# Rule of 78 Method

The rule of 78 method is used to analyse loan payments between interest and principal. The method uses the sum of years digits and the name arises from the fact that for a 12 period loan the sum of digits (1 + 2 + 3 … etc) is equal to 78.

The method takes the total interest over the term of the loan and allocates this to each period in proportion to the remaining term of the loan at the beginning of the period. The effect of using the rules of 78 method is to produce an interest allocation which is higher in the earlier years than in the later years of the loan term.

## Rule of 78 Method Formula

Although the rule of 78 implies a loan of 12 periods, in can in fact be applied for any number of periods. In general, the interest charge for a particular period is given by the rule of 78 formula as follows:

Rule of 78 interest = Total interest x Remaining term / SYD loan term

Where:
Total interest is the total loan repayments less the loan principal.
Remaining term is the term of the loan remaining at the beginning of the period.
SYD loan term is the sum of the years digits for the term of the loan

Since the remaining term is declining each period, the interest expense will decline each period.

## Calculation of SYD

The sum of years digits (SYD) is simply the sum of the period numbers. So for example, if there are 12 months then the sum of years digits is equal to 1+2+3 …+12 = 78, if there are 24 months then the SYD is equal to 1+2+3…+24 = 300. This process can be summarized in the sum of years digits formula as follows:

Sum of the years digits = n x (n + 1) / 2

Where:
n is the number of periods.

For example if there are 12 months as above,

```Loan term = 12 months
SYD = n x (n + 1) / 2
SYD = 12 x (12 + 1) / 2
SYD = 78
```

## Rule of 78 Method Example

Suppose a business has a loan of 1,000 which is repaid by 12 monthly payments of 95. The total payments on the loan are 12 x 95 = 1,140, and the total interest paid is therefore 1,140 – 1,000 = 140.

### Step 1: Calculate SYD

The first step is to calculate the sum of the years digits (SYD) for the term of the loan using the sum of years digits formula.

```Loan term = 12 months
SYD = n x (n + 1) / 2
SYD = 12 x (12 + 1) / 2
SYD = 78
```

### Step 2: Calculate Total Interest

The next step is to calculate the total interest expense over the lifetime of the loan, which is simply the total repayments less the loan principal.

```Loan principal = 1,000
Total interest = Repayments - Loan principal
Total interest = 12 x 95 - 1,000
Total interest = 140
```

### Step 3: Allocate Interest to Each Period

The final step is to allocate this total interest expense to each of the periods of the loan term in proportion to the remaining term of the loan at the beginning of the period using the rule of 78 method formula. The rule 78 interest calculation is as follows:

```Total interest expense = 140
SYD = 78
Month 1
Remaining life at start of the month = 12
Rule of 78 interest = Total interest x Remaining term / SYD loan term
Rule of 78 interest = 140 x 12 / 78 = 21.54
Principal = Repayment - Interest
Principal = 95.00 - 21.54 = 73.46
```
```Month 2
Remaining life at start of the month = 11
Rule of 78 interest = Total interest x Remaining term / SYD loan term
Rule of 78 interest = 140 x 11 / 78 = 19.74
Principal = Repayment - Interest
Principal = 95.00 - 19.74 = 75.26
```
```Month 3
Remaining life at start of the month = 10
Rule of 78 interest = Total interest x Remaining term / SYD loan term
Rule of 78 interest = 140 x 10 / 78 = 17.95
Principal = Repayment - Interest
Principal = 95.00 - 17.95 = 77.05
```

… this process is repeated for each period until month 12.

```Month 12
Remaining life at start of the month = 1
Rule of 78 interest = Total interest x Remaining term / SYD loan term
Rule of 78 interest = 140 x 1 / 78 = 1.79
Principal = Repayment - Interest
Principal = 95.00 - 1.79 = 93.21
```

Our rule of 78 calculator is available and can be used to calculate the rule of 78 interest expense by entering details of the loan principal, periodic repayments and the number of periods in the loan term.

Rule of 78 Method November 6th, 2016

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