Free · printable
CS2 formula sheet
41 formulae from Risk Modelling & Survival Analysis, gathered from the explainers that teach them. Print it, or tap any heading to read the page it came from.
S.MONK · CS2 formula sheet · 41 formulae · smonk.in/formula-sheet/cs2
The survival function and the hazard rate, explained
S(t) = P(T > t)
f(t) = −d/dt S(t)
h(t) = f(t) / S(t)
H(t) = ∫0t h(s) ds
S(t) = exp(−H(t)) = exp(−∫0t h(s) ds)
H = ∫0t Bcx+s ds = Bcx(ct − 1) / ln c
The Kaplan-Meier estimator, and why censoring needs it
S(t) = P(survive first stretch) × P(survive second | survived first) × …
Ŝ(t) = ∏tj ≤ t (nj − dj) / nj = ∏tj ≤ t (1 − dj/nj)
Ĥ(t) = ∑tj ≤ t dj / nj
The Cox proportional hazards model, and what a hazard ratio means
h(t; z) = h0(t) · exp(β1z1 + … + βpzp)
hazard ratio for a one-unit rise in zk = exp(βk)
S(t; z) = [S0(t)]exp(β′z)
contribution at tj = exp(β′z(j)) / ∑i ∈ R(tj) exp(β′zi)
L(β) = ∏j exp(β′z(j)) / ∑i ∈ R(tj) exp(β′zi)
ln(−ln S(t; z)) = ln(−ln S0(t)) + β′z
Markov chains: transition matrices, Chapman-Kolmogorov and the stationary distribution
P(Xn+1 = j | Xn = i, and the whole history before n) = P(Xn+1 = j | Xn = i)
p(2)ij = ∑k pik pkj
P(m+n) = P(m) P(n), so P(n) = Pn
π(n) = π(0) Pn, and π(n+1) = π(n) P
π P = π, with πi ≥ 0 and ∑i πi = 1
Central exposed to risk, and the two mortality rates it produces
Ecx = total time, in years, spent by all lives while aged x and under observation
Ecx = ∫ Px(t) dt
μ̂x+½ = Dx / Ecx
Ex = Ecx + ½Dx, and then q̂x = Dx / Ex
qx = 1 − exp(−μx+½)
Bühlmann credibility: how much to trust one risk's own experience
credibility premium = Z × (this risk’s own mean) + (1 − Z) × (the book’s mean)
credibility premium = (the book’s mean) + Z × (this risk’s mean − the book’s mean)
Z = n / (n + E[s2(θ)] / Var[m(θ)])
m̂ = the mean of the N risk means
E[s2]̂ = (1/N) ∑i [ 1/(n−1) ∑j (Xij − X̄i)2 ]
Var[m(θ)]̂ = [ 1/(N−1) ∑i (X̄i − m̂)2 ] − E[s2]̂ / n
Zi = Pi / (Pi + E[s2]/Var[m(θ)])
Extreme value theory: block maxima, thresholds and the generalised Pareto distribution
G(x) = exp( −[1 + ξ(x − μ)/σ]−1/ξ )
P(X − u ≤ y | X > u) = 1 − [1 + ξy/β]−1/ξ, for y > 0
P(X > x) = P(X > u) × [1 + ξ(x − u)/β]−1/ξ
xp = u + (β/ξ)[ (P(X > u)/p)ξ − 1 ]
E[X − u | X > u] = (β + ξu′) / (1 − ξ) for a threshold raised by u′, valid when ξ < 1
Copulas and Sklar's theorem: modelling dependence without touching the margins
C(u, v) = P(U ≤ u, V ≤ v)
H(x, y) = C( F(x), G(y) )
λU = limu→1 P(V > u | U > u) = limu→1 (1 − 2u + C(u, u)) / (1 − u)
C(u, v) = φ−1( φ(u) + φ(v) )
Free from S.MONK School of Actuaries · smonk.in · every formula here is explained, with a worked example, on the page it links to.