Question

One of the major risks in the economy is the oil price. Many financial assets are severely affected by the oil price. Consider a natural gas company and an automobile company. The natural gas company will be hit by a low oil price because the demand for natural gas will decrease as oil becomes cheaper. But the automobile company will benefit from a low oil price as more people can afford the cost of driving a car. The probability of the oil price next year is given by the following table:

Oil price	Probability
drops	25%
does not change	50%
arises	25%

Contingent on the change in the oil price next year, the annual returns to the companies are expected as follows:

If oil price	Natural gas	Automobile
drops	-8%	18%
does not change	6%	4%
arises	12%	-10%

You are planning to invest 70% of your investment in the natural gas company and the remaining in the automobile company. What is the variance of your portfolio?

	a.	3.00
	b.	16.08
	c.	10.32
	d.	5.88
	e.	1.52

Answer 1

x	y	f(x,y)	x*f(x,y)	y*f(x,y)	x^2f(x,y)	y^2f(x,y)	xy*f(x,y)
-8	18	0.25	-2	4.5	16	81	-36
6	4	0.5	3	2	18	8	12
12	-10	0.25	3	-2.5	36	25	-30
	Total	1	4	4	70	114	-54
	E(X)=ΣxP(x,y)=					4
	E(X2)=Σx2P(x,y)=					70
	E(Y)=ΣyP(x,y)=					4
	E(Y2)=Σy2P(x,y)=					114
	Var(X)=E(X2)-(E(X))2=					54
	Var(Y)=E(Y2)-(E(Y))2=					98
	E(XY)=ΣxyP(x,y)=					-54
	Cov(X,Y)=E(XY)-E(X)*E(Y)=					-70

hence Var(0.7x+0.3y)=0.49*Var(X)+0.09*Var(Y)+2*0.7*0.3*Cov(X,Y)=0.49*54+0.09*98+2*0.7*0.3*(-70)= 5.88