In actuarial science, particularly for casualty insurance, accurately setting premium rates is paramount. However, actuaries often face a challenge: limited historical data for new or niche lines of business. This is where credibility theory comes into play, providing a framework to blend limited specific experience with broader, more stable experience from similar risks. Bühlmann credibility is a cornerstone of this theory, offering a statistically sound method to achieve this blend.

Imagine a new type of cyber insurance policy or a specialized commercial auto policy for a unique industry. The available claims data for these specific risks might be sparse, making it unreliable to base rates solely on this limited experience. Relying only on broad industry data might not capture the unique risk characteristics of the specific policyholder or segment. Bühlmann credibility offers a solution by providing a weighted average of the specific (limited) experience and the general (broader) experience.

The Bühlmann model relies on two key variance components:

• V (Total Variance): This represents the variability of the expected claim rates across different risks. It measures how much the underlying rates differ from one risk to another.
• A (Average Variance): This represents the variability of claims within a single risk, given its underlying rate. It measures the random fluctuation of claims around the risk's expected rate.

The credibility factor (Z) for a specific risk with 'n' exposure units is calculated as:

$Z = \frac{n}{n + \frac{A}{V}}$

The credibility premium ($\hat{P}$) for a specific risk is then a weighted average of the specific premium ($P_{specific}$) and the general premium ($P_{general}$):

$\hat{P} = Z \cdot P_{specific} + (1 - Z) \cdot P_{general}$

Bühlmann credibility is widely used in:

• New Lines of Business: Setting initial rates for emerging insurance products.
• Small Portfolios: Adjusting rates for policyholders with limited claims history.
• Experience Rating: Modifying future premiums based on past performance.

Its benefits include:

• Improved Rate Accuracy: Blends specific and general data for more precise pricing.
• Statistical Rigor: Provides an objective, data-driven approach.
• Flexibility: Adapts to varying levels of data availability.

While powerful, the basic Bühlmann model has limitations. It assumes that the variance components (A and V) are constant across all risks and time. In reality, these can vary. Extensions like Bühlmann-Straub credibility address these limitations by allowing for varying exposure units and variances. Furthermore, the model assumes a specific distributional form for the underlying rates, which may not always hold true.