Question

   Stock A   Stock B
1   0.09   0.06
2   0.05   0.02
3   0.14   0.03
4   -0.03   0.02
5   0.08   -0.01

a. What are the expected returns of the two stocks?

b. What are the standard deviations of the returns of the two stocks?

c. If their correlation is 0.44, what are the expected return and standard deviation of a portfolio of 66% stock A and 34% stock B?

Answer 1

Stock A Return	Stock B Return
0.09	0.06
0.05	0.02
0.14	0.03
-0.03	0.02
0.08	-0.01

a. Expected Return

Expected Return is the mean of all given returns

Expected Return of Stock A = E[R_A] = (0.09+0.05+0.14+(-0.03)+0.08)/5 = 0.066

Expected Return of Stock B = E[R_B] = (0.06+0.02+0.03+0.02+(-0.01))/5 = 0.024

Answer a:

Expected Return of Stock A = E[R_A] = 0.066

Expected Return of Stock B = E[R_B] = 0.024

b. Standard Deviations of the returns of the two stocks

Standard Deviation of a sample is calculated using the Excel Function =STDEV.S(Range of Stock A return)

σ_A = 0.062689712

Similarly we can calculate the standard deviation from the given sample of Stock B return using the formula =STDEV.S(Range of Stock B return)

σ_B = 0.025099801

Alternatively, Standard deviation of a sample can be calculated using the below formula

The formula for calculating Sample Standard Deviation is

In this Example, n = 5

For Stock A

μ_A = E[R_A] = 0.066

Stock A return	(X-μA)2
0.09	0.000576
0.05	0.000256
0.14	0.005476
-0.03	0.009216
0.08	0.000196
Sum	0.015720

σ_A = (0.15720/4)^1/2 = 0.062689712

For Stock B

μ_B = E[R_B] = 0.024

Stock B return	(X-μB)2
0.06	0.001296
0.02	0.000016
0.03	0.000036
0.02	0.000016
-0.01	0.001156
Sum	0.00252

σ_B = (0.00252/4)^1/2 = 0.025099801

Answer b:

σ_A = 0.062689712

σ_B = 0.025099801

c. Expected Return and Standard Deviation of Portfolio

Expected Return of Portfolio

Weight distribution of the portfolio:

W_A = 66%

W_B = 34%

E[R_A] = 0.066

E[R_B] = 0.024

Expected return of portfolio =E[R_P] = W_A*E(R_A) + W_B* E(R_B) = 0.66*0.066 + 0.34*0.024 = 0.05172 = 5.172%

Standard Deviation of Portfolio

W_A = 66%, W_B = 34%

σ_A = 0.062689712, σ_B = 0.025099801

ρ = 0.44

Variance of the portfolio = W_A²* σ²_A + W_B²* σ²_B + 2 W_A*W_B *ρ* σ_{A *} σ_B

where, ρ is the correlation between Johnson&Johnson and Walgreen Company and the value of ρ is 0.44

putting these values in the above equation, we get:

Portfolio Variance = σ_p² = 0.66* (0.062689712)² + 0.34* (0.025099801)² + 2*0.66*0.34 *0.44*(0.062689712)*(0.025099801) = 0.001711908 + 0.000072828 + 0.000310722050814077 = 0.00209545805081408

Therefore, Standard Deviation of the portfolio = σ_p = Square root(Portfolio Variance) = (0.00209545805081408)^1/2 = 0.045776173 = 4.578%

Answer c:

Expected return of portfolio = E[R_P] = 0.05172 = 5.172%

Standard Deviation of the portfolio = σ_p = 0.045776173 = 4.578%