Loan Pricing

The loan_pricing function implements a risk-based loan pricing model that calculates the appropriate interest rate for a loan based on various risk factors and cost components. This model helps lenders determine fair and profitable loan terms while accounting for the borrower's risk profile.

Components of Loan Pricing

The loan pricing model considers several key components:

1. Expected Loss Component: Accounts for the probability of default and the expected loss given default
2. Funding Cost Component: Reflects the lender's cost of obtaining funds
3. Operating Cost Component: Covers the administrative costs of originating and servicing the loan
4. Capital Cost Component: Accounts for the required return on the capital allocated to support the loan

Usage in Pypulate

from pypulate.credit import loan_pricing

# Calculate risk-based loan pricing
result = loan_pricing(
    loan_amount=100000,           # $100,000 loan
    term=5,                       # 5-year term
    pd=0.02,                      # 2% annual probability of default
    lgd=0.4,                      # 40% loss given default
    funding_cost=0.03,            # 3% cost of funds
    operating_cost=0.01,          # 1% operating costs
    capital_requirement=0.08,     # 8% capital requirement
    target_roe=0.15               # 15% target return on equity
)

# Access the results
recommended_rate = result["recommended_rate"]
monthly_payment = result["monthly_payment"]
components = result["components"]

Parameters

Parameter	Type	Description	Default
loan_amount	float	The principal amount of the loan	Required
term	float	The loan term in years	Required
pd	float	Probability of default (annual rate, between 0 and 1)	Required
lgd	float	Loss given default (as a decimal, between 0 and 1)	Required
funding_cost	float	Cost of funds (annual rate)	Required
operating_cost	float	Operating costs as percentage of loan amount	Required
capital_requirement	float	Capital requirement as percentage of loan amount	Required
target_roe	float	Target return on equity (annual rate)	Required

Return Value

The function returns a dictionary with the following keys:

Key	Type	Description
recommended_rate	float	The calculated interest rate for the loan
effective_annual_rate	float	The effective annual rate (APR)
monthly_payment	float	The calculated monthly payment amount
total_interest	float	Total interest paid over the life of the loan
expected_profit	float	Expected profit after accounting for losses and costs
return_on_investment	float	Expected return on the allocated capital
components	dict	Dictionary containing the individual pricing components

The components dictionary includes: - expected_loss: Component accounting for credit risk - funding_cost: Component accounting for cost of funds - operating_cost: Component accounting for operational expenses - capital_cost: Component accounting for capital allocation - risk_premium: Combined risk-related components (expected_loss + capital_cost)

Risk Level Classification

The loan risk is categorized based on the expected loss rate (PD × LGD):

Expected Loss Rate Range	Risk Level
< 1%	Very Low
1% - 3%	Low
3% - 7%	Moderate
7% - 15%	High
> 15%	Very High

Comprehensive Example

Here's a complete example demonstrating how to calculate and analyze loan pricing for different risk profiles:

from pypulate.credit import loan_pricing
import matplotlib.pyplot as plt
import numpy as np

# Example 1: Low-risk borrower
low_risk_loan = loan_pricing(
    loan_amount=100000,           # $100,000 loan
    term=5,                       # 5-year term
    pd=0.01,                      # 1% probability of default
    lgd=0.3,                      # 30% loss given default
    funding_cost=0.03,            # 3% cost of funds
    operating_cost=0.01,          # 1% operating costs
    capital_requirement=0.08,     # 8% capital requirement
    target_roe=0.15               # 15% target return on equity
)

# Example 2: Medium-risk borrower
medium_risk_loan = loan_pricing(
    loan_amount=100000,           # $100,000 loan
    term=5,                       # 5-year term
    pd=0.03,                      # 3% probability of default
    lgd=0.4,                      # 40% loss given default
    funding_cost=0.03,            # 3% cost of funds
    operating_cost=0.01,          # 1% operating costs
    capital_requirement=0.08,     # 8% capital requirement
    target_roe=0.15               # 15% target return on equity
)

# Example 3: High-risk borrower
high_risk_loan = loan_pricing(
    loan_amount=100000,           # $100,000 loan
    term=5,                       # 5-year term
    pd=0.08,                      # 8% probability of default
    lgd=0.5,                      # 50% loss given default
    funding_cost=0.03,            # 3% cost of funds
    operating_cost=0.01,          # 1% operating costs
    capital_requirement=0.08,     # 8% capital requirement
    target_roe=0.15               # 15% target return on equity
)

# Print the results
print("Risk-Based Loan Pricing Analysis")
print("===============================")

print("\nExample 1: Low-Risk Borrower")
print(f"Recommended Interest Rate: {low_risk_loan['recommended_rate']:.2%}")
print(f"Effective Annual Rate: {low_risk_loan['effective_annual_rate']:.2%}")
print(f"Monthly Payment: ${low_risk_loan['monthly_payment']:.2f}")
print(f"Total Interest: ${low_risk_loan['total_interest']:.2f}")
print(f"Expected Profit: ${low_risk_loan['expected_profit']:.2f}")
print(f"Return on Investment: {low_risk_loan['return_on_investment']:.2%}")
print("Pricing Components:")
for component, value in low_risk_loan['components'].items():
    print(f"  {component}: {value:.2%}")

print("\nExample 2: Medium-Risk Borrower")
print(f"Recommended Interest Rate: {medium_risk_loan['recommended_rate']:.2%}")
print(f"Effective Annual Rate: {medium_risk_loan['effective_annual_rate']:.2%}")
print(f"Monthly Payment: ${medium_risk_loan['monthly_payment']:.2f}")
print(f"Total Interest: ${medium_risk_loan['total_interest']:.2f}")
print(f"Expected Profit: ${medium_risk_loan['expected_profit']:.2f}")
print(f"Return on Investment: {medium_risk_loan['return_on_investment']:.2%}")
print("Pricing Components:")
for component, value in medium_risk_loan['components'].items():
    print(f"  {component}: {value:.2%}")

print("\nExample 3: High-Risk Borrower")
print(f"Recommended Interest Rate: {high_risk_loan['recommended_rate']:.2%}")
print(f"Effective Annual Rate: {high_risk_loan['effective_annual_rate']:.2%}")
print(f"Monthly Payment: ${high_risk_loan['monthly_payment']:.2f}")
print(f"Total Interest: ${high_risk_loan['total_interest']:.2f}")
print(f"Expected Profit: ${high_risk_loan['expected_profit']:.2f}")
print(f"Return on Investment: {high_risk_loan['return_on_investment']:.2%}")
print("Pricing Components:")
for component, value in high_risk_loan['components'].items():
    print(f"  {component}: {value:.2%}")

# Visualize the results - Interest Rate Comparison
risk_profiles = ['Low Risk', 'Medium Risk', 'High Risk']
interest_rates = [
    low_risk_loan['recommended_rate'],
    medium_risk_loan['recommended_rate'],
    high_risk_loan['recommended_rate']
]

plt.figure(figsize=(10, 6))
bars = plt.bar(risk_profiles, interest_rates, color=['green', 'orange', 'red'])

# Add the rate values on top of the bars
for bar in bars:
    height = bar.get_height()
    plt.text(bar.get_x() + bar.get_width()/2., height + 0.005,
             f'{height:.2%}', ha='center', va='bottom')

plt.ylabel('Recommended Interest Rate')
plt.title('Risk-Based Interest Rate Comparison')
plt.grid(axis='y', linestyle='--', alpha=0.7)
plt.tight_layout()
plt.show()

# Visualize the pricing components
components = ['Expected Loss', 'Funding Cost', 'Operating Cost', 'Capital Cost']
low_risk_components = [
    low_risk_loan['components']['expected_loss'],
    low_risk_loan['components']['funding_cost'],
    low_risk_loan['components']['operating_cost'],
    low_risk_loan['components']['capital_cost']
]
medium_risk_components = [
    medium_risk_loan['components']['expected_loss'],
    medium_risk_loan['components']['funding_cost'],
    medium_risk_loan['components']['operating_cost'],
    medium_risk_loan['components']['capital_cost']
]
high_risk_components = [
    high_risk_loan['components']['expected_loss'],
    high_risk_loan['components']['funding_cost'],
    high_risk_loan['components']['operating_cost'],
    high_risk_loan['components']['capital_cost']
]

x = np.arange(len(components))  # the label locations
width = 0.25  # the width of the bars

fig, ax = plt.subplots(figsize=(12, 7))
rects1 = ax.bar(x - width, low_risk_components, width, label='Low Risk', color='green')
rects2 = ax.bar(x, medium_risk_components, width, label='Medium Risk', color='orange')
rects3 = ax.bar(x + width, high_risk_components, width, label='High Risk', color='red')

# Add some text for labels, title and custom x-axis tick labels, etc.
ax.set_ylabel('Rate Component')
ax.set_title('Loan Pricing Components by Risk Profile')
ax.set_xticks(x)
ax.set_xticklabels(components)
ax.legend()

# Add value labels
def autolabel(rects):
    for rect in rects:
        height = rect.get_height()
        ax.annotate(f'{height:.2%}',
                    xy=(rect.get_x() + rect.get_width() / 2, height),
                    xytext=(0, 3),  # 3 points vertical offset
                    textcoords="offset points",
                    ha='center', va='bottom')

autolabel(rects1)
autolabel(rects2)
autolabel(rects3)

fig.tight_layout()
plt.grid(axis='y', linestyle='--', alpha=0.7)
plt.show()

# Create a sensitivity analysis for PD
pd_values = np.linspace(0.01, 0.10, 10)  # Range of PD values from 1% to 10%
interest_rates = []
monthly_payments = []
expected_profits = []

for pd_value in pd_values:
    result = loan_pricing(
        loan_amount=100000,
        term=5,
        pd=pd_value,
        lgd=0.4,
        funding_cost=0.03,
        operating_cost=0.01,
        capital_requirement=0.08,
        target_roe=0.15
    )
    interest_rates.append(result['recommended_rate'])
    monthly_payments.append(result['monthly_payment'])
    expected_profits.append(result['expected_profit'])

# Plot interest rate sensitivity to PD
plt.figure(figsize=(12, 6))
plt.plot(pd_values * 100, np.array(interest_rates) * 100, 'b-', linewidth=2, marker='o')
plt.xlabel('Probability of Default (%)')
plt.ylabel('Recommended Interest Rate (%)')
plt.title('Interest Rate Sensitivity to Probability of Default')
plt.grid(True, linestyle='--', alpha=0.7)
plt.tight_layout()
plt.show()

# Plot monthly payment sensitivity to PD
plt.figure(figsize=(12, 6))
plt.plot(pd_values * 100, monthly_payments, 'g-', linewidth=2, marker='o')
plt.xlabel('Probability of Default (%)')
plt.ylabel('Monthly Payment ($)')
plt.title('Monthly Payment Sensitivity to Probability of Default')
plt.grid(True, linestyle='--', alpha=0.7)
plt.tight_layout()
plt.show()

# Plot expected profit sensitivity to PD
plt.figure(figsize=(12, 6))
plt.plot(pd_values * 100, expected_profits, 'r-', linewidth=2, marker='o')
plt.xlabel('Probability of Default (%)')
plt.ylabel('Expected Profit ($)')
plt.title('Expected Profit Sensitivity to Probability of Default')
plt.grid(True, linestyle='--', alpha=0.7)
plt.tight_layout()
plt.show()

Example Output

Risk-Based Loan Pricing Analysis
===============================

Example 1: Low-Risk Borrower
Recommended Interest Rate: 4.46%
Effective Annual Rate: 4.55%
Monthly Payment: $1862.48
Total Interest: $11749.02
Expected Profit: $6449.02
Return on Investment: 80.61%
Pricing Components:
  expected_loss: 0.06%
  funding_cost: 3.00%
  operating_cost: 0.20%
  capital_cost: 1.20%
  risk_premium: 1.26%

Example 2: Medium-Risk Borrower
Recommended Interest Rate: 4.64%
Effective Annual Rate: 4.74%
Monthly Payment: $1870.67
Total Interest: $12240.48
Expected Profit: $6040.48
Return on Investment: 75.51%
Pricing Components:
  expected_loss: 0.24%
  funding_cost: 3.00%
  operating_cost: 0.20%
  capital_cost: 1.20%
  risk_premium: 1.44%

Example 3: High-Risk Borrower
Recommended Interest Rate: 5.20%
Effective Annual Rate: 5.33%
Monthly Payment: $1896.30
Total Interest: $13778.00
Expected Profit: $4778.00
Return on Investment: 59.72%
Pricing Components:
  expected_loss: 0.80%
  funding_cost: 3.00%
  operating_cost: 0.20%
  capital_cost: 1.20%
  risk_premium: 2.00%

Pricing Component Analysis

Each component of the loan pricing model serves a specific purpose:

1. Expected Loss Component (pd × lgd / term)

   • Compensates for the expected credit losses
   • Directly proportional to both probability of default and loss severity
   • Higher risk borrowers have significantly higher expected loss components

2. Funding Cost Component

   • Represents the lender's cost of obtaining the funds to lend
   • Typically based on market interest rates
   • Generally consistent across borrowers regardless of risk

3. Operating Cost Component (operating_cost / term)

   • Covers origination, servicing, and administrative costs
   • Spread over the life of the loan
   • May vary slightly based on loan complexity

4. Capital Cost Component (capital_requirement × target_roe)

   • Compensates for the opportunity cost of capital allocated to the loan
   • Higher risk loans may require more capital allocation
   • Reflects the lender's required return on invested capital

5. Risk Premium (expected_loss + capital_cost)

   • The combined risk-related components
   • Represents the additional return required to compensate for risk
   • Primary differentiator in pricing between low and high-risk borrowers

Practical Applications

Risk-based loan pricing can be used for:

1. Consumer Lending: Setting appropriate rates for personal loans, auto loans, and mortgages
2. Commercial Lending: Pricing business loans based on company financial health
3. Credit Card Pricing: Determining APRs for different customer segments
4. Loan Portfolio Management: Ensuring adequate returns across a portfolio of loans
5. Competitive Analysis: Benchmarking pricing against market competitors

Limitations and Considerations

When using risk-based loan pricing, consider these limitations:

1. Model Assumptions: The accuracy depends on reliable estimates of PD and LGD
2. Market Constraints: Competitive pressures may limit the ability to charge risk-appropriate rates
3. Regulatory Considerations: Fair lending laws may restrict risk-based pricing in some markets
4. Customer Acceptance: Very high rates may lead to adverse selection or reduced demand
5. Economic Cycles: Risk parameters should be adjusted for changing economic conditions