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.
1---2name: lending3description: Analyze lending products including mortgages, HELOCs, and personal loans with amortization and comparison tools. Use when the user asks about mortgage comparison, fixed vs ARM rates, loan qualification, amortization schedules, extra payments, or buying points. Also trigger when users mention 'monthly payment calculation', '15-year vs 30-year mortgage', 'PMI', 'APR vs interest rate', 'HELOC', 'home equity', 'should I buy down the rate', 'biweekly payments', or ask how much house they can afford.4---5
6# Lending Analysis
7
8## Core Concepts
9
10### Fixed-Rate Mortgage
11The interest rate and monthly payment remain constant for the life of the loan:
12
13- **Advantages:** Predictable payments, protection against rising rates, simpler budgeting
14- **Disadvantages:** Higher initial rate than ARM, no benefit if rates decline (must refinance)
15- Most common terms: 30-year and 15-year fixed
16
17### Adjustable-Rate Mortgage (ARM)
18Rate is fixed for an initial period, then adjusts periodically based on an index plus a margin:
19
20- **Notation:** 5/1 ARM = fixed for 5 years, adjusts annually thereafter; 7/1, 10/1 similarly
21- **Fully indexed rate:** Index (e.g., SOFR, 1-year Treasury) + margin (e.g., 2.75%)
22- **Rate caps** protect against extreme adjustments:
23 - Initial adjustment cap (e.g., 2%): maximum first adjustment
24 - Periodic cap (e.g., 2%): maximum change per adjustment period
25 - Lifetime cap (e.g., 5%): maximum total increase over initial rate
26- **When ARM may be appropriate:** Planning to sell or refinance before the fixed period ends, expecting rates to decline, or comfortable with rate variability
27
28### Monthly Payment Calculation
29The standard amortization formula for a fixed-rate loan:
30
31- PMT = P × [r(1+r)^n] / [(1+r)^n - 1]
32- Where: P = principal (loan amount), r = monthly interest rate (annual rate / 12), n = total number of payments (term in months)
33- Each payment splits into interest (decreasing) and principal (increasing) components:
34 - Interest portion: remaining balance × monthly rate
35 - Principal portion: PMT - interest portion
36
37### Total Interest Paid
38- Total interest = (n × PMT) - P
39- For a $400K, 30-year loan at 6.5%: PMT = $2,528, total payments = $910,178, total interest = $510,178
40
41### Extra Payments
42Additional principal payments reduce the outstanding balance, shorten the loan term, and reduce total interest:
43
44- Each extra dollar goes entirely to principal reduction
45- Impact compounds: earlier extra payments save more interest than later ones
46- Methods: lump sum, fixed monthly extra, biweekly payments (26 half-payments = 13 full payments per year)
47
48### Mortgage Points
49Prepaid interest that reduces the loan's interest rate:
50
51- **1 point = 1% of the loan amount** (e.g., 1 point on $400K = $4,000)
52- Typically reduces the rate by approximately 0.25% (varies by lender and market)
53- **Breakeven calculation:** Points cost / monthly savings = months to recoup
54- Points make sense when: planning to hold the loan beyond breakeven, itemizing deductions (points may be tax-deductible in year of purchase)
55
56### APR vs Interest Rate
57- **Interest rate:** The cost of borrowing the principal, expressed annually
58- **APR (Annual Percentage Rate):** Includes the interest rate plus certain fees and costs (origination fees, points, PMI), annualized over the loan term
59- **APR > interest rate** (always, when there are fees)
60- APR is the better metric for comparing loan offers with different fee structures
61
62### HELOC (Home Equity Line of Credit)
63A revolving credit line secured by home equity:
64
65- **Combined LTV (CLTV):** (First mortgage balance + HELOC limit) / home value
66 - Most lenders require CLTV ≤ 80-90%
67- **Draw period** (typically 10 years): borrow and repay flexibly, often interest-only payments
68- **Repayment period** (typically 20 years): no new draws, fully amortizing payments
69- **Variable rate:** Typically prime rate + margin; rate fluctuates with market
70- **Use cases:** Home improvements, debt consolidation, emergency backup (but not as primary emergency fund)
71- **Risk:** Home is collateral — default means foreclosure
72
73### Loan Qualification Criteria
74- **FICO score:** 620+ for conventional, 580+ for FHA, 700+ for best rates
75- **DTI:** Front-end ≤ 28%, back-end ≤ 36-43% (varies by program)
76- **LTV (Loan-to-Value):** Loan amount / property value; lower LTV = lower risk = better terms
77- **Reserves:** Months of payments held in liquid assets after closing (2-6 months typical)
78- **Employment/income:** Stable income history, typically 2 years documentation
79
80### PMI (Private Mortgage Insurance)
81Required when conventional loan LTV exceeds 80%:
82
83- **Cost:** 0.5-1.5% of loan amount annually, added to monthly payment
84- **Removal:** Automatic at 78% LTV (based on original amortization), requestable at 80% LTV
85- **Avoidance strategies:** 20% down payment, piggyback loan (80/10/10), lender-paid PMI (higher rate), VA loan (no PMI)
86- PMI benefits the lender, not the borrower — it is pure cost to the borrower
87
88### 15-Year vs 30-Year Comparison
89- **15-year:** Higher monthly payment, lower interest rate (typically 0.5-0.75% less), dramatically less total interest, builds equity faster
90- **30-year:** Lower required payment, more flexibility, higher total interest cost
91- **Hybrid approach:** Take a 30-year for flexibility, make extra payments as if it were a 15-year
92
93## Key Formulas
94
95| Formula | Expression | Use Case |
96|---------|-----------|----------|
97| Monthly payment | PMT = P × [r(1+r)^n] / [(1+r)^n - 1] | Fixed-rate loan payment |
98| Total interest | n × PMT - P | Total cost of borrowing |
99| Remaining balance after k payments | B_k = P × [(1+r)^n - (1+r)^k] / [(1+r)^n - 1] | Balance at any point |
100| Points breakeven | Cost of points / monthly savings | Months to recoup points |
101| LTV | Loan amount / property value | Risk and PMI assessment |
102| CLTV | (First mortgage + HELOC) / home value | Combined leverage |
103| ARM fully indexed rate | Index + margin | Rate after fixed period |
104
105## Worked Examples
106
107### Example 1: 30-year vs 15-year mortgage comparison
108**Given:** Home price $500K, down payment $100K, loan amount $400K. 30-year rate: 6.5%. 15-year rate: 5.9%.
109**Calculate:** Monthly payments, total interest, and interest savings.
110**Solution:**
1111. **30-year monthly payment:** PMT = $400,000 × [0.005417 × (1.005417)^360] / [(1.005417)^360 - 1]
112 - r = 6.5%/12 = 0.005417, n = 360
113 - PMT = **$2,528/month**
1142. **30-year total interest:** 360 × $2,528 - $400,000 = **$510,178**
1153. **15-year monthly payment:** PMT = $400,000 × [0.004917 × (1.004917)^180] / [(1.004917)^180 - 1]
116 - r = 5.9%/12 = 0.004917, n = 180
117 - PMT = **$3,354/month**
1184. **15-year total interest:** 180 × $3,354 - $400,000 = **$203,694**
1195. **Payment difference:** $3,354 - $2,528 = **$826/month more** for 15-year.
1206. **Interest savings:** $510,178 - $203,694 = **$306,484 saved** by choosing 15-year.
1217. The 15-year costs 33% more per month but saves 60% in total interest.
122
123### Example 2: Extra payment impact
124**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.
125**Calculate:** Time saved and interest saved.
126**Solution:**
1271. **Without extra payments:** 360 months, total interest = 360 × $1,896.20 - $300,000 = **$382,633**.
1282. **New term (closed-form estimate):** n = -ln(1 - P × r / PMT_total) / ln(1+r) with r = 0.065/12 = 0.005417:
129 - n = -ln(1 - $300,000 × 0.005417 / $2,096.20) / ln(1.005417) ≈ 276.3 payments.
1303. **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.
1314. **Interest saved:** $382,633 - $279,185 = **$103,449**.
1325. **Time saved:** 360 - 277 = 83 months ≈ **6.9 years**.
1336. Total extra principal paid ≈ 276 × $200 = **$55,200**. Interest saved per dollar of extra payment: $103,449 / $55,200 ≈ **$1.87**.
134
135## Common Pitfalls
136- Comparing interest rate instead of APR — APR captures fees and gives a truer cost comparison
137- ARM teaser rates creating payment shock when the fixed period ends and rates adjust upward
138- Points breakeven: buying points is not worth it if selling or refinancing before the breakeven point
139- HELOC variable rate risk during rising rate environments — budget for rate increases
140- PMI costs making high-LTV loans more expensive than they appear — factor PMI into total monthly cost
141- Ignoring opportunity cost: extra mortgage payments at 3-4% vs investing at 7-10% expected return
142- Not shopping multiple lenders — rate quotes can vary 0.5%+ for the same borrower
143- Resetting to a 30-year term when refinancing — extends total payoff even if rate is lower
144- Ignoring closing costs in refinancing decisions (see debt-management refinance breakeven)
145
146## Cross-References
147- **debt-management** (wealth-management plugin): refinancing analysis, debt payoff vs investing decisions, DTI calculations
148- **emergency-fund** (wealth-management plugin): adequate reserves required for loan qualification and financial safety
149- **tax-efficiency** (wealth-management plugin): mortgage interest deductibility, points deduction
150- **savings-goals** (wealth-management plugin): down payment saving is a common goal-based savings target
151- **liquidity-management** (wealth-management plugin): mortgage payments are the largest fixed obligation in most household cash flow plans
152
153## Running the script
154Run the reference implementation directly:
155
156```
157uv run scripts/lending.py # PEP 723 header resolves dependencies automatically
158python3 scripts/lending.py # after: pip install numpy scipy
159```
160
161A 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.
joellewis