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CS110: Introduction to Computer Science: Lab8

Economics of Arithmetic and Geometric Growth Rates What is the difference between simple and compound interest and does it really matter?

There are two main methods for computing interest. Do you know the difference between them? Do you know what difference it makes in a savings account after a period of some years? How does this impact your analysis and investments in Stock Market?

Quantitative concepts and skills Arithmetic Growth Geometric Growth Forward Modeling Function, linear Function, exponential Adapted by Fred Annexstein Univ Cincinnati Graph, XY (scatter) Prepared for SSAC by Gary Franchy – Davenport University © The Washington Center for Improving the Quality of Undergraduate Education. All rights reserved. 2005 1

Overview of Module

Simple interest

is an example of

arithmetic growth

where the amount of interest generated each term is constant; it is based on

only

the starting amount. Year to year there is a

constant difference

in the value of the savings account, and so the successive values track a

linear function

Compound interest

is an example of

geometric growth

where the amount of interest generated each term increases because it is based on

both

the starting amount and the previously earned interest. Year to year there is a

constant ratio

of the values in the savings account, and so the successive values track an exponential function. Accordingly, compound interest is commonly said to exhibit

exponential growth

Slides 3-4 ask you to set up your worksheet and format the cells.

Slides 5-9 have you computing simple and compound interest for a set period of time and interest rate. You will perform the calculation, graph the increasing savings accounts, and draw trendlines through the plotted functions for the linearly increasing and geometrically increasing values. Slides 10-12 ask you to calculate the difference between the two types of accounts using a variety of interest rates.

Slides 13 has parts of assignment to hand in. Slides 14-16 asks to consider the stock market and has remainder of assignment to hand in.

Question 1 What is the difference in results between savings accounts that use simple and compound interest when you invest $100,000 at 8% for 25 years?

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 B Present Value = Interest Rate = C 100000 8% D E Year 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 F Simple G Compound

Type in the “%” symbol when entering percents; otherwise you must enter the decimal form.

One way to answer the question with a spreadsheet is to lay it out year by year.

Recreate this spreadsheet = Cell with a number in it = Cell with a formula in it 3

Question 1

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 B Present Value = Interest Rate = C $100,000 8% D E Year 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 F Simple G Compound Format these cells as currency rounded to the nearest dollar.

To format cells:

1. Select the cells.

2. Right-click the mouse.

Select “Format Cells”.

Choose “Number” tab.

Choose “Currency”.

Adjust “Decimal Places” to the desired number of 7.

places.

Select “OK”.

Question 1

2 3 9 10 11 12 4 5 6 7 8 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 B Present Value = Interest Rate = C $100,000 8% D E Year 1 7 8 9 10 2 3 4 5 6 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 F Simple G Compound Compute the results for

future value

for each year in the column labeled “Simple”. The Simple Interest Formula:



 1  Where

= Future Value ($)

= Present Value ($)

= Interest Rate

= Time (Years)

 5

Question 1

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 B Present Value = C Interest Rate = $100,000 8% D E Year 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 F Simple $108,000 $116,000 $124,000 $132,000 $140,000 $148,000 $156,000 $164,000 $172,000 $180,000 $188,000 $196,000 $204,000 $212,000 $220,000 $228,000 $236,000 $244,000 $252,000 $260,000 $268,000 $276,000 $284,000 $292,000 $300,000 G Compound Compute the results for

future value

for each year in the column labeled “Compound”.

The Compound Interest Formula:



 1 

r t

 Where

= Future Value ($)

= Present Value ($)

= Interest Rate

= Time (Years) 6

Question 1

2 3 19 20 21 22 23 24 25 26 27 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 B Present Value = C Interest Rate = $100,000 8% D E Year 1 17 18 19 20 21 22 23 24 25 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 F Simple $108,000 $116,000 $124,000 $132,000 $140,000 $148,000 $156,000 $164,000 $172,000 $180,000 $188,000 $196,000 $204,000 $212,000 $220,000 $228,000 $236,000 $244,000 $252,000 $260,000 $268,000 $276,000 $284,000 $292,000 $300,000 G Compound $108,000 $116,640 $125,971 $136,049 $146,933 $158,687 $171,382 $185,093 $199,900 $215,892 $233,164 $251,817 $271,962 $293,719 $317,217 $342,594 $370,002 $399,602 $431,570 $466,096 $503,383 $543,654 $587,146 $634,118 $684,848 Now create a single scatter graph where future value is on the

-axis and time is on the

-axis. Plot the values for both accounts on the same graph.

To draw a graph, you may either click on the chart wizard button or use “Insert



the menu.

Chart” from

Question 1

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 B Present Value = Interest Rate = C $100,000 8% D E Year 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 F Simple $108,000 $116,000 $124,000 $132,000 $140,000 $148,000 $156,000 $164,000 $172,000 $180,000 $188,000 $196,000 $204,000 $212,000 $220,000 $228,000 $236,000 $244,000 $252,000 $260,000 $268,000 $276,000 $284,000 $292,000 $300,000 G Compound $108,000 $116,640 $125,971 $136,049 $146,933 $158,687 $171,382 $185,093 $199,900 $215,892 $233,164 $251,817 $271,962 $293,719 $317,217 $342,594 $370,002 $399,602 $431,570 $466,096 $503,383 $543,654 $587,146 $634,118 $684,848 To add a trendline: 1.

Place mouse over any data point of 2.

the desired function.

Right-click the mouse.

Select “Add Trendline”.

Choose the function that resembles the pattern Add trendlines to each graph. $800,000 $700,000 $600,000 $500,000 $400,000 $300,000 $200,000 $100,000 $0 0 5 10

Years

15 20 8 25

Question 1

Years

15 20 25 The difference between them is almost $400,000.

The difference between them is exactly $384,848.

Question 2

What is the difference in results between savings accounts that use simple and compound interest when you invest $100,000 at

12%

for 25 years?

2 3 18 19 20 21 22 23 24 25 26 27 4 5 6 7 8 9 10 11 12 13 14 15 16 17 B Present Value = C 100000 D Interest Rate = 12% E Year 1 16 17 18 19 20 21 22 23 24 25 2 3 4 5 6 7 8 9 10 11 12 13 14 15 F Simple $112,000 $124,000 $136,000 $148,000 $160,000 $172,000 $184,000 $196,000 $208,000 $220,000 $232,000 $244,000 $256,000 $268,000 $280,000 $292,000 $304,000 $316,000 $328,000 $340,000 $352,000 $364,000 $376,000 $388,000 $400,000 G Compound $112,000 $125,440 $140,493 $157,352 $176,234 $197,382 $221,068 $247,596 $277,308 $310,585 $347,855 $389,598 $436,349 $488,711 $547,357 $613,039 $686,604 $768,997 $861,276 $964,629 $1,080,385 $1,210,031 $1,355,235 $1,517,863 $1,700,006 $1,800,000 $1,600,000 $1,400,000 $1,200,000

Note the change of scale on the y-axis

$1,000,000 $800,000 $600,000 $400,000 $200,000 $0 0 5 10

Years

15 20 25 10

Question 3

What is the difference in results between savings accounts that use simple and compound interest when you invest $100,000 at

for 25 years?

2 3 4 5 6 7 8 9 10 11 12 13 14 15 23 24 25 26 27 16 17 18 19 20 21 22 B Present Value = C 100000 D Interest Rate = 3% E Year 1 2 3 4 5 6 7 8 9 10 11 12 13 21 22 23 24 25 14 15 16 17 18 19 20 F Simple $103,000 $106,000 $109,000 $112,000 $115,000 $118,000 $121,000 $124,000 $127,000 $130,000 $133,000 $136,000 $139,000 $142,000 $145,000 $148,000 $151,000 $154,000 $157,000 $160,000 $163,000 $166,000 $169,000 $172,000 $175,000 G Compound $103,000 $106,090 $109,273 $112,551 $115,927 $119,405 $122,987 $126,677 $130,477 $134,392 $138,423 $142,576 $146,853 $151,259 $155,797 $160,471 $165,285 $170,243 $175,351 $180,611 $186,029 $191,610 $197,359 $203,279 $209,378 $250,000 $200,000

Note the change of scale on the y-axis

$150,000 $100,000 $50,000 $0 0 5 10

Years

15 20 25 11

Question 4

What is the difference in results between savings accounts that use simple and compound interest when you invest $100,000 at

20%

for 25 years?

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 B Present Value = Interest Rate = C 100000 20% D E Year 1 2 19 20 21 22 23 24 25 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 F Simple $120,000 $140,000 $160,000 $180,000 $200,000 $220,000 $240,000 $260,000 $280,000 $300,000 $320,000 $340,000 $360,000 $380,000 $400,000 $420,000 $440,000 $460,000 $480,000 $500,000 $520,000 $540,000 $560,000 $580,000 $600,000 G Compound $120,000 $144,000 $172,800 $207,360 $248,832 $298,598 $358,318 $429,982 $515,978 $619,174 $743,008 $891,610 $1,069,932 $1,283,918 $1,540,702 $1,848,843 $2,218,611 $2,662,333 $3,194,800 $3,833,760 $4,600,512 $5,520,614 $6,624,737 $7,949,685 $9,539,622 $8,000,000 $8,000,000 $6,000,000 $6,000,000 $4,000,000 $4,000,000 $2,000,000

Note the change of scale on the y-axis

$2,000,000 $0 0 $0 0 5 5 10 10

Years

15 15 20 20 25 25 12

Assignment: Part 1

1. Expand the number of years on your spreadsheet to 50 and redo the graph to include the new values. Does the compound interest graph for an interest rate of 3% still look linear? (Refer to Slide 11 for comparison.) 2. How long does it take for $10,000 to double at 5% using simple interest? 3. How long does it take for $10,000 to double at 5% using compound interest? (round answer to nearest year) 4. Redo questions #2 and #3 using interest rates of 10%, 15%, and 20%?

Investing in the Stock Market: Plotting the Dow Jones Industrial Index

Now let's consider one portion of the Dow Jones index, from 1980 to 2000, where dramatic growth occurred. We show that we can model this curve using an exponential function. The following are the DJIA closing values on the last trading day of the years indicated: Plot this data as before: observe that it produces a similar curve to our exponential growth curve that we met before. You are to add a trend line to this data and in the process, find a model. If you right-click on the graph (not the background) you will see the option "Add Trendline...".

Year 1980 1985 1990 1995 2000 DJIA 963 1546 2633 5117 10787

• In "Options" check "Display equation on chart" (this is the model, or equation, that we want). • • • • You should get the chart as above, showing the exponential graph and the model: That model function gives DJIA as a function of the year and is explained as follows:

2E-101

means

2 × 10 to the power -101

(This is a very small number indeed.) Recall "E" here means base 10 and

e0.1206x

is the exponential function with base

You can now use the model to make a prediction for any year. You can use the “Forecast” dialog under the "Options“ menu to extend the graph to year 2020.

Finally, in the same dialog, you should display the R-squared value, which indicates the goodness of fit. 15

Assignment: Part 2

5. Predict what will happen, given the same growth rate, out to year • 2020. How much will the Dow Jones Index Stock be worth in your prediction?

6. Now you will add a data point, and change the model. Add the closing value of the Dow at the end of 2005 (10,717). Describe the change in the model function, the R-squared value and the 2020 prediction.

7. Now add another data point, the value of the Dow today (about 13,583): Describe what happens to the model function and the R squared value and the 2020 predication.

8. Now investigate what happens to the model if the market dives in 2008 back to 10,000. What impact will it have on your 2020 predictions? Now use a linear model, what impact does it have on 2020 predictions? 9. Turn in your worksheet on Bb.

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Transcript Slide 1

Economics of Arithmetic and Geometric Growth Rates What is the difference between simple and compound interest and does it really matter?

Investing in the Stock Market: Plotting the Dow Jones Industrial Index

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