Final answer:
The question is about calculating the future value of Mr. Smith's annuity deposits using compound interest. By applying the compound interest formula for annuities to quarterly deposits of $100 over 15 years at a 5% interest rate, the total amount at the end of the period can be determined. This amount will be greater than that obtained through simple interest.
Step-by-step explanation:
Mr. Smith wants to know how much his compound interest savings plan will yield at the end of 15 years by depositing $100 each quarter with an interest rate of 5% compounded quarterly. To calculate this, we need to use the future value formula for an annuity, which is the type of deposit plan Mr. Smith is using.
Here's the formula for the future value of a series of annuity payments:
FV = P * [(1 + r)^n - 1] / r
Where:
- FV is the future value of the annuity.
- P is the payment amount per period ($100 in this case).
- r is the interest rate per period (5% compounded quarterly would be 0.05/4).
- n is the total number of payments (15 years * 4 quarters/year).
By plugging Mr. Smith's figures into this formula, we can calculate the exact amount that will be in the fund at the end of 15 years. The difference between simple interest and compound interest is that compound interest is calculated on the initial principal and also on the accumulated interest of previous periods.
Let's proceed step by step:
- Calculate the periodic interest rate: 5% annual interest compounded quarterly is 1.25% per quarter (0.05 / 4).
- Calculate the number of quarters over 15 years: 15 * 4 = 60 quarters.
- Substitute the values into the future value formula for annuities.
After performing the calculation, we'll have the total amount that will be available in the fund after 15 years, which will be significantly higher than the total accumulated by simple interest alone.