Finance and economics formulas

C = S·N(d₁) − K·e^(−rT)·N(d₂)
d₁ = [ln(S/K) + (r + σ²/2)T] / (σ√T),  d₂ = d₁ − σ√T

Closed-form price for European options.

D_mac = Σ (t · PV_t) / Price
D_mod = D_mac / (1 + y)

A bond's price sensitivity to interest-rate changes.

Practise it in Python: Bond Duration in Python: Macaulay and Modified

E(Rᵢ) = R_f + βᵢ · (E(R_m) − R_f)

Expected return as risk-free rate plus beta times market premium.

Practise it in Python: CAPM Beta in Python: Regression on Market Returns

Engle–Granger: regress A on B, test residuals for stationarity (ADF)

Two non-stationary prices whose linear combo is stationary.

ES_α = −E[ r | r ≤ −VaR_α ]

Average loss in the worst tail beyond VaR.

R − R_f = α + β·(R_m − R_f) + s·SMB + h·HML + ε

Extends CAPM with size (SMB) and value (HML) factors.

P₀ = D₁ / (r − g)      (requires r > g)

Values a stock as next dividend over (required return − growth).

IR = (R_p − R_benchmark) / TrackingError

Active return divided by tracking error.

Solve for r:  Σ_t  CF_t / (1 + r)^t = 0

The discount rate that makes a project NPV exactly zero.

Practise it in Python: NPV and IRR in Python: Is the Project Worth It?

f* = (b·p − q) / b        (discrete bets)
f* = μ / σ²              (continuous returns)

The bet size that maximises long-run logarithmic growth.

z = (xₜ − rolling_mean) / rolling_std
Trade when |z| exceeds a threshold (e.g. 2)

The tendency of a price or spread to return to its average.

NPV = Σ_t  CF_t / (1 + r)^t

Today's value of future cash flows, discounted for time and risk.

Practise it in Python: NPV and IRR in Python: Is the Project Worth It?

C − P = S − K·e^(−rT)

A no-arbitrage link between calls, puts, the stock, and a bond.

Sortino = (R_p − R_f) / σ_downside
σ_downside = sqrt( mean( min(R − target, 0)² ) )   (over ALL periods, target usually 0 or R_f)

Like Sharpe, but penalises only downside volatility.

PV = FV / (1 + r)^n      FV = PV · (1 + r)^n

A dollar today is worth more than a dollar tomorrow.

Practise it in Python: NPV and IRR in Python: Is the Project Worth It?

Treynor = (R_p − R_f) / β_p

Excess return per unit of market (systematic) risk.

VaR_α = −quantile(returns, 1 − α)

Worst expected loss over a horizon at a confidence level.

WACC = (E/V)·R_e + (D/V)·R_d·(1 − T_c)

Blended after-tax cost of a firm’s equity and debt.

Y = A · Kᵅ · Lᵝ      (constant returns when α + β = 1)

Output as a function of capital and labour with constant returns.

Specialise where opportunity cost is lowest:
OC(good A) = units of B forgone per unit of A

Gains from trade arise from differences in opportunity cost.

(1 + i) = (1 + r)(1 + π)   ⇒   r ≈ i − π

Links nominal interest rates, real rates, and inflation.

G = (Σ_i Σ_j |x_i − x_j|) / (2 n² x̄)

Single-number measure of income or wealth inequality (0–1).

Y = C + I + G + (X − M)
Real GDP = Nominal GDP / (Deflator / 100)

Total market value of goods & services produced in an economy.

Inflation_t = (CPI_t / CPI_{t−12}) − 1

Year-over-year percentage rise in a basket of consumer prices.

k = 1 / (1 − MPC)
ΔGDP = k × ΔSpending

How an initial spending injection cascades into larger GDP change.

No player i can improve by deviating:
uᵢ(sᵢ*, s₋ᵢ*) ≥ uᵢ(sᵢ, s₋ᵢ*)  ∀ sᵢ

A state where no player can gain by changing strategy alone.

(Y − Y*)/Y* ≈ −c · (u − u*)   with c ≈ 2

Empirical link between unemployment and the output gap.

Practise it in Python: What Is Machine Learning, Really?

PED = (%ΔQ) / (%ΔP)
Elastic if |PED| > 1, inelastic if |PED| < 1

How responsive quantity demanded is to a change in price.

S = P_domestic / P_foreign

Exchange rates should equalise the price of identical goods.

M · V = P · Q
⇒ %ΔP ≈ %ΔM + %ΔV − %ΔQ

Money supply × velocity equals price level × real output.

y = β₀ + β₁x + ε ;  β̂ = (XᵀX)⁻¹Xᵀy  minimises Σ(yᵢ − ŷᵢ)²

Fit a straight line (or hyperplane) that minimises the sum of squared errors: ordinary least squares (OLS).

Practise it in Python: CAPM Beta in Python: Regression on Market Returns

P(y=1 | x) = σ(β₀ + β₁x) = 1 / (1 + e^(−(β₀ + β₁x))) ;  log(p / (1 − p)) = β₀ + β₁x

Classification model that turns a linear score into a probability through the sigmoid.

Practise it in Python: Predict Market Direction with Logistic Regression