Original practice · fully worked
Original variant: capped equipment-repair layer
Equipment damage severity L is exponentially distributed with mean 2 repair units. A service contract pays the portion of a loss lying between 1 and 4 units: it pays nothing below 1, L-1 between 1 and 4, and a maximum of 3 units above 4. Calculate the expected contract payment per incident.
- A
0.1353 - B
0.4712 - C
0.6065 - D
0.9424 - E
1.2131
Answer and calculation
The capped layer can be valued by integrating its payment tail from zero to three. This gives 2(exp(−1/2)-exp(−2))=0.942391, so choice D is correct.
Setup
Setup
Represent the contract as a three-unit layer attaching when the severity reaches one.
L∼Exp(1/2) W=min((L−1)+,3) Model
Model
For a nonnegative capped payment, integrate the probability that the payment exceeds each level from zero up to its cap.
E[W]=∫03Pr(W>t)dt Pr(W>t)=Pr(L>1+t)=e−(1+t)/2 Compute
Compute
Evaluate the payment-tail integral across the full three-unit layer.
E[W]=∫03e−(1+t)/2dt E[W]=2(e−1/2−e−2)=0.9423907530… Answer
Answer
The expected capped-layer payment is approximately 0.9424 repair units.
0.9424(D) 