Formula sheet

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

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

Derivative of the inside, divided by the inside.

Negative curvature = peak. Positive curvature = trough.

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

Q

quantity demanded

P

price

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

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

Divide by the coefficient of x in the exponent.

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

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

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

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

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

Sum and product of eigenvalues — useful for quick verification.

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

Probability-weighted average of all possible outcomes.

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

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

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

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

The coefficient that minimises the sum of squared residuals.

Grows money forward n periods at rate r.

PV

present value (today's amount)

r

interest rate per period

n

number of periods

Discounts a future cash flow back to today.

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

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

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

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

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

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

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

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

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

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

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

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

Adding assets with correlation < +1 always reduces variance.

Yield y and price P always move in opposite directions.

No coupons — just one payout F at maturity t.

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

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

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

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

Every topic in corporate finance connects into this chain.