Finance Formula Handbook: Compound Interest, Mortgage, APR, and More
Personal finance runs on a surprisingly small set of mathematical formulas. Compound interest, mortgage amortization, future value, and the rule of 72 appear in credit card statements, investment accounts, mortgage disclosures, and savings projections. Understanding these formulas — not just plugging into a calculator — gives you the ability to sanity-check projections, compare offers, and make better decisions.
This handbook explains each core formula in plain language, shows you how to apply it with worked examples, and links to calculators that do the arithmetic for you. The goal is conceptual fluency: knowing what the formula is measuring and why the answer makes sense.
Simple Interest vs. Compound Interest
Simple interest grows linearly: Interest = Principal × Rate × Time. A $1,000 loan at 5% simple interest for 3 years accrues $150 interest (= 1000 × 0.05 × 3). Simple interest appears in short-term loans and some bonds where interest is paid out rather than reinvested.
Compound interest grows exponentially because interest earned in each period is added to the principal, which then earns interest in subsequent periods. Einstein is often (apocryphally) quoted calling compound interest 'the eighth wonder of the world.' The difference is dramatic over long time horizons: $10,000 at 7% simple interest for 30 years grows to $31,000. At 7% compounding annually, it grows to $76,123.
Compound Interest Formula
A = P × (1 + r/n)^(n×t)
A = final amount (principal + interest)
P = principal (starting amount)
r = annual interest rate (decimal: 6% → 0.06)
n = compounding periods per year
(1 = annual, 12 = monthly, 365 = daily)
t = time in years
Example: $5,000 at 6% compounded monthly for 10 years:
A = 5000 × (1 + 0.06/12)^(12×10)
A = 5000 × (1.005)^120
A = 5000 × 1.8194
A = $9,096.98
With continuous compounding: A = P × e^(r×t)
Same example: A = 5000 × e^(0.06×10) = 5000 × 1.8221 = $9,110.60Mortgage Payment Formula
M = P × [r(1+r)^n] / [(1+r)^n − 1] M = monthly payment P = loan principal r = monthly interest rate (annual rate ÷ 12) n = total number of payments (years × 12) Example: $300,000 mortgage, 7% annual rate, 30 years: r = 0.07 / 12 = 0.005833 n = 30 × 12 = 360 M = 300,000 × [0.005833 × (1.005833)^360] / [(1.005833)^360 − 1] M = 300,000 × [0.005833 × 8.1164] / [8.1164 − 1] M = 300,000 × 0.047348 / 7.1164 M ≈ $1,995.91 per month Total paid: $718,527 | Total interest: $418,527
APR vs. APY — Annual Percentage Rate vs. Annual Percentage Yield
APR (Annual Percentage Rate) is the yearly cost of borrowing money, expressed as a percentage, without accounting for compounding within the year. For mortgages and auto loans, federal law requires lenders to disclose the APR, which includes the interest rate plus fees, giving a more accurate cost comparison than the base interest rate alone.
APY (Annual Percentage Yield) accounts for the effect of compounding within the year. The formula is APY = (1 + APR/n)^n − 1, where n is the number of compounding periods per year. For a savings account paying 5% APR compounded monthly: APY = (1 + 0.05/12)^12 − 1 = 5.116%. Banks advertise APY for savings accounts (higher sounds better) and APR for loans (lower sounds better) — know which you are reading.
Future Value and Present Value
━━━ FUTURE VALUE (lump sum) ━━━ FV = PV × (1 + r)^t ($10,000 invested at 8% for 20 years: FV = 10,000 × (1.08)^20 = $46,610) ━━━ FUTURE VALUE (regular contributions) ━━━ FV = PMT × [(1+r)^t − 1] / r ($500/month at 7% annual (0.5833%/mo) for 30 years: FV = 500 × [(1.005833)^360 − 1] / 0.005833 = $567,764) ━━━ PRESENT VALUE ━━━ PV = FV / (1 + r)^t (PV of receiving $100,000 in 10 years at 6% discount: PV = 100,000 / (1.06)^10 = $55,839) ━━━ RULE OF 72 ━━━ Years to double = 72 / annual rate (%) At 6%: 72/6 = 12 years to double At 9%: 72/9 = 8 years to double At 12%: 72/12 = 6 years to double
Retirement Savings: How Much Do You Need?
The most common retirement benchmark is the '4% rule': in retirement, you can sustainably withdraw 4% of your portfolio per year without running out of money over a 30-year retirement. To find your target: divide your desired annual spending by 0.04. If you plan to spend $60,000/year, you need $60,000 / 0.04 = $1,500,000 in your portfolio at retirement.
To estimate whether you will reach that target: use the future value of regular contributions formula. If you have 25 years, invest $1,000 per month, and earn 7% annually, your future value is approximately $810,000 — below the $1.5M target. Increasing contributions to $1,850/month at the same rate reaches $1.5M. The earlier you start, the less you need to contribute monthly because time multiplies compounding dramatically.
CAGR — Compound Annual Growth Rate
CAGR measures the mean annual growth rate of an investment over a period longer than one year, assuming reinvestment. Formula: CAGR = (Ending Value / Beginning Value)^(1/years) − 1. If an investment grew from $10,000 to $18,000 over 6 years: CAGR = (18,000/10,000)^(1/6) − 1 = (1.8)^0.1667 − 1 = 1.1027 − 1 = 10.27%.
CAGR smooths out year-to-year volatility, giving a single growth rate that represents the steady growth needed to reach the same outcome. It is widely used to compare fund performance, business revenue growth, and investment returns. Note that CAGR does not reflect the actual year-by-year volatility — a fund with 10% CAGR may have had years of −20% and +40%.
Quick Tips
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The rule of 72 works in reverse too: at 3% inflation, prices double in 72/3 = 24 years. That $1 cup of coffee becomes $2 in 24 years.
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When comparing mortgage rates, compare APRs (which include fees), not just the quoted interest rates. A lower interest rate with high origination fees can cost more over the loan term.
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Compounding frequency matters most at higher rates. At 5%, daily vs. annual compounding adds less than 0.1% to APY. At 20% (credit card rates), the difference is significant.
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Max out tax-advantaged accounts (401k, IRA, HSA) before taxable accounts — the tax savings compound alongside the returns.
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A 1% difference in investment returns is worth far more than most people expect. $100,000 at 6% for 30 years = $574,349. At 7% = $761,226. That 1% difference is worth $187,000.
Frequently Asked Questions
What is compound interest?
Compound interest is interest calculated on both the initial principal and the accumulated interest from prior periods. Unlike simple interest (calculated only on principal), compound interest accelerates growth because each period's interest is added to the base that earns interest next period.
How is a mortgage payment calculated?
Using the annuity formula: M = P[r(1+r)^n]/[(1+r)^n − 1], where P is the loan amount, r is the monthly interest rate (annual rate ÷ 12), and n is the number of monthly payments. The formula distributes equal payments so that the loan is paid off exactly at the last payment.
What is the rule of 72?
A mental shortcut: divide 72 by the annual interest rate (as a percentage) to estimate how many years it takes to double your money. At 8%: 72/8 = 9 years. At 6%: 72/6 = 12 years. It is accurate within a few percent for rates between 2% and 20%.
What is the difference between APR and APY?
APR (Annual Percentage Rate) does not account for compounding within the year. APY (Annual Percentage Yield) does. For a savings account, APY shows what you actually earn after compounding. For a loan, APR includes fees in addition to the interest rate. APY > APR when compounding frequency > 1 per year.
How much should I save for retirement?
The common benchmark: save 15% of gross income throughout your career, targeting a portfolio of 25× your expected annual retirement spending (the 4% withdrawal rule). The earlier you start, the less percentage you need to save because compound growth does more work over a longer horizon.
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