answer to all parts of question 3 please don\'t answer otherwise Fies R Faes x F

Question

answer to all parts of question 3
please don't answer otherwise

Fies R Faes x Files x M Files x exercise set x C Chea ownloads exercise set 4 spring 2017 pd $10,000 15 0000 02 Should you purchase the extended warranty? 3. Probability ideas, including Joint probability distributiones, play a major role in financial activities. For ple, portfolio managers must allocate funds among competing assets Suppose there are two assets Microsoft (M) stock and StarbuFks (S) stock. For Microsoft stock, the following returns are possible -10%, 10%, 20%, and 30% Fot Starbucks slock, the following retums are possible: 0% and 20%. The joint probability distribution of the returns of the two assets is -10% 10% 20% 30% stock (s) 20% What is the expected return on Microsoft stock? b What is the expected return on Starbucks sock? c What is the standard deviat or volatility)of the returns on Microsoft stock' d What is the standard deviation (or volatility) of the returns on Startucks stock e Are the returns on Microsoft stock and the retams on Sunbecks stock slalistically independert? 4. A believer in the "random walk theory of stock markets thinks the value of an index of stock prices has a

Explanation / Answer

Solution

Back-up Theory

If X and Y are two discrete random variables having joint pmf, pxy(x, y), then

Marginal pmf of X = pX(X) sum of p(x, y) over all possible values of Y …………………..(1)

Marginal pmf of Y = py(Y) sum of p(x, y) over all possible values of X …………………..(2)

Mean of X = E(X) = sum of {x. pX(X)} over all possible values of X ……………………..(3)

Mean of Y = E(Y) = sum of {y. py(Y)} over all possible values of Y ……………………..(4)

E(X2) = sum of {x2. pX(X)} over all possible values of X …………………..……………..(5)

E(Y2) = sum of {y2. py(Y)} over all possible values of Y …………………………………(6)

V(X) = E(X2) – { E(X)}2 . ………………………………………………………………….(7)

V(Y) = E(Y2) – { E(Y)}2 . ………………………………………………………………….(8)

Standard Deviation of X = SD(X) = sq.rt of V(X) …………………………………………(9)

Standard Deviation of Y = SD(Y) = sq.rt of V(Y) …………………………………………(10)

If X and Y are independent, then joint probability = product of marginal probabilities ……(11)

Now, to work out solution,

Let X and Y represent the Microsoft and Starbucks stocks respectively.

The given joint probability table is as follows:

Y

X

Total

- 10%

10%

20%

30%

0%

0.05

0.20

0.10

0.05

0.40

20%

0.10

0.15

0.20

0.15

0.60

Total

0.15

0.35

0.30

0.20

1.00

[vide (1) under Back-up Theory], marginal distribution of X is:

X

- 10%

10%

20%

30%

Total

p(x)

0.15

0.35

0.30

0.20

1.00

[vide (2) under Back-up Theory], marginal distribution of Y is:

Y

0%

20%

Total

p(y)

0.40

0.60

1.00

Part (a)

Expected return of Microsoft = E(X)

[vide (3) under Back-up Theory],

E(X) = (-10% x 0.15) + (10% x 0.35) + (20% x 0.30) + (30% x 0.20) = 14% ANSWER

Part (b)

Expected return of Starbucks = E(Y)

[vide (4) under Back-up Theory],

E(Y) = (0% x 0.4) + (20% x 0.6) = 12% ANSWER

Part (c)

[vide (5) under Back-up Theory],

E(X2) = {(-10%)2 x 0.15} + {(10% )2x 0.35} + {(20%)2 x 0.30} + {(30%)2 x 0.20} = 350

[vide (7) under Back-up Theory], V(X) = 350 - 142 = 254

Standard Deviation of Microsoft = SD(x) = sqrt(350) [vide (9) under Back-up Theory],

= 18.71ANSWER

Part (d)

[vide (6) under Back-up Theory],

E(Y2) = {(0%)2 x 0.4} + {(20% )2x 0.6} = 240

[vide (8) under Back-up Theory], V(Y) = 240 - 122 = 96

Standard Deviation of Starbucks = SD(Y) = sqrt(96) [vide (10) under Back-up Theory],

= 9.80 ANSWER

Part (e)

P(X = -10%, Y = 0%) = 0.05. P(X = -10%) = 0.15 and P(Y = 0%) = 0.4

0.15 x 0.4 = 0.06 0.05. So, [vide (11) under Back-up Theory], X and Y are not independent.

i.e., Microsoft and Starbucks stocks are NOT statistically independent. ANSWER

Y

X

Total

- 10%

10%

20%

30%

0%

0.05

0.20

0.10

0.05

0.40

20%

0.10

0.15

0.20

0.15

0.60

Total

0.15

0.35

0.30

0.20

1.00

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