Finance Formulas

Money has its own short list of formulas. A few say how savings and debts grow over time, a few say what a future payment is worth today, a few turn a company's accounts into ratios you can compare, and a few weigh return against risk. Each page here works one example with plain numbers, so you can check a loan offer or a savings plan on a calculator.

40 formulas, in the order that makes them easiest to learn:

Simple Interest

$$I = P r t$$

Simple interest says the interest is the starting amount times the yearly rate times the number of years, and interest never earns interest of its own.

Compound Interest

$$A = P \left(1 + \frac{r}{n}\right)^{n t}$$

Compound interest says a savings balance grows faster over time, because each round of interest is added to the balance and then earns interest of its own.

Continuous Compounding

$$A = P e^{r t}$$

Continuous compounding says money grows as if interest were added every instant, and the number e, about 2.718, sets the size of that growth.

Rule of 72

$$t \approx \frac{72}{R}$$

The rule of 72 says that dividing 72 by a yearly percentage rate gives a quick estimate of the number of years it takes money to double.

Effective Annual Rate

$$\text{EAR} = \left(1 + \frac{r}{n}\right)^{n} - 1$$

The effective annual rate says what a stated yearly rate really adds up to over a year, once interest is added more than once and earns interest of its own.

Present Value

$$PV = \frac{FV}{(1 + r)^{n}}$$

Present value says how much money you must put aside today, at a steady interest rate, so that it grows to a known amount by a later date.

Future Value of an Annuity

$$FV = P \cdot \frac{(1 + r)^{n} - 1}{r}$$

The future value of an annuity says how much a row of equal payments adds up to by the end, once the interest earned on each payment is included.

Present Value of an Annuity

$$PV = P \cdot \frac{1 - (1 + r)^{-n}}{r}$$

The present value of an annuity says how much money you need today to pay a fixed amount every period for a set number of periods, with interest working in the background.

Loan Payment

$$M = P \cdot \frac{r (1 + r)^{n}}{(1 + r)^{n} - 1}$$

The loan payment formula gives the fixed amount you pay each month so that a loan is fully repaid, interest included, by the last payment.

Perpetuity

$$PV = \frac{C}{r}$$

The perpetuity formula gives the amount of money needed today to pay a fixed amount every period forever, using only the payment and the interest rate.

Gordon Growth Model

$$P_0 = \frac{D_1}{r - g}$$

The Gordon growth model says a share is worth its next dividend divided by the gap between the return you want and the rate the dividend grows.

Net Present Value (NPV)

$$\text{NPV} = \sum_{t=0}^{n} \frac{C_t}{(1 + r)^{t}}$$

Net present value says whether a plan is worth its cost: it adds every future cash flow, shrunk back to today’s money, and the total is the plan’s value in dollars today.

Internal Rate of Return (IRR)

$$0 = \sum_{t=0}^{n} \frac{C_t}{(1 + r)^{t}}$$

The internal rate of return is the yearly rate a plan earns on its money: the single rate at which all of its cash flows exactly balance in today’s money.

Return on Investment (ROI)

$$\text{ROI} = \frac{\text{gain} - \text{cost}}{\text{cost}}$$

Return on investment says how much profit you made for each dollar you spent, so you can compare a small investment with a large one.

Compound Annual Growth Rate (CAGR)

$$\text{CAGR} = \left(\frac{V_{\text{end}}}{V_{\text{start}}}\right)^{1/n} - 1$$

CAGR says how fast something grew each year, as if it grew by the same share every year, from its starting value to its ending value.

Fisher Equation

$$(1 + i) = (1 + r)(1 + \pi)$$

The Fisher equation says the real growth of your money is what is left of your interest after prices have risen.

Inflation Rate

$$\pi = \frac{\text{CPI}_{t} - \text{CPI}_{t-1}}{\text{CPI}_{t-1}}$$

The inflation rate says how much more prices rose this year than last, as a share of last year’s prices, measured with a price index.

Accounting Equation

$$\text{Assets} = \text{Liabilities} + \text{Equity}$$

The accounting equation says everything a business owns equals what it owes to others plus what belongs to its owners.

Break-Even Point

$$Q = \frac{F}{P - V}$$

The break-even point says how many items you must sell before the money from sales has paid back all your fixed costs, so you stop losing money.

Net Profit Margin

$$\text{Net profit margin} = \frac{\text{Net income}}{\text{Revenue}}$$

Net profit margin says how many cents of profit a business keeps from each dollar of sales, after every cost has been paid.

Markup

$$\text{Markup} = \frac{\text{Price} - \text{Cost}}{\text{Cost}}$$

Markup says how much a seller adds on top of what an item cost, measured as a share of that cost.

Straight-Line Depreciation

$$D = \frac{C - S}{L}$$

Straight-line depreciation spreads the cost of an item you use for years into equal yearly amounts of expense, so the whole cost is not counted in the year you buy it.

Earnings Per Share

$$\text{EPS} = \frac{\text{Net income} - \text{Preferred dividends}}{\text{Weighted average shares outstanding}}$$

EPS says how much profit a company earned for each share of its stock, after the preferred shareholders have been paid their dividends.

Price-to-Earnings Ratio

$$\text{P/E} = \frac{\text{Price per share}}{\text{Earnings per share}}$$

The P/E ratio says how many dollars investors pay today for each dollar of a company’s yearly earnings per share.

Dividend Yield

$$\text{Dividend yield} = \frac{\text{Annual dividends per share}}{\text{Price per share}}$$

Dividend yield says how much cash a company pays out each year, as a percentage of the price you would pay for one share.

Current Ratio

$$\text{Current ratio} = \frac{\text{Current assets}}{\text{Current liabilities}}$$

The current ratio tells you whether a business has enough short-term assets to cover its short-term debts, by dividing one total by the other.

Debt-to-Equity Ratio

$$\text{D/E} = \frac{\text{Total liabilities}}{\text{Shareholders' equity}}$$

The debt-to-equity ratio shows how much a business is funded by borrowing compared with money from its owners: a higher number means more debt.

Return on Equity (ROE)

$$\text{ROE} = \frac{\text{Net income}}{\text{Shareholders' equity}}$$

Return on equity tells you how many cents of yearly profit a business earns for each dollar its owners have in it.

Debt-to-Income Ratio (DTI)

$$\text{DTI} = \frac{\text{Monthly debt payments}}{\text{Gross monthly income}}$$

The debt-to-income ratio shows what share of a household’s monthly income, before tax, goes to paying debts each month.

Weighted Average Cost of Capital (WACC)

$$\text{WACC} = \frac{E}{V} R_e + \frac{D}{V} R_d (1 - T_c)$$

WACC is the average yearly price a company pays for all the money it uses, mixing the owners’ price and the lenders’ price by how much of each it has.

Capital Asset Pricing Model (CAPM)

$$E(R_i) = R_f + \beta_i \left(E(R_m) - R_f\right)$$

CAPM says the return you should expect from an investment is a safe rate plus extra return for the market risk it carries, scaled by its beta.

Beta

$$\beta = \frac{\operatorname{Cov}(R_i, R_m)}{\operatorname{Var}(R_m)}$$

Beta says how much a stock tends to move when the whole market moves: 2 means it moves about twice as far as the market, and 0.5 means about half as far.

Expected Return of a Portfolio

$$E(R_p) = \sum_{i=1}^{n} w_i \, E(R_i)$$

The expected return of a portfolio is the average of its parts’ expected returns, where each part counts in proportion to how much of the money it holds.

Portfolio Variance (Two Assets)

$$\sigma_p^2 = w_1^2 \sigma_1^2 + w_2^2 \sigma_2^2 + 2 w_1 w_2 \rho_{12} \sigma_1 \sigma_2$$

Portfolio variance says how much a two-asset mix’s risk comes from each part and from how the parts move together, and it shows that mixing assets can lower risk.

Volatility (Standard Deviation of Returns)

$$\sigma = \sqrt{\frac{1}{N - 1} \sum_{i=1}^{N} (R_i - \bar{R})^2}$$

Volatility says how far an investment’s returns usually land above and below their average, so a bigger number means a bumpier ride.

Sharpe Ratio

$$S = \frac{R_p - R_f}{\sigma_p}$$

The Sharpe ratio says how much extra return you earned for each unit of ups and downs, called risk, that you accepted to earn it.

Bond Price

$$P = \sum_{t=1}^{n} \frac{C}{(1 + y)^{t}} + \frac{F}{(1 + y)^{n}}$$

The bond price formula adds up what each future payment from a bond is worth today, and a higher market yield makes every one of those payments worth less.

Macaulay Duration

$$D = \frac{1}{P} \sum_{t=1}^{n} \frac{t \, CF_t}{(1 + y)^{t}}$$

Macaulay duration is the average wait, in years, until a bond pays you back, where each payment counts according to how much it is worth today.

Black-Scholes Formula for a Call Option

$$C = S_0 N(d_1) - K e^{-r T} N(d_2)$$

The Black-Scholes formula works out what the right to buy a stock later is worth today, using the stock price, the agreed price, the interest rate, the time left and the stock’s ups and downs.

Put-Call Parity

$$C - P = S_0 - K e^{-r T}$$

Put-call parity says that a call and a put on the same stock, with the same strike and expiry, are priced so that their gap equals the stock price minus the discounted strike.

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