Formula sheet

d/dx[xⁿ] = n · xⁿ⁻¹

Multiply by the exponent, then reduce the exponent by 1.

d/dx[f(g(x))] = f'(g(x)) · g'(x)

Derivative of the outer function (at the inner) times derivative of the inner.

d/dx[ln(u)] = u' / u

Derivative of the inside, divided by the inside.

f''(x) < 0 → local max f''(x) > 0 → local min

Negative curvature = peak. Positive curvature = trough.

ε_D = (dQ/dP) × (P/Q)

Percentage change in quantity demanded per 1% change in price.

Q

quantity demanded

P

price

lim(x→a) f(x)/g(x) = lim(x→a) f'(x)/g'(x)

Use when direct substitution gives 0/0 or ∞/∞.

∫ xⁿ dx = xⁿ⁺¹ / (n+1) + C

Works for all n ≠ −1. For n = −1: ∫(1/x)dx = ln|x| + C.

∫ eᵏˣ dx = eᵏˣ / k + C

Divide by the coefficient of x in the exponent.

∫ u dv = uv − ∫ v du

Choose u as the factor that gets simpler when differentiated (logs, polynomials).

|H| = f_xx · f_yy − (f_xy)²

|H|>0 + f_xx<0 → max; |H|>0 + f_xx>0 → min; |H|<0 → saddle point.

det(A) = ad − bc

For A = [[a,b],[c,d]]. Zero determinant means no unique solution.

A⁻¹ = (1/det(A)) · [[d, −b], [−c, a]]

Swap the diagonal, negate the off-diagonal, divide by the determinant.

det(A − λI) = 0

Eigenvalues λ are the solutions. For a diagonal matrix, they are just the diagonal entries.

Tr(A) = λ₁ + λ₂ det(A) = λ₁ · λ₂

Sum and product of eigenvalues — useful for quick verification.

L(x, y, λ) = f(x, y) + λ[c − g(x, y)]

λ is the shadow price: how much the objective improves per unit relaxation of the constraint.

E[X] = Σ xᵢ · pᵢ

Probability-weighted average of all possible outcomes.

Var(X) = E[X²] − (E[X])²

Also written as σ². Standard deviation σ = √Var(X) — same units as X.

Var(aX + b) = a² · Var(X)

Additive constants vanish. Multiplicative constants are squared. So Var(2X+3) = 4·Var(X).

ρ_XY = Cov(X, Y) / (σ_X · σ_Y)

Always between −1 and +1. Zero means linearly uncorrelated (not necessarily independent).

P(A|B) = P(B|A) · P(A) / P(B)

Updates the prior P(A) with the likelihood P(B|A) to get the posterior P(A|B).

β̂ = Cov(x, y) / Var(x)

The coefficient that minimises the sum of squared residuals.

FV = PV × (1 + r)ⁿ

Grows money forward n periods at rate r.

PV

present value (today's amount)

r

interest rate per period

n

number of periods

PV = FV / (1 + r)ⁿ

Discounts a future cash flow back to today.

PV = C / r

Present value of a constant cash flow C received forever, starting next period.

P₀ = D₁ / (r_e − g)

Growing perpetuity formula applied to dividends. Requires r_e > g.

D₁

dividend expected next period

r_e

cost of equity (required return)

g

constant perpetual growth rate of dividends

NPV = −C₀ + Σ CFₜ / (1 + r)ᵗ

Accept the project if NPV > 0. The discount rate r is the cost of capital.

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

After-tax cost of debt is R_d(1−T_c) — debt interest is tax-deductible.

E/V

equity weight in capital structure

D/V

debt weight in capital structure

T_c

corporate tax rate

ε_D = (% ΔQ) / (% ΔP) = (dQ/dP) × (P/Q)

|ε_D| < 1 = inelastic; |ε_D| > 1 = elastic; |ε_D| = 1 = unit elastic.

L = (P − MC) / P = 1 / |ε_D|

Measures monopoly power. L=0 in perfect competition; L=1 in pure monopoly.

MRS₁₂ = MU₁ / MU₂ = (∂U/∂x₁) / (∂U/∂x₂)

Rate at which the consumer is willing to exchange good 2 for good 1, holding utility constant.

MRS₁₂ = p₁ / p₂

Equivalently: MU₁/p₁ = MU₂/p₂ — equal marginal utility per euro spent.

GDP = C + I + G + NX

C = consumption, I = investment, G = government spending, NX = net exports.

Real GDP = (Nominal GDP / GDP Deflator) × 100

Holds prices fixed to measure genuine output change, not inflation.

s · f(k*) = (n + δ + g) · k*

Investment per worker equals effective capital dilution. k* is the steady-state capital per worker.

s

savings rate

n

labour force growth rate

δ

depreciation rate

g

technology growth rate

k*

steady-state capital per worker

ΔA/A = ΔY/Y − α(ΔK/K) − (1−α)(ΔL/L)

The part of GDP growth not explained by capital or labour accumulation. Measures productivity.

E[R_i] = R_f + β_i · (E[R_m] − R_f)

Expected return equals the risk-free rate plus a beta-scaled market risk premium.

R_f

risk-free rate (e.g. government bond yield)

β_i

sensitivity of asset i to market movements

E[R_m]−R_f

market risk premium

β_i = Cov(R_i, R_m) / Var(R_m)

β = 1 → moves with market; β > 1 → amplifies; β < 1 → defensive.

σ_p² = w₁²σ₁² + w₂²σ₂² + 2w₁w₂ · Cov(R₁, R₂)

Adding assets with correlation < +1 always reduces variance.

P = Σ C/(1+y)ᵗ + F/(1+y)ⁿ

Yield y and price P always move in opposite directions.

P = F / (1 + y)ᵗ

No coupons — just one payout F at maturity t.

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

After-tax cost of debt: R_d(1−T_c). V = E + D (total firm value).

E/V

equity weight

D/V

debt weight

R_e

cost of equity (from CAPM)

T_c

corporate tax rate

V_L = V_U

Without taxes, firm value is independent of capital structure. Leverage is irrelevant.

V_L = V_U + T_c · D

Each euro of debt saves T_c in taxes, permanently. Levered firm is worth more.

P₀ = D₁ / (r_e − g)

Growing perpetuity formula for stock valuation. Requires r_e > g.

D₁

next period's dividend

r_e

cost of equity / required return

g

constant perpetual dividend growth rate

CAPM → R_e → WACC → NPV discount rate

Every topic in corporate finance connects into this chain.