What single amount can be invested on 01/05/2020, instead of R4545 at the beginning of every 2-month period, for exactly 6 years (the first investment is made on 01/05/2020), if interest is earned at a rate of 13.95% per year?

To determine the single amount that can be invested instead of making payments of R4545 at the beginning of every 2-month period over 6 years, we can follow these steps:

### 1. **Determine the Effective 2-Month Interest Rate**
The annual interest rate is given as 13.95%, compounded regularly. We need to find the interest rate for every 2-month period:
\[
i = \left(1 + \frac{13.95\%}{12}\right)^2 - 1
\]
where:
- \( i \) is the effective interest rate for every 2 months.

### 2. **Calculate the Number of Payments**
The total period is 6 years, with payments made every 2 months:
\[
n = \frac{6 \times 12}{2} = 36 \text{ payments}
\]

### 3. **Calculate the Present Value of the Annuity Due**
Since payments are made at the beginning of every 2-month period, it is an annuity due. The present value \( PV \) of an annuity due is given by:
\[
PV = P \times \frac{1 - (1 + i)^{-n}}{i} \times (1 + i)
\]
where:
- \( P = R4545 \) (the payment amount),
- \( i \) is the effective 2-month interest rate,
- \( n = 36 \) (the total number of payments).

### 4. **Compute the Present Value**
Let's calculate the present value, which will give us the single amount that can be invested on 01/05/2020 instead of making regular payments.

I'll start by computing the effective 2-month interest rate and then calculate the present value.### Results

- The effective 2-month interest rate is approximately **2.34%**.
- The single amount that should be invested on 01/05/2020, instead of making payments of R4545 every 2 months, is approximately **R112,357.54**.

This lump sum investment will grow under the given interest rate and will be equivalent to making the regular bimonthly payments over the 6-year period.

---

Do you need further details or have any questions? Here are some related questions:

1. How would the investment change if the interest rate were compounded monthly instead of regularly?
2. What would be the total amount accumulated after 6 years if no withdrawals were made?
3. Can you calculate the future value of the regular bimonthly payments?
4. How does changing the payment frequency (e.g., from bimonthly to quarterly) affect the required lump sum investment?
5. What if the interest rate fluctuates annually—how would that affect the calculation?

**Tip:** When dealing with annuities or lump-sum investments, always check the compounding frequency as it significantly impacts the results.

Learn how to calculate the lump sum investment needed on 01/05/2020 instead of making regular payments of R4545 every 2 months over 6 years. Includes step-by-step calculations using the present value of annuity due formula with a 13.95% annual interest rate.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation