Math Problem Statement
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?
Solution
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: where:
- 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:
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 of an annuity due is given by: where:
- (the payment amount),
- is the effective 2-month interest rate,
- (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:
- How would the investment change if the interest rate were compounded monthly instead of regularly?
- What would be the total amount accumulated after 6 years if no withdrawals were made?
- Can you calculate the future value of the regular bimonthly payments?
- How does changing the payment frequency (e.g., from bimonthly to quarterly) affect the required lump sum investment?
- 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.
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Math Problem Analysis
Mathematical Concepts
Financial Mathematics
Annuities
Compound Interest
Formulas
Present Value of Annuity Due
Theorems
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Suitable Grade Level
Advanced High School
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