Question

Economic                                                                    Return on Asset

Conditions                  Probability                               A         B         C

Boom                              .30                                       60%     50%     10%

Normal                            .40                                       40        30        50

Bust                                 .30                                       20        10        90

Portfolio AB is formed by investing 50% of the funds in each of the assets A and B. A similar (equally weighted) portfolio has also been created from A and C, called AC. Find the rates of return not given for AB and AC.

Econ                                                       AB=                                               AC=

Conditions     Probability                   .50A+.50B                                      .50A+.50C

Boom                .30                                55                                                    35%

Normal             .40                                35 ?

Bust                  .30                                15 ?

Find the expected return and standard deviation of AB and AC. (The formulas are given in 1 above.) Do you see any evidence of risk reduction from the numbers you obtain? What do you think caused this?

Answer 1

	(in %)
State of economy	probability(p)	stock A(x)	stock B(y)	stock C(z)	p*x	p*y	p*z	p*((x-∑px)^2)	p*((y-∑py)^2)	p*(x-∑px)(y-∑py)	p*((z-∑pz)^2)	p*(x-∑px)(z-∑pz)
Boom	0.3	60	50	10	18	15	3	120.00	120.00	120.00	480.00	-240.00
Normal	0.4	40	30	50	16	12	20	-	-	-	-	0.00
bust	0.3	20	10	90	6	3	27	120.00	120.00	120.00	480.00	-240.00
Expected Return		40.00	30.00	50.00	240.00	240.00	240.00	960.00	-480.00
								stock A(x)	stock B(y)	stock C(z)
Expected Return =	∑px for stock A and ∑py for stock B and ∑pz for stock C		40.00	30.00	50.00
Standard deviation=	∑p*((x-∑px)^2)^(1/2)				15.49	15.49	30.98
								240^(1/2)	240^(1/2)	960^(1/2)
Expected Return of portfolio =		(Return of stock A* weight of stock A) + (Return of stock B* weight of stock B)
Portfolio AB		=	(40%*0.50)+(30%*0.50)		=	35%
Portfolio AC		=	(40%*0.50)+(50%*0.50)		=	45%
covariance (A,B)=	∑p*(x-∑px)(y-∑py)	covariance (A,C)=	∑p*(x-∑px)(z-∑pz)
=	240			=	-480
Standard Deviation of Portfolio=	[{(weight of A)^2 * (sigma of A)^2} + {(weight of B)^2 + (sigma of B)^2} + {2*(weight of A)*(weight of B)*Covariance of A&B}]^(1/2)
Portfolio AB	=	[{(0.50)^2*(15.49)^2} + {(0.50)^2*(15.49)^2} + (2*0.5*0.5*240)^(1/2)]
	=	15.49%
Portfolio AC		[{(0.50)^2*(15.49)^2} + {(0.50)^2*(30.98)^2} + (2*0.5*0.5*(-480))^(1/2)]
	=	7.74%