Question

Derive the standard deviation of the returns on a portfolio that is invested in stocks x, y, and z , where twenty percent of the portfolio is invested in stock x and 35 percent is invested in Stock z.

State of Economy	Probability of State of Economy	Rate of Return if State Occurs
		Stock x			Stock y			Stock z
Boom	.04		.17			.09			.09
Normal	.81		.08			.06			.08
Recession	.15	−	.24			.02		−	.13

	1.	6.31 percent
	2.	6.49 percent
	3.	7.38 percent
	4.	5.65 percent
	5.	7.72 percent

Answer 1

Answer is 5.65 percent

Weight of Stock X = 0.20
Weight of Stock Y = 0.45
Weight of Stock Z = 0.35

Boom:

Expected Return = 0.20 * 0.17 + 0.45 * 0.09 + 0.35 * 0.09
Expected Return = 0.1060

Normal:

Expected Return = 0.20 * 0.08 + 0.45 * 0.06 + 0.35 * 0.08
Expected Return = 0.0710

Recession:

Expected Return = 0.20 * (-0.24) + 0.45 * 0.02 + 0.35 * (-0.13)
Expected Return = -0.0845

Expected Return of Portfolio = 0.04 * 0.1060 + 0.81 * 0.0710 + 0.15 * (-0.0845)
Expected Return of Portfolio = 0.049075

Variance of Portfolio = 0.04 * (0.1060 - 0.049075)^2 + 0.81 * (0.0710 - 0.049075)^2 + 0.15 * (-0.0845 - 0.049075)^2
Variance of Portfolio = 0.003195

Standard Deviation of Portfolio = (0.003195)^(1/2)
Standard Deviation of Portfolio = 0.0565 or 5.65%