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:
First payment (1 year or 4 quarters away):
\[ PV_1 = \frac{100,000}{(1.02)^4} \approx 92,456 \]
Second payment (2 years or 8 quarters away):
\[ PV_2 = \frac{100,000}{(1.02)^8} \approx 85,564 \]
Third payment (3 years or 12 quarters away):
\[ PV_3 = \frac{100,000}{(1.02)^12} \approx 79,383 \]
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.
Answered
