Question

Compare the investment below to an investment of the same principal at the same rate compounded annually. Principal: $4000, annual interest: 5%, interest periods: 2, number of years: 13 After 13 years, the investment compounded periodically will be worth $____ more than the investment compounded annually.

A) $932.95
B) $981.53
C) $1067.15
D) $1189.23

Answer 1

None of the provided options correctly matches the calculated difference of $849.55, which is the additional amount the investment compounded semiannually is worth over the investment compounded annually after 13 years.

Step-by-step explanation:

To compare the future value of investments compounded semiannually and annually, we need to use the compound interest formula:

Future Value (FV) = P(1 + r/n)^(nt)

For the semiannual compounding:

FVsemi = $4000(1 + 0.05/2)^(2*13) = $4000(1 + 0.025)^(26) = $4000(1.025)^(26) ≈ $7,898.84

For the annual compounding:

FVannual = $4000(1 + 0.05/1)^(1*13) = $4000(1 + 0.05)^(13) = $4000(1.05)^(13) ≈ $7,049.29

The difference between the two future values:

Difference = FVsemi - FVannual ≈ $7,898.84 - $7,049.29 = $849.55

Therefore, none of the options (A) $932.95, (B) $981.53, (C) $1067.15, or (D) $1189.23 correctly state the difference. The investment compounded semiannually is worth $849.55 more than the investment compounded annually after 13 years.