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.

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?

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).

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