Question

A saving plan has started with an initial imvestment of K1200. Each month additional K200 is added to the plan and interest is paid at 24% p.a on the increasing balance for 3 years. Calculate the amount accumulated at the end of 3 years. Show working out.

Answer 1

To calculate the amount accumulated at the end of 3 years with an initial investment and monthly contributions, we will follow these steps:

1. Calculate the monthly interest rate.
2. Calculate the future value of the initial investment after 3 years.
3. Calculate the future value of the series of monthly contributions.

### 1. Calculate the Monthly Interest Rate

The annual interest rate is 24%, so the monthly interest rate (i) will be:

\[
i = \frac{24\%}{12} = \frac{24}{100} \div 12 = 0.02 \text{ (or 2% per month)}
\]

### 2. Calculate the Future Value of the Initial Investment

The formula for the future value of a lump sum investment is:

\[
FV = P \times (1 + i)^{nt}
\]

where:
- \(FV\) = future value
- \(P\) = principal amount (initial investment)
- \(i\) = monthly interest rate
- \(n\) = number of times the interest is compounded per year (12 for monthly)
- \(t\) = number of years

Given:
- \(P = 1200\)
- \(i = 0.02\)
- \(n = 12\)
- \(t = 3\)

Now, substitute the values:

\[
FV = 1200 \times (1 + 0.02)^{12 \times 3} = 1200 \times (1.02)^{36}
\]

Calculating \((1.02)^{36}\):

\[
(1.02)^{36} \approx 2.030377
\]

Thus:

\[
FV \approx 1200 \times 2.030377 \approx 2436.45
\]

### 3. Calculate the Future Value of Monthly Contributions

The future value of a series of monthly contributions can be calculated using the formula:

\[
FV = C \times \left( \frac{(1 + i)^{nt} - 1}{i} \right)
\]

where:
- \(FV\) = future value of the annuity (monthly contributions)
- \(C\) = monthly contribution
- \(i\) = monthly interest rate
- \(n\) = number of times the interest is compounded per year
- \(t\) = number of years

Given:
- \(C = 200\)

Substituting the known values:

\[
FV = 200 \times \left( \frac{(1 + 0.02)^{12 \times 3} - 1}{0.02} \right) = 200 \times \left( \frac{(1.02)^{36} - 1}{0.02} \right)
\]

Using \((1.02)^{36} \approx 2.030377\):

\[
FV = 200 \times \left( \frac{2.030377 - 1}{0.02} \right)
\]

Calculating:

\[
FV = 200 \times \left( \frac{1.030377}{0.02} \right) = 200 \times 51.51885 \approx 10303.77
\]

### Total Amount Accumulated

Now, just add the future values of the initial investment and the monthly contributions:

\[
\text{Total Amount} = 2436.45 + 10303.77 \approx 12740.22
\]

### Conclusion

The amount accumulated at the end of 3 years is approximately **K12,740.22**.