Risk Margin

Study guides: Exam 6C, Exam 7, Exam GI-301

A risk margin is an amount held on top of a best (central) estimate to allow for the uncertainty in an insurance liability or in an insurer's risks. It carries two exam meanings. In unpaid-claim estimation (Exam 7) it is the loading above the central estimate of unpaid claims that brings the provision to a target probability of adequacy. In Canadian solvency (Exam 6C) the risk margins are the risk-by-risk capital requirements of the MCT.

$\text{Risk margin} = V_p - \text{Central estimate}$

$\text{Target capital} = \text{IR} + \text{MR} + \text{CR} + \text{OR} - \text{DC}$

$V_p$ is the $p$ -th percentile of the liability's distribution, where $p$ is the probability of adequacy. IR, MR, CR and OR are the MCT's insurance, market, credit and operational risk margins, and DC is the Diversification Credit.

Exam 7: a margin over the central estimate (Marshall et al.)

Sources of uncertainty.
- Independent risk is the random part of process and parameter risk. It is measured with Mack, bootstrap, GLM or Bayesian models.
- Internal systemic risk is specification error, parameter selection error and data error. It is scored against best practice on a balanced scorecard, with each indicator rated 1 to 5, and the weighted score is mapped to a CoV.
- External systemic risk covers seven categories: economic and social; legislative, political and claims inflation; claim management process change; expense; event; latent claim; and recovery. Each is assessed directly.
Consolidation. CoVs are set by valuation class, separately for outstanding claims and premium liabilities. They are combined across classes with judgmental correlations (nil, low $25\%$ , medium $50\%$ , high $75\%$ , full). The three sources are taken as independent, so $\mathrm{CoV}^2 = \mathrm{CoV}_{\text{ind}}^2 + \mathrm{CoV}_{\text{int}}^2 + \mathrm{CoV}_{\text{ext}}^2$ .
From CoV to margin. Under a normal distribution the margin is $z_p \times \mathrm{CoV} \times$ the central estimate. A lognormal gives a smaller margin at $75\%$ and can give a larger one at $90\%$ .
Checks on the result. The margin is then tested with sensitivity testing, scenario testing, internal and external benchmarking, and hindsight analysis.
Stochastic models capture independent risk well but systemic risk poorly. That is why a bootstrap percentile alone understates the margin (see Stochastic Reserving and Parameter Risk).

Exam 6C: the MCT risk margins (Canada)

The insurance risk margin covers the liability for incurred claims, unexpired coverage, unregistered reinsurance, and earthquake and nuclear catastrophe reserves (see the Earthquake Exposure Risk Margin). The LIC factors are applied to the best estimate excluding the IFRS 17 risk adjustment.
The market risk margin covers interest rate, foreign exchange, equity, real estate and other exposures. The credit risk margin covers counterparty default on and off the balance sheet, including collateral for unregistered reinsurance.
The operational risk margin is a formula in capital required (before operational risk and diversification) and premium volumes, capped at $30\%$ of that capital required.
The diversification credit recognises that asset risk and insurance risk do not peak together. With $A$ = market + credit and $I$ = insurance, it is $A + I - \sqrt{A^2 + I^2 + 2RAI}$ with $R = 50\%$ .
Calibration. The margins are set at a target level of 99% CTE over one year, including a terminal provision. Their sum less the credit (the second block) is divided by $1.5$ to give minimum capital required. The MCT ratio is capital available over minimum capital required: $100\%$ is the minimum and $150\%$ the supervisory target, and each insurer also sets an internal target.

The accounting margin: IFRS 17

The Canadian financial-statement margin is the risk adjustment for non-financial risk. It is entity-specific and is commonly set by a confidence-level, cost-of-capital or CTE technique. Whichever is used, its equivalent confidence level must be disclosed. It replaced the MfADs. Solvency II's risk margin is instead a cost-of-capital amount.

Consolidating CoVs into a Risk Margin For a casualty portfolio with a net central estimate of $\$200$ million, the analysis selects CoVs of $6\%$ for independent risk, $5\%$ for internal systemic risk and $8\%$ for external systemic risk. The target probability of adequacy is $75\%$ .

Compute the risk margin under a normal and a lognormal distribution.
Answer
 $\begin{align*} \mathrm{CoV} &= \sqrt{0.06^2 + 0.05^2 + 0.08^2} \\ &= 11.18\% \\ \text{Normal} &= 0.6745 \times 11.18\% \\ &= 7.54\% \end{align*}$ 
The normal margin is $\$15.1$ million. For the lognormal, $\sigma^2 = \ln(1 + 0.1118^2) = 0.01242$ and $\sigma = 0.1115$ :
 $\begin{align*} \text{Lognormal} &= e^{-\sigma^2/2 + 0.6745\sigma} - 1 \\ &= 7.14\% \end{align*}$ 
The lognormal margin is $\$14.3$ million. The external systemic CoV is the largest single input, yet it came from judgment rather than from the triangle. Dropping it would cut the CoV to $7.8\%$ and the normal margin to $5.3\%$ . That sensitivity is exactly what the framework's sensitivity testing is meant to expose.

MCT Risk Margins to a Ratio A Canadian P&C insurer's target-level margins are: insurance $\$90$ M, market $\$40$ M, credit $\$30$ M and operational $\$17$ M. Capital available is $\$250$ M.

Compute the diversification credit, minimum capital required and the MCT ratio.
Answer With $A = 40 + 30 = 70$ and $I = 90$ :
 $\begin{align*} \text{DC} &= 70 + 90 - \sqrt{70^2 + 90^2 + 2(0.5)(70)(90)} \\ &= 160 - 138.92 \\ &= 21.08 \\ \text{Target} &= 90 + 40 + 30 + 17 - 21.08 \\ &= 155.92 \\ \text{Minimum} &= 155.92 / 1.5 \\ &= 103.95 \end{align*}$ 
$\text{MCT} = 250 / 103.95 = 240.5\%$

At the $150\%$ supervisory target the insurer would hold exactly the target-level capital of $\$155.9$ M. The target is where capital meets the 99% CTE calibration, and the $100\%$ minimum is two-thirds of it. The insurer has about $\$94$ M above the supervisory target, to be judged against its own internal target.