Formula Cheat Sheet

36 formulas · last-minute reference for Module 2

Linear Algebra

4 formulas

Diagonalization

A` diagonalizable ⇔ sum of geometric multiplicities = `n

\(A` diagonalizable ⇔ sum of geometric multiplicities = `n\)

A` symmetric (`Aᵀ = A`) ⇒ orthogonally diagonalizable: `B…

\(A` symmetric (`Aᵀ = A`) ⇒ orthogonally diagonalizable: `BᵀAB = Λ`, `BᵀB = I\)

Aᵏ = PDᵏP⁻¹`; `Dᵏ = diag(λᵢᵏ)

\(Aᵏ = PDᵏP⁻¹`; `Dᵏ = diag(λᵢᵏ)\)

2×2 Shortcut

det(A) = ac − b²`, `tr(A) = a + c

\(det(A) = ac − b²`, `tr(A) = a + c\)

Differential Calculus

5 formulas

Taylor Expansion (2nd order)

f(x) = f(x₀) + ∇f(x₀)ᵀ(x − x₀) + ½(x − x₀)ᵀH_f(x₀)(x − x₀) +

\(f(x) = f(x₀) + ∇f(x₀)ᵀ(x − x₀) + ½(x − x₀)ᵀH_f(x₀)(x − x₀) + o(‖x − x₀‖²)\)

Convexity / Concavity (on convex domain C)

f` convex ⇔ `H_f(x)` PSD for all `x ∈ C

\(f` convex ⇔ `H_f(x)` PSD for all `x ∈ C\)

f` concave ⇔ `H_f(x)` NSD for all `x ∈ C

\(f` concave ⇔ `H_f(x)` NSD for all `x ∈ C\)

Chain Rule

z = f(x(t), y(t))` ⇒ `dz/dt = f_x·x' + f_y·y'

\(z = f(x(t), y(t))` ⇒ `dz/dt = f_x·x' + f_y·y'\)

Dini's Implicit Function Theorem

φ'(x₀) = − (∂g/∂x)(x₀, y₀) / (∂g/∂y)(x₀, y₀)

\(φ'(x₀) = − (∂g/∂x)(x₀, y₀) / (∂g/∂y)(x₀, y₀)\)

Integral Calculus

3 formulas

Improper Integrals

∫₁^∞ 1/xᵖ dx` converges iff `p > 1

\(∫₁^∞ 1/xᵖ dx` converges iff `p > 1\)

∫₀¹ 1/xᵖ dx` converges iff `p < 1

\(∫₀¹ 1/xᵖ dx` converges iff `p < 1\)

∫₁^∞ 1/(x(ln x)^p) dx` converges iff `p > 1

\(∫₁^∞ 1/(x(ln x)^p) dx` converges iff `p > 1\)

Probability

12 formulas

Basic Properties

P(∅) = 0`; `P(Eᶜ) = 1 − P(E)

\(P(∅) = 0`; `P(Eᶜ) = 1 − P(E)\)

Conditional Probability

P(A|B) = P(A ∩ B)/P(B)`, `P(B) > 0

\(P(A|B) = P(A ∩ B)/P(B)`, `P(B) > 0\)

Independence

A ⊥ B ⇔ P(A ∩ B) = P(A)P(B)

\(A ⊥ B ⇔ P(A ∩ B) = P(A)P(B)\)

Measures / Set Function Hierarchy

M(A ∪ B) + M(A ∩ B) = M(A) + M(B)

\(M(A ∪ B) + M(A ∩ B) = M(A) + M(B)\)

Linearity / Variance / Covariance

V(X) = E[X²] − (E[X])²

\(V(X) = E[X²] − (E[X])²\)

V(αX + β) = α²V(X)

\(V(αX + β) = α²V(X)\)

V(X + Y) = V(X) + V(Y) + 2Cov(X, Y)

\(V(X + Y) = V(X) + V(Y) + 2Cov(X, Y)\)

Cov(X, Y) = E[XY] − E[X]E[Y]

\(Cov(X, Y) = E[XY] − E[X]E[Y]\)

Cov(αX + β, γY + δ) = αγ·Cov(X, Y)

\(Cov(αX + β, γY + δ) = αγ·Cov(X, Y)\)

X ⊥ Y ⇒ E[XY] = E[X]E[Y]`, `Cov = 0`, `V(X+Y) = V(X) + V(Y)

\(X ⊥ Y ⇒ E[XY] = E[X]E[Y]`, `Cov = 0`, `V(X+Y) = V(X) + V(Y)\)

|ρ| = 1` ⇔ linear relation `Y = aX + b

\(|ρ| = 1` ⇔ linear relation `Y = aX + b\)

Standard Normal / Tips

Z = (X − μ)/σ ~ N(0,1)` when `X ~ N(μ, σ²)

\(Z = (X − μ)/σ ~ N(0,1)` when `X ~ N(μ, σ²)\)

Mathematical Finance

12 formulas

NPV / IRR

NPV(r) = −C₀ + Σ_{k=1}^n CFₖ/(1+r)ᵏ

\(NPV(r) = −C₀ + Σ_{k=1}^n CFₖ/(1+r)ᵏ\)

Classify stationary point `x₀` (after `∇f(x₀) = 0`)

Compute Hessian H = ∇²f(x₀)

\(Compute Hessian H = ∇²f(x₀)\)

Classify quadratic form `q(x) = xᵀAx`

Compute eigenvalues of A (or principal minors)

\(Compute eigenvalues of A (or principal minors)\)

2×2 fast path (`A = [[a,b],[b,c]]`, `Δ = ac − b²`, `T = a + c`)

Δ > 0, T > 0 → PD

\(Δ > 0, T > 0 → PD\)

Improper integral convergence (positive `f`)

Find asymptotic equivalent f(x) ~ C/xᵖ (or C/(x−a)ᵖ)

\(Find asymptotic equivalent f(x) ~ C/xᵖ (or C/(x−a)ᵖ)\)

IRR localization (from sample `G(i)` values, investment)

G is strictly decreasing.

\(G is strictly decreasing.\)

Convex vs concave optimization

f on convex domain C

\(f on convex domain C\)

Bond price under yield shift

Know D(y), P(y), proposed Δy:

\(Know D(y), P(y), proposed Δy:\)

Annuity formula selection

Payment timing?

\(Payment timing?\)

Bayes workflow

Given causes {Eᵢ} partitioning Ω, event A observed.

\(Given causes {Eᵢ} partitioning Ω, event A observed.\)

Expected value / variance shortcuts

Sum of independent Xᵢ:

\(Sum of independent Xᵢ:\)

§8 Numerical Benchmarks

Φ(1) ≈ 0.841`, `Φ(1.96) ≈ 0.975`, `Φ(2) ≈ 0.977

\(Φ(1) ≈ 0.841`, `Φ(1.96) ≈ 0.975`, `Φ(2) ≈ 0.977\)