Financial Math Made Simple

Most financial decisions require some math, and most people avoid that math by either guessing or delegating. Neither is necessary. The concepts underlying personal finance math are not complicated — they are just rarely explained in terms that connect to actual decisions. Here are the calculations that matter most.

The Rule of 72: How Long Until Your Money Doubles?

The Rule of 72 is the single most useful financial calculation shortcut you can memorize. Divide 72 by an annual interest rate to get the approximate number of years it takes for money to double.

At 6% annual return — approximately what diversified stock index funds have averaged after inflation historically — your money doubles in 72 ÷ 6 = 12 years. At 8%, it doubles in 9 years. At 3% (roughly the current high-yield savings rate environment), your money doubles in 24 years.

The rule also works for debt. If you carry credit card debt at 24% annual interest, the balance doubles roughly every 3 years if you make no payments. That framing makes the urgency of high-interest debt visceral in a way that abstract interest rate numbers do not.

A companion insight: the rule compounds. If your money doubles every 9 years, then over 36 years it doubles four times: $10,000 becomes $20,000, then $40,000, then $80,000, then $160,000. Starting early is not just advisable — it is the dominant variable. Getting 30 years of compounding at 8% is worth significantly more than getting 20 years of compounding at 10%.

Understanding Compound Interest

Compound interest means you earn returns on your returns, not just on your original principal. The formula is A = P(1 + r/n)^(nt), where P is principal, r is annual interest rate, n is compounding frequency per year, and t is time in years.

In practice, the frequency of compounding matters less than people think for most personal finance decisions. Daily compounding vs. annual compounding on $10,000 at 5% over 10 years produces a difference of about $90 — not nothing, but far less important than the rate itself or the time horizon.

What does matter is the difference between earning compound interest (savings and investments) and paying compound interest (debt). The same mechanism that grows your wealth through investing compounds against you in debt situations. Understanding this asymmetry motivates paying down high-interest debt aggressively and investing as early as possible.

Estimating Monthly Loan Payments

The precise mortgage payment formula requires a calculator, but there is a useful approximation. For a 30-year fixed mortgage at around 7% interest: the monthly payment is approximately $6.65 per $1,000 borrowed. Borrow $400,000: payment is roughly $2,660 per month.

At 6%: about $6.00 per $1,000. Borrow $400,000: roughly $2,400 per month. At 8%: about $7.34 per $1,000, giving $2,936 per month for the same loan.

For car loans at typical 5-year (60-month) terms around 6% interest: the payment is approximately $19.33 per $1,000 borrowed. A $30,000 car loan: roughly $580 per month.

These are approximations, not exact calculations — for actual loan commitments, use a precise calculator. But these rough numbers let you quickly evaluate whether a proposed payment is in the right ballpark without plugging numbers into formulas.

APR vs. APY: The Number That Actually Matters

APR (Annual Percentage Rate) and APY (Annual Percentage Yield) represent the same interest from different angles. APR does not include compounding effects within the year. APY does.

When you are borrowing money, lenders are required to disclose the APR, which is the more meaningful comparison point for loans. When you are saving money, banks advertise the APY, which reflects the actual annual yield after compounding.

The difference is most significant at higher interest rates and when compounding is frequent. At 12% APR compounded monthly, the APY is 12.68% — not dramatically different for a savings account, but meaningful when evaluating credit card offers where the "APR" is the only disclosed rate.

For deposits, always compare APYs, not APRs. For loans, compare APRs (which must legally be disclosed) and be skeptical of any offer that leads with a rate that turns out to be monthly rather than annual.

The Rent vs. Buy Ratio

The price-to-rent ratio is a quick way to assess whether buying or renting makes more financial sense in a given housing market. Divide the purchase price of a home by the annual rent for a comparable property. A ratio below 15 suggests buying is likely favorable; above 20 suggests renting is likely favorable; between 15 and 20 is context-dependent.

A house priced at $400,000 in a market where comparable rentals go for $2,500 per month ($30,000 per year) has a price-to-rent ratio of about 13.3 — suggesting ownership may make financial sense. The same house price in a market where rentals are $1,500 per month ($18,000 per year) has a ratio of 22.2 — renting is probably more financially sensible until you have high confidence about staying for many years.

This ratio does not account for mortgage rates, tax deductions, or opportunity cost — factors that can shift the calculation meaningfully. But it provides a quick directional signal that is more useful than either raw price or rent alone.