Question

If you have 3 stocks (A), (B), & (C). The following are the rates of returns in the last 5 years for the three stocks:

Stock A	Stock B	Stock C
30	60	20
20	35	30
40	20	40
25	25	35
35	20	40

Required

1. Calculate the expected returns for individual stocks A, B & C.
2. Calculate the expected risk for individual stocks A, B & C.
3. Calculate the covariance and correlation between A&B stocks and A&C stocks
4. If you decide to construct a portfolio that consist of 40 % for stock (A) & 60% for the selected stocks (B or C). Which portfolio would you choose? Why?

Answer 1

1.Expected returns for individual stocks

ER(Stock A)=Sum of Returns/No.of yrs. Of data

ie.(30+20+40+25+35)%/5=

30.00%

ER(Stock B)=Sum of Returns/No.of yrs. Of data

(60+35+20+25+20)%/5=

32.00%

ER(Stock C )=Sum of Returns/No.of yrs. Of data

(20+30+40+35+40)%/5=

33.00%

2. Expected risk/Standard deviation of returns for individual stocks A, B & C

Std. deviation= Sq. rt. Of (sum of (Return-Exp.return^2)/(n-1))

Std. deviation(Stock A)=

ie.(((30%-30%)^2+(20%-30%)^2+(40%-30%)^2+(25%-30%)^2+(35%-30%)^2)/(5-1))^(1/2)

0.0791

Std. deviation(Stock B)=

((60%-32%)^2+(35%-32%)^2+(20%-32%)^2+(25%-32%)^2+(20%-32%)^2)/(5-1))^(1/2)=

0.1681

Std. deviation(Stock C)=

(((20%-33%)^2+(30%-33%)^2+(40%-33%)^2+(35%-33%)^2+(40%-33%)^2)/(5-1))^(1/2)=

0.0837

3…

Covariance between A& B stocks

CoVariance(A,B)=Sum ((Return A-ER(A))*(Return B-ER(B)))/(Sample size-1)

ie.(((30%-30%)*(60%-32%))+((20%-30%)*(35%-32%))+((40%-30%)*(20%-32%))+((25%-30%)*(25%-32%))+((35%-30%)*(20%-32%)))/(5-1)=

-0.00438

Correlation(A,B)=Covariance(A,B)/(Std. Devn.A*Std. devn. B)

ie.-0.00438/(0.0791*0.1681)

-0.33

CoVariance(A,C)=Sum ((Return A-ER(A))*(Return C -ER(C)))/(Sample size-1)

ie.(((30%-30%)*(20%-33%))+((20%-30%)*(30%-33%))+((40%-30%)*(40%-33%))+((25%-30%)*(35%-33%))+((35%-30%)*(40%-33%)))/(5-1)=

0.003125

Correlation(A,C)=Covariance(A,C)/(Std. Devn.A*Std. devn. C)

ie.0.003125/(0.0791*0.0837)=

0.47

4..

While constructing a portfolio that consist of 40 % for stock (A) & 60% for the selected stocks (B or C)---we select a portfolio that results in lesser standard deviation --calculated as follows:

2 assets Portfolio std. deviation= Sq. rt of (Wt A^2*Std. Devn. A^2)+(Wt.B^2*Std. devn.B^2)+(2*Wt.A*Wt.B*Covariance A,B))

Std. devn.(A,B)=((40%^2*0.0791^2)+(60%^2*0.1681^2)+(2*40%*60%*-0.00438))^(1/2)=

9.52%

Std. Devn.(A,C)==((40%^2*0.0791^2)+(60%^2*0.0837^2)+(2*40%*60%*0.003125))^(1/2)=

7.09%

Based on std. deviation, ie. A standard measure for risks, we can select A &C as it has lesser std. deviation

Going by the Covariance metric

A& B inversely related , move in the opp. Direction --as negative covariance--so, If A's returns are negative , atleast B's return will be positive

A& C positively related , move in the same Direction --as positive covariance --- both A & C's returns will be either positive together or negative together.

So,a portfolio manager will select stock B to go with stock A

As for Correlation metric

A& B's returns move in the opposite direction as correlation coefficient is between 0 & -1

A&C 's returns move in the same direction as corr. Coeff. Is between 0 & 1

So, going by correlation coefficient also, a portfolio manager will select stock B to go with stock A

so that risks are mitigated to a certain extent.

So, in the above case, covariance & correlation metrics are giving the same results, that if the investor wants the returns to move in opposite directions, so that all will not be lost even if one performs badly, he will select stock B to go with stock A

But, considering standard deviation,combined portfolio std. devn. Is less for A&C than A&B --- deviation of returns from the mean is more for A&B combination.