Question

Here are the returns on two stocks.

		Digital Cheese			Executive Fruit
January			+19			+8
February			−3			+1
March			+5			+4
April			+7			+17
May			−4			+2
June			+3			+4
July			−2			−3
August			−8			−2

a-1. Calculate the variance and standard deviation of each stock. (Do not round intermediate calculations. Round your answers to 2 decimal places.)

a-2. Which stock is the riskier if held on its own?

• Digital Cheese

• Executive Fruit

b. Now calculate the returns in each month of a portfolio that invests an equal amount each month in the two stocks. (Negative amounts should be indicated by a minus sign. Do not round intermediate calculations. Round your answers to 2 decimal places.)

c. Is the standard deviation more or less than half way between the variance of the two individual stocks?

Answer 1

a-1

	Digital Cheese	Executive Fruit
January	19	8
February	-3	1
March	5	4
April	7	17
May	-4	2
June	3	4
July	-2	-3
August	-8	-2

Variance and Standard Deviation of each stock

Standard deviation of a sample (sample size n) can be calculated using the below formula

In this example n = 8

Variance is the Square of Standard Deviation

Variance = Standard_Deviation²

For Digital Cheese

E[R_D] = μ_D = (19+(-3)+5+7+(-4)+3+(-2)+(-8))/8 = 2.125

so Variance of Digital Cheese is:

σ_D² = [(19-2.125)²+(-3-2.125)²+(5-2.125)²+(7-2.125)²+(-4-2.125)²+(3-2.125)²+(-2-2.125)²+(-8-2.125)²]/(8-1)

σ_D² = 500.875/7 = 71.55357

so standard deviation of Digital Cheese = σ_D = (71.55357)^1/2 = 8.458934%

For Executive Fruit

E[R_E] = μ_E = (8+1+4+17+2+4+(-3)+(-2))/8 = 3.875%

So, From the above formula, Variance of Executive Fruit is:

σ_E² = [(8-3.875)²+(1-3.875)²+(4-3.875)²+(17-3.875)²+(2-3.875)²+(4-3.875)²+(-3-3.875)²+(2-3.875)²]/(8-1)

σ_E² = 282.875/7 = 40.41071

so standard deviation of Executive Fruit = σ_E = (40.41071)^1/2 = 6.356942%

Answer:

	Variance	Standard Deviation
Digital Cheese	71.55	8.46
Executive Fruit	40.41	6.36

a-2 - Since the standard deviation of Digital Cheese is greater than that of Executive Fruit. So, Digital Cheese is more riskier if held in its own.

Answer -> Digital Cheese

b. Now, portfolio contains equal weights of Digital Cheese and Executive Fruit i.e. W_D = 0.5; W_E = 0.5

E[R_D] = μ_D = 2.125%; E[R_E] = μ_E = 3.875%

Portfolio Return is given by:

E[R_p] = W_D*E(R_D) + W_E* E(R_E) = 0.5*2.125+0.5*3.875 = 3%

Answer -> Return on Portfolio = 3

c. Standard Deviation of the portfolio of stocks

We need to find the covariance between these two stocks first

The formula to calculate the variance between two variables X and Y (sample size - n) is given by the below formula:

Cov(D,E) = [(19-2.125)*(8-3.875) +(-3-2.125)*(1-3.875)+(5-2.125)*(4-3.875)+(7-2.125)*(17-3.875)+(-4-2.125)*(2-3.875)+(3-2.125)*(4-3.875)+(-2-2.125)*(-3-3.875)+(-8-2.125)*(-2-3.875)]/7 = 35.44643

Formula for calculating variance of a portfolio is given by

σ_p² = W_D²* σ²_D + W_E²* σ²_E + 2*Cov(D,E)*W_D*W_E

σ_p² = (0.5²*71.55357) +(0.5²*40.41071)+(2*35.44643*0.5*0.5) = 17.888392857+ 10.10267857+ 17.7232142 = 45.7142857

Therefore, Standard Deviation of the portfolio is σ_p = 45.7142857^1/2 = 6.761234038 ~ 6.76%

The average Standard deviation of the two stocks is (σ_D + σ_E)/2 = (8.458934414+9.356942)/2 = 7.407938313

Answer -> Hence the standard deviation of the portfolio (σ_p=6.76) is less than half way between the variance of the two individual stocks which is around 7.41