Question

You work for a large investment management firm. The analysts with your firm have made the following forecasts for the returns of stock A and stock B:

	Probability	Stock A	Stock B
very very weak	10.00%	65.00%	-65.00%
Weak	15.00%	30.00%	-20.00%
Moderate	30.00%	25.00%	35.00%
strongly Moderate	20.00%	15.00%	40.00%
Strong	15.00%	-20.00%	45.00%
Very Very Strong	10.00%	-55.00%	65.00%
	100.0%

Answer the following questions:

1. Calculate the expected returns, variance and the standard deviations for stock A and B.
2. What is the covariance of returns for Stock A and Stock B? What is the correlation coefficient for Stock A and Stock B?
3. What is the expected return and standard deviation of a portfolio where 40% of the portfolio is in stock A and 60% of the portfolio is in stock B?
4. Create a table that has the expected return and standard deviation for different weights in each stock. This can be done using an excel data table. Start with 100% in A and zero in B, and increments of 10%, complete the table. The last row, will have 0% in A and 100% in B. Then chart (or graph) your results.

weight in B	weight in A	Portfolio standard deviation	portfolio expected return
		Link to the answer for 30% and 70%, let the weights change…	Link to the answer for 30% and 70%, let the weights change…
0%	100%
10%	90%
20%	80%
30%	70%
40%	60%
50%	50%
60%	40%
70%	30%
80%	20%
90%	10%
100%	0%

Note: All calculations should be rounded to one decimal place if you are using percentages, if you are using decimals then the answer should be rounded to three decimal places.

Answer 1

a) Below is the calculation of Expected return and standard deviation for each asset:

Asset A

• Probability Weighted Return = Return x Probability
  • Very Very Weak Probability Weighted Return = 0.10 x 65% = 6.5%
  • Weak Probability Weighted Return = 0.15x30%=4.5%
  • Moderate Probability Weighted Return = 0.3x25%=7.5%
  • Strongly Moderate Probability Weighted Return = 0.2x15%=3%
  • Strong Probability Weighted Return = 0.15x-20%=-3%
  • Very Very Strong Probability Weighted Return = 0.1x-55%=-5.5%
• Expected Return = sum of Probability Weighted Return      
  • Expected Return =6.5%+4.5%+7.5%+3%-3%-5.5% = 13.000%
• To calculate variance and standard deviation we need to calculate the probability weighted squared deviations or P(X - Expected return of A)^2
  • Very Very Weak P(X - Expected return of A)² = 0.1(0.65-0.13)²=2.704%
  • Weak P(X - Expected return of A)² = 0.15(0.3-0.13)²=0.4335%
  • Moderate P(X - Expected return of A)² = 0.3(0.25-0.13)²=0.432%
  • Strongly Moderate P(X - Expected return of A)² = 0.2(0.15-0.13)²=0.00799999999999999%
  • Strong P(X - Expected return of A)² = 0.15(-0.2-0.13)²=1.6335%
  • Very Very Strong 0.1(-0.55-0.13)²=4.624%
• Variance = Sum of Probability weighted squared deviations
  • Variance = 2.704+ 0.4335+ 0.432 + 0.00799999999999999+1.6335+4.624
    • Variance = 8.802%
• Standard deviation = Square root of variance          
  • Standard deviation = Square root of 8.802%
  • Standard deviation = 29.667%

Similarly all calculations of Asset B are also shown in the above screenshot

Below is the summary table:

Asset	Expected return	Standard deviation
A	13.000%	29.667%
B	22.250%	40.340%

b) Covariance and correlation are calcualted as follows

Covariance between A and B is -9.12%

Correlation between A and B is -0.76

c)

d)       

Weight B	Weight A	Expected Return	Standard Deviation
0%	100%	13.000%	8.8015000%
10%	90%	13.925%	5.6547569%
20%	80%	14.850%	3.3733240%
30%	70%	15.775%	1.9572014%
40%	60%	16.700%	1.4063891%
50%	50%	17.625%	1.7208871%
60%	40%	18.550%	2.9006954%
70%	30%	19.475%	4.9458139%
80%	20%	20.400%	7.8562427%
90%	10%	21.325%	11.6319819%
100%	0%	22.250%	16.2730313%