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Past Due Interest

The current chapter describes formulas for calculating the late payment interest (hereinafter in this article referred to as “the past due interest”) charged according to the monthly late payment interest (hereinafter in this article referred to as “the overdue interest rate”) specified in a credit product. In the system, the past due interest is charged on the current total bad debt (hereinafter in this article referred to as “the outstanding balance”) or on the amount outstanding for the current installment (hereinafter in this article referred to as “the current debt”).

The core principle of charging the past due interest consists in the following: the past due interest is calculated daily as a percentage of a current outstanding amount. In the credit product, the percentage is specified as the “overdue interest rate” and the current outstanding amount is expressed by the parameter “overdue interest type”. More precisely, on any overdue day, the total past due interest is calculated by the following formula:

OI = Debt x DOIR                                           (1)

where

OI - past due interest rate charged for the current overdue day,

Debt - current outstanding amount (as of the current date) on which the past due interest is charged,

DOIR - daily late payment interest (hereinafter “the daily overdue interest rate”).

Formula (1) is a basic rule according to which the past due interest is calculated in the system. Depending on a selected overdue interest type, only the current outstanding amount (Debt) can be calculated by different methods and is then inserted into formula (1).

The follow-up paragraphs cover two methods used in the system for charging the past due interest. When the Payday Loan lending scheme is used, these two methods become similar, since the entire payment schedule consists of one installment and the total outstanding amount of the current installment equals the outstanding balance.

Past due interest charged on the outstanding balance

According to this method, the past due interest is computed daily on the basis of the outstanding balance. The outstanding balance on the first overdue day OB comprises the total outstanding principal amount, interest outstanding as of the current date (i.e. excluding interest outstanding for all next scheduled installments that are not due yet) and, if provided, commission. On the first overdue day (n = 1), the past due interest is calculated as follows: OI₁ = OB x DOIR, where DOIR is the daily overdue interest rate. Then, on the same day, the outstanding balance gets updated OB₁ = OB + OI_1.  On the second overdue day (n = 2), the past due interest is charged on the updated outstanding balance OI₂ = OB₁ x DOIR, whereupon the outstanding balance gets also updated: OB₂ = OB₁ + OI_2. Proceeding with these considerations, we conclude that the past due interest on the n- th overdue day can be calculated by the following formula:

Image Added                               (2)

where

n - number of overdue days,

OI_n - past due interest charged for the n-th overdue day,

OB_n-1 - outstanding balance (comprising the principal amount, interest, past due interest and, if provided, commission) on the n-1-th overdue day,

DOIR - daily overdue interest rate determined on the basis of the overdue interest rate by formula (10).

The outstanding balance on the n-th overdue day gets updated according to the following formula:

Image Added                                 (3)

Taking into account equation (2), formula (3) can be written as follows:

Image Added                   (4)

Thus, formulas (2) – (3) form a recursive algorithm for calculating the interest for the n-th overdue day and updating the outstanding balance. It is only necessary to know the daily overdue interest rate and the outstanding balance on the previous (n-1) day.

Formula (4) for calculating the outstanding balance on the n-th overdue day can be transformed (using properties of a geometric progression) into the following formula:

Image Added                       (5)

where

OB = OB₀ - outstanding balance (comprising the principal amount, interest and, if provided, commission) on the due date (one day before the first overdue day, n=0).

Formula (5) implies that the past due interest charged for all n overdue days can be calculated as follows:

Image Added             (6)

where

PDI_n - past due interest for all n overdue days.

It is worth noting that when deriving formulas (2) – (6) it has been supposed that during the whole overdue period only the past due interest is charged and the loan is not repaid. The outstanding balance will therefore change (increase) over time only due to the past due interest charged. In the general case, the late fee can be charged or the installment can be repaid on any day of the overdue period and that will affect the outstanding balance and, consequently, the further increase in the past due interest. Therefore, in the general case, the past due interest for the current overdue day is calculated by formula (2), where the outstanding balance OB_n-1 is updated daily taking into account all payments received (including the repayment of the past due interest and fees), while the total past due interest charged for all n overdue days is calculated by the following formula:

Image Added  (7)

Thus, formulas (2) and (7) are valid in any case, while formulas (3) – (6) describe only a particular case without late fees and made repayments. 

Past due interest charged on the current debt

According to this method, the past due interest is charged daily on the current debt. On the first overdue day, the current debt includes the principal and interest outstanding for the current installment and, if provided, commission.

Based on considerations similar to those described in the previous paragraph 1.1, we obtain the following formulas for:

1) calculating the past due interest charged for the n-th overdue day

Image Added                            (8)

2) calculating the past due interest charged for all n overdue days

Image Added            (9)

In formulas (8) – (9), the following notation is used:

OI_n - past due interest charged for the n-th overdue day,

СD = СD₀ - outstanding amount for the current installment (principal amount, interest and, if provided, commission) on the due day (one day before the first overdue day, n = 0),

СD_n - outstanding amount for the current installment on the n-th overdue day,

DOIR - daily overdue interest rate determined on the basis of the overdue interest rate by formula (10),

n - number of overdue days.

As in paragraph 1.2, it should be noted that formula (9) is valid when during the whole overdue period only the past due interest is charged and other changes in the current debt (related to charged fees or received payments) do not take place. In the general case, the past due interest for the current overdue day is calculated by formula (8), where the current debt CD_n-1 is updated daily taking into account all received payments (including the repayment of the past due interest and fees), while the total past due interest for all n overdue days is calculated by formula (7).

It should also be noted that the past due interest is calculated by formulas (8) and (9) before the first overdue day of the next scheduled installment. On the first overdue day of the next installment, the amount outstanding for the current installment gets updated and the amount outstanding for the next installment (principal amount, interest and, if provided, commission) gets added to the amount outstanding for the current installment. Then the past due interest is calculated by formulas (8) – (9) taking into account the updated current debt that now takes into account all overdue installments. The system always updates the current debt in this way on the first overdue day of the next scheduled installment.

In the payment schedule, each overdue installment is charged a corresponding amount of the outstanding debt that has been calculated within the settlement period.

Transforming the overdue interest rate into the daily overdue interest rate

To charge the interest for overdue payments, the monthly late payment interest (the overdue interest rate) is applied in the system, while the arrears of interest are calculated on the basis of the daily late payment interest (the daily overdue interest rate).  The daily overdue interest rate can be computed on the basis of the overdue interest rate by the following formula:

Image Added                                 (10)

where

DOIR - daily overdue interest rate,

OIR - overdue interest rate specified by the user when creating a credit product.

Equation (10) is used for all methods of charging the overdue interest.

Late Fees

The current paragraph describes formulas and rules according to which a late fee is charged.

Some types of credit products provide for the late fee. The late fee is charged once for a time period stipulated in the agreement. There can be several time periods and different amounts of the late fee can be charged for each time period. The late fee can be charged in two ways:

1) absolute fees are stipulated in the agreement for each overdue period irrespective of the outstanding balance;

2) the agreement stipulates interest rates according to which late fees are charged for each overdue period depending on the outstanding balance.

Consequently, in any overdue period, the fee amount can comprise two parts: absolute part (does not depend on the outstanding balance) and relative part (depends on the outstanding balance). The late fee is calculated by the following formula:

Image Added                                     (11)

where

F - total late fee charged in the overdue period,

F_a - absolute fee stipulated in the agreement for the overdue payment in the overdue period,

OB - outstanding balance at the time of charging the current late fee,

F_r - interest stipulated in the agreement in percentage terms for each overdue period.

The late fee charged gets added to the outstanding balance or the current debt. If the payment(s) falls (completely or partially) under several different overdue periods, the total fee amount will equal the total fee amount for all overdue periods to which the overdue payment relates.

The late fee is charged from the first day after the late grace period expires.

If on any overdue day, the borrower makes payment in the amount covering the current outstanding installment (including the past due interest and late fee charged for all overdue days), the installment becomes repaid and the new past due interest is not charged.

Otherwise, the made payment S covers partially the current installment (the commission is covered first, then the late fee, past due interest, interest, and principal amount) and after the outstanding balance is updated (on a day a redemption amount is received), the past due interest is charged again according to formula (2) or (8) and the late fee is charged by formula (11).

Late Grace Days

The current paragraph describes the rules of applying late grace days specified by the user.

The term “late grace days” refers to a period that starts on the next day after the installment due date and during which the borrower can fulfil their loan obligations without being charged any late fees or past due interest.

In the system, the user can specify the late grace days (LGD) during which neither the past due interest nor late fees will be charged after the installment due date.

If the borrower repays the full amount of the current installment during the late grace days (before the LGD+1-th day after the installment due date), the system closes the current installment without charging any past due interest or late fees. Otherwise, the system will charge the past due interest according to formulas (2), (7) or (8), (7) and/or the late fee according to formula (11) for the entire overdue period, including all late grace days, at corresponding interest rates.

Thus, taking into account the late grace days, formulas (6), (7) and (9) are adjusted as follows:

Image Added           (12)

Formula (11) for the late fees changes similarly:

Image Added                         (13)

Updating the outstanding balance

The current paragraph describes the simultaneous calculation of the past due interest, late fees and update of the outstanding balance.

Once the overdue days start, the system charges the predefined past due interest and late fee every day and updates the outstanding balance or the current debt accordingly (i.e. taking into account the past due interest and late fee charged).

Calculation method using the outstanding balance

In case the past due interest is charged on the outstanding balance on the n-th overdue day, taking into account the outstanding balance OB_n-1 on the previous n-1-th day, the system makes the following calculations:

1) Formula (2) is used to calculate the past due interest for the n-th overdue day

Image Added                           (14)

2) If the late fee is provided for this day, formula (11) is used to calculate the late fee on the n-th overdue day

Image Added                            (15)

3) The outstanding balance gets updated taking into account the past due interest and late fees charged for the n-th overdue day

Image Added                    (16)

where

n - current overdue day,

OI_n - past due interest charged for the n-th overdue day,

OB_n-1 - outstanding balance (comprising the principal amount, interest, past due interest, late fee and commission) on the n-1 -th overdue day,

DOIR - daily overdue interest rate,

F_n - total late fee amount charged on the n-th overdue day,

F_a - absolute fee amount charged on the n-th overdue day,

F_r - interest charged on the outstanding debt to determine the late fee amount on the n-th overdue day,

ОB_n - outstanding balance (comprising the principal amount, interest, past due interest, late fee and commission) on the n-th overdue day.

Calculation method using the current debt

In case the past due interest is charged on the current debt on the n-th overdue day, the system makes the following calculations:

1) The past due interest is calculated for the n-th overdue day

Image Added                          (17)

2) If the late fee is provided for this day, formula (12) is used to calculate the late fee on the n-th overdue day

Image Added                           (18)

3) The outstanding balance gets updated taking into account the past due interest and late fees charged for the n-th overdue day

Image Added                   (19)

4) The total outstanding amount for the current installment gets updated taking into account the past due interest and fees charged for the n-th overdue day

Image Added                   (20)

where

СD_n - outstanding amount for the current overdue installment (comprising the principal amount, interest, past due interest, late fee and commission) on the n-1 -th overdue day.

If the late grace days are available, the system charges neither the past due interest nor fees during these days. However, once the overdue period exceeds the late grace days, the system charges the past due interest and late fee for all overdue days by using recursive algorithm (14) – (16) or (17) – (20) depending on a method used for calculating the past due interest.

Using formulas (6) and (9) for charging late fees

To calculate the past due interest and late fees, formulas (6) or (9) can be used in addition to recursive algorithms (14) – (16) or (17) - (20). Formulas (6) and (9) or their generalization in case of applying the late grace days (12) are convenient to use for the direct (without recursion) calculation of the past due interest for all n overdue days. However, as described in paragraph 1, these formulas have been derived on condition that only the past due interest is charged for the overdue installment each day, i.e. when deriving formulas (6), (9) it has not been taken into account that late fees can be additionally charged on any overdue days. In case the past due interest and fees are provided for the overdue installment, the outstanding balance and the current debt must be updated every day, taking into account the past due interest and fees charged.

To illustrate the aforesaid, we will consider the case when the payment is m days overdue and the past due interest is charged every day on the basis of the outstanding balance. Besides, on the k-th overdue day (k<m), the late fee must be charged.

Image Added

Let us make calculations. From the first to the k-1-th overdue day, only the past due interest is charged and formula (6) can be used. According to this formula, the past due interest for the first k-1 overdue days is calculated:

Image Added.

Using formula (3), the outstanding balance is updated on the k-1-th overdue day:

Image Added.

Next, the past due interest is charged on the k-th overdue day by using formula (2):

Image Added

And the late fee is calculated by formula (12):

Image Added.

Then the outstanding balance is updated on the k-th overdue day:

Image Added.

After that, formula (6) can be used again. Thus, the past due interest charged by the system for all m overdue days taking into account the late fee charged on the k-th overdue day can be calculated as follows:

Image Added.

The example of charging the past due interest and late fee

Assume a loan has been taken out for 3 months in the amount of 5 000 USD. Late grace days are not available in a credit product, the monthly interest rate equals 1.2 %. The overdue interest rate equals 3%. The percentage of the outstanding balance for charging the late fee on the first and second days equals 2% and 5% correspondingly. The commission amounts to 20 USD.

Initial payment schedule

Suppose that the annuity calculation method is used and the loan origination date is May 18. Then according to the payment schedule calculation rules described in the articles “Annuity” and “/wiki/spaces/TKL/pages/38211”, the initial payment schedule will look as follows:

 Calculating the past due interest and late fee

Let us consider the case when the first installment is 4 days overdue. The installment due date is June 18, i.e. the borrower fails to repay the loan till June 22 inclusive. According to the method of calculating the past due interest on the basis on the current debt, the system charges the past due interest and late fees daily in accordance with formulas (17) – (20). On June 18, the current debt comprises all payments due for the first installment:

CD₀ = P₁ + I₁ + AF = 1646.83 + 60.00 + 20.00 = 1726.83 USD

The outstanding balance comprises all outstanding payments:

OB₀ = 3 x Total = 3 x 1726.83 = 5180.48 USD

The following past due interest is charged for the first overdue day according to formula (17):

Image Added

The late fee must be also charged for this day. The late fee can be calculated by formula (18):

Image Added

Then the outstanding balance gets updated according to formula (19):

Image Added

and formula (20) is used to calculate the total outstanding amount for the current installment:

Image Added

Thus, as of June 19, the payment schedule looks as follows:

On June 20, on the second overdue day, the past due interest is charged by formula (17) on the basis of the updated amounts of the outstanding balance and current debt:

Image Added

and formula (18) is used to calculate the late fee:

Image Added

Then the outstanding balance and the current debt are updated according to formulas (19) – (20):

Image Added

Thus, as of June 20, the payment schedule looks as follows:

Next, on June 21 and 22, only the past due interest is charged. According to formula (9), the system charges the following past due interest for the third and the forth overdue days:

Image Added

As of June 22, the payment schedule looks as follows: