Question

There are only two possible states of the economy. State 1 has a 75% chance of occurring. In State 1, Asset A returns 5.50% and Asset B returns 8.50%. In-State 2, Asset A returns -3.20% and Asset B returns -6.20%. A portfolio of just these two assets is invested 35% in Asset A (with Asset B comprising the remainder without any negative weights). What is the standard deviation of the portfolio's returns?

	5.46%
	5.59%
	5.73%
	5.87%
	6.00%

Answer 1

Answer: 5.46%

Working Note

Asset A

Expected Return = (0.75*5.50) + (0.25*-3.20) = 3.33%

Probability	Return (X)	A = X- Expected Return	B = A* A	C= Probability * B
0.75	5.50	2.18	4.73	3.55
0.25	-3.20	-6.53	42.58	10.64

Variance = 3.55+10.64= 14.19
Standard Deviation = square root of variance = 3.77

Asset B

Expected Return = (0.75 *8.50) + (0.25 *-6.20) = 4.83%

Probability	Return (Y)	A= Y- Expected Return	B= A* A	C= Probability * B
0.75	8.5	3.67	13.51	10.13
0.25	-6.2	-11.03	121.55	30.39

Variance = 10.13+30.39 = 40.52
Standard Deviation = square root of variance = 6.37

Now, we will calculate covariance

Probability	A = X- Expected Return	B= Y- Expected Return	Probability * A*B
0.75	2.18	3.68	6.02
0.25	-6.53	-11.03	18.01
			24.02

Therefore, the covariance between both the stocks is 24.02

Standard Deviation = σ_P = √(w_A²σ_A² + w_B² σ_B² + 2w_Aw_Bcov_AB )

Standard Deviation = √ 0.35*0.35*3.77*3.77 + 0.65*0.65*6.37*6.37 + 2*0.35*0.65*24.02
Standard Deviation = √ 29.8140 = 5.46%