How much will $8,000 grow to in five years, assuming an interest rate of 8% compounded quarterly? note: use tables, excel, or a financial calculator. (fv of $1, pv of $1, fva of $1, pva of $1, fvad of $1 and pvad of $1) multiple choice $11,755 $10,989 $12,013 $11,888
The Correct Answer and Explanation is:
To determine how much $8,000 will grow to in five years at an interest rate of 8% compounded quarterly, we can use the future value (FV) formula for compound interest, which is expressed as:
[
FV = PV \times (1 + r/n)^{nt}
]
Where:
- (FV) = future value
- (PV) = present value (initial investment)
- (r) = annual interest rate (decimal)
- (n) = number of compounding periods per year
- (t) = number of years
Given:
- (PV = 8000)
- (r = 0.08) (8% as a decimal)
- (n = 4) (quarterly compounding)
- (t = 5)
Step 1: Calculate the interest rate per period and total number of periods.
- Interest rate per quarter:
[
r/n = 0.08/4 = 0.02
] - Total number of compounding periods over 5 years:
[
nt = 4 \times 5 = 20
]
Step 2: Substitute the values into the FV formula.
[
FV = 8000 \times (1 + 0.02)^{20}
]
Step 3: Calculate.
[
FV = 8000 \times (1.02)^{20}
]
Calculating ( (1.02)^{20} ):
[
(1.02)^{20} \approx 1.485947
]
Thus,
[
FV \approx 8000 \times 1.485947 \approx 11887.58
]
Conclusion:
The future value of $8,000 after five years at an 8% interest rate compounded quarterly is approximately $11,888.
Selection of Answer:
Among the provided options, $11,888 is the correct answer.
Explanation
The concept of compound interest significantly impacts the growth of investments over time. By compounding quarterly, we leverage the interest earned in previous periods, leading to exponential growth rather than linear. This example illustrates how an initial investment can increase substantially over a relatively short time due to the effects of compounding. When choosing investments or savings plans, understanding the power of compound interest and the frequency of compounding is crucial for maximizing returns. Financial calculators and Excel can efficiently perform these calculations, enabling investors to plan and project future earnings effectively. The results highlight the importance of starting to save early and taking advantage of favorable interest rates.