You are given:

1. The number of claims follows a negative binomial distribution with parameters [math]r[/math] and [math]\beta = 3 [/math].
2. Claim severity has the following distribution:

Claim Size	Probability
1	0.4
10	0.4
100	0.2

Calculate the expected number of claims needed for aggregate losses to be within 10% of expected aggregate losses with 95% probability.

• Less than 1200
• At least 1200, but less than 1600
• At least 1600, but less than 2000
• At least 2000, but less than 2400
• At least 2400

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

Key: E

For claim severity,

[math]\mu_S = 1(0.4) + 10(0.4) + 100(0.2) = 24.4, [/math]

[math] \sigma_S^2 = 1^2 (0.4) + 10^2 (0.4) + 100^2 (0.2) − 24.4^2 = 1, 445.04. [/math]

For claim frequency,

[math] \mu_F = r \beta = 3r , \sigma^2_F = r \beta (1 + \beta ) = 12r , [/math]

For aggregate losses,

[math] \mu = \mu_S\mu_F = 24.4(3r ) = 73.2r , \sigma^2 = \mu_S^2\sigma_F^2 + \sigma_S^2 \mu_F = 24.4^2 (12r ) + 1, 445.04(3r ) = 11, 479.44r. [/math]

For the given probability and tolerance, [math]\lambda_0 = (1.96 / 0.1) 2 = 384.16. [/math]

The number of observations needed is

[math] \lambda_0 \sigma^2 / \mu^2 = 384.16(11, 479.44r ) / (73.2r ) 2 = 823.02 / r. [/math]

The average observation produces 3r claims and so the required number of claims is (823.02 / r )(3r ) = 2, 469.

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