Question

Consider 2 scenarios: Boom Economy and Normal Economy. The Boom economy has 20% chance of happening, while Normal economy has 80% chance of happening. For each scenario (Boom and Normal), stock ABC has a return of 25%, and 4%, respectively; stock XYZ has a return of 10% and 6.5%, respectively; the market portfolio has a return of 12% and 5% respectively. 1) Calculate Expected return, Variance and Standard deviation for stock ABC and XYZ 2) Based on your results in part (1), can you decide which stock to invest? 3) Calculate Beta for stock ABC and XYZ 4) If the T-bill rate is 3%, what does the CAPM say about the fair expected rate of return on the two stocks? How does this result influence your investment decision?

Answer 1

1)	For ABC -
	State of the Economy	Probability	Fund Return	P x X	X-Mean	(X- Mean)^2	P x (X- Mean)^2
		(p)	(x)		(x - 37)
	Boom	0.2	25	5	-12	144	28.8
	Normal	0.8	40	32	3	9	7.2
				37			36
	Expected return = Sum Px =	37
	(Mean)
	Variance =	Sum p x (x -mean)^2
		36
	Standard deviation =	Sq. root of (Sum p x (x -mean)^2)
		Sq root (36)
		6
	For XYZ -
	State of the Economy	Probability	Fund Return	P x X	X-Mean	(X- Mean)^2	P x (X- Mean)^2
		(p)	(x)		(x - 7.2)
	Boom	0.2	10.0	2.0	2.8	7.8	1.6
	Normal	0.8	6.5	5.2	-0.7	0.5	0.4
				7.2			2.0
	Expected return = Sum Px =	7.2
	(Mean)
	Variance =	Sum p x (x -mean)^2
		2.0
	Standard deviation =	Sq. root of (Sum p x (x -mean)^2)
		Sq root (2)
		1.40
2)	Coefficient of variance =	SD/Mean x 100
	Lower the Better
	For ABC =	6/37 x 100 =	16.22%
	For XYZ =	1.4/7.2 x 100 =	19.44%
	ABC has low volatility per unit of return then XYZ, So we should invest in ABC.
3)	For ABC -
	State of the Economy	Probability	Fund Return	Market Return	P x X	P x Y	X-Mean	Y-Mean	P x (Y- Mean)^2	P x (x-Mean)(Y-Mean)
		(p)	(x)	(y)			(x - 37)	(y - 6.4)
	Boom	0.2	25.0	12.0	5.0	2.4	-12.0	5.6	6.3	-13.4
	Normal	0.8	40.0	5.0	32.0	4.0	3.0	-1.4	1.6	-3.4
					37.0	6.4			7.8	-16.8
	COV (X,Y) =	Sum of Px(X-Mean)(Y-Mean) =	-16.80
	Variance Of Market return =	Sum Of P x (Y- Mean)^2 =	7.8
	Beta =	COV (X,Y)/Variance Of Market return
		-16.8/ 7.8
		-2.14
	For XYZ -
	State of the Economy	Probability	Fund Return	Market Return	P x X	P x Y	X-Mean	Y-Mean	P x (Y- Mean)^2	P x (x-Mean)(Y-Mean)
		(p)	(x)	(y)			(x - 37)	(y - 6.4)
	Boom	0.2	10.0	12.0	2.0	2.4	2.8	5.6	6.3	3.1
	Normal	0.8	6.5	5.0	5.2	4.0	-0.7	-1.4	1.6	0.8
					7.2	6.4			7.8	3.9
	COV (X,Y) =	Sum of Px(X-Mean)(Y-Mean) =	3.92
	Variance Of Market return =	Sum Of P x (Y- Mean)^2 =	7.8
	Beta =	COV (X,Y)/Variance Of Market return
		3.9/ 7.8
		0.50
4)	Rate of return as per CAPM =	RF + (Rm-Rf) x Beta
		ABC	XYZ
	Rf	3	3
	Rm	6.4	6.4
	Rm-Rf	3.4	3.4
	Beta	-2.14	0.50
	RF + (Rm-Rf) x Beta	-4.29	4.70
	As per CAPM ABC has a negative rate of return, we should not invest in ABC.