Final answer:
Shaylea's annual savings of $4,500 at 7% interest yields different totals over 20, 30, and 40 years due to compound interest. Switching to monthly savings of $375 with monthly compounding also changes the future value of her investments. The compounding frequency's impact is evident, with monthly compounding generally offering better returns over long periods.
Step-by-step explanation:
Understanding Compound Interest in Savings
Shaylea is making an astute decision by starting to save early, allowing her to harness the potential of compound interest to build her retirement savings. When compounded annually at 7% interest, Shaylea's $4,500 yearly investment would grow to different amounts over 20, 30, and 40 years due to the exponential nature of compound interest.
To calculate the future value of her investment, we use the compound interest formula:
A = P (1 + r/n)^(nt), where:
- A is the amount of money accumulated after n years, including interest.
- P is the principal amount (the initial amount of money).
- r is the annual interest rate (decimal).
- n is the number of times that interest is compounded per year.
- t is the time the money is invested for, in years.
With an annual deposit, the future value is also affected by the number of contributions, which must be summed over the investment period.
For monthly investment with monthly compounding, the same formula is applied but with different parameters for the frequency of compounding and contributions. Let us assume Shaylea switches to a monthly deposit of $375 with monthly compounding; we would use a monthly interest rate and the number of months as the time period.
The difference over time between annual and monthly compounding shows the impact of increased compounding frequency, often leading to higher returns over long periods.