4. Melissa has the following loss distribution: x | p(x) 0 | 0.44 80 | 0.56 Liz

Question

4. Melissa has the following loss distribution:
x | p(x)
0 | 0.44
80 | 0.56
Liz has the following loss distribution:
y | p(y)
80 | 0.62
160 | 0.38

The loss experiences for Melissa and Liz are independent.
(a) Melissa and Liz decide to pool their loss experience. Let S represent the pooled loss random variable. Calculate the probability that S is within 0.50 standard deviations of its mean.
(b) Melissa and Liz pool their loss experience. They purchase an insurance policy that covers their pooled loss. Under this policy, the deductible is 35 and the maximum reimbursement is 215. Calculate the actuarially fair premium for this policy.

Explanation / Answer

(a)

S = x + y with the below PMF.

P(S = 80) = P(X=0, Y = 80) = P(X=0) (Y = 80) = 0.44 * 0.62 = 0.2728

P(S = 160) = P(X=80, Y = 80) + P(X=0, Y = 160) = P(X=80) (Y = 80) + P(X=0) (Y = 160)= 0.56 * 0.62 + 0.44 * 0.38 = 0.5144

P(S = 240) = P(X=80, Y = 160) = P(X=80) (Y = 160) = 0.56 * 0.38 = 0.2128

E[S] = 0.2728 * 80 + 0.5144 * 160 + 0.2128 * 240 = 155.2

E[S2] = 0.2728 * 802 + 0.5144 * 1602 + 0.2128 * 2402 = 27171.84

Var[S] = E[S2] - E[S]2 = 27171.84 - 155.22 = 3084.8

Standard deviation of S = sqrt(3084.8) = 55.54

Range of within 0.50 standard deviations of its mean = (155.2 - 0.5 * 55.54, 155.2 + 0.5 * 55.54)

= (127.43, 182.97)

From the PMF of S, P(127.43 < S < 182.97) = P(S = 160) = 0.5144

(b)

For S = 80, 160 and 240, the payment is (80-35), (160-35) , (240-35)

= 45, 125, 205

Actuarially fair premium for this policy = 0.2728 * 45 + 0.5144 * 125 + 0.2128 * 205 = 120.2

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