You put \( 20 \% \) down on a home with a purchase price of \( \$ 250,000 \). The down payment is thus \( \$ 50,000 \), leaving a balance owed of \( \$ 200,000 \). The bank will loan the remaining balance at \( 3.91 \% \) APR. You will make annual payments with a 30 - year payment schedule. What is the annual annuity payment under this schedule? A. \( \$ 11,009.49 \) B. \( \$ 6,666.67 \) C. \( \$ 18,100.23 \) D. \( \$ 11,439.96 \)

To find the annual annuity payment under the given schedule, we can use the formula for the annual payment of an annuity: \[ A = \frac{P \times r \times (1 + r)^n}{(1 + r)^n - 1} \] where: - \( A \) is the annual payment - \( P \) is the principal amount (the balance owed after the down payment) - \( r \) is the annual interest rate (APR) - \( n \) is the number of years in the payment schedule Given: - Principal amount \( P = \$200,000 \) - Annual interest rate \( r = 3.91\% = 0.0391 \) - Number of years \( n = 30 \) Substitute the values into the formula to find the annual annuity payment. Calculate the value by following steps: - step0: Calculate: \(\frac{200000\times 0.0391\left(1+0.0391\right)^{30}}{\left(\left(1+0.0391\right)^{30}-1\right)}\) - step1: Remove the parentheses: \(\frac{200000\times 0.0391\left(1+0.0391\right)^{30}}{\left(1+0.0391\right)^{30}-1}\) - step2: Add the numbers: \(\frac{200000\times 0.0391\times 1.0391^{30}}{\left(1+0.0391\right)^{30}-1}\) - step3: Add the numbers: \(\frac{200000\times 0.0391\times 1.0391^{30}}{1.0391^{30}-1}\) - step4: Convert the expressions: \(\frac{200000\times 0.0391\left(\frac{10391}{10000}\right)^{30}}{1.0391^{30}-1}\) - step5: Convert the expressions: \(\frac{200000\times 0.0391\left(\frac{10391}{10000}\right)^{30}}{\left(\frac{10391}{10000}\right)^{30}-1}\) - step6: Multiply: \(\frac{7820\times \frac{10391^{30}}{10000^{30}}}{\left(\frac{10391}{10000}\right)^{30}-1}\) - step7: Subtract the numbers: \(\frac{7820\times \frac{10391^{30}}{10000^{30}}}{\frac{10391^{30}-10000^{30}}{10000^{30}}}\) - step8: Evaluate the power: \(\frac{\frac{391\times 10391^{30}}{20^{59}\times 25^{30}}}{\frac{10391^{30}-10000^{30}}{10000^{30}}}\) - step9: Multiply by the reciprocal: \(\frac{391\times 10391^{30}}{20^{59}\times 25^{30}}\times \frac{10000^{30}}{10391^{30}-10000^{30}}\) - step10: Rewrite the expression: \(\frac{391\times 10391^{30}}{20^{59}\times 25^{30}}\times \frac{400^{30}\times 25^{30}}{10391^{30}-10000^{30}}\) - step11: Rewrite the expression: \(\frac{391\times 10391^{30}}{20^{59}\times 25^{30}}\times \frac{20^{60}\times 25^{30}}{10391^{30}-10000^{30}}\) - step12: Reduce the numbers: \(\frac{391\times 10391^{30}}{25^{30}}\times \frac{20\times 25^{30}}{10391^{30}-10000^{30}}\) - step13: Rewrite the expression: \(\frac{391\times 10391^{30}}{25^{30}}\times \frac{5\times 4\times 25^{30}}{10391^{30}-10000^{30}}\) - step14: Rewrite the expression: \(\frac{391\times 10391^{30}}{5^{60}}\times \frac{5\times 4\times 25^{30}}{10391^{30}-10000^{30}}\) - step15: Reduce the numbers: \(\frac{391\times 10391^{30}}{5^{59}}\times \frac{4\times 25^{30}}{10391^{30}-10000^{30}}\) - step16: Rewrite the expression: \(\frac{391\times 10391^{30}}{5^{59}}\times \frac{4\times 5^{60}}{10391^{30}-10000^{30}}\) - step17: Reduce the numbers: \(391\times 10391^{30}\times \frac{4\times 5}{10391^{30}-10000^{30}}\) - step18: Multiply: \(\frac{391\times 10391^{30}\times 20}{10391^{30}-10000^{30}}\) - step19: Multiply: \(\frac{7820\times 10391^{30}}{10391^{30}-10000^{30}}\) The annual annuity payment under the given schedule is approximately \( \$11,439.96 \). Therefore, the correct answer is D. \( \$11,439.96 \).