Original practice · fully worked
Original variant: compound expected payment with a discrete deductible
The annual number of device losses is Poisson with mean 0.08. Individual severities independently equal 2, 4, 6, or 8 with probability proportional to severity. A policy pays the amount above a deductible of 3 for every loss. Find the expected aggregate annual payment.
- A
0.160 - B
0.248 - C
0.310 - D
0.800 - E
3.100
Answer and calculation
Severity weights normalize by 20. Expected payment per loss is 3.1, and the compound-Poisson mean is 0.08×3.1=0.248, choice B.
Setup
Setup
Normalize the four severity probabilities, which are proportional to severity and therefore have total weight 20.
Pr(X=x∣L)=x/(2+4+6+8)=x/20,x∈{2,4,6,8} Model
Model
For each severity, apply the deductible of three and weight the positive excess by the normalized severity probability. The expected payment per loss is 3.1.
E[(X−3)+∣L]=200(2)+1(4)+3(6)+5(8)=3.1 Compute
Compute
A compound-Poisson mean equals the frequency mean times the mean payment per loss. Multiplying 0.08 by 3.1 gives 0.248.
E[S]=E[M]E[Y]=0.08(3.1)=0.248 Answer
Answer
The expected aggregate annual payment is 0.248, selecting choice B.
0.248(B) 