Question

Problem 01: Calculate individual stock average, standard deviation, variance, portfolio variance

Given are the daily returns for two stocks X and Y for five business days in last week. If you have 1000 dollars to invest and you are considering to invest 600 dollars in stock X and 400 dollars in stock Y. Calculate the following

A. average stock return for individual stocks

B. variance and standard deviations for the individual stocks

C. calculate the covariance and correlation between the two stocks

D. weights of the stocks in the portfolio

E. portfolio expected return

F. portfolio standard deviation and portfolio variance

Day	Stock X (return in %)	Stock Y (return in %)
1	10	- 3
2	6	5
3	8	7
4	1	4
5	-5	2

Answer 1

A.

Average return for Stock X = (10 + 6 + 8 + 1 - 5) / 5 = 4%

Average return for Stock Y = (-3 + 5 + 7 + 4 + 2) / 5 = 3%

B.

Variance of Stock = (stock return on day i - Average return)² / (n - 1)

where i = 1,2,3,4,5

n = total number of days

Variance of Stock X = [ (10 - 4)² + (6 - 4)² + (8 - 4)² + (1 - 4)² + (-5 - 4)² ] / (5 - 1)

= [6² + 2² + 4² + (-3)² + (-9)²] / 4

= 36.5%

Variance of Stock Y = [ (-3 - 3)² + (5 - 3)² + (7 - 3)² + (4 - 3)² + (2 - 3)² ] / (5 - 1)

= [(-6)² + 2² + 4² + (1)² + (-1)²] / 4

= 14.5%

Standard deviation = (Variance) ^1/2

Standard deviation of Stock X = (36.5) ^1/2 = 6.04%

Standard deviation of Stock Y = (14.5) ^1/2 = 3.81%

C.

Covariance = [(Stock X return on day i - Average return) * (Stock Y return on day i - Average return) ] / (n - 1)

= [ (10 - 4) * (-3 - 3) + (6 - 4) * (5 - 3) + (8 - 4) * (7 - 3) + (1 - 4) * (4 - 3) + (-5 - 4) * (2 - 3)] / (5 - 1)

= [ 6 * (-6) + 2 * 2 + 4 * 4 + (-3) * 1 + (-9) * (-1) ] / 4

= [ -36 + 4 + 16 - 3 + 9 ] / 4

= (-10) / 4

= -2.5

Correlation = Covariance / (Standard deviation of X * Standard deviation of Y)

= -2.5 / (6.04 * 3.81)

= - 0.1086

D. Weights of stocks in the portfolio

Total Investment = $ 1,000

Investment in Stock X = $ 600

Weight of Stock X in portfolio = 600 / 1,000 = 0.6 or 60%

Investment in Stock X = $ 400

Weight of Stock X in portfolio = 400 / 1,000 = 0.4 or 40%

E.

Expected return on portfolio = (Weight of stock X * Average return of stock X) + (Weight of stock Y * Average return of stock Y)

= 0.6 * 4 + 0.4 * 3

= 3.6%