A Comprehensive Guide to the Bornhuetter–Ferguson Method

The Bornhuetter–Ferguson method is an actuarial technique used to estimate the ultimate loss development for a given portfolio of claims. This method is widely used in the insurance industry.

It was developed by Fred Bornhuetter and Robert Ferguson in the 1980s. They introduced this method as an alternative to traditional loss development methods.

The Bornhuetter–Ferguson method is based on the concept of separating the ultimate loss development into two components: the initial loss development and the remaining loss development. This separation allows for a more accurate estimation of the ultimate loss.

The initial loss development is estimated using a combination of the initial loss and the average loss development. The remaining loss development is then estimated using a combination of the initial loss development and the average loss development.

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The Bornhuetter-Ferguson method is a technique used to estimate outstanding claim reserves. It's a key tool for insurance companies to set aside funds for unpaid claims. One important aspect of the BF method is its ability to estimate skewness, which is a measure of the asymmetry of the distribution of claim frequencies.

Skewness is estimated using the Fleishman polynomial, which approximates a standard normal random variable. The polynomial is used to estimate the skewness of each risk value, which is then aggregated over all accident years. The BF method assumes a lognormal distribution for the underlying data, which allows for the estimation of skewness. The parameters of the lognormal distribution are derived from the expected value and variance of the data.

The BF method uses the following formula to estimate skewness: SKEW(Ri) = K(URi) - K(Ri), where K is the cumulant generating function. The cumulant generating function is estimated using the Fleishman polynomial, which is calibrated using the method of moments. The coefficients of the polynomial are solved using Cardano's formula, which is a cubic equation.

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The Chain Ladder method is a technique used by insurers to estimate IBNR, or Incurred But Not Reported, losses. It examines the point in time when a claim is reported or paid, and uses this information to "budget" for future losses.

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This method relies on past loss experience to make concrete estimates of future claims. The actuary essentially swaps past estimates with actual claims to get a more accurate picture of what's to come.

The Chain Ladder method is most useful when actual reported losses provide a good indicator of IBNR, which is often the case with high frequency, low severity claims.

Skewness and reserving are closely related concepts in the actuarial science field. Skewness measures the asymmetry of a distribution, while reserving involves estimating future claims payments. In this article, we'll explore the concept of skewness and its application in reserving.

The mean skewness of a reserve estimate is defined as \( \text{SKEW}\left( {\widehat{R}}_{i} \right) = E\left\lbrack \left( {\widehat{R}}_{i} - R_{i} \right)^{3}│S_{i,1},...,S_{i,n + 1 - i} \right\rbrack \). This formula calculates the expected value of the difference between the estimated reserve and the actual reserve, raised to the power of 3.

The BF method assumes that the estimated reserve and the actual reserve are independently distributed from the underlying data. This assumption allows us to simplify the skewness formula to \( \text{SKEW}\left( {\widehat{R}}_{i}^{\text{BF}} \right) = K\left( {\widehat{R}}_{i}^{\text{BF}} \right) - K\left( R_{i} \right) \). This formula highlights the relationship between the skewness of the estimated reserve and the skewness of the actual reserve.

The BF method also provides a formula for estimating the skewness of the estimated reserve: \( K\left( {\widehat{R}}_{i}^{\text{BF}} \right) = - K\left\lbrack {\widehat{U}}_{i} \right\rbrack K\left( {\widehat{z}}_{n + 1 - i} \right) + K\left\lbrack {\widehat{U}}_{i} \right\rbrack E\left( 1 - {\widehat{z}}_{n + 1 - i} \right)\left\lbrack 3Var\left( {\widehat{z}}_{n + 1 - i} \right) + E\left( 1 - {\widehat{z}}_{n + 1 - i} \right)^{2} \right\rbrack \). This formula involves several components, including the skewness of the underlying data and the variance of the estimated reserve.

To estimate the skewness of the estimated reserve, we need to make an assumption about the underlying distribution of the data. A commonly used distribution in reserving contexts is the lognormal distribution. This distribution is characterized by two parameters: μ and σ. The skewness of the lognormal distribution can be estimated using the following formula: \( K\left\lbrack {\widehat{U}}_{i} \right\rbrack = \left( 2 +\text{exp}\left( \sigma^{2} \right) \right)\frac{{\text{Var}\left( {\widehat{U}}_{i} \right)}^{2}}{E\left( {\widehat{U}}_{i} \right)} \).

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In summary, the BF method provides a framework for estimating the skewness of the estimated reserve. This framework involves several components, including the skewness of the underlying data and the variance of the estimated reserve. By making an assumption about the underlying distribution of the data and using the BF method, we can estimate the skewness of the estimated reserve with reasonable accuracy.

Outstanding Reserves are a crucial aspect of insurance companies' financial management. They represent the estimated amount of claims that have not been settled, and can be interpreted to include only claims that have been reported or all claims not yet settled.

These reserves are essentially cash reserves held by insurance companies to cover unpaid claims. This set aside amount is for insurance benefits that have not been paid out yet.

Outstanding Claim Reserves are used to estimate the amount of claims that have not been settled. This can be done by estimating individual outstanding claims or using statistical methods for the totals.

IBNR, or Incurred But Not Reported, is a type of reserve used to provide for claims and/or events that have occurred but claims related to which have not yet been reported to the insurance company.

The IBNR reserve is estimated using various methods, including the Chain Ladder Method and the Bornhuetter-Ferguson Technique. These methods help insurers to "budget" for future losses and estimate the ultimate loss for certain risk exposures.

The Chain Ladder Method examines the point in time when a claim is reported or paid, allowing insurers to make concrete claim estimates based on loss experience. This method is useful for making accurate estimates, especially when actual reported losses provide a good indicator of IBNR.

In contrast, the Bornhuetter-Ferguson Technique estimates IBNR by estimating the ultimate loss for certain risk exposures and then estimating the percent of this ultimate loss that was not reported at the time. This method is particularly useful in cases where actual reported losses do not provide a good indicator of IBNR.