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

  • Most accurate method
  • Used by lenders
  • Complicated formulas

Constant-ratio
method

  • Simple formula
  • Overstated EAR
  • Higher quoted rate, more overstated EAR

EAR = 2 x M x C ÷ [P x (N + 1)]

M is the number of payment periods per year
C is the cost of credit (finance charges)
P is the original proceeds
N is the number of scheduled payments

Direct-ratio
method

  • Simple formula
  • More complicated than constant-ratio method but less complicated than actuarial method
  • Slightly understates effective annual interest rate

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
C is the cost of credit (finance charges)
P is the original proceeds
N is the number of scheduled payments

N-ratio method

  • More accurate than constant-ratio or direct-ratio methods
  • Effective annual interest rate is slightly overstated or understated depending on the nominal rate and the maturity of the loan

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
C is the cost of credit (finance charges)
P is the original proceeds
N is the number of scheduled payments

If the amount of payment or time between payments varies from period to period (e.g., balloon payments), the constant-ratio, direct-ratio, and N-ratio 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)

 Direct-ratio method:

EAR =

6 x 12 x $1,400

= 0.2073

3 x 12,000 x (12 + 1) +1,400 x (12+1)

N-ratio 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 constant-ratio method overstated the effective annual interest rate, while the direct-ratio method slightly understated the effective annual interest rate on the installment loan.

There is another method used to approximate this rate on one-year 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

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