ACCA Paper 3

Financial Mathematics April 11^th 2000

1 SIMPLE INTEREST

• Interest accrues only on the initial amount invested.

Illustration 1

If £100 is invested at 10% pa simple interest

• A single principal sum, P invested for n years at an annual rate of interest, r (as a decimal) will amount to S.

Where S = P (1 + nr)

2 COMPOUND INTEREST

• Interest is reinvested alongside the principal.

2.1 Single sum

• Compounding calculates the future value of a given sum invested now (at a fixed interest rate).

Illustration 2

If Georgia placed £100 in the bank today (t₀) earning 10% interest per annum, what would this sum amount to in three years time (t₃)?

Solution

In 1 year’s time, £100 would have increased by 10% to £110

In 2 years’ time, £110 would have grown by 10% to £121

In 3 years’ time, £121 would have grown by 10% to £133.10

£133.10 is known as the terminal value of a single sum (S), and can be calculated using the formula

S = P (1 + r)ⁿ

where

P	=	initial principal
r	=	annual rate of interest (as a decimal)
n	=	number of years for which the principal is invested

£133.10 receivable in three years time has an equivalent value to £100 received today.

Conversely, the present value of £133.10 receivable in 3 years time is £100.

Example 1

£500 is invested in a fund on 1.1.02. Calculate the amount on deposit by 31.12.05 if the interest rate is

(a) 7% per annum simple

(b) 7% per annum compound.

Solution

The £500 is invested for a total of 4 years

(a)	Simple interest	S = P (1 + nr)
		S =
(b)	Compound interest	S = P (1 + r)^n
		S =

Example 2

£1,000 is invested in a fund earning 5% per annum on 1.1.00. £500 is added to this fund on 1.1.01 and a further £700 is added on 1.1.02. How much will be on deposit by 31.12.X2?

Solution

Date	Amount invested		Compound interest factor	=	Compounded cashflow
	£				£
1.1.00	1,000
1.1.01	500
1.1.02	700
					————
	Amount on deposit			=
					————

2.2 Annuities

• Many saving schemes involve the same amount being invested annually – ie annual payments.

• Formulae for the terminal value of an annuity, TV

(i) =

(ii)

where	A	= annuity (ie annual sum)
		(i) first sum paid/received at the end of each year
		(ii) first sum paid/received at the beginning of each year
	R	= interest rate (interest payable annually in arrears)
	N	= number of years annuity is paid/invested

Note: You do not need to learn these formulae – if asked for, you will always be able to work out TVs "manually". However, for the purpose of discounting annuities (later in this Session) it is important that you should appreciate that a formula (and annuity tables) removes the need for manual calculations.

Illustration 3

Andrew invests £3,000 per annum in a high interest account offering 7% pa. How much will he have to spend after a fixed 5 year term?

Solution

Assume £3,000 would be invested at the beginning of each year (ie interest accrues at the end of each year on balance brought forward at the beginning of each year plus the £3,000 invested at the beginning of the year).

TV = £3,000 = £3,000 6.153 = £18,460

3 EFFECTIVE ANNUAL INTEREST RATE (EAIR)

3.1 Nominal interest rates

• A nominal annual interest rate of 12% compounded
• quarterly is effectively 3% per quarter
• monthly is effectively 1% per month.

3.2 Effective annual interest rates (EAIR)

Illustration 4

Borrow £100 at a cost of 2% per month. How much (principal + interest) will be owed after a year?

* EAIR is 26.82%

Formula

1 + R = (1 + r)ⁿ

R = annual rate

r = rate per period (month/quarter)

n = number of periods in year

Example 3

£300 is invested in a fund every 6 months for three years, the first instalment being made on 1.1.X0. The fund earns 8% per annum nominal. How much will be on deposit by 31.12.X2?

Solution

Date	Cashflow	Compound interest	Compounded
		factor	cashflow
	£		£
1.1.X0	300
1.7.X0	300
1.1.X1	300
1.7.X1	300
1.1.X2	300
1.7.X2	300
			————
	Amount on deposit =

4 DISCOUNTING

4 "Compounding in reverse"

• Discounting calculates the sum which must be invested now (at a fixed interest rate) in order to receive a given sum in the future.

Illustration 5

If Will needed to receive £251.94 in three years time (t₃), what sum would he have to invest today (t₀) at an interest rate of 8% per annum?

Solution

Rearranging the compounding formula P = S

or alternatively PV = CF

where	PV	= the present value of a future cash flow (CF)
	r	= annual rate of interest
	n	= number of years before the cash flow arises

In this case PV = £251.94 = £200

The present value of £251.94 receivable in three years time is £200.

4.2 Points to note

• 

is known as the "simple discount factor" and gives the present value of £1 in n years at a discount rate, r.

• A present value table is provided in the exam

• For a cash flow arising now (at t₀) the discount factor will always be 1.

• t₁ is defined as a point in time exactly one year after t₀.

• Always assume that cash flows arise at the end of the year to which they relate (unless told otherwise).

Example 4

Find the present value of

(a) £250 received or paid in 5 years time, r = 6% pa

(b) £30,000 received or paid in 15 years time, r = 9% pa.

5 DISCOUNTED CASH FLOW (DCF) TECHNIQUES

5.1 Time value of money

• Individuals and businesses would prefer to receive £1,000 now than to receive £1,000 in one year’s time.

DCF techniques take account of the time value of money by restating each cash flow in terms of its equivalent value now.

5.2 Two methods for project appraisal

6 NET PRESENT VALUE (NPV)

6.1 Calculation

• Discount each cash flow (receipt or payment) to its present value
• If PV of receipts is greater than PV of payments, NPV is positive (ie sum PVs to give the NPV of the project as a whole).
• Accept projects with positive NPV.

Example 5

Edgar has £10,000 to invest for a five-year period. He could deposit it in a building society, earning 8% pa compound interest.

He has been offered an alternative: investment in a project that is expected to yield net cash inflows of £3,000 for each of the first three years, £5,000 in the fourth year and £1,000 in the fifth.

Required

Calculate the net present value of the project.

Solution

Time	Description	Cash flow	8% DF	PV
		£		£
0	Investment	(10,000)
1	Net inflow	3,000
2	Net inflow	3,000
3	Net inflow	3,000
4	Net inflow	5,000
5	Net inflow	1,000
				_____
			NPV =
				_____

6.4 Annuities

• An annuity is a stream of identical cash flows arising each year for a finite period of time.
• The present value of an annuity is given as

CF

where CF is the cash flow received each year commencing at t₀.

• is known as the "annuity factor" or "cumulative discount factor". It is simply the sum of a geometric progression.
• The formula may be presented as
• If required, both a formula and tables will be given in the exam

6.5 Perpetuities

• A perpetuity is a stream of identical cash flow arising each year forever.

The present value of a perpetuity is given as CF

where CF is the cash flow received each year.

• To show a perpetuity time is written as t_1–¥ .

Illustration 7

Calculate the present value of £1,000 receivable each year in perpetuity if interest rates are 10%.

Solution

Time	Description	Cash flow	10% Annuity factor	PV
		£		£
t1–¥	Perpetuity	1,000	= 10	10,000

Example 7

Calculate the present value of £2,000 receivable in perpetuity commencing in 10 years time. Assume interest at 7%.

Solution: Work out PV of 2,000 receivable from years 1-9 and deduct from Year now to forever calculation.

7 INTERNAL RATE OF RETURN (IRR)

7.1 Definition

• NPV tells us whether a project gives a greater or lower return than the discount rate.

• IRR is the rate of return inherent in the project, and is the discount rate at which the NPV is zero.

7.2 Perpetuities

• If a project has equal annual cash flows receivable in perpetuity (or for a considerable period of time) then

IRR = 100%

Illustration 8

An investment of £1,000 gives income of £140 per annum indefinitely, the return on the investment is given by

IRR = 100% = 14%

Example 8

An investment of £15,000 now will provide a £2,400 annuity in perpetuity.

Required

Calculate the return inherent in the investment.

Solution

IRR =

7.3 Annuities (Equal annual cash flows)

• To give an NPV of nil, the present value of the cash inflows must equal the initial cash outflow.
• ie Cash inflow Annuity factor = Cash outflow

Annuity factor =

• Once the cumulative discount factor is known the discount rate can be established from the appropriate table.

Illustration 9

An investment of £6,340 will yield an income of £2,000 for four years.

Calculate the internal rate of return of the investment.

Solution

Year	Description	CF	DF	PV
0	Initial investment	(6,340)	1	(6,340)
1-4	Annuity	2,000	AF1-4 years	6,340
				_____
	NPV			Nil
				_____

AF_1-4 years = = 3.17

From the annuity table, the rate with a four year annuity factor closest to 3.17 is 10% and this is therefore the IRR for this investment.

Example 9

An immediate investment of £10,000 will secure an annuity of £1,000 for the next 15 years.

Required

Calculate the internal rate of return of the investment.

Solution

Time	Description	Cash flow	Discount factor	PV
		£		£
0	Investment	(10,000)	1	(10,000)
1-15	Annuity	1,000		10,000
				______
				Nil
				______

From the annuity table the rate with a 15 year annuity factor of ?

Thus if £10,000 could be otherwise invested for a return of ?

7.4 Uneven cash flows

Method

• Calculate the NPV of the project at any estimated rate (say 10%).
• If NPV is positive, calculate another NPV at a higher discount rate (ie to get closer to IRR).
• If NPV is negative, recalculate at a lower discount rate.
• The IRR can be estimated using the formula:

Where	A	=	Lower discount rate
	B	=	Higher discount rate
	NA	=	NPV at rate A
	NB	=	NPV at rate B

Illustration 10

The NPVs of a project with uneven cash flows are as follows.

Estimate the IRR of the investment.

Solution

IRR ~ A + (B – A)

IRR ~ 10% + (20 – 10)%

IRR ~ 19%

Graphically

Note: The direction of rounding is important. In this case, it must be down.

Example 10

An investment opportunity with uneven cash flows has the following net present values

Required

Estimate the IRR of the investment.

Financial Mathematics always looks more complicated than it is. For this reason the focus will be on learning how to apply this to exam style questions. The selected questions are:

20, 21, 30, 40, 54, 86 from Revision Kit 2000.