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\( B(t) \quad \text{or} \quad B(\tau) \) Loan balance vs time
\( B_i \) Loan balance after the ith payment
\( B_0\) Principal or loan balance at \(t=0\)
\( \phi_i \) Fraction of payment to interest during the ith payment \(\phi = \frac{rB_i}{P} \)
\( \phi \) Fraction of payment to interest \(\phi = \frac{rB}{P} \)
\( r \) Interest rate
\( t_\text{term} \) Loan term
\( n \) Number of loan payments
\( \Delta t \) Time between loan payments\( \Delta t = \frac{t_\text{term}}{n} \)
\( rt_\text{term} \) Loan product, important parameter which fully specifies a loan
\( R \) Helpful collection of variables\( R = 1+r\Delta t = 1+\frac{rt_\text{term}}{n} \)
\( P \) Repayment rate (dollars per time)
\( \tau \) Fraction of loan term \(\tau= \frac{t}{t_\text{term}}\)
\( I \) Total payment to interest
\( V \) Sum of all payments \(V = B_0+I \)
\( \frac{V}{B_0} \) Overpay ratio \(\frac{V}{B_0} = \frac{B_0+I}{B_0} \)

The error between the finite difference and continuous solutions is less than 1% in all practical cases

Photo credit to George Becker

Goal of this article

When I present the continuous solutions to loans, I meet push back because those equations do not exactly match the computations done by the bank. Nonetheless, the continuous solutions are important because unlike the discrete "real" solutions, the continuous solutions show us the functional form of the parameters for which we are solving. In this section, we will answer the question, "if the continuous solution is not exactly correct, how much error is there and is that error significant?" The answer will depend mostly on the number of repayment periods, \(n\), and slightly on the loan product, \(rt_\text{term}\), but when computing the interest on a loan, it will be less than 5% for \(n>\) 6 and less than 1% for \(n>\) 36 regardless of the loan product.

Continuous solutions are useful for forming an understanding of the shape of the functional form of the equations, for estimating values (repayment rates, interest or overpay, etc) with simple closed-form equations that do not require extensive sums, and for comparing loans of different terms and interest rates. They are not useful for comparing against bank statements after computing payments or balances to the nearest penny and should not be used as such.

All of the equations of the continuous form were derived here while all the equations of the discrete form were derived here.

View of loan balance

We have already seen in the discrete solution article that a plot of \(\phi(\tau)\) approaches the continuous solution as \(n \rightarrow\) 0. For completeness here is a plot of \(\frac{B(\tau)}{B_0}\) for various \(n\) and \(rt_\text{term}\).

Continuous solution Discrete solution
\( \displaystyle \frac{B(\tau)}{B_0} = \frac{1-\phi^{-1} }{\left(1-\phi\right)^{\tau}}+\phi^{-1} \) \( \displaystyle \frac{B_{i}}{B_0} = R^{i} - \frac{ R^{n}}{ \sum^{n-1}_{k=0} R^k } \sum_{j=1}^{i} R^{j-1} \)
\(\phi_0 = 1-e^{-rt_\text{term}} \) \( R = 1+ \frac{rt_\text{term}}{n} \)
Plots of B/B_0 vs tau for various rt and n

The superior plot for comparison shows \(\phi\) vs \(\tau\). Again, as \(n \rightarrow \infty\), the plot of \(\phi_i\) approaches the continuous \(\phi(\tau)\). This is the same figure shown in the article on finite difference solutions.

Continuous solution Discrete solution
\( \displaystyle \phi(\tau) = 1 - \frac{1-\phi_0}{(1-\phi_0)^{\tau}} \) \( \displaystyle \phi_i = \frac{rt_\text{term}}{n}\left( \frac{\sum_{j=1}^{n} R^{j-1}}{ R^n} R^{i} - \sum_{j=1}^{i} R^{j-1} \right) \)
\( R = 1+ \frac{rt_\text{term}}{n} \)
Plots of phi vs tau for various rt and n

Error in overpay ratio and repayment rate

For a given loan we can compute the error ratio between the continuous and discrete ODE solutions when solving for the repayment rate.

$$ \frac{\left. \frac{P}{B_0} \right\vert _\text{continuous}}{\left. \frac{P}{B_0} \right\vert _\text{discrete}}= \frac{\left. P \right\vert _\text{continuous}}{\left. P \right\vert _\text{discrete}}= \frac{r}{1-e^{-rt_\text{term}}} \frac{\Delta t\sum^{t_\text{term} /\Delta t-1}_{i=0} (1+r\Delta t)^i}{(1+r\Delta t)^{t_\text{term} /\Delta t}} $$

This expression is exactly equivalent to the error in the overpay ratio. Just multiply the top and bottom of the fraction by \(t_\text{term}\) and you have the expression for the error in the overpay ratio. For most loans, this error ratio will be close to one, typically within 0.5%.

$$ \text{Error function} = \frac{\left. P \right\vert _\text{continuous}}{\left. P \right\vert _\text{discrete}}= \frac{\left. V \right\vert _\text{continuous}}{\left. V \right\vert _\text{discrete}}= \frac{rt_\text{term}}{1-e^{-rt_\text{term}}} \frac{ \Delta t \sum^{t_\text{term} /\Delta t-1}_{i=0} (1+r\Delta t)^i}{t_\text{term}(1+r\Delta t)^{t_\text{term} /\Delta t}} $$

This is better stated as a function of the number of loan discretization steps (\(n = t/\Delta t\)) and the loan product (\(rt_\text{term}\)).

$$ \text{Error function}(n,rt_\text{term}) = \frac{\left. V \right\vert _\text{continuous}}{\left. V \right\vert _\text{discrete}}= \frac{rt_\text{term}}{1-e^{-rt_\text{term}}} \frac{ \sum^{n-1}_{i=0} (1+\frac{rt_\text{term}}{n})^i}{n(1+\frac{rt_\text{term}}{n})^{n}} $$

This is a complicated function of \(n\) and \(rt_\text{term}\) but when plotted it is easy to understand. It's much more a function of \(n\) than \(rt_\text{term}\) and for most reasonable \(n\), say \(n>\) 36 which is a 3-year loan with monthly payments, the continuous and discrete solutions never differ by more than 1%.

Plot of error in overpay ratio vs n and rt

Alternatively, we can reproduce the graphs from the article on overpay ratios and repayment rates using various \(n\). The lines quickly become indistinguishable for \(n>\) 20. For the overpay ratio, the continuous and discrete equations are given below.

Continuous solution Discrete solution
\( \displaystyle \frac{V}{B_0} = \frac{rt_\text{term}}{1- e^{-rt_\text{term}}} \) \( \displaystyle \frac{V}{B_0} = \frac{ n (1+\frac{rt_\text{term}}{n})^{n} }{\sum^{n-1}_{i=0} (1+\frac{rt_\text{term}}{n})^i }\)

The repayment rate is the overpay ratio divided by the loan term (for both the continuous and discrete equations).

Continuous solution Discrete solution
\( \displaystyle \frac{P}{B_0} = \frac{r}{1- e^{-rt_\text{term}}} \) \( \displaystyle \frac{P}{B_0} = \frac{ nR^{n}}{t_\text{term} \sum^{n-1}_{i=0} R^i } \)
\( R = 1+ \frac{rt_\text{term}}{n} \)
Plots of V/B_0 and P/B_0 vs rt for various n
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