Questions: What is the present value (PV) of an investment that pays 100,000 every year for four years if the interest rate is 8% APR, compounded quarterly? A. 428,256 B. 395,313 C. 362,370 D. 329,428

To find the present value (PV) of an investment that pays \$100,000 every year for four years with an interest rate of 8% APR compounded quarterly, we need to use the formula for the present value of an annuity. The formula is:

$$PV = P \times \left(1 - (1 + r)^{-n}\right) / r$$

Where:

• $P$ is the annual payment (\$100,000).
• $r$ is the effective quarterly interest rate.
• $n$ is the total number of periods.

First, we need to calculate the effective quarterly interest rate. Since the APR is 8% compounded quarterly, the quarterly interest rate is:

$$r = \frac{0.08}{4} = 0.02$$

Next, we need to determine the total number of periods. Since the payments are annual and the interest is compounded quarterly, we have:

$$n = 4 \text{ years} \times 4 \text{ quarters per year} = 16 \text{ quarters}$$

Now, we can substitute these values into the present value formula for an annuity:

$$PV = 100,000 \times \left(1 - (1 + 0.02)^{-16}\right) / 0.02$$

Calculating the expression inside the parentheses:

$$(1 + 0.02)^{-16} = (1.02)^{-16} \approx 0.7248$$

$$1 - 0.7248 = 0.2752$$

Now, calculate the present value:

$$PV = 100,000 \times \frac{0.2752}{0.02}$$

$$PV = 100,000 \times 13.76$$

$$PV = 1,376,000$$

However, this calculation seems incorrect because it doesn't match any of the given options. Let's re-evaluate the approach:

The correct approach is to calculate the present value of each individual payment and sum them up. The formula for the present value of a single future payment is:

$$PV = \frac{FV}{(1 + r)^n}$$

Where:

• $FV$ is the future value (\$100,000).
• $r$ is the quarterly interest rate (0.02).
• $n$ is the number of quarters until the payment.

Calculate the present value for each of the four payments:

1. First payment (1 year or 4 quarters away): $$PV_1 = \frac{100,000}{(1.02)^4} \approx 92,456$$

2. Second payment (2 years or 8 quarters away): $$PV_2 = \frac{100,000}{(1.02)^8} \approx 85,564$$

3. Third payment (3 years or 12 quarters away): $$PV_3 = \frac{100,000}{(1.02)^12} \approx 79,383$$

4. Fourth payment (4 years or 16 quarters away): $$PV_4 = \frac{100,000}{(1.02)^16} \approx 73,886$$

Sum these present values:

$$PV_{\text{total}} = 92,456 + 85,564 + 79,383 + 73,886 \approx 331,289$$

This value is closest to option D. However, there might be a slight discrepancy due to rounding or calculation errors. The closest answer is:

The answer is D: \$329,428

Explanation for each option:

• A. \$428,256: This is too high for the given interest rate and compounding.
• B. \$395,313: This is also too high.
• C. \$362,370: This is closer but still higher than the calculated value.
• D. \$329,428: This is the closest to the calculated present value.