Korea Insurance Actuary cheat sheet

Present value of a life annuity-immediate: ä_x = Σ_{k=0}^∞ v^k * _kp_x; annuity-due adds one payment at time 0: ä_x = 1 + a_x (using v=1/(1+i)).

Net premium for a whole life insurance: P_x = A_x / ä_x, where A_x = Σ_{k=0}^∞ v^{k+1} * kp_x * q{x+k} (assuming benefits at end of year of death).

Reserve at time t (prospective): tV = A{x+t} - P_x * ä_{x+t}; recursive relation: (tV + P)(1+i) = q{x+t} + p_{x+t} * _{t+1}V.

Gross premium includes loading for expenses: G * ä_x = A_x + e_0 + e_1 * ä_x + e_2 * (a_x - ä_x) etc., where e_0=initial expense, e_1=per-policy renewal, e_2=per-premium expense.

Mortality law: Makeham's law μ_{x} = A + B c^x; Gompertz's law μ_{x} = B c^x. Under constant force μ, _tp_x = exp(-μ t).

For two lives, joint life status (xy) has survival probability tp{xy} = _tp_x * _tp_y (independent). Last-survivor status (xȳ) has tp{xȳ} = _tp_x + _tp_y - tp{xy}.

Fractional age assumptions: UDD gives sq{x} = s * q_x for 0≤s≤1; constant force gives _sp_x = (p_x)^s. For deaths, UDD: _tq_x = t q_x for integer t plus fractional.

Discount function: v = 1/(1+i); d = i/(1+i) = 1 - v. Relationship: i - d = i*d. Force of interest δ = ln(1+i).

Net level premium reserve can be computed retrospectively: tV = [P * ä{x:t} - A^1_{x:t}] / _tp_x (accumulated value of premiums minus cost of insurance).

Whole life insurance net single premium: A_x = Σ_{k=0}^∞ v^{k+1} * kp_x * q{x+k}; under constant force μ, A_x = μ/(μ+δ).

Term life insurance: A^1_{x:n} = Σ_{k=0}^{n-1} v^{k+1} * kp_x * q{x+k}; endowment insurance: A_{x:n} = A^1_{x:n} + _np_x * v^n.

Continuous whole life insurance: Ā_x = ∫_0^∞ v^t * tp_x * μ{x+t} dt; under constant force, Ā_x = μ/(μ+δ); relationship: Ā_x = (i/δ) A_x under UDD.

Annual premium for endowment insurance: P_{x:n} = A_{x:n} / ä_{x:n}; for term insurance: P^1_{x:n} = A^1_{x:n} / ä_{x:n}.

Variance of present value for whole life insurance: Var(Z) = ²A_x - (A_x)², where ²A_x uses v² (i.e., discount at rate (1+i)²-1).

For increasing/decreasing benefits, use commutation functions: (IA)_x = Σ (k+1) v^{k+1} * kp_x * q{x+k}; (DA)_x = Σ (n-k) v^{k+1} * kp_x * q{x+k} for n-year decreasing.

Premium conversion: P_x = (1/ä_x) - d (since A_x = 1 - d * ä_x). Similarly, P^1_{x:n} = (A^1_{x:n} / ä_{x:n}).

Reserve recursion for term insurance: (tV + P)(1+i) = q{x+t} * S + p_{x+t} * _{t+1}V, where S is sum assured (if payable at end of year).

For joint life annuities, ä_{xy} = Σ v^k * kp{xy}; relationship: ä_{xy} = ä_x + ä_y - ä_{xȳ} (where xȳ is last-survivor).

Collective risk model: S = Σ_{i=1}^N X_i, where N is claim count (frequency) and X_i are individual claim amounts (severity). E[S] = E[N] * E[X]; Var(S) = E[N] * Var(X) + Var(N) * (E[X])².

Compound Poisson model: N ~ Poisson(λ), then E[S] = λ * E[X], Var(S) = λ * E[X²]. Probability generating function: P_S(z) = exp(λ (P_X(z) - 1)).

Lundberg's inequality: ψ(u) ≤ e^{-R u}, where R is the adjustment coefficient (root of λ + δ = λ * M_X(r) for compound Poisson with premium loading).

Cramér-Lundberg model: surplus U(t) = u + c t - S(t), where c = premium rate. Ruin probability ψ(u) = P(U(t) < 0 for some t ≥ 0).

Adjustment coefficient R solves: 1 + (1+θ) μ r = M_X(r), where θ is safety loading and μ = E[X]. For exponential claims with mean β, R = θ/(β(1+θ)).

Stop-loss premium: E[(S - d)_+] = ∫_d^∞ (1 - F_S(x)) dx. For compound Poisson, stop-loss premium can be computed via Panjer recursion.

Panjer recursion for compound Poisson: f_S(x) = (λ/x) ∫0^x (x-y) f_X(x-y) f_S(y) dy for continuous case; discrete version: f_S(x) = (λ/x) Σ{y=1}^x y f_X(y) f_S(x-y).

Risk measures: VaR_α(S) = inf{x: P(S ≤ x) ≥ α}; TVaR_α(S) = E[S | S > VaR_α(S)] + (1-α)(VaR_α(S) - E[S | S ≤ VaR_α(S)])? Actually TVaR_α = (1/(1-α)) ∫_α^1 VaR_u(S) du.

Individual risk model: S = Σ_{i=1}^n X_i, where X_i = B_i * Y_i, B_i ~ Bernoulli(q_i). Then E[S] = Σ q_i μ_i, Var(S) = Σ [q_i Var(Y_i) + q_i(1-q_i) μ_i²].

Fisher equation: (1 + i) = (1 + r)(1 + π), where i = nominal rate, r = real rate, π = inflation. Approximation: i ≈ r + π.

Present value of a perpetuity: PV = C / r; growing perpetuity: PV = C / (r - g), valid for r > g.

Net present value (NPV) = Σ_{t=0}^n CF_t / (1+r)^t; accept project if NPV > 0. Internal rate of return (IRR) solves NPV = 0.

Duration (Macaulay) = Σ t * PV(CF_t) / Price; modified duration = Macaulay duration / (1 + y). Price change approximation: ΔP/P ≈ -D_mod * Δy.

Yield to maturity (YTM) is the discount rate that equates bond price to present value of coupons and principal. For a bond with annual coupon C, face F, n years: P = Σ C/(1+y)^t + F/(1+y)^n.

Utility theory: expected utility = Σ p_i U(w_i). Risk aversion implies U''(w) < 0. Certainty equivalent CE satisfies U(CE) = E[U(W)]. Risk premium = E[W] - CE.

Sharpe ratio = (E[R_p] - R_f) / σ_p. Capital Asset Pricing Model (CAPM): E[R_i] = R_f + β_i (E[R_m] - R_f), where β_i = Cov(R_i, R_m) / Var(R_m).

Law of one price: identical cash flows must have same price. Arbitrage-free pricing: price = expected discounted payoff under risk-neutral measure.

Elasticity: price elasticity of demand = (% change in quantity) / (% change in price). For insurance, elasticity of demand affects premium volume and solvency.

Accounting equation: Assets = Liabilities + Equity. Balance sheet presents financial position at a point in time.

Income statement: Revenues - Expenses = Net Income. Insurance premiums are recognized as revenue over the coverage period (unearned premium reserve).

Insurance reserves: Premium reserves (net level premium reserve) and claim reserves (IBNR, outstanding claims). Under IFRS 17, insurance contracts measured using building blocks: fulfilment cash flows + contractual service margin (CSM).

Solvency II / K-ICS: Risk-based capital requirement. K-ICS (Korean Insurance Capital Standard) ratio = Available Capital / Required Capital; regulatory minimum 100%, supervisory intervention below 150% (as per FSC guidelines).

Duration gap analysis: ΔE ≈ -D_A * A * Δy + D_L * L * Δy, where D_A and D_L are durations of assets and liabilities. Immunization requires D_A * A = D_L * L and convexity matching.

Profit testing: Profit = Premium - Claims - Expenses - Increase in reserves + Investment income. Return on equity (ROE) = Net Income / Equity.

Embedded value (EV) = Net asset value (NAV) + Value of in-force business (VIF). VIF = Present value of future profits from in-force policies.

Loss ratio = Incurred claims / Earned premiums. Expense ratio = Underwriting expenses / Written premiums. Combined ratio = Loss ratio + Expense ratio; < 100% indicates underwriting profit.

Fair value hierarchy: Level 1 (quoted prices), Level 2 (observable inputs), Level 3 (unobservable inputs). Insurance liabilities often classified as Level 3.

Korean Insurance Business Act (IBA) Article 4: Insurance contracts must be based on the principle of utmost good faith. Insurer must explain key contract terms (설명의무) and provide a copy of the policy.

Duty of disclosure (고지의무): The policyholder must disclose material facts asked by the insurer. Breach allows insurer to cancel the contract within 1 month of knowledge and 3 years of contract inception (IBA Article 651).

Insurable interest: In life insurance, the policyholder must have insurable interest in the insured at contract inception. In non-life, insurable interest required at time of loss.

Premium payment and grace period: If premium is not paid, insurer may terminate after giving notice (typically 15 days). Policyholder can reinstate within 2 years (부활) if insurable.

Third-party liability insurance: Direct action against insurer allowed for bodily injury (자동차손해배상보장법).

Insurance contract standard terms (약관) must be filed with the Financial Services Commission (FSC) and can be changed with prior approval. Unfair terms are void.

Solvency regulation: K-ICS introduced in 2023, replacing RBC. Minimum capital requirement based on risk factors (insurance, market, credit, operational).

Policyholder protection: Depositor Protection Act covers insurance claims up to 50 million KRW per person per insurer (as of 2024, some exceptions).

Anti-discrimination and suitability: Insurers must assess suitability and appropriateness when selling complex products; cooling-off period (청약철회) of 15 days for life insurance.

Life insurance products: Whole life, term life, endowment, annuity, variable life, universal life. Non-life: property, casualty, health, motor, liability.

Pricing components: Expected claims (mortality/morbidity), expenses, profit margin, and risk margin. Gross premium = Net premium + expense loading + risk loading.

Underwriting: Selection of risks based on age, health, occupation, etc. Substandard risks charged extra premium or excluded.

Reserving: For life insurance, net level premium reserve; for non-life, unearned premium reserve (UPR) and outstanding claims reserve (OCR) including IBNR.

Asset-liability management (ALM): Matching duration and convexity of assets and liabilities to minimize interest rate risk. Immunization and cash flow matching.

Reinsurance: Proportional (quota share, surplus) and non-proportional (excess of loss, stop loss). Reinsurer's share of claims and premiums.

Product development process: Market research, design, pricing, filing with regulator, launch, monitoring. Profit testing and sensitivity analysis.

Distribution channels: Direct, agency, bancassurance, online. In Korea, bancassurance allowed for life and non-life with restrictions (e.g., 25% cap on sales).

Experience monitoring: Track actual vs expected mortality, morbidity, lapse, expense. Adjust pricing and reserves accordingly.

Probability distributions: Binomial(n,p): mean np, variance np(1-p). Poisson(λ): mean λ, variance λ. Exponential(λ): mean 1/λ, variance 1/λ².

Normal distribution: X ~ N(μ, σ²). Standardize: Z = (X-μ)/σ. Central Limit Theorem: sum of independent variables tends to normal.

Lognormal: If Y ~ N(μ, σ²), then X = e^Y has mean exp(μ + σ²/2) and variance (exp(σ²) - 1) exp(2μ + σ²).

Maximum likelihood estimation: Choose parameters that maximize likelihood function L(θ) = Π f(x_i|θ). Log-likelihood often used.

Hypothesis testing: Type I error (α) = reject true null; Type II error (β) = fail to reject false null. Power = 1-β.

Confidence interval for mean (σ known): x̄ ± z_{α/2} * σ/√n. For σ unknown, use t-distribution with n-1 degrees of freedom.

Goodness-of-fit: Chi-square test statistic = Σ (O_i - E_i)² / E_i. Degrees of freedom = categories - 1 - parameters estimated.

Bayesian inference: Posterior ∝ Likelihood × Prior. Conjugate priors: Beta-Binomial, Gamma-Poisson, Normal-Normal.

Copulas: Model dependence between variables. Gaussian copula: C(u,v) = Φ_ρ(Φ^{-1}(u), Φ^{-1}(v)). Used in risk aggregation.