MLC at Boise State Logarithms Activity 6 Week #8

Transcription

1 Logarithms Activity 6 Week #8 In this week s activity, you will continue to look at the relationship between logarithmic functions, exponential functions and rates of return. Today you will use investing as an example. The chart below looks at the value of IBM stock over their history. This chart was taken from YAHOO finance, but there are many free locations on the web where one can obtain stock information and charts such as this. In a contest (imaginary winner wins an imaginary $1,000,000), you are asked to circle three time periods where an owner of IBM stock would earn the greatest return on their investment. The time period must be at least two years in length. In your group, circle the three time periods where you think the greatest return is: Explain your reasoning. Before examining your choices, let s create a greater understanding of how exponential functions will assist in analyzing interest rates. pg. 1

2 You have two friends who bring you information regarding their investments and they try to convince you to join their investment group. Both friends give you the balances of their accounts every ten years for a thirty year period. Friend A: = 0 Amount $ $ $ $ Friend B: = 0 Amount $ $ $ $ To begin the process of making a decision, graph both sets of data on the same graph. Place time on the horizontal axis and amount on the vertical axis then, draw a smooth curve connecting the related points. Based on your graph, which friend s investment group do you want to join? Why? pg. 2

3 All the sudden, you remember you can use Excel and the data to find an exponential trendline. Compute the exponential trendline for Friend A s investment and write it down, explicitly stating the rate of return. Compute the exponential trendline for Friend B s investment and write it down, explicitly stating the rate of return. Now which Friend s investment group do you want to join and why? This is the second time you have answered this question. Even if the answer is the same as above, why is this answer the best answer? Next, let s turn to logarithms as a means to help us visualize the difference in rates. Use the data provided by the two friends and apply the natural log (ln) function to all the values. Three values have been computed for you; compute the rest to complete the tables below. Friend A: Amount Friend B: Amount = 0 ln($200.00)= = 0 ln($ )= ln($518.75)= ln($ )= ln($ )= ln($ )= ln($ )= ln($ )= Very carefully graph these new values on the graph below. The computed values indicate that all of the vertical values will fall between 5.3 and 9.0. Thus, the axis has been labeled to help with your precision. pg. 3

4 $ After carefully plotting the points and connecting the dots, you will notice that this process changes exponential curves into straight lines. Thus the slope represents a relative percentage increase. By relative percent, we mean that the slope can be used to compare rates of return but will not state the exact rate of return. Hence in this case, you should see that Friend A s slope is slightly greater than Friend B s slope. Thus Friend A has a higher rate of return. pg. 4

5 On the right side of the graph, there are some lines, with the bottom line having the value $ This value is the value of that line on the graph. It was computed by finding the inverse of the natural log of 5.2 (which means solve e 5.2 = ). You can compute the value of the 5.6 line similarly by computing e 5.6 = ) Compute the values for all of the lines and describe how the values change. This type of chart is called a semi-log chart as one axis has been changed to a log scale. However, in business this type of chart is often just referred to as a log chart. If you return to the YAHOO chart for IBM, you will notice a chart setting tab at the top of the chart. In the chart setting pull down menu, there is a chart scale menu which includes both log and linear. Linear is the e standard scale as given above. By choosing the Log scale, the following chart is obtained. To determine the time periods where the greatest return is, you just need to look at the slope. The log graph has been copied again and some lines have been added in order to locate the lines of greatest slope. pg. 5

6 While you may find some time frames with a steeper slope, remember the original problem asked for at least a two year time period. Thus, the next chart goes back to the linear scale and circles the three winning time periods. How did you do? While looking at investments, we only used the chart settings menu. If you wish to explore any of the online investment tools, you will find several mathematical models which help investors make good pg. 6

7 decisions. The YAHOO charts have a pull down menu marked technical indicators which includes many of these tools. Many of these tools require math beyond our current level. pg. 7