FINANCIAL MATHEMATICS : GRADE 12

Topics:

1 Simple Interest/decay

2 Compound Interest/decay

3 Converting between nominal and effective

4 Annuities

4.1 Future Value

4.2 Present Value

5 Sinking Funds

6 Loan Repayments:

6.1 Repayments with Future Value Formula.

6.2 Present value formula

7 Balance on a loan

8 Calculation of Time Period “n”

9 Microlenders

10 Pyramid Schemes

Introduction:

No business can exist without the information given by figures. Borrowing, using and making money is the heart of the commercial world thus the principle of interest and interest rate calculations are extremely important. This leads into an examination of the principles involved in assessing the value of money over time and how this Information can be utilized in the evaluation of alternate financial decisions. Remember that the financial decision area is a minefield in the real world, full of tax implications, depreciation allowances, investment and capital allowances etc. The basic principles in financial decision making are established through the concept of interest and present value: – Definition of interest:

Interest is the price paid for the use of borrowed money

Interest is paid by the user of the money to the supplier of it. It is calculated as a fraction of the amount borrowed or saved over a certain period of time. This fraction is also known as interest rate and is expressed as a percentage per year (per annum).

Present Value of money is the value of the initial investment:

i.e. PV (Present Value) = P (Principle)

r = interest

PV or P t = term

FV or S = P(1 + rt)

PV = Present Value or Principle

FV = Future Value or Sum

Assured

Simple interest

Simple interest: is computed on the principle for the entire term of the loan and is thus due at the end of

term.

Growth Decay

S I = Prt )1( niPA )1( niPA

where:

I is the interest paid or earned

P is the principle or Present value

r is the interest rate per annum

t is the time or term of loan

n is the number of years

Compound Interest

Compound interest arises when, in a transaction over an Extended period of time, interest due at the end of a payment period is not paid, but added to the principal. Thus interest also earns interest i.e. it is compounded. The amount due at the end of transaction period is referred to as the compounded amount or accrued principal. Interest periods can vary : daily, monthly, quarterly, half-yearly or yearly.

Formula

Compound Growth: Compound Decay:

niPA )1( niPA )1(

OR OR

F = Amount or Future Value

P = Principal or Initial value

r = rate of interest per annum

n = number of years invested

s = number of time periods interest is calculated

( annum, quarterly, half yearly, monthly or daily)

ns

s

rPFv )

1001( ns

s

rPFv )

1001(

Further Formulae

1. Finding Principle: niAP )1( or

2. Finding the interest rate:

NB: To get rate (r) , multiply i by 100.

Nominal Interest rates:

1.1. In cases where interest is calculated more than once a year, the annual rate quoted is the nominal annual rate or nominal rate.

Effective Interest rates:

1.2. If the actual interest earned per year is calculated and expressed as a percentage of the relevant principal , then the so-called effective rate is obtained. This is the equivalent annual rate of interest – that is, the rate of interest earned in one year if compounding is done on a yearly basis.

Converting Nominal Rate to Effective Rate:

Formula

reff =100[(1+𝑟𝑛𝑜𝑚

100𝑠) – 1 ]

where:

reff : effective rate (percentage) r : nominal rate (percentage) s : no. of periods in one year e.g. Calculate the effective rate corresponding to nominal rate of interest of 22% p.a. compounded biannually. r nom =22 s = 2 (biannually)

reff = 100[(1+22

100(2))2-1]

= 23.21 %

ns

s

rFvP )

1001(

1

1

ns

P

Fvsi

Annuities

Definition:

An annuity is a sequence of equal payments at equal intervals of time. The payment interval of an annuity is the time between successive Payments while term is the time from the beginning of the first payment interval to the end of the last payment interval. Types of Annuities: Ordinary annuity- annuity where payments are made at the end of each payment interval. Annuity due- annuity where payments are made at the beginning of the payment interval.

Future Value Annuities

Definition: Regular payments into a saving account. Interest grows the investment. The formula for the sum of a geometric series is used in financial maths to calculate values of annuities.

1

)1(

r

raS

n

n

series.the in terms of no or payments of number the is n

i) 1 ( ratio common the is r

made payment or term first the is a

Future value formula As well as the sum of geometric series formula, a more useful formula is:

𝐹 =𝑥[(1 + 𝑖)𝑛 − 1]

𝑖

where: x : payment amount i : interest rate n : number of payments The above formula ‘F’ can only be used if there is a final payment at the end which does not earn interest (ordinary annuity).

Example:

1) Suppose R1000 is invested every month, starting one month from now for 10 months. Interest rate of 18% p.a. compounded monthly, calculate the accumulated amount.

𝐹 =𝑥[(1+𝑖)𝑛−1]

𝑖 i =

18

100= 0.18

=1000[(1 +

0.1812

)10 − 1]

0.1812

= 10702.72

2) Kimi deposits R2000 immediately into a savings account, continuing to make monthly payments at

the end of each month for 10 years. Interest rate is 24% p.a. compounded monthly.

Note: There will be a total of 121 payments. Why? n will be 12 months X 10 years =120 but one more

payment (“the immediately” payment) must be added. Payment starts immediately, so this is the extra

payment at T0. There will be an immediate payment and additional payments at the end of each month.

Therefore there will be 121 payment in total.

𝐹 =𝑥[(1+𝑖)𝑛−1]

𝑖 i =

24

100= 0.24

1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 | | | | | | | | | | | T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 Last payment does not earn interest Payment starts 1 month from now

2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 | | | | | | | | | | | ……………………………………………… | | | T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 ………………………………………………. T118 T119 T120 Last payment does not earn interest Payment immediately

=2000[(1 +

0.2412

)121 − 1]

0.2412

= R 998 046.63

Summary: Cases when 𝐹 =𝑥[(1+𝑖)𝑛−1]

𝑖 formula may be used

Case 1: When payments made at the end of the month. Case 2: When payments made one month from now (i.e. One time period from now). Case 3: When payment immediately and continued at the end of each time period. (NB: One extra payment) Case 4: Start and end on birthday. NB: One extra payment. All these cases have one thing in common: the final payment does not earn interest. Whenever this is the case we can use the above formula. When the final payment does earn interest, we use a different formula. See ‘Gaps’ below. Gaps Rule: When payment begins immediately and when there is a gap between the last payment and the end of the year i.e. when payments made at the beginning of the time period (Annuity due), then the following formula is used:

𝐹 =𝑥(1+𝑖)[(1+𝑖)𝑛−1]

𝑖

Example Ten equal payments of R8000 are made into a savings account annually at the beginning of each year (

effective immediately). Calculate the total accumulated amount at the end of 10 years if an interest rate of

9% compounded annually is applied.

1T represents the beginning of the 2nd year and the end of the 1st year. Similarly 9T represents the

beginning of the 10th year as well as the end of the 9th year. 10 payments are made at the beginning of each

year (starting immediately), the last payment will therefore be made at the beginning of the 10th year, but

interest will be accumulated until the end of the 10th year.

8000 8000 8000 8000 8000 8000 8000 8000 8000 8000 | | | | | | | | | | | T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 Gap at the end, last payment earns interest Payment starts immediately

Therefore there is a gap at the end between the last payment and until the interest stops accumulating.

Because of this “gap” we use the formula: 𝐹 =𝑥(1+𝑖)[(1+𝑖)𝑛−1]

𝑖

The solution to the previous example will therefore be:

𝐹 =𝑥(1+𝑖)[(1+𝑖)𝑛−1]

𝑖 i=

9

100= 0.09

= (8000(1+0.09)[(1+0.09)10−1])

0.09

= R 132482,35

Present Value Annuities Definition: Regular payments into a loan account, interest accumulated is the ‘enemy’. Present Value Formula:

𝑃 =𝑥[1− (1 + 𝑖)−𝑛 ]

𝑖

where: P: present value i : interest rate n: number of payments x: payments of annuity Note: The ‘P’ formula can only be used if there is a gap between the loan and the first payment. The gap must be one period.

Sinking Funds

Companies often purchase equipment and use it for a specified time period. The old equipment is then sold at scrap value and new, upgraded equipment is bought. In order to finance the purchasing of the new equipment, the company will, in advance, have set up an annuity called a sinking fund. Important terms: Book Value of an asset is its value after depreciation has been taken into account.

Scrap Value is the book value of an asset at the end of its useful life.

A sinking fund is a fund set up to replace an asset at the end of its useful life.

Example A school buys a photocopying machine that cost R150 000. It depreciates at 22% p.a. reducing balance. Its useful life is 5 years, and a new machine will inflate at 19% p.a. effective. Old machine will be sold at scrap value in 5 years and the proceeds will be used together with a sinking fund to buy a new machine. The school will pay monthly and earn at a 14.4 % p.a. compounded monthly rate. The first payment will be made immediately and the last at the end of the 5 year period.

1) Scrap value (Use compound decay formula and depreciate machine for 5 years).

A=P(1-i)n =150 000(1-0.22)5 = R43307.62

2) Cost of new machine in 5 years(Use compound interest to inflate machine for 5 years)

A=P(1+i)n

=150 000(1+0.19)5 = R357953.05

3) Amount required in sinking fund in 5 years sinking fund= cost of new equipment – scrap value = 357953.05-43307.62 =314645.43 4) Find equal monthly payments(NB: There will be one extra “immediate” payment) No. of payments: 61 i=14.4/100=0.144 Solve for x

𝐹 =𝑥[(1 + 𝑖)𝑛 − 1]

𝑖

314645.43 =𝑥[(1+

0.144

12)61−1]

0.144

12

x = R 3528.09

Loan Repayments

Loan repayments using the Future-Value formula:

When using the future value:

Load(plus interest)= Repayment(plus interest)

James buys a car for R120 000,00 and takes a loan from the bank. Calculate his monthly repayments if the

loan is for 5 yrs. The bank charges 9% interest p.a. compounded monthly. Repayments start a month after

the loan is drawn.

In this solution we calculate the loan with the interest accrued at the end of the 5 yrs. The repayments are

also calculated at 60T together with the interest earned. The repayments plus the interest earned are then

equal to the loan plus the interest at the end of the 5 yrs.

Loan ( plus interest) = repayments (plus interest)

112

09,01

112

09,01

12

09,01120000

60

60

x

= R 2 491,00

i.e

112

09,01

112

09,01

12

09,01120000

60

60

x

In general :

11

111

m

m

m

i

m

i

m

iP

x Where:

repaymentsmonthlyx

amountloanP

periodstimem

eresti

int

Notes:

1 In general loans start one month after loan has been drawn.

2 In some situations , such as home loans, it can be arranged that the repayments start 3 months

after loan is drawn etc.

Loan repayments using the Present-Value formula:

niAP )1( or ni

AP

)1( or n

vv iAP )1(

0T1T 2T 59T 60T

R120 000 x x x x

It is more common to use the above formula to calculate repayments and balances than the Future-

value formula.

Example:

A loan is taken out to buy a TV set with surround sound. The loan is repaid with two equal payments of

R5000,00. The first payment is made one year after he bought the set, and the second one year later.

Interest is calculated at 6% p.a. effective. Calculate the initial value of the loan.

Method 1:

Use the Present-value formula.

The repayments represent the future value and include the interest needed to repay the loan.

The present value of the first repayment is:

98,4716)06,01(5000)1( 1 RiAP n

The Pv of the 2nd loan repayment is:

98,4449)06,01(5000)1( 2 RiAP n

Total value of the loan = R4716,98 + R4449,98 = R9166,96

In one calculation:

1)06,01(5000 + 2)06,01(5000 = R9166,96

Method 2:

Using the future value formula.

96,9166

)06,01(

)06,01(500050002

Rx

x

Example 2:

A loan of R90 000,00 is taken out and repaid by equal monthly installments over a 3 year period. Interest

is set at 7,5% p.a. compounded monthly. Calculate the monthly repayment.

11

111

n

n

i

iiPx

112

075,01

112

075,01

12

075,0190000

36

36

x

= R2799,56

Balance on a loan

balance outstanding =loan (plus interest) – repayments(plus interest) Example:

A car is purchased for R180 000,00. 10% deposit is paid and the balance is financed through a bank loan.

The loan is for 6 years at an interest rate of 10,5% compounded monthly.

Task:

a) Determine the monthly repayments.

b) Determine the balance owed on the loan at the end of two years immediately after the 24th

payment.

Solution:

a) Loan is for R162 000,00 (i.e 90% of R180 000,00.)

paymentsxxx 72......12

105,01

12

105,01

12

105,01162000

2172

0T1T 2T

R162 000 x x x x

71T 72T

112

105,01

112

105,01

12

105,01162000

72

72

x

= R3042,19

b)

07,118820

309,8085337,199673

112

105,01

112

105,0119,3042

12

105,01162000

24

24

R

RR

NB: The calculation can be split into two sections:

1.

24

12

105,01162000

= R199 673,37

2.

112

105,01

112

105,0119,3042

24

=R80 853,309

Final Answer: 1 – 2 = R118 820,07

If you have a new calculator i.e the Casio ESfx 82 or later version then the calculation can be

done as one calculation BUT it would be prudent to write down the equation used to show the

examiner.

In long term loans (i.e.Home Loans) the repayments in the early stages cover mostly interest with a

small amount towards capital reduction. This reverses in the latter stages of the loan.

Calculation of time period ‘n’ (see log rules for more on logs)

Example 1:

To calculate the time period logarithms are used.

An investment of R24500 at an interest rate of 9%.

The investment grows to R 48817,78 after n years.

Calculate n:

niPA )1(

yrsn

n

n

n

n

n

8

99999,7

09,1log

9925,1log

9925,1log09,1log

9925,109,1

)09,01(24500

78,48817

)09,01(2450078,48817

Example 2:

R200 000,00 is deposited into a savings account . Interest is paid at 8,5% p.a. Compounded annually. How

long will it take for the principle to double?

niPA )1(

P = 200 000 thus A = 400 000 i = 0,085

monthsyrsn

daysyrsn

n

n

n

68

1828

22.1813654965,0

4965,8

085,1log

2log

2log085,1log

Microlenders

They offer short term loans at very high interest rates.

The interest is calculated upfront using simple interest based on the full amount of the loan over the

repayable period.

There is no advantage in early settlement.

The following is from an advert for instant finance:

Do you need to borrow money urgently?

You could have instant cash in your hand within 24 hours. You have been specially selected to receive this

loan from FSP. Whether you want to add value to your homewith some renovations, or spoil yourself to a

dream holiday, instant cash up to R25 000 is available to you right now.

Amount 24 months 36 months 48 months 60 months

R8000 R500 R388 R333 R300

R16000 R960 R737 R626 R560

R20000 R1183 R905 R767 R683

R25000 R1479 R1132 R958 R854

Use this handy installment table to choose the loan that will suit your budget and circumstances. Find the

loan amount you need and choose the repayment period that offers you a monthly repayment you feel

comfortable with.

Choice of loan term: Up to 5 years to repay your loan.

Fixed interest rate: For the full term of your loan.

Cash to use as you choose: For anything that is important to you.

BEWARE OF THESE TYPES OF SCHEMES AS THEY CAN COST YOU AN ARM AND A LEG.

EG: R25000 over 5 yrs = R854 X 60 = R51240

Interest rate:

niPA )1(

%21

0496,15

510496,2

5125000

51240

)51(2500051240

i

i

i

i

i

Pyramid Schemes

A pyramid scheme is a scam that relies on new investors to provide money to those above them in the

pyramid. These schemes have been around for a long time and exist in one form or another all over the

world. They rely on gullibility and greed.

The following is a theoretical example of how a pyramid scheme works.

Each investor pays R5000 and recruits only 2 others to invest into the scheme.

Once the investor gets 2 more recruits the fund will be a further R10000. He is then refunded his

R5000 that he paid into the scheme and the other R5000 is shared by the other investors above him

in the pyramid. As the pyramid grows below him so does his share of the spoils.