Question

year stock A stock B

2015 3 % 40%

2016 5 % -5%

2017 1 % 30%

2018 10 % -10%

2019 6 % 35 %

(1) Determine the correlation coefficient of returns of stocks A and B. Can you reduce risk by creating a portfolio of the combination of both stocks? why or why not?

(2) if you invest 80% of money in stock A and another 20% in stock B, calculate expected rate of return and standard deviation of this portfolio. Is this portfolio better than the individual stock A and B? why

Answer 1

Year	Stock A	Stock B
2015	3%	40%
2016	5%	-5%
2017	1%	30%
2018	10%	-10%
2019	6%	35%

(1) Expected Return of Stock A = E[R_A] = (3%+5%+1%+10%+6%)/5 = 5%

Expected Return of Stock B = E[R_B] = (40%+(-5%)+30%+(-10%)+35%)/5 = 18%

If n is the sample size, then the variance of the sample is calculated using below formula:

where R_i is the return in different years and E[R] is the expected return as calculated above.

Variance of Stock A = σ_A² = (1/4)*[(3%-5%)²+(5%-5%)²+(1%-5%)²+(10%-5%)²+(6%-5%)²]

σ_A² = (0.004+0+0.0016+0.0025+0.0001)/4 = 0.00115

Therefore, standard deviation of Stock A = σ_A = (0.00115)^1/2 = 0.03391165 = 3.39%

Variance of Stock B = σ_B² = (1/4)*[(40%-18%)²+(-5%-18%)²+(30%-18%)²+(-10%-18%)²+(35%-18%)²]

σ_B² = (0.0484+0.0529+0.0144+0.0784+0.0289)/4 = 0.05575

Therefore, standard deviation of Stock B = σ_B = (0.05575)^1/2 = 0.236114 = 23.61%

E[R_A] = 5%, σ_A = 3.39%

E[R_B] = 18%, σ_B = 23.61%

Covariance between returns of Stock A and Stock B is calculated using below formula:

Cov(A,B) = [(3%-5%)*(40%-18%)+(5%-5%)*(-5%-18%)+(1%-5%)*(30%-18%)+(10%-5%)*(-10%-18%)+(6%-5%)*(35%-18%)]/4 = -0.0215/4 = -0.005375

Relation between Correlation coefficient(ρ)and covariance between A & B (Cov(A,B)) is given below:

Cov(A,B) = ρ* σ_A* σ_B

Therefore, correlation coefficient = ρ = Cov(A,B)/( σ_A* σ_B) = -0.005375/(0.03391165*0.236114) correlation coefficient = ρ = -0.671285262235802 [Answer(1)]

As the correlation coefficient is negative, so creating a portfolio of the combination of both stocks will reduce the risk. Negative correlation means that when the return of Stock A decreases, then the return of stock A will see a positive trend. In this way we can manage the risk by creating a portfolio of stocks A and stock b.

(2) 80% of money is invested in stock A and 20% is invested in stock B

Therefore, weight of Stock A in the portfolio = W_A = 0.8

weight of Stock B in the portfolio = W_B = 0.2

E[R_A] = 5%, σ_A = 3.39%, W_A = 0.8

E[R_B] = 18%, σ_B = 23.61%, W_B = 0.2

ρ = -0.671285262235802

Expected return of portfolio = E[R_P] = W_A*E[R_A] + W_B* E[R_B] = 0.8*5% + 0.2*18% = 7.6%

Variance of the portfolio = σ²_p = W_A²* σ²_A + W_B²* σ²_B + 2*W_A*W_B *ρ* σ_{A *} σ_B

σ²_p = 0.8²* 0.00115 + 0.2²* 0.05575 + 2*0.8*0.2*(-0.671285)* 3.39% _* 23.61% σ²_p = 0.000736+0.00223+(-0.00172) = 0.001246

Therefore, Standard Deviation of the portfolio = (0.001246)^1/2 = 0.03529873 = 3.53%

Answer(2)

Expected return of the portfolio = 7.6%

Standard Deviation of the portfolio = 3.53%

We know that the standard deviation of A is around 3.39% and that of B is 23.61% but the standard deviation of the portfolio is 3.53%. We will calculate the reward to risk ration of A, B and the portfolio. Reward to rist ratio is the ratio of Expected Return and standard Deviation. Higher Reward to risk ratio is always better.

	Reward or Expected Return	Standard deviation or Risk	Reward/Risk ratio
A	5%	0.03391165	1.474419562
B	18%	0.236114379	0.76234239
Portfolio	7.60%	0.035298725	2.153052259

We see that the Reward to risk ratio of the portfolio is greater than the individual stocks A and B. So, the portfolio is better than the individual stocks A and B.