Suppose that R 12 500 is invested in an account paying the following interest rates:

Question

16% p.a. compounded semi-annually fFor the first 18 months
14.4% p.a. compounded quarterly for the next 27 months.
13% p.a. compounded monthly, thereafter…
How long (to the nearest month) in total will it take for the fund to accumulate to at least a value of R 54 678?


131


130


86


18

Answer 1

We can use the formula for compound interest to calculate the future value of the investment:

A = P(1 + r/n)^(nt)

Where:
A = the future value of the investment (R 54 678)
P = the initial investment (R 12 500)
r = annual interest rate
n = number of compounding periods per year
t = number of years

For the first 18 months at 16% p.a. compounded semi-annually:
r = 0.16
n = 2
t = 1.5 years

A1 = 12 500(1 + 0.16/2)^(2*1.5)
A1 ≈ 12 500(1 + 0.08)^3
A1 ≈ 12 500(1.08)^3
A1 ≈ 12 500(1.259712)
A1 ≈ R 15 746.40

For the next 27 months at 14.4% p.a. compounded quarterly:
r = 0.144
n = 4
t = 2.25 years

A2 = 15 746.40(1 + 0.144/4)^(4*2.25)
A2 ≈ 15 746.40(1 + 0.036)^9
A2 ≈ 15 746.40(1.036)^9
A2 ≈ 15 746.40(1.412202)
A2 ≈ R 22 262.78

Now, we need to calculate how many months at 13% p.a. compounded monthly to reach R 54 678:

r = 0.13
n = 12
t = unknown

54 678 = 22 262.78(1 + 0.13/12)^(12*t)
2.457965178 ≈ (1 + 0.013)^t
t ≈ log(2.457965178) / log(1.013)
t ≈ 130 months

Therefore, it will take approximately 130 months in total for the fund to accumulate to at least a value of R 54 678. So the nearest month is 130.