Examenvragen januari 2014

1a) why is countermonotonicity the perfect negative dependence
b) if X1 and X2 are countermonotonic prove that the joint cdf is given by: max{Fx1+Fx2-1,0}
c)give an example of a countermonotonic vector without the variables being mutually exclusive
d) prove that for a countermonotonic vector which has the following proporty:
P[x1>0]+P[x2>0]>=1
the variables are mutual exclusive
2a) take a loss variable L=uniform(0,90000)
b) calculate fair premium P=phi1(L) with phi1(L)= stoploss with K=60000 P=E[phi1(L)]=(L-K)+
c) consider phi2(L)=alpha*L
Determine alpha so that P=E[phi2(L)]
d)proof that all risk averse decision makers wille prefer phi1 over phi2 (hint: use crossing theorem)
3a) Consider 2 risks: P[X=0]=0.6, P[X=1]=0.37, P[X=5]=0.03
P[Y=0]=0.6, P[Y=1]=0.39, P[Y=11]=0.01
Calculate VaR0.95 for X and Y
b)calculate TVaR0.95 for X and Y
c) calculate the Wang Transform risk measure WT0.95 for X and Y
d)which risk measure do you prefer and why
4a)consider the vector (X1,X2) where both variables are indepentently standard normally distributed.
Now assume vector (Y1,Y2)=(X1,V*X1) where P[V=-1]=0.5 and P[V=1]=0.5
Prove that Y1 and Y2 are standard normally distributed
b)prove that corr(Y1,Y2)=0. Dies this mean that Y1 and Y2 are independent?
c)prove that VaRp[X1+X2]=squareroot(2)*PHI^(-1)(p)
And that VaRp[Y1+Y2]=2PHI^(-1)(2p-1)
d)why is it dangerous to rely the pearson's correlation coefficient for dependence?