Quick Definitions
- Simple interest: Interest is calculated only on the original principal, not on previously accrued interest.
- Compound interest: Interest accrues on the principal plus past interest (interest on interest).
- Amortization: A repayment structure with fixed periodic payments where each payment covers interest for the period and some principal, driving the balance to zero by the end of the term.
Core Formulas
Simple Interest
Interest = P · r · t
Amount = P(1 + r·t)
Compound Interest
Amount = P · (1 + r/m)^(m·t)
Set m=1 for annual compounding.
Amortization Payment
Payment = [P · i] / [1 - (1 + i)^(-n)]
Where i = r/m and n = m·t. Interest each period is balance × i.
Worked Example 1: One-Year at 12% on $10,000
Assumptions: principal $10,000.00, nominal annual rate 12%, time 1 year.
| Method | Ending Amount / Total Paid | Total Interest |
|---|---|---|
| Simple Interest (1 year) | $11,200.00 | $1,200.00 |
| Compound Interest (annual) | $11,200.00 | $1,200.00 |
| Compound Interest (monthly) | $11,268.25 | $1,268.25 |
| Compound Interest (daily, 365) | $11,274.75 | $1,274.75 |
| Amortized Loan (12× monthly) | $10,661.85 | $661.85 |
Amortization looks cheaper here because you begin repaying principal immediately. In lump-sum simple/compound examples, the full principal stays outstanding all year.
APY and Compounding Frequency
| Compounding Frequency | APY (Effective Annual Rate) |
|---|---|
| Annual (m = 1) | 12.0000% |
| Semiannual (m = 2) | 12.3600% |
| Quarterly (m = 4) | 12.5509% |
| Monthly (m = 12) | 12.6825% |
| Daily (m = 365) | 12.7475% |
What Amortization Looks Like Month to Month
Using $10,000.00, 12% APR, and 12 monthly payments, the fixed payment is $888.49 per month.
| Month | Payment | Interest | Principal | Ending Balance |
|---|---|---|---|---|
| 1 | $888.49 | $100.00 | $788.49 | $9,211.51 |
| 2 | $888.49 | $92.12 | $796.37 | $8,415.14 |
| 3 | $888.49 | $84.15 | $804.34 | $7,610.80 |
Worked Example 3: Savings Growth at 8% for 3 Years on $5,000
| Method | Final Value |
|---|---|
| Simple Interest | $6,200.00 |
| Compound Interest (annual) | $6,298.56 |
| Compound Interest (monthly) | $6,351.19 |
Conceptual Differences at a Glance
| Dimension | Simple Interest | Compound Interest | Amortization |
|---|---|---|---|
| What it is | Interest on original principal only | Interest on principal plus accumulated interest | Repayment schedule with fixed periodic payments |
| Balance movement | Usually unchanged until maturity | Grows faster as interest compounds | Drops each payment as principal is repaid |
| Common uses | Short-term notes, quick quotes | Savings, investments, credit cards | Mortgages, auto loans, personal installment loans |
| Cost for borrowers | Higher than amortization if principal sits unchanged | Highest if you defer payments and let interest compound | Usually lowest over same horizon |
| Predictability | Low | Low/medium | High (fixed payments, known payoff date) |
| Key metric | Rate r and time t | APY reflects frequency | APR and payment formula |
Decision Guide
| Your Goal / Situation | Best Fit | Why |
|---|---|---|
| Predictable payment and definite payoff date | Amortized loan | Fixed payment; principal shrinks; total cost is transparent |
| Maximize growth of savings/investments | Compound interest with frequent compounding | Interest on interest; APY captures true effective return |
| Short-term borrowing or quick evaluation | Simple interest | Easy to compute and explain; good for brief horizons |
| Lower the cost of an existing amortized loan | Prepay principal | Shrinks balance earlier and reduces future interest |
Frequently Asked Questions
Is amortization better than compound interest?
They solve different problems. Amortization is a loan repayment structure; compound interest is a growth rule. For borrowers, amortization often costs less because the balance declines during the term.
Why do simple and annual compound interest match for one year?
With exactly one compounding period and a one-year holding period, P(1+rt) equals P(1+r). More periods or time make compound interest diverge upward.
How much does frequency matter?
Compounding more often increases the effective annual return (APY). At 12% nominal, monthly is about 12.68% APY and daily is about 12.75% APY.
Why is amortized total interest smaller?
Because each payment reduces principal, so subsequent interest applies to a smaller balance.
APR vs APY—what should I compare?
Use APR for loans and APY for deposit accounts. APR standardizes borrowing cost; APY includes compounding frequency.
Can simple-interest loans penalize late payments?
Yes. Many compute interest daily; paying late adds days of interest and shifts more of your payment to interest.
Should I always pick the lowest total interest?
Usually—but consider cash flow needs, fees, and flexibility. A slightly higher cost may bring features you need.
Final Takeaways
- Simple interest is straightforward but costly if you defer repayment.
- Compound interest accelerates growth for savers and increases costs on unpaid balances.
- Amortization typically minimizes total interest for borrowers by reducing principal throughout the term.