Continuous-repayment mortgage

The purpose of this article is to extend the Mathematics of continuously compounded interest to include a very common application of compound interest: namely fixed repayment loans more commonly referred to as mortgages or annuities.

In the same way that it is possible to derive the formula for continuous compounding by taking to infinity the frequency (N) of compounding periods (N = 12 for monthly compounding), we can derive a formula for a 'continuous repayment mortgage'.

In this article the derivation is presented and the result compared with a well known physical system which exhibits the same mathematical characteristics. The time continuous mortgage function obeys a first order linear differential equation and an alternative derivation thereof is obtained by solving the equation using Laplace transforms.

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:

$P_{v}(n)={\frac {x(1-(1+i)^{-n})}{i}}$

The formula may be re-arranged to determine the monthly payment x on a loan of amount P taken out for a period of n months at a monthly interest rate of i%:

$x={\frac {Pi}{1-(1+i)^{-n}}}$

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. It should be clear that as N increases, x_N decreases but the product Nx_N approaches a limiting value as will be shown:

$x_{N}={\frac {Pr}{N(1-(1+{\frac {r}{N}})^{-NT})}}$

$Nx_{N}={\frac {Pr}{1-(1+{\frac {r}{N}})^{-NT}}}$

Note that Nx_N is simply the amount paid per year - in effect an annual repayment rate $M_{a}$. It is well established that:

$\lim _{N\to \infty }\left(1+{\frac {r}{N}}\right)^{Nt}=e^{rt}$

Applying the same principle to the formula for annual repayment, we can determine a limiting value:

$M_{a}=\lim _{N\to \infty }Nx_{N}=\lim _{N\to \infty }{\frac {Pr}{1-(1+{\frac {r}{N}})^{-NT}}}={\frac {Pr}{1-e^{-rT}}}$

Having determined $M_{a}$ - the annual repayment rate - the original present value formula may be re-written as a time dependent function:

$P_{v}(t)={\frac {M_{a}}{r}}(1-e^{-rt})$

Figure 1

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 (ie T-t), we may write:

$P(t)={\frac {M_{a}}{r}}(1-e^{-r(T-t)})={\frac {P_{(0)}(1-e^{-r(T-t)})}{1-e^{-rT}}}$

The graph(s) in the diagram are a comparision 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.

Comparision with Similar Physical Systems

Define the 'reverse time' variable z = T - t. (t=0, z=T and t=T, z=0). Then:

$P(z)={\frac {M_{a}}{r}}(1-e^{-rz})$

This may be recognized as a solution to the 'reverse time' differential equation:

${\frac {M_{a}}{r}}={\frac {1}{r}}{\frac {dP(z)}{dz}}+P(z)$

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:

$V_{0}=RC{\frac {dV(t)}{dt}}+V(t).$

The key characteristics of such equations are explained in detail in the Wiki page on 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 r the annual interest rate. 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.

Ordinary Time Differential Equation

The concept of a 'reverse time variable' is somewhat abstract: it was employed above to demonstrate the similarity between the equations governing mortgage balance and physical systems such as the RC circuit referred to.

In ordinary time - given interest rate r and annual repayment rate $M_{a}$ - the mortgage balance function satisfies the following differential equation:

${\frac {dP(t)}{dt}}=rP(t)-M_{a}$

From this we may obtain the Laplace transform P(s):

$P(s)={\frac {M_{a}}{s(r-s)}}={\frac {M_{a}}{r}}{\frac {(-r)}{s(s+(-r))}}$

Using a table of LaPlace transforms and their time domain equivalents, P(t) may be determined:

$P(t)={\frac {M_{a}}{r}}(1-e^{rt})$

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:

$P(t)={\frac {M_{a}}{r}}(1-e^{r(t-T)})$

$P(0)={\frac {M_{a}}{r}}(1-e^{-rT})\Rightarrow {\frac {M_{a}}{r}}={\frac {P(0)}{(1-e^{-rT})}}$

$\Rightarrow P(t)={\frac {M_{a}}{r}}(1-e^{-r(T-t)})={\frac {P(0)(1-e^{-r(T-t)})}{1-e^{-rT}}}$

Note that both the original solution and 'time shifted' version satisfy the original differential equation whence both are derived.

'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:

${\frac {P(t)}{P_{0}}}={\frac {1}{2}}={\frac {1-e^{-r(T-t)}}{1-e^{-rT}}}$

Solving for t we obtain:

$t_{\frac {1}{2}}={\frac {1}{r}}ln{\frac {1+e^{rT}}{2}}$

For example applying the formula to some test data (loan of 1 million at 10% for 20 years) we obtain the half life as 14.34 years. If in practice the loan is being repaid via monthly instalments, the decimal portion can be converted to months and rounded so this answer would equate to 172 months.

Concluding Remarks

The concept of a 'continuous repayment' mortgage is a somewhat theoretical construct. Whether it has practical value or not is a question that would need to be carefully considered by economists and actuarials. An obvious question which may be asked is what is the 'real world' meaning of the limiting annual payment $M_{a}$ referred to in many of the calculations effected above? However the 'time continuous' 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.

References

• Hackman, Steve, Financial Engineering: ISyE 4803A Course Notes (Georgia Institute of Technology)

Fixed Rate Mortgages

Bibliography

• Kreyszig, Erwin, Advanced Engineering Mathematics (1998, Wiley Publishers, USA), ISBN 0471154962.
• Munem, M.A and Foulis D.J, Algebra and Trigonometry with Applications (1986, Worth Publishers, USA), ISBN 0879012811