3. Growth of money

When money is invested and interest is paid, the amount is seen to grow. Any money you save can be made to do this as you saw in Section 2.1, and this section indicates the types of calculation involved. Two situations are considered: the growth of a fixed sum and the growth of an amount which is increased regularly.

3.1 Growth of a fixed amount

If you invest some money, what does the amount you have at the end of the year depend upon? Can you see any connection between this question and the problems of the Law of Natural Growth?

3.2 Growth of regular investment

Here w show how a regular investment of Ј10 per month grows. This is perhaps the easiest way of saving. Ј10 per month is an easy figure to choose for the calculations of this chapter but it is more than most people can manage, especially at the beginning of their careers. Consider how much you hope to be able to save and work out the final totals from this starting point.

If Јx have been invested, then, at the end of the month, Јx - Јkx. The following pattern emerges when the mappings are shown down the page:

after 0 months, there is x,

after 1 month, there is kx +x

after 2 months, there is k²x+kx+x

after 3 months, there is k³x+k²x+kx+x

after 4 months, there is k⁴x+k³x+k²x+kx+x

after n months, there is kⁿx +kⁿ¹x +kⁿ²x + ... k²x +kx

= xk(kⁿ¹+kⁿ²+kⁿ³+ ... k + l)

x is added in at the end of each month, ready for the next, so it has not been added to the last month.

k measures the rate of growth per month.

Now consider the sum 1 +k+k²+k^s+...+kⁿ¹; call this S_n (S for sum and n because there are n terms). (1)

Exercise G

In Questions 1-3 use Figure 3—remember these growth curves are exponential.

1. How much will an investment of Ј50 become in 10 years at 6 % p.a.?

2. How much will an investment of Ј10 become in 20 years at 6 % p.a.?

3. How much will an investment of Ј100 become in 15 years at 4 % p.a.?

4. Estimate what an investment of Ј40 will become in 30 years at a rate of interest of 6 % p.a. by using the formula and assuming interest is added annually. (1-06¹⁰ = 1-791)

5. Repeat Question 4 but assume that interest is added monthly. (1-005¹²⁰ = 1-819).

4. Hire purchase

In Section 2, you saw that when government or industries borrow money, they have to pay interest in order to obtain it. You also have to pay interest, if you borrow money. When entering a hire purchase agreement, you are, in effect, borrowing money. How much interest do you think is reasonable to pay; up to 1 %, 5 %, 10 %, 100 % or more? Does it matter if the interest is 5 % per year or 5 % per month?

A recent advertisement from a reputable firm offered a scooter either for Ј80 cash, or for Ј8 deposit and Ј88 to be paid over a period of two years. This total of Ј96 includes the interest charges. This is an honest statement: you can see that, if you cannot wait until you have saved the money, then, for another Ј16, you can have the machine now.

When a company borrows money, it will have to pay a very high rate of interest if it is engaged in an enterprise of some risk; for example, in some mining venture or in the development of some new invention.

An individual borrower might borrow from a bank or from a moneylender or hire-purchase firm. The bank will not lend money unless it is sure that the borrower is a reliable person who will do his utmost to repay the loan. The hire purchase firm has to take greater risks, as it does not know a great deal about its borrowers and therefore charges more for the loan. One way in which this is done is by calculating the loan charges (the interest you have to pay) on the initial loan, whereas a bank would charge interest only on the loan that is outstanding at any time, and this gets less and less as you make your monthly repayments.

Consider how much it would cost in interest charges to borrow Ј72 from a bank at a rate of 6 % on the outstanding loan if the loan is repaid at the rate of Ј3 a month together with interest charges. A bank would actually use compound interest, but the difference is negligible over this short period of time. This loan will be gradually reduced over the 2 years and Figure 6 shows how much of the loan would be outstanding during each month of this period. For example, the height of the first column represents Ј72 and its width one month. A loan of Ј72 for one month is represented by the area of this first column.

Rotating the shaded portion of the graph about the mid-point of the stepped edge, S, gives the rectangle ABCO which represents a loan of Ј75 for one year and at 6 % per annum.

Compare this with the stated loan charge of Ј16; it is 16/4-5 = 3-55 times as much, or a rate of interest on the outstanding loan of 21-3 %.

Example 4

An old jalopy is bought for Ј55. Ј7 is paid in cash and the balance of Ј48 over a period of 12 months at 10 % per annum on the original loan. What annual rate of interest is this on the outstanding loan ?

Ј48 is owed for the first month, Ј4 for the last month; equivalent to Ј52 for 1 month.

Ј44 is owed for the second month and Ј8 for the 11th month; again equivalent to Ј52 for one month and similarly for a total of 6 pairs of months.

Thus the total loan is effectively Ј52 for 6 months.

Exercise H

1. A television set, cash price Ј80, is bought by paying a 10 % deposit and then 10 % per annum interest on the initial loan for 2 years. What is paid each month in repayment of loan and loan charges ? How much does the set cost you altogether to buy ?

2. Determine to the nearest pound the amount that would have to be paid if 90 % of the cost of the same television set was borrowed from a bank and repaid over 2 years with interest charged at 6 % of the outstanding loan.

3. A guitar costs Ј27 for cash. Instead, a deposit of Ј3 is paid and interest of 12% p.a. is charged on the balance (that is, on Ј24), the repayments being made in 36 monthly instal­ments. What is the total loan charge paid? What is the effective rate of interest on the outstanding loan ?

4. A record player costs Ј53 for cash. If instead a deposit is paid of Ј5, and 10 % interest is charged on the balance, what is the loan charge and repayment of loan per month if the loan is repaid in monthly instalments over 1 year ? What is the effective rate of interest on the outstanding loan ?