Problem 3 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

A Boston Globe article on January 1, 1997, said that the best stock of 1996 was Information Analysis, Incorporated, which closed the year at a price of \(\$ 63\) per share, an increase of \(1525 \%\) during the year. The worst stock of 1996 was Mobilemedia Corporation, which closed the year at \(\$ 7 / 16\) per share, a decrease of \(97.6 \%\). What was the price of each of these stocks at the beginning of the year?

Step-by-Step Solution

Answer

The original price of each of the stocks at the beginning of the year were: Information Analysis Inc. at approximately $3.89 and Mobilemedia Corporation at approximately $18.75

Step 1: Calculate the original price of Information Analysis, Incorporated

Given the final price of \( \$ 63 \) per share and an increase of \( 1525 \% \), to find the original price of the stock at the beginning of the year, one could use the formula \[ Original\ Price = \frac{{Final\ Price}}{(1 + \frac{{Percentage\ Increase}}{100})} \] Replace the 'Final Price' with \$ 63 and 'Percentage Increase' with 1525 then compute.

Step 2: Calculate the original price of Mobilemedia Corporation

Given the final price of \( \$ 7 / 16 \) per share and a decrease of \( 97.6 \% \), to find the original price of the stock at the beginning of the year the formula \[ Original\ Price = \frac{{Final\ Price}}{(1 - \frac{{Percentage\ Decrease}}{100})} \] can be used. Replace 'Final Price' with \$ 7/16 and 'Percentage Decrease' with 97.6 then compute.

Key Concepts

Stock Price Analysis

Stock price analysis involves evaluating the performance of a stock over a period of time.
This can help investors decide whether to buy, sell, or hold a stock.
In the exercise we discussed, we looked at the best and worst performing stocks of 1996. To analyze these stocks, it is crucial to understand their end-of-year prices, the percentage increase or decrease they experienced, and their prices at the beginning of the year.

By understanding these key data points, investors can gauge the volatility of these stocks and the risks involved.
For instance, Information Analysis, Inc. showed a massive improvement in its stock price; however, Mobilemedia Corporation suffered a significant fall.
Such information is vital for making informed investment decisions.

Mathematical Formulas

Mathematical formulas are tools used for calculating various financial metrics.
They allow us to derive valuable insights from raw data.
In stock price analysis, formulas help us determine the original stock price from the price at the end of the year and the percentage change over the year.

One of the fundamental formulas used here is:

• For a price increase: \[Original\ Price = \frac{{Final\ Price}}{(1 + \frac{{Percentage\ Increase}}{100})}\]
• For a price decrease:\[Original\ Price = \frac{{Final\ Price}}{(1 - \frac{{Percentage\ Decrease}}{100})}\]

Utilizing these formulas allows investors and analysts to accurately track back to the original costs to understand irregular price movements or substantial changes.

Percent Increase

Percent increase is a statistical measure that calculates the extent of increase in the value of an asset or item.
It reveals how much a stock or asset has grown in comparison to its initial value.
In our given exercise, Information Analysis, Inc. experienced an impressive increase of 1525%, signaling a remarkable return on investment for current stockholders.

This percentage is calculated by taking the difference between the final and the original prices, dividing it by the original price, and then multiplying by 100 to convert it to a percentage.
The formula is as follows:

• \[Percent\ Increase = \left(\frac{{Final\ Price - Original\ Price}}{{Original\ Price}}\right) \times 100\]

Understanding percent increase can help investors to quickly assess how well a stock has performed over a specific period and compare different investment opportunities easily.