## Understanding the Variance in Loan Amortization Calculations

Loan amortization calculations can vary significantly depending on the tools and methods used. This article delves into the discrepancies encountered when implementing French and Italian amortization methods in ** Python** using the numpy_financial library compared to results obtained from Excel.

Despite using identical conditions such as interest rate, loan amount, duration, and payment frequency, the results from Python calculations differ from those in Excel. Understanding these differences is crucial for developing accurate financial applications.

Command | Description |
---|---|

dateutil.relativedelta | Provides a way to compute relative deltas for date arithmetic, allowing flexible handling of date calculations such as adding months or years. |

numpy_financial.pmt | Calculates the fixed payment required to fully amortize a loan over a given number of periods, considering a fixed interest rate. |

numpy_financial.ipmt | Returns the interest portion of a payment for a given period of a loan or investment based on constant periodic payments and a constant interest rate. |

numpy_financial.ppmt | Returns the principal portion of a payment for a given period of a loan or investment based on constant periodic payments and a constant interest rate. |

pandas.DataFrame | A two-dimensional labeled data structure in pandas, used to store and manipulate tabular data efficiently. |

cumsum() | Computes the cumulative sum of array elements, often used to calculate running totals, such as the remaining loan balance after each payment. |

dt.datetime.fromisoformat() | Parses a string representing a date in ISO format and returns a datetime object, enabling easy manipulation of date values. |

### Understanding Loan Amortization Calculation Discrepancies

The Python script provided is designed to calculate loan amortization schedules using both the French and Italian methods. The script leverages the **numpy_financial** library to compute payments, interest, and principal amounts. The **Loan** class is initialized with parameters such as interest rate, loan term, loan amount, amortization type, and payment frequency. The class calculates the total number of payment periods using the **calculate_periods** method, which adjusts based on whether the frequency is monthly, quarterly, semi-annual, or annual. It also calculates the period-specific interest rate using the **calculate_period_rate** method. The core calculation for the payment amount is done using **numpy_financial.pmt**, which determines the fixed payment amount required to amortize the loan over the given periods.

The method **loan_table** generates the amortization schedule. It constructs a list of payment dates based on the payment frequency and creates a table of payments, interest, and principal amounts. For the French amortization method, the script uses **numpy_financial.ipmt** to calculate the interest portion of each payment and **numpy_financial.ppmt** to calculate the principal portion. These values are then combined into a Pandas DataFrame for easy manipulation and visualization. For the Italian method, the script calculates the interest as a fixed percentage of the remaining loan balance and the principal as a fixed amount. This schedule is also stored in a Pandas DataFrame. Despite the correct implementation, discrepancies arise when comparing the Python results to those from Excel, where the **PMT** function provides different payment values under identical conditions.

### Resolving Discrepancies in Loan Amortization Calculations

Python Backend Script for Loan Amortization Calculation

`import datetime as dt`

`from dateutil.relativedelta import relativedelta`

`import numpy_financial as npf`

`import pandas as pd`

`class Loan:`

`def __init__(self, rate, term, loan_amount, amortization_type, frequency, start=dt.date.today().isoformat()):`

`self.rate = rate`

`self.term = term`

`self.loan_amount = loan_amount`

`self.start = dt.datetime.fromisoformat(start).replace(day=1)`

`self.frequency = frequency`

`self.periods = self.calculate_periods()`

`self.period_rate = self.calculate_period_rate()`

`self.pmt = npf.pmt(self.period_rate, self.periods, -self.loan_amount)`

`self.amortization_type = amortization_type`

`self.table = self.loan_table()`

`def calculate_periods(self):`

`if self.frequency == 'monthly':`

`return self.term * 12`

`elif self.frequency == 'quarterly':`

`return self.term * 4`

`elif self.frequency == 'semi-annual':`

`return self.term * 2`

`elif self.frequency == 'annual':`

`return self.term`

`else:`

`raise ValueError("Unsupported frequency")`

`def calculate_period_rate(self):`

`if self.frequency == 'monthly':`

`return self.rate / 12`

`elif self.frequency == 'quarterly':`

`return self.rate / 4`

`elif self.frequency == 'semi-annual':`

`return self.rate / 2`

`elif self.frequency == 'annual':`

`return self.rate`

`else:`

`raise ValueError("Unsupported frequency")`

### Excel Formula Approach to Loan Amortization

Excel Formula for French Amortization

`=PMT(4.5%/1, 10*1, -1500000)`

`=IPMT(4.5%/1, A2, 10*1, -1500000)`

`=PPMT(4.5%/1, A2, 10*1, -1500000)`

`=A2-P2`

`for each period`

### Implementing Amortization Schedule Calculation in Python

Python Code for Amortization Schedule

`def loan_table(self):`

`if self.frequency == 'monthly':`

`periods = [self.start + relativedelta(months=x) for x in range(self.periods)]`

`elif self.frequency == 'quarterly':`

`periods = [self.start + relativedelta(months=3*x) for x in range(self.periods)]`

`elif self.frequency == 'semi-annual':`

`periods = [self.start + relativedelta(months=6*x) for x in range(self.periods)]`

`elif self.frequency == 'annual':`

`periods = [self.start + relativedelta(years=x) for x in range(self.periods)]`

`else:`

`raise ValueError("Unsupported frequency")`

`if self.amortization_type == "French":`

`interest = [npf.ipmt(self.period_rate, month, self.periods, -self.loan_amount, when="end") for month in range(1, self.periods + 1)]`

`principal = [npf.ppmt(self.period_rate, month, self.periods, -self.loan_amount) for month in range(1, self.periods + 1)]`

`table = pd.DataFrame({'Payment': self.pmt, 'Interest': interest, 'Principal': principal}, index=pd.to_datetime(periods))`

`table['Balance'] = self.loan_amount - table['Principal'].cumsum()`

`elif self.amortization_type == "Italian":`

`interest = [self.loan_amount * self.period_rate]`

`principal_payment = self.loan_amount / self.periods`

`principal = [principal_payment]`

`payment = [interest[0] + principal[0]]`

`for month in range(1, self.periods):`

`interest_payment = (self.loan_amount - (month) * principal_payment) * self.period_rate`

`interest.append(interest_payment)`

`principal.append(principal_payment)`

`payment.append(interest_payment + principal_payment)`

`principal[-1] = self.loan_amount - sum(principal[:-1])`

`payment[-1] = interest[-1] + principal[-1]`

`table = pd.DataFrame({'Payment': payment, 'Interest': interest, 'Principal': principal}, index=pd.to_datetime(periods))`

`table['Balance'] = self.loan_amount - table['Principal'].cumsum()`

`else:`

`raise ValueError("Unsupported amortization type")`

`return table.round(2)`

### Exploring Interest Calculation Differences in Loan Amortization

One key aspect contributing to discrepancies between Python and Excel calculations is the way interest is compounded and handled over periods. Excel's **PMT**, **IPMT**, and **PPMT** functions are designed to work with a specific compounding method, often aligning with financial industry standards. However, when these calculations are replicated in Python using the **numpy_financial** library, slight differences in the handling of interest accrual and rounding can lead to varied results. Understanding these nuances is crucial for ensuring consistency across different platforms.

Additionally, differences in the underlying algorithms used by Python and Excel can result in varying amortization schedules. Excel's functions are optimized for quick, accurate calculations but may use approximations that differ from Python's more granular calculations. The Python script provided uses a class structure to define the loan parameters and methods to calculate the amortization schedule. This allows for greater flexibility and customization but also requires careful handling of each calculation step to avoid discrepancies. Ensuring that both platforms use the same compounding frequency, interest calculation methods, and rounding practices is essential for achieving matching results.

#### Frequently Asked Questions about Loan Amortization Calculations

- Why do my Python and Excel amortization schedules differ?
- Differences can arise from varying compounding methods, interest calculation practices, and rounding differences. Ensuring consistency in these aspects is crucial.
- What is the
**numpy_financial**library used for in loan calculations? **numpy_financial**provides financial functions such as**PMT**,**IPMT**, and**PPMT**to calculate payments, interest, and principal for loans and investments.- How can I ensure my Python results match Excel?
- Verify that the compounding frequency, interest rates, and rounding methods are consistent between Python and Excel.
- What does the
**PMT**function do? - The
**PMT**function calculates the fixed payment required to fully amortize a loan over a given number of periods with a constant interest rate. - Why is compounding frequency important?
- Compounding frequency affects how interest is calculated and can significantly impact the total payment amount and amortization schedule.
- What is the difference between French and Italian amortization methods?
- French amortization involves constant payments with varying principal and interest portions, while Italian amortization has fixed principal payments with decreasing interest amounts.
- How does the
**cumsum()**function help in amortization schedules? - The
**cumsum()**function calculates the cumulative sum, useful for determining the remaining loan balance after each payment. - Can rounding differences affect loan calculations?
- Yes, even small rounding differences can lead to noticeable discrepancies over multiple periods. Consistent rounding practices are essential.
- What are
**IPMT**and**PPMT**functions used for? **IPMT**calculates the interest portion of a payment, while**PPMT**calculates the principal portion for a given period of a loan.

#### Final Thoughts on Amortization Discrepancies

Addressing the discrepancies between loan amortization calculations in ** Python** and Excel requires a detailed understanding of the underlying methods used by each platform. By ensuring consistent compounding frequencies,

**interest**calculation practices, and rounding methods, it is possible to achieve matching results. This exploration highlights the importance of meticulous attention to detail in financial calculations to ensure accuracy and reliability across different tools and software.