Final answer:
Amelia's initial investment with annual compounding is approximately $14,801.89, while with semiannual compounding, it is approximately $14,878.56. The correct answer is option b).
Step-by-step explanation:
The question involves the calculation of the initial investment made by Amelia in a corporate bond based on the future value of the investment and differing compound interest periods (annually and semiannually).
Calculation for Annual Compounding:
To calculate the size of Amelia's initial investment with annual compounding, we apply the formula for compound interest:
A = P(1 + r/n)nt
If we know Amelia's investment has grown to $15,180, with an interest rate of 2.68% compounded annually over 7 years, we can rearrange the formula to solve for P (the principal amount):
P = A / (1 + r)t
Substituting the values, we get:
P = $15,180 / (1 + 0.0268)7
Therefore, the initial investment P is approximately $14,801.89.
Calculation for Semiannual Compounding:
For semiannual compounding, the interest is compounded twice a year. The formula is the same, but we need to adjust for the semiannual compounding period:
P = A / (1 + r/2)2t
Substituting the values, we get:
P = $15,180 / (1 + 0.0268/2)2 * 7
Therefore, the initial investment P with semiannual compounding is approximately $14,878.56.
Based on these calculations, the correct answer is option b) $14,801.89 for annual compounding and $14,878.56 for semiannual compounding.