Loan math

1.1 · Loan-to-value

How much of the house the lender owns the risk on

LTV is the loan amount divided by the property value. On a purchase, the value is the lower of the sales price and the appraised value.

LTV = loan amount ÷ lesser of (price, appraised value)

Why: The lender is lending against what the house would fetch, and the lower number is the safer estimate of that.

$360,000 ÷ $400,000 = 90%. The other 10% is the borrower's equity, the down payment, shown on top of the bar.

Watch out: If the house appraises at $390,000, the LTV is $360,000 ÷ $390,000 = 92.3%, not 90%. Use the lower of price and appraisal, never the higher.

1.2 · Combined LTV

Every lien on the property, added up

CLTV adds every mortgage secured by the property, first and second liens alike, and divides by the same value.

CLTV = (first lien + second lien + …) ÷ value

Why: If the borrower defaults, every lien holder is competing for the same house.

Split the same purchase 80-10-10: a $320,000 first mortgage, a $40,000 second, and $40,000 down.

The first mortgage is now at 80% LTV, so it needs no private mortgage insurance. The lender's total exposure has not changed: CLTV is still 90%.

Watch out: LTV looks at the first lien only. A question that mentions a second mortgage or a HELOC is almost always asking for CLTV.

1.3 · Where the payment goes

Same payment every month, a different split each time

A fixed-rate payment stays the same for the whole term. Each month, interest is charged on the balance still owed, and whatever is left of the payment pays down principal.

interest this month = balance × annual rate ÷ 12

Why: As the balance falls, each month's interest falls with it, so more of the same payment reaches principal.

The principal and interest payment on $360,000 at 6.5% for 30 years is $2,275.

Month 1: interest is $360,000 × 6.5% ÷ 12 = $1,950, so only $325 goes to principal.

Principal does not overtake interest until month 233 (19 yr 5 mo). Over the full term the borrower pays $459,160 in interest, more than the amount borrowed.

Watch out: Divide the annual rate by 12 before multiplying by the balance. Using the full annual rate overstates the first month's interest twelvefold.

1.4 · When PMI comes off

Two thresholds on the original value

Under the Homeowners Protection Act, the borrower may ask to cancel PMI once the balance reaches 80% of the original value. The servicer must end it automatically when the balance is scheduled to reach 78%, if the borrower is current.

original value = lesser of sales price or appraisal at closing

Why: The thresholds are fixed at closing, so both dates can be read off the original amortization schedule.

80% of $400,000 is $320,000. Paying as scheduled, the balance reaches it in month 95 (7 yr 11 mo): the earliest the borrower can request cancellation.

78% is $312,000, reached in month 109 (9 yr 1 mo). The servicer must drop PMI then without being asked.

Watch out: Both thresholds use the original value, not today's appraisal. Rising home prices do not move the automatic 78% date; cancelling early on a new appraisal is an investor policy, not the HPA rule.