Original practice · fully worked
Original variant: environmental cleanup reimbursement
An environmental response fund reimburses 80% of the portion of a chemical-cleanup cost above a trigger. The cleanup cost C is exponential with mean θ, the trigger is 0.2 θ, and the expected reimbursement per incident is 1000 dollars. Determine θ.
- A
819 - B
1250 - C
1527 - D
1908 - E
6896
Answer and calculation
Coinsurance multiplies the exponential excess-loss expectation by 0.8. Solving the resulting equation gives θ approximately 1526.753, so choice C is correct.
Setup
Setup
Write the fund's reimbursement using the trigger and the 80% reimbursement rate.
R=0.8(C−0.2θ)+ Model
Model
For an exponential cost, the expected excess above 0.2 times the mean is the mean multiplied by the corresponding survival factor.
E[R]=0.8θe−0.2 Compute
Compute
Set the modeled reimbursement equal to 1000 and isolate the mean.
1000=0.8θe−0.2 θ=0.81000e0.2=1526.753448… Answer
Answer
Rounding to the listed whole-dollar values selects choice C.
θ≈1527(C) 