CIMA - C3 Fundamentals Of Business Mathematics

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250 9: Discounting and basic investment appraisal ⏐ Part D Financial mathematics

5 Linking compounding and discounting

Compounding and discounting are directly linked to each other. Make sure that you understand clearly the

relationship between them.

5.1 Sinking funds

In the previous chapter we introduced you to sinking funds. You will remember that a sinking fund is an investment into

which equal annual instalments (an

annuity) are paid in order to earn interest, so that by the end of a given period, the

investment is large enough to pay off a known commitment at that time (

final value).

5.2 Example: A sinking fund (1)

Jamie wants to buy a Porsche 911. This will cost him $45,000 in two years' time. He has decided to set aside an

equal amount each quarter until he has the amount he needs. Assuming he can earn interest in his building society

account at 5% pa how much does he need to set aside each year? Assume the first amount is set aside one period

from now.

(a) Calculate the amounts using the annuity formula.

(b) Calculate the amounts using annuity tables.

Solution

If Jamie needs $45,000 in two years' time, the present value that he needs is

PV =

)05.01(

000,45$

= $40,816

(a)

Using the annuity formula

The annuity factor =

()

⎟

⎠

⎞

⎜

⎝

⎛

−

Where r = 0.05

n = 2

Annuity factor =

()

⎟

⎠

⎞

⎜

⎝

⎛

−

05.01

05.0

= 1.8594

The amount to save each quarter is an annuity. We can therefore use the formula

Annuity =

ctorAnnuity fa

8594.1

816,40$

= $21,951

Therefore Jamie must set aside $21,951 per annum.

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Part D Financial mathematics ⏐ 9: Discounting and basic investment appraisal 251

(b) Using annuity tables

When n = 2 and r = 5%, the annuity factor (from cumulative present value tables) is 1.859.

Annuity =

factor Annuity

859.1

816,40$

= $21,956

The difference of $5 ($21,956

− $21,951) is due to rounding.

5.3 Example: A sinking fund (2)

At this point it is worth considering the value of the fund that would have built up if we had saved $21,956 pa for

two years at an interest rate of 5%, with the first payment at the end of year 1.

Solution

The situation we are looking at here can be shown on the following time line.

Saving

NOW 1

(END)

0 1 2

$21,956

× (1.05)

$21,956

× 1

The value of the fund at the end of year 2 is:

21,956 + 21,956(1.05)

This is a geometric progression with:

A = $21,956

R = 1.05

n = 2

If S =

)1R(A

−

105.1

)105.1(956,21

−

= $45,000 (to the nearest $100)

Therefore, if we were to save $21,956 for two years at 5% per annum we would achieve a final value of $45,000.

Can you see how compounding and discounting really are the reverse of each other? In our first example, we

calculated that Jamie needed to save $21,956 pa for two years at a cost of capital of 5%. In the second example,

we demonstrated that using the equation for the sum of a geometric progression, saving $21,956 pa for two years

would result in a sinking fund of $45,000.

252 9: Discounting and basic investment appraisal ⏐ Part D Financial mathematics

Work through these two examples again if you are not totally clear: it is vitally important that you understand how

compounding and discounting are linked.

5.4 Mortgages

We also considered mortgages in Chapter 8. You will remember that the final value of a mortgage must be equal to

the sum of the repayments. If the repayments are regular, they can be treated as an annuity, in which case the

annuity formula may be used in mortgage calculations.

When an individual takes out a mortgage, the present value of the mortgage is the amount of the loan taken out.

Most mortgages will be taken out at a given rate of interest for a fixed term.

Annuity =

tables) or formula (from factor Annuity

mortgage)ofvaue(originalannuity of valuePresent

The annuity is the regular repayment value.

Let's have a look at an example.

5.5 Example: Mortgages

Tim has taken out a $30,000 mortgage over 25 years. Interest is to be charged at 12%. Calculate the monthly

repayment.

Solution

Present value of mortgage = $30,000

Annuity factor =

⎟

⎠

⎞

⎜

⎝

⎛

−

)12.01(

12.0

= 7.843

Annuity (annual repayments) =

factor annuity

843.7

000,30$

= $3,825

Monthly repayment = $3,825 ÷ 12 = $318.75

Did you recognise any of these figures? Look back at Paragraph 5.7 in the previous chapter. We have used the same

information but used the annuity formula method rather than the sum of a geometric progression.

5.6 Example: Interest rate changes

After nine years, the interest rate on Tim's mortgage changes to 10%. What is the new monthly repayment?

Solution

In the solution in Paragraph 5.7 in Chapter 8, it was established that the value of the mortgage after 9 years was

$26,675.

Important!

Part D Financial mathematics ⏐ 9: Discounting and basic investment appraisal 253

After 9 years, our annuity factor changes to:

Annuity factor =

⎟

⎠

⎞

⎜

⎝

⎛

−

)1.01

(

1.0

7.8237

Annuity (annual repayments) =

factor annuity

8237.7

675,26$

$3,410

∴ The monthly repayments = $3,410 ÷ 12 = $284

This is the same as the answer that we calculated in Chapter 8, Paragraph 5.7 when we used the sum of a

geometric progression formula.

Sinking funds are an example of

saving whilst mortgages are an example of borrowing.

5.7 Borrowing versus saving

(a) Borrowing

The chief advantage of borrowing money via a loan or mortgage is that the asset the money is used

to purchase can be

owned now (and therefore be put to use to earn money) rather than waiting. On

the other hand, borrowing money

takes some control away from the business's managers and

makes a business venture more risky. Because an obligation is owed to the lender the managers

may have

less freedom to do what they like with their assets. If the business is not successful the

debt will still be owed, and if the lender demands that it is repaid immediately the business might

collapse.

(b)

Saving

The advantages of saving up are that no interest has to be paid and the business does not have

to surrender any control to a third party

. The savings will earn interest. However the money will

not be available for other, potentially more profitable, uses. Also, the business cannot be

sure in

advance that it will be able to generate the cash needed over the timescale envisaged.

Note that both annuity tables and the formulae used in this chapter assume that the first payment or receipt is

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.

6 Using spreadsheets

Excel has a number of built-in functions such as =NPV and =IRR that can be used in investment appraisal.

6.1 Investment appraisal using Excel

The following spreadsheet will be used to demonstrate how to use financial mathematics functions in Excel.

Assessment

focus point

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254 9: Discounting and basic investment appraisal ⏐ Part D Financial mathematics

A B

Year Cashflow

0 -18000

1 6000

2 8000

3 5000

4 1000

Cost of capital % 10

NPV -1,494

IRR 5%

13 Rate of return %

27.78%

(a) The NPV function is used to calculate the net present value. The formula to use is =NPV(cost of

capital, cash inflows) – initial investment. In the spreadsheet above =NPV(B10,B4:B7)+B3 is entered

into cell B11. The initial investment is added as it has been entered as a negative figure.

(b) The

IRR function is used to calculate the internal rate of return. This requires you to include an initial

guess for the IRR to give the function a starting point. The formula to use is =IRR(cash movements,

guess) In the spreadsheet above =IRR(B3:B7,0.1) is entered into cell B12.

AVERAGE function is used to calculate the rate of return. The rate of return (also known as

return on investment or

ROI) is the average of the cash inflows divided by the original investment,

expressed as a percentage. In the spreadsheet above =AVERAGE(B4:B7)/-B3 is entered into cell B13.

(-B3 is entered as the initial investment has been entered as a negative figure).

Question

Excel functions

A B C

Project cash out 275000

Net project benefits Year 1 61200

Year 2 84600

Year 3 95600

Year 4 112500

Year 5 117800

Cost of capital % 15

ROI

NPV

Required

(a) What formula is required to calculate the ROI in cell C9? (Use the Excel function 'AVERAGE')

Part D Financial mathematics ⏐ 9: Discounting and basic investment appraisal 255

(b) What formula is required to calculate the NPV in cell C10? (Use the Excel function 'NPV')

Answe

(a) =AVERAGE(C2:C6)/C1

(b) =NPV(C8,C2:C6)-C1

256 9: Discounting and basic investment appraisal ⏐ Part D Financial mathematics

Chapter Roundup

• 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 have to be invested today to produce a given amount at some future date

• Discounting is the reverse of compounding. The discounting formula is X = S × 1/(1+r)

which is a rearrangement

of the compounding formula.

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

NPV method and the IRR 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.

• 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 =

⎡⎤

+−

⎢⎥

−

⎣⎦

NPV

a(ba)%

NPV NPV

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

• 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 =

()

⎛⎞

⎜⎟

−

⎜⎟

⎝⎠

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.

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

Annuity =

Present value of an annuity

Annuity factor

• A perpetuity is an annuity which lasts forever, 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.

• Compounding and discounting are directly linked to each other. Make sure that you understand clearly the

relationship between them.

• Excel has a number of built-in functions such as =NPV and =IRR that can be used in investment appraisal.

Part D Financial mathematics ⏐ 9: Discounting and basic investment appraisal 257

Quick Quiz

1 The discounting formula is X = S

r)1(

Where S =

X =

r =

n =

(a) the rate of return (as a proportion)

(b) the sum to be received after n time periods

(d) the number of time periods

2 An annuity is a sum of money received every year.

True

False

3 If Fred were to save $7,000 per annum, and we used the formula for the sum of a geometric progression to

calculate the value of the fund that would have built up over ten years at an interest rate of 20%, what is the value

of A to be used in the formula if:

(a) the first payment is now

(b) the first payment is in one year's time

4 Which Excel function should be used to calculate the return on an investment?

5 A machine will cost $25,000 to replace in 25 years time. The rate of interest is 8.5% per annum.

What is the present value of the replacement cost of the machine? (to the nearest $)

6 A project requiring an investment of $120,000 is expected to generate returns of $40,000 in years 1 and 2 and

$35,000 in years 3 and 4. If the NPV is $22,000 at 9% and –$4,000 at 10%, what is the IRR for the project? (to 2

decimal places)

7 What is the present value of an annuity of $5,000 per annum discounted at 7% if it starts at the end of the third

year and finishes at the end of the 10

year?

258 9: Discounting and basic investment appraisal ⏐ Part D Financial mathematics

Answers to Quick Quiz

1 S = (b)

X = (c)

r = (a)

n = (d)

2 False. It is a

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

3 (a) A = $7,000

× 1.2

(b) A = $7,000

4 =AVERAGE

(1.085)

25,000

6 IRR = a +

⎥

⎦

⎤

⎢

⎣

⎡

−

)ab(

NPVNPV

NPV

= 9% +

⎥

⎦

⎤

⎢

⎣

⎡

)000,4000,22(

000,22

= 9% + 0.85%

= 9.85%

7 PV of annuity from year 3-10 = PV from year 1-10 – PV from yr 1-2

= $5,000 × (7.024 – 1.808)

= $26,080

Now try the questions below from the Exam Question Bank

Question numbers Pages

71-78 341-344

3,252

26,080

259

Part E

Inter-relationships between

variables