# The Math Behind Simple Interest and the Rule of 78s

Should you use simple interest or the Rule of 78s when calculating interest on a loan? Let's compare both to see why it matters.

We will start with a simple interest loan, and run the math behind the amortization schedule. Then we will do the same loan (same balance, APR, and term) but with the interest calculated using the rule of 78s. Finally, we will do a side-by side comparison between the two.

## Simple Interest Calculation

- Term: 12 months.
- Interest Rate: 10% periodic interest.
- Loan Amount: $1000.00
- The monthly payment calculates out to be $87.92.
- Interest for the first month may be calculated by the following equation: interest = principal x rate/12 (1000 x 10%)/12 = 8.33
- The principal portion of the first month's payment is: Principal Payment = Total Payment - Interest Payment. 87.92 - 8.33 = 79.59
- Principal balance at the start of month #2 = Principal Balance (month #1) - Principal Payment (month #1). 1000.00 - 79.59 = 920.41

Using these three equations over and over, we derive the following amortization schedule for the twelve month term of the loan:

Payment # | Payment | Interest | Principal | Remaining Balance |
---|---|---|---|---|

1,000.00 | ||||

1 | 87.92 | 8.33 | 79.59 | 920.41 |

2 | 87.92 | 7.67 | 80.25 | 840.16 |

3 | 87.92 | 7.00 | 80.92 | 759.24 |

4 | 87.92 | 6.33 | 81.59 | 677.65 |

5 | 87.92 | 5.64 | 82.28 | 595.37 |

6 | 87.92 | 4.97 | 82.95 | 512.42 |

7 | 87.92 | 4.27 | 83.65 | 428.77 |

8 | 87.92 | 3.57 | 84.35 | 344.42 |

9 | 87.92 | 2.87 | 85.05 | 259.37 |

10 | 87.92 | 2.16 | 85.56 | 173.61 |

11 | 87.92 | 1.45 | 86.47 | 87.14 |

12 | 87.86 | 0.72 | 87.14 | 0.00 |

Grand Totals |
1,054.98 | 54.98 | 1,000.00 |

## Rule of 78's Calculation

Now, we can see from this amortization schedule that the total interest accrued and paid over the life of this loan will be $54.98 if all payments are made on their due dates. We can now use that number as our add-on for a Rule of 78s loan with a 10% APR, a 12 month term, and a monthly payment of $ 87.86. The interest has been pre-computed to be $54.98. The Rule of 78s is just used to determine how much of each month's payment shall go to interest and how much to principal.

We start by taking the summation of the number of payments. 1 + 2 + 3 + ..... + 11 + 12 = 78. (That's why it is called Rule of 78s.)

In the first month, our interest will be 12/78 x 54.98. In the second month, 11/78 x 54.98, etc. Over the total life of the loan, interest will be 78/78 x 54.98.

## Side By Side Comparison of Rule of 78s vs Simple Interest

Now for the side-by-side comparison based on these calculations. Keep in mind, the Total Payment is the same. Although it is traditional to state the balance of a Rule of 78s loan to include the remaining pre-computed interest, showing it without will help to show the difference in payoff at any point along the term of the two loans.

Payment # | Int (s) | Prin (s) | Bal (s) | Int (R78) | Prin (R78) | Bal(R78) |
---|---|---|---|---|---|---|

1000.00 | 1000.00 | |||||

1 | 8.33 | 79.59 | 920.41 | 8.46 | 79.46 | 920.54 |

2 | 7.67 | 80.25 | 840.16 | 7.75 | 80.17 | 840.37 |

3 | 7.00 | 80.92 | 759.24 | 7.05 | 80.87 | 759.50 |

4 | 6.33 | 81.59 | 677.65 | 6.34 | 81.58 | 677.92 |

5 | 5.64 | 82.28 | 595.37 | 5.63 | 82.29 | 595.63 |

6 | 4.97 | 82.95 | 512.42 | 4.93 | 82.99 | 512.64 |

7 | 4.27 | 83.65 | 428.77 | 4.23 | 83.69 | 428.95 |

8 | 3.57 | 84.35 | 344.42 | 3.52 | 84.40 | 344.55 |

9 | 2.87 | 85.05 | 259.37 | 2.82 | 85.10 | 259.45 |

10 | 2.16 | 85.76 | 173.61 | 2.11 | 85.81 | 173.64 |

11 | 1.45 | 86.47 | 87.14 | 1.41 | 86.51 | 87.13 |

12 | 0.72 | 87.14 | 0 | 0.73 | 87.13 | 0 |

TOTALS | 54.98 | 1000.00 | 54.98 | 1000.00 |

## What does this side-by-side comparison tell us?

Since we used the simple interest amortization schedule to pre-compute the interest that we were going to charge on the Rule of 78s loan, the total interest and principal paid over both loans had to be equal. Notice, however, that with the exception of the final payment, the payoff of the Rule of 78s loan is always higher than it would be for the corresponding simple interest loan. On the rule of 78s loan, the same interest is "front loaded," which means that the principal is paid back slower. Extra profit is made on these loans whenever they are paid off early.

Of course, our loan servicing software can handle Rule 78's with ease. Learn more by speaking to our sales team.