Lending

Lending Analysis

Core Concepts

Fixed-Rate Mortgage

The interest rate and monthly payment remain constant for the life of the loan:

• Advantages: Predictable payments, protection against rising rates, simpler budgeting
• Disadvantages: Higher initial rate than ARM, no benefit if rates decline (must refinance)
• Most common terms: 30-year and 15-year fixed

Adjustable-Rate Mortgage (ARM)

Rate is fixed for an initial period, then adjusts periodically based on an index plus a margin:

• Notation: 5/1 ARM = fixed for 5 years, adjusts annually thereafter; 7/1, 10/1 similarly
• Fully indexed rate: Index (e.g., SOFR, 1-year Treasury) + margin (e.g., 2.75%)
• Rate caps protect against extreme adjustments:
• Initial adjustment cap (e.g., 2%): maximum first adjustment
• Periodic cap (e.g., 2%): maximum change per adjustment period
• Lifetime cap (e.g., 5%): maximum total increase over initial rate
• When ARM may be appropriate: Planning to sell or refinance before the fixed period ends, expecting rates to decline, or comfortable with rate variability

Monthly Payment Calculation

The standard amortization formula for a fixed-rate loan:

• PMT = P × [r(1+r)^n] / [(1+r)^n - 1]
• Where: P = principal (loan amount), r = monthly interest rate (annual rate / 12), n = total number of payments (term in months)
• Each payment splits into interest (decreasing) and principal (increasing) components:
• Interest portion: remaining balance × monthly rate
• Principal portion: PMT - interest portion

Total Interest Paid

• Total interest = (n × PMT) - P
• For a $400K, 30-year loan at 6.5%: PMT = $2,528, total payments = $910,178, total interest = $510,178

Extra Payments

Additional principal payments reduce the outstanding balance, shorten the loan term, and reduce total interest:

• Each extra dollar goes entirely to principal reduction
• Impact compounds: earlier extra payments save more interest than later ones
• Methods: lump sum, fixed monthly extra, biweekly payments (26 half-payments = 13 full payments per year)

Mortgage Points

Prepaid interest that reduces the loan's interest rate:

• 1 point = 1% of the loan amount (e.g., 1 point on $400K = $4,000)
• Typically reduces the rate by approximately 0.25% (varies by lender and market)
• Breakeven calculation: Points cost / monthly savings = months to recoup
• Points make sense when: planning to hold the loan beyond breakeven, itemizing deductions (points may be tax-deductible in year of purchase)

APR vs Interest Rate

• Interest rate: The cost of borrowing the principal, expressed annually
• APR (Annual Percentage Rate): Includes the interest rate plus certain fees and costs (origination fees, points, PMI), annualized over the loan term
• APR > interest rate (always, when there are fees)
• APR is the better metric for comparing loan offers with different fee structures

HELOC (Home Equity Line of Credit)

A revolving credit line secured by home equity:

• Combined LTV (CLTV): (First mortgage balance + HELOC limit) / home value
• Most lenders require CLTV ≤ 80-90%
• Draw period (typically 10 years): borrow and repay flexibly, often interest-only payments
• Repayment period (typically 20 years): no new draws, fully amortizing payments
• Variable rate: Typically prime rate + margin; rate fluctuates with market
• Use cases: Home improvements, debt consolidation, emergency backup (but not as primary emergency fund)
• Risk: Home is collateral — default means foreclosure

Loan Qualification Criteria

• FICO score: 620+ for conventional, 580+ for FHA, 700+ for best rates
• DTI: Front-end ≤ 28%, back-end ≤ 36-43% (varies by program)
• LTV (Loan-to-Value): Loan amount / property value; lower LTV = lower risk = better terms
• Reserves: Months of payments held in liquid assets after closing (2-6 months typical)
• Employment/income: Stable income history, typically 2 years documentation

PMI (Private Mortgage Insurance)

Required when conventional loan LTV exceeds 80%:

• Cost: 0.5-1.5% of loan amount annually, added to monthly payment
• Removal: Automatic at 78% LTV (based on original amortization), requestable at 80% LTV
• Avoidance strategies: 20% down payment, piggyback loan (80/10/10), lender-paid PMI (higher rate), VA loan (no PMI)
• PMI benefits the lender, not the borrower — it is pure cost to the borrower

15-Year vs 30-Year Comparison

• 15-year: Higher monthly payment, lower interest rate (typically 0.5-0.75% less), dramatically less total interest, builds equity faster
• 30-year: Lower required payment, more flexibility, higher total interest cost
• Hybrid approach: Take a 30-year for flexibility, make extra payments as if it were a 15-year

Key Formulas

| Formula | Expression | Use Case |

|---------|-----------|----------|

| Monthly payment | PMT = P × [r(1+r)^n] / [(1+r)^n - 1] | Fixed-rate loan payment |

| Total interest | n × PMT - P | Total cost of borrowing |

| Remaining balance after k payments | B_k = P × [(1+r)^n - (1+r)^k] / [(1+r)^n - 1] | Balance at any point |

| Points breakeven | Cost of points / monthly savings | Months to recoup points |

| LTV | Loan amount / property value | Risk and PMI assessment |

| CLTV | (First mortgage + HELOC) / home value | Combined leverage |

| ARM fully indexed rate | Index + margin | Rate after fixed period |

Worked Examples

Example 1: 30-year vs 15-year mortgage comparison

Given: Home price $500K, down payment $100K, loan amount $400K. 30-year rate: 6.5%. 15-year rate: 5.9%.

Calculate: Monthly payments, total interest, and interest savings.

Solution:

• 30-year monthly payment: PMT = $400,000 × [0.005417 × (1.005417)^360] / [(1.005417)^360 - 1]
• r = 6.5%/12 = 0.005417, n = 360
• PMT = $2,528/month
• 30-year total interest: 360 × $2,528 - $400,000 = $510,178
• 15-year monthly payment: PMT = $400,000 × [0.004917 × (1.004917)^180] / [(1.004917)^180 - 1]
• r = 5.9%/12 = 0.004917, n = 180
• PMT = $3,354/month
• 15-year total interest: 180 × $3,354 - $400,000 = $203,694
• Payment difference: $3,354 - $2,528 = $826/month more for 15-year.
• Interest savings: $510,178 - $203,694 = $306,484 saved by choosing 15-year.
• The 15-year costs 33% more per month but saves 60% in total interest.

Example 2: Extra payment impact

Given: $300K 30-year mortgage at 6.5% (payment = $1,896.20/month). Borrower adds $200/month extra to principal, making the total payment $2,096.20/month.

Calculate: Time saved and interest saved.

Solution:

• Without extra payments: 360 months, total interest = 360 × $1,896.20 - $300,000 = $382,633.
• New term (closed-form estimate): n = -ln(1 - P × r / PMT_total) / ln(1+r) with r = 0.065/12 = 0.005417:
• n = -ln(1 - $300,000 × 0.005417 / $2,096.20) / ln(1.005417) ≈ 276.3 payments.
• Exact amortization (month-by-month simulation, final payment partial): payoff in 277 months = 23 years 1 month (~23.1 years), total interest = $279,185. The closed form and the exact schedule agree closely.
• Interest saved: $382,633 - $279,185 = $103,449.
• Time saved: 360 - 277 = 83 months ≈ 6.9 years.
• Total extra principal paid ≈ 276 × $200 = $55,200. Interest saved per dollar of extra payment: $103,449 / $55,200 ≈ $1.87.

Common Pitfalls

• Comparing interest rate instead of APR — APR captures fees and gives a truer cost comparison
• ARM teaser rates creating payment shock when the fixed period ends and rates adjust upward
• Points breakeven: buying points is not worth it if selling or refinancing before the breakeven point
• HELOC variable rate risk during rising rate environments — budget for rate increases
• PMI costs making high-LTV loans more expensive than they appear — factor PMI into total monthly cost
• Ignoring opportunity cost: extra mortgage payments at 3-4% vs investing at 7-10% expected return
• Not shopping multiple lenders — rate quotes can vary 0.5%+ for the same borrower
• Resetting to a 30-year term when refinancing — extends total payoff even if rate is lower
• Ignoring closing costs in refinancing decisions (see debt-management refinance breakeven)

Cross-References

• debt-management (wealth-management plugin): refinancing analysis, debt payoff vs investing decisions, DTI calculations
• emergency-fund (wealth-management plugin): adequate reserves required for loan qualification and financial safety
• tax-efficiency (wealth-management plugin): mortgage interest deductibility, points deduction
• savings-goals (wealth-management plugin): down payment saving is a common goal-based savings target
• liquidity-management (wealth-management plugin): mortgage payments are the largest fixed obligation in most household cash flow plans

Running the script

Run the reference implementation directly:

uv run scripts/lending.py          # PEP 723 header resolves dependencies automatically
python3 scripts/lending.py         # after: pip install numpy scipy

A bare run prints a demo covering payment calculation, extra-payment impact, LTV/PMI, DSCR, balloon payments, refinance analysis, points breakeven, and ARM resets. Use --verify to recompute the demo figures and assert they match this skill's worked examples (prints PASS/FAIL, exits nonzero on mismatch), and --help to list the available classes and functions. The file is primarily meant to be imported as a module (from lending import LendingAnalysis) rather than run standalone.