Question

Consider the following information on Stocks I and II:

State of Economy	Probability of State of Economy	Rate of Return if State Occurs
		Stock I	Stock II
Recession	.20	.02	−.20
Normal	.55	.32	.12
Irrational exuberance	.25	.18	.40

The market risk premium is 7 percent, and the risk-free rate is 4 percent. Calculate the beta and standard deviation for both stocks Also include which one has more systematic risk, and which is riskier?

Answer 1

Solution : Calculation of Beta : For the pupose of calculation of beta we first need to calculate Expected Return

Calculation of Expected Return of Stock 1 = Sum of ( Probability of each stock * Rate of Return )

= 0.20 * 2% + 0.55 * 32% + 0.25 * 18%

= 0.40 + 17.6 + 4.5

= 22.5 %

Note : Rate of Return has been converted into percentages.

Calculation of Expected Return of Stock 2 = Sum of ( Probability of each stock * Rate of Return )

= [0.20 * (-20)%] + 0.55 * 12% + 0.25 * 40%

= (-4) + 6.6 + 10

= 12.6 %

Expected Return of Stock 1 = Risk free rate + (beta of stock 1 * Market Risk Premium)

22.5% = 4 % + Beta * 7%

18.5% = 7% Beta

Beta of Stock 1= 18.5 / 7

= 2.64

Expected Return of Stock 2 = Risk free rate + (beta of stock 2 * Market Risk Premium)

12.6 % = 4 % + Beta * 7%

8.6% = 7% Beta

Beta of Stock 2= 8.6 / 7

= 1.23

Calculation of Standard Deviation :

Below is the table showing calculation of Standard deviation:

Economy	Probability	Rate of Return of Stock 1	Rate of return of stock 2	Deviation from expected Return of Stock 1	Deviation from Expected Return of Stock 2	Probability * Deviation square of stock 1	Deviation square of stock 2
Recession	0.20	2%	-20%	2-22.5 = -20.5	(-20-12.6) =32.60	0.20 * (-20.5)2 =84.05	0.20*(-32.60)2 =212.552
Normal	0.55	32%	12%	32 - 22.5 = 9.5	12-12.6 =(-0.6)	0.55 * (9.5)2=49.6375	0.55 * (-0.6)2=0.198
Irrational excurbence	0.25	18%	40%	18 - 22.5 = -4.5	40-12.6 = 27.40	0.25 * (-4.5)2=5.0625	0.25 * (27.40)2= 187.69
					Sum of deviation	138.75	400.44

Standard deviation of Stock 1=(Sum of deviation)1/2

= (138.75)1/2

= 11.78%

Standard deviation of Stock 2=(Sum of deviation)1/2

= (400.44)1/2

= 20.01%

The stock I has the most systematic risk because it has the highest beta.

The stock 2 is more riskier because it has higher standard deviation.