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\( B_0\) Principal or loan balance at \(t=0\)
\( P \) Repayment rate (dollars per time)
\( r \) Interest rate
\( t_\text{term} \) Loan term
\( rt_\text{term} \) Loan product, important parameter which fully specifies a loan
\( n \) Number of loan payments
\( \Delta t \) Time between loan payments\( \Delta t = \frac{t_\text{term}}{n} \)
\( I \) Total payment to interest
\( V \) Sum of all payments \(V = B_0+I \)
\( \frac{V}{B_0} \) Overpay ratio \(\frac{V}{B_0} = \frac{B_0+I}{B_0} \)

You do not need calculus to use the loan equations

Photo credit to Jp Valery


All of the equations of the continuous form were derived here while all the equations of the discrete form were derived here. The comparison article explains the error in more detail and has some extremely aesthetic graphs showing just how small the error is.

Goals of this article

Loans are an interesting thing. In theory, the borrower is interested in two things, (1) the repayment rate which must be compatible with their cash flow, and (2) the amount of interest the loan issuer will charge which must not be so large that the borrower decides to forgo the loan altogether. The borrower is offered a loan on the basis of 2 or 3 parameters where only 2 of the 3 are required to fully specify the loan. These are the interest rate, \(r\), the loan term, \(t_\text{term}\), and the repayment rate \(P/B_0\). The problem of loans is in computing the amount of interest charged over the life of the loan or the overpay ratio, \(V/B_0\), and whichever parameter of the three listed earlier is not given by the lender.

In practice, the borrower is often ignorant of all of the math presented here and simply accepts a loan for anything they have decided to finance through debt if a lender is willing to lend them the money at all. The marginally more informed understand that a higher interest rate loan will incur more interest and a longer loan term will reduce the monthly payments a bit but for even most "savvy" consumers, that is about the extent of understanding. Ignorance of the equations presented here leaves the borrower wandering through the financial system like blind man feeling his way through a rattlesnake pit wondering what all the buzz is about and if perhaps it has anything to do with the considerable pain. The purpose of this series of articles is to show not just the relevant equations, but also the origins of the equations which govern loans thereby restoring the blind man's sight and leading him out of the snake pit.

Because most loans are quoted with a loan term (\(t_\text{term}\)) and an interest rate (\(r\)), in this article I will present the equations for the repayment rate, \(P/B_0\), and overpay ratio, \(V/B_0\), as a function of \(r\) and \(t_\text{term}\).

Interest and repayment

Because most loans are quoted with a loan term (\(t_\text{term}\)) and an interest rate (\(r\)), we will assume that information is given and we would like to solve for the repayment rate and interest. It is mathematically more convenient to solve for the overpay (interest plus principal) than for the interest alone.

$$\begin{align} \frac{P}{B_0} &=\frac{n(1+\frac{rt_\text{term}}{n} )^{n}}{t_\text{term}\sum^{n-1}_{i=0} (1+\frac{rt_\text{term}}{n} )^i}\\ \frac{V}{B_0} &= \frac{I+B_0}{B_0} =\frac{n(1+\frac{rt_\text{term}}{n})^{n}} { \sum^{n-1}_{i=0} (1+\frac{rt_\text{term}}{n})^i} \end{align} $$

These are well approximated by the continuous solutions which are plotted below. The error between the discrete and continuous formulas is a function of the number of repayment periods, \(n\), and slightly of the loan product, \(rt_\text{term}\), but it is less than 5% for \(n>\) 6 and less than 1% for \(n>\) 36 regardless of the loan product.

$$\begin{align} \frac{P}{B_0} &= \frac{r}{1-e^{-rt_\text{term}}}\\ \frac{V}{B_0} &= \frac{t_\text{term}P}{B_0} = \frac{rt_\text{term}}{1-e^{-rt_\text{term}}} \end{align} $$

In these formulas, the solution for \(P\) will be in $ per time with that time unit matching \(r\). Because \(r\) is usually given in % per year, \(P\) will be in $ per year. Normally, we want it in $ per month which requires the following unit conversion.

$$ P \left[\frac{{\$}}{\text{year}}\right] = P \left[\frac{{\$}}{\text{year}}\right] \left[\frac{\text{1 year}}{\text{12 month}}\right] = \frac{P}{12} \left[\frac{{\$}}{\text{month}}\right] $$ Plots of V/B_0 and P/B_0 for various rt
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