Krzys’ Ostaszewski: http://www.krzysio.net
Author of the “Been There Done That!” manual for Course P/1
http://smartURL.it/krzysioP (paper) or http://smartURL.it/krzysioPe (electronic)
Instructor of online P/1 seminar: http://smartURL.it/onlineactuary
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November 2001 Course 1 Examination, Problem No. 7, also P Sample Exam
Questions, Problem No. 107, and Dr. Ostaszewski’s online exercise posted January
2, 2010
Let X denote the size of a surgical claim and let Y denote the size of the associated
hospital claim. An actuary is using a model in which E ( X ) = 5, E X 2 = 27.4,
( )
( )
E (Y ) = 7, E Y 2 = 51.4 and Var ( X + Y ) = 8. Let C1 = X + Y denote the size of the
combined claims before the application of a 20% surcharge on the hospital portion of the
claim, and C2 denote the size of the combined claims after the application of that
surcharge. Calculate Cov ( C1 ,C2 ) .
A. 8.80
B. 9.60
C. 9.76
D. 11.52
E. 12.32
Solution.
First, let us note that:
Cov ( C1 ,C2 ) = Cov ( X + Y , X + 1.2Y ) =
= Cov ( X, X ) + Cov (Y , X ) + Cov ( X,1.2Y ) + Cov (Y ,1.2Y ) =
= Var ( X ) + Cov ( X,Y ) + 1.2Cov ( X,Y ) + 1.2Var (Y ) =
= Var ( X ) + 2.2Cov ( X,Y ) + 1.2Var (Y ) .
Furthermore,
2
Var ( X ) = E X 2 − ( E ( X )) = 27.4 − 5 2 = 2.4,
( )
Var (Y ) = E (Y ) − ( E (Y ))
2
2
= 51.4 − 7 2 = 2.4,
and because
Var ( X + Y ) = Var ( X ) + Var (Y ) + 2Cov ( X,Y ) ,
we have
1
1
Cov ( X,Y ) = ( Var ( X + Y ) − Var ( X ) − Var (Y )) = ⋅ ( 8 − 2.4 − 2.4 ) = 1.6.
2
2
Hence
Cov ( C1 ,C2 ) = 2.4 + 2.2 ⋅1.6 + 1.2 ⋅ 2.4 = 8.8.
Answer A.
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