Question

Here are the returns on two stocks.

		Digital Cheese			Executive Fruit
January			+18			+6
February			−3			+2
March			+5			+4
April			+7			+16
May			−4			+3
June			+3			+6
July			−2			−3
August			−8			−2

Required:

a-1. Calculate the variance and standard deviation of each stock.

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

b. Now calculate the returns in each month of a portfolio that invests an equal amount each month in the two stocks.

c. Is the variance more or less than halfway between the variance of the two individual stocks?

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

b. Now calculate the returns in each month of a portfolio that invests an equal amount each month in the two stocks.

c. Is the variance more or less than halfway between the variance of the two individual stocks?

Answer 1

(1a)

	Digital Cheese (x)	Executive Fruits (y)	X - x̅	Y - ȳ	(X - x̅)2	(Y - ȳ)2
January	18	6	16	2	256	4
February	-3	2	-5	-2	25	4
March	5	4	3	0	9	0
April	7	16	5	12	25	144
May	-4	3	-6	-1	36	1
June	3	6	1	2	1	4
July	-2	-3	-4	-7	16	49
August	-8	-2	-10	-6	100	36
	16	32			468	242

x̅ = ΣX/n = 16/8 = 2

ȳ = Σȳ/n = 32/8 = 4

Total Variance of Digital Cheese = Σ(X - x̅)²/n = 468/8 = 58.5

Total Variance of Executive Fruits = Σ(Y - ȳ)²/n = 242/8 = 30.25

Standard deviation of Digital Cheese = {Σ(X - x̅)²/n}^1/2 = (468/8)^1/2 = (58.5)^1/2 = 7.65

Standard deviation of Executive Fruits = {Σ(Y - ȳ)²/n}^1/2 = (242/8)^1/2 = (30.25)^1/2 = = 5.50

(2a)

Since the Standard deviation of Digital Cheese is higher than that of Executive Fruits i.e. (7.65>5.50), therefore Digital Cheese is higher than Executive Fruits.

(2b)

Calculation of return of Portfolio consisting equal amount of stock in each month:

	Digital Cheese (x)	Executive Fruits (y)	Portfolio (50% of X and 50% of Y) (Z)	Z - Z̄̄	(Z - Z̄̄)2
January	18	6	12	9	81
February	-3	2	-0.5	-3.5	12.25
March	5	4	4.5	1.5	2.25
April	7	16	11.5	8.5	72.25
May	-4	3	-0.5	-3.5	12.25
June	3	6	4.5	1.5	2.25
July	-2	-3	-2.5	-5.5	30.25
August	-8	-2	-5	-8	64
	16	32	24		276.5

(2c)

Variance of the Portfolio = Σ(Z - Z̄̄)²/n = 276.5/8 = 34.56

Variance of the Digital Cheese = 58.5

Variance of the Executive Fruits = 30.25

The Variance of the Portfolio is less than that of Digital Cheese and more than that of Executive Fruits.