Credibility

Question 1

Classical Credibility/Limited Fluctuation credibility

Full Credibility

Answer 1

of exposures needed for full credibility nc:

Full credibility for aggregate claims:
nc=((Z(1+p)/2)/k%)(CV^2)

Full credibility of aggregate claims :
nc=((Z(1+p)/2)/k%)((sigma^2 n )/un + CV^2)

• full credibility for claim frequency : set CV^2 =0
• full credibility for claim severity : set sigma^2n /un =0

**full credibility requires the confidence interval to be +- nc(k%) = sigma* Z(1+p)/2

Question 2

Classical Credibility/Limited Fluctuation credibility

Partial Credibility

Answer 2

Credibility Premium Pc=Z*average+(1-Z)M

where M=manual premium
Z=credibility factor

Square Root Rule : Z=(n/nc)^0.5

Question 3

Bayesian Credibility

Model Distribution

Answer 3

Distribution of model conditioned on a parameter

model Density function : f(x/theta)

Question 4

Bayesian Credibility

Prior Distribution

Answer 4

Inital distribution of the parameter

Prior Density function : pi(theta)

Question 5

Bayesian Credibility

Posterior Distribution

Answer 5

Revised distribution of the parameter
Posterior Density function : pi(theta/data)

posterior= (f(data/theta)pi(theta))/ (integral of f(data/theta)pi(theta) dtheta )

posterior mean = estimate the number of claims

Question 6

Bayesian Credibility

Predictive Distribution

Answer 6

Revised unconditional distribution of the model

Predictive Density function : f(x/data)
**Predictive mean= Bayesian Premium*

Question 7

Bülhmann Credibility

Steps

Answer 7

1. Expected Hypothetical Mean (EHM) = E(E(x/theta))
2. Expected Process Variance (EPV):v = E(Var(x/theta))
3. Variance of the Hypothetical Mean (VHM) :a = Var(E(x/theta))
4. k=v/a
5. Bülhmann Credibility Factor = n/(n+k)

all policies must have an equal number of exposure units

Question 8

Bülhmann Credibility Estimate

Bülhmann as Least Square Estimate of Bayesian

Answer 8

Z(x)+(1-Z)u where x is the actual value

need to minimise sum of (1/n)(BCE-BayesCE)^2

Question 9

Properties of the Bayesian/ Bülhmann Graph

Answer 9

• Bülhmann estimate are on a straight line
• Bayesian estimate are within the range of hypothetical means
• There are Bayesian estimâtes above and bellow the Bülhmann line
• Bülhmann estimâtes are between the sample mean and theorical mean

Question 10

Conjugate Priors

Poisson/Gamma

Answer 10

Model: Poisson (lambda)
Prior: Gamme(alpha, theta)

Posterior (lamdba/ data) follows Gamma(alpha, theta)
alpha= alpha + sum (xi)
theta = (1/theta+n)^(-1)

Predicitve = Neg. Binomial (r=alpha, beta=theta)

Question 11

Conjugate Priors

Binomial/Beta

Answer 11

Model: x/q Binomial(m,q)
Prior : Beta (a,b,1)

Posterior (q/data) follows a Beta(a, b, 1)
a=a+sum (xi)
b= beta + (n(m) -sum(xi)

Predictive = no formula

Question 12

Conjugate Priors

Exponential/ inv. Gamma

Answer 12

Model x/theta Exponential (theta)
Prior theta Inv. Gamma (alpha, beta)

Psterior Theta/data follows Inv. Gamma( alpha, beta)

alpha* = alpha + n 
beta* = beta +sum(xi)

Predictive Pareto (alpha, Theta=Beta)

Question 13

Conjugate Priors

Normal/Normal

Answer 13

Model x/theta : normal (theta, v(sigma^2))
Prior: normal (u,a(sigma^2))

Posterior theta/data follows normal (u,a)
u=Zaverage+(1-Z)u
a*=(1-Z)a
Z=na/(na+v)

Predictive follows a normal(u=u, sigma^2=v+a)

Question 14

Conjugate Priors

Uniform/S-P Pareto

Answer 14

Model x/theta Uniform (0, theta)
Prior S-P Pareto (aplha, beta)

Posterior x/theta S-P Pareto (alpha, beta)
alpha=alpha +n
Beta=max(beta, x1,…xn)