Why Regulators May Permit an Internal Model
A prescribed formula gives insurers a common basis for calculating regulatory capital. An internal model takes a different route: it uses an insurer’s own data, risk profile, and modeling methods to estimate the capital required under the regulatory standard. This can represent the insurer’s exposures in more detail, but it does not give the insurer freedom to choose a weaker capital target.
For a Solvency II-style regulatory calculation, the central target is a one-year risk horizon and a 99.5% Value-at-Risk measure for the change in basic own funds. The model may use the insurer’s own distributions, scenario structure, and methods to estimate that measure; it must still meet the required regulatory calibration and supervisory approval requirements. The regulatory result is not automatically the same as the insurer’s preferred economic-capital measure, which might use a different horizon or risk standard.
Scope: Model the Insurer’s Material Risks
Model scope begins with the insurer’s risk profile, not with the risks that are easiest to quantify. The model must represent material exposures that could affect basic own funds over the one-year horizon. Depending on the insurer, these may arise from market movements, insurance losses and reserves, counterparty default, operational events, or other material sources of loss. The model needs to represent the exposures themselves and the ways they can affect assets and liabilities.
For example, a hypothetical insurer writing property coverage might need to reflect catastrophe losses, changes in the value of invested assets, and the possibility that a reinsurer fails to pay an expected recovery. If a risk is material, omitting it merely because its data are limited does not make the omission acceptable. The insurer needs a defensible way to measure it, such as a suitable scenario or other justified modeling method, and must explain important limitations.
A model’s scope also includes how risk mitigation and management actions are reflected. A reinsurance recovery should not be treated as certain if the contract does not provide that protection in the modeled circumstances. Any assumed future management action needs to be realistic and consistent with the insurer’s governance and ability to carry it out. The model must describe its treatment of these features rather than silently relying on them to reduce capital.
| Scope question | What the model assessment should establish |
|---|---|
| Which risks are included? | All material risks relevant to the regulatory capital calculation are represented. |
| What is measured? | The effect of modeled events and changes on basic own funds over the required horizon. |
| How are risk mitigation and actions treated? | Assumptions reflect the actual protection and actions that can reasonably be relied upon. |
| What are the limitations? | Data gaps, simplifications, and material assumptions are identified and supported. |
Calibration: The Regulatory Target Is Fixed
The model’s method is insurer-specific, but its regulatory calibration is not. In the one-year framing, let \(L\) denote the modeled loss in basic own funds over the next year. A loss is positive when basic own funds fall; it is not necessarily the same as a single claim payment or an accounting expense. Let \(F_L(x)=P(L\le x)\) be the cumulative distribution function of that loss.
The distinction between the required standard and the model’s mechanics matters. An insurer can use a distributional model, simulation, or a justified combination of methods; the method must estimate the same regulatory quantity. A one-year 99.5% target cannot be replaced by a multi-year horizon or another percentile just because that alternative is easier to estimate. Nor does using a more detailed model automatically justify a lower capital result.
Worked Example: Converting Own-Funds Outcomes to Regulatory Capital
A hypothetical insurer has basic own funds of $120 million at the valuation date. Its model projects the following one-year ending basic own funds. Treat the outcomes and probabilities as the complete modeled distribution for this illustration. Calculate the 99.5% VaR of the one-year loss.
| Ending basic own funds | Probability |
|---|---|
| $110 million | 0.950 |
| $90 million | 0.040 |
| $75 million | 0.008 |
| $50 million | 0.002 |
Method. For each outcome, calculate the loss as starting basic own funds minus ending basic own funds. Then order the losses and find the smallest loss whose cumulative probability is at least 0.995.
Step 1: Convert each outcome to a loss. The four losses are \(120-110=10\), \(120-90=30\), \(120-75=45\), and \(120-50=70\) million dollars. Their probabilities are respectively 0.950, 0.040, 0.008, and 0.002.
Step 2: Accumulate probability. At a loss of $10 million, the cumulative probability is 0.950. At $30 million it is \(0.950+0.040=0.990\). At $45 million it is \(0.990+0.008=0.998\).
Step 3: Find the quantile. Since 0.990 is below 0.995 and 0.998 is at least 0.995, the 99.5% quantile is $45 million.
Check and common mistake. The probability of a loss strictly greater than $45 million is 0.002, which is no greater than 0.005. The probability of a loss equal to or greater than $45 million is \(0.008+0.002=0.010\). It would be incorrect to claim that the latter probability must be no greater than 0.005: the 0.008 probability at the quantile itself is included in “equal to or greater than,” but not in “strictly greater than.”
Why the Quantile Condition Holds
The definition of a quantile explains why the regulatory tail statement uses a strict inequality. For a loss distribution with cumulative distribution function \(F_L\), choose \(q=\inf\{x:F_L(x)\ge p\}\). At the quantile, the cumulative probability is at least \(p\), so the probability of a loss beyond it is at most \(1-p\):
Setting \(p=0.995\) gives \(P(L>q)\le 0.005\). If the distribution is continuous and has no probability mass at \(q\), then \(P(L\ge q)=P(L>q)\), and the tail probability at the quantile is 0.005 when the quantile lies in a strictly increasing part of the distribution. A discrete distribution can instead have a jump at \(q\). The quantile definition still applies, but probability at the quantile makes \(P(L\ge q)\) potentially larger than 0.005.
Worked Example: Applying a Continuous Model’s Calibration
For a deliberately simplified one-year illustration, an insurer models its loss in basic own funds as \(L\sim N(8,10^2)\), with amounts in millions of dollars. Assume the normal model is the chosen model for this calculation and use the standard normal 99.5th percentile \(z_{0.995}=2.5758\). Calculate the modeled 99.5% VaR.
Step 1: Identify the parameters. The mean loss is \(\mu=8\) million and the standard deviation is \(\sigma=10\) million; the second parameter in \(N(8,10^2)\) is the variance.
Step 2: Convert the standard-normal percentile. Under the stated normal assumption, \(q=\mu+z_{0.995}\sigma\). Thus,
Step 3: Report and check. Rounded to the nearest cent, the modeled capital is $33,758,000.00. In a continuous normal distribution, the probability of a loss strictly greater than this quantile is 0.005. The result is a calculation under the stated normal assumption, not evidence that a normal distribution is suitable for a particular insurer’s tail risks.
Common mistake avoided. Multiplying the variance \(100\) by the normal percentile would be wrong. The quantile formula uses the standard deviation, \(\sqrt{100}=10\), not the variance.
Estimating the Tail From a Model
An insurer may estimate a tail quantile from simulated outcomes. The empirical quantile depends on the ordered outcomes and the chosen quantile convention; under the generalized-inverse convention, it is the smallest loss at which the empirical cumulative probability reaches the target. Simulation is a computational method, not a relaxation of the regulatory standard. The insurer still needs to consider whether the data, assumptions, and model structure can represent rare but severe outcomes adequately.
Worked Example: Finding a Quantile From Simulated Losses
A hypothetical insurer runs 10,000 one-year simulations of loss in basic own funds. The ordered results can be summarized as follows. Each simulation has equal weight. Use the empirical generalized-inverse convention and calculate the 99.5% VaR.
| Loss | Number of simulations at that loss |
|---|---|
| $0.4 million | 9,800 |
| $1.2 million | 120 |
| $2.0 million | 50 |
| $4.0 million | 30 |
Step 1: Check the simulation count. The counts sum to \(9{,}800+120+50+30=10{,}000\).
Step 2: Find the target rank. The cumulative target is \(0.995(10{,}000)=9{,}950\), so the 9,950th ordered result is the relevant order statistic.
Step 3: Locate that result. The cumulative count is 9,800 at $0.4 million and \(9{,}800+120=9{,}920\) at $1.2 million. It reaches \(9{,}920+50=9{,}970\) at $2.0 million. Therefore, the 9,950th result is $2.0 million, and the empirical 99.5% VaR is $2.0 million.
Step 4: Check the tail interpretation. Thirty simulations have losses strictly greater than $2.0 million, so the empirical probability is \(30/10{,}000=0.0030\). Eighty simulations have losses equal to or greater than $2.0 million, so that probability is \(80/10{,}000=0.0080\). The difference is the 50 simulations exactly at the quantile.
Common mistake avoided. Counting all 80 outcomes at or above $2.0 million as though they were strictly beyond the quantile gives the wrong tail check. Also, 10,000 simulations do not by themselves demonstrate that the model is adequate: the results depend on the model and assumptions, and rare-tail estimates can be uncertain.
Approval and Evidence
A regulator approves the internal model for regulatory capital only after assessing whether it meets the applicable requirements. The insurer must be able to explain the model’s scope, data, assumptions, and methods and show how the output is produced. Relevant requirements include statistical quality, calibration, profit-and-loss attribution, validation, documentation, and governance. These are connected safeguards: a precisely computed percentile is not persuasive if the underlying risks are missing or the data and assumptions are not justified.
The supervisory authority’s approval is not a certification that every future result is correct. Models remain subject to oversight as the insurer’s business, exposures, data, and methods change. A material change can affect whether the model continues to reflect the insurer’s risk profile and satisfy the required standard. The insurer must therefore maintain evidence that supports the model and its ongoing use for the regulatory calculation.
A useful review separates three questions:
Check that all material risks affecting the regulatory result are represented, with limitations and mitigation assumptions identified.
Examine whether the data, assumptions, and modeling methods support an estimate of the relevant one-year tail, rather than just a central forecast.
Confirm that the final capital measure is calibrated to the one-year 99.5% VaR standard and that its tail interpretation is correct.
Common Exam Traps
- Changing the standard along with the model. An insurer-specific method does not permit an insurer-specific confidence level or time horizon for the required regulatory result.
- Confusing business loss with loss in own funds. A claim amount alone may not capture the net effect on assets and liabilities that the regulatory loss measure is intended to represent.
- Omitting a material risk because its data are weak. Limited data call for a justified modeling treatment and disclosed limitations, not an unexplained omission.
- Using “at or above” as the tail probability statement. The quantile definition bounds the probability of a loss strictly greater than the quantile. A discrete mass at the quantile can make the probability of reaching or exceeding it larger.
- Treating approval as proof of certainty. Supervisory approval is conditional on requirements and evidence; it does not eliminate model risk or the need for ongoing oversight.
Check Your Understanding
For each problem, apply the stated assumptions and quantile convention.
- A hypothetical full internal model includes market and insurance risks but omits a material operational exposure because little historical data are available. Is that scope acceptable as described? (Answer: No; a material risk cannot be omitted solely because data are limited. It needs a justified modeling treatment and disclosed limitations.)
- A one-year loss is modeled as \(L\sim N(12,8^2)\) million dollars, and \(z_{0.995}=2.5758\). What is the 99.5% VaR, rounded to the nearest cent? (Answer: \(12+2.5758(8)=32.6064\) million dollars, or $32,606,400.00.)
- A discrete loss distribution has losses of $0, $20, and $60 million with probabilities 0.960, 0.025, and 0.015. Find the 99.5% quantile and the probability of a loss strictly greater than it. (Answer: The quantile is $60 million; \(P(L>60)=0.0000\).)
- Among 2,000 equally weighted simulations, 1,950 losses are $1 million, 30 are $5 million, and 20 are $12 million. Using the empirical generalized-inverse convention, find the 99.5% VaR. (Answer: The target rank is \(0.995(2{,}000)=1{,}990\); cumulative counts reach 1,980 at $5 million and 2,000 at $12 million, so VaR is $12 million.)
- A model estimates a 99.5% loss quantile of $45 million. At that value, 0.8% of modeled outcomes equal $45 million and 0.2% exceed it. What are the probabilities of a loss strictly greater than $45 million and of a loss equal to or greater than $45 million? (Answer: Strictly greater: 0.2%; equal to or greater than: 1.0%.)