This is the first of a number of post teaching statistics and probability with stock market applications.
Question One: A broker calls his client in 2014 and gives
four possible dates for an appointment to go over possible transactions. The client purchases shares of QQQ (a high
tech ETF) right after the appointment on one of the four equally likely
dates. The potential purchase dates are
presented below.
Price of QQQ on Four
Dates


Date

Probability of Purchase
Date

Purchase Price QQQ

20May14

0.25

88.0

7Jul14

0.25

95.1

7Aug14

0.25

94.2

10Sep14

0.25

100.1

What is the expected purchase price if each conference date
is equally likely?
What is the variance of the purchase price if each
conference date is equally likely?
What is the number of potential shares purchased, at each
date if the person spends $25,000?
What is the range of potential shares purchased over the
four dates?
Methodological Note: I use a spreadsheet to calculate the expected
value and the variance of share price for the four equally likely outcomes.
The expected value of the share price is the weighted
average of the share price where the weights are the likelihood of each outcome. (in this case each outcome has a probability
of 0.25.)
I calculate the variance two ways. The first method is directly from the
definition of the variance.
Var(QQQ) =E((QQQ)E(QQQ))^{2}
The second method involves the computational formula.
Var(QQQ) = E(QQQ)^{2 }– E(QQQ^{2})
Tabulations: The calculations of expected value and the
variance are laid out in the tables below.
E(QQQ) is the dot product of the probability vector and the price of QQQ
vector. E(QQQ^{2}) is the dot
product of the probability vector and the price of QQQ^{2 }vector.
Tabulation of E(QQQ),
E(QQQ2)


Date

Col A

Col B

Col C

Col D

Col E

Date

Probability Of Purchase
Date

Purchase Price QQQ

Purchase Price QQQ
Squared

E(QQQ)

(QQQE(QQQ))2

20May13

0.25

88.0

7744.0

94.4

40.32

7Jul13

0.25

95.1

9044.0

94.4

0.56

7Aug13

0.25

94.2

8873.6

94.4

0.02

10Sep13

0.25

100.1

10020.0

94.4

33.06

Tabulation of Variance
Explained


E(QQQ)

94.4

sumproduct of col A &
Col B

E(QQQ2)

8920.4

sumproduct of col A &
Col C

E(QQQ2) E(QQQ)2

18.5

E(QQQ2)E(QQQ)2

E(QQQE(QQQ))2

18.5

sumproduct of Col A &
Col E

It is nice to see that the formula for the definition of
variance and the computational formula give the same result.
The number of shares purchased at a particular price with
$25,000 is simply $25,000/Price per share.
The results for the four share prices are presented below.
Information on Quantity Purchased


Date

Purchase Price QQQ

Quantity purchased $25,000/Price

20May13

88.0

284.1

7Jul13

95.1

262.9

7Aug13

94.2

265.4

10Sep13

100.1

249.8

Min

249.8


Max

284.1


Range

34.3

I am planning a lot more problems that use statistics and
probability to solve financial problems over the next few weeks.
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