Continuous-repayment mortgage

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.

[2] Within the timespan of the loan the time continuous mortgage balance function obeys a first order linear differential equation (LDE)[3] 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.

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.

For home owners with mortgages the important parameter to keep in mind is the time constant of the equation which is simply the reciprocal of the annual interest rate r. So (for example) the time constant when the interest rate is 10% is 10 years and the period of a home loan should be determined – within the bounds of affordability – as a minimum multiple of this if the objective is to minimise interest paid on the loan.

Given an annual interest rate r and a borrower with an annual payment capability MN (divided into N equal payments made at time intervals Δt where Δt = 1/N years), we may write: If N is increased indefinitely so that Δt → 0, we obtain the continuous time differential equation: Note that for there to be a continually diminishing mortgage balance, the following inequality must hold: P0 is the same as P(0) – the original loan amount or loan balance at time t = 0.

We begin by re-writing the difference equation in recursive form: Using the notation Pn to indicate the mortgage balance after n periods, we may apply the recursion relation iteratively to determine P1 and P2: It can already be seen that the terms containing MN form a geometric series with common ratio 1 + rΔ t. This enables us to write a general expression for Pn: Finally noting that r Δ t = i the per-period interest rate and

the per period payment, the expression may be written in conventional form: If the loan timespan is m periods, then Pm = 0 and we obtain the standard present value formula: One method of solving the equation is to obtain the Laplace transform P(s): Using a table of Laplace transforms and their time domain equivalents, P(t) may be determined: In order to fit this solution to the particular start and end points of the mortgage function we need to introduce a time shift of T years (T = loan period) to ensure the function reaches zero at the end of the loan period: Note that both the original solution and "time-shifted" version satisfy the original differential equation whence both are derived.

Similar to the expression derived above for Pn in the difference equation, the expression for P(t) may be written in the following algebraically equivalent form: Re-arranging the original differential equation we obtain: Integrating both sides of the equation yields: The first integral on the right hand side determines the accumulated interest payments from time of inception to time t whilst the second determines the accumulated principal payments over the same period.

The product rT is an easily obtained but important parameter in determining loan cost according to the equation C=P0xC(s).

For a fixed term loan of t years, we may compare the above loan cost factor against an equivalent simple interest cost factor 1+se where se=ret and re is the equivalent simple interest rate: It is straightforward to determine se in terms of s. Dividing by loan time period t will then give the equivalent simple interest rate.

The above example is adapted from the one given in Dr Hahn's book in which he employs the Newton-Raphson algorithm to solve the same problem albeit for a discrete interval (i.e. monthly) repayment loan over the same time period (3 years).

As with many similar examples the discrete interval problem and its solution is closely approximated by calculations based on the continuous repayment model - Dr Hahn's solution for interest rate is 40.8% as compared to the 41.6% calculated above.

We will use the letter k to denote minimum payment ratio: Now we may consider a small re-arrangement of the equation for loan period T: Plotting s(k) against k gives a very graphic demonstration of why it is a good idea to keep the k value well below the asymptote at k = 1 since in the vicinity thereof, s(k) increases sharply and therefore so does loan cost which is in turn an increasing function of parameter s (rT product).

In the discrete time interval model, calculation of a mortgage based interest rate given the remaining parameters has not been possible using analytic methods.

Given: we may write: In order to visualise the above as a function of r (for which we wish to determine zeroes), it will be helpful to select numerical values of P0, Ma and T as 10000, 6000 and 3 respectively and plot as shown at right.

The function has a minimum value which can be determined by differentiation: Since the function is approximately parabolic between the roots at r = 0 and the sought value, we may estimate the required root as: Using this as a starting point, increasingly accurate values for the root may be determined by repeated iterations of the Newton–Raphson algorithm:[14] Some experimentation on Wolfram Alpha reveals that an exact analytical solution employing the Lambert-W or "product log" function can be obtained.

The following table shows calculation of an initial estimate of interest rate followed by a few iterations of the Newton–Raphson algorithm.

Newton–Raphson iterations Corresponding to the standard formula for the present value of a series of fixed monthly payments, we have already established a time continuous analogue: In similar fashion, a future value formula can be determined: In this case the annual rate Ma is determined from a specified (future) savings or sinking fund target PT as follows.

It will be noted that as might be expected: Another way to calculate balance due P(t) on a continuous-repayment loan is to subtract the future value (at time t) of the payment stream from the future value of the loan (also at time t): The following example from a school text book[18] will illustrate the conceptual difference between a savings annuity based on discrete time intervals (per month in this case) and one based on continuous payment employing the above future value formula: On his 30th birthday, an investor decides he wants to accumulate R500000 by his 40th birthday.

For the sake of brevity, we will solve the "discrete interval" problem using the Excel PMT function: The amount paid annually would therefore be 26082.57.

If the error margin is acceptable, the hourly payment rate can be more simply determined by dividing Ma by 365×24.

The (hypothetical) lending institution would then need to ensure its computational resources are sufficient to implement (when required) hourly deductions from customer accounts.

In short cash "flow" for continuous payment annuities is to be understood in the very literal sense of the word.

In particular the meaning of an annual repayment rate must be clearly understood as illustrated in the above example.

However the "continuous payment" model does provide some meaningful insights into the behaviour of the discrete mortgage balance function – in particular that it is largely governed by a time constant equal to the reciprocal of r the nominal annual interest rate.

And if a mortgage were to be paid off via fixed daily amounts, then balance due calculations effected using the model would – in general – be accurate to within a small fraction of a percent.

Finally the model demonstrates that it is to the modest advantage of the mortgage holder to increase frequency of payment where practically possible.

This illustrates the characteristic curve of mortgage balance vs time over a given loan timespan.