IFoA CS2 flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Survival function S(x)

Probability a newborn survives to age x: S(x) = P(X > x). Non-increasing, S(0)=1, S(∞)=0. Related to force of mortality by S(x) = exp(−∫₀ˣ μ(t)dt).

Force of mortality μ(x)

Instantaneous hazard of death at age x given survival to x: μ(x) = f(x)/S(x) = −d/dx ln S(x). Under UDD between integer ages, μ(x+t) = qₓ/(1 − t·qₓ).

Curtate expectation of life eₓ

Expected number of complete future years lived: eₓ = Σₖ≥₁ ₖpₓ. Complete expectation ėₓ = ∫₀^∞ ₜpₓ dt. Approximation ėₓ ≈ eₓ + ½ under UDD.

UDD fractional survival

Uniform distribution of deaths: for 0 ≤ t ≤ 1, ₜpₓ = 1 − t·qₓ and ₜqₓ = t·qₓ. Deaths are spread evenly across each year of age.

Constant force of mortality

Assumes μ(x+t) = μ constant for 0 ≤ t ≤ 1, giving ₜpₓ = e^(−μt) = (pₓ)ᵗ. Fractional survival is exponential within the year of age.

Balducci assumption

Assumes ₁₋ₜqₓ₊ₜ = (1−t)·qₓ, so survival within year is ₁₋ₜpₓ₊ₜ = 1 − (1−t)qₓ. Force of mortality decreases within each year of age.

Whole life annuity äₓ

Expected present value of 1 per year payable in advance while alive: äₓ = Σₖ≥₀ vᵏ ₖpₓ. Identity äₓ = (1 − Aₓ)/d links it to whole life insurance.

Whole life insurance Aₓ

Expected present value of 1 payable at end of year of death: Aₓ = Σₖ≥₀ vᵏ⁺¹ ₖpₓ qₓ₊ₖ. For continuous payment use Āₓ with force of interest δ.

Net premium reserve

Prospective reserve ₜV = EPV(future benefits) − EPV(future net premiums) at duration t, conditional on survival. Equals accumulated past premiums minus benefits under equivalence principle.

Thiele's differential equation

d/dt V(t) = δ·V(t) + P − b·μ(t) − (S − V(t))·μ(t), where b is death benefit, S survival benefit. Continuous-time reserve recursion used for profit and risk analysis.

Recursive reserve formula

(ₜV + P)(1+i) = qₓ₊ₜ·S + pₓ₊ₜ·ₜ₊₁V, where S is death benefit. Allows forward or backward computation of reserves from terminal value.

Equivalence principle

Premium set so EPV of premiums equals EPV of benefits plus expenses at policy inception, i.e. reserve at duration 0 is zero. Basis for net and gross premium calculation.

Gross premium

Premium covering benefits plus expenses (initial, renewal, claim) and profit margin. Gross premium reserve uses the same premium basis; expense loadings are added to benefit outgo in the recursion.

Profit signature

Sequence of expected profits at each duration multiplied by survival probability to start of that duration: Πₜ = profitₜ · ₜ₋₁p. Used to compute NPV at the risk discount rate.

Risk discount rate

Rate r used to discount the profit signature: NPV = Σₜ Πₜ·vᵗ. Reflects the shareholders' required return; higher r reduces NPV and lowers the internal rate of return margin.

Internal rate of return

Rate i* at which NPV of profit signature equals zero. Compared with the risk discount rate to judge profitability; multiple roots may occur when cash flows change sign more than once.

Multiple decrement table

Model with several causes of exit. Dependent rate (aq)ᵈₓ is the probability of exit due to cause d in the year; independent rate qᵈₓ assumes other causes removed.

Associated single decrement

Table for one cause with all other decrements removed. Independent rate qᵈₓ relates to dependent rate (aq)ᵈₓ; under UDD in the multiple table, (aq)ᵈₓ = qᵈₓ·(1 − ½Σⱼ≠d qʲₓ).

Markov multi-state model

States with transition intensities μᵢⱼ(t); probabilities satisfy Kolmogorov forward equations dpᵢⱼ/dt = Σₖ pᵢₖ μₖⱼ − pᵢⱼ Σₖ≠ⱼ μⱼₖ. Used for disability, critical illness and long-term care.

Gompertz law

Mortality law μ(x) = B·cˣ, giving exponentially increasing force of mortality with age. Makeham extends it to μ(x) = A + B·cˣ, adding an age-independent component.

Lee-Carter model

Mortality projection model: ln m(x,t) = aₓ + bₓ·kₜ + εₓₜ, where aₓ is average log mortality, bₓ age sensitivity, kₜ time trend. kₜ typically modelled as a random walk with drift.

Select mortality

Mortality in years immediately after underwriting is lower than ultimate mortality at the same age, due to selection. Denoted q₍ₓ₎₊ₜ for select duration t; converges to ultimate qₓ₊ₜ after the select period.

Value at Risk (VaR)

The α-quantile of the loss distribution: smallest l such that P(L ≤ l) ≥ α. Simple to communicate but not subadditive and ignores the size of losses beyond the threshold.

Tail Value at Risk (TVaR)

Conditional expected loss beyond VaR: TVaR_α = E[L | L > VaR_α], also called Expected Shortfall. Coherent, subadditive, and captures the severity of tail losses.

Coherent risk measure axioms

A risk measure ρ is coherent if it satisfies monotonicity, translation invariance, positive homogeneity, and subadditivity: ρ(X+Y) ≤ ρ(X) + ρ(Y). TVaR is coherent; VaR is not.

Stochastic simulation for reserves

Simulate future mortality, interest and lapse paths; compute reserves or profit for each path; summarise distribution. Used to quantify reserve variability and capital requirements under stress.

Deterministic profit test

Project expected cash flows using best-estimate assumptions for mortality, interest, expenses and lapses. Produces the profit signature and NPV or IRR for a given premium and basis.

Asset-liability modelling

Joint projection of asset and liability cash flows to test solvency, liquidity and interest-rate mismatch. Immunisation requires matching duration and convexity of assets and liabilities.

Duration and convexity

Macaulay duration = Σ t·PV(CFₜ)/Σ PV(CFₜ); modified duration = D/(1+i). Convexity adjusts for curvature. Used to immunise reserves against interest-rate changes.

Premium basis vs reserving basis

Premium basis sets premiums at inception; reserving basis (often more prudent) values future liabilities. A stronger reserving basis increases reserves and reduces reported profit.

Surrender value

Amount paid on lapse; modelled as a decrement with its own rate. Affects reserve recursion: (V+P)(1+i) = qᵈ·Sᵈ + qˡ·SV + p·ₜ₊₁V where qᵈ, qˡ are decrement rates.

Mortality improvement

Long-term reduction in mortality rates over time. Projected via Lee-Carter or similar; ignoring it understates annuity costs and overstates life insurance profitability.