$$B_0$$ 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
$$I$$ Total interest paid over the loan term
$$V$$ Sum of all payments $$V = B_0+I$$
$$\frac{V}{B_0}$$ Overpay ratio $$\frac{V}{B_0} = \frac{B_0+I}{B_0}$$
$$n$$ Number of payment periods

# A graphical loan calculator

Photo credit to Jp Valery

Disclaimer: This calculator can use either the continuous or discrete solutions to the loan equations. As discussed in the solution comparison article, the continuous solution is important for understanding loans and the discrete solution is what the banks use in practice. As the number of payment periods, $$n$$, increases the two solutions converge.

There are thousands of loan graphing utilities online which show the loan balance vs time or total interest paid over time. Awareness of how that part of the loan works is important - most of the interest is paid at the beginning of the loan, most of the principal is paid at the end, etc. The first two loan articles explain those concepts from first principles. However, to make purposeful decisions about loans and to compare loan products properly, we need to visualize how total interest and repayment rate depend on the loan parameters ($$r$$ and $$t_\text{term}$$) selected. For convenience, the total interest and monthly payment are computed from the loan parameters and the principal, $$B_0$$.

Graphs will display if 2 $$\leq n \leq$$ 650, $$r >$$ 0, and $$t_\text{term} >$$ 0. Any inputs which do not allow the computation of the values below will result in 'NvN', or 'Not a Valid Number.' For a 5, 8, 10, 15 or 30 year loan with monthly payments, $$n =$$ 60, 96, 120, 180, or 360 respectively.

## Inputs

Method:
$$B_0 =$$ \$
$$r =$$ [%/year]
$$t_\text{term} =$$ [years]
$$n =$$

## Outputs

$$rt_\text{term} =$$ NvN
$$\displaystyle \frac{V}{B_0} = \frac{rt_\text{term}}{1-e^{-rt_\text{term} }} =$$ $$\displaystyle \frac{V}{B_0} = \frac{ n (1+\frac{rt_\text{term}}{n})^{n} }{\sum^{n-1}_{i=0} (1+\frac{rt_\text{term}}{n})^i} =$$ NvN $$\frac{\text{\ repaid}}{\text{\ borrowed}}$$
$$\displaystyle \frac{P}{B_0} =\frac{r}{1-e^{-rt_\text{term} }}\left[ \frac{1 \text{ year}}{12 \text{ months}}\right] =$$ $$\displaystyle \frac{P}{B_0} = \frac{ n (1+\frac{rt_\text{term}}{n})^{n} }{t_\text{term} \sum^{n-1}_{i=0} (1+\frac{rt_\text{term}}{n})^i} =$$ NvN $$\frac{\text{¢ per month}}{\text{\ borrowed}}$$
Total interest NvN
Monthly payment NvN