📊 Loan Installments

📝 Introduction

This document explains how loan installments work, detailing the lifecycle of an installment from creation through repayment, including how interest accumulates, penalties are calculated, and payments are allocated.

🔍 Basic Concepts

Loan Structure

A loan is created with:

• A disbursement amount - the principal sum provided to the borrower
• A monthly interest rate - the rate at which interest accrues (e.g., 0.1 = 10%)
• A schedule of installments - each with a due date and expected amount

Installment Independence

• Each installment is independent from all others
• Penalties and interest apply only to overdue amounts within a specific installment
• Clients can pay installments in any order they choose

The Present Value Curve

Before the Due Date

When a loan is disbursed, each installment's value follows a "curve" from disbursement to due date. This only applies for CCBs:

1. At disbursement, the present value (PV) of a future installment is lower than its face value
2. As time progresses, the PV increases following a compound interest formula
3. At the due date, the PV equals the full installment amount

This curve is calculated using:

PV = Future Value / (1 + monthly_rate)^(days_to_maturity/30)

Early Payment Benefits

• If a client pays before the due date, they benefit from the present value curve, but only if the installment is settled
• The earlier the payment, the less they pay
• This represents the time value of money (a payment today is worth more than the same payment in the future)

Due Date Mechanics

Working Day Adjustments

• If a due date falls on a non-working day, it automatically moves to the next working day
• However, if the installment isn't fully paid on this adjusted date, penalties will be calculated from the original due date
• This aligns with common banking practices

What Happens After the Due Date?

If an installment is not fully paid by the due date, three types of charges begin to accumulate:

1. Remunerative Interest

• This is the standard loan interest that continues to apply to any outstanding principal
• It compounds daily at a rate derived from the monthly interest rate
• Formula: daily_remunerative_rate = (1 + monthly_rate)^(1/30) - 1

2. Default Interest

• Additional interest charged on the overdue amount (typically 1% per month)
• Applied daily to the entire outstanding balance (not including the current's date remunerative interest)
• Compounds along with the regular interest

3. Contractual Penalty

• A one-time penalty (typically 2%) applied when the first payment after the due date is made
• Applied only once per installment, regardless of how many late payments are made
• Calculated based on the outstanding balance at the time of payment (including accrued interest)

Payment Allocation Order

When a payment is received, it's applied in the following priority order:

1. Remunerative Interest
2. Default Interest
3. Contractual Penalty
4. Principal

This order ensures that debt service charges are covered first before reducing the principal balance.

Detailed Examples

Example 1: Early Payment

Scenario:

• Loan disbursed on January 1
• Installment amount: $1,000
• Due date: March 1
• Monthly interest rate: 1%
• Client pays on February 1 (30 days early)

Calculation:

1. Present value on February 1: $1,000 / (1 + 0.01)^(30/30) = $990.10
2. Client pays: $990.10
3. Installment is fully settled

Example 2: Late Payment with Penalties

Scenario:

• Same loan as above
• Client pays $500 on March 5 (4 days late)
• Remaining amount paid on March 15

Calculation on March 5:

1. Outstanding principal: $1,000
2. Remunerative interest (4 days): $1,000 × [(1 + 0.01)^(4/30) - 1] ≈ $1.33
3. Default interest (4 days): $1,000 × daily_default_rate × 4 ≈ $1.33
4. Contractual penalty: ($1,000 + $1.33 + $1.33) × 0.02 = $20.05
5. Total due: $1,022.71
6. Payment of $500 is applied:
   • First to remunerative interest: $1.33
   • Then to default interest: $1.33
   • Then to contractual penalty: $20.05
   • Remaining $477.29 to principal
7. New outstanding principal: $522.71

Calculation on March 15 (10 more days):

1. Remaining principal: $522.71
2. Additional remunerative interest (10 days): Approximately $1.74
3. Additional default interest (10 days): Approximately $1.74
4. Total due: $526.19
5. Client pays $526.19
6. Installment is fully settled

Handling Special Cases

Partial Payments

• Interest calculations adjust proportionally to the new outstanding balance after each partial payment
• This prevents overcharging interest on portions already paid

Overpayments

• The client is responsible for paying the right amount using the available QRs
• If there's an overpayment, the amount could be returned or applied to future installments, but this would be done manually

Installment Settlement

An installment is considered fully settled when:

• The outstanding principal is zero
• All penalties and interest charges have been paid

Glossary of Terms

• Present Value (PV): The current value of an installment, considering time value of money
• Outstanding Amount: The total amount required to settle an installment today (Present Value + penalties)
• Remunerative Interest: Standard loan interest that continues to apply on the outstanding balance
• Default Interest: Additional interest charged on overdue amounts
• Contractual Penalty: A one-time penalty applied on the first payment after the due date
• Disbursement Date: The date when the loan amount is provided to the borrower
• Due Date: The date by which an installment payment is expected to be made
• Adjusted Due Date: The due date moved to the next working day if the original date falls on a non-working day