Question

Is it possible to get a example of how to complete these steps? It would be greatly appreciated.

Using the following 3 securities calculate: 1. Expected return 2. Variance 3. Standard deviation 4. Correlation between all possible pairs 5. Covariance between all possible pairs Probability Stock A .10 .10 .30 .20 30 5% 5% 12% 6% 18% Stock B 35% 31% 30% 25% 17% Stock C 2% 6% 10% 15% 20% Using the following percentages, calculate the portfolio variance and expected return for each portfolio. Stock C Stock A 70% 40% 25% Portfolio Stock B 30% 60% 30% 45%

Answer 1

1. Expected Return for all 3 Securities

Probability	Stock A	Stock B	Stock C
0.10	5%	35%	2%
0.10	5%	31%	6%
0.30	12%	30%	10%
0.20	6%	25%	15%
0.30	18%	17%	20%

The sum of the probability = 0.10+0.10+0.30+0.20+0.30 = 1

To calculate the Expected return, you have to multiply the probabilities with the return of that probability.

Expected Return of Stock A = (0.10*5%) + (0.10*5%) + (0.30*12%) + (0.20*6%) + (0.30*18%)

Expected Return of Stock A = 11.20%

Expected Return of Stock B = (0.10*35%) + (0.10*31%) + (0.30*30%) + (0.20*25%) + (0.30*17%)

Expected Return of Stock B = 25.70%

Expected Return of Stock C = (0.10*2%) + (0.10*6%) + (0.30*10%) + (0.20*15%) + (0.30*20%)

Expected Return of Stock C = 12.80%

2. & 3. Variance and Standard Deviation of all 3 Securities

Variance of a stock is calculated with respect to it's mean/expected return. The formula is:

Variance of Stock A = {0.1*[(5%-11.20%)^2]} + {0.1*[(5%-11.20%)^2]} + {0.3*[(12%-11.20%)^2]} + {0.2*[(6%-11.20%)^2]} + {0.3*[(18%-11.20%)^2]}

Variance of Stock A = 0.002716

Standard deviation of Stock A = Square root of Variance = 5.21%

Similarly, do this for Stock B and C and we will get the following answer:

Particulars	Stock A	Stock B	Stock C
Variance	0.002716	0.003981	0.003516
Standard Deviation	5.21%	6.31%	5.93%

5. Covariance between all possible pairs

There are 3 possible Covariance pairs possible. The 3 pairs are Cov (A,B), Cov (B,C), Cov(A,C).

The formula for Covariance is given below:

For example, Cov (A,B) = {0.1*[(5%-11.20%)*(35%-25.70%]} + {0.1*[(5%-11.20%)*(31%-25.70%]} + {0.3*[(12%-11.20%)*(30%-25.70%]} + {0.2*[(6%-11.20%)*(25%-25.70%]} + {0.3*[(18%-11.20%)*(17%-25.70%]}

Cov (A,B) = -0.002504

Cov (Stock A, Stock B)	Cov (Stock B, Stock C)	Cov (Stock A, Stock C)
-0.002504	-0.003636	0.002264

4. Correlation between all possible pairs

There are 3 possible Correlation pairs possible. The 3 pairs are Correl (A,B), Correl (B,C), Correl (A,C).

Using Covariance and Standard deviation, the correlation formula can stated as that below:

For example, Correlation between Stock A and Stock B = Cov (A,B)/[(Stock A std dev)*(Stock B Std dev)]

Correl (A,B) = -0.002504/ (0.0521*0.0631)

Correl (A,B) = -0.76

Correl (Stock A, Stock B)	Correl (Stock B, Stock C)	Correl (Stock A, Stock C)
-0.76	-0.97	0.73

Portfolio 1

Stock A = 70%

Stock B = 30%

Expected Portfolio Return = 0.7*(11.2%) + 0.3*(25.7%)

Expected Portfolio Return = 15.55%

Portfolio Variance can be found by the given formula:

Thus, Portfolio 1 Variance = [(0.7*0.0521)^2] + [(0.3*0.0631)^2] + 2*0.7*0.3*-0.002504

Portfolio 1 Variance = 0.0006374 % square

Particulars	Portfolio 1	Portfolio 2	Portfolio 3
Expected Return	15.55%	12.16%	16.27%
Variance (% square)	0.0006374	0.002787	0.0003921