Let's go step by step as you requested, **with clear explanations for each part**: --- ### **Step 1: Introduction of the Concept** This problem deals with **mortgage loans** and the calculation of the actual amount disbursed by the lender, the APR for the borrower, the effective interest rate if the loan is paid off early, and the impact of a prepayment penalty. The calculations involve understanding how loan fees and penalties affect the true cost of borrowing. --- ### **Step 2: Answer to Part (a)** **How much will the lender actually disburse?** - **Loan amount requested:** $85,000 - **Loan fee:** $3,600 **Amount actually disbursed = Loan amount - Loan fee** \[ \text{Disbursement} = \$85,000 - \$3,600 = \$81,400 \] **Explanation:** The lender deducts the loan fee from the total loan amount, so the borrower receives $81,400. --- ### **Step 3: Answer to Part (b)** **What is the APR for the borrower, assuming the mortgage is paid off after 30 years (full term)?** - **APR** (Annual Percentage Rate) is the true interest rate accounting for all costs. - The borrower receives $81,400 but repays the loan as if he received $85,000 at 6% for 30 years. The APR is the interest rate that equates the present value of all payments ($85,000 loan at 6% over 30 years) to the amount received ($81,400). This is usually calculated using a financial calculator or Excel's `RATE` function: \[ PMT = \frac{0.06}{12} \times 85,000 / [1 - (1 + 0.06/12)^{-360}] \approx \$509.52 \] Find the rate \( r \) that solves for: \[ 81,400 = \sum_{t=1}^{360} \frac{509.52}{(1 + r/12)^t} \] The **APR ≈ 6.37%** (calculated using a financial calculator or Excel). --- ### **Step 4: Answer to Part (c)** **If John pays off the loan after five years, what is the effective interest rate?** - **Loan paid off in 5 years (60 months)** - Calculate remaining balance after 60 payments: - Use amortization formula or Excel: `=FV(6%/12, 300, -509.52, -85,000)` after 60 payments - Remaining balance after 5 years ≈ $79,073 John pays 60 monthly payments of $509.52 plus the remaining balance: \[ \text{Total outflow} = 60 \times 509.52 + 79,073 = \$109,644 \] Calculate the IRR for cash flows: - Outflow at time 0: $81,400 (amount received) - Outflows: $509.52/month for 60 months, + $79,073 at month 60 **Effective interest rate ≈ 6.57%** --- ### **Step 5: Answer to Part (d)** **With a 2% prepayment penalty if paid within 8 years, what is the effective interest rate if John repays after 5 years?** - **Prepayment penalty**: 2% of $79,073 = $1,581.46 - Total final payment: $79,073 + $1,581.46 = $80,654.46 So, cash flows: - Outflow at time 0: $81,400 - 60 monthly payments of $509.52 - Lump sum of $80,654.46 at month 60 Calculate the IRR for these cash flows. **Effective interest rate ≈ 6.77%** --- ### **Final Step: Summary of the Final Answer** - **Disbursed amount:** $81,400 - **APR (full term):** ≈ 6.37% - **Effective rate (paid off in 5 years):** ≈ 6.57% - **Effective rate (5 years + penalty):** ≈ 6.77% --- ### **What We Did in Each Step** 1. **Introduced the concept** of mortgage loan calculations. 2. **Computed the lender's actual disbursement** after fees. 3. **Calculated the APR** over the full loan term, considering the fee. 4. **Determined the effective interest rate** if paid off after 5 years. 5. **Included a prepayment penalty** and recalculated the effective interest rate for early payoff. --- **In short:** We calculated how much the borrower actually receives, the true cost of borrowing over 30 years, and how early repayment (with and without penalties) increases the effective interest rate.