Question

Returns for Stocks A and Stock B have the following distribution:

Probability	Rate of Return Stock A	Rate of Return Stock B
0.20	+16%	-10%
0.30	+10%	-6%
0.50	-30%	+40%

a) What is the Expected Return for Stock A?

b) What is the Standard Deviation for Stock A?

c) What is the Expected Return for Stock B?

d) What is the Standard Deviation for Stock B?

e) What is the Expected Return for a Portfolio with an equal 50% investment of each Stock A and Stock B?

f) What is the Standard Deviation for a Portfolio with an equal 50% investment of each Stock A and Stock B?

g) What is the Correlation Coefficient of Stock A and Stock B?

Answer 1

Solution:

a) Expected Return for Stock A

Probability(Pi)	Return(ri)	riPi
0.20	16%	3.2
0.30	10%	3
0.50	-30%	-15
Expected Return	-8.8%

Thus, the expected return of Stock A is -8.8%

b) Standard Deviation for Stock A

ri	ri - Expected Return	(ri - Expected Return)2	Pri	Pri(ri - Expected Return)2
16%	7.2	51.84	0.20	10.368
10%	1.2	1.44	0.30	0.432
-30%	-38.8	1505.44	0.50	752.72
Variance	763.52

Standard Deviation = √Variance

= √763.52

= 27.6318%

Thus, Standard Deviation for Stock A is 27.6318% or 27.63%

c)  Expected Return for Stock B   

Probability(Pi)	Return(ri)	riPi
0.20	-10%	-2
0.30	-6%	-1.8
0.50	40%	20
Expected Return	16.2

Thus, the expected return for Stock B is 16.2%

d) Standard Deviation for Stock B

ri	ri - Expected Return	(ri - Expected Return)2	Pri	Pri(ri - Expected Return)2
-10%	-26.2	686.44	0.20	137.288
-6%	-22.2	492.84	0.30	147.852
40%	23.8	566.44	0.50	283.22
Variance	568.36

Standard Deviation = √Variance

= √568.36

= 23.8403%

Thus, Standard Deviation for Stock B is 23.8403% or 23.84%

e) Expected Return for a Portfolio with an equal 50% investment of each Stock A and Stock B

Stock	Return (Calculated above) (ri)	Proportion (xi)	xiri
A	-8.8%	0.50	-4.4
B	16.2%	0.50	8.1
Portfolio Return	3.7%

Thus, the expected return for a portfolio with an equal 50% investment of each Stock and Stock B is 3.7%

f) Standard Deviation for a Portfolio with an equal 50% investment of each Stock A and Stock B

Particulars	Stock A	Stock B
Standard Deviation (Calculated above)	27.63%	23.84%
Proportion of investment	0.50	0.50

Portfolio Variance = ({Proportion of stock A}2 * {Standard deviation of Stock A}2) + ({Proportion of stock B}2 * {Standard deviation of Stock B}2) + (2 * Proportion of Stock A * Proportion of Stock B * Correlation Coeefficient * Standard deviation of Stock A * Standard deviation of Stock B)

Correlation between A and B = (Co-variance between A and B) / (Standard deviation of Stock A * Standard deviation of Stock B)

Calculation of Co-Variance

Return of Stock A	Return of Stock B
16%	-10%
10%	-6%
-30%	40%

Mean:

Stock A = -4 / 3 Stock B = 24/3

= -1.3 = 8

Return of Stock A - Mean of Stock A	Return of Stock B - Mean of Stock B	(Return of Stock A - Mean of Stock A) * (Return of Stock B - Mean of Stock B)
17.3	-18	-311.4
11.3	-14	-158.2
-28.7	32	-918.4
Total	-1388

Co-variance between A and B = -1388 / 3

= -462.67

Correlation = -462.67 / ( 27.63 * 23.84 )

= -0 .7024 or -0.70

Portfolio Variance = ({.50}2*{27.63}2 )+ ({.50}2 *{ 23.84}2) + (2*.50*.50*-.70*27.63*23.84)

= (.25 * 763.4169) + (.25 * 568.3456) - 230.54472

= 190.854225 + 142.0864 - 230.54472

= 102.395905

Standard Deviation = √Portfolio Variance

= √102.395905

= 10.1191%

Thus, Standard Deviation for a Portfolio with an equal 50% investment of each Stock A and Stock B is 10.1191% or 10.12%

g) Correlation Coefficient of Stock A and Stock B

Correlation between A and B = (Co-variance between A and B) / (Standard deviation of Stock A * Standard deviation of Stock B)

Calculation of Co-Variance

Return of Stock A	Return of Stock B
16%	-10%
10%	-6%
-30%	40%

Mean:

Stock A = -4 / 3 Stock B = 24/3

= -1.3 = 8

Return of Stock A - Mean of Stock A	Return of Stock B - Mean of Stock B	(Return of Stock A - Mean of Stock A) * (Return of Stock B - Mean of Stock B)
17.3	-18	-311.4
11.3	-14	-158.2
-28.7	32	-918.4
Total	-1388

Co-variance between A and B = -1388 / 3

= -462.67

Correlation = -462.67 / ( 27.63 * 23.84 )

= -0 .7024 or -0.70

Thus, Correlation Coefficient of Stock A and Stock B is -0.7024 or -0.70 which shows a negative correlation between Stock A and Stock B