Chain ladder and Bornhuetter-Ferguson methods are key techniques for estimating ultimate claims in insurance. These approaches help actuaries project future claims based on historical patterns, providing crucial insights for setting reserves and managing risk.
Both methods rely on claims development triangles, but differ in their assumptions and inputs. Chain ladder uses only historical data, while Bornhuetter-Ferguson incorporates external information through an a priori loss ratio, offering stability for immature business lines.
Chain ladder method
- Deterministic reserving technique used to estimate ultimate claims based on historical claims development patterns
- Relies on the assumption that past claims development is indicative of future claims development
- Involves organizing claims data into a development triangle and calculating age-to-age factors to project ultimate claims
Assumptions of chain ladder
- Claims will continue to develop in each development period as they have historically
- The proportional increases in the amount of claims from one development period to the next is the same for all origin periods
- Claims are fully developed after a certain number of development periods (varies by line of business)
- No material changes in the claims handling process, policy benefits, or reinsurance arrangements
Development factors in chain ladder
- Measure the growth in cumulative claims from one development period to the next
- Calculated as the ratio of cumulative claims in a given development period to the previous development period
- Example: $f_{2,3} = \frac{C_{2,3}}{C_{2,2}}$ where $C_{i,j}$ is the cumulative claims for origin period $i$ at development period $j$
- Selected by averaging historical age-to-age factors, potentially excluding outliers or unusual years
Calculating ultimate claims with chain ladder
- Project cumulative claims for future development periods by multiplying the latest cumulative claims by the appropriate development factors
- $\hat{C}{i,n} = C{i,n-i+1} \times f_{n-i+1,n-i+2} \times \cdots \times f_{n-1,n}$ where $n$ is the number of development periods
- Ultimate claims for each origin period are estimated as the projected cumulative claims at the latest development period
- Total ultimate claims is the sum of the ultimate claims for each origin period
Advantages vs disadvantages of chain ladder
- Advantages:
- Simple and intuitive method that is widely used in practice
- Requires minimal assumptions and inputs (only historical claims data)
- Can quickly provide an estimate of ultimate claims
- Disadvantages:
- Sensitive to changes in claims development patterns over time
- Does not incorporate exposure or premium information
- Relies heavily on the assumption that the future will resemble the past
- Can produce unreliable estimates for immature origin periods with limited claims experience
Bornhuetter-Ferguson method
- Reserving technique that combines an a priori estimate of ultimate claims with actual claims experience
- Uses expected claims based on an assumed loss ratio to supplement the chain ladder method, particularly for more recent origin periods
- Responsive to actual claims experience as it emerges while providing stability from the use of an a priori estimate
Assumptions of Bornhuetter-Ferguson
- The a priori loss ratio is a reasonable estimate of the expected claims as a percentage of earned premium
- Actual claims experience is credible and should be incorporated into the ultimate claims estimate
- The percentage of ultimate claims that have emerged by a given development period is consistent across origin periods
- No material changes in the mix of business, claims handling practices, or policy benefits
A priori loss ratio in Bornhuetter-Ferguson
- Represents the expected claims as a percentage of earned premium for a given line of business
- Determined based on historical loss ratios, pricing assumptions, or industry benchmarks
- Example: If the a priori loss ratio is 70% and earned premium is $1,000, expected ultimate claims would be $700
Expected vs actual claims in Bornhuetter-Ferguson
- Expected claims are calculated by multiplying the earned premium for each origin period by the a priori loss ratio
- $E_i = P_i \times LR$ where $P_i$ is the earned premium for origin period $i$ and $LR$ is the a priori loss ratio
- Actual claims are the cumulative claims that have emerged for each origin period as of the valuation date
- The Bornhuetter-Ferguson method combines expected claims and actual claims to estimate ultimate claims
Calculating ultimate claims with Bornhuetter-Ferguson
- Estimate the percentage of ultimate claims that have emerged by each development period based on historical claims development patterns
- $%{i,j} = \frac{C{i,j}}{C_{i,n}}$ where $C_{i,j}$ is the cumulative claims for origin period $i$ at development period $j$ and $C_{i,n}$ is the ultimate claims for origin period $i$
- Calculate the expected unreported claims by multiplying the expected claims by the percentage of claims unreported (1 minus the percentage reported)
- $UERC_i = E_i \times (1 - %{i,n-i+1})$ where $%{i,n-i+1}$ is the percentage reported for origin period $i$ at the latest development period
- Estimate ultimate claims as the sum of actual reported claims and expected unreported claims
- $\hat{C}{i,n} = C{i,n-i+1} + UERC_i$
Advantages vs disadvantages of Bornhuetter-Ferguson
- Advantages:
- Incorporates external information (a priori loss ratio) to improve estimates for immature origin periods
- More stable estimates than the chain ladder method, particularly for recent origin periods
- Responsive to actual claims experience as it emerges
- Disadvantages:
- Relies on the accuracy of the selected a priori loss ratio
- Assumes consistency in the percentage of claims reported across origin periods
- May not capture changes in claims development patterns over time
- Requires more inputs and assumptions than the chain ladder method
Comparison of methods
Similarities of chain ladder and Bornhuetter-Ferguson
- Both methods rely on historical claims development patterns to project ultimate claims
- Assume that claims will continue to develop in a similar manner as they have historically
- Organize claims data into a development triangle and estimate age-to-age factors or percentages reported
- Provide a deterministic estimate of ultimate claims based on a set of assumptions
Differences between chain ladder and Bornhuetter-Ferguson
- Chain ladder relies solely on actual claims experience, while Bornhuetter-Ferguson incorporates an a priori estimate of ultimate claims
- Bornhuetter-Ferguson is more stable for immature origin periods, as it is less sensitive to limited claims experience
- Chain ladder assumes consistency in the proportional increases in claims across origin periods, while Bornhuetter-Ferguson assumes consistency in the percentage of claims reported
- Bornhuetter-Ferguson requires additional inputs (earned premium and a priori loss ratio) compared to chain ladder
Criteria for selecting chain ladder vs Bornhuetter-Ferguson
- Maturity of the line of business and origin periods
- Chain ladder may be more appropriate for mature lines with stable claims development patterns
- Bornhuetter-Ferguson may be preferred for immature lines or origin periods with limited claims experience
- Stability and reliability of historical claims data
- Chain ladder may be more suitable when historical data is stable and credible
- Bornhuetter-Ferguson may be preferred when historical data is volatile or lacks credibility
- Availability and credibility of external information (industry benchmarks, pricing assumptions)
- Bornhuetter-Ferguson may be more appropriate when reliable external information is available to set the a priori loss ratio
- Desired responsiveness to actual claims experience
- Chain ladder is more responsive to actual claims experience, while Bornhuetter-Ferguson provides a more stable estimate
Reserving applications
Estimating IBNR with chain ladder and Bornhuetter-Ferguson
- IBNR (Incurred But Not Reported) represents claims that have occurred but have not yet been reported to the insurer
- Chain ladder: IBNR is the difference between projected ultimate claims and reported claims as of the valuation date
- $IBNR_i = \hat{C}{i,n} - C{i,n-i+1}$
- Bornhuetter-Ferguson: IBNR is the expected unreported claims based on the a priori loss ratio and percentage unreported
- $IBNR_i = UERC_i = E_i \times (1 - %_{i,n-i+1})$
Setting reserves using chain ladder and Bornhuetter-Ferguson
- Reserves represent the amount of money an insurer must set aside to pay for future claims and expenses
- Chain ladder: Reserves are the sum of IBNR and case reserves (known claims that have been reported but not yet paid)
- $Reserves_i = IBNR_i + Case Reserves_i$
- Bornhuetter-Ferguson: Reserves are the sum of IBNR and case reserves
- $Reserves_i = IBNR_i + Case Reserves_i$
- Total reserves are the sum of reserves across all origin periods
Monitoring claims development with chain ladder and Bornhuetter-Ferguson
- Actual vs expected analysis: Compare actual claims development to the expected development based on the selected age-to-age factors (chain ladder) or percentages reported (Bornhuetter-Ferguson)
- Investigate and understand the reasons for significant deviations
- Update estimates: As new claims data emerges, update the development factors, a priori loss ratio, and ultimate claims estimates
- Assess the impact of new data on the estimated reserves and adjust as needed
- Monitor key assumptions: Regularly review and validate the assumptions underlying the reserving methods
- Test the sensitivity of the estimates to changes in the assumptions
Limitations and considerations
Data requirements for chain ladder and Bornhuetter-Ferguson
- Sufficient volume and history of claims data to establish credible development patterns
- Consistent claims reporting and settlement practices over time
- Accurate and complete claims data, including claim counts and amounts, reported and paid dates, and case reserves
- Appropriate segmentation of data by line of business, coverage, or other relevant factors
Adjustments for unusual claims in chain ladder and Bornhuetter-Ferguson
- Large or catastrophic claims may distort development patterns and require separate treatment
- Exclude large claims from the development triangle and estimate their impact separately
- Apply a large claims loading to the ultimate claims estimates
- Changes in claims handling practices, policy benefits, or reinsurance arrangements may impact claims development
- Adjust historical data to reflect the impact of these changes on claims development patterns
- Use judgment to select development factors or a priori loss ratios that are appropriate for the current environment
Sensitivity testing of assumptions in chain ladder and Bornhuetter-Ferguson
- Assess the impact of alternative assumptions on the ultimate claims estimates
- Select different age-to-age factors or tail factors for the chain ladder method
- Use alternative a priori loss ratios or percentage reported assumptions for the Bornhuetter-Ferguson method
- Identify the assumptions that have the greatest impact on the estimates and focus on refining those assumptions
- Consider using a range of reasonable assumptions to develop a range of potential outcomes
Uncertainty in chain ladder and Bornhuetter-Ferguson estimates
- Recognize that the ultimate claims estimates are based on assumptions and are subject to uncertainty
- Use techniques such as bootstrapping or stochastic modeling to quantify the variability in the estimates
- Bootstrapping involves resampling the historical claims data to create a distribution of potential outcomes
- Stochastic modeling incorporates random variables and probability distributions to simulate potential future claims development
- Communicate the uncertainty in the estimates to stakeholders and consider the implications for decision-making
- Set reserves at a level that provides an appropriate margin for adverse development
- Regularly monitor actual claims development against the estimates and adjust as needed