Analogous to continuous compounding, a continuous annuity - Entry on continuous annuity, p. 86 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.Strictly speaking compounding occurs momentarily before payment is deducted so that interest is calculated on the balance as it was before deduction of the period payment.
Within the timespan of the loan the time continuous mortgage balance function obeys a first order linear differential equation (LDE)Beckwith p. 116: "Technically speaking, the underlying equation is known as an ordinary, linear, first order, inhomogenous, scalar differential equation with a boundary condition." 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:
Beckwith p.115Munem and Foulis p.273
Applying the same principle to the formula for annual repayment, we can determine a limiting value:
Beckwith: Equation (29) p. 123.
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:
See also: ps. 470–471
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:
Beckwith: Equation (31) p. 124.
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 here.
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.
The key characteristics of such equations are explained in detail at RC circuits. 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.
Mortgage difference and differential equation
The conventional difference equation for a mortgage loan is relatively straightforward to derive - balance due in each successive period is the previous balance plus per period interest less the per period fixed payment.
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:
Beckwith: Equation (25) p. 123Hackman: Equation (2) p.1
Note that for there to be a continually diminishing mortgage balance, the following inequality must hold:
Where equality holds, the mortgage becomes a perpetuity.
P0 is the same as P(0) – the original loan amount or loan balance at time t = 0.
Solving the difference equation 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:
Solving the differential equation
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:
Calculation of accumulated interest and principal payments
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 sum of these interest and principal payments must equal the cumulative fixed payments at time t i.e. Mat. Evaluating the first integral on the right we obtain an expression for I(t), the interest paid:
Unsurprisingly the second integral evaluates to P0 − P(t) and therefore:
The reader may easily verify that this expression is algebraically identical to the one above.
Loan cost factor
The cost of a loan is simply the annual rate multiplied by loan period:
Let s = rT. Then we may define loan cost factor C(s) such that C = P0C(s) i.e.: C(s) is the cost per unit of currency loaned.
The function C(s) is characterised by having a limiting value of 1 when s is close to zero since for small values of s, exp(−s) ≈ 1 − s and the denominator simplifies to s. Also when s is very large, exp(−s) is small so C(s) ≈ s and thus loan cost C ≈ P0rT (rT >> 0).
By way of example, consider a loan of 1000000 at 10% repaid over 20 years. Then s = 0.1 × 20 = 2.
The product rT is an easily obtained but important parameter in determining loan cost according to the equation C=P0xC(s). This is best illustrated by plotting the cost factor function for s values in domain [0;5]. The linear behaviour of the function for higher values of s is clear.
Equivalent simple interest cost factor 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. More challenging is the reverse determination of s given se.
In his book Problem Solving with True Basic,Hahn p. 247 Dr B.D. Hahn has a short section on certain 'hire purchase' schemes in which interest is calculated in advance in one lump sum, which is added to the capital amount, the sum being equally divided over the repayment period. The buyer, however, is often under the impression that the interest is calculated on a reducing balance.
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.
Period of a loan
If a borrower can afford an annual repayment rate Ma, then we can re-arrange the formula for calculating Ma to obtain an expression for the time period T of a given loan P0:
Minimum payment ratio
The minimum payment ratio of a loan is the ratio of minimum possible payment rate to actual payment rate. The minimum possible payment rate is that which just covers the loan interest – a borrower would in theory pay this amount forever because there is never any decrease in loan capital. 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).
"Half-life" of a loan
A useful parameter of the mortgage model is the "half-life" of the loan which is the time it takes for the balance on the loan to reach half its original value. To determine the "half-life" we may write: