Loan repayment structures

In this post, we will discuss the two main types of loan repayment structures and build sample loan schedules in Python using the cashflower package.

List of content:

Repayment structures
Equal payment loan schedule
Equal principal loan schedule

Repayment structures

There are two main types of loan repayment structures:

equal payment loan (annuity loan),
equal principal loan (fixed principal).

Equal payment loan (annuity loan)

The total monthly payment is fixed (the same amount every month). Initially, a larger portion of the payment goes towards interest and a smaller portion towards the principal. Over time, as the outstanding loan balance decreases, more of the payment goes towards the principal. This type of loan is commonly used in mortgages and consumer loans.

Equal principal loan (fixed principal)

The principal repayment is fixed each month, meaning the amount applied to reducing the loan stays the same. The interest payment decreases over time as the loan balance shrinks, so the total monthly payment also decreases over the life of the loan. This type of loan is less common but can be used for loans where the borrower prefers to reduce the principal at a steady rate.

We will build cash flow models for both types of loans using an example.

Loan characteristics:

loan amount: EUR 100,000
interest rate: 7%
term: 20 years

We will keep this information in the model point.

# input.py
from cashflower import ModelPointSet
import pandas as pd

loan = ModelPointSet(data=pd.DataFrame({
    "amount": [100_000],
    "interest_rate": [0.07],
    "term": [20],
}))

Equal payment loan schedule

Next, let's define the model variables.

We'll start by converting the input data to months.

# model.py
from cashflower import variable
from input import loan

@variable()
def term_in_months():
    return loan.get("term") * 12


@variable()
def monthly_interest_rate():
    return loan.get("interest_rate") / 12

The outstanding loan decreases each month by the value of the principal.

# model.py
@variable()
def outstanding_loan(t):
    if t == 0:
        return loan.get("amount")
    else:
        return outstanding_loan(t-1) - principal(t)

We can use the annuity formula to calculate the fixed monthly payment.

# model.py
@variable()
def payment(t):
    if 0 < t <= term_in_months():
        i = monthly_interest_rate()
        n = term_in_months()
        return loan.get("amount") * (i * (1 + i) ** n) / ((1 + i) ** n - 1)
    else:
        return 0

The interest payment is calculated by multiplying the outstanding loan by the monthly interest rate.

# model.py
@variable()
def interest(t):
    if t == 0:
        return 0
    else:
        return outstanding_loan(t-1) * monthly_interest_rate()

The principal is simply the difference between the payment and the interest.

# model.py
@variable()
def principal(t):
    return payment(t) - interest(t)

Output:

# output
  t  outstanding_loan  payment  principal  interest
  0         100000.00     0.00       0.00      0.00
  1          99808.03   775.30     191.97    583.33
  2          99614.95   775.30     193.09    582.21
  3          99420.74   775.30     194.21    581.09
  4          99225.39   775.30     195.34    579.95
  5          99028.91   775.30     196.48    578.81
  6          98831.28   775.30     197.63    577.67
...
235           3809.57   775.30     748.71     26.59
236           3056.49   775.30     753.08     22.22
237           2299.02   775.30     757.47     17.83
238           1537.13   775.30     761.89     13.41
239            770.80   775.30     766.33      8.97
240              0.00   775.30     770.80      4.50

Notes:

the payment remains constant throughout the loan term,
the interest payment decreases over time because the outstanding loan decreases,
the principal increases as the interest payment decreases, while the total payment stays fixed.

Equal principal loan schedule

The calculations for the term in months, monthly interest rates, interest and the outstanding loan are the same as those in the equal payment loan schedule.

# model.py
from cashflower import variable
from input import loan

@variable()
def term_in_months():
    return loan.get("term") * 12


@variable()
def monthly_interest_rate():
    return loan.get("interest_rate") / 12

@variable()
def outstanding_loan(t):
    if t == 0:
        return loan.get("amount")
    else:
        return outstanding_loan(t-1) - principal(t)

@variable()
def interest(t):
    if t == 0:
        return 0
    else:
        return outstanding_loan(t-1) * monthly_interest_rate()

In this type of loan, the principal payment remains constant each month.

# model.py
@variable()
def principal(t):
    if 0 < t <= term_in_months():
        return loan.get("amount") / term_in_months()
    else:
        return 0

The total monthly payment is the sum of the fixed principal and the interest.

# model.py
@variable()
def payment(t):
    return principal(t) + interest(t)