IAI CM2 cheat sheet

Survival function S(x)=P(T>x)=1-F(x); it is non-increasing, right-continuous, with S(0)=1 and S(∞)=0.

Force of mortality μ(x)=f(x)/S(x)= -d/dx ln S(x), so S(x)=exp(-∫₀ˣ μ(t)dt) and f(x)=μ(x)S(x).

Curiate expectation: e̊ₓ = ∫₀^∞ S(x+t)/S(x) dt; complete expectation of life equals the integral of tPx over t.

tPx = S(x+t)/S(x); tQx = 1 - tPx; deferred probability t|uQx = tPx · uQx+t.

Fractional-age assumptions: UDD gives tQx = t·Qx for 0≤t≤1; constant force gives tPx = (Px)^t; Balducci gives 1/(1-t+t/Px).

Under UDD, μ(x+t)=Qx/(1-t·Qx) and e̊ₓ ≈ eₓ + 0.5; under constant force, μ is piecewise constant and e̊ₓ = eₓ + (1-Px)/(Qx·(-ln Px)) adjustment.

Life table columns: lx survivors, dx=lx-lx+1 deaths, qx=dx/lx, px=1-qx, Lx≈∫₀¹ lx+t dt, Tx=ΣLx, e̊ₓ=Tx/lx.

Select mortality: l[x]+r reflects underwriting; select period ends when q[x]+r = qx+r; ultimate table applies thereafter.

Crude central death rate mₓ = Dₓ/Eₓᶜ where Eₓᶜ is central exposed to risk; qₓ ≈ mₓ/(1+0.5mₓ) under UDD.

Exposed to risk: exact age x to x+1 uses Eₓᶜ=∫ lx+t dt; initial exposed Eₓ = Eₓᶜ + ½Dₓ under UDD.

Binomial model: Dₓ ~ Binomial(Eₓ, qₓ); Poisson approximation when qₓ small: Var(Dₓ)≈Eₓqₓ.

Standardised mortality ratio SMR = observed deaths / expected deaths = ΣDₓ / ΣEₓᶜ·mₛₓ.

Graduation checks: preserve total deaths (ΣEᶜ·q̂ = ΣD), smoothness of q̂, and goodness of fit via chi-square Σ(D-Eq̂)²/(Eq̂(1-q̂)).

Gompertz: μ(x)=Bcˣ; Makeham: μ(x)=A+Bcˣ; both give log qₓ roughly linear in x over adult ages.

Lee-Carter: ln m(x,t)=aₓ + bₓ·kₜ + ε; kₜ fitted as random walk with drift, aₓ as time-average of ln m.

Heligman-Pollard law models infant, accident hump and senescent components with parameters A-H; useful for full-age-range graduation.

Whole life assurance: Ax = E[v^{Kx+1}] = Σ v^{k+1}·kPx·qx+k; term assurance restricts sum to n years.

Endowment assurance Ax:n = Ax:n¹ + nEx where nEx = vⁿ·nPx is the pure endowment.

Annuity-due äx = Σ vᵏ·kPx; immediate annuity ax = äx - 1; continuous āx = ∫ vᵗ·tPx dt.

Relations: Ax = 1 - d·äx (d = i/(1+i)); under UDD Ax = (i/δ)·Ax¹ (accelerated) and Ax:n = 1 - d·äx:n.

Increasing/decreasing benefits: (IA)x = Σ(k+1)v^{k+1}kPx·qx+k; term version relates to äx:n via commutation functions.

Commutation functions: Dx=vˣlx, Cx=v^{x+1}dx, Nx=ΣDx, Mx=ΣCx; Ax=Mx/Dx, äx=Nx/Dx.

Variance of PV: Var(Z)=²Ax-(Ax)², computed with a second rate of interest i' where v'²=v, i.e. (1+i')²=1+i.

Mortality gain/loss on assurance arises because death benefit paid earlier or later than expected; annuity gain reverses the sign.

Equivalence principle: P·äx = Ax for whole life, so P = Ax/äx; gross premium adds expenses to the equation of value.

Prospective reserve tV = Ax+t - P·äx+t; retrospective reserve = (P·äx - Ax¹)/ nEx-type accumulation, equal under equivalence.

Recursive (Fackler) relation: (tV + P)(1+i) = qx+t·S + px+t·t+1V for unit sum assured S, ignoring expenses.

Reserve as expected loss: tV = A_{x+t} - P·ä_{x+t}; for endowment use Ax+t:n-t and äx+t:n-t.

Gross premium reserve uses gross premium P^g and includes future expenses; net premium reserve omits expenses.

Thiele's differential equation: d/dt(tV) = δ·tV + P - μ(x+t)(S - tV) - expense intensity.

Modified reserves (e.g. Zillmer, full preliminary term) reduce early reserves; Zillmer deducts α = acquisition expense from first-year reserve.

Profit testing: model cash flows with expenses, mortality, lapse and interest; profit signature = expected profit per policy in force at outset.

Final salary scheme benefit often = n/60 × final pensionable salary; CARE schemes accrue on each year's pensionable earnings.

Defined benefit funding: normal cost plus amortisation of surplus/deficit; contribution rate set via projected unit or aggregate method.

Defined contribution: benefit = accumulated fund / annuity factor at retirement; investment and longevity risk borne by member.

Actuarial liability for active members uses projected benefits and salary growth; past service liability vs future service normal cost.

Pension increase rules (RPI/CPI caps) and commutation factors materially affect liability valuation.

Withdrawal, ill-health and early retirement decrements modelled as multiple decrement table; independent rates convert via μᵢⱼ.

Pension annuity factor äx at retirement uses post-retirement mortality, possibly with spouse's reversion and escalation.

Social security / state scheme: pay-as-you-go financing; dependency ratio affects sustainability.

Model is a simplification of reality; the actuary balances accuracy, transparency, and cost of implementation.

Deterministic vs stochastic: stochastic models produce distributions (e.g. percentiles, VaR) not just means.

The Principle of Equivalence underpins premium calculation; the Principle of Consistency links models across uses.

Sensitivity analysis varies one parameter; scenario testing varies several; stress testing picks adverse combinations.

Model points: representative cells chosen to reflect the portfolio; grouping reduces run time with controlled error.

Calibration vs validation: calibration fits parameters; validation checks out-of-sample performance (back-testing).

Data quality issues: missing data, heterogeneity, credibility weighting, and spurious correlation.

Communicating results: state assumptions, limitations, and uncertainty; regulators require documentation of model risk.

Standard Brownian motion W(t): W(0)=0, independent increments, W(t)-W(s)~N(0,t-s), continuous paths.

Quadratic variation of W over [0,t] is t; (dW)²=dt, dW·dt=0, (dt)²=0 in Itô calculus.

Itô's lemma: for f(t,X), df = (∂f/∂t + μ∂f/∂x + ½σ²∂²f/∂x²)dt + σ∂f/∂x dW when dX=μdt+σdW.

Geometric Brownian motion dS=μS dt+σS dW solves S(t)=S(0)exp((μ-½σ²)t+σW(t)).

Itô integral ∫H dW is a martingale when H is adapted and square-integrable; Itô isometry E[(∫H dW)²]=E[∫H² dt].

Risk-neutral measure Q: discounted asset prices are Q-martingales; Girsanov changes drift via dQ/dP = exp(-∫θ dW - ½∫θ² dt).

First passage time of driftless BM to level a has density a/(t√(2πt))exp(-a²/(2t)); expectation infinite.

Ornstein-Uhlenbeck: dX=κ(θ-X)dt+σdW; mean-reverting, Gaussian stationary distribution with variance σ²/(2κ).

Martingale: E[M(t)|F(s)]=M(s) for s<t; submartingale ≥, supermartingale ≤.

Optional stopping: for bounded stopping time τ and suitable M, E[M(τ)]=E[M(0)].

Doob-Meyer decomposition: submartingale = martingale + increasing predictable compensator.

Predictable quadratic variation ⟨M⟩(t) is the compensator of M²; for BM, ⟨W⟩(t)=t.

Stochastic integral ∫₀ᵗ H(s)dM(s) is a martingale when H is predictable and E[∫H² d⟨M⟩]<∞.

Itô product rule: d(XY)=X dY + Y dX + d⟨X,Y⟩; cross-variation d⟨W₁,W₂⟩=ρ dt for correlated BMs.

Martingale representation theorem: any Q-martingale can be written as M(0)+∫H dW under Brownian filtration.

Change of measure: Z(t)=E[dQ/dP|Fₜ] is a P-martingale; under Q, W̃=W+∫θ ds is a Brownian motion.

Black-Scholes PDE: ∂V/∂t + ½σ²S²∂²V/∂S² + rS∂V/∂S - rV = 0, with payoff boundary condition.

Call price C = S·N(d₁) - K e^{-rT}N(d₂); put via put-call parity C - P = S - Ke^{-rT}.

d₁ = [ln(S/K)+(r+½σ²)T]/(σ√T); d₂ = d₁ - σ√T.

Risk-neutral pricing: V = e^{-rT}E^Q[payoff]; delta call = N(d₁), put = N(d₁)-1; gamma = n(d₁)/(Sσ√T).

Greeks: vega = S·n(d₁)√T; theta for call = -S n(d₁)σ/(2√T) - rK e^{-rT}N(d₂).

Implied volatility solves BS price = market price; volatility smile/skew reflects departure from lognormal.

Binomial tree: risk-neutral p = (e^{rΔt}-d)/(u-d); CRR sets u=e^{σ√Δt}, d=1/u; converges to BS as Δt→0.

Dividend yield q: replace S by Se^{-qT} and r by r-q in d₁; early exercise of American calls possible only with dividends.

Short rate models: Vasicek dr=κ(θ-r)dt+σdW (Gaussian, allows negative rates); CIR dr=κ(θ-r)dt+σ√r dW (positive, non-central chi-square).

Affine term structure: bond price P(t,T)=A(t,T)exp(-B(t,T)r(t)) for Vasicek and CIR.

Hull-White dr=(θ(t)-ar)dt+σdW fits the initial term structure exactly; extended Vasicek.

No-arbitrage HJM: df(t,T)=α(t,T)dt+σ(t,T)dW; drift condition α=σ∫σ du under risk-neutral measure.

Forward rate and discount: P(t,T)=exp(-∫ₜᵀ f(t,u)du); instantaneous forward f(t,T)=-∂ln P/∂T.

Short-rate model calibration to caps/swaptions and to initial yield curve; LMM/BGM models forward LIBOR rates as lognormal.

Duration and convexity: Macaulay duration = Σ t·PV(CF)/Price; modified duration = D/(1+y); ΔP/P ≈ -D_mod·Δy + ½·C·(Δy)².

Immunisation: match duration and convexity of assets and liabilities; Redington immunisation requires PV, duration and convexity conditions.