Final answer:
To calculate how much you will have after 4 years of saving $4,500 annually at a 6.3% interest rate, use the future value of an annuity formula, adjusting for the immediate first deposit. This will show you the power of compound interest on your savings over the specified period.
Step-by-step explanation:
The question involves calculating the future value of an annuity with regular deposits in an account that pays compound interest. To determine how much you will have 4 years from today after saving $4,500 per year at a 6.3% annual interest rate, you can use the future value of an annuity formula:
FV = P×(ⁱ (¹ + r)^t - 1) / r
Where FV is the future value of the annuity, P is the annual deposit, r is the annual interest rate (expressed as a decimal), and t is the number of years.
Plugging in the values, you get:
FV = $4,500×{ⁱ (¹ + 0.063)^4 - 1} / 0.063
This calculation would yield the total amount saved after 4 years, taking into account the compound interest earned.
However, since the interest is being compounded annually and payments are made at the start of each period, we must adjust the formula to account for the immediate first deposit:
FV_immediate = FV×(1 + r)
In this case:
FV_immediate = [($4,500×{ⁱ (¹ + 0.063)^4 - 1} / 0.063)]×(1 + 0.063)
Performing this calculation gives us the final amount that will be available 4 years from today. This result should be rounded to two decimal points as instructed in the question.
Compound interest can substantially increase savings over time, as seen in examples which highlight the long-term growth of investments. This financial concept is incredibly powerful for both savings and loans, as illustrated by various scenarios including saving for retirement or paying off a car loan.