Suppose that R 15 000 is invested in an account paying the following interest rates:

Question

16% p.a. compounded semi-annually for 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 75 000?


99


144


98


None of the above.

Answer 1

The formula for compound interest is:

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

Where:
A = Final amount
P = Initial principal (R 15 000)
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, and t = 1.5 years
A1 = 15,000(1 + 0.16/2)^(2*1.5) = 15,000(1.08)^3 = 21,964.44

For the next 27 months at 14.4% p.a. compounded quarterly:
r = 0.144, n = 4, and t = 2.25 years
A2 = 21,964.44(1 + 0.144/4)^(4*2.25) = 21,964.44(1.036)^9 = 33,816.19

For the rest of the period at 13% p.a. compounded monthly, let's assume it takes x years:
r = 0.13, n = 12, and t = x years
75,000 = 33,816.19(1 + 0.13/12)^(12x)
(75,000/33,816.19) = (1.0034882)^(12x)
ln(75,000/33,816.19) = 12xln(1.0034882)
x = ln(75,000/33,816.19) / (12*ln(1.0034882))
x ≈ 4.69 years

Total time = 1.5 + 2.25 + 4.69 = 8.44 years

Therefore, it will take approximately 8 years and 5 months to accumulate at least a value of R 75,000.