Effective interest rate in the context of loans
4. Effective annual interest rate on installment loans
An installment (amortized) loan is a loan that is periodically paid off in equal installments. Examples may include car loans, commercial loans, and mortgages.
There are four methods used to calculate the effective annual interest rate on installment loans (refer to the table below).
Illustration 2: Effective interest rates on installment loans
Actuarial method 

Constantratio 

EAR = 2 x M x C ÷ [P x (N + 1)] M is the number of payment periods per year 

Directratio 

EAR = 6 x M x C ÷ [3 x P x (N+1) + C x (N+1)] M is the number of payment periods per year 

Nratio method 

EAR = M x C x (95 x N + 9) ÷ [12 x N x (N+1) x (4P+C)] M is the number of payment periods per year 
If the amount of payment or time between payments varies from period to period (e.g., balloon payments), the constantratio, directratio, and Nratio methods cannot be used. If a lender charges a credit investigation, loan application, or life insurance fee, such a cost should be added to the cost of credit (finance charge).
Let us look at a simple example to see how the effective annual interest rate is calculated on installment loans.
Installment Loans – Example 1:
Company ABC borrows $12,000 to be repaid in 12 months. The monthly installments are $1,116 each. The finance charge is $1,400. What is the approximate value of effective annual interest rate?
Constant ratio method:
EAR = 
2 x 12 x $1,400 
= 0.2154 
12,000 x (12+1) 
Directratio method:
EAR = 
6 x 12 x $1,400 
= 0.2073 
3 x 12,000 x (12 + 1) +1,400 x (12+1) 
Nratio method:
EAR = 
12 x $1,400 (95 x 12 + 9) 
= 0.2104 
12 x 12 (12 + 1)(4 x 12,000 + 1,400) 
Using the actuarial method, the effective annual interest rate is likely to be close to 21.04%.
As we can see from this example, the constantratio method overstated the effective annual interest rate, while the directratio method slightly understated the effective annual interest rate on the installment loan.
There is another method used to approximate this rate on oneyear installment loans to be paid in equal monthly installments. The effective interest rate is determined by dividing the interest by the average amount outstanding for the year. If the loan is discounted, the average loan balance equals the average of proceeds (i.e., principal less interest).
Installment Loans – Example 2:
Company ABC borrows $10,000 at a 10% interest rate to be paid in 12 monthly installments.
In this example, the EAR could be approximated as follows:
Interest = $12,000 x 0.10 = $1,200
Average Loan Balance = $12,000 ÷ 2 = $6,000
Effective Annual Interest Rate = $1,200 ÷ $6,000 = 0.20
If this loan is discounted, the effective annual interest rate will be calculated as follows:
Interest = $1,200
Proceeds = $12,000  $1,200 = 10,800
Average Loan Balance: $10,800 ÷ 2 = $5,400
Effective Annual Interest Rate = $1,200 ÷ $5,400 = 0.2222