Starting January 1, 2020, five equal deposits will be made into a fund. Task: Using the relevant tables, calculate the amount of each deposit required to achieve $200,000 in the fund by the target date, with interest at 10% compounded annually. Round your answers to two decimal places. January 1, 2024: $27,707.38 January 1, 2025: $29,781.36
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To calculate the amount of each deposit required to achieve a total of $200,000 in the fund by the target date, we need to understand that these deposits are made at the beginning of each year and are compounding annually at a 10% interest rate.
Let's break down the problem.
1. A deposit is made on January 1, 2020. This deposit will earn interest for four years (2020, 2021, 2022, and 2023). 2. A deposit is made on January 1, 2021. This deposit will earn interest for three years (2021, 2022, and 2023). 3. A deposit is made on January 1, 2022. This deposit will earn interest for two years (2022 and 2023). 4. A deposit is made on January 1, 2023. This deposit will earn interest for one year (2023). 5. The final deposit is made on January 1, 2024. This deposit will not earn any interest by January 1, 2024, as it is the target date.
To find the amount of each deposit (let's call it D), we can use the future value of an annuity due formula, but since you have provided the future values for January 1, 2024, and January 1, 2025, we can work backward using those amounts.
The total future value needed by January 1, 2024, is $200,000. Given that on January 1, 2024, the fund already has $27,707.38 (which is the value of the deposit made on January 1, 2020, compounding for four years), we need to calculate the deposits that would have compounded from 2021, 2022, and 2023 to contribute to the remainder of the $200,000.
By January 1, 2024: - The deposit on January 1, 2020, grew to: $27,707.38 - The deposit on January 1, 2021, will have compounded for 3 years. - The deposit on January 1, 2022, will have compounded for 2 years. - The deposit on January 1, 2023, will have compounded for 1 year. - The deposit on January 1, 2024, has just been made and has not earned interest.
Now, let's calculate the deposit required using the given future value for January 1, 2025. On this date, the deposit made on January 1, 2024, will have earned one year of interest at 10%.
Given future value (FV) = P * (1 + r)^n Where P is the principal (the amount deposited), r is the annual interest rate (0.10 for 10%), and n is the number of years the money is compounded.
We can solve for P: P = FV / (1 + r)^n
Using the future value for January 1, 2025: P = $29,781.36 / (1 + 0.10)^1 P = $29,781.36 / 1.10 P = $27,074.87
This means that each annual deposit must be $27,074.87 to accumulate to $200,000 by January 1, 2024, with the last deposit earning one year of interest to become $29,781.36 by January 1, 2025.
Extra: When dealing with compound interest, it's important to understand how it differs from simple interest. Compound interest is earned not only on the initial principal but also on the accumulated interest from previous periods. This means that the investment grows at an increasing rate if the interest is left to accumulate.
In an annuity, like in this scenario, regular deposits are made (annuity due means at the beginning of each period) and the compounding effect works on each of those individual deposits for different lengths of time, depending on when they were made relative to the calculation date. It is essential to keep in mind the timing of each deposit to accurately calculate the total future value of the annuity.
Modern financial calculators and spreadsheet software can greatly simplify these calculations by using built-in functions for future value (FV) and present value (PV) of annuities.