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Analogous to continuous compounding, a continuous annuity is an ordinary annuity in which the payment interval is narrowed indefinitely. A continuous repayment mortgage is a mortgage loan paid by means of a continuous annuity.
Mortgages are generally settled over a period of years by a series of fixed regular payments commonly referred to as an annuity. Each payment accumulates compound interest from time of deposit to the end of the mortgage timespan at which point the sum of the payments with their accumulated interest equals the value of the loan with interest compounded over the entire timespan. Given loan P0, per period interest rate i, number of periods n and fixed per period payment x, the end of term balancing equation is:
Summation can be computed using the standard formula for summation of a geometric sequence.
In a continuous-repayment mortgage the payment interval is narrowed indefinitely until the discrete interval process becomes continuous and the fixed interval payments become—in effect—a literal cash "flow" at a fixed annual rate. In this case, given loan P0, annual interest rate r, loan timespan T and annual rate Ma, the infinitesimal cash flow elements Maδt accumulate continuously compounded interest from time t to the end of the loan timespan at which point the balancing equation is: