# Flat Interest Rate

The term flat interest rate is sometimes used in relation to flat rate finance loan agreements (particularly car loans) to show the rate of interest based on the original principal loan amount (PV). The flat rate does not take account of the reducing loan balance as payments are made, and therefore does not provide a true reflection of the actual interest rate being charged and can be misleading.

## Flat Interest Rate Example 1

Suppose for example, a car loan for 8,640 (PV), has monthly payments of 200 (Pmt), for a term of 48 months (n), then the flat interest rate would be calculated as follows:

```Principal repayment each month = Principal / Term
Principal repayment each month = 8,640 / 48 = 180
Monthly interest = Pmt - Principal repayment
Monthly interest = 200 - 180 = 20
Annual interest = 20 x 12 = 240
Flat interest rate = Interest / Principal (PV)
Flat interest rate = 240 / 8,640
Flat interest rate = 2.778%
```

The flat interest rate is calculated by dividing the annual interest by the original loan amount

## Flat Interest Rate Formula

In general the flat interest rate can be calculated using the flat interest rate formula

`i = (Pmt - PV / n) / PV`
Variables used in the formula
i = Flat interest rate per period
n = Number of periods
PV = Present value
Pmt = Periodic payment

Using the values in the example above, the flat interest is calculated using the formula as follows:

```i = Flat interest rate = (Pmt - PV / n) / PV
n = Number of periods = 48
PV = Present value = 8,640
Pmt = Periodic payment = 200
Flat interest rate = (200 - 8,640 / 48) / 8,640
Flat interest rate = 0.2315% per month
Annual rate = 12 x 0.2315% = 2.778%
```

## Flat Rate vs APR

Of course in practice, the principal loan balance is reduced by the payment each month, and the interest is calculated on the principal balance at the start of each month.

The interest rate (APR) is given by the present value of an annuity formula or alternatively by the Excel RATE function.

Using the same values as in the flat interest rate example

```n = 48
Pmt = 200
PV = 8,640
i = RATE(48,-200,8640)
i = 0.4385% per month
APR = 12 x 0.4385% = 5.262%
```

This calculation shows that the flat rate of interest of 2.778% is equivalent to an APR of 5.262%. To show this is the case, we can compare the total interest for each rate.

### Total Interest Using the Flat Interest Rate

Using the flat interest rate the total interest is:

```Interest = Principal x Rate X Term
Interest = 8,640 x 2.778% x 4 = 960
```

### Total Interest Using the APR Interest Rate

Using the APR interest rate the total interest is:

```Pmt = PV x i / (1 - 1 / (1 + i)n)
Pmt =8640 x (5.262%/12)/(1-1/(1+5.262%/12)48
Pmt = 200
Interest = Pmt x n - PV
Interest = 200 x 48 - 8,640
Interest = 960
```

## Flat Interest Rate Example 2

To show the effect of the two interest rates, consider another example of a loan of 3,000 paid off over 4 months with payments of 780 a month. The two rates are calculated as before using the formulas discussed above.

```Flat interest rate = (780 - 3,000 / 4) / 3,000
Flat interest rate = 1% or 12% a year
APR = RATE(4,-780,3000) = 1.5875% = 19.05% a year
```

If we now look at the payment schedules for each interest rate we get the following:

Flat Interest Rate (1%) Payment Schedule
Month 1 Month 2 Month 3 Month 4
Opening 3,000.00 2,250.00 1,500.00 750.00
Interest 30.00 30.00 30.00 30.00
Payment -780.00 -780.00 -780.00 -780.00
Closing 2,250.00 1,500.00 750.00 0
APR Interest Rate (1.5875%) Payment Schedule
Month 1 Month 2 Month 3 Month 4
Opening 3,000.00 2,267.63 1,523.62 787.81
Interest 47.63 36.00 24.19 12.19
Payment -780.00 -780.00 -780.00 -780.00
Closing 2,267.63 1,523.62 787.81 0

The flat interest rate schedule calculates interest at 1% on the opening principal balance of 3,000, whereas the APR interest rate schedule calculates interest at 1.5875% on the reducing balance at the start of each month. Both schedules show the loan reducing from 3,000 to zero over the 4 month period with monthly payments of 780, and a total interest charge of 120.

## Flat Rate to Effective Interest Rate

Finally, the effective annual interest rate can be calculated using the APR with the standard formula as follows:

```EAR = (1 + r / m )m - 1
r = Annual nominal rate of interest = 1.5875%
m = Number of compounding periods in a year = 12
EAR = (1 + 1.5875% )12 - 1
EAR = 20.805%
```

To summarize for this example, a flat rate of 12% is equivalent to an APR of 19.05% which is equivalent to an effective annual rate (EAR) of 20.805%.

Flat Interest Rate November 6th, 2016

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