$$B_0$$ principal or initial loan balance
$$r$$ Interest rate
$$t_\text{term}$$ Loan term
$$rt_\text{term}$$ Loan product, important parameter which fully specifies a loan
$$P$$ Repayment rate (dollars per time) $$P = \frac{rB_0}{1- e^{-rt_\text{term}}}$$
$$\phi_0$$ Fraction of initial payment to interest$$\phi_0 = \frac{rB_0}{P} = 1-e^{-rt_\text{term}}$$
$$I$$ Total payment to interest
$$V$$ Sum of all payments $$V = B_0+I$$
$$\frac{V}{B_0}$$ Overpay ratio $$\frac{V}{B_0} = \frac{rt_\text{term}}{1- e^{-rt_\text{term}}}$$

Loan interest is a function of one parameter but the repayment rate is a function of two

Photo credit to Jp Valery

Because most loans are advertised by specifying $$r$$ and $$t_\text{term}$$, we will focus this discussion assuming those values are given. This is not always true but it is the most common case (otherwise loans are usually quoted with the repayment rate and loan term). For reasons of mathematical simplicity, we will focus on the overpay ratio $$V/B_0 = (B_0 +I)/B_0$$ instead of directly solving for interest, $$I$$. We will show that $$V/B_0$$ is linear for very large or very small loan products, $$rt_\text{term}$$, but non-linear over the range of most loans. We will show that the repayment rate, $$P/B_0$$, goes to infinity as the loan product goes to 0, but levels out to a constant for large loan terms.

The limiting cases of the overpay ratio

In the article on the continuous solution to the loan problem, we solved for the overpay ratio (and by extension, the total interest paid).

$$\frac{B_0+I}{B_0} = \frac{V}{B_0} = \frac{rt_\text{term}}{1- e^{-rt_\text{term}}}$$

Now we want to know how this function behaves in the limit of very large and very small loan products, $$rt_\text{term} \rightarrow \infty$$ and $$rt_\text{term} \rightarrow$$ 0.

\begin{align} \frac{\partial\left( \frac{V}{B_0}\right)}{\partial \left(rt_\text{term}\right)} &= \frac{(1-e^{-rt_\text{term}}) -(rt_\text{term})^2 e^{-rt_\text{term}} }{(1- e^{-rt_\text{term}})^2}\\ \left. \frac{ \partial\left( \frac{V}{B_0}\right)}{\partial \left(rt_\text{term}\right)} \right|_{rt_\text{term}\rightarrow \infty} &= \frac{(1-e^{-rt_\text{term}}) -(rt_\text{term})^2 e^{-rt_\text{term}} }{(1- e^{-rt_\text{term}})^2} = 1 \end{align}

The derivative as $$rt_\text{term} \rightarrow 0$$ is found by two applications of L'Hôpital's rule after multiplying top and bottom by $$e^{2rt_\text{term}}$$ for simplicity.

\begin{align} \label{eq:overpayprime0} \left. \frac{ \partial\left( \frac{V}{B_0}\right)}{\partial \left(rt_\text{term}\right)} \right|_{rt_\text{term}\rightarrow 0} &= \frac{e^{rt_\text{term}}(e^{rt_\text{term}}-1-rt_\text{term}) }{(e^{rt_\text{term}}-1)^2} \\\label{eq:Lhop1} &=\frac{ \frac{\partial}{\partial \left(rt_\text{term}\right)}\left[ e^{rt_\text{term}}(e^{rt_\text{term}}-1-rt_\text{term})\right]}{\frac{\partial}{\partial \left(rt_\text{term}\right)} (e^{rt_\text{term}}-1)^2}\\ &= \frac{ e^{rt_\text{term}}(e^{rt_\text{term}}-1-rt_\text{term}) + e^{rt_\text{term}}(e^{rt_\text{term}}-1)}{2(e^{rt_\text{term}}-1)e^{rt_\text{term}} }\\ &= \frac{2(e^{rt_\text{term}}-1)-rt_\text{term}}{2(e^{rt_\text{term}}-1)}\\\label{eq:Lhop2} &= \frac{\frac{\partial}{\partial \left(rt_\text{term}\right)}\left[ 2(e^{rt_\text{term}}-1)-rt_\text{term}\right]}{\frac{\partial}{\partial \left(rt_\text{term}\right)}\left[2(e^{rt_\text{term}}-1)\right]}\\ &= \frac{2e^{rt_\text{term}}-1}{2e^{rt_\text{term}}}\\ \left. \frac{ \partial\left( \frac{V}{B_0}\right)}{\partial \left(rt_\text{term}\right)} \right|_{rt_\text{term}\rightarrow 0} &= \frac{2-1}{2} = \frac{1}{2} \end{align}

At extreme loan products, the overpay ratio behaves as follows:

for $$rt_\text{term}$$ near 0 (in practice $$rt_\text{term} < 0.4$$) $$\displaystyle \frac{V}{B_0} \approx 1+\frac{rt_\text{term}}{2}$$
for large $$rt_\text{term}$$ (in practice $$rt_\text{term} > 4$$) $$\displaystyle \frac{V}{B_0} \approx rt_\text{term}$$

This is the first order Taylor series for the function $$\frac{V}{B_0} = f(rt_\text{term})$$ centered on $$rt_\text{term} =$$ 0 and $$rt_\text{term} \rightarrow \infty$$, respectively. Technically, Taylor series about $$\infty$$ is not the right way to describe it but for the non-mathematicians, it communicates the idea.

The limiting cases of the repayment rate

Begin with the equation for the repayment rate from the continuous solution.

$$\frac{P}{B_0} = \frac{r}{1- e^{-rt_\text{term}}}$$

In the limit of a 0 interest loan, $$P = \frac{B_0}{t_\text{term}}$$, while in the limit of a long loan term $$t_\text{term}$$ the repayment rate just covers the interest, $$P = rB_0$$. As the loan term approaches 0, the repayment rate goes to $$\infty$$ regardless of interest rate.

\begin{align}\lim_{r \to 0} \frac{P}{B_0} &= \lim_{r \to 0} \frac{r}{1- e^{-rt_\text{term}}} = \lim_{r \to 0} \frac{\frac{d}{dr} r}{\frac{d}{dr} \left( 1- e^{-rt_\text{term}}\right) } = \lim_{r \to 0} \frac{1}{t_\text{term} e^{-rt_\text{term}}} = \frac{1}{t_\text{term}} \\ \lim_{t_\text{term} \to \infty} \frac{P}{B_0} &= r\\ \lim_{t \to 0} \frac{P}{B_0} &= \lim_{t \to 0} \frac{r}{1- e^{-rt_\text{term}}} = \infty \end{align}

Graphically, here is what these limits look like.

The ultimate loan graph

Below is the ultimate loan graph. It describes all the loans a reasonable person is likely to consider during their lifetime, which is to say loan products from 0 to 2. Normal mortgages are between 2% and 6% for 15 to 30 years ($$rt_\text{term}$$ between 0.3 and 1.8). A subprime auto loan may be up to 25% or 30% and last up to 8 years giving a loan product of just over 2 in the worst case scenario. If the loan you are considering is outside this range, it is highly inadvisable under all circumstances. Loans with loan products above 2 require the borrower to pay back over twice as many dollars as they borrowed over the life of the loan. Usually, that situation results in default which is why loans with the highest interest rates, like payday loans, are only given over short loan terms. Loan products higher than 2 are almost always predatory in that they are typically issued to borrowers who are incapable of understanding the terms of the contract and unlikely to ever be able to pay them in full. The business model is built on the idea that enough harassment and legal bullying will eventually yield a decent return for the lender despite knowing full well they are unlikely to ever collect on the full agreed-upon amount.

The graph below also shows where the loan is on the repayment curve for a given interest rate.

[Caption] [Red line] Left axis, overpay ratio as a function of $$rt_\text{term}$$. [Viridis lines labeled by $$r$$] Right axis, repayment rate as a function of $$rt_\text{term}$$ for a given $$r$$. [x-axis] Because either $$rt_\text{term}$$ or $$\phi_0$$ can be used to fully specify a loan, both are given. ($$rt_\text{term} =$$ 0.75 is equivalent to $$\phi_0 =$$ 0.528, $$rt_\text{term} =$$ 1.25 is equivalent to $$\phi_0 =$$ 0.713, etc.) Both the red overpay line and the viridis repayment rate lines may be correctly interpreted from either axis.