9.2 Loan Mathematics: LTV Ratios, DTI Ratios, Discount Points & Loan Fees

Loan-to-Value (LTV) Ratio Mechanics & Equity Calculations

The Loan-to-Value (LTV) ratio is the primary metric institutional mortgage lenders use to assess credit risk and determine whether private mortgage insurance (PMI) is required on conventional financing.

The Fundamental LTV Rule

$\text{LTV Ratio} = \frac{\text{Loan Amount}}{\text{Lower of Purchase Price or Appraised Value}}$

[!IMPORTANT] The Conservative Underwriting Rule: Lenders never calculate LTV against an inflated purchase price if the independent appraisal comes in lower. The denominator is always the purchase price or appraised value, whichever is less. If the appraisal falls short of the contract price, the borrower must either negotiate a price reduction, increase their cash down payment, or the seller must make concessions.

• Down Payment Calculation: $\text{Down Payment} = \text{Purchase Price} - \text{Loan Amount}$

• Homeowner Equity Formula: $\text{Equity} = \text{Current Market Value} - \text{Total Outstanding Liens}$

Worked Example: Appraisal Deficit and Required Down Payment

• Scenario: A buyer signs a contract to purchase a home in Summit for $620,000. The buyer applies for an 80% conventional first mortgage. The lender's appraisal evaluates the property at $600,000.

1. Determine Maximum Loan Amount Permitted: $\text{Underwriting Base} = \min(\$620,000, \$600,000) = \$600,000$ $\text{Maximum Loan Amount} = \$600,000 \times 0.80 = \$480,000$

2. Calculate Buyer's Required Cash Down Payment: $\text{Required Down Payment} = \$620,000 - \$480,000 = \$140,000$

   • Note on Percentages: The buyer must provide $140,000 in cash, which represents $22.58% of the $620,000 contract price ($120,000 base down payment on $600,000 plus the $20,000 appraisal shortfall).

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Debt-to-Income (DTI) Qualifying Ratios

Mortgage underwriters evaluate a borrower's capacity to repay debt using two standard Debt-to-Income (DTI) ratios: the Front-End (Housing Expense) Ratio and the Back-End (Total Debt Service) Ratio.

┌────────────────────────────────────────────────────────────────────────┐
│                     GROSS MONTHLY INCOME (GMI)                         │
└────────────────────────────────────────────────────────────────────────┘
                     │                                 │
                     ▼                                 ▼
       ┌───────────────────────────┐     ┌───────────────────────────┐
       │      FRONT-END RATIO      │     │      BACK-END RATIO       │
       │       (PITI / GMI)        │     │  ([PITI + Debt] / GMI)    │
       └───────────────────────────┘     └───────────────────────────┘

1. Front-End Ratio (Housing Expense Ratio)

The front-end ratio measures total housing expenses as a percentage of gross monthly income:

$\text{Front-End DTI} = \frac{\text{Total Monthly Housing Expense (PITI)}}{\text{Gross Monthly Income (GMI)}}$

Where PITI consists of:

• P: Principal portion of monthly mortgage payment
• I: Interest portion of monthly mortgage payment
• T: Real estate property taxes (Annual taxes / 12)
• I: Hazard/homeowner insurance premiums (Annual premium / 12)
• Additions: Homeowners Association (HOA) dues, condo fees, and Private Mortgage Insurance (PMI) premiums.

2. Back-End Ratio (Total Debt Service Ratio)

The back-end ratio measures all recurring long-term debt obligations plus total housing expenses against gross monthly income:

$\text{Back-End DTI} = \frac{\text{PITI} + \text{Recurring Monthly Debt Payments}}{\text{Gross Monthly Income (GMI)}}$

Recurring Monthly Debt Includes:

• Auto loan installment payments
• Student loan payments
• Minimum monthly credit card payments
• Alimony, child support, or separate maintenance obligations
• Personal loans or other installment debts with 10 or more remaining payments
• Exclusions: Utility bills, cell phone service, groceries, and medical insurance premiums.

Standard Underwriting Guidelines by Loan Type

Computational Example: Maximum Qualifying Mortgage Payment

• Borrower Profile:
  • Annual Gross Income: $132,000 (Gross Monthly Income = $$132,000 / 12 = $11,000$)
  • Car Payment: $450/month
  • Student Loans: $350/month
  • Minimum Credit Card Payments: $200/month
  • Total Recurring Debt: $$450 + $350 + $200 = $1,000/\text{month}$
  • Loan Program: Conventional Conforming (28% / 36%)

1. Step 1: Test Front-End Maximum: $\text{Max Housing Allowed} = \$11,000 \times 0.28 = \$3,080$

2. Step 2: Test Back-End Maximum: $\text{Max Total Debt Allowed} = \$11,000 \times 0.36 = \$3,960$ $\text{Max Housing Permitted via Back-End} = \$3,960 - \$1,000 = \$2,960$

3. Step 3: Identify Qualifying Benchmark:

   • The lender limits the borrower to the lesser of the two thresholds.
   • Maximum allowable monthly PITI payment is $2,960 (constrained by back-end debt).

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Discount Points, Origination Fees & Lender Yield

Points are upfront percentage fees paid to the lender at loan origination. Each point equals exactly 1% of the loan amount—never the property sales price.

$\text{Point Cost} = \text{Loan Amount} \times (\text{Number of Points} \times 0.01)$

• Loan Origination Fee: An administrative processing charge assessed by the lender to establish the loan file.
• Discount Points: Prepaid interest paid upfront to buy down the contractual note interest rate below par.

Impact on Lender Yield

As a standardized rule of thumb on real estate licensing examinations:

$\text{Yield Increase} \approx 1\text{ Discount Point} = \frac{1}{8}\text{ of } 1\% = 0.125\%$

• Conversely, to lower the borrower's note rate by $0.25%$ (1/4 of 1%), the lender generally charges 2 discount points ($2 \times 0.125% = 0.250%$).

Break-Even Analysis on Discount Points

A broker must be able to advise clients on whether paying upfront points makes financial sense:

$\text{Break-Even Period (Months)} = \frac{\text{Upfront Cost of Discount Points}}{\text{Monthly Payment Savings}}$

• Worked Problem: A buyer secures a $400,000 mortgage. Par rate is 6.5% with a monthly principal and interest (P&I) payment of $2,528.27. By paying 2 discount points, the borrower buys down the rate to 6.0%, reducing monthly P&I to $2,398.20.

1. Upfront Cost of 2 Points: $\$400,000 \times 0.02 = \$8,000$

2. Monthly Savings: $\$2,528.27 - \$2,398.20 = \$130.07/\text{month}$

3. Break-Even Horizon: $\frac{\$8,000}{\$130.07} = 61.5\text{ months (approx. 5.1 years)}$

   • If the buyer expects to own the property longer than 62 months, purchasing the points provides long-term net savings; if refinancing or moving within 5 years, paying points results in a net financial loss.

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Amortization Calculations & Simple Interest Allocation

Fully amortizing fixed-rate mortgages feature equal monthly payments consisting of fluctuating allocations to interest and principal reduction.

Monthly Interest Calculation Formula

Interest on residential mortgages is calculated in arrears using simple interest on the unpaid principal balance:

$\text{Monthly Interest Due} = \frac{\text{Current Principal Balance} \times \text{Annual Interest Rate}}{12}$

$\text{Principal Reduction} = \text{Total Monthly Principal and Interest Payment} - \text{Monthly Interest Due}$

$\text{New Principal Balance} = \text{Current Principal Balance} - \text{Principal Reduction}$

• Amortization Demonstration Table: Loan Amount: $300,000 | Interest Rate: 6.0% | Monthly P&I: $1,798.65

• Calculation Details for Month 1:

  • Interest: $($300,000 \times 0.06) / 12 = $1,500.00$
  • Principal: $$1,798.65 - $1,500.00 = $298.65$
  • New Balance: $$300,000 - $298.65 = $299,701.35$

• Calculation Details for Month 2:

  • Interest: $($299,701.35 \times 0.06) / 12 = $1,498.51$
  • Principal: $$1,798.65 - $1,498.51 = $300.14$
  • New Balance: $$299,701.35 - $300.14 = $299,401.21$