Basic chain ladder method (BCL)
Definition
The BCL method is a statistical method of estimating outstanding claims, specifically estimating the ultimate value of a set of development data. It projects the future by calculating and applying the weighted average of past claim development (known as link ratios or development factors) into the future. This method can be applied to many different categories of data, including premiums, paid claims, incurred claims, and numbers of claims.
BCL Method
Step by step method
- Tabulate Claims: Tabulate the claims data ( eg cumulative paid or incurred amounts) in a run-off triangle format, group by origin year (eg accident year or underwriting year) and development year.
- Calculate Development Ratios (Link Ratios): Calculate the ratios of accumulated claims in successive development periods (link ratios).
- Apply Ratios: Apply these ratios to the cumulative claims data to complete the triangle (i.e project the data to ultimate maturity).
- Find Future payments: From the completed cumulative results, find the expected future payments in each cell (usually by disaccumulating the ultimate cumulative claims and then calculating the outstanding amount by subtracting claims paid to date)
BCL Method
Assumptions
- Stability of development pattern: it assumes that the future claims development will be in line with the past claims development. The core assumption is that the claim development pattern is stable for different cohorts.
- Inflation implication: the basis method implicitly assumes that past inflation will continue into the future.
- Homogeneity: the method assumes that the data is homogenous and that the run-off pattern is the same for each origin period.
BCL Method
Advantages and disadvantages
Advantages
- Simplicity and Clarity: Conceptually straightforward and easy to relate results back to the pattern of development
- Wide Applicability: Can be applied to a wide variety of sets of data (paid, incurred, or claim numbers).
- Flexibility: The basic method can easily be modified to allow for data distortions.
- Foundation for Other Methods: It can be developed to serve as a starting point for a number of other methods, such as the Bornhuetter-Ferguson method.
Disadvantages
- Distortion by Anomalies: Results can be distorted by unusual experience (e.g., very good or very bad claims experience).
- Volatility: If development factors (link ratios) are volatile or unstable, projections may be difficult.
- Dependence on Experience to Date: It projects the claims experience to date. If experience is light or heavy, applying the development factor could result in an under- or over-estimation of the ultimate value without adjustment.
- Ignores Claim Numbers/Amounts: On its own, it doesn’t separate the effects of changes in claim frequency versus claim severity.
Bornhuetter Ferguson (BF) Method
Definition
The Bornhuetter-Ferguson (BF) method is a reserving method that can be viewed as a credibility estimate. It is a weighted average that combines an expected level of claims (usually estimated using the Expected Loss Ratio approach) with a projection of ultimate claims based on experience to date (usually estimated using the Chain Ladder method).
Bornhuetter Ferguson (BF) Method
Step by Step Method
- Determine Initial Ultimate Estimate (Loss Ratio): Determine an initial expected ultimate claim amount (LR) for the relevant origin period. This loss ratio is typically derived from the previous year’s best estimate results, adjusted for inflation, rate changes, large claims, and other factors.
- Estimate Developed Proportion: Estimate the proportion of claims that are currently incurred (p) or paid for the period. This proportion often comes from the Chain Ladder run-off pattern.
- Derive Undeveloped Proportion: Derive the proportion of claims yet to be incurred or paid, calculated as (1−p).
- Calculate Expected Future Claims (IBNR): Determine the expected value of the undeveloped claims (IBNR/IBNER) by multiplying the initial expected ultimate claims (LR) by the undeveloped proportion (1−p): LR×(1−p).
- Calculate Final Ultimate Claims: Calculate the final expected ultimate claims for the origin period by summing the current value of the incurred or paid claims (C) and the expected value of the undeveloped claims (from Step 4): C+LR×(1−p).
- Determine Reserve: To calculate the required reserve, deduct the amount of claims paid to date from the final ultimate claims estimate.
Bornhuetter Ferguson (BF) Method
Assumptions
- A Priori Estimate Credibility: The method relies on the assumption that the initial expected ultimate claims (loss ratio) used as the prior estimate is correct.
- Assumed Run-off Pattern: It relies on an assumed run-off pattern, usually derived from the Chain Ladder technique. This pattern (p) dictates the speed at which claims are expected to emerge.
Bornhuetter Ferguson (BF) Method
Advantages and Disadvantages
Advantages
- Stability: It provides a result that is less dependent on the volatility of claims experience to date, unlike the Chain Ladder method.
- Utility with Sparse Data: It is very useful where the available data for a cohort is sparse (e.g., recent cohorts, long-tailed business).
- Incorporates External Information: Combines stable external estimates (loss ratio) with internal data patterns (Chain Ladder factors).
Disadvantages
- Subjective Prior Estimate: It can be difficult to gather information for the prior loss ratio estimate.
- Sensitivity: The result, particularly at early stages of development, can be heavily dependent on the prior loss ratio estimate.
- Accuracy Risk: If the selected loss ratio is inappropriate (e.g., too low or too high), the resulting estimate will be similarly biased.
Expected Loss Ratio (ELR) Method
Definition
The Expected Loss Ratio method can be used to estimate future claims. It works by applying historical loss ratios (based on the company’s or industry data) to current premiums to obtain future claim estimates. It is a simple, high-level approach and is often used to derive the a priori estimate required by the Bornhuetter-Ferguson method.
Expected Loss Ratio (ELR) Method
Step by step Method
- Gather Premium Data: Collect earned premium data (or ultimate premiums) for the cohort being reserved.
- Determine Historical Loss Ratio: Determine historical loss ratios, adjusting them for current factors such as claims inflation and rate changes.
- Select Loss Ratio: Select the estimated loss ratio (LR expected) for the period in question.
- Estimate Ultimate Claims: Calculate the ultimate claim estimate by multiplying the earned premium (P) by the selected loss ratio: Ultimate Claims=P×LR expected.
- Calculate Reserve: Determine the required outstanding claims reserve by subtracting claims paid to date from the estimated ultimate claims.
Expected Loss Ratio (ELR) Method
Assumptions
- Loss Ratio Credibility: Assumes that the selected historical or a priori loss ratio is correct.
- Underlying Consistency: Assumes that the underlying premium rates and claims data used to calculate the historical loss ratio are relevant to the current business being reserved.
Expected Loss Ratio (ELR) Method
Advantages and Disadvantages
Advantages
- Simplicity: The method is simple to apply.
- Utility in Sparse Data: Especially useful when data is sparse, unreliable, missing altogether, or when a company has started a new line of business.
- Benchmarking: Gives a useful reference point against which to compare results from other, more complex methods.
Disadvantages
- Subjectivity: The underlying assumptions, particularly when based on underwriter opinion or external benchmarks, can be subjective.
- Model Risk: It ignores claim number development and payment patterns, relying solely on an assumed ratio.
- Obscured Trends: Trends in frequency and severity are difficult to identify separately, as the method relies only on the aggregate ratio.
Average Cost Per Claim (ACPC) Method
Definition
The Average Cost per Claim (ACPC) method is a reserving approach that relies on the average cost of claims paid or incurred. It is a member of the family of statistical methods often employed in reserving work using triangulations.
The method estimates the expected loss cost by analyzing two separate components for each origin year: claims frequency (the number of claims) and claims severity (the average size of those claims). It is not uniquely defined, and many variations are possible regarding whether it is applied to incurred, reported, settled, or pure Incurred But Not Reported (IBNR) claims, or whether the average size is based on origin, development, or calendar years.
Average Cost Per Claim (ACPC) Method
Step by Step Method
The ACPC method requires development tables for both total claim amounts and claim numbers. A common approach, where the average ultimate claim size is estimated, involves the following steps:
- Tabulate Data: Create two separate run-off triangles (tabulated by origin year and development year): one for the number of claims (claim frequency) and one for the total claim amounts (paid or incurred).
◦ Note: If using incurred claims data, each cell typically comprises claims paid to date plus case estimates of outstanding reported claims. - Ensure Consistency: Verify that the aggregate claim amounts used correspond accurately to the claim numbers used (i.e., the claim frequency and claim severity are consistent). Special care must be taken with claims involving partial payments (which should typically be aggregated if the claim is fully settled to avoid understating the average cost) or zero claims.
- Project Claim Numbers (Frequency): Apply a projection technique (such as the basic Chain Ladder method) to the claim number triangle to estimate the ultimate number of claims for each origin period.
- Calculate Average Cost per Claim (Severity): Calculate the historical average claim size (severity) for each cell in the triangle by dividing the total claim amount by the number of claims in that corresponding cell.
- Estimate Ultimate Severity: Apply trending or projection methods to the historical average claim sizes to arrive at an estimated average ultimate claim size for each origin period.
- Calculate Ultimate Claims: Calculate the estimated ultimate claim amount for the origin period by multiplying the estimated ultimate number of claims (Step 3) by the estimated average ultimate claim size (Step 5).
- Determine Reserve: Calculate the required outstanding claims reserve by subtracting claims paid to date (obtained from a separate source if working with incurred or reported claims data) from the estimated ultimate claims amount.
Alternative Reserve Calculation: If the average claim size used in Step 5 is the average of future payments rather than the ultimate average claim, the required claims reserve can be calculated directly by multiplying the estimated number of future claims by the estimated average claim size.
Average Cost Per Claim (ACPC) Method
Assumptions
- Stability of Average Cost per Claim: The method relies on the fundamental assumption of the assumed stability of the average cost per claim.
- Consistency of Definition: It assumes that the definition of a claim and the average cost of a claim remain consistent over time.
- Accuracy of Frequency and Severity Projections: It relies on the ability to accurately project the development patterns for both claim numbers and average claim sizes.
Average Cost Per Claim (ACPC) Method
Advantages and Disadvantages
Advantages
- Separates Frequency and Severity: Enables more accurate adjustments to be made if expected changes only affect frequency or only affect severity.
- Utility in Protocols Changes: Can be applied to settled claims, which can be useful when claims reserving protocols have changed over the development history, potentially invalidating other methods.
- Latent Claims: Can be useful for estimating latent claims (reported long after occurrence) because it allows the actuary to make explicit assumptions about the average claim size, long-term inflation effects, and the expected number of claims.
- Treatment of Large Losses: Easily allows for the adjustment of data for large claims (by excluding them or capping amounts) to focus the projection on attritional claims.
Disadvantages
- Data Intensive: Requires reliable data for both claim amounts and claim counts.
- High Sensitivity: Results can be easily invalidated by a change in the definition of a claim or the average cost of a claim.
- Complexity in Payment Structures: Changes in how partial payments or zero claims are processed can disturb the method.
- Potential Instability: The calculation can be influenced by acceleration or slowing in claims payments or settlements.
Stochastic Reserving Methods
Step by step method
Stochastic claims reserving models can be broadly categorized as analytical, simulation, or Bayesian methods. Simulation methods, particularly bootstrapping, are commonly used:
- Fit a Model and Calculate Residuals: Fit a deterministic model (such as the Chain Ladder/ODP GLM) to the historical claims triangle. Calculate the expected values (fitted values) that would have occurred if the data perfectly followed the model, and then calculate the residuals (the differences between actual values and fitted values).
- Create Pseudo-Data Sets (Sampling): Re-sample (with replacement) from the residual distribution and combine these residuals with the fitted values to create a large number of alternative or pseudo-data sets.
- Calculate Projections: Apply the original deterministic reserving method (e.g., Chain Ladder) to each of the generated pseudo-data sets to obtain a revised reserve estimate for each set.
- Collate and Analyze Distribution: Collate the resulting reserve projections to determine the distribution, moments (mean, variance), and percentiles of the predicted reserve distribution.
Stochastic Reserving Methods
Definition
Stochastic reserving refers to methods used to assess the uncertainty surrounding reserve estimates. While traditional (deterministic) methods like the Chain Ladder produce a single best estimate of the claims reserve, stochastic methods provide a confidence interval or a full distribution of the reserves.
By modelling the random variation around the chosen development pattern, stochastic methods quantify the likely error involved in using a best estimate, thus providing information about the distribution of possible outcomes (e.g., the variance).
The Mack Method
Definition and Methodology
The Mack method is an analytical stochastic reserving model that builds upon the assumptions of the basic Chain Ladder method. It is effectively distribution-free, focusing on the first two moments (mean and variance) of the ultimate reserve outcome, rather than specifying the precise underlying distribution. The mean outcome matches the result derived by the standard Chain Ladder method.
The methodology involves calculating the variance of the forecast error (prediction error) by calculating and combining the variance parameter (σj^2) derived from the observed run-off data.
The ODP Model (Over-Dispersed Poisson)
Definition and Methodology
The Over-Dispersed Poisson (ODP) model is a Generalised Linear Model (GLM) applied to claims triangles in a stochastic context. It can be used in either a deterministic or a stochastic (bootstrap) form.
The model assumes that incremental claims follow an ODP distribution, meaning the variance of the claim amount is proportional to the mean, but typically greater (hence over-dispersed). The expected values obtained from fitting the ODP model (a special case of GLM) are exactly the same as the basic Chain Ladder estimates.
The ODP model is often used in conjunction with bootstrapping to obtain a full distribution of possible outcomes:
- Fit the GLM: Fit the GLM assuming an ODP distribution to the incremental claims data, resulting in parameters and expected values.
- Calculate Residuals: Calculate the Pearson residuals (a measure of noise).
- Bootstrap: Sample repeatedly from these residuals to create pseudo-data sets.
- Refit and Forecast: Refit the model (or the corresponding Chain Ladder mechanics) to each pseudo-data set to obtain a distribution of parameters and a forecast distribution of reserve estimates.
Accident Year (AY) Cohort
Definition
An Accident Year grouping combines all claims related to loss events that occurred within a specific 12-month period. This grouping is applied irrespective of when the claim was reported or paid, and regardless of when the period of cover started. This basis is consistent with a losses-occurring policy (LOD).
Accident Year (AY) Cohort
Assumptions
Assumptions
1. Consistency of Exposure: It is assumed that all claims stem from the same exposure cohort (the period of loss occurrence).
2. Date Identifiability: It assumes that the date of loss is identifiable.
3. Run-off Stability: Assumes that the future development pattern will be stable and in line with past claims development.
Accident Year (AY) Cohort
Advantages and Disadvantages
Advantages
- All claims stem from the same exposure cohort.
- Variations between cohorts can be easily related to external influences operating at that time, such as legislative changes or changes in business volume.
- Can be aligned with the accounting year for comparing emerging losses with the financial charges made for that period.
- The appropriate exposure measure, earned premium, naturally aligns with the accident year approach.
Disadvantages
- The full ultimate number or amount of claims is not known until the last claim is reported.
- The date of loss is not always known for certain classes (e.g., subsidence, asbestos, or some London Market business), making the approach potentially impractical.
- Rules must be established to consistently treat claims where the event date cannot be precisely determined (e.g., asbestos claims allocated over the period of exposure).
Underwriting Year (UY) Cohort
Definition
An Underwriting Year grouping (also known as the year of account or policy year) combines all claims relating to loss events that can be attributed to all policies that commenced cover within a specific calendar year. This grouping applies regardless of when the claim occurred, was reported, or was paid. This basis is consistent with a risks-attaching policy.
-
Insurance Types29
-
Risk Characteristics and Policy Design6
-
Reinsurance and alternative risk transfer6
-
Data and Reserving15
-
Risk and Volatility6
-
Regulatory and financial management5
-
Operations and delays5
-
Investment strategy6
-
Pricing12
-
Reserving36
-
Reinsurance37
-
Capital Modelling19
-
Investments11
-
Actuarial investigations23
-
Products21
