an asset and the repayment of a mortgage.

Assessment

focus point

Part D Financial mathematics ⏐ 8: Compounding 231

Quick Quiz

1 A sum of $60,000 is invested for 10 years at 6% per annum. What is its final value (to two decimal places) if

interest is:

Simple

Compound

2 A depreciation rate of 20% equates to a value for r of

A + 20%

−20%

C +0.2

−0.2

3 The effective annual rate of interest is the same as the annual percentage rate which is the same as the compound

annual rate.

True

False

4 An investor has been offered two deals:

Option 1 1.1% compounded every three months

Option 2 3.2% compounded every six months

Which option offers the best APR?

5 If Smita Smitten invests $250

now and a further $250 each year for five more years at an interest rate of 20%,

which of the following are true if the final investment is calculated using the formula for the sum of a geometric

progression?

A = n =

A $250

× 1.2 5

B $250

× 1.2 4

C $250 5

D $250 4

6 A shopkeeper wishes to refurbish his store in five years' time. At the end of five years, the refurbishment will cost

$50,000, and the storekeeper has decided to provide for this future refurbishment by setting up a sinking fund into

which equal annual investments will be made, starting one year from now. The fund will earn interest at 10%.

Using the formula for the sum of a geometric progression, calculate the annual investment.

7 What is the formula used for calculating the sum of the repayments of a mortgage?

232 8: Compounding ⏐ Part D Financial mathematics

Answers to Quick Quiz

1 60,000 + (10 ×

× 60,000)

60,000 × (1.06)

2 D A depreciation rate of 20% equates to a negative rate of interest of

−20% where r = −0.2.

3 True. Effective annual rate = APR = CAR

4 APR of option 1 = (1+ 0.011)

–1

= 0.0447

= 4.47%

APR of Option 2 = (1 + 0.032)

– 1

= 0.065

= 6.50%

Option 2 therefore offers the best APR.

5 A S =

)1A (R

−

Where A = the first term

= $250

× 1.2 (as investment is made now)

n = 5 years (the number of periods)

6 Using S =

)1A (R

−

Where S = final value of fund = $50,000

A = annual investment = ?

R = common ratio = 1.1

n = number of periods = 5

$50,000 =

11.1

1)(1.1 A

−

∴ A =

1)(1.1

1)(1.1$50,000

−

−×

= $8,189.87

7 S =

)1A (R

−

(the sum of a geometric progression formula)

Now try the questions below from the Exam Question Bank

Question numbers Pages

64-70 340-341

96,000.00

107,450.86

233

Topic list Syllabus references

1 The concept of discounting F, (iii), (5)

2 The Net Present Value (NPV) method F, (vii), (5)

3 The Internal Rate of Return (IRR) method F, (vii), (5)

4 Annuities and perpetuities F, (iv), (2)

5 Linking compounding and discounting F, (iv), (v), (vi), (2), (4)

6 Using spreadsheets G (i), (iii), (1), (3)

Discounting and

basic investment

appraisal

Introduction

Discounting is the reverse of compounding, the topic of the previous chapter. Its major

application in business is in the evaluation of investments, to decide whether they offer a

satisfactory return to the investor. We will be looking at two methods of using discounting to

appraise investments, the net present value (NPV) method and the internal rate of return (IRR)

method.

234 9: Discounting and basic investment appraisal ⏐ Part D Financial mathematics

1 The concept of discounting

1.1 Present value

The concept of present value can be thought of in two ways.

• It is the value today of an amount to be received some time in the future

• It is the amount which would be invested today to produce a given amount at some future date

The term 'present value' simply means the amount of money which must be invested now for n years at an

interest rate of r%, to earn a given future sum of money at the time it will be due.

1.2 The basic principles of discounting

Discounting is the reverse of compounding. The discounting formula is X = S ×

()

The basic principle of compounding is that if we invest $X now for n years at r% interest per annum, we should

obtain $X (1 + r)

in n years' time.

Thus if we invest $10,000 now for four years at 10% interest per annum, we will have a total investment worth

$10,000 × 1.10

= $14,641 at the end of four years (that is, at year 4 if it is now year 0).

The basic principle of

discounting is that if we wish to have $S in n years' time, we need to invest a certain sum

now (year 0) at an interest rate of r% in order to obtain the required sum of money in the future.

1.3 Example: Discounting formula

For example, if we wish to have $14,641 in four years' time, how much money would we need to invest now at

10% interest per annum? This is the reverse of the situation described in Paragraph 1.2.

Using our corresponding formula, S = X(1 + r)

Where X = the original sum invested

r = 10%

n = 4

S = $14,641

$14,641 = X(1 + 0.1)

$14,641 = X × 1.4641

∴X =

4641.1

641,14$

= $10,000

$10,000 now, with the capacity to earn a return of 10% per annum, is the equivalent in value of $14,641 after four

years. We can therefore say that $10,000 is the present value of $14,641 at year 4, at an interest rate of 10%.

Key term

T F

RWAR

T F

RWAR

Part D Financial mathematics ⏐ 9: Discounting and basic investment appraisal 235

The discounting formula is:

X = S ×

(1 r)+

Where S is the sum to be received after n time periods

X is the present value (PV) of that sum

r is the rate of return, expressed as a proportion

n is the number of time periods (usually years)

The rate r is sometimes called a cost of capital.

Note that this equation is just a rearrangement of the compounding formula.

1.4 Example: Discounting

(a) Calculate the present value of $60,000 at year 6, if a return of 15% per annum is obtainable.

(b) Calculate the present value of $100,000 at year 5, if a return of 6% per annum is obtainable.

Solution

The discounting formula, X = S ×

r)1(

is required.

(a) S = $60,000

n = 6

r = 0.15

PV = 60,000

1.15

= $25,939.66

(b) S = $100,000

n = 5

r = 0.06

PV = 100,000 ×

1.06

= $74,725.82

n = 2 or 3

r = 0.12

PV =

⎟

⎠

⎞

⎜

⎝

⎛

×+

⎟

⎠

⎞

⎜

⎝

⎛

12.1

000,4

12.1

000,4

= 3,188.78 + 2,847.12

= $6,035.90

Formula to

learn

236 9: Discounting and basic investment appraisal ⏐ Part D Financial mathematics

This calculation can be checked as follows.

Year 0

6,036.00

Interest for the first year (12%)

724.32

6,760.32

Interest for the second year (12%)

811.24

7,571.56

Less: withdrawal

(4,000.00

)

3,571.56

Interest for the third year (12%)

428.59

4,000.15

Less: withdrawal

(4,000.00

)

Rounding error

0.15

1.5 Present value tables

Present value tables are provided in your exam (and at the end of this study text), and give the present value factor

or discount factor for given values of n and r. They can only be used for whole numbers up to 20% and are

rounded so lose some accuracy, but they simplify and speed up your calculations.

Look up the discount factor in the table and multiply the value of 'S' by the discount factor.

1.6 Example: Discounting and present value tables

Calculate the present values in (a) and (b) in example 1.4 using present value tables.

Solution

(a) Present value = 60,000 × 0.432

= $25,920

(b) Present value = 100,000

× 0.747

= $74,700

Question

Present value

The present value at 7% interest of $16,000 at year 12 is $

rounded to the nearest whole number.

Answe

$7,104

Working

Using the discounting formula, X = S ×

r)(1

Where S = $16,000

n = 12

r = 0.07

X = Present Value

Part D Financial mathematics ⏐ 9: Discounting and basic investment appraisal 237

X = $16,000 ×

1.07

= $7,104

1.7 Investment appraisal

Discounted cash flow techniques can be used to evaluate capital expenditure projects. There are two methods: the

NPV method and the IRR method.

Discounted cash flow (DCF) involves the application of discounting arithmetic to the estimated future cash flows

(receipts and expenditures) from a project in order to decide whether the project is expected to earn a satisfactory

rate of return.

The cost of capital is the minimum return required by the owners of a company and this is used as the discount

factor in present value calculations.

2 The Net Present Value (NPV) method

The Net Present Value (NPV) method works out the present values of all items of income and expenditure related

to an investment at a given cost of capital, and then works out a net total. If it is positive, the investment is

considered to be acceptable. If it is negative, the investment is considered to be unacceptable.

2.1 Example: The net present value of a project

Dog Co is considering whether to spend $5,000 on an item of equipment. The 'cash profits', the excess of income

over cash expenditure, from the project would be $3,000 in the first year and $4,000 in the second year.

The company will not invest in any project unless it offers a return in excess of 15% per annum.

Required

Assess whether the investment is worthwhile, or 'viable'.

Solution

(a) In this example, an outlay of $5,000 now promises a return of $3,000 during the first year and $4,000

during the second year. It is a convention in DCF, however, that cash flows spread over a year are assumed

to occur at the end of the year, so that the cash flows of the project are as follows.

Year 0 (now) (5,000)

Year 1 (at the end of the year) 3,000

Year 2 (at the end of the year) 4,000

The NPV method takes the following approach.

(i) The project offers $3,000 at year 1 and $4,000 at year 2, for an outlay of $5,000 now.

(ii) The company might invest elsewhere to earn a return of 15% per annum.

Key term

T F

RWAR

T F

RWAR

238 9: Discounting and basic investment appraisal ⏐ Part D Financial mathematics

(iii) If the company did invest at exactly 15% per annum, how much would it need to invest now, at 15%,

to earn $3,000 at the end of year 1 plus $4,000 at the end of year 2?

(iv) Is it cheaper to invest $5,000 in the project, or to invest elsewhere at 15%, in order to obtain these

future cash flows?

(b) If the company did invest elsewhere at 15% per annum, the amount required to earn $3,000 in year 1 and

$4,000 in year 2 would be as follows.

Year

Cash flow

Discount factor

Present value

15%

3,000

0.870

2,610

4,000

0.756

3,024

5,634

cash flows. We can therefore reach the following conclusion.

• It is cheaper to invest in the project, by $634

• The project offers a return of over 15% per annum

(d) The net present value is the difference between the present value of cash inflows from the project ($5,634)

and the present value of future cash outflows (in this example, $5,000 × 1/1.15

= $5,000).

(e) An NPV statement could be drawn up as follows.

Year

Cash flow

Discount factor

Present value

15%

(5,000)

1.000

(5,000)

3,000

0.870

2,610

4,000

0.756

3,024

Net present value

+634

The project has a positive net present value, so it is acceptable.

Question

NPV method

A company is wondering whether to spend $18,000 on an item of equipment, in order to obtain cash profits as

follows.

Year $

1 6,000

2 8,000

3 5,000

4 1,000

The company requires a return of 10% per annum.

Required

Use the NPV method to assess whether the project is viable.

Part D Financial mathematics ⏐ 9: Discounting and basic investment appraisal 239

Answe

Year

Cash flow

Discount factor

Present value

10%

(18,000)

1.000

(18,000)

6,000

0.909

5,454

8,000

0.826

6,608

5,000

0.751

3,755

1,000

0.683

683

Net present value

(1,500)

The NPV is negative. We can therefore draw the following conclusions.

(a) It is cheaper to invest elsewhere at 10% than to invest in the project

(b) The project would earn a return of less than 10%

2.2 Project comparison

The NPV method can also be used to compare two or more investment options. For example, suppose a

business can choose between the investment outlined in the previous question above or a second investment,

which costs $28,000 but which would earn $6,500 in the first year, $7,500 in the second, $8,500 in the third,

$9,500 in the fourth and $10,500 in the fifth. Which one should the business choose?

The decision rule is to choose the option with the highest NPV. We therefore need to calculate the NPV of the

second option.

Year

Cash flow

Discount factor

Present value

11%

(28,000)

1.000

(28,000)

6,500

0.901

5,857

7,500

0.812

6,090

8,500

0.731

6,214

9,500

0.659

6,261

10,500

0.593

6,227

NPV =

2,649

The business should therefore invest in the second option since it has the higher NPV.

2.3 Expected values and discounting

Future cash flows cannot be predicted with complete accuracy. To take account of this uncertainty an expected net

present value

can be calculated which is a weighted average net present value based on the probabilities of

different sets of circumstances occurring

. Let us have a look at an example.