Investment Accumulation with Different Compounding

Find the accumulated value of an investment of $ 20 comma 000 for 7 years at an interest rate of 1.65 % if the money is a. compounded​ semiannually; b. compounded​ quarterly; c. compounded monthly d. compounded continuously. LOADING... Click the icon to view some finance formulas. Question content area bottom Part 1 a. What is the accumulated value if the money is compounded​ semiannually? ​$    22,438.06 ​(Round to the nearest cent as​ needed.) Part 2 b. What is the accumulated value if the money is compounded​ quarterly? ​$    enter your response here ​(Round to the nearest cent as​ needed.)

To calculate the accumulated value of an investment with different compounding frequencies, we can use the formula for compound interest:

[ A = P \left(1 + \frac{r}{n}\right)^{nt} ]

Where:

• (A) = the accumulated amount (future value)
• (P) = the principal amount (initial investment)
• (r) = annual interest rate (as a decimal)
• (n) = number of times interest is compounded per year
• (t) = number of years the money is invested

Given:

• (P = 20,000)
• (r = 1.65% = 0.0165)
• (t = 7)

Part 2: Compounded Quarterly

For quarterly compounding, (n = 4).

Using the formula:

[ A = 20000 \left(1 + \frac{0.0165}{4}\right)^{4 \times 7} ]

Calculating step-by-step:

1. Calculate (\frac{r}{n}): [ \frac{0.0165}{4} = 0.004125 ]

2. Calculate (nt): [ 4 \times 7 = 28 ]

3. Calculate the accumulated value: [ A = 20000 \left(1 + 0.004125\right)^{28} ] [ A = 20000 \left(1.004125\right)^{28} ] [ A = 20000 \times 1.119576 ] [ A \approx 22391.52 ]

So, the accumulated value if the money is compounded quarterly is approximately:

[ \text{Accumulated Value} \approx $22,391.52 ]

Part 3: Compounded Monthly

For monthly compounding, (n = 12).

Using the formula:

[ A = 20000 \left(1 + \frac{0.0165}{12}\right)^{12 \times 7} ]

Calculating step-by-step:

1. Calculate (\frac{r}{n}): [ \frac{0.0165}{12} = 0.001375 ]

2. Calculate (nt): [ 12 \times 7 = 84 ]

3. Calculate the accumulated value: [ A = 20000 \left(1 + 0.001375\right)^{84} ] [ A = 20000 \left(1.001375\right)^{84} ] [ A = 20000 \times 1.119576 ] [ A \approx 22391.52 ]

So, the accumulated value if the money is compounded monthly is approximately:

[ \text{Accumulated Value} \approx $22,391.52 ]

Part 4: Compounded Continuously

For continuous compounding, we use the formula:

[ A = Pe^{rt} ]

Where (e) is the base of the natural logarithm (approximately 2.71828).

Using the formula:

[ A = 20000 \times e^{0.0165 \times 7} ]

Calculating step-by-step:

1. Calculate (rt): [ 0.0165 \times 7 = 0.1155 ]

2. Calculate (e^{0.1155}): [ e^{0.1155} \approx 1.1223 ]

3. Calculate the accumulated value: [ A = 20000 \times 1.1223 \approx 22446.00 ]

So, the accumulated value if the money is compounded continuously is approximately:

[ \text{Accumulated Value} \approx $22,446.00 ]

Summary of Results:

• Compounded Semiannually: $22,438.06
• Compounded Quarterly: $22,391.52
• Compounded Monthly: $22,391.52
• Compounded Continuously: $22,446.00