Credibility Guided Tutorials

Let (X_1,X_2…X_n) be iid. Actuaries use the phrase full credibility to mean that we can use sample mean (bar{X}) for pricing. To achieve this standard, it is common to require the sample size n to be large enough so that (bar{X}) is within 5% of the mean ((mu)) at least (90%) of the time. Using (r=0.05) and (p=0.9), then we require n large enough so that
$$Prleft((1-r)muleqoverline{X}leq(1+r)muright)geq p .$$ To find the smallest value of n, replace the inequality by equality and use the central limit to solve for n as follows:
$$begin{array}{ll}
p
&=Prleft(frac{(1-r)mu-mu}{sigma/sqrt{n}}leq frac{overline{X}-mu}{sigma/sqrt{n}}leq frac{(1+r)mu-mu}{sigma/sqrt{n}}right) \
&=Prleft(frac{-rmu}{sigma/sqrt{n}}leq frac{overline{X}-mu}{sigma/sqrt{n}}leq frac{rmu}{sigma/sqrt{n}}right) \
& =Prleft(-y_pleq N(0,1) leq y_pright) .end{array}$$ With this, ( y_p={Phi}^{-1}left((1+p)/2right) ) is a quantile from the standard normal distribution. For example, if p = 0.90, then ( y_{0.9}={Phi}^{-1}left(0.95right) = 1.645.) From the solution, you see that $y_p=frac{rmu}{sigma/sqrt{n}}.$ Solving for n yields
$$ n_f geq left(frac{y_p sigma}{rmu}right)^2= left(frac{y_p}{r}right)^2 CV^2 quad quad … (1)$$
Here, (n_f) is the exposure needed for full credibility and (CV = sigma/mu) is the coefficient of variation.

Note: If you are establishing the standard for full credibility for pure premium or aggregate losses, the formulae for the expected number of claims needed for full credibility is :
$$E(N)times n_f quad quad … (2)$$ Where (E(N)) is the expected number of claims and (n_f) is the number of exposures required for full credibility calculated from equation (1) above.

When there is inadequate experience for full credibility , we calculate the credibility factor (Z) by
$$Z=
left{begin{array}{ll}
sqrt{frac{n}{n_f}} & text{if} quad nleq n_f \
1 & text{otherwise}end{array} right. … (3)$$
We use (Z) to calculate the credibility premium (P_c) given by;
$$P_c=Zoverline{X} + (1-Z)M quad quad … (4)$$ Here, (M) is the manual premium which is generally part of the problem givens.

To apply these rules,

• Using the question, identify the distribution of the claims, (p) and (r).
• Using equation (1) above calculate the standard for full credibility in terms of exposure.
• If the question requires you to establish the standard for full credibility for claim sizes(severity) in terms of exposure units, the exposure unit is a claim, so the the standard expressed in exposure units is the same as the standard for number of claims and it is given by equation (1). But if you are required to establish the standard for full credibility for aggregate losses in terms of expected number of claims, its is given by equation (2) above.
• If the question requires you to calculate the credibility factor, it is given by equation (3).
• If the question requires you to calculate the credibility premium, it is given by equation (4).

The Bühlmann credibility is sometimes known as greatest accuracy credibility. As before, (X_1,X_2, ldots, X_n) are from a policyholder. Let (theta) be a latent random variable associated with a policyholder and ({X_i|theta}) are iid, Then:
$$begin{array}{lll}
mu(theta) & =E(X_i| theta) &=text{“hypothetical mean”} \
v(theta) & =Var(X_i| theta) &=text{“process variance”} end{array}$$
The credibility premium is :
$$P_c=Zoverline{X} + (1-Z)M quad quad (1)$$As before, (M) is the manual premium which is often given to be (mu=E(X)). In this case, the credibility factor (Z) is
$$begin{array}{ll}
Z & =frac{n}{n+k} & & quad quad (2)\
end{array}$$ where k is the ratio of “expectation of the process variance” to the “variance of the hypothetical mean”
$$ k = frac{mathrm{E~} v(theta)}{mathrm{Var~}mu(theta)}
= frac{mathrm{E}left(mathrm{Var}(X_i| theta)right)}{mathrm{Var}left(mathrm{E}(X_i| theta)right)} quad quad (3)
$$ As before, n is the number of observations.

To apply these rules,

• Use the problem to identify the conditional model distribution of each observation given (theta), (f_{X|theta}(x|theta)) and the prior distribution of (theta), (pi(theta)).
• Using the conditional model distribution (f_{X|theta}(x|theta)), find the hypothetical mean (E(X|theta)) and process variance (Var(X|theta)).
• Using the results above and the prior distribution (pi(theta)), Calculate (k) using equation (3) above.
• To get credibility factor (Z) use equation (2) above.
• If the question requires you to calculate the credibility premium, it is given by equation (1).

The Bühlmann model assumes one exposure in every period but the Bühlmann-Straub model generalizes Bühlmann model to a case where there are $m_j$ exposures in period $j$. Then for the Bühlmann-Straub model:

$$
Var(X_j| theta)= frac{Var(X| theta)}{m_j} ;quad quad m_j=text{exposure} quad quad (1)
$$

This is because the variance of the sample mean is the distribution variance divided by the number of observations. It follows that the credibility factor (Z) is given by ;

$$begin{array}{ll}
Z & =frac{sum m_j}{sum m_j+k} & & quad quad (2)\
end{array}$$

When (m_j=1) it implies that (sum m_j =n) and the formulae is the same as that of the Bühlmann model. As before k is the ratio of “expectation of the process variance” to the “variance of the hypothetical mean”.

$$ k = frac{mathrm{E~} v(theta)}{mathrm{Var~}mu(theta)}
= frac{mathrm{E}left(mathrm{Var}(X_i| theta)right)}{mathrm{Var}left(mathrm{E}(X_i| theta)right)} quad quad (3)
$$
and the

To apply these rules,

• Use the problem to identify the conditional model distribution of each observation given (theta), (f_{X|theta}(x|theta)) and the prior distribution of (theta), (pi(theta)).
• Using the conditional model distribution (f_{X|theta}(x|theta)), find the hypothetical mean (E(X|theta)) and process variance (Var(X|theta)).
• Using the results above and the prior distribution (pi(theta)), Calculate (k) using equation (3) above.
• To get credibility factor (Z) use equation (2) above.
• If the question requires you to calculate the credibility premium, it is the same as that for the Bühlmann model.

For Bayesian Premium problems:

• Use the problem to identify the conditional model distribution of each observation given (theta), (f_{X|theta}(x|theta)). Then get the likelihood of the model distribution, which is typically the product of the model distributions for all observations,
  $$f_{X|theta}(x_1…x_n|theta)=prod_{i=1}^N f_{X|theta}(x_i|theta) .$$
• From the question identify the prior distribution of (theta), (pi(theta)).
• Now obtain the posterior distribution using the Bayes theorem;
  $$pi(theta|x_1…x_n)=frac{f_{X|theta}(x_1…x_n|theta)pi(theta)}{int f_{X|theta}(x_1…x_n|theta)pi(theta) dtheta}$$Note that the denominator of this expression ( f(x) = int f_{X|theta}(x_1…x_n|theta)pi(theta) dtheta) is the marginal distribution of (x) and does not involve ( theta ).
• With this, to calculate the posterior probability that (a lt theta lt b);
  $$pi(a lt theta lt b|x_1…x_n)=int_{a}^{b} pi(theta|x_1…x_n) dtheta$$
• For a new data (y) , the predictive distribution is
  $$f(y|x_1…x_n)=int f(y|theta)pi(theta|x_1…x_n)dtheta .$$Here, (f(y|theta)) is obtained from the conditional model distribution

There are several ways that one can use the predictive distribution. For some problems, you may wish to calculate Bayesian prediction for a new data (y). Specifically, given the past observations (x_1…x_n), you can use
$$E(y|x_1…x_n)=int E(y|theta)pi(theta|x_1…x_n)dtheta. $$Here, (E(y|theta)) is the expected value of the conditional model distribution.