PJENL Actuario Aptitude Exam cheat sheet

Law of total probability: P(A) = Σ P(A|B_i)P(B_i) over a partition {B_i}; Bayes' theorem flips the conditioning: P(B_j|A) = P(A|B_j)P(B_j) / Σ_i P(A|B_i)P(B_i).

For independent events P(A∩B) = P(A)P(B); for mutually exclusive events P(A∪B) = P(A) + P(B). Independence and mutual exclusivity are different conditions and cannot be assumed interchangeable.

Variance identities: Var(aX+b) = a²Var(X); Var(X) = E[X²] − (E[X])²; Var(X+Y) = Var(X) + Var(Y) + 2Cov(X,Y), which reduces to Var(X)+Var(Y) only when Cov = 0.

Common moment generating functions: Poisson(λ) gives M(t) = exp(λ(e^t − 1)); exponential(λ) gives M(t) = λ/(λ − t) for t < λ; normal(μ,σ²) gives M(t) = exp(μt + σ²t²/2).

Poisson process with rate λ: N(t) ~ Poisson(λt), so E[N(t)] = Var[N(t)] = λt; interarrival times are exponential with mean 1/λ, and the waiting time to the k-th event is Gamma(k, λ).

Maximum likelihood: maximize the log-likelihood ℓ(θ) = Σ ln f(x_i|θ); set ℓ'(θ) = 0 and verify the second derivative is negative. The MLE is invariant under reparameterization: if θ̂ estimates θ, then g(θ̂) estimates g(θ).

Central limit theorem: for iid variables with mean μ and finite variance σ², (X̄ − μ)/(σ/√n) converges in distribution to standard normal; use the sample variance for σ when it is unknown and n is large.

Confidence interval for a normal mean with unknown variance: X̄ ± t_{α/2, n−1} · s/√n; replace t by z when σ is known or n is large. The interval widens with confidence level and narrows as √n grows.

Accumulation with compound interest: a(t) = (1+i)^t; with simple interest a(t) = 1+it; with discount rate d, (1−d)(1+i) = 1, so d = i/(1+i) and i = d/(1−d).

Present value of an annuity-immediate: a-angle-n = (1 − v^n)/i, where v = 1/(1+i); annuity-due: ä-angle-n = (1 + i)·a-angle-n = (1 − v^n)/d.

Perpetuity-immediate has present value 1/i; a growing perpetuity with first payment 1 growing at rate g has present value 1/(i − g) when i > g.

Nominal rates convert by compounding frequency: (1 + i^(m)/m)^m = 1 + i; the effective rate rises with m at fixed nominal rate, and continuous compounding gives force of interest δ = ln(1+i).

Net present value rule: accept a project when NPV = Σ CF_t·v^t − initial outlay > 0; the internal rate of return is the rate making NPV zero, and IRR ranking can conflict with NPV when project scales differ.

Duration (Macaulay) D = Σ t·CF_t·v^t / Σ CF_t·v^t; modified duration D* = D/(1+i) approximates the percentage price change for a small yield change: ΔP/P ≈ −D*·Δi.

Immunization of a liability stream requires matching present value and duration (and convexity for stronger protection) between assets and liabilities, with the asset portfolio positioned on both sides of the liability horizon.

Loan amortization: the outstanding balance after k payments equals the present value of remaining payments, and the interest portion of payment k+1 equals i times the balance just after payment k.

Survival notation: _t p_x = probability a life aged x survives t years; _t q_x = 1 − _t p_x; under constant force of mortality μ, _t p_x = e^(−μt) for all t.

Whole life insurance payable at the moment of death: Ā_x = ∫_0^∞ v^t · t p_x · μ{x+t} dt; with constant force μ and δ, Ā_x = μ/(μ + δ).

Whole life annuity-due: ä_x = (1 − A_x)/d, where A_x is the insurance payable at end of year of death and d is the discount rate; this relation links annuity and insurance values.

Under UDD within each year of age, fractional-age identities hold: _t q_x = t·q_x, and A_x = (i/δ)·Ā_x for the same underlying mortality.

Recursive (retrospective) relation: A_x = v·q_x + v·p_x·A_{x+1}; the same recursion applied to ä_x gives ä_x = 1 + v·p_x·ä_{x+1}.

Net premium for a fully discrete whole life policy: P_x = A_x / ä_x; the equivalence principle sets expected present value of premiums equal to expected present value of benefits.

Net level premium reserve at duration t: t V = A{x+t} − P_x·ä_{x+t}; reserves are prospective, and the recursion t V + P = v·q·(benefit) + v·p·{t+1}V must hold.

Multiple decrement: the independent (associated) single-decrement rate q' relates to the dependent rate by q' = −ln(1 − q^(τ)) when other decrements operate uniformly, so crude dependent rates understate true forces.

Collective risk model: aggregate loss S = Σ X_i with N claims of sizes X_i; E[S] = E[N]·E[X] and Var(S) = E[N]·Var(X) + Var(N)·(E[X])².

Compound Poisson special case: if N ~ Poisson(λ), then Var(S) = λ·E[X²] = λ(Var(X) + (E[X])²), so variance depends on the second moment of claim size.

Classical credibility: Z = √(n/(n+k)) with k = E[σ²]/Var(μ) for full credibility based on claim counts; the credibility premium is Z·(observed mean) + (1−Z)·(prior mean).

Bühlmann credibility: Z = n/(n+k) with k = EPV/VHM, where EPV is the expected value of the process variance and VHM is the variance of the hypothetical means; the estimate is Z·X̄ + (1−Z)·μ.

Bühlmann-Straub model with weights m_i: Z = Σm_i / (Σm_i + k), k = EPV/VHM; equal weights reduce to the Bühlmann formula.

Limited fluctuation (classical) credibility requires n ≥ (z_{α/2}/r)²·(σ²/μ²) claim counts for full credibility at relative error r, where σ²/μ² is the coefficient of variation squared.

Panjer recursion for compound distributions: p_k = Σ_{j=1}^{k} (a + bj/k)·p_{k−j}·f_j, applicable when the claim-count distribution belongs to the (a,b,0) class such as Poisson, binomial, or negative binomial.

Stop-loss reinsurance with retention d pays (S − d)+; its expected cost is E[(S−d)+] = ∫_d^∞ (1 − F_S(x)) dx for nonnegative severity, which links directly to the survival function.

Present value under certainty: PV = Σ CF_t/(1+r)^t; the discount rate should reflect the opportunity cost of capital, and a higher rate lowers the value of distant cash flows more heavily.

Capital asset pricing model: E[R_i] = r_f + β_i(E[R_m] − r_f), where β_i = Cov(R_i, R_m)/Var(R_m); only systematic risk earns a premium in the model.

Arbitrage pricing theory: expected return is a linear combination of factor exposures, E[R] = r_f + Σ β_ik·λ_k, with no assumption that the market portfolio is mean-variance efficient.

Put-call parity for European options on a non-dividend stock: C − P = S − K·e^(−rT); violations permit arbitrage by combining the option with a bond and the underlying.

Black-Scholes inputs: raising volatility raises both call and put values; raising the risk-free rate raises call value and lowers put value; raising the strike lowers call value and raises put value.

Duration-based bond risk: price change ≈ −D*·Δy + ½·Convexity·(Δy)²; convexity is positive for plain bonds, so the price rise for a yield fall exceeds the fall for an equal yield rise.

Efficient market hypothesis forms: weak (prices reflect past prices), semi-strong (public information), strong (all information including private); each stronger form implies the weaker ones.

Modigliani-Miller without taxes: firm value is independent of capital structure; with corporate taxes and deductible interest, value rises by the tax shield, approximately t_c·D.

Integration by parts: ∫u dv = uv − ∫v du; useful for products such as x·e^(−x) and for expectations of functions of exponential variables.

L'Hôpital's rule applies to 0/0 or ∞/∞ forms: lim f/g = lim f'/g' when the latter limit exists; rewrite 0·∞ and ∞−∞ forms before applying it.

Taylor expansion about a: f(x) = Σ f^(n)(a)(x−a)^n/n!; the first-order term gives linear approximation and the second-order term gives the convexity correction.

A square matrix is invertible iff det(A) ≠ 0 iff its rows (and columns) are linearly independent iff zero is not an eigenvalue.

Eigenvalues solve det(A − λI) = 0; the trace equals the sum of eigenvalues and the determinant equals their product, which gives a quick consistency check.

A symmetric real matrix has real eigenvalues and an orthogonal set of eigenvectors; a positive definite matrix has all eigenvalues positive and all leading principal minors positive.

For a linear system Ax = b, a unique solution exists iff A is invertible; otherwise solutions exist only when b lies in the column space of A, and the solution set is an affine subspace.

Gradient conditions for unconstrained optimization: solve ∇f = 0, then classify with the Hessian - positive definite gives a local minimum, negative definite a local maximum, indefinite a saddle point.