Question

The followings are the Stock X and Stock Y information: Rate of Return if State Occurs State of the Economy Recession Normal Boom Probability of State of Economy 0.15 0.70 0.15 Stock X 0.20 0.21 0.06 Stock Y -0.25 0.09 0.44 a. b. What are the expected return and standard deviation of Stock X and Stock Y? (12 points) If you invest 65% in Stock X and 35% in Stock Y, what are the expected return and standard deviation of your portfolio? Also calculate the correlation between Stock X and Stock Y returns. (12 points)

Answer 1

1. Expected return= R1P1+ R2P2….+ RnPn

Where , R= expected return on that scenario

P= probability of that return

N= period or number of scenario

Here

		given rate of return		Expected Return
Scenario	probability	stock X	Stock Y	stock X	Stock Y
recession	0.15	0.20	-0.25	0.0300	-0.0375
normal	0.70	0.21	0.09	0.1470	0.0630
boom	0.15	0.06	0.44	0.0090	0.0660
Total				0.1860	0.0915

Expected return X= 0.15*0.20+0.70*0.21+0.15*0.06= 0.1860 or 18.60%

Similarly , Expected return of Y= 0.0915 or 9.15%

Standard deviation = (( actual return-expected return)* ( probability))^1/2

So,

		given rate of return		Expected Return		A= (Given return - expected return)^2		D= A*P
	probability	stock X	Stock Y	stock X	Stock Y	stock X	Stock Y	stock X	Stock Y
recession	0.15	20.00	-25.00	3.00	-3.75	1.96	1,166.22	0.29	174.93
normal	0.70	21.00	9.00	14.70	6.30	5.76	0.02	4.03	0.02
boom	0.15	6.00	44.00	0.90	6.60	158.76	1,214.52	23.81	182.18
				18.60	9.15			28.14	357.13

Here , For stock X, A would be = (20-18.60)^2=1.96 and so on

For Y= (-25-9.18)^2= 1166.22

D for X= 0.15*1.96+0.70*5.76+ 0.15*158.76= 28.14

For Y= 0.15*1166.22+0.70*0.02+0.15*1214.52= 357.13

Now,

	stock X	Stock Y
Variance = from above	28.14	357.13
Standard deviation = Variance1/2	14.07	178.56

b.

Expected Return=WA×RA+WB×RB+WC×RC

where:

WA = Weight of security X= 65%

RA = Expected return of security X= 18.60%

WB = Weight of security Y= 35%

RB = Expected return of security Y= 9.15%

​               

Expected return = 0.65*0.1860 + 0.35*0.0915

= 0.152925 or 15.29%

Standard deviation of portfolio= (W1²* D1² + W2²*D2²+ 2 W1*W2*D1*D2* P12)^1/2                            

Where,

W1= weight of X

W2= weight of Y

D1= std deviation of X

D2= standard deviation of Y

P12= correlation coefficient of X and Y

Covariance =

	A= (Given return - expected return)		C= P* A of X*A of Y
probability	stock X	Stock Y
0.15	20-18.60= 1.40	-25-9.15=-34.15	0.15*1.40*-34.15= -7.17
0.7	21-18.60= 2.40	9-9.15=-0.15	0.70*2.40*-0.15= -0.25
0.15	6-18.60= -12.60	44-9.15= 34.85	0.15*-12.60*34.85= -65.87
		Total	-73.29

So Cov= -73.29

Correlation =Cov/ D1*D2

= -73.29/ (14.07*178.56)= -0.0292

Std. Deviation= (0.65²*14.07² + 0.35²*178.56² + 2*0.65*0.35*14.07*178.56*-0.0292)^1/2

= (83.64+ 3905.75 -33.3789)^1/2

= 62.89