CAS Exam 9 cheat sheet

Risk is the uncertainty of outcomes; pure risk has only loss or no-loss outcomes, speculative risk has gain or loss outcomes. Insurable risks are typically pure, accidental, measurable, and independent in large pools.

Expected loss = frequency × severity. For a Poisson frequency N with mean λ and independent severity X, aggregate expected loss = λ·E[X] and Var = λ·E[X²] = λ(Var(X) + E[X]²).

Coefficient of variation = standard deviation / mean; it measures relative risk and lets pools of different size or scale be compared. Higher frequency and lower severity reduce relative volatility of aggregate losses.

Value at Risk (VaR) at confidence p is the smallest loss threshold exceeded with probability 1−p; VaR is not subadditive in general, so VaR(A+B) can exceed VaR(A)+VaR(B).

Tail Value at Risk (TVaR) = E[L | L > VaR_p]; it is coherent, satisfies subadditivity, and for continuous distributions equals the average of losses beyond the p quantile.

Ruin probability for a surplus process with premium loading: ψ(u) ≤ e^(−Ru), where R is the adjustment coefficient solving the Lundberg equation; larger initial surplus u lowers the bound exponentially.

Risk control hierarchy: avoidance, loss prevention (reduce frequency), loss reduction (reduce severity), separation and duplication (reduce correlation), and transfer (insurance, reinsurance, derivatives).

Moral hazard changes behavior after coverage; adverse selection changes the composition of insureds before coverage. Deductibles, coinsurance, and experience rating mitigate both.

Portfolio expected return = Σ wᵢ·E[Rᵢ]. Portfolio variance for two assets = w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρσ₁σ₂; only covariance terms diversify, not individual variances.

Minimum-variance portfolio weight for two assets: w₁ = (σ₂² − ρσ₁σ₂)/(σ₁² + σ₂² − 2ρσ₁σ₂). The efficient frontier is the upper branch of the minimum-variance set.

CAPM: E[Rᵢ] = R_f + βᵢ(E[R_m] − R_f), with βᵢ = Cov(Rᵢ,R_m)/Var(R_m). Beta measures systematic risk; idiosyncratic risk is not priced in equilibrium.

Sharpe ratio = (E[R_p] − R_f)/σ_p; the tangency portfolio maximizes it. Treynor ratio uses beta in the denominator and is appropriate for well-diversified portfolios.

Duration approximates bond price sensitivity: ΔP/P ≈ −D·Δy + ½·C·(Δy)², where D is modified duration and C is convexity. Convexity is positive for plain vanilla bonds and benefits the holder.

Black-Scholes call: C = S·N(d₁) − K·e^(−rT)·N(d₂), with d₁ = [ln(S/K) + (r + σ²/2)T]/(σ√T) and d₂ = d₁ − σ√T. Put-call parity: C − P = S − K·e^(−rT).

Delta = ∂V/∂S, gamma = ∂²V/∂S², vega = ∂V/∂σ, theta = ∂V/∂t, rho = ∂V/∂r. Delta-hedging removes first-order price risk but leaves gamma and vega exposure.

Credit risk: expected loss = PD × LGD × EAD. Unexpected loss ≈ EAD × √(PD·σ²_LGD + LGD²·σ²_PD) and is covered by economic capital rather than reserves.

ERM aggregates risks across silos to measure portfolio-level risk; diversification benefit = Σ standalone capital − consolidated capital, and it is limited when tail correlations rise in stress.

Risk appetite is the amount and type of risk the firm accepts; risk tolerance sets quantitative boundaries; risk limits translate tolerance into line-level constraints monitored against triggers.

COSO ERM components: governance and culture, strategy and objective-setting, performance, review and revision, and information/communication/reporting. It is a process, not a checklist.

Three lines model: business units own risk (first line), risk and compliance functions oversee and set frameworks (second line), internal audit provides independent assurance (third line).

Economic capital is the capital needed to keep insolvency probability below a target over a horizon; it is often measured at a 99.5% one-year VaR or TVaR and allocated to lines by contribution to tail risk.

Risk-adjusted performance: RAROC = (expected return − expected loss − expenses)/economic capital; compare to the hurdle rate. EVA = net income − cost of capital × capital employed.

Key risk indicators are leading, measurable, and tied to limits; a KRIs breach should trigger escalation, not automatically force a transaction. Scenario and stress testing complement models, which fail in regime shifts.

Concentration risk (single name, sector, geography, counterparty) and correlation assumptions dominate tail outcomes; model risk is managed through validation, backtesting, and conservative overlays.

Loss ratio = incurred losses / earned premium; expense ratio = underwriting expenses / written premium; combined ratio = loss ratio + expense ratio; a combined ratio above 100% means an underwriting loss.

Reserve adequacy: IBNR covers incurred-but-not-reported claims and development on known claims; case reserves are set claim by claim. Reserve strengthening reduces current income and surplus.

Policyholders' surplus = admitted assets − liabilities. Writing premium consumes surplus because acquisition costs are paid before premium is earned; the capacity ratio limits premium to a multiple of surplus.

NAIC IRIS ratios (e.g., premium-to-surplus, change in surplus, liabilities-to-liquid-assets) flag anomalies, not violations; RBC ratios compare total adjusted capital to risk-based capital and trigger action levels at 200%, 150%, 100%, and 70%.

RBC charges apply factors to asset risk (R0–R2), credit risk (R3), reserve risk (R4), and premium risk (R5); the covariance formula R = √(R0² + R1² + R2² + R3² + R4² + R5²) assumes independence across categories.

Reinsurance reduces net retained risk and required capital but introduces credit risk and ceded premium; net leverage = net written premium / surplus measures retained exposure.

Investment leverage = invested assets / surplus; total leverage = (invested assets + net premium written)/surplus. Higher leverage raises return on surplus and insolvency risk simultaneously.

Cash flow analysis: operating cash flow = premium collected − losses paid − expenses; a growing insurer can show negative operating cash flow while remaining solvent because of the timing of claim payments.