To solve these problems, we use the future value formula:

$FV = PV \times (1 + r)^n$

Where:

• $FV$ = Future Value
• $PV$ = Present Value ($8,000)
• $r$ = Interest rate per period (annual or semiannual rate)
• $n$ = Total number of compounding periods

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Part a: Future Value with 6% Annual Interest Rate, 5 Years

Given:
$PV = 8,000$, $r = 0.06$, $n = 5$
$FV = 8,000 \times (1 + 0.06)^5 = 8,000 \times 1.338225 = 10,705.80$
Future Value = $10,705.80

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Part b: Future Value with 6% Annual Rate, Semiannual Compounding

Given:
$PV = 8,000$, semiannual rate $r = 0.06 / 2 = 0.03$, and $n = 5 \times 2 = 10$:
$FV = 8,000 \times (1 + 0.03)^{10} = 8,000 \times 1.343916 = 10,751.33$
Future Value = $10,751.33

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Part c: Future Value with 12% Annual Rate

Annual Compounding (12%)

Given:
$PV = 8,000$, $r = 0.12$, $n = 5$:
$FV = 8,000 \times (1 + 0.12)^5 = 8,000 \times 1.7623 = 14,098.40$
Future Value = $14,098.40

Semiannual Compounding (12%)

Given:
$PV = 8,000$, semiannual rate $r = 0.12 / 2 = 0.06$, and $n = 5 \times 2 = 10$:
$FV = 8,000 \times (1 + 0.06)^{10} = 8,000 \times 1.790847 = 14,326.78$
Future Value = $14,326.78

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Part d: Future Value with 6% Annual Rate, 12 Years

Given:
$PV = 8,000$, $r = 0.06$, $n = 12$:
$FV = 8,000 \times (1 + 0.06)^{12} = 8,000 \times 2.012196 = 16,097.57$
Future Value = $16,097.57

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Summary of Results:

a. Annual $6\%$, 5 years: $10,705.80
b. Semiannual $6\%$, 5 years: $10,751.33
c. Annual $12\%$, 5 years: $14,098.40
Semiannual $12\%$, 5 years: $14,326.78
d. Annual $6\%$, 12 years: $16,097.57

Would you like further clarification or detailed breakdowns?

5 Related Questions:

1. How does increasing the compounding frequency affect future value?
2. What is the difference between simple and compound interest?
3. How does the time horizon influence the growth of an investment?
4. What would happen to the future values if the principal were doubled?
5. Can you derive a formula for continuous compounding?

Tip:

The higher the compounding frequency, the faster your investment grows due to the effects of compounding within a shorter interval.