Loan payment

A fully amortizing installment loan uses a fixed payment that retires principal over n months.

M = P · r · (1+r)^n / ((1+r)^n − 1) with r = annual rate / 12, n = months. If r = 0, M = P / n.

Convert the advertised annual rate to a monthly rate r.
Compute the payment M that makes the present value of n payments equal P.
Each month, interest = balance × r; principal = M − interest; new balance = balance − principal.
APR is the rate that sets the present value of the payments equal to proceeds (P minus fees).

Mortgage (PITI + schedule)

The loan is amortized as above. Housing cost adds tax, insurance, HOA, and PMI. UK-style splits keep an interest-only balloon.

PITI = M + tax/12 + insurance/12 + HOA + PMI. IO payment = IO balance × r. Balloon = IO balance at term.

Loan amount = price − down payment. LTV = loan / price.
If an interest-only amount is set, only the remainder amortizes.
PMI (when enabled) is charged while scheduled LTV is above the threshold (default 80%).
Total cost sums interest, fees, and the recurring housing add-ons over the payoff period.

How much can I borrow?

We cap the housing payment using the selected country's DTI and/or income-multiple rules, then search for the most expensive home whose PITI still fits.

max housing = min(income × front DTI, income × back DTI − other debts). Optional: loan ≤ income × LTI. Qualify at stressed rate when the country uses a buffer.

Compute the maximum monthly housing budget from DTI (and other debts).
If the country stresses rates, use rate + buffer (or the floor) to size the loan.
Binary-search home price so PITI (including tax, insurance, PMI, HOA) fits the budget.
Apply an income-multiple cap when it binds (UK, IE, NO, …).

Refinance savings

Compare the remaining amortization of the current loan with a new loan after costs.

Break-even months = closing costs / monthly payment reduction. Lifetime savings = remaining cost of old loan − (cost of new loan + cash costs).

Build the remaining schedule on the current balance, rate, and term.
New principal = balance + cash-out (+ costs if rolled in).
Difference in monthly P&I is the cash-flow change; costs delay the benefit.
A lower payment from a longer term can still lose on lifetime interest.

Extra payments

Any amount above the contractual payment is applied to principal the same month, which reduces future interest and shortens the term.

Same amortization as the base loan, with extra_t added to principal in month t. Biweekly ≈ one extra contractual payment per year (M / 12 each month).

Compute the contractual payment.
Each month add extra monthly, a yearly bonus in month 12, 24, …, and an optional one-time lump sum.
Stop when the balance hits zero; the last payment is smaller.
Months saved and interest saved are versus the no-extra schedule.

Rent vs buy

Both paths start with the same cash. The buyer puts it into the house; the renter invests it. Each year we compare net worth.

Buyer NW = home value − mortgage balance − selling costs. Renter NW = invested upfront + reinvested (buyer housing cost − rent) at the investment return.

Buyer cash outlay: down payment + buy closing, then PITI + maintenance each month.
A simplified tax deduction reduces buyer cost by interest × marginal rate.
Renter pays rent (growing) and renter insurance, and keeps the invested stake growing.
Break-even is the first year buyer net worth exceeds renter net worth.

Debt consolidation

Pay off several amortizing debts with one new loan and compare interest and time.

For each debt, remaining interest comes from its own schedule. New loan amortizes the combined principal (and fees if rolled in) at the new rate and term.

If a debt has a payment but no term, infer remaining months; if it has a term but no payment, compute the payment.
Sum balances and remaining interest.
Amortize the new loan; subtract fees from 'savings' when they are paid in cash.