## Sunday, November 22, 2015

### Expected value and variance of share purchases on four equally likely dates

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 20-May-14 0.25 88.0 7-Jul-14 0.25 95.1 7-Aug-14 0.25 94.2 10-Sep-14 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(QQQ2)

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(QQQ2) is the dot product of the probability vector and the price of QQQ2 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) (QQQ-E(QQQ))2 20-May-13 0.25 88.0 7744.0 94.4 40.32 7-Jul-13 0.25 95.1 9044.0 94.4 0.56 7-Aug-13 0.25 94.2 8873.6 94.4 0.02 10-Sep-13 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(QQQ-E(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 20-May-13 88.0 284.1 7-Jul-13 95.1 262.9 7-Aug-13 94.2 265.4 10-Sep-13 100.1 249.8 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.