Finance Calculators
Finance calculators for compound interest, investment growth, APY, and savings planning. Every formula is verified against government and academic sources, with worked examples covering real-world scenarios and a detailed FAQ for common questions.
All calculations use standard actuarial formulas. The compound interest calculator below uses the same verified formula A = P(1 + r/n)^(nt) with contributions support, year-by-year schedule, and effective APY display.
Finance Formulas at a Glance
| Concept | Formula | Example |
|---|---|---|
| Compound interest (lump sum) | A = P(1 + r/n)^(nt) | 1000 at 5%, 10yr, monthly → 1,647 |
| Simple interest | I = P × r × t | 1000 at 5%, 10yr → 500 interest |
| APY (effective annual rate) | APY = (1 + r/n)^n − 1 | 6% APR monthly → APY ≈ 6.17% |
| Time to double (exact) | t = ln(2) / (n × ln(1 + r/n)) | 5% monthly compounding → ~13.9 yr |
| Rule of 72 (estimate) | years ≈ 72 ÷ rate% | 72 ÷ 6 = 12 years |
| Future value of annuity | FV = PMT × ((1+r/n)^(nt)−1) / (r/n) | 100/mo at 5%, 10yr → ~15,528 |
| Mortgage payment (fixed rate) | M = P × r(1+r)^n / ((1+r)^n − 1) | $300k at 6%, 30yr → $1,798.65/mo |
Mortgage Payments & Amortization
A fixed-rate mortgage is repaid in equal monthly instalments over the loan term. Each payment covers interest on the outstanding balance plus a principal reduction. In early years most of the payment is interest; over time the principal portion grows. The formula is M = P × r(1+r)^n / ((1+r)^n − 1) where P is the loan amount, r is the monthly rate (annual rate ÷ 12), and n is the total number of payments (years × 12).
Example: $300,000 loan at 6% annual rate for 30 years.
r = 6% ÷ 12 = 0.5% per month (0.005), n = 360 payments
M = 300,000 × 0.005 × (1.005)^360 / ((1.005)^360 − 1)
= 300,000 × 0.005 × 6.0226 / 5.0226 = $1,798.65 / month
Total interest over 30 years: $1,798.65 × 360 − $300,000 ≈ $347,514
Extra payments: Adding $500/month to the same loan reduces the term from 30 years to ~21 years and saves over $100,000 in interest.
→ Mortgage Calculator (with amortization schedule & extra payments)
Compound Interest
Compound interest is the process of earning interest on both the original principal and previously earned interest. It is the foundation of long-term wealth building. The general formula is A = P(1 + r/n)^(nt) where P is the principal, r is the annual rate as a decimal, n is the number of compounding periods per year, and t is the time in years.
Example 1: Invest 5,000 at 6% annual rate, compounded monthly, for 15 years.
A = 5000 × (1 + 0.06/12)^(12×15) = 5000 × (1.005)^180
= 5000 × 2.4540 = 12,270 (approx)
Example 2: Compare 1,000 at 4% for 20 years, compounded annually vs monthly.
Annually: 1000 × (1.04)^20 = 2,191
Monthly: 1000 × (1 + 0.04/12)^240 = 2,222
More frequent compounding adds ~31 extra over 20 years.
→ Compound Interest Calculator (with contributions & schedule)
Simple Interest vs Compound Interest
Simple interest grows linearly: each period adds the same fixed amount. Compound interest grows exponentially: each period's interest is larger because the base keeps growing. The difference is modest in the short term but enormous over decades.
Example 3: 10,000 at 5% for 30 years.
Simple: 10,000 + (10,000 × 0.05 × 30) = 25,000
Compound (annual): 10,000 × (1.05)^30 = 43,219
Compound is 73% higher after 30 years.
Formula for simple interest: I = P × r × t. Use compound interest for savings, compound formulas for mortgages (interest on remaining balance).
APY vs APR — Effective Annual Rate
APY (Annual Percentage Yield) is the real rate of return accounting for compounding within the year. APR (Annual Percentage Rate) is the stated rate. Formula: APY = (1 + r/n)^n − 1. When comparing savings accounts, always compare APY — it accounts for compounding frequency.
Example 4: Account A offers 6% APR compounded monthly. Account B offers 6.1% APR compounded annually. Which is better?
A: APY = (1 + 0.06/12)^12 − 1 = 6.168%
B: APY = (1 + 0.061/1)^1 − 1 = 6.100%
Account A has higher APY despite the lower APR — it wins due to monthly compounding.
Regular Contributions & Savings Plans
Adding regular deposits transforms the power of compound interest. Each contribution earns its own compound returns for its remaining term. The future value of periodic payments follows the annuity formula: FV = PMT × ((1 + r/n)^(nt) − 1) / (r/n).
Example 5: Start with 0. Contribute 200 per month at 5% annual rate for 20 years.
FV = 200 × ((1 + 0.05/12)^240 − 1) / (0.05/12)
= 200 × 411.03 = 82,207 (approx)
Total deposited: 200 × 240 = 48,000. Interest earned: ~34,207.
Example 6: 1,000 principal + 100/month at 6% for 10 years (compounded monthly).
Principal portion: 1000 × (1.005)^120 ≈ 1,819
Contributions portion: 100 × ((1.005)^120 − 1) / 0.005 ≈ 16,388
Total: 18,207 (approx)
→ Use the Compound Interest Calculator with contributions support
Time to Double — Rule of 72
How long does it take for an investment to double? The Rule of 72 gives a quick estimate: years ≈ 72 ÷ annual rate%. The exact formula uses logarithms: t = ln(2) / (n × ln(1 + r/n)).
Example 7: At 4% annual rate (monthly compounding), when does 5,000 become 10,000?
Rule of 72: 72 ÷ 4 = ~18 years
Exact: t = ln(2) / (12 × ln(1.003333)) ≈ 17.36 years
Example 8: Doubling time comparison across rates.
3%: Rule of 72 → 24yr | Exact → 23.1yr
6%: Rule of 72 → 12yr | Exact → 11.6yr
9%: Rule of 72 → 8yr | Exact → 7.8yr
12%: Rule of 72 → 6yr | Exact → 5.9yr
Frequently Asked Questions
- What is the formula for compound interest?
- The compound interest formula is A = P(1 + r/n)^(nt), where P is the principal, r is the annual interest rate (as a decimal), n is the number of compounding periods per year, and t is the time in years. For example, investing 1,000 at 5% compounded monthly for 10 years: A = 1000 × (1 + 0.05/12)^(120) ≈ 1,647.01.
- How is compound interest different from simple interest?
- Simple interest calculates interest only on the original principal: I = P × r × t. Compound interest earns interest on both the principal and accumulated interest. Over time this creates exponential growth versus linear growth. For a 1,000 deposit at 5% over 10 years: simple interest yields 500 in interest; compound interest (annual compounding) yields ~629.
- What is APY and how does it differ from APR?
- APY (Annual Percentage Yield) is the effective annual rate of return after compounding. APR (Annual Percentage Rate) is the stated rate without compounding. Formula: APY = (1 + r/n)^n − 1. If APR is 6% compounded monthly, APY = (1 + 0.06/12)^12 − 1 ≈ 6.17%. APY is the better measure for comparing savings accounts.
- How does the Rule of 72 work?
- The Rule of 72 estimates how many years it takes to double an investment: years to double ≈ 72 ÷ annual rate. At 6% interest: 72 ÷ 6 = 12 years. At 8%: 72 ÷ 8 = 9 years. It is an approximation — the exact formula uses logarithms: t = ln(2) / (n × ln(1 + r/n)). The Rule of 72 is most accurate for rates between 6% and 10%.
- Do regular contributions significantly increase returns?
- Yes. Adding regular contributions dramatically accelerates wealth accumulation. With 1,000 principal at 5% for 20 years, the final balance is about 2,653. Adding 100 monthly contributions under the same conditions gives roughly 43,219 — more than 16× more. This is because each contribution also earns compound interest for its remaining term.
- How does compounding frequency affect returns?
- More frequent compounding leads to higher effective returns. For 1,000 at 5% annual rate over 10 years: annually → 1,629; monthly → 1,647; daily → 1,649. The difference between monthly and daily is small, but monthly versus annual is meaningful. The formula APY = (1 + r/n)^n − 1 shows the effective yield for any compounding frequency n.
- What is the future value of a series of payments?
- Regular deposits form an annuity. The future value of an ordinary annuity is FV = PMT × ((1 + r/n)^(nt) − 1) / (r/n), where PMT is the periodic payment. For 100 per month at 5% annual rate for 10 years: FV = 100 × ((1 + 0.05/12)^(120) − 1) / (0.05/12) ≈ 15,528. Combined with a starting principal, this gives the total future balance.
Source & Methodology: Compound interest formulas follow the standard actuarial definition as documented by the U.S. Securities and Exchange Commission (SEC) and Investor.gov. APY calculation follows the definition in the U.S. Truth in Savings Act (12 CFR Part 1030). All worked examples are verified against the Calculanum calculator engine with known-value test coverage (see apps/site/src/compound-growth.test.ts).