Question

A prescription drug plan provides that the insured has the following annual payment re- sponsibilities: (i) All costs up to a deductible of 250. (ii) 25% of costs between 250 and 2250. (iii) All costs above 2250 until the total insured's payment reaches 3600. (iv) 5% of all remaining costs. The plan's costs are modeled by a Pareto distribution with a 2 and 0- 2000. Find the expected annual payment made by the plan.

Answer 1

Solution: When X- 2250 the insured's payment equals 750 [250+0.25(2250 250) 750] After this point, the payment by insured would be 100% of the costs above 2250 until he pays a total amount of 3600 in total. However it indicates that insured would make payment of $2850 ($3600 - 50 = $2850) that past the 2250 mark. Thus it means that the insured reaches the total payment amount or $3,600 when X 5100 (2250+2850 5100) Y can be expressed as below 0 X 250 0, 0.75(Х — 250), 0.75(2250 250), 1500 0.95(X- 5100), 250 X 2250 Y = 2250 X5100 X 5100 X —х, X - 250 2250 250, 2250 < X < 5100 0 X 250 250< X < 2250 0.75(2250 250), 2250 < X <5100 0 X 250 250< X< 2250 0.75(X 250) = 0.75 — 0.75(Х л 2250 — х^250). Therefore Y can be expressed as below: Y 0.75(XA 2250 X A 250) 0.95(X - XA 5100) Е(Y) -0.75 E (X л 2250) — Е(Х л 250) + 0.95 |Е(X) - Е(Х Л5100)] {0) 2000 2000 2000 |-) 2000 2000 2000 2000 +0.95 E(Y)0.75 2000+250 20005100 20002250 E(Y) 1163 =