How Loan Interest Works: Simple vs Compound

Whether you're taking out a mortgage, car loan, or saving for retirement, understanding how interest works is essential. The difference between simple and compound interest can mean thousands of dollars over the life of a loan. Here's a practical guide.

Simple Interest

Simple interest is calculated only on the original principal.

Formula

Interest = Principal × Rate × Time
A = P(1 + rt)

Where:
A = Final amount
P = Principal (original amount)
r = Annual interest rate (as decimal)
t = Time in years

Example

$10,000 at 5% simple interest for 3 years:

Interest = $10,000 × 0.05 × 3 = $1,500
Final amount = $10,000 + $1,500 = $11,500

When Simple Interest Is Used

• Some personal loans
• Car loans (sometimes)
• Treasury bonds
• Short-term loans between individuals

Compound Interest

Compound interest is calculated on principal PLUS accumulated interest. Interest earns interest.

Formula

A = P(1 + r/n)^(nt)

Where:
A = Final amount
P = Principal
r = Annual interest rate (as decimal)
n = Number of times compounded per year
t = Time in years

Compounding Frequencies

Example

$10,000 at 5% compounded monthly for 3 years:

A = $10,000 × (1 + 0.05/12)^(12×3)
A = $10,000 × (1.004167)^36
A = $10,000 × 1.1614
A = $11,614.72

Compare to simple interest: $114.72 more from compounding.

The Power of Compounding

Over longer periods, the difference becomes dramatic:

After 30 years, compound interest nearly doubles simple interest returns!

APR vs APY

These terms often confuse people:

APR (Annual Percentage Rate)

• The stated annual rate
• Does NOT account for compounding
• Used for loans (mortgages, credit cards)
• Required by law to be disclosed

APY (Annual Percentage Yield)

• The effective annual rate
• INCLUDES the effect of compounding
• Used for savings accounts and investments
• Also called "effective annual rate"

Converting APR to APY

APY = (1 + APR/n)^n - 1

Example: 12% APR compounded monthly
APY = (1 + 0.12/12)^12 - 1
APY = (1.01)^12 - 1
APY = 1.1268 - 1
APY = 12.68%

The more frequently interest compounds, the higher the APY for the same APR.

Loan Payments (Amortization)

Most loans (mortgages, car loans) use amortized payments—fixed monthly payments that gradually pay off both interest and principal.

Monthly Payment Formula

M = P × [r(1+r)^n] / [(1+r)^n - 1]

Where:
M = Monthly payment
P = Principal (loan amount)
r = Monthly interest rate (annual rate / 12)
n = Total number of payments (years × 12)

Example: Mortgage Payment

$300,000 mortgage at 6% for 30 years:

P = $300,000
r = 0.06 / 12 = 0.005
n = 30 × 12 = 360

M = 300,000 × [0.005(1.005)^360] / [(1.005)^360 - 1]
M = 300,000 × [0.005 × 6.0226] / [6.0226 - 1]
M = 300,000 × 0.03011 / 5.0226
M = 300,000 × 0.005996
M = $1,798.65

Total Interest Paid

Total paid = $1,798.65 × 360 = $647,514
Interest = $647,514 - $300,000 = $347,514

You pay more in interest than the original loan!

Amortization Schedule

Early payments are mostly interest; later payments are mostly principal:

Example: First vs Last Payment (30-year mortgage at 6%)

Payment #1:

• Interest: $300,000 × 0.005 = $1,500.00
• Principal: $1,798.65 - $1,500.00 = $298.65

Payment #360 (last):

• Remaining balance: ~$1,789
• Interest: $1,789 × 0.005 = $8.95
• Principal: $1,789.70

Code Examples

JavaScript: Compound Interest

function compoundInterest(principal, rate, years, compoundsPerYear = 12) {
    return principal * Math.pow(1 + rate / compoundsPerYear, compoundsPerYear * years);
}

function monthlyPayment(principal, annualRate, years) {
    const monthlyRate = annualRate / 12;
    const payments = years * 12;
    return principal * 
        (monthlyRate * Math.pow(1 + monthlyRate, payments)) / 
        (Math.pow(1 + monthlyRate, payments) - 1);
}

// Examples
compoundInterest(10000, 0.05, 3);    // $11,614.72
monthlyPayment(300000, 0.06, 30);    // $1,798.65

Python: Loan Amortization

def amortization_schedule(principal, annual_rate, years):
    monthly_rate = annual_rate / 12
    num_payments = years * 12
    
    # Calculate monthly payment
    payment = principal * (monthly_rate * (1 + monthly_rate)**num_payments) / \
              ((1 + monthly_rate)**num_payments - 1)
    
    balance = principal
    schedule = []
    
    for month in range(1, num_payments + 1):
        interest = balance * monthly_rate
        principal_paid = payment - interest
        balance -= principal_paid
        
        schedule.append({
            'month': month,
            'payment': round(payment, 2),
            'principal': round(principal_paid, 2),
            'interest': round(interest, 2),
            'balance': round(max(0, balance), 2)
        })
    
    return schedule

# Generate first 3 months
schedule = amortization_schedule(300000, 0.06, 30)
for payment in schedule[:3]:
    print(payment)

Rule of 72

Quick estimation: How long for money to double?

Years to double ≈ 72 / Interest Rate

At 6%: 72 / 6 = 12 years
At 8%: 72 / 8 = 9 years
At 12%: 72 / 12 = 6 years

Strategies for Loans

1. Extra Payments

Adding extra to principal reduces total interest dramatically:

$300,000 mortgage at 6%, 30 years:
- Regular: $1,798.65/month → $347,514 interest
- Extra $200/month: $1,998.65 → $253,282 interest + paid off 6 years early

Savings: $94,232

2. Bi-Weekly Payments

Pay half the monthly payment every two weeks:

• 26 half-payments = 13 full payments per year
• One extra payment annually
• Can shorten a 30-year mortgage by 4-6 years

3. Refinancing

When rates drop, refinancing can save money:

$300,000 at 6% vs 4%:
Monthly: $1,798.65 vs $1,432.25
30-year savings: $131,904

But consider closing costs (2-5% of loan)

Key Takeaways

Summary

• Simple interest: Calculated on principal only
• Compound interest: Calculated on principal + accumulated interest
• APR: Stated rate (doesn't include compounding)
• APY: Effective rate (includes compounding)
• Amortization: Fixed payments split between interest and principal
• Extra payments: Significantly reduce total interest

Understanding these concepts helps you make better financial decisions—whether you're saving, investing, or borrowing.

Need to calculate loan payments? Try our Loan Calculator!