Exam 7 Notes - Brehm et al.

Introduction: Quantitative Analysis of Insurance and Financial Risk

These notes are based on the Exam 7 Syllabus reading Enterprise Risk Analysis for Property and Liability Insurance Companies by Brehm, Gluck, Kreps, Major, Mango, Shaw, Venter, White, and Witcraft.

easypackages::packages("data.table", "DT", "ggplot2")
options(scipen = 999)

Definition of ERM and key elements of consideration

Enterprise Risk Management (ERM) is defined to be “the process of systematically and comprehensively identifying critical risks, quantifying their impacts, and implementing integrated strategies to maximize enterprise value.” Key elements of ERM include:

1. It is a regular process.
2. It is enterprise-wide; for an insurer, this includes not just insurance risk, but financial risk, strategic risk, and operational risk.
3. It focuses on the most significant risks.
4. Risk exists due to outcomes that differ from expectations, regardless of whether the departures are adverse.
5. Risks are quantified to the extent possible. This is done portfolio-wide and considers correlations among risks.
6. Development of strategies and plans to avoid, mitigate, or exploit risks.
7. Risk versus return evaluation is performed on risk management strategies.

The major categories of risk facing an insurer are:

1. Insurance hazard risk: includes risk associated with in-force exposures (catastrophic and underwriting risk) and past exposure (reserve deterioration). This is a speculative risk because it has a potential positive outcome for the insurer, and is the firm’s reason for existence.
2. Financial risk: market risk is high for insurers, given the high proportion of assets that are invested. Credit risk relates to recoveries from reinsurers, and liquidity risk arises from the need to pay claims at times that are not known in advance.
3. Operational risks relate to the ability of the company to execute on its plans.
4. Strategic risks relate to the company choosing the wrong plan of action.

ERM can be used to support decisions such as:

• Identifying sources of risk, and determining the amount of capital needed to support them
• Selecting a reinsurance program
• Managing the mix of assets
• Planning growth
• Valuing companies for mergers or acquisitions

ERM process and risk management

The ERM process can be divided broadly into four phases, which repeat:

1. Diagnosis: identify threats that pose a potentially serious threat to the value of the firm. These can include:
   • General environment risks (political, macroeconomic, social, natural)
   • Industry-specific risks (supply and demand markets, competition)
   • Firm-specific risks (operational, liability, R&D, credit, behavoural)
2. Analysis: create probability distributions of potential outcomes, considering correlations among risk factors. Risk factors that contribute to adverse scenarios, or metrics above critical thresholds, must be prioritized.
3. Imlementation: apply traditional risk management techniques:
   • Avoidance, e.g. do not write a line of business
   • Reduction in frequency, e.g. write only low-risk policies within a line
   • Mitigation of severity given occurrence, e.g. with high deductibles or low limits
   • Elimination / transfer, e.g. reinsurance
   • Retention
   • For speculative risks, plan to exploit the risk.
4. Monitoring: assess process relative to expectations. ERM is not viewed as a “project” to be completed, but the plans should be updated and expectations should change as conditions change.

ERM risk models evaluation

Characteristics of good ERM models include:

• The model shows a realistic balance between risk and reward from a range of different strategies. In particular, it reflects the relative importance of various risks.
• The model recognizes its own imperfection, e.g. due to parameter uncertainty. Note that the ERM model often uses other models (catastophe, macroeconomic, credit risk), so the ERM should take into account the uncertainty associated with these models as well.
• The model is built by individuals with knowledge of the fundamentals of those risks.
• The model uses mathematical techniques to reflect dependencies between risks.
• The modellers have a trusted relationship with senior management.

Sources of risk and modelling of dependencies

Underwriting Risk

• Loss frequency and severity are used to quantify loss potential. Statistical methods are used to estimate the parameters of the distributions and assess the quality of the fit.
• Pricing risk arises from a mis-estimation of projected losses, or competitive pressure. It may be difficult to detect underpricing for a number of years, with the accumulated losses resulting in a reserve deficiency. The underwriting cycle is a major source of pricing risk.
• Parameter risk arises from the imperfection in the frequency and severity distributions themselves, which includes the following risks:
  • Estimation risk: there is never enough data to determine the “true” parameters of a distribution, so statistical methods should be used to determine how fare the estimates are likely to be from the true parameters; this can be used in generating scenarios with alternate parameter values.
  • Projection risk: result from changes over time, such as trend and development. These often have dependencies with macroeconomic factors, such as inflation. Unexpected changes in risk conditions (e.g. increased driving, more extreme weather) can also produce projection risk.
  • Event risk: occurs when there is a causal link between an unpredicted external event and the insurer’s losses. Examples include court or regulatory decisions, latent exposures (asbestos), new causes of loss, new entrants in the market.
  • Systematic risk: non-diversifying risks that operate on a large number of policies, such as inflation. Do not improve with added volume.
• Catastrophe model uncertainty: typically external models are used, and there is significant variation both between models, and within a single model as it gets updated over time. Both frequency and severity of catastrophes are sources of uncertainty. Data quality, such as a mis-match between company data and cat model requirements, is another source of uncertainty. Use of more than one model for each peril is recommended.

Reserving Risk

• Both reserve estimate and reserve uncertainty can be understated.
• Bigger risk for long-tailed lines of business.
• Reserve uncertainty impacts both the amount of capital that must be held, and the duration for which it must be held.
• Models for reserve uncertainty are an essential part of the enterprise risk model.

Asset Risk

• Asset modelling generates scenarios that are consistent with historical patterns (“probabilistic reality”).
• More probable scenarios are given more weight
• Key concern is balancing asset and underwriting risk, i.e. optimize the use of capital by offsetting insurance risks with investment risk.
• Bond market scenarios should be based on arbitrage-free models; this avoids a result in which the model misdirects the user toward a failed arbitrage-based strategy.
• Equities are often correlated with bonds, and this should be incorporated into the scenario generation.
• Exchange rates are correlated with the interest rates between two countries, and the interest rates across two countries are themselves correlated.

Dependencies

Without a model for dependencies, the enterprise as a whole may appear unrealistically stable. Sources of dependency include:

• Simultaneous impact of macroeconomic factors on many risks (e.g. asset values, claim inflation, underwriting cycle)
• UW cycles / loss trends / reserve development correlate across lines of business
• Catastrophes can cross lines of business, e.g. an earthquake can cause large losses for property, workers compensation, and auto lines.

A crucial component is tail dependency, which is the degree of dependency in extreme events. For example, inflation may be generally uncorrelated across two lines of business, unless inflation overall is extremely high. Later sections show how tail dependency can be modelled using copulas.

ERM in setting captial requirements

ERM can be used to help the insurer find the optimum level of capital, balanced between two competing forces:

• Policyholders, regulators, and rating agencies want capital to be adequate to guarantee that the insurer will be able to pay claims.
• Shareholders want capital allocation to be efficient so that they achieve an attractive return on their investment.

Examples of criteria for setting capital requirements include:

• Sufficient capital so that probability of default is remote. This is an approach that focuses on protecting policyholders; shareholders would still be concerned with a significant partial loss of capital that does not cause a default.
• Amount of capital needed to avoid a rating agency downgrade. This is essential because financial strength is a key component of an insurer’s value, which would be diminished by a downgrade.
• Maintaining sufficient capital to continue to service renewals.
• Maintaining sufficient capital to survive a major catastrophe and thrive in its aftermath.
• Value-at-risk, tail-value-at-risk, and other measures can be used. Details are in a later section.

A key consideration is that extreme reference points, such as a total loss of capital, are where the models are least reliable; there is little data to support modelling the tail.

Managing Insurance and Financial Risks

Broadly speaking, processes for making decisions in the context of uncertainty fall into three categories:

1. Deterministic project analysis: a single forecast (present value, internal rate of return) is produced, possibly along with sensitivity to critical variables. Uncertainty is handled judgmentally.
2. Risk analysis: a Monte Carlo simulation, using distributions of forecasts of critical variables, produces a distribution of present values of cash flows. Risk judgment is still applied judgmentally.
3. Certainty equivalent: extends risk analysis by formalizing risk judgment into a utility function representing corporate risk preference.

Current best practice is at the risk analysis level (dynamic financial analysis, or DFA). It is debatable whether there is value in progressing to a certainty equivalent approach:

• Argument against: investors can diversify away firm-specific risk (which is diversifiable), so managers should be indifferent to firm-specific risk as well.
• Argument for: there is no way to identify which risks are firm-specific and which are systematic. Market-based risk-signals such as risk-adjusted discount rate reflect risk only if there is a time lag, and cannot reflect instantaneous risks (“jump risks”).

IRM and other capital adequacy models

An internal risk model (IRM) is a DFA model that produces an aggregate loss distribution for the firm. Once this has been produced, there are three ways in which it is used for decision making.

Corporate Risk Tolerance

Broadly speaking, there are four steps in the corporate decision-making process, with an understanding of risk tolerance required at steps 2 and 3:

1. Determine an aggregate loss distribution with many sources of risk
2. Quantify the impact of possible losses on the corporation
3. Assign a cost to each impact
4. Attribute the cost back to its sources

Considerations in determining risk tolerance include:

• Size
• Financial resources
• Ability and willingness to tolerate volatility

Approaches to determining explicit risk preferences may be either implicit (e.g. derived from capital markets) or explicit (e.g. determined by management). Examples include:

1. Normative approach: quantify the minimum decrease in risk sufficient to justify a given mitigation cost, through a series of experiments involving indifference.
2. Portfolio theory approach: identify an efficient frontier which minimizes risk (measured as standard deviation of NPV) for a given return, subject to a constraint on total funds available. The current portfolio is suboptimal if there exist either portfolios with the same return at lower risk, or higher return at the same risk. Key questions for the firm include:
   • How much risk are we willing to tolerate?
   • How much reward are we willing to give up for a given reduction in risk?
   • Are the risk/reward tradeoffs along the efficient frontier acceptable?

Cost of Capital Allocated

For insurers, allocation of capital pertains to the allocation of the cost of risk capital, in contrast to the capital allocations for manufacturers that involve actual cash transfers. Risk capital allocations are purely theoretical, and no actual capital is transferred.

• Risk capital is a measure of the firm’s total risk-bearing capacity; it is an aggregate measure that only has meaning for the firm in total.
• Non-linear dependence between the sources of risk makes a proportional (linear) allocation of the total amount back to individual elements complicated.
• RORAC (Return on Risk Adjusted Capital) is determined by multiplying risk-adjusted capital by a “hurdle rate.” Risk capital may still be allocated as an interim step in assigning the cost of risk capital to individual elements.
• Alternate approach: risk capital is the amount required to guarantee performance of contractual obligations at a default-free level.
• For insurance, the entire pool of risk capital is treated as a common resource.
• Decisions can be based on the economic value added, which is the net present value of the return, less the cost of capital. If this is positive, the activity adds value, and if it is negative, it destroys value.

Cost-benefit analysis on Mitigation and Hedging

• Benefit of mitigation projects can be quantified as a reduction in capital cost.
• Decisions can be made based on whether a mitigation project has a positive or negative economic value added.
• Decisions should not be based on a single risk metric. Instead, a combination of distinct and independent metrics should be used. They should be responsive to different dynamics.
• In practice, corporate decisions may require fewer / simpler metrics, so a balance must be struck.

Regulatory and Rating Agency Capital Adequacy models

Leverage Ratios

Leverage ratios are a simple measure of capital adequacy that have been used historically. (Today, they are still in use, but other approaches are given more weight.) The most notable ratios are:

• Net written premium to surplus
• Reserves to surplus
• Net leverage: sum of the above ratios

The ratios are compared to fixed values for all companies, regardless of the risks to which the company is exposed. Furthermore, it only considers underwriting risks.

Examples provided by the Insurance Regulatory Information System (IRIS) include:

• Written premium to surplus: Can be on a gross or net basis. Net ratio should be less than 3.
• Change in writings
• Surplus aid to surplus
• Two-year operating ratio
• Investment yield
• Change in surplus
• Liabilities to liquid assets
• Agents’ balances to surplus
• Reserve development to surplus: on a one-year or two-year basis
• Estimated current reserve deficiency to surplus

The IRIS test compares these ratios to a range of reasonable values, and if four or more fall outside the range, then additional regulatory scrutiny is warranted.

Risk-Based Capital Models

In a risk-based capital model, various accounting balances are multiplied by a factor to produce a risk charge as an amount of required capital.

• These models combine different aspects of risk (premium and reserve, asset, credit, accumulation, operational) into a single number.
• Factors vary by the quality and type of asset, or by the line of business.
• They are currently used by many regulators as well as A.M. Best and Standard and Poor’s
• They often include explicit recognition of accumulation risk (exposure to catastrophic natural hazards). An example is a capital charge for a 1-in-500 year earthquake. Treatment of accumulation risk is one of the biggest sources of differences between models.
• Stress tests can also be performed, notably testing the impact of a second severe event after the first.
• The size of the factors depends on the use case. A regulatory model concerned with the one-year likelihood of insolvency will use smaller factors than a model concerned with the long-term viability of the company.
• A covariance adjustment reduces the capital requirement to reflect the independence of the various risk components. The use of such an adjustment may mean that the factors are larger than they would be otherwise, because the total capital requirement is reduced by the covariance adjustment.
• A major source of credit risk is reinsurance recoverables. Charges typically vary based on the rating of the reinsurer. Excessive dependence on reinsurance may be subject to an additional risk charge.
• These models may be used to inform decisions about capitalization strategies:
  • Identify a target rating from a rating bureau.
  • Determine and quantify options that will allow the company to achieve the target rating, such as ceding $X in reinsurance or issuing a $Y surplus note.
  • For each option that allows the company to meet its target rating, calculate the present value of its cost to determine which option should be pursued.

Scenario Testing

Scenario testing, using either static or stochastic scenarios, is emerging as a tool for monitoring solvency.

• Risk-based capital formula is used as a base metric.
• Insurer performs a self-assessment of its risk-based capital needs, typically using scenario testing or stochastic modelling.
• Regulator may review the self-assessment and provide an alternate capital requirement.
• Typically, the internal capital requirement is higher than the formula-based minimum.
• Typical approach is to produce multi-year financial projections, using models for probability distributions for sources of uncertainty.
• Incorporation of correlations among risks is a critical feature.
• Multi-year models should reflect management responses to adverse financial results.
• Thresholds for evaluating capital adequacy are typically phrased in terms of a maximum probability of ruin over a given timeframe.

Implementation of Internal Risk Models

A correctly executed IRM will impact risk decision making in planning, pricing, reinsurance purchase, capacity allocation, and interactions with rating agencies. Given the wide-ranging impact of IRM, it requires careful planning. Major steps in the process are listed below.

Planning Staffing and Scope

Expectation management is an important aspect of IRM implementation, since it runs the risk of becoming “all things to all people.” Considerations include:

• Organizational chart: reporting lines (solid versus dotted), multi-function oversight committee. Key recommendation is to ensure the IRM team reports to a leader with a reputation for fairness and balance.
• Functions represented: actuarial (reserving / pricing), finance, planning underwrtiing, risk.
• Resource committment: should be viewed as the establishment of a new competency, which suggests full-time employee commitment (either transfers or outside hires). Mix of skill set is important.
• Critical roles and responsibilities: control of inputs and outputs should be similar to that used with general ledger or reserving systems.
• Purpose: initial focus on quantifying variation around plan; can move on to providing a view of distribution of results
• Scope: initially, prospective underwriting period. Can later expand to include reserves, assets, and operational risk. Key question is whether to provide low detail on the whole company, or high detail on a pilot segment.

Parameter Development

This stage often extends past timelines due to data quality, unique product lines, and involvement of many functional areas. Key considerations include:

• Modelling software: assess what is pre-built versus user-built, and ensure alignment with IRM team capabilities. Scalability, integration with other systems, and output management are key technical considerations.
• Developing input parameters: expert opinion plays a significant role, because parameter estimation methods rarely produce results that can be used without applying informed judgment. As such, a systematic way to capture expert risk opinion should be developed. Sources of expert opinion include claims, planning, actuarial, and underwriting.
• Correlations: are potentially politically sensitive because they have a major impact on the aggregate risk profile, and impact capital allocation. IRM team should recommend assumptions which are ultimately owned by a C-level executive.
• Validation: testing should be done over an extended period. Challenge is that there is no “existing IRM” to compare to.

Implementation and Rollout

An “internal sell” approach involves first convincing key opinion leaders, who become advocates convincing others. Key considerations include:

• Priority setting: managers must choose which initiatives get priority, and which do not. Opinion leaders can slow the adoption process, leading to inefficiency in rollout. Top management should set the priority for implementation.
• Interest and impact: use regular communications to broad audiences. Education is also a component.
• Pilot testing: can ease the transitition, allowing effective preparation for the magnitude of the change. Should be based on real data and analysis, involve a multipdisciplinary team, and initial model indications should receive no weight (focus is on learning and familiarization).
• Education: bring leadership to a similar base level of understanding. Topics include probability, statistics, and risk model evaluation and usage.

Integration and Maintenance

Techniques to incorporate the IRM into the organization include:

• Cycle: input and output stages should be incorporated into the corporate calendar around planning, reinsurance purchasing, porfolio reviews, or capacity allocation. IRM output should be a mandatory part of decision support.
• Updating: no more frequent than semi-annually. Minor changes may be performed through scale adjustments for impacted segments.
• Centralized control of input and output sets, inculuding date stamping. Analytical templates that manipulate IRM outputs for application purposes should be controlled to maintain credibility of the IRM.

An asset-liability modelling approach

Asset-liability management (ALM) is the analysis of the asset portfolio in light of current liabilities and future cash flows, on a going-concern basis.

• It is a contrast with the simpler asset matching approach, which attempts to maintain a fixed-income asset portfolio with the same cash flow patterns as the liabilities.
• ALM considers risk factors such as interest rate changes, inflation risk, market risk, and credit risk.
• ALM considers investments other than fixed income, particularly those that can be used as hedges. For example, equities can hedge against inflation risk, and reinsurance can hedge underwriting risk.
• It integrates the underwriting portfolio (not liquid) and the investment portfolio (more liquid); focus is on adjustments to the investment portfolio given the constraints of the UW portfolio (e.g. loss reserves).
• The approach is not limited to assets, since it can inform future underwriting considerations or reinsurance.
• Key findings of a study by Venter et al include:
  • Combinations of risk-free and risky assets should be assessed by their expected mean and variance, and alternatives along an efficient frontier identified.
  • Holding shorter-term investments creates a reinvestment risk: if rates drop, then assets may get reinvested at a lower-than-expected rate. Conversely, if rates rise, then longer-term investments may need to be liquidated to fund liabilities. This risk can be reduced through duration matching.
  • Liabilities that are variable in amount and timing make precise duration matching impossible. Inflation-sensitive liabilties make the situation even more complicated.
  • Incorporating premium flow from a going-concern company provides greater flexibility: when conditions for liquidation are unfavourable, the company can pay claims out of premium cash flows. This requires an enterprise-wide model that includes premium income, losses (including catastrophes), and expenses.
• Taxation is an additional consideration, since shifting between tax-exempt and taxable securities over the course of the underwriting cycle may be one of the key drivers of investment strategy, because underwriting losses absorb taxable investment income.
• A study by the CAS Valuation, Finance, and Investment Committee (VFIC) tested the optimality of duration-matching investment strategies. The investigation considered a variety of risk measures, on both a GAAP and statutory accounting basis. Key findings include:
  • The results are dependent on the choice of risk metrics, return metrics, and risk-return preferences.
  • Statutory accounting metrics suggest little hedging is achieved through duration matching, since bonds are amortized and liabilities are undiscounted.
  • Similar conclusions for GAAP accounting metrics.
  • Using metrics based on true economics (bonds marked to market, and liabilities discounted at market rates), duration matching produced low interest rate risk. Short-term investment strategies increase risk and lower return; longer-term strategies increase both risk and return and the decision to pursue them is a value judgment.

The steps in an asset-liability modelling approach include:

1. Start with models of assets, existing liabilities, and current business operations.
2. Define risk metrics for the analysis. Considerations include:
   • Statutory, GAAP, or economic basis?
   • Income-based or balance-sheet based?
   • Selection of the risk measure itself, e.g. standard deviation, probability of falling below a threshold, VaR, etc.
3. Define what constitutes “return.” Accounting basis should be consistent with the risk measure. Examples of returns include periodic earnings or terminal value of surplus.
4. Determine the time frame of the analysis. This is a tradeoff between reflecting the nature of the business through a cycle (requires a longer period) and difficulty / complexity of the analysis (single period analysis is easier).
5. Identify constraints, such as regulatory limits on asset classes, or investments that drive capital requirements too high.
6. Run the model for a variety of investment, underwriting, and reinsurance strategies that are under consideration. Risk and return metrics are calculated over these simulations.
7. Construct an efficient frontier that describes the maximum return for a given level of risk. Plot the frontier (mean return vs. standard deviation) along with the company’s current portfolio. Moves that can increase return without increasing risk, or risk can be decreased without decreasing return, should be explored.
8. Although insurance liabilities are illiquid, they can be hedged through reinsurance; this should be considered, particularly in multiperiod models. Combinations of different reinsurance structures and asset portfolio options should be considered.
9. Review simulations to identify situations in which even the preferred portfolios performed poorly. This may reveal asset hegdging strategies to further reduce risk. It can also identify conditions that lead to substandard performance (e.g. persistent soft market, catastrophe that forces liquidation) and inform the establishment of monitoring mechanisms.

Areas of continuing research include:

• Correlations between lines of insurance, between assets and liabilities, and over time.
• Models of unpaid losses have not been developed as explanatory models – they do not predict future loss payments with parameters linking losses to economic indices. An example would be incorporating inflation from an economic scenario generator.

Reinsurance and Risk Optimization

Although this section appears earlier in the learning objectives, I have placed it after the discussion of risk measures in these notes, since some risk measures are used in the discussion.

Risk Measures and Application to Strategic Management

Economic capital is the amount of capital that a firm must hold in order to continue operating under a worst-case scenario; for non-insurers, this is typically determined as VaR at a remote probability. For insurers, a wider range of risk metrics are used and compared to actual capital. Risk measures can be used to provide comparison points to actual capital held. Considerations when selecting a capital level include:

• There is generally no single efficient capital level, given variation in the insurance market: some consumers may be more price-sensitive, while others may be more concerned about the security of the insurance guarantee. As a result, insurers can find different niches in the market based on different levels of capital.
• Increasing an insurer’s rating generally has a small impact on growth, but a decrease in rating can see a rapid decline in business. The reason is because after a rating drop, customers can easily leave, but all a rating increase does is provide an opportunity to compete with other insurers that already have the business.
• Rating agencies may also be a source of capital requirements.
• The split between new and renewal business should be considered, given that renewal business is generally more profitable than new business. A possible approach is that the insurer should aim to retain enough capital to service the renewal book, e.g. if renewals comprise 80% of the book, it should have enough capital that 20% of capital can cover an adverse scenario.

To demonstrate the use of risk measures, and in particular, the techniques for alloctaing risk to indivdual business units, the following simulated data can be used. This simulation assumes that an insurer has two lines of business, with some correlation between the losses in the two lines. Assumptions underlying the simulation include:

• Aggregate losses in each line follow a lognormal distribution.
• The first line of business has losses with parameters \(\mu_1 = 8\) and \(\sigma_1 = 0.25\)
• The second line of business has losses with parameters \(\mu_2 = 7\) and \(\sigma_2 = 0.5\)
• In each simulation, \(\mu\) is adjusted based on a random normal variable with mean 0 and standard deviation \(tau = 0.25\)

number.of.simulations = 10000
mu_1 = 8
sigma_1 = 0.25
mu_2 = 7
sigma_2 = 0.5
tau = 0.25
set.seed(12345) # Fix random seed to allow replicability of results
simulation_number = 1:number.of.simulations
adjustment = rnorm(number.of.simulations, mean = 0, sd = tau)
line_1_loss = rlnorm(number.of.simulations, meanlog = mu_1 + adjustment, sdlog = sigma_1)
line_2_loss = rlnorm(number.of.simulations, meanlog = mu_2 + adjustment, sdlog = sigma_2)
company_loss = line_1_loss + line_2_loss
simulation.results = data.table(simulation_number, adjustment, line_1_loss, line_2_loss, company_loss)

Furthermore, for purposes of assessing loss of capital under each simulation, assume that the company charges premium equal to the expected value in each line, with a 12% profit margin:

line.1.premium = 1.12 * exp(mu_1 + sigma_1^2/2)
line.2.premium = 1.12 * exp(mu_2 + sigma_2^2/2)
premium.collected = line.1.premium + line.2.premium
simulation.results[, impact_on_capital := company_loss - premium.collected]
simulation.results[, line_1_capital := line_1_loss - line.1.premium]
simulation.results[, line_2_capital := line_2_loss - line.2.premium]
datatable(simulation.results)

A key consideration is allocating risk to different business units within the company, which may be needed for various reasons:

• To set capacity controls such as premium targets and limits for each line
• For computing risk-adjusted profitability by line
• Calcultaing the denominator for return-on-capital by allocating capital in proportion to a risk measure
• Allocating the cost of capital to provide a minimum target profit (profit above this is considered economic value added)

Note that it is the cost of capital that is allocated among the units, rather than the capital itself – this is more realistic, since all business units have access to the entirety of the firm’s capital. A simple approach is to calculate the risk measure individually for each business unit, and then spreading overall risk proportionally. An alternate approach applies to risk measures that can be expressed as sums of co-measures, which exist when the risk measure is of the form \[ \rho(Y) = E[h(Y)L(Y) | g(Y)] \] where

• \(h(Y)\) is an additive function, i.e. \(h(U+V) = h(U) + h(V)\)
• \(L(Y)\) is any function for which the conditional expectation exists
• \(g(Y)\) is some condition on \(Y\)

In this case, when \(Y = \sum_j X_j\), the co-measure \(r\) is defined as \[ r(X_j) = E[h(X_j)L(Y)|g(Y)] \] By the additivity condition, \[ \rho(Y) = \sum_j r(X_j) \] Note that for any given risk measure, the functions \(L\) and \(h\) are not unique, so this does not uniquely define a decomposition into co-measures.

An alternative approach is to use a marginal method, which quantifies the change in overall company risk due to a small change in a business unit’s volume. If a unit with an above-average ratio of profit to risk increases its volume, then the overall ratio of profit to risk will increase. (This is easiest to achieve when a business unit can uniformly change its volume, e.g. with a quota share treaty.) This approach is most effective when the risk measure is proportional to the market value of the risk. The marginal method can be defined as a directional derivative: \[ r_j(Y) = \lim_{\epsilon \rightarrow 0} \frac{\rho(Y + \epsilon X_j) - \rho(Y)}{\epsilon} \]

If a risk measure is scalable (homogeneous of degree 1), then multiplying a random variable by a scalar multiplies the risk measure by the same amount. This typically occurs for measures that are expressed in units of currency. In this case, a marginal risk attribution method exists, and it is a co-measure as well. One way to calculate this is to apply l’H^{o}pital’s Rule: take the derivative of \(\rho(Y + \epsilon X_j)\) with respect to \(\epsilon\), and set \(\epsilon = 0\).

VaR, TVaR, and XTVaR

Tail-based measures assess the risk associated with large losses only. Value at Risk (VaR) is just a percentile of the loss distribution. In other words, if \(F(x)\) is the cumulative distribution function for the loss distribution, then \[ \mathrm{VaR} = F^{-1}(p) \] at probability level \(p\). For example, to calculate VaR at the 99th percentile:

VaR_99 = quantile(simulation.results$impact_on_capital, probs = 0.99)
VaR_99_line_1 = quantile(simulation.results$line_1_capital, probs = 0.99)
VaR_99_line_2 = quantile(simulation.results$line_2_capital, probs = 0.99)
print(paste0("VaR-99 for the company is ", VaR_99))

## [1] "VaR-99 for the company is 4661.1768865033"

print(paste0("VaR-99 for line 1 alone is ", VaR_99_line_1))

## [1] "VaR-99 for line 1 alone is 3433.49978735007"

print(paste0("VaR-99 for line 2 alone is ", VaR_99_line_2))

## [1] "VaR-99 for line 2 alone is 2650.04451047233"

We can express VaR as a sum of co-measures: \[ r(X_j) = E[X_j | F(Y) = p] \] However, this is not a very practical approach since there is exactly one simulation for which \(F(Y) = p\), so this is does not produce a stable result. VaR is scalable, and the directional derivative gives the same result as above.

Tail Value at Risk is the average value of losses above a given probability level: \[ TVaR = E[X | X \geq \mathrm{VaR}] = \frac{\int_p^{\infty} xf(x)dx}{1-p} \] Using the simulated data:

TVaR_99 = simulation.results[impact_on_capital >= VaR_99, mean(impact_on_capital)]
TVaR_99_line_1 = simulation.results[line_1_capital >= VaR_99_line_1, mean(line_1_capital)]
TVaR_99_line_2 = simulation.results[line_2_capital >= VaR_99_line_2, mean(line_2_capital)]
print(paste0("TVaR-99 for the company is ", TVaR_99))

## [1] "TVaR-99 for the company is 5690.81822767097"

print(paste0("TVaR-99 for line 1 alone is ", TVaR_99_line_1))

## [1] "TVaR-99 for line 1 alone is 4332.5867110888"

print(paste0("TVaR-99 for line 2 alone is ", TVaR_99_line_2))

## [1] "TVaR-99 for line 2 alone is 3528.84379408856"

print(paste0("Proportion of risk allocated to line 1 is ", TVaR_99_line_1 / (TVaR_99_line_1 + TVaR_99_line_2)))

## [1] "Proportion of risk allocated to line 1 is 0.55111938065667"

TVaR is generally more useful than VaR, because we don’t see a VaR-99 loss once every 100 years. Instead, once every 100 years, we see an average loss equal to TVaR.

We can decompose TVaR as a sum of the co-measure \[ r(X_j) = E[X_j | Y > \mathrm{VaR}] \] since in this case, \(\mathrm{TVaR} = E[\sum_j X_j | Y > \mathrm{VaR}] = \sum_j r(X_j)\). In order to allocate TVaR among lines of business, one approach is to calculate the average contribution to the loss, conditional on the overall company results exceeding the 99th percentile:

co_TVaR_99_line_1 = simulation.results[impact_on_capital >= VaR_99, mean(line_1_capital)]
print(paste0("co-TVaR-99 for line 1 is ", co_TVaR_99_line_1))

## [1] "co-TVaR-99 for line 1 is 3407.91649867727"

co_TVaR_99_line_2 = simulation.results[impact_on_capital >= VaR_99, mean(line_2_capital)]
print(paste0("co-TVaR-99 for line 2 is ", co_TVaR_99_line_2))

## [1] "co-TVaR-99 for line 2 is 2282.9017289937"

print(paste0("Proportion of risk allocated to line 1 is ", co_TVaR_99_line_1 / TVaR_99))

## [1] "Proportion of risk allocated to line 1 is 0.598844728883917"

Note that the alternate TVaR approach does not apply well in practice to VaR, because there is only one simulation that occurs exactly at the 99th percentile of the overall company performance.

A closely-related metric is Excess Tail Value at Risk (XTVaR), which is equal to TVaR less the mean. This measure reflects the idea that the mean is already financed by other funding, so capital is only needed for losses above the mean:

XTVaR_99 = TVaR_99 - simulation.results[, mean(impact_on_capital)]
XTVaR_99_line_1 = TVaR_99_line_1 - simulation.results[, mean(line_1_capital)]
XTVaR_99_line_2 = TVaR_99_line_2 - simulation.results[, mean(line_2_capital)]
print(paste0("XTVaR-99 for the company is ", XTVaR_99))

## [1] "XTVaR-99 for the company is 6069.59787861799"

print(paste0("XTVaR-99 for line 1 is ", XTVaR_99_line_1))

## [1] "XTVaR-99 for line 1 is 4605.54371673404"

print(paste0("XTVaR-99 for line 2 is ", XTVaR_99_line_2))

## [1] "XTVaR-99 for line 2 is 3634.66643939035"

Similarly to TVaR, we can decompse XTVaR as a sum of the co-measure \[ r(X_j) = E[X_j - E[X_j] | Y > \mathrm{VaR}] \]

mean.line.1 = simulation.results[, mean(line_1_capital)]
mean.line.2 = simulation.results[, mean(line_2_capital)]
co_XTVaR_99_line_1 = simulation.results[impact_on_capital >= VaR_99, mean(line_1_capital - mean.line.1)]
co_XTVaR_99_line_2 = simulation.results[impact_on_capital >= VaR_99, mean(line_2_capital - mean.line.2)]
print(paste0("co-XTVaR-99 for line 1 is ", co_XTVaR_99_line_1))

## [1] "co-XTVaR-99 for line 1 is 3680.87350432251"

print(paste0("co-XTVaR-99 for line 2 is ", co_XTVaR_99_line_2))

## [1] "co-XTVaR-99 for line 2 is 2388.72437429548"

This is also a marginal risk measure, though the directional derivative is not obvious.

Expected Policyholder Deficit

Expected Policyholder Deficit (EPD) is the expected value of defaulted losses, if capital available were set at VaR: \[ \mathrm{EPD} = (\mathrm{TVaR} - \mathrm{VaR}) (1-p) \] Note that \(\mathrm{TVaR} - \mathrm{VaR}\) is the conditional expected loss, given the loss exceeds \(\mathrm{VaR}\). Multiplying by the probability of exceeding \(\mathrm{VaR}\) converts this into an unconditional expectation:

EPD_99 = (TVaR_99 - VaR_99) * 0.01
EPD_99_line_1 = (TVaR_99_line_1 - VaR_99_line_1) * 0.01
EPD_99_line_2 = (TVaR_99_line_2 - VaR_99_line_2) * 0.01
print(paste0("The expected policyholder deficit is ", EPD_99))

## [1] "The expected policyholder deficit is 10.2964134116767"

print(paste0("The expected policyholder deficit for line 1 is ", EPD_99_line_1))

## [1] "The expected policyholder deficit for line 1 is 8.99086923738736"

print(paste0("The expected policyholder deficit for line 2 is ", EPD_99_line_2))

## [1] "The expected policyholder deficit for line 2 is 8.78799283616238"

To define a co-measure, let \(B = F_Y^{-1}(p)\) and express EPD as \[ \rho(Y) = (1-p) E[Y-B | Y > B] \] This is only scalable when \(B\) is a fixed quantile of \(Y\). In this case, the marginal decomposition is \[ r(X_j) = \alpha[\mathrm{coTVaR - coVaR}] \] This follows from the definition of EPD, and linearity of the directional derivative.

Probability transforms

A probability transform measures risks by shifting probability toward the unfavourable outcomes, and then computing one of the risk measures with the transformed probabilities.

• Typical example: expected loss under the transformed probabilities. The Black-Scholes option pricing formula is an example of this.
• Key benefit is that it can incorporate the market value of the risk it is measuring.
• Transformed probabilities can be used with other risk measures as well, producing weighted analogues (WVaR, WTVaR, WXTVaR).
• This can be used to address concerns that TVaR treats all large losses linearly, which is not consistent with most risk preferences.

Generalized moments

A simple moment-based measure is the variance of the decrease in capital: Typically, standard deviation is preferred over variance because it is in units of currency rather than currency squared:

sd_capital = simulation.results[, sd(impact_on_capital)]
sd_capital_1 = simulation.results[, sd(line_1_capital)]
sd_capital_2 = simulation.results[, sd(line_2_capital)]
print(paste0("The standard deviation of the change in capital is ", sd_capital, " and the variance is ", sd_capital^2))

## [1] "The standard deviation of the change in capital is 1572.86167913589 and the variance is 2473893.86169418"

print(paste0("The standard deviation of the change in capital for line 1 only is ", sd_capital_1, " and the variance is ", sd_capital_1^2))

## [1] "The standard deviation of the change in capital for line 1 only is 1162.73013844201 and the variance is 1351941.37484137"

print(paste0("The standard deviation of the change in capital for line 2 only is ", sd_capital_2, " and the variance is ", sd_capital_2^2))

## [1] "The standard deviation of the change in capital for line 2 only is 772.313690179503 and the variance is 596468.436038681"

Note that due to the correlation between the lines, we would underestimate the standard deviation if we were to measure each line individually and combine them:

sqrt(sd_capital_1^2 + sd_capital_2^2)

## [1] 1395.855

Since we can express \[ \mathrm(Var)(Y) = E[(Y - E[Y])(Y - E[Y])] \] by setting \(h(Y) = L(Y) = Y - E[Y]\), we obtain the co-variance as a co-measure: \[ \mathrm{Cov}(X_j, Y) = r(X_j) = E[(X_j - E[X_j])(Y - E[Y])] \] In this example:

mean_X_1 = simulation.results[, mean(line_1_capital)]
mean_X_2 = simulation.results[, mean(line_2_capital)]
mean_Y = mean_X_1 + mean_X_2
mean_X_1_Y = simulation.results[, mean(line_1_capital * impact_on_capital)]
mean_X_2_Y = simulation.results[, mean(line_2_capital * impact_on_capital)]
cov_X_1 = mean_X_1_Y - mean_X_1 * mean_Y
cov_X_2 = mean_X_2_Y - mean_X_2 * mean_Y
print(paste0("The covariance of line 1 with the overall company result is ", cov_X_1))

## [1] "The covariance of line 1 with the overall company result is 1614521.93190841"

print(paste0("The covariance of line 2 with the overall company result is ", cov_X_2))

## [1] "The covariance of line 2 with the overall company result is 859124.540399601"

Variance is not scalable, since it is in units of currency squared, so the directional derivative approach does not apply.

For standard deviation, there are two reasonable candidates for a co-measure. The first is \[ r_1(X_j) = E\left[X_j \frac{\mathrm{sd}(Y)}{E[Y]}\right] \] The second is \[ r_2(X_j) = E\left[(X_j - E[X_j]) \frac{Y - E[Y]}{\mathrm{sd}(Y)}\right] = \frac{\mathrm{Cov}(X_j, Y)}{\mathrm{sd}(Y)} \] The marginal risk measure can be determined through differentiation of \[ \rho(Y + \epsilon X_j) = (\mathrm{Var}(Y + \epsilon X_j))^{1/2} \] First, re-express this as \[ \rho(Y + \epsilon X_j) = (\mathrm{Var}(Y) + 2\epsilon\mathrm{Cov}(X_j, Y) + \epsilon^2 \mathrm{Var}(X_j)^2)^{1/2} \] The derivative with respect to \(\epsilon\) is \[ \frac{d\rho(Y + \epsilon X_j)}{d\epsilon} = \frac12 (\mathrm{Var}(Y) + 2\epsilon\mathrm{Cov}(X_j, Y) + \epsilon^2 \mathrm{Var}(X_j)^2)^{-1/2}(2 \mathrm{Cov}(X_j, Y) + 2\epsilon\mathrm{Var}(X_j)^2) \] Setting \(\epsilon = 0\), we obtain \[ r(X_j) = \frac{\mathrm{Cov}(X_j, Y)}{\mathrm{sd}(Y)} \] indicating that the function \(r_2\) defined above is the unique marginal co-measure.

A disadvantage of this approach is that it treats favourable deviations the same as unfavourable ones. The semistandard deviation uses only the unfavourable deviations:

semi_sd_capital = simulation.results[impact_on_capital > 0, sd(impact_on_capital)]
semi_sd_capital_1 = simulation.results[line_1_capital > 0, sd(line_1_capital)]
semi_sd_capital_2 = simulation.results[line_2_capital > 0, sd(line_2_capital)]
print(paste0("The semistandard deviation of the change in capital is ", semi_sd_capital))

## [1] "The semistandard deviation of the change in capital is 1282.5678877533"

print(paste0("The semistandard deviation of the change in capital for line 1 only is ", semi_sd_capital_1))

## [1] "The semistandard deviation of the change in capital for line 1 only is 953.399528035576"

print(paste0("The semistandard deviation of the change in capital for line 2 only is ", semi_sd_capital_2))

## [1] "The semistandard deviation of the change in capital for line 2 only is 762.407926475597"

Quadratic risk measures may not fully reflect market attitudes toward risk. This can be addressed by using higher moments (e.g. skewness), or by using exponential moments: \[ E[Y \exp(cY/E[Y])] \] Setting \(c=0.5\), this value may be calculated as follows:

E_Y = simulation.results[, mean(impact_on_capital)]
simulation.results[, exponential_moment_contribution := impact_on_capital * exp(0.5 * impact_on_capital / E_Y)]
exponential.moment = simulation.results[, mean(exponential_moment_contribution)]
print(paste0("The exponential moment is ", exponential.moment))

## [1] "The exponential moment is -12128.6536034656"

The exponential moment has the advantage that it reflects all losses, but responds more to larger losses, in contrast to tail-based measures that do not consider intermediate-sized losses. A simple co-measure in this case is \[ r(X_j) = E[X_j \exp(cY / E[Y])] \] Since the exponential moment is scalable, then it has a marginal decomposition. A straightforward (but messy) calculation of the derivative of \(\rho(Y+\epsilon X_j)\) gives \[ \frac{\partial \rho(Y + \epsilon X_j)}{\partial \epsilon}|_{\epsilon = 0} = E[X_j \exp(cY/E[Y])] + \frac{c E[X_j]}{E[Y]}E\left[y\exp(cY/E[Y]) \left(\frac{X_j}{E[X_j]} - \frac{Y}{E[Y]}\right) \right] \]

A generalized moment is an expectation that is not expressable as a power of the variable. For example, if we let \((Y>X)\) denote the function that is equal to 1 when \(Y>X\), and 0 otherwise, then we can express TVaR as a generalized moment: \[ TVaR = E[Y(F_Y(Y) > p)] \] A spectral measure is a generalized moment that can be written in the form \[ \rho = E[Y \eta(F(Y))] \] Therefore, TVaR is an example of a spectral measure. A variation on VaR can be defined by weighting based on the distance from the target percentile: \[ \eta(p) = \exp(-\theta(p - 0.99)^2) \]

Summary of Advantages / Disadvantages

Depending on the context, each of the following could be either an advantage or disadvantage:

• Variance is in units of squared currency, so standard deviation is generally preferred. (Necessary condition for a risk measure to be scalable is that it is in units of currency.)
• Traditional quadratic risk measures treat favouable deviations the same as unfavourable ones; semistandard deviation uses only unfavourable deviations.
• Quadratic risk measures do not generally capture market attitudes to risk; higher moments such as skewness or exponential moments address this.
• Tail measures (VaR, TVaR, XTVaR, EPD, default option) focus on large losses only and ignore intermediate-valued losses. This can be partly addressed by applying these measures at lower probability thresholds.
• Exponential moments reflect all losses, but respond more to large losses.
• Tail measures treat large losses linearly, which does not correspond to market risk preferences. Weighted versions of tail measures can shift probability toward the unfavourable outcomes to address this.
• Tail measures are only scalable when expressed in term of a fixed percentile of the distribution.
• VaR focusses on only one point in the distribution. This results in the contributions of individual business units being unstable in simulations. (Only one simulation at that exact probability level.)
• TVaR is easier to allocate among business lines (average contribution of the line when the company’s loss exceeds the threshold).
• TVaR is a better representative of return time than VaR at the same percentile. For example, a 1-in-10 event is not equal to the 90th percentile VaR. Instead, once in ten years, we expect a random draw from the 90th percentile and above; TVaR represents the average in this case.

Converting Risk Measures into a Captial Requirement

There is no one optimal capital level, since insurers can pursue different niches (e.g. customers concerned with the quality of the insurance guarantee vs. customers who are price-sensitive). Determining a capital requirement involves establishing a capital adequacy reference point, and comparing that to various risk measures.

• As an example, consider a strategy in which the company wants to ensure that it can continue to service renewals following a 1-in-10 event.
• Assume that renewals compose 80 percent of the book.
• Therefore, the company can lose at most 20 percent of its capital.
• Therefore, 20 percent of capital should cover the 90th percentile TVaR.
• Therefore, the total capital should equal five times the 90th percentile TVaR.

Reinsurance and Risk optimization

Given the illiquid nature of insurance risks, reinsurance is a valuable tool for hedging insurance risk. A simple cost-benefit analysis that subtracts ceded premiums from recoveries and ceding commissions will generally provide a negative value, and this is not an appropriate way to measure the value of the reinsurance contract, because it does not quantify the benefits of the following:

• Stability of results, including protection of surplus from erosion from adverse fluctuations
• Reinsurance frees up risk capital that would otherwise be required for the cedant. Value of reinsurance can be measured as the ratio of income forgone to purchase the cover to the amount of capital freed up. This is the ROE of the cost of reinsurance purchase, and as long as it is better than the firm’s target ROE, the purchase is a good financial decision. In this case, “better” means higher if the change involves a capital outlay, and lower if it involves recovering capital.
• Viewed in reverse: if an insurer want to remove a reinsurance contract, additional risk capital needs to be invested to replace it, and the net cost of the reinsurance is the return on this investment.
• Reinsurance adds value to the firm

Quantifying Stability and its Value

The general approach to quantifying stability is to calculate the company’s average results under various reinsurance structures, and stability measures such as standard deviation, Value-at-Risk, skewness, etc. The approach will be illustrated using the simulated data from the previous section (note that this is not the same simulation appearing in the syllabus reading), under three possible reinsurance structures:

• Option 1: No reinsurance
• Option 2: Line 1 has a proportional excess treaty, covering 80 percent of 4000 excess of 4000, and Line 2 has an aggregate excess treaty covering 3000 excess of 4000. The premium for this reinsurance policy is 240.
• Option 3: A stop-loss treaty covering the entire company, at 5000 excess of 7000. The premium for this reinsurance policy is 110.

option.2.premium = 240
option.3.premium = 110
reinsurance.simulation = simulation.results[, .(simulation_number, line_1_loss, line_2_loss, company_loss)]
reinsurance.simulation[, recoverable_2 := 0.8 * pmax(0, pmin(line_1_loss, 8000) - 4000) + pmax(0, pmin(line_2_loss, 7000) - 3000)]
reinsurance.simulation[, recoverable_3 := pmax(0, pmin(company_loss, 12000) - 7000)]
reinsurance.simulation[, net_results_1 := line.1.premium + line.2.premium - company_loss]
reinsurance.simulation[, net_results_2 := line.1.premium + line.2.premium - company_loss + recoverable_2 - option.2.premium]
reinsurance.simulation[, net_results_3 := line.1.premium + line.2.premium - company_loss + recoverable_3 - option.3.premium]
reinsurance.summary = reinsurance.simulation[, .(net_results_1 = mean(net_results_1), 
                                                 sd_1 = sd(net_results_1), 
                                                 min_1 = min(net_results_1),
                                                 max_1 = max(net_results_1),
                                                 safety_1 = quantile(net_results_1, probs = 0.99),
                                                 net_results_2 = mean(net_results_2), 
                                                 sd_2 = sd(net_results_2), 
                                                 min_2 = min(net_results_2),
                                                 max_2 = max(net_results_2),
                                                 safety_2 = quantile(net_results_2, probs = 0.99),
                                                 net_results_3 = mean(net_results_3), 
                                                 sd_3 = sd(net_results_3),
                                                 min_3 = min(net_results_3),
                                                 max_3 = max(net_results_3),
                                                 safety_3 = quantile(net_results_3, probs = 0.99))]
reinsurance.summary

##    net_results_1     sd_1     min_1   max_1 safety_1 net_results_2
## 1:      378.7797 1572.862 -9934.364 3737.36 2951.696       322.735
##        sd_2    min_2   max_2 safety_2 net_results_3    sd_3     min_3
## 1: 1230.044 -5344.04 3497.36 2711.696      354.3525 1352.56 -5044.364
##      max_3 safety_3
## 1: 3627.36 2841.696

Observations about this simulation include:

• In both reinsurance options, the overall net result is lower, but so is the standard deviation. Option 2 reduces the standard deviation much more than Option 3, but at a higher net cost.
• The “Safety Value” is the 99th percentile of net results, indicating how each treaty performs in a very good year; Option 3 and removal of reinsurance perform well here, illustrating the tradeoff between stability and the opportunity to benefit during favourable years.
• Overall, Option 2 appears to provide the most protection.

Visualizing probability densities can also be helpful:

ggplot(data = reinsurance.simulation, aes(x = net_results_1)) + geom_density(color = "red") + geom_density(aes(x = net_results_2), color = "green") + geom_density(aes(x = net_results_3), color = "blue")

Notice that the treaties that provide more protection are shifted to the left, indicating a lessened ability to benefit in good years. Cumulative distribution plots may be more insightful:

ggplot(data = reinsurance.simulation, aes(x = net_results_1)) + stat_ecdf(geom = "step", color = "red") + stat_ecdf(aes(x = net_results_2), geom = "step", color = "green") + stat_ecdf(aes(x = net_results_3), geom = "step", color = "blue")

At a given probability level, the curve furthest to the right performs best. Notice that for most of the graph, Option 3 outperforms Option 2, though there is a range of probabilities in the 5% to 20% range where Option 2 performs better. We can also look at these values in tabluar format:

reinsurance.cdfs = data.table(probability = 1:50 * 0.01)
reinsurance.cdfs$option_1 = apply(reinsurance.cdfs, 1, function(r) quantile(reinsurance.simulation$net_results_1, probs = r['probability']))
reinsurance.cdfs$option_2 = apply(reinsurance.cdfs, 1, function(r) quantile(reinsurance.simulation$net_results_2, probs = r['probability']))
reinsurance.cdfs$option_3 = apply(reinsurance.cdfs, 1, function(r) quantile(reinsurance.simulation$net_results_3, probs = r['probability']))
datatable(reinsurance.cdfs)

Option 2 outperforms Option 3 between the 3-in-100 year level and the 28-in-100 year level. Note that these do not necessarily correspond to the same events, since the CDFs were calculated independently for each reinsurance option. This is essential: we are ranking results according to their own ending probability distributions, not ranking them by the difference with another program. Variations on this approach include:

• Produce a graph of losses at various probabilities, against the cost of the program. A more expensive program is efficient at a selected probability if it has a lower loss level at that probability.
• Financial measures other than net UW income can be used as well, such as pre-tax net income (will consider differences in expenses and investment income).
• Combined ratio is not recommended as a measure, since recoveries are subtracted from net losses and ceded premium is subtracted from gross premium, but if expenses are not ceded, this will overstate the expense portion of the ratio.
• When large numbers of programs are being considered, scatterplots of some risk metric against some reward metric may be used at various probability levels. The idea is to identify programs that are clearly inefficient, and look toward structures that either reduce risk or increase reward.
• In general, competing programs may not be “better” or “worse” but instead represent risk / reward tradeoffs that can only be distinguished by company preferences, budgetary, or other constraints. Examples include looking at the probability that:
  • Surplus drops below a regulatory or internal requirement
  • Surplus drops below that required for a target BCAR rating
  • Percentage drop in quarterly earnings per share

Reinsurance as Capital

By improving stability, reinsurance should reduce needed capital. Since capital carries a cost, this provides a basis for assessing the benefit of reinsurance relative to the cost (net recoverables). There are two approaches:

• Subtract net recoverables from the reduction in cost of capital for a net benefit in dollar terms.
• Divide the cost of reinsurance by the change in required capital to obtain an ROE measure. Structures with higher ROE are preferred, and structures with marginal ROE above the company’s cost of capital are preferable to no reinsurance.

With the ROE approach, we need to determine the reduction in capital required. Two approaches include:

• Theoretical approach: derive required capital and determine how it changes under an ERM model, based on some risk measure.
• Practical approach: derive required capital from rating agency formulas or regulatory requirements. Re-calculate the required capital with each of the various reinsurance structures in place.
  • Easier to implement
  • Relies on proxies, such as premium, to quantify risk. Some structures may show little impact on required capital as a result.
  • Can avoid disadvantages by building rating agency and regulatory capital models into an ERM model.
  • A multiple of XTVaR at lower percentiles is generally preferred to VaR, since VaR limits the definition of risk to a single point on the loss distribution.

These ideas can be illustrated using the simulated data, setting a capital requirement of XTVaR at the 90th percentile:

option.1.mean.loss = mean(reinsurance.simulation$company_loss)
option.2.mean.loss = mean(reinsurance.simulation$company_loss - reinsurance.simulation$recoverable_2)
option.3.mean.loss = mean(reinsurance.simulation$company_loss - reinsurance.simulation$recoverable_3)
option.1.VaR = quantile(reinsurance.simulation$company_loss, probs = 0.9)
option.2.VaR = quantile(reinsurance.simulation$company_loss - reinsurance.simulation$recoverable_2, probs = 0.9)
option.3.VaR = quantile(reinsurance.simulation$company_loss - reinsurance.simulation$recoverable_3, probs = 0.9)
option.1.TVaR = reinsurance.simulation[company_loss > option.1.VaR, mean(company_loss)]
option.2.TVaR = reinsurance.simulation[company_loss - recoverable_2 > option.2.VaR, mean(company_loss - recoverable_2)]
option.3.TVaR = reinsurance.simulation[company_loss - recoverable_3 > option.3.VaR, mean(company_loss - recoverable_3)]
option.1.XTVaR = option.1.TVaR - option.1.mean.loss
option.2.XTVaR = option.2.TVaR - option.2.mean.loss
option.3.XTVaR = option.3.TVaR - option.3.mean.loss
option.1.capital.req = option.1.XTVaR
option.2.capital.req = option.2.XTVaR
option.3.capital.req = option.3.XTVaR
option.2.ROE = (mean(reinsurance.simulation$recoverable_2) - option.2.premium) / (option.2.capital.req - option.1.capital.req)
print(paste0("The ROE for option 2 is ", option.2.ROE))

## [1] "The ROE for option 2 is 0.0536912923887152"

option.3.ROE = (mean(reinsurance.simulation$recoverable_3) - option.3.premium) / (option.3.capital.req - option.1.capital.req)
print(paste0("The ROE for option 3 is ", option.3.ROE))

## [1] "The ROE for option 3 is 0.0317172414690246"

Since options 2 and 3 result in a recovery of captial relative to option 1, based on the above, option 3 is preferred. However, the ROE should still be compared to the company’s cost of capital: if it is less than 3.1%, then it would be better to not purchase reinsurance.

The above approach only considered capital requirements over a single year. For long-tailed business, the reserves typically absorb capital for more than one year, and this should be considered when analyzing the value of reinsurance. An additional concern is the accumulation of risks that are correlated across accident years.

• As-if reserves for an accident year are the loss reserves that would exist at the beginning of the year if that business had been written in steady state, aside from trend, in all prior years.
• Captial absorbed in the curent year is the combination of the accident year losses and the as-if loss reserves. This represents the capital absorbed over time.

Reinsurance and Market Value

Extending the idea in the previous section, since the stability obtained from reinsurance is a substitute for capital, it should increase the ultimate value of the firm, which is a more advanced level of cost-benefit analysis. Recent studies have found the following:

• Insureds demand price discounts between 10 and 20 times the expected cost of an insurer default.
• A 1 percent decrease in capital leads to a 1 percent loss in pricing, and a 1 percent increase in the standard deviation in earnings leads to a 0.33 percent decrease in pricing.
• A ratings upgrade is worth 3 percent in growth, but a ratings downgrade can produce a 5 to 20 percent drop in business.

Enterprise-wide risk modelling and aggregation

Incorporating the use of correlation

Copulas provide a method for combining individual distributions into a single multivariate distribution. A key consideration is not only how much correlation there is, but where in the distribution it takes place. A copula is a function \(C(u,v)\) that expresses a joint probability distribution \(F(x,y)\) in terms of the marginal distributions \(F_X(x)\) and \(F_Y(y)\): \[ F(x,y) = C(F_X(x), F_Y(y)) \] Note that this requires that \(C\) be a distribution function on the unit square. Any multivariate distribution can be expressed in this way for some copula. Provide the conditional distribution function \(C(v|u)\) can be inverted, copulas can be simulated by first simulating a random draw of \(u\in [0,1]\), then generating a random \(p \in [0,1]\), then calculating \(C^{-1}(p|u)\). Then, the marginal distributions can be inverted to get \(x = F_X^{-1}(u)\) and \(y = F_Y^{-1}(v)\). Examples of copulas include:

Frank Copula

Given a parameter \(a\), this is defined by \[ C(u,v) = -a^{-1} \log(1 + g_ug_v / g_1) \] where \(g_z = e^{-az} - 1\). The Kendall \(\tau\) value for this copula is \[ \tau = 1 - \frac4a + \frac{4}{a^2} \int_0^a \frac{t}{e^t-1}dt \] This can be visualized as follows:

g = function(z, a) {
  return(exp(-a * z) - 1) 
}
frank.copula = function(u, v, a) {
  return(- log(1 + g(u, a) * g(v, a) / g(1, a)) /a)
}
u = 0:100
v = 0:100
copula.illustration = data.table(merge(u, v, by = NULL))
colnames(copula.illustration) = c("u_integer", "v_integer")
copula.illustration[, u := u_integer / 100]
copula.illustration[, v := v_integer / 100]
copula.illustration$frank_copula_CDF = apply(copula.illustration, 1, function(r) frank.copula(u = r['u'], v = r['v'], a = 5))
ggplot(copula.illustration, aes(x = u, y = v, fill = frank_copula_CDF)) + geom_raster() + scale_fill_gradientn(colours = terrain.colors(10)) + geom_contour(aes(z = frank_copula_CDF))

For comparison, look at a the cumulative distribution function for a copula in which \(X\) and \(Y\) are independent:

copula.illustration$independent_CDF = apply(copula.illustration, 1, function(r) r['u'] * r['v'])
ggplot(copula.illustration, aes(x = u, y = v, fill = independent_CDF)) + geom_raster() + scale_fill_gradientn(colours = terrain.colors(10)) + geom_contour(aes(z = independent_CDF))

Properties of the Frank copula include:

• The conditional distribution can be inverted, so it can be used in simulations.

Gumbel Copula

The Gumbel copula is defined for \(a\geq 1\): \[ C(u,v) = \exp(-((-\log(u))^a + (- \log(v))^a)^{1/a}) \] The Kendall \(\tau\) value for this copula is \[ \tau = 1 - \frac1a \]

gumbel.copula = function(u, v, a) {
  return(exp(-((-log(u))^a + (-log(v))^a)^(1/a)))
}
copula.illustration$gumbel_copula_CDF = apply(copula.illustration, 1, function(r) gumbel.copula(u = r['u'], v = r['v'], a = 1.5))
ggplot(copula.illustration, aes(x = u, y = v, fill = gumbel_copula_CDF)) + geom_raster() + scale_fill_gradientn(colours = terrain.colors(10)) + geom_contour(aes(z = gumbel_copula_CDF))

Properties of the Gumbel copula include:

• It has more probability concentrated in the tails than the Frank copula does
• The conditional distribution function is not invertible, so alternative methods must be used for simulation, e.g. solving numerically

Heavy Right Tail (HRT)

The Heavy Right Tail copula, for \(a>0\), is \[ C(u,v) = u + v - 1 + ((1 - u)^{-1/a} + (1 - v)^{-1/a} - 1)^{-a} \] The Kendall \(\tau\) value for this copula is \[ \tau = \frac{1}{2a + 1} \]

hrt.copula = function(u, v, a) {
  return(u + v - 1 + ((1 - u)^(-1/a) + (1 - v)^(-1/a) -1)^(-a))
}
copula.illustration$hrt_copula_CDF = apply(copula.illustration, 1, function(r) hrt.copula(u = r['u'], v = r['v'], a = 1))
ggplot(copula.illustration, aes(x = u, y = v, fill = hrt_copula_CDF)) + geom_raster() + scale_fill_gradientn(colours = terrain.colors(10)) + geom_contour(aes(z = hrt_copula_CDF))

Properties of the heavy right tail copula include:

• It has high correlation in the right tail (e.g. for large losses), but not in the left tail. As a result, it is well-suited to actuarial applications.
• The conditional distribution function is invertible, so this copula is easy to simulate.
• It produces a joint Burr distribution when both marginal distributions are Burr distributions with the same parameter as the copula; it is an analogue of the joint normal distribution.

Normal Copula

Let \(N(x)\) denote the CDF for the standard normal distribution, and let \(B(x, y;a)\) denote the CDF for the bivariate normal distribution. The Normal copula is \[ C(u,v) = B(N^{-1}(u), N^{-1}(v); a) \]

Properties of the normal copula include:

• It is easy to simulate and generalizes to multiple dimensions. (The \(t\)-copula is another multivariate option, and it has a parameter that provides more control over the tail weight.)
• It is lighter in the right tail than the Gumbel or HRT, but heavier than the Frank copula

Evaluation and selection of copulas

The general approach to evaluation of copulas involves calculating functions of the copulas, then looking at these functions for the data. Several copulas are compared, and the one that is the best fit for the data is selected. For example, the right and left tail concentrations are defined as \[ R(z) = \mathrm{Pr}(U> z | V>z) \] and \[ L(z) = \mathrm{Pr}(U < z | V < z) \] The right-tail concentration can be re-expressed as \[ R(z) = \mathrm{Pr}(U>z, V>z) / (1-z) = \frac{1 - 2z + C(z,z)}{1-z} \] and the left tail concentration as \[ L(z) = \mathrm{Pr}(U<z, V<z) / z = \frac{C(z,z)}{z}. \] To illustrate these ideas, compare the Gumbel and Heavy Right Tail copulas. (Note that the parameters selected above ensure that \(\tau = 1/3\) for both copulas, so the comparison will be fair.)

#right.tail.concentration = function(z, copula_data) {
#  cp.numerator = copula_data[u_integer == z & v_integer >= z, .(sum(1 - copula_value))][[1]]
#  cp.denominator = copula_data[v_integer == z, .(sum(1 - copula_value))][[1]]
#  return(cp.numerator / cp.denominator)
#}
gumbel.right = function(z, a) {
  (1 - 2 * z + gumbel.copula(z, z, a)) / (1 - z)
}
hrt.right = function(z, a) {
  (1 - 2 * z + hrt.copula(z, z, a)) / (1 - z)
}
independent.right = function(z) {
  (1 - 2 * z + z^2) / (1 - z)
}
#left.tail.concentration = function(z, copula_data) {
#  cp.numerator = copula_data[u_integer == z & v_integer <= z, .(sum(copula_value))][[1]]
#  cp.denominator = copula_data[v_integer == z, .(sum(copula_value))][[1]]
#  return(cp.numerator / cp.denominator)
#}
gumbel.left = function(z, a) {
  gumbel.copula(z, z, a) / z
}
hrt.left = function(z, a) {
  hrt.copula(z, z, a) / z
}
independent.left = function(z) {
  z
}
tail.concentration.data = data.table(z = 0.001 * 1:999)
tail.concentration.data$gumbel_right = apply(tail.concentration.data, 1, function(r) gumbel.right(r['z'], a = 1.5))
tail.concentration.data$hrt_right = apply(tail.concentration.data, 1, function(r) hrt.right(r['z'], a = 1))
tail.concentration.data$gumbel_left = apply(tail.concentration.data, 1, function(r) gumbel.left(r['z'], a = 1.5))
tail.concentration.data$hrt_left = apply(tail.concentration.data, 1, function(r) hrt.left(r['z'], a = 1))
tail.concentration.data$independent_right = apply(tail.concentration.data, 1, function(r) independent.right(r['z']))
tail.concentration.data$independent_left = apply(tail.concentration.data, 1, function(r) independent.left(r['z']))
ggplot(data = tail.concentration.data, aes(x = z, y = gumbel_right)) + geom_line(colour = "blue") + geom_line(aes(y = hrt_right), colour = "red") + geom_line(aes(y = gumbel_left), colour = "blue") + geom_line(aes(y = hrt_left), colour = "red") + geom_line(aes(y = independent_left), colour = "green") + geom_line(aes(y = independent_right), colour = "green")

Although both the left and right tail concentrations are shown above, they are often graphed so that the left tail concentration is shown below 50%, and the right tail concentration is shown above 50%. These graphs can be compared to the graphs of data to select the copula that best captures the correlation. Note that for many (but not all) copulas, \(\lim_{z\rightarrow 1} R(z) = 0\). As a result, these graphs may be misleading since for some copulas the slope is very steep near 1; there can be significant tail dependence even if the limit is zero. Investigating \(R(z)\) for values close to 1 indicates copula influence on large loss relationships. Other metrics that can be used to select copulas include the overall correlation (e.g. Kendall \(\tau\)), and the \(J\) and \(\chi\) functions.

Tail dependence and tail correlations

Low frequency / high severity events

Parameter, projection, estimation, and model risk

Parameter Risk

Let \(N\) denote the number of claims, and \(S_i\) denote the amount of claim \(i\), assuming all claims are independent and identically distributed, with \(S\) denoting this common distribution. Aggregate losses are given by \[ A = \sum_{1\leq i \leq N} S_i. \] As a result, the expected value of aggregate losses is \[ E[A] = E[E[A|N]] = E\left[\sum_{1\leq i \leq N} E[S_i]\right] = E[N]E[S] \] The variance of aggregate losses is \[ \mathrm{Var}(A) = E[\mathrm{Var}(A|N)] + \mathrm{Var}[E[A|N]] = E[N\mathrm{Var}(S)] + \mathrm{Var}(N E[S]) = E[N]\mathrm{Var}(S) + E[S]^2\mathrm{Var}(N) \] Therefore, the square of the coefficient of variation is \[ \mathrm{CV}(A)^2 = \frac{1}{E[N]}\left(\frac{\mathrm{Var}(S)}{E[S]^2} + \frac{\mathrm{Var}(N)}{E[N]}\right) \] From the factor of \(1 /E[N]\), it is clear that this risk is larger for small companies than for larger ones. However, the larger company will be more susceptible to projection risk, as the following example illustrates. Assume the following:

• Frequency distribution is Poisson
• Severity coefficient of variation is 7
• Multiply aggregate losses by a factor \(1+J\) that is random, but constant for all claims in a given year. The mean of \(1+J\) is 1.

Let \(T = 1+J\). Since \(E[1+J]=1\), then \(E[A] = E[T]\). Assuming \(J\) is independent of \(A\), then \[ \mathrm{Var}((1+J)A) = \mathrm{Var}(1+J)\mathrm{Var}(A) + \mathrm{Var}(1+J) E[A]^2 + \mathrm{Var}(A) E[(1+J)]^2 \] Therefore, \[ \frac{\mathrm{Var}((1+J)A)}{E[(1+J)A]^2} = \frac{\mathrm{Var}(A)}{E[A]^2}(1 + \mathrm{Var}(1+J)) + \mathrm{Var}(1+J) \] As the coefficient of variation of \(J\) increases, we can assess how the overall coefficient of variation for the company changes as follows:

CV.poisson = function(severity_CV, expected_frequency, cv_J = 0) {
  original_cv_squared = 1 / expected_frequency * (severity_CV^2 + 1)
  revised_cv_squared = original_cv_squared * (1 + cv_J^2) + cv_J^2
  return(sqrt(revised_cv_squared))
}
expected.frequencies = data.frame(expected_frequency = c(2000, 20000, 200000))
CV.J.assumptions = data.frame(CV_assumption = c(0, 0.01, 0.03, 0.05))
CV.example = merge(expected.frequencies, CV.J.assumptions, by = NULL)
CV.example$aggregate_cv = apply(CV.example, 1, function(r) CV.poisson(severity_CV = 7, expected_frequency = r['expected_frequency'], cv_J = r['CV_assumption']))
CV.example = as.data.table(CV.example)
datatable(CV.example)

Note that the relative impact on the larger company is greater – although the introduction of \(J\) increases the volatility of all sizes of companies, the small company was already fairly volatile. Assuming a normally-distributed loss ratio of 65%, we can determine percentiles of the loss ratio as follows:

mean.LR = 0.65
CV.example[, LR_percentile_90 := mean.LR + mean.LR * aggregate_cv * qnorm(0.9)]
CV.example[, LR_percentile_95 := mean.LR + mean.LR * aggregate_cv * qnorm(0.95)]
CV.example[, LR_percentile_99 := mean.LR + mean.LR * aggregate_cv * qnorm(0.99)]
datatable(CV.example)

Without considering parameter risk, the large company is unrealistically stable. Smaller companies are impacted less.

Projection Risk

This is the risk resulting from error in projecting past trends into the future.

• In a simple trend model, a trend line is fit to loss cost history.
  • Prediction intervals around the projection are produced. This provides some quantification of projection risk, specifically, the portion resulting from the uncertainty in the fitted trend line.
  • A problem with this approach is that the history itself is based on estimates of past claims that have not yet settled, so there is additional uncertainty associated with development on open claims and IBNR. This can be addressed by incorporating uncertainty (e.g. a random factor of \((1+J)\) in the analysis of historical data.
  • More advanced techniques can correct prediction intervals for the uncertainty associated with historical data points.
• Claim severity trend often differs from general inflation, which provides another source of uncertainty:
  • Social inflation or superimposed inflation is the excess of severity trend over general inflation.
  • Claim severity trend is often modelled directly from insurance data, without reference to general inflation.
  • Alternatively, specific inflation (e.g. auto collision) indices may be used
  • Inflation can be incorporated into trend by correcting payment data using general inflation indices, then modelling the residual inflation. This reflects a dependancy between severity trend and general inflation, which is essential in an ERM that includes a macro-economic model for future inflation rates. This allows inflation uncertainty to be incorporated into projection risk.
• Time series can be used to avoid making the implicit assumption that there is a single underlying trend rate that has been constant throughout the period of historical data.
  • Simple example is a first-order autocorrelated time series (AR-1), a simple mean-reverting time series.
  • Underlying mean is unknown and is estimated from the data.
  • Two sources of uncertainty: uncertainty in estimated trend rate, and uncertainty of the AR-1 process itself.
  • Generally, time series do a better job of quantifying projection risk than simple trend models, especially over long time frames.
  • A drawback is that a substantial number of data points is needed; a wide range of behavours must be exhibited in the data, or the model will understate risk potential. (May need to include non-insurer specific data.)

Estimation Risk

This is the risk resulting from only having a sample of the universe of possible claim outcomes.

• Maximum likelihood estimation is generally preferred, since for large datasets it is a minimum variance unbiased estimator.
• Uncertainty about the estimate depends on the shape of the likelihood surface; if it is very flat near the maximum, then a wide range of parameters have similar likelihoods, increasing the uncertainty about the parameters. Conversely, a sharply-peaked surface at the maximum has less uncertainty.
• Information matrix (second derivatives of the negative log likelihood) is used to quantify uncertainty; its inverse is the covariance matrix for the parameters of the distribution. For large datasets, parameters will have a multivariate normal distribution with this covariance matrix.
• The normality assumption is not always appropriate, e.g. if data volume is small, or the distribution is one where the parameters themselves have heavy tails.
• When available, an explicit expression for the MLE can help assess the distribution of the parameter estimates. For example, the MLE estimator for a simple Pareto distribution is the reciprocal of the average of the log of the observations. Since the logarithm of a Pareto random variable is exponential, this means its sum is Gamma, hence the estimator is Inverse Gamma, which has a heavy tail.
• A lognormal assumption may be a better fit for the parameter distribution, especially if it is heavy-tailed.
• Visualizing the likelihood surface for a 2-parameter model can help identify situations where there is ambiguity in explaining the data, e.g. because two ranges of parameters have similar likelihood. In this case, there will be correlation between the parameters.

These ideas can be illustrated using simulated data.

set.seed(12345)
random.data = data.table(observation = rgamma(1000, shape = 20, scale = 500))
head(random.data)

##    observation
## 1:   11085.668
## 2:   11379.376
## 3:    9510.158
## 4:    9801.123
## 5:   15535.713
## 6:   11660.322

gamma.loglikelihood = function(observations, shape, scale) {
  logdensity = sapply(observations, function(x) dgamma(x, shape = shape, scale = scale, log = TRUE))
  return(sum(logdensity))
}
shape.parameter.range = data.frame(shape = 5 + 0:10 * 3)
scale.parameter.range = data.frame(scale = 100 + 0:8 * 100)
parameter.ranges = merge(shape.parameter.range, scale.parameter.range, by = NULL)
parameter.ranges$loglikelihood = apply(parameter.ranges, 1, function(r) gamma.loglikelihood(observations = random.data$observation, shape = r['shape'], scale = r['scale']))

ggplot(data = parameter.ranges, aes(x = shape, y = scale, fill = loglikelihood)) + geom_raster() + scale_fill_gradientn(colours = terrain.colors(10)) + geom_contour(aes(z = loglikelihood))

The wide range of high values for the loglikelihood function suggests that there is uncertainty in the parameter estimates. In particular, it is difficult to distinguish from the data whether this is from a high-shape, low-scale Gamma distribution, or a low-shape, high-scale Gamma distribution.

datatable(as.data.table(parameter.ranges)[order(-loglikelihood)])

Model Risk

This is the risk associated with having the wrong models to start with.

• Model selection rules typically involve the likelihood function with penalties for the number of parameters
• Hannan-Quinn Information Criterion (HQIC): penalty is the log of the log of the number of observations, per parameter. It is a compromise between other methods.
• Method to quantify model risk: assign a probability of being “right” to all the better-fitting distributions, e.g. based on exponentiation of HQ metric. Simulation model selects a distribution that would be used throughout the scenario, then select parametrs from joint lognormal distribution. Make different selections in the next scenario.
• Tests of validity for projection models are likely to be inconclusive, and judgment is needed to ensure that the model is structurally consistenty with the process it is modelling.
• Model parsimony improves the stability of a projection, but in risk modelling our objective is to determine a realistic spread of potential outcomes; excessive parsimony can lead to projections that are too stable.
• A parsimonious model has embedded assumptions, and the inaccuracy of these assumptions is not generally reflected. Adding parameters transfers assumptions from the model structure to the parameters, where it is easier to assess uncertinty.

Operational Risk: Mitigation and Quantification

Types of operational risk

Operational risk is “the risk of loss resulting from inadequate or failed internal processes, people, and systems or from external events.” Legal risk is included, but strategic and reputational risk are excluded. The Basel Committee has identified categories of operational risk loss events; examples of insurer exposures in each category are as follows:

• Internal fraud: employee theft, claim falsification
• External fraud: claim fraud, policy fraud (misrepresentation)
• Employment practices and workplace safety: discrimination, repetitive stress
• Clients, products, and business practices: policyholder privacy breach, redlining, bad faith
• Damage to physical assets: exposures include own office and own automobile fleets
• Business disruption and system failures: processing centre downtime, system interruptions
• Execution, delivery, and process management: policy and claim processing errors

A.M. Best has identified the following as the top 5 causes of P&C Company impairments:

1. Deficient loss reserves (though technically this is a proximate cause in almost all impairments)
2. Rapid growth
3. Alleged fraud
4. Overstated assets
5. Catastrophic losses

A report from the U.S. Congress identified the following causes of insurer failure:

1. Underreserving
2. Underpricing
3. Insufficient supervision of underwriting authority
4. Rapid expansion
5. Reckless management
6. Abuse of reinsurance

Internal controls exist to achieve several objectives:

• Reliability of information
• Compliance with laws, regulations, etc.
• Safeguarding of assets
• Efficient use of resources
• Accomplishment of stated goals and objectives

The effectiveness of these controls can be assessed through an internal audit, which addresses:

• Effectiveness and efficiency of operations
• Reliability of financial reporting
• Compliance with laws and regulations

Plan Loss Ratios

The process for determining the plan loss ratio is viewed as the “fulcrum” of operational risk for an insurer.

• Premium targets are driven by the forecasted loss ratio, e.g. by growing in a line with a low forecasted loss ratio.
• Companies often always meet these targets because bonus schemes are based on them.
• Plan loss ratios are typically developed based on ultimate loss ratios in prior years – as a result, appropriate reserving is essential. Some reserving methods (e.g. Bornhuetter-Ferguson) rely on the plan loss ratio as an assumption. This can introduce interdependence and autocorrelation among prior-year ultimate loss ratios.
• The result is that a series of overly-optimistic forecasts can be strung together.
• As older accident years begin to deteriorate, the results will propagate forward in a process called reserve conflagration. This is unanticipated model risk, and a type of operational risk.

For both loss ratio and reserve models, there are three possible explanations for their failure.

1. The model could not produce accurate forecasts. This is generally only a valid explanation if other insurers experienced the same problem.
2. The model could have produced accurate forecasts, but was improperly used. This is an example of a “people failure.”
3. The model did produce accurate forecasts, but they were unpopular and were ignored. This is a failure of process and governance.

An inadequate plan loss ratio can incentivize several negative behaviours:

• Irresponsible underwriting could occur, given that the line is viewed as more profitable than it actually is.
• It could encourage inappropriate claims handling and inadequate reserving, since incentives may be based on “making plan”.

Underwriting Cycle Management

An assessment of an insurer’s ability to manage the underwriting cycle revolves around three questions:

1. Does the company have a proactive cycle management policy?
2. Does the company know where in the cycle the market stands at any given point?
3. Are underwriters making decisions that are consisten with the above?

The effectiveness of a cycle management policy can be assessed on the insurer’s ability to achieve the following goals:

• Stability
• Availability
• Reliability
• Affordability

As an example, adopting a “maintain market share” strategy during a soft market will result in a situation in which premium volume remains the same, but the company takes on more exposure. Once the company can no longer avoid recognition of its growing claims exposure, it may face a rating downgrade or insolvency. A ratings downgrade may force policyholders to switch carriers (failure of stability and availability). Insolvency means policyholders will only see partial recoveries on claims (failure of reliability and affordability, since customers now have to pay a portion of the claim).

A cycle management strategy generally consists of four components:

1. Intellectual property: an insurer’s franchise value is predominantly a result of its intellectual property: expertise of personnel, databases, forecasting systems, and market relationships. Core assets must be maintained throughout the cycle, which entails:
   • Retaining top talent through periods of retraction
   • Maintain a presence in core market channels
   • Maintain a consistent pattern of investment in systems, models, and databases
2. Underwriter incentives: hard-coded bonuses tied to making “the plan” does not provide flexibility. Incentive plans should be based on how well underwriters supported portfolio goals throughout the year, and these should change based on market conditions. As an example, if market prices drop to unacceptable levels, underwriters need to be able to stop writing new business while being confident that their bonuses and employment prospects remain intact.
3. Market overreaction: the industry tends to have an aggregate overreaction in both directions, which should be anticipated when determining how to manage capacity. Firms with the most available capacity during the price-improvement phase will reap profits that can offset the years of small underwriting losses.
4. Owner education: owners must understand that a drop in revenue, while a bad sign for most companies, may be a sign of good cycle management for an insurer; owners must know not to call for top-line growth at the worst point of the cycle. As a result, the overhead expense ratio will also be volatile, and this is to be expected (ties in to first point: must maintain people and systems).

Agency theory attempts to determine ways to align management and owner interests, and to understand the implications of possible divergence. For example, a management incentive plan tied to market capitalization may lead management to be more risk-seeking than the owners, essentially gambling with someone else’s money. On the other hand, stock options may make management more risk-averse than diversified shareholders, resulting in a company run more like a family enterprise. As mentioned above, in insurance this is not just an issue for management incentives, but also underwriter incentives.

Key risk indicators and operational risk modelling

Key Risk Indicators (KRIs) are leading indicators of risk, such as:

• Business production: conversion and retention ratios, pricing levels
• Internal controls: audit results and frequency
• Staffing: employee turnover, training budget, premium / policies per employee
• Claims: frequency, severity, new classes of loss

The main steps in modeling operational risk are:

1. Identify exposure bases for each risk source, which are typically KRIs. Examples include payroll, head count, policy count, claim count, and premium volume.
2. Measure the exposure level for each business unit – may include exposure or experience modelling, industry data, and large loss scenarios.
3. Estimate loss potential per unit of exposure, on a frequency and severity basis
4. Combine 2 and 3 to produce loss frequency and severity distributions for each business unit
5. Estimate the impact of loss mitigation on the frequency and severity distributions. This is a step where significant expert opinion is required, and is analogous to making adjustments to losses for changes due to tort reform, for example.

Note that at a high level this is essentially the same process as standard actuarial analysis, except that the exposures are typically not covered by any insurance program.

Types and Examples of strategic risks

There is no consensus on a definition of strategic risk, but generally focus on three aspects:

1. Intentional risk-taking as an essential part of executing a company’s strategy
2. Unintentional risks resulting from poor strategic decisions or improper execution of those decisions
3. The steps a company is taking to mitigate risks

Discussions of strategic risk often make a distinction between uncertainty as a source of risk, and the risks themselves as the effect of uncertainty. Types of strategic risks, with insurer-specific examples, include:

1. Industry risks: capital intensiveness, overcapacity, commoditization, deregulation, cycle volatility are very high risks for an insurer.
2. Technology risks: low for an insurer, with the exception of changes in distribution for personal lines via the Internet
3. Brand: moderate risk for insurers; key concern is a reputation for fair claims handling, which can be damaged through adverse press coverage or lawsuits
4. Competitor: moderate risk, since it is easy for a competitor to gain market share by writing coverage at a discount (predatory pricing). This risk is also present when the company entres into new lines or territories with inadequate underwriting expertise, pricing systems, claims handling systems, etc.
5. Customer: moderate risk, especially for large commercial insurance, due to shifts in priorities, power, and concentration
6. Project: high risk for insurers. The industry has a poor track record of value-destroing mergers and acquisitions, and insurers are small investors in research, development, and information technology, despite their high dependence on intellectual property. Mergers and acquisitions may be entered without considering intergration costs or timelines, cultural incompatibilies, reserve deficiencies, etc.
7. Stagnation: high risk of flat or declining volume / price, highly correlated with the underwriting cycle. May be flaws in the organization’s response to market cycles, such as maintaining premium volume and market share during declines, and improper incentives for underwriters.
8. Planning: major concern is with the process of setting the plan loss ratio – may fail to update based on internal financial indicators and external benchmarks, and may be susceptible to systematic optimism.

Scenario planning

Key features of scenario planning include:

• A limited number of possible future states (scenarios) are considered
• Scenarios should be internally consistent and plausible
• Scenarios consider the interactions between uncertainties
• Scenarios change several variables at one time
• Scenarios include subjective intprepretations of factors that cannot be explicitly modelled

Key steps in the scenario planning process include:

1. Define the time frame and scope of the analysis
2. Identify stakeholders
3. Identify basic trends that affect the organization
4. Identify key uncertainties
5. Construct initial scenarios
6. Check the scenarios for consistency and plausibility. Key considerations are the consistency of the trends and time frame, whether the outcomes fit together, and whether stakeholders are in realistic positions
7. Develop “learning scenarios” to identify strategically relevant themes
8. Identify research needs, e.g. building understanding of trends and uncertainties
9. Quantitative modelling of interactions
10. Develop “decision scenarios” that will be used to test strategies and generate new ideas

The key distinction between this approach and a traditional business plan is that it is a set of less detailed plans, rather than one highly-detailed (and almost certainly wrong) plan.

In an insurance context, a key application of scenario planning is in determining the planned mix of business, i.e. the planned combination of written premium, price changes, and loss ratios by line of business. The approach assumes that the company can monitor price changes on renewal business, and that loss ratio is a function of price. An example provided in the textbook assumes the following:

base.loss.ratio = 0.8
cost.trend = 1.06
plan.loss.ratio = base.loss.ratio * cost.trend
print(paste0("The plan loss ratio is ", plan.loss.ratio*100, "%"))

## [1] "The plan loss ratio is 84.8%"

A scenario analysis might have produced the following three scenarios:

scenario.table = data.table(Scenario = c("Optimistic", "Realistic", "Pessimistic"), price_change = c(0.05, 0, -0.1), likelihood = c(0.1, 0.5, 0.4))
scenario.table

##       Scenario price_change likelihood
## 1:  Optimistic         0.05        0.1
## 2:   Realistic         0.00        0.5
## 3: Pessimistic        -0.10        0.4

Under each scenario, we can calculate the actual loss ratio under each price change to see how it deviates from the plan.

scenario.table[, actual_loss_ratio := plan.loss.ratio / (1 + price_change)]
scenario.table

##       Scenario price_change likelihood actual_loss_ratio
## 1:  Optimistic         0.05        0.1         0.8076190
## 2:   Realistic         0.00        0.5         0.8480000
## 3: Pessimistic        -0.10        0.4         0.9422222

The company can now develop response plans for each scenario, e.g. writing more premium in the optimistic scenario, and less premium under the pessimistic scenario. This contrasts with an approach in which the company “sticks to plan” regardless of the price change, which could result in the company writing signifciant premium volume at a high loss ratio. A key benefit of this approach include:

• The company has already though through its responses to the scenarios beforehand, and can save time during crises by having action plans ready to go.
• Organizational inertia is reduced because flexibility is now built into the system.
• Politically challenging decisions, such as allotment of underwriting capacity across the organization, are made during the scenario planning process rather than in the heat of a market crisis.

More advanced approaches include:

• Advanced scenario testing involves simulation through an ERM model, including rules that specify the company’s responses to market environment changes. Key steps in this process include:
  • Captrure environmental variables that describe the “state of the world”
  • Define performance quality in terms of desirable goals and undesirable downsides
  • Specify action rules that describe responses to environmental changes
• Agent-based Modelling (ABM): Modelling the impact of the interaction of many market participants executing strategies simultaneously. The system can have emergent properties that arise from the interactions between participants, and which cannot be predicted by aggregating the individual participants. A typical example is that a profitable market segment will attract competition, lowering the profitability of that segment.

Modelling the Underwriting Cycle

Definition, characteristics, and drivers of the underwriting cycle

The underwriting cycle is the recurring pattern of increases (hardening) and decreases (softening) in insurance prices and profits. It is illustrated in the following dataset of industry combined ratios by year. (Original data source is A.M. Best; however, since exact values were not provided in the reading I ``eyeballed’’ it for the purpose of having sample data to illustrate concepts.)

industry.COR <- fread("./Brehm_UW_cycle.csv")
ggplot(data = industry.COR, aes(x = Year, y = COR)) + geom_point() + geom_line() + labs(title = "U.S. Property and Casualty Insurance Industry Combined Ratios")

The ratio between premiums and expected losses is viewed as the “price” of insurance for economic purposes, since it accounts for variations in premium that result from coverage changes (limits, retentions, terms and conditions). Key features of the cycle include:

• It is less of a predictable pattern, and more the result of a dynamical system with feedback, external shocks, and slow adjustment.
• Each line of business theoretically has its own cycle; however, the dominance of the automobile line tends to drive the cycle.
• Linkages across business lines can be caused by common economic / social factors, and the rise of multiline companies (since capital is shared across lines).

There are four major stages in the evolution of a line of insurance business:

• Emergence: new line, little historical data, high demand, and erratic prices. Solvency crisis leads to price correction and elimination of weaker competitors. Dynamics are primarily competitive in nature.
• Control: stabilization achieved through rating bureaux, insurance departments, etc. Dynamics are driven primarily by statistical reporting lags.
• Breakdown: control regime breaks down due to technological and social changes. Dynamics are a result of both competitive pressure and statistical lags.
• Reorganization: return to the first phase as a new “version” of the line emerges.

Note that underwriting cycle dynamics vary by stage, which makes it difficult to quantify overall.

Drivers of the underwriting cycle include:

• Institutional factors: pricing involves forecasting based on historical results; reporting and regulatory delays contribute to the cycle.
• Competition: the “winners’ curse” pushes the group toward lower rates. Firm strategies alternate between aggressive growth and price maintenance, with the change in strategy announced.
• Supply and demand: insurance needs capital to support it, so shocks that drain capital (e.g. catastrophe) reduce capacity and raise prices. Profits may decline if favourable business exits first. Capital market friction also introduces a delay factor.
• Economic linkages: variation in economic background activity, e.g. inflation affecting expected losses, investment income, cost of capital.

Soft, behavioural, and techncial modeling approaches

The response variable in underwriting cycle analysis is typically a profitability measure, such as loss ratio and combined ratio. Predictor variables vary depending on the theory of the cycle being used, but may include previous time period values of the response variable, other internal financial variables, econometric variables, and financial market variables. Competitor analysis may also be conducted through field staff, customer surveys, trade publicatinos, and rate filings. The key objective is to identify leading indicators to foretell a “turn” in the underwriting cycle. There are broadly three styles of modelling:

• Soft approaches focus on human factors, and variety / complexity of data. Mathematical rigour is demphasized.
• Technical approaches focus on mathematical formalism and rigor
• Behavioural approaches are in between

Soft

Soft approaches rely primarily on data gathering and intelligence, and focus on human judgment. Terminology used in this approach includes:

• A scenario is a detailed written statement describing a possible future state of the world, with the intention of organizing thinking around possible responses. It differs from a scenario in the context of simulations, in that soft analysis scenarios are intended to be fewer in number but more detailed.
• The Delphi method involves providing experts with core background material and a questionnaire. Answers are collated and summaries are sent back to the experts, who may reconsider their opinions or provide reasons for disagreement. Process iterates until consensus emerges.
• The Delphi method may be used to construct scenarios, or to evaluate scenarios that have already been constructed.
• Competitive analysis is a major component, by considering key financials and news items about major competitors. Distinctions are made between normal and abnormal statistics, and predicting turns in the cycle by identifying distressed financial conditions over a large number of firms.

Technical

This approach involves viewing the profitability measure as a time series, and fitting an autoregressive model to the data. Let \(X_t\) denote the response variable at time \(t\). An \(AR(n)\) model of the form \[ X_t = a + \sum_{1\leq i \leq n} b_x X_{t-i} + \sigma \epsilon_i \] is fit to the data, where \(\epsilon_i\) are independent and identically distributed unit normal random variables. Typically \(n = 2\) or \(n = 3\) is sufficient to model the underwriting cycle. We can fit an \(AR(2)\) model to the data as follows:

ar.2.model <- ar(industry.COR$COR, order.max = 2)

The autoregressive coefficients of the model are

ar.2.model$ar

## [1]  0.8847058 -0.2855619

Note that R fits a model of the form \[ X_t - m = a_1 (X_{t-1} - m) + a_2 (X_{t-2} - m) + \epsilon_t \] where \(m\) is the overall mean. To relate this to the form given in the reading, we’ll collect all the intercept terms on the right side:

TS.intercept = ar.2.model$x.mean * (1 - ar.2.model$ar[1] - ar.2.model$ar[2])
TS.intercept

## [1] 0.4228504

Model accuracy on in-time data can be assessed by calculating one-step-ahead forecasts:

industry.COR[, cor_lag_1 := shift(COR, type = "lag")]
industry.COR[, cor_lag_2 := shift(COR, n = 2, type = "lag")]
industry.COR[, OSE_forecast := TS.intercept + cor_lag_1 * ar.2.model$ar[1] + cor_lag_2 * ar.2.model$ar[2]]

The predictions for the next three points are:

ar.2.predict = predict(ar.2.model, n.ahead = 3)
ar.2.predict$pred

## Time Series:
## Start = 39 
## End = 41 
## Frequency = 1 
## [1] 1.002872 1.030247 1.047934

To visualize these predictions along with the actual data, convert this to a data table, then append it to the original dataset.

prediction.table = data.table("Year" = 2004:2006, "OSE_forecast" = ar.2.predict$pred, "SE" = ar.2.predict$se)
prediction.table[, lower_ci := OSE_forecast - 2 * SE]
prediction.table[, upper_ci := OSE_forecast + 2 * SE]
data.with.predictions = rbind(industry.COR, prediction.table, fill = TRUE)

We can plot the one-step-ahead forecasts along with the predictions to visualize the results. A 95% confidence interval around the predictions is shown in purple.

ggplot(data = data.with.predictions, aes(x = Year, y = COR)) + geom_point(color = "blue") + geom_line(color = "blue") + geom_point(aes(x = Year, y = OSE_forecast), color = "red") + geom_line(aes(x = Year, y = lower_ci), color = "purple") + geom_line(aes(x = Year, y = upper_ci), color = "purple") + labs(title = "U.S. Property and Casualty Insurance Industry Combined Ratios With Time Series Predictions")

## Warning: Removed 3 rows containing missing values (geom_point).

## Warning: Removed 3 rows containing missing values (geom_path).

## Warning: Removed 2 rows containing missing values (geom_point).

## Warning: Removed 38 rows containing missing values (geom_path).

## Warning: Removed 38 rows containing missing values (geom_path).

An important use case for these models is to simulate possible future values of the target variable, which can be used as part of a larger ERM or strategic planning model of the firm.

Behavioural (Econometric)

This approach is based on economic and behavioural theory, and represents a middle ground between the soft and technical approaches.

• Can be performed either an an aggregate industry level, or a firm-level.
• Firm-level modelling is more complicated because interactions between firms need to be tracked.
• Individual behaviours, when aggregated, can lead to “emergent” collective behaviour as a result of interactions.

Aggregate approaches based on economic considerations are described below.

Supply and Demand

• Supply and demand curves relate price (\(P\)) to quantity (\(Q\)). Graphing convention is that \(Q\) is on the horizontal axis.
• Supply curve slopes upward, indicating that in order to supply a greater quantity, a greater price is demanded.
  • Forces that increase supply (technology, new entrants) shift the curve right
  • Forces that decrease supply (capital restrictions) shift the curve left
• Demand curve slopes downward, indicating that customers will demand more at a lower price. (Typically, this is fairly flat for individual firms, but not for the industry as a whole.)
  • Increase in capital results in an overall improvement in quality, by lowering the probability of default. This shifts the demand curve to the right.
  • Conversely, after a capital reduction, the demand curve will shift to the left to recognize overall lower quality.
• Equilibrium price is the point at which the curves intersect.
• Note that by affecting both curves, a change in capital can result in both a decrease in availability and increase in price.
• In practice, linear supply curves may not be realistic. In a Gron supply curve:
  • Supply is flat below a minimum price of insurance that reflects expected losses and expenses.
  • The curve increases once capacity is reached, and the industry must acquire more capital.
  • Once the premium is high enough to supply sufficient capital on its own, the curve flattens out asymptotically.
  • Capital shocks shift the curve left or right.
  • Competitive drift is reflected by a lowering of the minimum price.

Capital Flows

• Captial infusions are likely to occur when capacity is limited and expected profits are high
• Withdrawal of capital can be caused by firms exiting the business
• In between, there is a “normal” region in which steady dividends are paid
• Sources of uncertainty: point at which a surge in new capital occurs, and the size of the discontinuity