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CS2 formula cheat sheet

The formulas CS2 keeps coming back to, on one page: what each one says, when it applies, and the plain Memori syntax that types it. The grey text under each name doubles as a reference for writing your own cards.

Survival analysis

Estimating survival from data with censoring, and the hazard language it is written in.

Survival from the hazard

The integrated hazard is the cumulative hazard Λ(t)

S(t) = e^{-int_0^t mu(s) ds}

S(t) = e-t∫0 μ(s) ds

Kaplan-Meier estimate

Product over event times; censored lives just leave the risk set

S(t) = prod_{t_j <= t} (1 - (d_j)/(n_j))

S(t) = ∏t_j ≤ t (1 - d_jn_j)

Nelson-Aalen estimate

Cumulative hazard; exp(−Λ) gives a survival estimate

Lambda(t) = sum_{t_j <= t} (d_j)/(n_j)

Λ(t) = ∑t_j ≤ t d_jn_j

Cox proportional hazards

Covariates scale a baseline hazard multiplicatively

lambda(t) = lambda_0(t) e^{beta z}

λ(t) = λ₀(t) eβ z

Mortality laws

The parametric hazards fitted in graduation questions.

Gompertz

Exponential ageing

mu_x = B c^x

μ_x = B cx

Makeham

Gompertz plus an age-independent accident term

mu_x = A + B c^x

μ_x = A + B cx

Weibull hazard

γ > 1 increasing, γ < 1 decreasing

lambda(t) = c gamma t^{gamma - 1}

λ(t) = c γ tγ - 1

Markov processes

Jump processes are specified by intensities; everything else is solved from them.

Kolmogorov forward equations

μ_kj are the transition intensities (generator entries)

(d)/(dt) p_{ij}(t) = sum_k p_{ik}(t) mu_{kj}

ddt p_ij(t) = ∑k p_ik(t) μ_kj

Sojourn (holding) time

Time in state i is exponential with the total exit intensity

p_{ii}(t) = e^{-mu_i t}

p_ii(t) = e-μ_i t

Stationary distribution normalisation

With πP = π (discrete) or πQ = 0 (continuous)

sum_j pi_j = 1

∑j π_j = 1

Time series

The identification results the correlogram questions test.

AR(1) autocorrelation

Geometric decay; stationary when |φ| < 1

rho_k = phi^k

ρ_k = φk

MA(1) autocorrelation

And ρ_k = 0 for k ≥ 2: the ACF cuts off dead

rho_1 = (theta)/(1 + theta^2)

ρ₁ = θ1 + θ2

AR(1) process

PACF cuts off after lag 1; ACF tails off

X_t = mu + phi(X_{t-1} - mu) + e_t

X_t = μ + φ(X_t-1 - μ) + e_t

Loss distributions and reinsurance

Lognormal moments

Set k = 1, 2 for mean and second moment

E[X^k] = e^{k mu + (k^2 sigma^2)/(2)}

E[Xk] = ek μ + k2 σ22

Pareto mean excess

Linear and increasing in u: heavy tail

e(u) = (lambda + u)/(alpha - 1)

e(u) = λ + uα - 1

Insurer's payout under excess of loss

Reinsurer pays X − M above retention M

Y = min(X, M)

Y = min(X, M)

Extreme value theory

Generalised Pareto distribution

Threshold exceedances; ξ > 0 is the heavy-tailed case

G(x) = 1 - (1 + (xi x)/(sigma))^{-1/xi}

G(x) = 1 - (1 + ξ xσ)-1/ξ

GPD mean excess

Linear in u — the mean excess plot's threshold diagnostic

e(u) = (sigma + xi u)/(1 - xi)

e(u) = σ + ξ u1 - ξ

Ruin theory

Adjustment coefficient equation

Smallest positive root R; c is the premium rate

lambda + c r = lambda M_X(r)

λ + c r = λ M_X(r)

Lundberg's inequality

Initial surplus u buys exponentially less ruin probability

psi(u) <= e^{-R u}

ψ(u) ≤ e-R u

Copulas

Clayton lower tail dependence

Upper tail dependence is zero

lambda_L = 2^{-1/theta}

λ_L = 2-1/θ

Gumbel upper tail dependence

Lower tail dependence is zero

lambda_U = 2 - 2^{1/theta}

λ_U = 2 - 21/θ

Clayton's Kendall tau

Gaussian: τ = (2/π) arcsin ρ

tau = (theta)/(theta + 2)

τ = θθ + 2

Every one of these has an interactive tool. Try the Kaplan-Meier playground, multi-state models, time series, ruin theory or copulas. Memori is a flashcard app built by actuarial students, with a ready-made CS2 set in the shop. Join the beta.