A risk manager wants to study the behavior of a portfolio that depends on only 2 economic variables, X and Y ,X is uniformly distributed between 4 and 7,and Y is uniformly distributed between 5and 8. The risk manager wants to model their joint distribution, H(X,Y). The theorem of Sklar proves that, for any joint distribution H, there is a copula C such that :
A. H(3X+4,3Y+5) is equal to C[X,Y].
B. H(X,Y) is equal to C[u,d] where u is the density marginal distribution of X and d is the density marginal distribution of Y
C. H(X, Y) is equal to C[X-4/3,Y-5/3].
D. H[X-4/3,Y-5/3] is equal to C(X, Y).
Answer:C
Explanation: Sklar’s theorem proves that if F(x,y) is a joint distribution function with continuous marginalsFx(X)=u and Fy(y)=v, then F(x,y) can be written in terms of a unique function C(u,v) such as F(x,y)=C(u,v). in this case u=(X-4)/3 (the cumulative marginal function of X, which is uniformly distributed between 4 and 7) and v=(Y-5)/3.
Note: The candidate is not responsible for knowing the properties of the uniform distribution.