Final answer:
The future value of a $1000 investment at a 9% annual interest rate compounded semi-annually for 12 years is approximately $2820.21, while the same investment compounded annually yields approximately $2797.56, showing that more frequent compounding results in higher yields.
Step-by-step explanation:
To compare the investment with a principal of $1000 at an annual interest rate of 9% with interest compounded semi-annually over 12 years, to an investment with the same principal and rate but compounded annually, we need to calculate the future value of both investments and then compare the results.
Compound Interest Formula
The compound interest formula is 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 in years.
Semi-Annually Compounded Interest
For the given investment, n is 2 (since interest is compounded semi-annually), r is 0.09, P is $1000, and t is 12. Plugging these into the compound interest formula, we get:
A = $1000(1 + 0.09/2)^(2*12)
A = $1000(1.045)^24
After calculating, we find A ≈ $2820.21
Annually Compounded Interest
If the interest is compounded annually, then n is 1. The formula becomes:
A = $1000(1 + 0.09/1)^(1*12)
A = $1000(1.09)^12
After calculating, we find A ≈ $2797.56
The investment compounded semi-annually yields more due to the effect of more frequent compounding periods.