Casualty Actuarial Society Credentials (ACAS/FCAS) Exam 7 — Estimation of Policy Liabilities, Insurance Company Valuation, and Enterprise Risk Management Syllabus

Define the increased limits factor (ILF) and the loss elimination ratio (LER) in terms of limited expected values.

$\text{ILF}(L)=\dfrac{E[X\wedge L]}{E[X\wedge B]}$ relative to a basic limit $B$ (assuming losses-only / no risk load). $\text{LER}(d)=\dfrac{E[X\wedge d]}{E[X]}$, the fraction of ground-up losses eliminated by a deductible $d$.

What is a consistency requirement among increased limits factors, and why?

ILFs must increase at a decreasing rate as the limit rises (the marginal premium for each additional layer must be non-increasing), i.e., the ILF curve is concave. Otherwise an insured could buy a higher limit cheaper by combining layers — an arbitrage / inconsistency.

Compare proportional (quota share, surplus share) versus non-proportional (excess-of-loss, stop-loss) reinsurance in one sentence each.

Proportional: reinsurer shares a fixed percentage of premiums and losses (quota share = constant %; surplus share = % varies by policy size relative to retention line). Non-proportional: reinsurer pays only losses above a retention—per-risk/per-occurrence XOL, or aggregate stop-loss on the whole book.

What is the difference between a working-layer, exposed, and clash (or higher) excess reinsurance treaty regarding loss frequency and credibility?

Working-layer treaties attach low, with frequent losses and credible experience (experience-rated). Higher/clash layers attach far above per-risk limits, are loss-free most years, driven by multi-claim/multi-policy events, and must be exposure-rated since experience is non-credible.

List the main reasons (functions) an insurer purchases reinsurance.

(1) Capacity—write larger limits; (2) Stabilization of results (smooth volatility); (3) Catastrophe protection; (4) Surplus relief / financing (improve leverage ratios); (5) Underwriting/market entry & exit assistance; (6) Access to reinsurer expertise.

In experience rating an excess treaty, what adjustments must be applied to historical losses before they can be projected to the future layer?

Trend each loss to the future cost level, apply loss development (to ultimate) including the leveraged excess development, restate to the prospective retention/limit, and adjust exposures (on-level for rate/exposure changes) before computing the burning cost and loading for expenses and risk.

What is the burning cost (experience) rate and how is it converted to a treaty rate?

Burning cost $=\dfrac{\text{trended, developed excess losses}}{\text{subject premium}}$. The treaty rate $=\dfrac{\text{expected excess losses}}{\text{subject premium}}\div(1-\text{expense/profit loading})$, i.e., burning cost loaded for expenses, risk margin, and brokerage.

Define ALAE treatment options 'pro rata' versus 'in addition to' the limit in an excess reinsurance contract.

'Pro rata' (in proportion): ALAE is shared in the same ratio as the loss falls in the layer (reinsurer pays ALAE × layer loss / total loss). 'In addition': ALAE is added to indemnity and the sum is applied against the retention/limit, increasing reinsurer exposure. They produce different recoveries for losses near the attachment.

What distinguishes economic value, fair (market-consistent) value, and statutory/GAAP book value in insurance company valuation?

Economic/intrinsic value discounts expected future cash flows at a risk-appropriate rate; fair value is the market-consistent exchange price (often using risk-neutral valuation); statutory/GAAP book value follows accounting rules with conservatism (statutory) or matching (GAAP) and may not equal economic value.

State the appraisal (actuarial) value of an insurer and its three components.

Appraisal value $=$ Adjusted net worth (ANW) $+$ Value of existing/in-force business (PV of future earnings from current book) $+$ Value of future new business (goodwill). It represents the economic worth to an acquirer.

What is embedded value (EV) and how does it differ from appraisal value?

Embedded value $=$ Adjusted net worth $+$ Value of in-force business (present value of future shareholder earnings from existing policies, net of cost of capital). It excludes the value of future new business, which appraisal value includes as goodwill.

Define the Dividend Discount Model (DDM) valuation of an insurer's equity.

$V_0 = \sum_{t=1}^{\infty} \dfrac{D_t}{(1+r_e)^{t}}$, where $D_t$ are distributable earnings (dividends to shareholders, constrained by required capital) and $r_e$ is the cost of equity. Often split into an explicit forecast period plus a terminal value.

Write the Capital Asset Pricing Model (CAPM) and explain its use in setting an insurer's cost of equity.

$r_e = r_f + \beta\,(E[r_m] - r_f)$, where $r_f$ is the risk-free rate, $\beta$ the systematic risk of the insurer's equity, and $(E[r_m]-r_f)$ the market risk premium. It supplies the discount rate / target return for pricing and valuation.

State the insurance CAPM (underwriting-beta) formula for the fair underwriting profit margin.

$r_{uw} = -k\,r_f + \beta_{uw}\,(E[r_m]-r_f)$, where $k$ is the funds-generating (premium-to-surplus reserve) coefficient, $r_f$ the risk-free rate, and $\beta_{uw}$ the underwriting beta. The $-k\,r_f$ term credits investment income on policyholder-supplied funds.

Give the weighted average cost of capital (WACC) formula.

$\text{WACC} = \dfrac{E}{V}\,r_e + \dfrac{D}{V}\,r_d\,(1-\tau)$, where $E$ and $D$ are market values of equity and debt, $V=E+D$, $r_e$ and $r_d$ the costs of equity and debt, and $\tau$ the tax rate.

What is the risk-adjusted discounted cash flow (DCF) approach to pricing, and what cash flows are discounted?

Project the policyholder and shareholder cash flows—premiums, losses, expenses, taxes, and the cost of holding required capital—and discount them at a risk-adjusted rate (or risk-neutral with risk-free). The premium is set so the NPV of shareholder cash flows equals zero (target return achieved).

Describe the internal rate of return (IRR) pricing model for insurance.

Treat the policy from the equityholder's view: capital is contributed (outflow) and released over time with underwriting and investment earnings (inflows). The IRR is the rate making the NPV of equity flows zero; the premium is adequate if IRR $\geq$ the target cost of equity.

In the risk-adjusted return on capital framework, define RAROC and the decision rule.

$\text{RAROC} = \dfrac{\text{risk-adjusted return (e.g., expected profit net of expected loss)}}{\text{economic (risk) capital}}$. Accept/retain the business or allocate more capital when $\text{RAROC} \geq$ the hurdle rate (cost of capital); it puts all units on a common risk-adjusted footing.

What is Return on Equity (ROE) and how is it decomposed via DuPont for an insurer?

$\text{ROE}=\dfrac{\text{Net income}}{\text{Equity}}$. DuPont: $\text{ROE}=\text{Profit margin}\times\text{Asset turnover}\times\text{Leverage}=\dfrac{NI}{\text{Rev}}\times\dfrac{\text{Rev}}{\text{Assets}}\times\dfrac{\text{Assets}}{\text{Equity}}$. For insurers leverage is largely premium-to-surplus and reserve leverage.

Define the combined ratio and operating ratio, and what each tells you.

$\text{Combined ratio}=\dfrac{\text{Losses+LAE}}{\text{Earned premium}}+\dfrac{\text{Underwriting expenses}}{\text{Written (or earned) premium}}$ = loss ratio + expense ratio. $\text{Operating ratio}=\text{Combined ratio}-\dfrac{\text{Investment income}}{\text{Earned premium}}$. Combined $<100\%$ means underwriting profit; operating ratio adds investment income.

Why can economic value added (EVA) be preferred to accounting ROE for performance measurement?

$\text{EVA}=\text{NOPAT}-r\times\text{Capital}$, charging for the cost of capital used. Unlike ROE it is a dollar measure that rewards growth only when returns exceed the cost of capital, discourages capital-destroying growth, and aligns with shareholder value creation.

What is the COSO definition of Enterprise Risk Management (ERM)?

ERM is a process, effected by an entity's board, management, and personnel, applied across the enterprise and in strategy-setting, designed to identify potential events that may affect the entity, manage risk within its risk appetite, and provide reasonable assurance regarding achievement of objectives.

List the four broad categories of risk in an ERM risk classification (commonly used by rating agencies/regulators).

(1) Market risk; (2) Credit risk; (3) Insurance/underwriting risk (incl. reserving and pricing/cat risk); (4) Operational risk. Some frameworks add strategic risk and liquidity risk as distinct categories.

Distinguish a pure risk-management view from a value-based (strategic) view of ERM.

The traditional/pure view treats ERM as loss control—reducing the probability and severity of adverse events to protect solvency. The value-based view treats risk as a resource to be optimized, balancing risk and return to maximize firm value and inform strategy (taking the 'right' risks, not just avoiding risk).

What are the common methods of risk identification in an ERM program?

Risk workshops/brainstorming, structured questionnaires and surveys, risk registers/checklists, process flow analysis, scenario analysis and stress testing, loss-event databases, SWOT/PESTLE, and reverse stress testing (start from failure and find causes).

Define Value at Risk (VaR) and Tail Value at Risk (TVaR/CTE).

$\text{VaR}_p(X)=$ the $p$-quantile of the loss distribution, $\inf\{x: F_X(x)\geq p\}$. $\text{TVaR}_p(X)=E[X\mid X>\text{VaR}_p(X)]$ (for continuous $X$), the average loss in the worst $(1-p)$ tail. TVaR is more sensitive to extreme tail severity than VaR.

What is a coherent risk measure, and which of VaR and TVaR is coherent?

A coherent measure $\rho$ satisfies monotonicity, translation invariance, positive homogeneity, and subadditivity ($\rho(X+Y)\leq\rho(X)+\rho(Y)$). TVaR is coherent; VaR is not, because it can violate subadditivity (it fails to always reward diversification).

In risk aggregation, how does the choice of copula affect tail dependence and required capital?

A copula links marginals into a joint distribution; the Gaussian copula has zero tail dependence (extremes nearly independent), while the t-copula and Gumbel (Archimedean) copulas exhibit upper-tail dependence (joint extremes more likely), producing fatter aggregate tails and higher diversified capital requirements.

What is risk-based capital (RBC) in the U.S. and the formula relationship among its risk components?

RBC is a regulatory minimum capital based on a company's risk profile. The authorized control level uses a square-root covariance/aggregation: $\text{RBC} = R_0 + \sqrt{R_1^{2}+R_2^{2}+R_3^{2}+R_4^{2}+R_5^{2}}$, recognizing diversification among asset, credit, reserving, premium, and off-balance-sheet risks.

Define the RBC ratio and the regulatory action levels.

$\text{RBC ratio}=\dfrac{\text{Total Adjusted Capital}}{\text{Authorized Control Level RBC}}$. Action levels (as % of ACL or of the ratio): Company Action ($<200\%$ of ACL, i.e. ratio $<2$), Regulatory Action ($<150\%$), Authorized Control ($<100\%$), Mandatory Control ($<70\%$).

Contrast the Solvency II three-pillar structure with U.S. solvency regulation.

Solvency II Pillar 1: quantitative capital (SCR via standard formula or internal model, calibrated to 99.5% VaR over 1 year, plus MCR); Pillar 2: governance, ORSA, supervisory review; Pillar 3: disclosure/reporting. U.S. uses factor-based RBC, statutory accounting, and ORSA but is less market-consistent and not a single VaR calibration.

What is ORSA and what is its purpose?

The Own Risk and Solvency Assessment is an internal process (required by the NAIC ORSA Model Act and Solvency II) in which the insurer assesses its own risk profile, risk management adequacy, and current and prospective solvency under normal and stressed conditions, reporting the results to its board and regulator.

Define the Solvency II Solvency Capital Requirement (SCR) calibration and the Risk Margin concept.

SCR is the economic capital to absorb losses at a $99.5\%$ VaR (1-in-200) over a one-year horizon. Technical provisions $=$ best-estimate liabilities $+$ risk margin, where the risk margin is a cost-of-capital margin: $\text{RM}=\text{CoC}\times\sum_t \dfrac{\text{SCR}_t}{(1+r_{t+1})^{t+1}}$ (CoC typically $6\%$).

What is economic capital and how does it differ from regulatory capital?

Economic capital is the amount of capital a firm internally determines it needs to remain solvent at a chosen risk tolerance (e.g., a target probability of ruin or rating) over a defined horizon, using the firm's own models. Regulatory capital is the externally-prescribed minimum (RBC, SCR); economic capital is internal, risk-sensitive, and often used for allocation and pricing.

Describe the cost-of-capital method for determining a risk margin (market value margin) on reserves.

Project the capital required to support the runoff of the liabilities each future year, multiply each by a cost-of-capital rate (e.g., $6\%$) representing the return demanded above risk-free, and discount: $\text{Risk margin}=\sum_t \dfrac{\text{CoC}\times C_t}{(1+r)^{t+1}}$, where $C_t$ is required capital at time $t$.

What is the leverage (premium-to-surplus) ratio and why do regulators monitor it?

$\text{Premium-to-surplus ratio}=\dfrac{\text{Net written premium}}{\text{Policyholder surplus}}$. High leverage (commonly a benchmark of $\approx 3{:}1$) signals that a small adverse loss or reserve shock can impair surplus, so it is a key solvency early-warning (IRIS) ratio.

Define the cost-of-capital (risk load) approach to allocating capital by line via the marginal/Myers-Read method.

Capital is allocated so each line's marginal contribution to the firm's default value (option value of default) is covered. Myers-Read allocates the total default cost by each line's marginal effect on the insurer's overall default option, summing exactly to total capital (additive surplus allocation).

What is the difference between the chain-ladder and Cape Cod (Stanard-Bühlmann) methods for estimating ultimate losses?

Chain ladder relies solely on development factors applied to reported losses (heavy weight on immature data). Cape Cod estimates an a priori expected loss ratio from the data itself—$\hat{ELR}=\dfrac{\sum \text{reported losses}}{\sum (\text{premium}\times \% \text{reported})}$—then applies BF logic, producing a more stable result by using exposure (used-up premium).

What is model risk in reserving and how can actuaries mitigate it?

Model risk is the risk that the selected model is structurally wrong or mis-specified (wrong distribution, missing trends, inappropriate assumptions). Mitigation: use multiple methods/models, back-test and validate against actual emergence, perform sensitivity testing, apply expert judgment, and reconcile differences across methods.

State Stone's criteria (desirable properties) for a good risk measure used in capital/risk-load contexts.

A useful risk measure should reflect the relevant features of the loss distribution (e.g., capture tail and dispersion), be consistent/sub-additive to reward diversification, be objective and computable, and align with the firm's risk preferences and the decision being made (pricing vs. solvency).

How do you compute the discounted (present value) reserve, and why might a risk margin then be added?

$\text{PV reserve}=\sum_t \dfrac{\text{expected payment}_t}{(1+r)^{t}}$ using the payment pattern and a discount rate $r$. Because discounting removes the implicit conservatism of nominal reserves and ignores uncertainty, a risk/market-value margin is added so the carried liability reflects the cost of bearing the runoff risk.