Compounding periodically yields a $35.09 higher future value than compounding annually after 13 years.
In comparing the investment compounded periodically to an investment with the same principal at the same annual interest rate compounded annually, we can use the compound interest formula to analyze the growth of each investment over time.
For the investment compounded annually, the formula is A = P * (1 + (r/n))^(nt), where A is the future value, P is the principal, r is the annual interest rate, n is the number of times interest is compounded per year, and t is the number of years.
Given that the principal (P) is $5,000, the annual interest rate (r) is 3%, the interest is compounded 6 times per year (n = 6), and the investment period is 13 years (t = 13), we can calculate the future value of the investment compounded periodically.
A_periodic = 5000 * (1 + (0.03/6))^(6 * 13) is approximately $5747.43.
For the investment compounded annually:
A_annual = 5000 * (1 + 0.03)^13 is approximately $5712.34.
The difference between the two future values is $5747.43 - $5712.34 = $35.09.
After 13 years, the investment compounded periodically will be worth $35.09 more than the investment compounded annually.