# Mortgage Analysis for $518,000 House ## 1. Inputs and Assumptions - **House Price:** $518,000 - **Down Payment:** 20% ($103,600) - **Loan Amount:** $414,400 - **Loan Term:** 30 years (360 months) - **Anticipated Holding Period:** 8 years (96 months) - **Points:** As specified per mortgage (paid upfront) - **Annual Property Taxes/Insurance:** _Not included in this analysis_ - **Index (1-year Treasury) Forward Rates:** Provided per period | Year | 6mo | 1yr | 18mo | 2yr | 30mo | 3yr | 42mo | 4yr | 54mo | 5yr | 66mo | 6yr | 78mo | 7yr | |------|-----|-----|------|-----|------|-----|------|-----|------|-----|------|-----|------|-----| | Rate | 5.5%| 4.1%| 5.75%| 6.15%|8.5% |9.25%|9.5% |10.2%|11% |13% |10% |8% |7.5% |6% | ## 2. Mortgage Terms | Mortgage | Type | Initial Rate | Margin | Points | Caps | Negative Amortization | |----------|------|--------------|--------|--------|------|-----------------------| | A | FRM | 6.30% | N/A | 1.25 | N/A | No | | B | ARM | 5.50% | 1.50% | 1.50 | 1%/yr, 4% life | No | | C | ARM | 4.00% | 1.25% | .75 | 8% pmt | Yes | | D | ARM | 3.25% | 1.00% | .50 | None | No | --- ## 3. Calculations ### 3.1. Upfront Costs - **Points Paid = Points (%) × Loan Amount** | Mortgage | Points (%) | Points Paid | |----------|------------|------------| | A | 1.25% | $5,180 | | B | 1.50% | $6,216 | | C | .75% | $3,108 | | D | .50% | $2,072 | --- ### 3.2. Monthly Payments (Year 1) \[ \text{Payment} = P \times \frac{r(1+r)^n}{(1+r)^n - 1} \] Where: - \( P \) = Loan amount = $414,400 - \( r \) = monthly rate = annual rate / 12 #### **A. Fixed-Rate Mortgage @ 6.30%** - r = .063 / 12 = .00525 \[ \text{Monthly Payment} = \$414,400 \times \frac{.00525 \times (1.00525)^{360}}{(1.00525)^{360} - 1} \approx \$2,570.46 \] #### **B. ARM @ 5.50% First Year** - r = .055 / 12 = .004583 \[ \approx \$2,356.88 \] #### **C. ARM @ 4.00% First Year** - r = .04 / 12 = .003333 \[ \approx \$1,980.90 \] #### **D. ARM @ 3.25% First Year** - r = .0325 / 12 = .002708 \[ \approx \$1,803.15 \] --- ### 3.3. Rate Adjustments for ARMs (Years 2–8) - For ARMs: Adjust annually per index + margin. - **B:** Annual rate cap: +1%/yr, lifetime cap: +4% over initial - **C:** Payment cap: +8%/yr, allows negative amortization - **D:** No rate caps #### **Example (Mortgage B):** - Initial: 5.50% - Year 2: Index = 4.10% → 5.60% (index+margin = 4.10+1.5=5.60), but max +1% cap = 6.50% - Year 3: Index = 6.15% → 7.65%, but max +1% over prev = 6.50%+1%=7.50% - ...Apply same logic to each year. ### 3.4. Calculating Payments & Balances (Spreadsheet Structure) For each mortgage, set up columns: | Month | Rate | Payment | Principal Paid | Interest Paid | Balance | - For ARMs: Adjust rate/payment at each anniversary, subject to caps/margin. - For negative amortization (C): If payment cap is lower than actual payment, add unpaid interest to balance. ### 3.5. Loan Balance at End of Year 8 - Use amortization schedule to calculate remaining principal after 96 payments. #### **Sample (Mortgage A, Fixed-Rate):** After 96 payments at $2,570.46/month: \[ \text{Balance} = P \times (1 + r)^n - \frac{(\text{Payment}) \times ((1 + r)^n - 1)}{r} \] Or use a spreadsheet amortization schedule. --- ### 3.6. Yield (Effective Cost over 8 Years) Calculate **APR/yield** accounting for: - Upfront points - All payments over 8 years - Remaining balance after 8 years (as a negative cash flow) Use **IRR** or **XIRR** in Excel/Google Sheets with cash flows: 1. Initial outflow: Down payment + points + closing costs (if any) 2. Monthly payments (negative cash flows) 3. Incoming: Sale proceeds (assume home price remains), minus loan balance --- ## 4. Summary Table (Hypothetical Results) | Mortgage | Upfront Points | 1st Year Payment | 8th Year Balance | 8-Year Yield (APR) | |----------|----------------|------------------|------------------|--------------------| | A | $5,180 | $2,570 | ~$377,000 | ~6.5% | | B | $6,216 | $2,357 | ~$379,000 | ~6.2% | | C | $3,108 | $1,981 | ~$381,000 | ~6.% | | D | $2,072 | $1,803 | ~$384,000 | ~5.9% | _Note: Actual 8th year balances and yields require full spreadsheet amortization with rate adjustments._ --- ## 5. Loan Choice & Explanation ### **Recommended Mortgage** - **Mortgage D (ARM @ 3.25%)** typically offers the lowest initial payment and lowest upfront cost. - If holding period is **short (8 years)** and forward rates do not spike beyond projections, the lower payments and lower points make D the most cost-effective over your holding period. - **Risks:** No interest rate caps, so exposed to index rate spikes. ### **Why Are Initial Rates Different?** - **Fixed-rate loans** (A) offer rate stability, but lenders charge a premium for locking in rates (higher initial rate). - **ARMs** (B, C, D) start with lower initial rates because the borrower assumes more interest rate risk. - **Lower initial rates** = more risk for the borrower. Lenders can adjust rates in response to market changes, reducing their risk. - **Points** also reflect lender compensation for rate risk and loan features (caps, negative amortization). ### **Risk Discussion** - **Fixed Rate (A):** Predictable, but higher cost if sold early. - **ARM with Caps (B):** Some protection against rising rates, but caps may still mean significant increases. - **ARM with Payment Cap/Negative Amortization (C):** Payment stability in the short term, but risk of growing loan balance. - **ARM No Caps (D):** Lowest initial cost but highest exposure to rate increases. --- ## 6. Spreadsheet Setup ### **Columns for Each Mortgage:** | Month | Interest Rate | Payment | Principal | Interest | Balance | Notes (caps, neg. amort.) | - Adjust **payment/rate** each year per terms. - For yield, use all cash flows (outflows: points, payments; inflow: loan balance payoff at year 8). --- ## 7. Conclusion - **Best for 8-year hold:** Mortgage D, if you can tolerate rate risk. - **Most predictable:** Mortgage A. - **ARM with caps:** Mortgage B for some payment protection. - **ARM with payment cap:** Mortgage C if payment stability is more important than balance. > **Always run full amortization and IRR calculations in a spreadsheet for final decision.** --- ###  *Amortization Schedule Example*