Revision as of 13:29, 14 May 2023 by Admin (Created page with "For an insurance: #The number of losses per year has a Poisson distribution with <math>\lambda = 10 </math> . #Loss amounts are uniformly distributed on (0, 10). #Loss amount...")
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
ABy Admin
May 14'23

Exercise

For an insurance:

  1. The number of losses per year has a Poisson distribution with [math]\lambda = 10 [/math] .
  2. Loss amounts are uniformly distributed on (0, 10).
  3. Loss amounts and the number of losses are mutually independent.
  4. There is an ordinary deductible of 4 per loss.

Calculate the variance of aggregate payments in a year.

  • 36
  • 48
  • 72
  • 96
  • 120

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

ABy Admin
May 14'23

Key: C

Since loss amounts are uniform on (0, 10), 40% of losses are below the deductible (4), and 60% are above. Thus, claims occur at a Poisson rate [math]\lambda^* = 0.6(10) = 6 . [/math]

Since loss amounts were uniform on (0, 10), claims are uniform on (0, 6).

Let N = number of claims; X = claim amount; S = aggregate claims.

[[math]] \begin{aligned} & \operatorname{E}[ N ] = \operatorname{E}[ N ] = \lambda^* = 6\\ & \operatorname{E}[ X ] = (6 − 0) / 2 = 3\\ &\operatorname{E}[ X ] = (6 − 0)2 /12 = 3 \\ &\operatorname{E}[ S ] = \operatorname{E}[ N ]\operatorname{E}[ X ] + \operatorname{E}[ X ] 2\operatorname{E}[ N ] = 6(3) + 32 (6) = 72 \end{aligned} [[/math]]

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

00