The current article describes the classic payment scheme and exemplifies the calculation of the payment schedule for this scheme.

The classic payment scheme is based on a differential payment consisting of two parts. One part is the principal that is a constant value in the payment schedule during a whole loan term. The second part is the interest obtained as a result of charging of a fixed monthly interest rate on the total outstanding principal amount. The interest reduces for each installment, since the total outstanding principal amount reduces after repayment of corresponding installments. Correspondingly, the reduction in the interest results in the reduction of the total amount of each successive monthly installment.

Calculating the monthly differential payment

The principal amount is a constant value during the whole loan term and can be calculated as the loan amount - number of installments ratio:

                                 (1)

where

P – constant monthly principal amount,

A – loan amount,

NP – number of installments.

The total amount of the monthly installment comprises the principal and interest.

To calculate the interest and the total amount of the monthly installment, a recursive algorithm is used. In the recursive algorithm, each iteration corresponds to a next installment, i.e. a total number of iterations of the recursion equals NP. In each i-th iteration (each iteration corresponds to the i-th installment), the i-th interest amount and the i-th installment amount are calculated sequentially and the current balance is updated. Calculations are performed according to the following formulas:

                          (2)

                             (3)

                       (4)

where

I_i - interest amount of the i-th installment,

B_i - total outstanding principal amount (hereinafter “the balance”) at the moment of the i-th installment,

IR - monthly interest rate (hereinafter ”the interest rate”),

T_i - total amount of the monthly installment,

i - integer index that corresponds to an installment number in the payment schedule and takes on values from 1 to NP sequentially.

Formulas (2)-(4) are used sequentially for calculating the monthly interest amount I_i  and the total amount of the monthly installment T_i (i = 1, 2, ..., NP), based on a specified interest rate IR and an initial balance that originally equals a loan amount, i.e. B1 = A. It should be noted that the interest rate in formula (2) is taken in hundredth. Therefore, the interest rate expressed as a percentage must be divided by 100. For example, the interest rate of 1% corresponds to IR = 0.01.

Thus, each month, the principal amount remains unchanged, while the interest amount reduces for each successive installment in the payment schedule. Consequently, the total amount of each successive installment also gradually reduces.

Rounding values in the payment schedule

The payment schedule is calculated by formulas (1)- (4) with the machine precision (TOL = 1e-16) but outcomes of calculations are displayed round to the hundredth. The rounding may cause discrepancies related to total amounts of monthly installments. Namely, the total amount of the monthly installment may not be equal to the sum of the principal and interest. To eliminate these discrepancies, values are rounded in the payment schedule as follows:

1) The constant value of the principal (P) of each monthly installment is rounded by the following formula: 

               (5)

where

P_round – round value of the principal amount constant for each monthly installment,

round - rounding function.

2) The difference between the loan amount and a sum of round principal amounts of all installments is calculated:

      (6)

where

Diff - difference between the loan amount and the sum of round principal amounts of all installments.

3) The principal amount in the last installment is updated by the formula:

   (7)

where

P_roundNP - round principal amount in the last installment.

4) The calculated interest amounts are rounded to the hundredth by the following formula:

              (8)

where

I_{round i} - round interest amounts.

5) The total amount of the monthly installment is calculated:

   (9)

where

T_{round i} - round total amounts of monthly installments.

Example of the payment schedule calculated according to the classic scheme 

Suppose that the loan was taken out in the amount of 1000 USD at the monthly interest rate of 1% for 3 months, i.e. A = 1000, IR = 0.01 and NP = 3.

Calculating the payment schedule

First, we use formula (1) and calculate the constant amount of the principal:

Then, taking into account that the initial balance equals the loan amount B₁ = А = 1000 and the number of installments NP = 3, we use formulas (2) – (4) to calculate the interest (I_i), total amount of the monthly installment (Т_i) and balance (B_i). All calculations are performed with the machine precision and output values contain only two last digits after the point, taking into account the rounding used.

The final balance equals zero, i.e. B₄ = 0.00 USD.

Rounding values in the payment schedule

According to the rounding procedure stated in paragraph 2, the system performs the following actions: 

1) Rounds the principal amount according to formula (5): P_round = 333.33 USD.

2) Calculates the difference between the loan amount and the sum of round principal amounts of all installments according to formula (6): Diff = A – NP x P_round = 1000.00 – 3 х 333.33 = 0.01 USD.

3) Updates the principal amount by formula (7): P_roundNP = P_round + Diff = 333.33 + 0.01 = 333.34 USD.

4) Rounds all calculated interest amounts to the hundredth according to formula (8).

5) According to formula (9), calculates total amounts of monthly installments by using round values.

As a result, the system creates the following payment schedule with all values rounded to the hundredth: