Let's solve the problem step by step: ### Given Data - **Loan Amount (\(PV\))**: $80,200 - **Interest Rate (\(r\))**: 10% per year (monthly = 10%/12 = 0.8333% per month) - **Loan Term**: 25 years (300 months) - **Regular Monthly Payment (\(PMT\))**: $728.78 #### Builder's Offer (Buydown Plan): - **Year 1**: Payments reduced by 50% ($728.78 × 0.5 = $364.39 per month for first 12 months) - **Year 2**: Payments reduced by 25% ($728.78 × 0.75 = $546.585 per month for months 13–24) - **Year 3 onward**: $728.78 per month (months 25–300) #### Question (a): **How much would you expect the builder to have to give the bank to buy down the payments as indicated?** --- ## Step 1: Calculate the Actual Payments - **Months 1–12:** $364.39 per month - **Months 13–24:** $546.585 per month - **Months 25–300:** $728.78 per month --- ## Step 2: Calculate the Present Value (PV) of the Reduced Payments We need to find the **present value of the actual payments** (which is less than $80,200), then the **difference** is the amount the builder must pay the bank upfront to compensate for the lower payments. ### PV Formula for Each Period \[ PV = \sum \left( \frac{PMT}{(1 + r)^n} \right) \] where - \(PMT\) = monthly payment in that period - \(r\) = monthly interest rate = 0.10/12 = 0.008333 - \(n\) = number of months #### 1. First Year (Months 1-12): \[ PV_1 = PMT_1 \times \frac{1 - (1 + r)^{-12}}{r} \] \[ PV_1 = 364.39 \times \frac{1 - (1 + 0.008333)^{-12}}{0.008333} \] \[ (1 + 0.008333)^{-12} = (1.008333)^{-12} = 0.90438 \] \[ 1 - 0.90438 = 0.09562 \] \[ 0.09562 / 0.008333 = 11.48 \] \[ PV_1 = 364.39 \times 11.48 = \$4,184.10 \] #### 2. Second Year (Months 13-24): *The present value at month 0 of $546.585 for months 13-24 is:* First, find the present value at month 12, then discount it back 12 months. At month 12: \[ PV_{2@12} = 546.585 \times \frac{1 - (1 + r)^{-12}}{r} \] \[ = 546.585 \times 11.48 = \$6,284.33 \] Discount back to present (month 0): \[ PV_2 = 6,284.33 \times (1 + r)^{-12} \] \[ = 6,284.33 \times 0.90438 = \$5,684.42 \] #### 3. Remaining Payments (Months 25–300): From month 24, there are \(300 - 24 = 276\) payments of $728.78 per month. First, find the present value at month 24: \[ PV_{3@24} = 728.78 \times \frac{1 - (1 + r)^{-276}}{r} \] \[ (1 + r)^{-276} = (1.008333)^{-276} \approx 0.09462 \] \[ 1 - 0.09462 = 0.90538 \] \[ 0.90538 / 0.008333 = 108.68 \] \[ PV_{3@24} = 728.78 \times 108.68 = \$79,242.18 \] Discount back 24 months: \[ (1 + r)^{-24} = (1.008333)^{-24} = 0.81800 \] \[ PV_3 = 79,242.18 \times 0.81800 = \$64,820.05 \] --- ### Step 3: Add Up the Present Values \[ PV_{total} = PV_1 + PV_2 + PV_3 = 4,184.10 + 5,684.42 + 64,820.05 = \$74,688.57 \] --- ### Step 4: Calculate the Buydown Amount \[ \text{Buydown Amount} = \text{Original Loan Amount} - PV_{total} \] \[ = \$80,200 - \$74,688.57 = \$5,511.43 \] **Rounded to nearest whole dollar:** \[ \boxed{\$5,511} \] --- ## Part (a) Final Answer **The builder would have to give the bank \$5,511 to buy down the payments as described.** --- ## Part (b): Would you recommend buying the property if it was selling for \$5,000 more than similar properties that do not have the buydown? Why or why not? ### Analysis - The **value of the buydown** is \$5,511 (present value). - **Extra price** over comparable properties = \$5,000. - Thus, the buyer pays \$5,000 extra but receives a benefit worth \$5,511 in present value. ### Recommendation Since the present value of the payment reduction (\$5,511) **exceeds** the extra cost (\$5,000), the buyer **gains** \$511 in present value by choosing this property with the buydown. --- ## Final Answers **a. The builder would have to give the bank \$5,511 to buy down the payments.** **b. Yes, I would recommend purchasing the property for \$5,000 more, because the present value of the payment reduction (\$5,511) exceeds the additional price (\$5,000).**