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Interest, loans and time value of money, one page per formula

The interest, loan, savings and investment formulas people search for, each with the direct answer, the formula, a worked example with exact numbers, the Excel function, the mistakes that produce the wrong payment, and the Ratedrill problems that practice it.

40 formulas, each with the direct answer, syntax, worked examples, the mistakes people make and the drills that practice it. Updated November 2026.

Interest basics

  • Simple interestSimple interest is principal times rate times time (I = P x r x t). Worked examples for years, months and days, the Excel formula and the mistakes to avoid.
  • Compound interestThe compound interest formula A = P(1 + r/n)^(nt), what each symbol means, examples for yearly and monthly compounding, the Excel FV function and errors.
  • Future valueHow to calculate the future value of a lump sum with FV = PV x (1 + r)^n, step-by-step examples, the Excel FV function with its sign convention, and errors.
  • Present valueHow to calculate present value with PV = FV / (1 + r)^n, what discounting means, examples with yearly and quarterly rates, the Excel PV function and mistakes.
  • Rule of 72The rule of 72 estimates doubling time as 72 divided by the rate. Examples at 6%, 9% and 18%, how accurate it is, the exact ln(2) formula and Excel NPER.
  • APR vs APYAPR is the yearly rate without compounding; APY includes it. Convert with APY = (1 + APR/n)^n - 1, worked examples and the Excel EFFECT and NOMINAL functions.
  • Effective annual rateThe effective annual rate turns a nominal rate with monthly, quarterly or daily compounding into the true yearly rate: (1 + i/m)^m - 1. Examples and Excel.
  • Continuous compoundingContinuous compounding uses A = P x e^(rt), the limit of compounding infinitely often. Examples, the effective rate e^r - 1, doubling time and Excel EXP.

Annuities and savings

  • Annuity formula (future value)The future value of an ordinary annuity is PMT x ((1 + r)^n - 1) / r. Examples for yearly and monthly deposits, the Excel FV function and common errors.
  • Present value of an annuityHow to find the present value of equal payments with PV = PMT x (1 - (1 + r)^-n) / r. Worked examples, Excel PV and why this is the loan formula in disguise.
  • Annuity dueAn annuity due has payments at the start of each period, so it is worth the ordinary annuity value times (1 + r). PV and FV examples, Excel type = 1, errors.
  • PerpetuityA perpetuity pays a fixed amount forever and is worth PV = C / r. The growing perpetuity formula C1 / (r - g), examples for preferred stock and dividends.
  • Growing annuityThe present value of a growing annuity is C1 / (r - g) x (1 - ((1 + g) / (1 + r))^n). Examples for rising rents, the special case r = g, and mistakes.
  • Monthly deposit for a savings goalThe monthly deposit needed to reach a target is PMT = FV x r / ((1 + r)^n - 1). Examples for 10,000 in 5 years and 50,000 in 10 years, Excel PMT and errors.
  • How long savings will lastHow many months a balance lasts with fixed withdrawals: n = -ln(1 - PV x r / PMT) / ln(1 + r). Retirement examples, Excel NPER and the withdrawal that lasts.

Loans and mortgages

  • Mortgage payment formulaHow to calculate a monthly mortgage payment with M = P x r / (1 - (1 + r)^-n). Examples for 200,000 at 6% and 300,000 at 5%, Excel PMT and common mistakes.
  • Car loan paymentHow to calculate a car loan payment with PMT = P x r / (1 - (1 + r)^-n). Examples for 20,000 at 7% over 60 months and 15,000 at 5%, total interest, Excel.
  • Amortization scheduleAn amortization schedule splits each payment into interest and principal and tracks the balance. Worked schedule for a 10,000 loan, Excel IPMT and PPMT.
  • Remaining loan balanceThe balance after k payments is the present value of the payments still to come: B_k = PMT x (1 - (1 + r)^-(n-k)) / r. Mortgage and car loan examples, Excel.
  • Total interest paid on a loanTotal interest on an amortizing loan is the payment times the number of payments minus the principal. Examples for 30-year and 15-year loans, Excel CUMIPMT.
  • Extra payments and payoff timePaying extra each month shortens the loan: n = -ln(1 - P x r / PMT) / ln(1 + r) with the higher payment. Example: 200 extra on 200,000 saves 9 years, 79,800.
  • Balloon paymentA balloon loan has smaller regular payments and a large final lump sum. How to find the payment given a balloon, or the balloon given a payment, with Excel.
  • How much can I borrowThe loan you can afford for a given payment is the present value of an annuity: P = PMT x (1 - (1 + r)^-n) / r. Mortgage and car loan examples and Excel PV.

Investment appraisal

  • NPV (net present value)Net present value is the sum of discounted cash flows minus the initial investment. Examples at 10% and 8%, the accept rule, Excel NPV and the sign mistake.
  • IRR (internal rate of return)The IRR is the discount rate that makes NPV zero. How to find it by trial, what Excel IRR returns, worked examples, the accept rule and where IRR fails.
  • Payback periodPayback period is the time it takes cumulative cash inflows to equal the initial investment. Formula for even and uneven cash flows, examples, interpolation.
  • Discounted payback periodThe discounted payback period counts the years until the present value of inflows recovers the investment. Examples at 10% and 8% versus simple payback.
  • Profitability indexThe profitability index is the present value of future cash flows divided by the initial investment, or 1 + NPV / investment. Examples and the accept rule.

Bonds, returns and business math

  • Bond priceA bond price is the present value of its coupons plus the present value of its face value at the yield. Annual and semiannual examples, premium and discount.
  • Yield to maturity (YTM)YTM is the discount rate that makes a bond price equal the present value of its cash flows. The approximation formula, examples, Excel RATE and YIELD, errors.
  • Current yieldCurrent yield is the bond annual coupon divided by its current price. Examples for discount and premium bonds, how it compares with YTM and mistakes to avoid.
  • Holding period returnHolding period return is (ending value - beginning value + income) / beginning value. Stock and bond examples, how to annualize an HPR and common mistakes.
  • CAGRCAGR = (ending value / beginning value)^(1/years) - 1. Examples for investments and revenue growth, Excel RRI, and why CAGR differs from the average return.
  • Real vs nominal returnThe real return strips inflation out of a nominal return: real = (1 + nominal) / (1 + inflation) - 1. Examples, the quick approximation and its errors.
  • WACCWACC = E/V x Re + D/V x Rd x (1 - T): the blended after-tax cost of a company equity and debt. Step-by-step examples, where each input comes from, errors.
  • After-tax cost of debtThe after-tax cost of debt is the interest rate times one minus the tax rate, because interest is deductible. Examples, finding the pre-tax rate, mistakes.
  • Break-even pointBreak-even units = fixed costs / (price - variable cost per unit); break-even revenue = fixed costs / contribution margin ratio. Examples, margin of safety.
  • Margin vs markupMargin is profit divided by price; markup is profit divided by cost. How to convert between them, set a price from a target margin, examples and the error.
  • Cost of trade creditSkipping an early payment discount such as 2/10 net 30 is an expensive loan: annual cost = d / (1 - d) x 365 / extra days, about 37%. Examples, when to pay.
  • Depreciation methodsThree depreciation methods with worked examples on the same 10,000 asset: straight-line, double-declining balance and sum-of-the-years-digits, plus Excel.

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