An insurance policy pays a total medical benefit consisting of two parts for eac

Question

An insurance policy pays a total medical benefit consisting of two parts for each claim.
Let X represent the part of the benefit that is paid to the surgeon, and let Y represent the
part that is paid to the hospital. The variance of X is 5000, the variance of Y is 10,000, and the variance of the total benefit, X + Y, is 17,000. Due to increasing medical costs, the company that issues the policy decides to increase X by a flat amount of 100 per claim and to increase Y by 10% per claim.
Calculate the variance of the total benefit after these revisions have been made.

Explanation / Answer

Each claim consists of two parts = X + Y

where X = the benefit that is paid to the surgeon and Y = benefit that is paid to the hospital

V(X) = 5000, V(Y) = 10000 and V(X+Y) = 17000

So V(X+Y) = V(X) + V(Y) + 2cov(X,Y)

17000 = 5000 + 10000 +2 cov(X,Y)

17000 -15000 = 2cov(X,Y)

2000 = 2cov(X,Y)

cov(X,Y) = 1000

Now X is increased by flat Rs. 100 per claim and Y by 10% per claim

total benefit = X+100+Y+0.1Y = X+100 + 1.1Y

V(total benefit) = V(X) + 1.12V(Y) +2(1.1)cov(X,Y) [ V(aX+bY) = a2V(X) +b2V(Y) +2abcov(X,Y) and V(X+c) = V(X)]

= 5000 + (1.21*10000) + (2.2*1000)

= 5000 + 12100 + 2200

= 19300

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