You are given the following information about a general liability book of business comprised of 2500 insureds:

1. [math]X_i = \sum_{j=1}^{N_i}Y_{ij} [/math] is a random variable representing the annual loss of the i^th insured.
2. [math]N_1,N_2,\ldots,N_{2500}[/math] are independent and identically distributed random variables following a negative binomial distribution with parameters [math]r = 2[/math] and [math]\beta = 0.2[/math].
3. [math]Y_{i1},Y_{i2},\ldots,Y_{iN}[/math] are independent and identically distributed random variables following a Pareto distribution with [math]\alpha = 3[/math] and [math]\theta = 1000 [/math].
4. The full credibility standard is to be within 5% of the expected aggregate losses 90% of the time.

Calculate the partial credibility of the annual loss experience for this book of business using limited fluctuation credibility theory.

• 0.34
• 0.42
• 0.47
• 0.50
• 0.53

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Key: C

[math]\operatorname{E}(N)=r \beta=0.4[/math]

[math]\operatorname{Var}(N)=r \beta(1+\beta)=0.48[/math]

[math]\operatorname{E}(Y)=\theta /(\alpha-1)=500[/math]

[math]\operatorname{Var}(Y)=\frac{\theta^{2} \alpha}{(\alpha-1)^{2}(\alpha-2)}=750,000[/math]

Therefore, [math]\operatorname{E}(X)=0.4(500)=200[/math]

[math]\operatorname{Var}(X)=0.4(750,000)+0.48(500)^{2}=420,000[/math].

The full credibility standard is [math]n=\left(\frac{1.645}{0.05}\right)^{2} \frac{420,000}{200^{2}}=11,365, \mathrm{Z}=\sqrt{2,500 / 11,365}=0.47[/math].

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.