The company EquiCola’s stock price fluctuates in the following way from one day
ID: 3357911 • Letter: T
Question
The company EquiCola’s stock price fluctuates in the following way from one day to the next.
With probability 0.5, the price remains the same the next day. With probability 0.3, the price increases by one dollar. With probability 0.2, the price decreases by one dollar.
What is the expected change in value for the stock from one day to the next?
What is the expected change in value for the stock over the course of a week (7 days)? What principle allows you to make this conclusion?
What is the probability that the stock does not have a single decline over the course of the whole week? (To two decimal places.)
What is the probability that the stock declines no more than once over the course of the week? (To two decimal places.)
What is the probability that the price changes every day for the full week, and ends a dollar higher than it started? (To two significant digits.)
Explanation / Answer
What is the expected change in value for the stock from one day to the next?
The expected change in value for the stock from one day to the next = 0.5 * 0 + 0.3 * 1 - 0.2 * 1 = 0.1 dollar
What is the expected change in value for the stock over the course of a week (7 days)?
The expected change in value for the stock over the course of a week (7 days) =
= Number of days * Expected change in value for the stock from one day = 7 * 0.1 = 0.7 dollar
Principle used - The expected value of the sum of any finite number of random variables is the sum of the expected values of the individual random variables.
What is the probability that the stock does not have a single decline over the course of the whole week?
The stock declines with with probability 0.2. So, the stock does not decline is 1 - 0.2 = 0.8
So, he probability that the stock does not have a single decline over the course of the whole week = 0.87 = 0.21
What is the probability that the stock declines no more than once over the course of the week?
By binomial theorem, the probability that the stock declines no more than once over the course of the week (probability that stock declines for 0 or 1 day)
= 0.87 + 7C1 * 0.86 * (1 - 0.8)
= 0.21 + 7 * 0.2621 * 0.2 = 0.58
What is the probability that the price changes every day for the full week, and ends a dollar higher than it started?
If the price changes everyday (increases or decreases), and ends a dollar higher means that the stock increases for 4 days and decreases for 3 days.
So, by binomial distribution formula, probability that the price changes every day for the full week, and ends a dollar higher than it started is
7C4 * 0.34 * 0.23 = 35 * 0.0081 * 0.008 = 0.0022