Every formula, from first principles.

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

Effective Annual Rate (EAR)

EAR = \left(1 + \frac{r}{m}\right)^{m} - 1

Continuous Compounding

FV = PV \cdot e^{r t}

Future Value of an Ordinary Annuity

FV = PMT \times \frac{(1+r)^n - 1}{r}

Present Value of an Ordinary Annuity

PV = PMT \times \frac{1 - (1+r)^{-n}}{r}

Future Value of an Annuity Due

FV_{due} = PMT \times \frac{(1+r)^n - 1}{r} \times (1+r)

Present Value of a Perpetuity

PV = \frac{PMT}{r}

Growing Perpetuity (Gordon Growth)

PV = \frac{PMT_1}{r - g}

Net Present Value (NPV)

NPV = -CF_0 + \sum_{t=1}^{n} \frac{CF_t}{(1+r)^t}

Quantitative Methods

Returns, dispersion, probability and the statistics that describe risk.

Holding Period Return (HPR)

HPR = \frac{P_1 - P_0 + D}{P_0}

Arithmetic Mean Return

\bar{R} = \frac{1}{n}\sum_{i=1}^{n} R_i

Geometric Mean Return

R_G = \left[\prod_{i=1}^{n}(1+R_i)\right]^{1/n} - 1

Weighted Mean (Portfolio Return)

\bar{R}_w = \sum_{i=1}^{n} w_i R_i, \qquad \sum w_i = 1

Sample Standard Deviation

s = \sqrt{\frac{\sum_{i=1}^{n}(R_i - \bar{R})^2}{n-1}}

Coefficient of Variation (CV)

CV = \frac{s}{\bar{X}}

Sharpe Ratio

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

Roy's Safety-First Ratio

SF = \frac{R_p - R_L}{\sigma_p}

Correlation from Covariance

\rho_{AB} = \frac{Cov_{AB}}{\sigma_A \sigma_B}

Expected Value

E(X) = \sum_{i=1}^{n} p_i X_i

Total Probability Rule

P(B) = P(B\mid A)\,P(A) + P(B\mid A^{c})\,P(A^{c})

Bayes' Formula

P(A\mid B) = \frac{P(B \mid A)\,P(A)}{P(B)}

Combinations (nCr)

{}_{n}C_{r} = \binom{n}{r} = \frac{n!}{(n-r)!\,r!}

Permutations (nPr)

{}_{n}P_{r} = \frac{n!}{(n-r)!}

Economics

Elasticity, money and inflation — the macro identities behind the markets.

Price Elasticity of Demand

E_d = \frac{\%\Delta Q}{\%\Delta P} = \frac{\Delta Q / \bar{Q}}{\Delta P / \bar{P}}

Fisher Equation

(1 + i) = (1 + r)(1 + \pi^e)

Quantity Theory of Money

MV = PY

Financial Statement Analysis

The ratio machinery that turns raw statements into judgments about a business.

Current & Quick Ratios

\text{Current} = \frac{CA}{CL} \qquad \text{Quick} = \frac{CA - \text{Inventory}}{CL}

Net Profit Margin

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

Return on Equity

ROE = \frac{\text{Net income}}{\text{Average shareholders' equity}}

DuPont Decomposition (3-Way)

ROE = \underbrace{\frac{NI}{Rev}}_{\text{margin}} \times \underbrace{\frac{Rev}{Assets}}_{\text{turnover}} \times \underbrace{\frac{Assets}{Equity}}_{\text{leverage}}

DuPont Decomposition (5-Way)

ROE = \underbrace{\frac{NI}{EBT}}_{\text{tax burden}} \times \underbrace{\frac{EBT}{EBIT}}_{\text{interest burden}} \times \underbrace{\frac{EBIT}{Rev}}_{\text{operating margin}} \times \underbrace{\frac{Rev}{Assets}}_{\text{turnover}} \times \underbrace{\frac{Assets}{Equity}}_{\text{leverage}}

Straight-Line vs. Double-Declining-Balance Depreciation

SL = \frac{\text{Cost} - \text{Salvage}}{\text{Life}} \qquad DDB_t = \text{Book value}_{t-1} \times \frac{2}{\text{Life}}

Debt-to-Equity, Debt-to-Capital & Interest Coverage

D/E = \frac{\text{Total debt}}{\text{Total equity}} \qquad D/C = \frac{\text{Total debt}}{\text{Total debt} + \text{Total equity}} \qquad \text{Interest coverage} = \frac{EBIT}{\text{Interest expense}}

Cash Conversion Cycle

CCC = DOH + DSO - DPO

Sustainable Growth Rate

g = b \times ROE, \qquad b = 1 - \text{payout ratio}

Basic Earnings per Share

EPS = \frac{NI - D_{pref}}{\text{weighted-average common shares}}

Receivables Turnover & Days Sales Outstanding

\text{Receivables turnover} = \frac{\text{Revenue}}{\text{Average receivables}} \qquad DSO = \frac{365}{\text{Receivables turnover}}

Inventory Turnover & Days of Inventory on Hand

\text{Inventory turnover} = \frac{COGS}{\text{Average inventory}} \qquad DOH = \frac{365}{\text{Inventory turnover}}

Payables Turnover & Days Payables Outstanding

\text{Payables turnover} = \frac{\text{Purchases}}{\text{Average payables}} \qquad DPO = \frac{365}{\text{Payables turnover}}

Total Asset Turnover

\text{Total asset turnover} = \frac{\text{Revenue}}{\text{Average total assets}}

Return on Assets

ROA = \frac{\text{Net income}}{\text{Average total assets}}

Diluted Earnings per Share (Treasury Stock Method)

\text{Incremental shares} = N\left(1 - \frac{Ex}{P}\right) \qquad \text{Diluted EPS} = \frac{NI - D_{pref}}{S + \text{Incremental shares}}

Free Cash Flow to the Firm

FCFF = NI + NCC + Int(1-t) - FCInv - WCInv

Free Cash Flow to Equity

FCFE = FCFF - Int(1-t) + \text{Net borrowing}

Corporate Issuers

What capital costs a company — the WACC cluster and leverage measures.

After-Tax Cost of Debt

r_d^{after} = r_d(1 - t)

Cost of Preferred Stock

r_p = \frac{D_p}{P_p}

Cost of Equity (Dividend Discount)

r_e = \frac{D_1}{P_0} + g

Weighted-Average Cost of Capital

WACC = w_d\,r_d(1-t) + w_e\,r_e

Degree of Operating Leverage

DOL = \frac{S - VC}{S - VC - F} = \frac{\text{contribution}}{\text{operating income}}

Equity Investments

Valuing stocks: justified multiples, enterprise value and margin mechanics.

Enterprise Value & EV/EBITDA

EV = \text{Market cap} + \text{Debt} - \text{Cash}

Margin Call Price

P_{call} = P_0\,\frac{1 - IM}{1 - MM}

Justified Leading P/E

\frac{P_0}{E_1} = \frac{p}{r - g}

Two-Stage Dividend Discount Model

V_0 = \sum_{t=1}^{n} \frac{D_0(1+g_1)^t}{(1+r)^t} + \frac{1}{(1+r)^n}\left[\frac{D_0(1+g_1)^n(1+g_2)}{r-g_2}\right]

Price-to-Book, Price-to-Sales & Price-to-Cash-Flow

P/B = \frac{P_0}{BVPS} \qquad P/S = \frac{P_0}{SPS} \qquad P/CF = \frac{P_0}{CFPS}

Fixed Income

Bond pricing, yields and the duration machinery that measures rate risk.

Current Yield

CY = \frac{\text{annual coupon}}{\text{price}}

Bond Price & Yield to Maturity

P = C \times \frac{1-(1+y)^{-n}}{y} + \frac{F}{(1+y)^{n}}

Accrued Interest & Full (Dirty) Price

AI = C \times \frac{t}{T} \qquad \text{Full price} = \text{Flat price} + AI

Zero-Coupon Bond & Spot Rates

P = \frac{F}{(1+z_n)^n}

Macaulay Duration

MacDur = \frac{\sum_{t=1}^{n} t \cdot \frac{CF_t}{(1+y)^t}}{P}

Modified Duration

ModDur = \frac{MacDur}{1+y}

Money Duration & PVBP

MoneyDur = ModDur \times P, \qquad PVBP = MoneyDur \times 0.0001

Price Change from Duration & Convexity

\frac{\Delta P}{P} \approx -ModDur \cdot \Delta y + \tfrac{1}{2} \cdot Conv \cdot (\Delta y)^2

Implied Forward Rate

(1+z_2)^2 = (1+z_1)(1+f_{1,1})

Derivatives

Forwards, options and the no-arbitrage logic that prices every contract.

Forward Price

F_0 = S_0 (1 + r)^T

Put–Call Parity

S_0 + p_0 = c_0 + \frac{X}{(1+r)^T}

One-Period Binomial Option Value

c_0 = \frac{\pi c_u + (1-\pi) c_d}{1+r}, \qquad \pi = \frac{(1+r) - d}{u - d}

Alternative Investments

Real estate cap rates and the fee arithmetic of hedge funds and private funds.

Cap-Rate Valuation

V = \frac{NOI}{\text{cap rate}}

Hedge Fund Fees (2-and-20)

\text{fees} = m \cdot A_1 + i \cdot \max(A_1 - A_0 - m A_1,\, 0)

Portfolio Management

Diversification, beta and the CAPM — how risk is priced and performance judged.

Two-Asset Portfolio Return & Risk

\sigma_p = \sqrt{w_A^2\sigma_A^2 + w_B^2\sigma_B^2 + 2 w_A w_B \rho_{AB}\sigma_A\sigma_B}

Beta

\beta_i = \frac{Cov(R_i, R_m)}{\sigma_m^2} = \rho_{im}\,\frac{\sigma_i}{\sigma_m}

CAPM (Security Market Line)

E(R_i) = R_f + \beta_i\,[E(R_m) - R_f]

Jensen's Alpha

\alpha_p = R_p - \left[R_f + \beta_p(R_m - R_f)\right]

Treynor Ratio