Let [math]X[/math] denote the size of a surgical claim and let [math]Y[/math] denote the size of the associated hospital claim. An actuary is using a model in which

• [math]\operatorname{E}[X] = 5 [/math]
• [math]\operatorname{E}[X^2] = 27.4 [/math]
• [math]\operatorname{E}[Y] = 7 [/math]
• [math]\operatorname{E}[Y^2] = 51.4 [/math]
• [math]\operatorname{Var}[X + Y] = 8 [/math]

Let [math]C_1 = X + Y [/math] denote the size of the combined claims before the application of a 20% surcharge on the hospital portion of the claim, and let [math]C_2[/math] denote the size of the combined claims after the application of that surcharge.

Calculate [math]\operatorname{Cov}(C_1,C_2)[/math] .

• 8.80
• 9.60
• 9.76
• 11.52
• 12.32

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Solution: A

[[math]] \begin{align*} \operatorname{Cov}( C_1 , C_2 ) &= \operatorname{Cov}( X + Y , X + 1.2Y ) \\ &= \operatorname{Cov}( X , X ) + \operatorname{Cov}(Y , X ) + \operatorname{Cov}( X ,1.2Y ) + \operatorname{Cov}( Y,1.2Y ) \\ &= \operatorname{Var}(X) + \operatorname{Cov}( X , Y ) + 1.2\operatorname{Cov}( X , Y ) + 1.2\operatorname{Var}(Y) \\ &= \operatorname{Var}(X) + 2.2 \operatorname{Cov}( X , Y ) + 1.2\operatorname{Var}(Y) \end{align*} [[/math]]

[[math]] \operatorname{Var}(X) = \operatorname{E}( X^2 ) − ( \operatorname{E}( X ) )^2 = 27.4 − 5^2 = 2.4 [[/math]]

[[math]] \operatorname{Var}(Y) = \operatorname{E}( Y^2 ) − ( \operatorname{E}( Y ) )^2 = 27.4 − 5^2 = 2.4 [[/math]]

[[math]] \operatorname{Var}(X+Y) = \operatorname{Var}(X) + \operatorname{Var}(Y) + 2 \operatorname{Cov}( X , Y ) [[/math]]

[[math]] \operatorname{Cov}( C_1 , C_2 ) = 2.4 + 2.2 (1.6 ) + 1.2 ( 2.4 ) = 8.8 [[/math]]

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.