Introduction

These study notes are based on the Exam 7 Syllabus reading A Framework for Assessing Risk Margins by Karl Marshall, Scott Collings, Matt Hodson, and Conor O’Dowd. The paper proposes a framework for assessing risk margins that incorporates both qualitative and quantitative considerations.

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Terminology

Current Approach

The general approach to assessing risk margins is as follows:
  1. Determine coefficients of variation for individual portfolios of relatively homogeneous risks
  2. Develop a correlation matrix with assumed coefficients. Typically, this is done judgmentally: correlation is categorized as high, medium, or low, with each category having a given correlation coefficient.
  3. The above are determined separately for claim liabilities and premium liabilities, and assumptions about the correlation between these two liabilities are made.
  4. A distributional assumption is made (typically lognormal or normal), and the aggregate risk margin is determined as a specified probability of adequacy.

This is referred to as a bolt-on approach in that the analysis is conducted separately from the determination of the central estimate.

Valuation Classes

A valuation class is a portfolio that will be considered individually as part of the risk margin analysis.
  • Examples include Home, Auto
  • May be aligned with the valuation portfolios used for producing central estimates
  • Are subdivided into homogeneous claim groups, such as attritional claims, liability claims, catastrophe claims. A key consideration here is that claim groups should be created when different types of claims have materially different development patterns.
Considerations in defining valuation classes include:
  • Matching the split used to produce central estimates is preferable, since it allows the risk margin analysis to be aligned with the central estimate analysis, and decisions about volatility can be made in the contexct of that analysis.
  • Achieving the same level of granualarity as the central estimate analysis is generally not needed, especially if the central estimate valuation classes are too small for credibile analysis of risk margins. As a result, analysis may be performed on aggregated valuation classes.
  • Cost of performing analysis at a highly granular level, and the materiality on the final outcome, should be considered.
  • Subdivision of valuation classes into homogeneous claim groups may also be needed, balancing benefits against cost.
  • For each valuation portfolio, consider whether or not the full risk assessment framework should apply, especially if it has little historical data.

Sources of Uncertainty

Broadly speaking, sources of uncertainty fall into one of two categories:
  1. Systemic risks are potentially common across valuation classes or claim groups. It is further subdivided into two groups:
    • Internal systemic risks are internal to the insurance liability valuation process, and reperesents the extend to which the actuarial valuation approach is imperfect. It is also referred to as model specification risk
    • External systemic risks are risks external to the acturial modelling process, such as future trends in claim costs that are external to the modelling process.
  2. Independent risks arise due to the randomness inherent in the insurance process, and arises from two sources:
    • Parameter risk is the extent to which randomness of the insurance process compromises the ability to select appropriate parameters in the valuation models.
    • Process risk is the pure effect of randomness associated with the insurance process, which would still exist even if models were perfectly calibrated.

While indpendent risks can be assessed quantitatively, qualitative approaches are required to assess systemic risks.

Assessing Independent Risk

Typically, independent risk is assessed quantitatively, using stochastic modelling techniques (e.g. chain ladder method, bootstrapping, GLMs, Bayesian techniques). However, adjustments may still be made to reflect an the fact that it might not have fully captured all sources of uncertainty. Reasons for this may include: Considerations in assessing independent risk include:

Assessing Systemic Risk

Internal Systemic Risk

Sources of internal systemic risk include:
  • Specification error: results from inability to build a model that is fully representative of the insurance process. Model structure may be simplified based on limitations on available information
  • Parameter selection error: model is unable to adequately measure all the predictors of claim coust outcomes, or trends in these predictors.
  • Data error: arises from poor data or unavailablity of data. Also includes inadequate knowledge of the portfolio being analyzed.
A balanced scorecard approach is used to assess internal systemic risk:
  1. Risk indicators are developed and aspects of the modelling process scored against the adopted criteria, on a scale from worst to best practice. Examples, with corresponding best practices, include:
    • Specification error:
      • Number of models used: many approaches considered
      • Separate analysis of claim / payment types: separate into homogeneous groups
      • Range of results: low variation between models
      • Reasonability checks: significant checks conducted, including acceptance by business and peer review
      • Confidence in model goodness of fit: actual and expected are close
      • Few subjective adjustments to models
      • Continuous monitoring and review of model and assumption performance
      • Ability to detect trends in the past
      • Performance of superimposed inflation analysis
      • Unit record data is available for modelling
    • Parameter selection error:
      • Best predictors identified, regardless of whether they are used: include internal and external variables
      • Best predictors are stable over time or respond well to process changes
      • Value of predictors used: close to best predictors, lead rather than lag, modelled rather than subjectively selected, unimpaired by past systemic events
    • Data error:
      • Knowledge of past processes / changes to past processes that effect predictors. Note that this is considered a risk associated with understanding the data, not parameter selection error.
      • Timeliness, consistency, and reliability of information from business: regular two-way communication between valuation actuary and portfolio managers / claims staff
      • Data are subject to reconciliations and quality control
      • Processes for obtaining and processing data are robust and replicable
      • No past instances of data revision (if so, consider frequency and severity)
      • Current data issues and possible impact on predictors
  2. Weighted averages are calculated for each valuation class. The weights are selected subjectively to reflect the importance of the risk indicators to the portfolio.
  3. Aggregated scores are mapped to a quantitative measure (CoV) of the variation arising from internal systemic risk.
    • Low scores map to high CoVs, and high scores to low CoVs. This mapping is selected subjectively.
    • CoVs may differ between portfolios and between claims and premium liabilities.

Although the decisions in this process are mostly subjective, quantitative techniques may inform some of these decisions.

External Systemic Risk

Analysis based on past claim experience can only assess the uncertainty resulting from past episodes of external systemic risk. Using these techniques in isolation requires us to assume that future external systemic risk is similar to what has been experienced in the past. Generally, this assumption does not hold: in general, future external systemic risk will exhibit different characteristics from actual past episodes. General considerations when assessing external systemic risk include:
  • Should consider all aspects of the portfolio management process, including underwriting and risk selection, pricing, claims management, expense management, emerging trends, and the environment within which the portfolio operates.
  • Consider the entire distribution, to the extent possible. Some risks may not be material to a 75th percentile risk margin, but would be for higher probabilities of adequacy.
  • Impact of each risk may vary between outstanding claim and premium liabilities.
  • To simplify analysis, risk categories should be consolidated so that each category is independent.
The main external systemic risks for each valuation class can be categorized as follows. Note that the importance of each risk varies depending on the valuation class.
  • Economic and social risks: inflation, general economic conditions, fuel prices, driving patterns, and other social trends.
    • Driving patters are more relevant for premium liabilities rather than outstanding claim liabilities
    • Inflation affects all classes, but has a bigger impact on long-tailed classes
    • Concern is not with random volatility of inflation, but with the extent to which is is impacted by systemic events
  • Legislative, political, and claim inflation risks: known or unknown changes to the legislative or political environment that shifts the level of claim settlements.
    • This category includes superimposed inflation. Analysis of this may impact not only CoVs, but central estimates as well.
    • High importance for both outstanding claim and premium liabilities.
    • Much more material for long-tailed valuation classes.
    • Subgroups include: recent legislation (or future retroactive legislation), court precedents, changes in medical technology costs, changes in legal costs, systemic shifts in large claim frequency or severity
    • For short-tailed lines: includes risk that inflation increases at a level different from that used for the central estimate. Includes uncertainty over claim cost reduction initiatives.
  • Claim management process change risk: changes to the processes relating to claim reporting, payment, etc.
    • Potential impacts: reporting patterns, payment patterns, finalization and reopening rates, case estimation
    • Sensitivity testing of key valuation assumptions can be performed (i.e. re-apply the reserving methodology with different assumptions)
    • More relevant for outstanding claim liabilities than for premium liabilities. For claim liabilities the main concern is with changes to the pattern of emergence of credible claim estimates, but for premium liabilities the main concern is the impact on the magnitude of claim costs.
  • Expense risk: uncertainty associated with the expense of managing run-off of insurance liabilities, or the cost of maintaining the unexpired risk until the date of loss.
    • May be correlated with event risk (e.g. catastrophe increases claim handling expenses), so there may be value in grouping these categories.
  • Event risk: uncertainty associated with either natural peril or man-made events that give rise to large numbers of claims.
    • Highly important for premium liabilities. Outstanding claims are only affected to the extend to which there are outstanding events.
    • Material for property; less so for auto.
    • Can also arise in medical malpractice where a large number of claims arises from a single doctor.
    • Past experience with event claims may have been conducted by pricing actuaries, and can provide frequency and severity assumptions.
    • Output from proprietary catastrophe models, which may be provided by reinsurance intermediaries, can also inform the range of possible outcomes.
  • Latent claim risk: results from claims arising from a source that is not currently considered to be covered
    • Material for workers compensation, but the most difficult risk to quantify
    • Generally these are very low probability, but extremely high severity events
  • Recovery risk: results from reinsurance and non-reinsurance recoveries
    • May be most relevent to auto classes, where subrogation is common
    • Catastrophic events or a market downturn may substantially reduce the ability to recover from reinsurers.

Consideration of these risks is often considered as part of the central estimate valuation, so it is recommended that the analysis of external systemic risk be done in conjunction with this process. This will ensure that both processes have a consistent and complete view of external systemic risk. Risks should be ranked within each valuation class so that more effort can be spent on quantifying the most material risks.

Correlation

Correlation coefficients are generally determined subjectively, rather than quantitatively, for several reasons: Instead, correlations are selected based on a small number of correltaion bands, e.g. none, low, medium, high, and full, which correspond to 0%, 25%, 50%, 75%, and 100%. Too many categories can lead to spurious accuracy. Several independence assumptions allow us to limit the number of correlations considered:

The independence assumptions reduce the required size of the correlation matrix considerably.

Given the information about relative weightings \(w_X\) of each valuation class, coefficients of variation \(c_X\) for each, and correlation coefficients \(\rho_{X,Y}\), the results can be aggregated into a single coefficient of variation by assuming a linear correlation structure. This approach is considered reasonable up to probabilities of adequacy of 90%. In that case, aggregations can be calculated using the equation \[ \mathrm{Var}(X+Y) = \mathrm{Var}(X) + 2 \mathrm{Cov}(X,Y) + \mathrm{Var}(Y) \] relating variance to coefficients of variation via \[ \mathrm{Var}(X) = w_X^2 c_X^2 \] and covariance to correlation coefficients via \[ \mathrm{Cov}(X,Y) = \rho_{X,Y} w_Xw_Yc_Xc_Y \] As an example, Marshall et al aggregate the following independent risk CoVs:

independent.risk = data.table(valuation_class = c("Motor", "Motor", "Home", "Home", "CTP", "CTP"), liability_type = c("Claims", "Premium", "Claims", "Premium", "Claims", "Premium"))
independent.risk$weight = c(0.05, 0.25, 0.05, 0.25, 0.3, 0.1)
independent.risk$CoV = c(0.07, 0.05, 0.06, 0.05, 0.06, 0.15)
datatable(independent.risk)
independent.variance = independent.risk[, sum(weight^2 * CoV^2)]
independent.sd = sqrt(independent.variance)
independent.cv = independent.sd / independent.risk[, sum(weight)]
print(paste0("The independent CoV is ", independent.cv))
## [1] "The independent CoV is 0.0297111090334912"

To illustrate internal systemic risk, assume a correlation of 75% between the claims and premium liabilities for the Motor valuation class. The assumptions in this case are:

w_X = 0.05
w_Y = 0.25
c_X = 0.055
c_Y = 0.05
rho_XY = 0.75
motor.internal.variance = w_X^2 * c_X^2 + w_Y^2 * c_Y^2 + 2 * rho_XY * w_X * c_X * w_Y * c_Y
motor.internal.CV = sqrt(motor.internal.variance) / (w_X + w_Y)
print(paste0("The CoV for Internal Risk for the Motor valuation class is ", motor.internal.CV))
## [1] "The CoV for Internal Risk for the Motor valuation class is 0.0489188670714639"

Once the results have been aggregated to the firm level, the CoV can be converted to a risk margin by specifying a probaility of adequacy and a distributional assumption. A typical example is the 75th percentile of the normal distribution:

print(paste0("For a CoV of ", 0.087, " the corresponding risk margin is ", 0.087*qnorm(0.75)))
## [1] "For a CoV of 0.087 the corresponding risk margin is 0.0586806082670591"

The LogNormal distribution can also be used to determine risk margins. This produces a situation in which risk margins reduce significantly as a percentage of CoV as the CoV increases, while the Normal risk margins are a constant percentage of CoV. Generally, lognomral is used for high probabilities of adequacy, while the normal is used for lower.

Reasonability Checks

Given the amount of judgment involved in applying this framework, additional reasonability checks are recommended: A mechanical hindsight analysis involves applying the chain ladder method on smaller and smaller triangles to assess how the estimate changes over time:
  1. Start by applying the chain ladder method to cumulative paid claims, using an objective approach (e.g. average of last 3 years) to select factors. Outstanding claim payments using all data are referred to as the current estimate
  2. Delete the most recent diagonal of the triangle and re-calculate the unpaid claim estimate using this data. Repeat to produce a number of past estimates.
  3. Compare each of the past unpaid claim estimates with the current estimate.
The approach can assess the following: