Credibility theory

Credibility theory is a branch of <a href="/facts/Actuarial_Mathematics/KxmCE0ME">actuarial mathematics</a> concerned with determining <a href="/facts/Risk_premium/ngjVLPpF">risk premiums</a>. To achieve this, it uses mathematical models in an effort to forecast the (<a href="/facts/Expected_value/1XV0JKL8">expected</a>) number of insurance claims based on past observations. Technically speaking, the problem is to find the best linear approximation to the mean of the <a href="/facts/Bayesian_predictive_density/I6hSVx66">Bayesian predictive density</a>, which is why credibility theory has many results in common with <a href="/facts/Filtering_problem_(stochastic_processes)/BNuNppLf">linear filtering</a> as well as <a href="/facts/Bayesian_statistics/9w7b1Bw4">Bayesian statistics</a> more broadly.
For example, in <a href="/facts/Group_insurance/9zjJzcCy">group health insurance</a> an insurer is interested in calculating the risk premium, 
 
 
 
 R
 P
 
 
 {\displaystyle RP}
 
, (i.e. the theoretical expected claims amount) for a particular employer in the coming year. The insurer will likely have an estimate of historical overall claims experience, 
 
 
 
 x
 
 
 {\displaystyle x}
 
, as well as a more specific estimate for the employer in question, 
 
 
 
 y
 
 
 {\displaystyle y}
 
. Assigning a credibility factor, 
 
 
 
 z
 
 
 {\displaystyle z}
 
, to the overall claims experience (and the reciprocal to employer experience) allows the insurer to get a more accurate estimate of the risk premium in the following manner:

 
 
 
 R
 P
 =
 x
 z
 +
 y
 (
 1
 −
 z
 )
 .
 
 
 {\displaystyle RP=xz+y(1-z).}
 
The credibility factor is derived by calculating the <a href="/facts/Maximum_likelihood_estimate/0Yq2dpQD">maximum likelihood estimate</a> which would minimise the error of estimate. Assuming the variance of 
 
 
 
 x
 
 
 {\displaystyle x}
 
 and 
 
 
 
 y
 
 
 {\displaystyle y}
 
 are known quantities taking on the values 
 
 
 
 u
 
 
 {\displaystyle u}
 
 and 
 
 
 
 v
 
 
 {\displaystyle v}
 
 respectively, it can be shown that 
 
 
 
 z
 
 
 {\displaystyle z}
 
 should be equal to:

 
 
 
 z
 =
 v
 
 /
 
 (
 u
 +
 v
 )
 .
 
 
 {\displaystyle z=v/(u+v).}
 
Therefore, the more uncertainty the estimate has, the lower is its credibility.

Credibility theory open-in-new

Credibility theory