Profit and risk when there are four random purchase dates and four random sale dates
Question: In 2013 a person buys QQQ the high tech ETF) on one of four randomly selected dates determined by when the broker arranges a meeting. I
The person who bought the QQQ shares in 2014 got fired in 2015. As soon as the person was fired he realized he needed cash so he called his broker and said “SELL QQQ” The firing is a random event independent of the market and out of control of the person, which occurred on one of four dates.
The four potential purchase and four potential sales dates for the QQQ transactions are presented below.
| Information on Potential Purchases and Sales of QQQ | ||||
| Potential Purchase Date | Purchase Price QQQ | Quantity purchased $25,000/Price | Potential Sale Date | Sale Price |
| 20-May-14 | 88.0 | 284.1 | 5-Jan-15 | 101.4 |
| 7-Jul-14 | 95.1 | 262.9 | 8-Aug-15 | 110.5 |
| 7-Aug-14 | 94.2 | 265.4 | 24-Aug-15 | 98.5 |
| 10-Sep-14 | 100.1 | 249.8 | 5-Nov-15 | 114.7 |
The person spends $25,000 on the purchase of QQQ in 2014 and sells all shares in 2015.
Assume no dividends are paid.
What are all possible profit outcomes from the purchase and sale of the QQQ securities?
What is the expected profit?
What is the variance of profit?
Analysis: The number of share purchased is $25,000 divided by the purchase price; hence the purchase price determines the number of shares purchased.
| Tabulation of Number of Shares Purchased | ||
| Potential Purchase Date | Purchase Price QQQ | Number of shares purchased |
| 20-May-14 | 88.0 | 284.1 |
| 7-Jul-14 | 95.1 | 262.9 |
| 7-Aug-14 | 94.2 | 265.4 |
| 10-Sep-14 | 100.1 | 249.8 |
Revenue received after the sale is price at time of sale times the number of shares owned.
Profit after the sale is revenue minus the $25,000 initial investment.
There are four possible purchase dates and four possible sale dates. The purchase and sale dates are independent so there are a total of 16 possible equally likely combinations of sale and purchase dates. The probability of each purchase/sale combination is 0.0625 (0.25*0.25).
The profit calculation for the 16 purchase-sale combinations is presented in the table below.
| Potential Profit Calculation for Four Purchase Dates and Four Sale Dates | ||||||
| Comb # | Probability | Purchase Date | Sale Date | Number of Shares Owned | Sale Price | Profit |
| 1 | 0.0625 | 20-May-14 | 5-Jan-15 | 284.1 | 101.4 | $3,807 |
| 2 | 0.0625 | 20-May-14 | 8-Aug-15 | 284.1 | 100.5 | $3,552 |
| 3 | 0.0625 | 20-May-14 | 24-Aug-15 | 284.1 | 98.5 | $2,984 |
| 4 | 0.0625 | 20-May-14 | 5-Nov-15 | 284.1 | 114.7 | $7,586 |
| 5 | 0.0625 | 7-Jul-14 | 5-Jan-15 | 262.9 | 101.4 | $1,656 |
| 6 | 0.0625 | 7-Jul-14 | 8-Aug-15 | 262.9 | 100.5 | $1,420 |
| 7 | 0.0625 | 7-Jul-14 | 24-Aug-15 | 262.9 | 98.5 | $894 |
| 8 | 0.0625 | 7-Jul-14 | 5-Nov-15 | 262.9 | 114.7 | $5,152 |
| 9 | 0.0625 | 7-Aug-14 | 5-Jan-15 | 265.4 | 101.4 | $1,911 |
| 10 | 0.0625 | 7-Aug-14 | 8-Aug-15 | 265.4 | 100.5 | $1,672 |
| 11 | 0.0625 | 7-Aug-14 | 24-Aug-15 | 265.4 | 98.5 | $1,141 |
| 12 | 0.0625 | 7-Aug-14 | 5-Nov-15 | 265.4 | 114.7 | $5,441 |
| 13 | 0.0625 | 10-Sep-14 | 5-Jan-15 | 249.8 | 101.4 | $325 |
| 14 | 0.0625 | 10-Sep-14 | 8-Aug-15 | 249.8 | 100.5 | $100 |
| 15 | 0.0625 | 10-Sep-14 | 24-Aug-15 | 249.8 | 98.5 | -$400 |
| 16 | 0.0625 | 10-Sep-14 | 5-Nov-15 | 249.8 | 114.7 | $3,646 |
| Min | -$400 | |||||
| Max | $7,586 | |||||
| Range | $7,986 | |||||
The minimum profit is -$400. The maximum profit is $7,985.
The expected profit is obtained by taking the dot product or the sumproduct of the probability vector with the profit vector. The variance was obtained from the computational formula.
Var (Profit) = E(profit2) – E(Profit)2
For a discussion of these calculations see the previous post.
http://dailymathproblem.blogspot.com/2015/11/expected-value-and-variance-of-share.html
The expected value and variance or profit from the purchase of QQQ on one of four dates in 2014 and the sale of QQQ on one of four dates in 2015 are presented below.
| Expected Profit and Variance of Profit Calculations | |
| E(PROFIT) | 2555.4 |
| E(PROFIT2) | 11036765.0 |
| E(PROFIT2)-E(PROFIT)2 | 4506556.2 |
| E(PROFIT-E(PROFIT))2 | 4506556.2 |
Financial Discussion:
The purchaser of QQQ or any stock that buys randomly and is forced to sell because of random events unrelated to the market bears substantial risk compared to an investor with enough liquid assets who will not need to sell in an emergency. Investors would be wise to consider the level of the market and their ability to hold through downturns prior to selling. The experts say that stock market returns beat returns on other securities over the long haul. But this investor was only able to hold for a year.
Outcomes could have been worse. The broker put the investor in QQQ a relatively diversified ETF that focuses on tech stocks. Had the broker put his client in one particular stock (say IBM) and the investor was forced to sell he would have realized a large loss.