Analogous to continuous compounding, a continuous annuity is an ordinary annuity in which the payment interval is narrowed indefinitely. A (theoretical) continuous repayment mortgage is a mortgage loan paid by means of a continuous annuity.
Mortgages (i.e., mortgage loans) 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 (theoretical) 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 (years) 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:
Summation of the cash flow elements and accumulated interest is effected by integration as shown. It is assumed that compounding interval and payment interval are equal—i.e., compounding of interest always occurs at the same time as payment is deducted.
Within the timespan of the loan the time continuous mortgage balance function obeys a first order linear differential equation (LDE) and an alternative derivation thereof may be obtained by solving the LDE using the method of Laplace transforms.
Application of the equation yields a number of results relevant to the financial process which it describes. Although this article focuses primarily on mortgages, the methods employed are relevant to any situation in which payment or saving is effected by a regular stream of fixed interval payments (annuity).
Derivation of time-continuous equation
The classical formula for the present value of a series of n fixed monthly payments amount x invested at a monthly interest rate i% is:
The formula may be re-arranged to determine the monthly payment x on a loan of amount P0 taken out for a period of n months at a monthly interest rate of i%:
We begin with a small adjustment of the formula: replace i with r/N where r is the annual interest rate and N is the annual frequency of compounding periods (N = 12 for monthly payments). Also replace n with NT where T is the total loan period in years. In this more general form of the equation we are calculating x(N) as the fixed payment corresponding to frequency N. For example, if N = 365, x corresponds to a daily fixed payment. As N increases, x(N) decreases but the product N·x(N) approaches a limiting value as will be shown:
Note that N·x(N) is simply the amount paid per year – in effect an annual repayment rate Ma.
It is well established that:
Applying the same principle to the formula for annual repayment, we can determine a limiting value:
At this point in the orthodox formula for present value, the latter is more properly represented as a function of annual compounding frequency N and time t:
Applying the limiting expression developed above we may write present value as a purely time dependent function:
Noting that the balance due P(t) on a loan t years after its inception is simply the present value of the contributions for the remaining period (i.e. T − t), we determine:
The graph(s) in the diagram are a comparison of balance due on a mortgage (1 million for 20 years @ r = 10%) calculated firstly according to the above time continuous model and secondly using the Excel PV function. As may be seen the curves are virtually indistinguishable – calculations effected using the model differ from those effected using the Excel PV function by a mere 0.3% (max). The data from which the graph(s) were derived can be viewed
Comparison with similar physical systems
Define the "reverse time" variable z = T − t. (t = 0, z = T and t = T, z = 0). Then:
This may be recognized as a solution to the "reverse time" differential equation:
Electrical/electronic engineers and physicists will be familiar with an equation of this nature: it is an exact analogue of the type of differential equation which governs (for example) the charging of a capacitor in an RC circuit.