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 Personal Finance Decisions 

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Our second application topic is Personal Finance Decisions. This topic is trickier. Unlike with health decisions, some of the subtopics look too much alike. Each subtopic has its own formula, but until you get used to things it might seem hard to know when to use each formula.

Real life will continue to be more complicated than any math formula. In this topic we see this in many different ways.

Mortgages are complicated only because they have lots of steps. Each step is small and easy. But there are so many steps!

Saving for retirement is complicated because in real life people never save the same amount each year. Almost everyone saves less when they are younger and just starting to earn money, and more when they are older and their career has grown. But the formulas involve situations that are the same from year to year. So the formulas can help estimate whether a household is "on track" for retirement, and we use them to invent examples that show which saving habits are sufficient. The formulas can give us hope and guidelines, but will never match our lives.

As we study this topic, work on making helpful and organized notes, so you have handy the comments, formulas, and example problems you need.

Typicality

Ambiguity

pattern block shapes

Pattern Blocks

Pattern blocks come in several shapes.

The green triangle is smallest.

Two green triangles make a blue diamond. (If you want, be especially mathy and call it a rhombus.)

Three green triangles make a red trapezoid.

Six green triangles make a yellow hexagon.

Pattern blocks can be used for many kinds of math activities.

If we call a green triangle "one", then we can teach multiplication. We can physically model 2 × 6 = 12 by asking how many green triangles are needed to cover two yellow hexagons.

If we call a yellow hexagon "one", then we can teach fractions. We can physically model 12 × 13 = 4 by asking how many yellow hexagons are covered by twelve blue diamonds.

But we are going to use pattern blocks while being a bit more philosophical. Instead of explaining arithmetic, we want to investigate language.

Three Kinds of Typicality

My scale tells me that a green triangle pattern block weighs 1.5 grams.

green weighs one and one-half grams

1. What is the typical weight of a block in this group?

a group of pattern blocks

Hm. What do we mean by typical? Discuss this with your classmates and develop an answer that you are prepared to defend before the class, using the pattern blocks as props if that helps.

There is no right answer. The definition of the word typical is somewhat slightly vague language issue.

However, people's answers tend to focus on three slightly different ideas:

Some people focus on what appears the most often.

Those people pick the blue block, because we have six of those. So a "typical" block would be blue, and would weigh 1.5 grams × 2 = 3 grams.

Some people throw out items that only appear once or twice. Those items do not appear often enough to be considered representative of the group. They focus on what appears the most representative.

Those people also pick the blue block. But not because of how many blue blocks there were. Instead, because they threw out the green and red blocks. Again a "typical" block would be blue, and would weigh 1.5 grams × 2 = 3 grams.

Some people feel a need for either inclusiveness or precision, and take an average. They sum the values and then divide by how many values there were.

Those people would not pick a kind of block. Instead they do more calculation. The total weight is (1.5 × 2 greens) + (3 × 6 blues) + (4.5 × 1 red) = 25.5 total grams. Then dividing by how many blocks there gives an average weight of 25.5 total grams ÷ 9 blocks ≈ 2.8 grams.

Note that the first and second kind of people would say a blue block is typical. But the third kind of people would say that a blue block is a bit more than typical. Does that matter?

2. Someone added two big blocks! Now what is the typical weight of a block in the group?

another group of pattern blocks

2a. The people that focus on most often would still pick the blue block, because we have six of those, and still say 3 grams.

2b. The people that focus on most representative would still pick the blue block, because we have one or two of the other kinds, and still say 3 grams.

2c. The people that calculate the average would now get a total weight of (1.5 × 2 greens) + (3 × 6 blues) + (4.5 × 1 red) + (9 × 2 yellows) = 43.5 total grams, which would change the average weight to 43.5 total grams ÷ 11 blocks ≈ 4 grams.

There is no right answer. We can disagree about definition of the word typical.

Math to the Rescue

It was perhaps Benjamin Disraeli who first said, "There are three kinds of lies: lies, damned lies, and statistics." That quotation is often said to imply that statistics can be purposefully used to mislead.

But the point to our investigation of the word typical is how people can get different answers without any purposeful deceit. Statistics can be naturally uncertain because language is naturally ambiguous.

Fortunately, mathematicians have tools to remove the uncertainly and ambiguity.

Certainly we want some math terms to help us recognize when someone might be misleading us with statistics.

But more important is to develop an intuition about what happens when real life is not as typical as the formulas expect.

The formulas and guidelines are accurate and reliable! But the ones about home ownership assume a typical home, and the ones about saving for retirement assume a typical life progression from younger (with little income) to older (with more income). In reality, homes and lives are all unique, and future home prices and salaries are guesses. We plug fuzzy numbers into reliable formulas.

Good news! Math can still help us, even when we do not fully trust the numbers. Let's see how.

Words About Averages

I see that the life of this place is always emerging beyond expectation or prediction or typicality, that it is unique, given to the world minute by minute, only once, never to be repeated. That this is when I see that this life is a miracle, absolutely worth having.
    - Wendell Berry

That is the crowning unlikelihood, the thermodynamic miracle...Come, dry your eyes, for you are life, rarer than a quark and unpredictable beyond the dreams of Heisenberg; the clay in which the forces that shape all things leave their fingerprints most clearly.
    - Watchmen, Chapter IX

Mean

The average you we used above with pattern blocks is formally called the mean.

The Mean

To find the mean of a group of numbers, first add up all numbers and then divide by how many numbers are in the group.

When people say "average" it should be safe to assume they are talking about the mean, unless they say otherwise.

The mean is the most commonly used average because in many everyday situations the mean does what we want. It looks at a set of numbers and provides an answer close to most often but more accurate, and close to most representative but more inclusive.

3. In a certain preschool classroom, the students' weights are 34, 36, 36, 36, 37, 37, 38, 39, 39, 40, 40, and 41 pounds. What is the average weight of the twelve students in the room?

weights of preschoolers

3. First we add up all the numbers.

34 + 36 + 36 + 36 + 37 + 37 + 38 + 39 + 40 + 40 + 41 = 453 pounds

Then we divide that total by 12, because there are 12 students.

453 pounds ÷ 12 = 37.75 ≈ 38 pounds

Notice why the answer to the previous problem feels right. It is close to what number appears most often. Also, looking at the histogram it seems most representative.

That last point is important. The mean is where a histogram balances if it were a measuring scale.

4. Double check that this picture of four blocks on a balance scale is accurate, and the mean really is where the balance point is drawn at 5.

mean on a balance scale

4. First we add up all the numbers.

2 + 2 + 6 + 10 = 20

Then we divide that total by 4, because there are 4 blocks.

20 ÷ 4 = 5

(We can also see this illustration in a different way. The two blocks on the left side are each 3 spots from the center, for a total left hand weight of 6. The two blocks on the right side are 1 and 5 spots from the center, for a total right hand weight of 6. The balance spot is accurate because the left and right hand weights both total six.)

Histograms

We should pause for a moment to talk about the type of graph we just used.

Histogram

A histogram is a bar chart in which each bar counts a different category of things.

In a histogram, the categories can be numeric values, such as ages, weights, heights, test scores, etc. When the categories are numeric they will have a natural order from smallest to greatest.

Not all histograms have numeric categories. Common examples include populations for different countries, costs of living in different cities, popularity of different foods, etc.

In a histogram, the height of the bar counts how many things are in each category. So the height is always numeric.

Below is a chart that shows midterm scores for a Math 25 class back in Winter term 2016.

math class test histogram

Notice that each student in the class has his or her own piece of bar height somewhere on the histogram. Bars that are 1 high are each representing a certain student. Bars that are 2 high are each counting two students. And so on. We could point to any unit of bar height and sensibly ask, "Which student in the class is this?"

5. How many students scored between 80% and 90% (inclusive) on the histogram above?

5. The word "inclusive" tells us to include both the 80% bar and the 90% bar. So we are being asked to total the heights of all the yellow bars.

1 + 1 + 2 + 3 + 2 + 1 = 10 students

6. How many students took that midterm?

6. We are being asked to total the heights of all the bars. Let's group them by color just to help avoid making a careless mistake.

(1 + 1 + 1 + 1) + (1 + 1 + 1) + (1 + 1 + 2 + 3 + 2 + 1) + (3 + 2 + 1) = 23 students

The histogram above was also color-coded to group the categories into D, C, B, and A grades. But color-coding is not normally a part of histograms.

All histograms are designed. The histogram above uses categories of size 1.7% (starting at the lowest score of 48%). This visually sorts the students a certain way. The color coding emphasizes that there is a green A subgroup that did amazing, a big yellow B subgroup that did well, three orange C students who squeaked by, and four blue D students who definitely need improvement.

If the instructor instead used wider categories of size 5% the histogram would change. Below is exactly the same set of test scores, but with wider categories and altered color-coding.

math class test histogram

7. Kasara and Ray scored 85.1 and 85.3% on the test. The first histogram put them in a "84.5% to 86.2%" category. The second histogram put them in a "83% to almost 88%" category. How would they feel about that?

7. Their new category appears to be a low A, which is a big improvement from the histogram above where their category appeared to be a middle B. They would like the second histogram much better!

position in first histogram position in second histogram

8. Jordan scored 79.7% on the test. The first histogram put him in a "79.6% to 81.3%" category, the lowest score in the yellow B category. What does the second histogram say about his score?

8. In the new histogram he is in the category "78% to almost 83%". This puts him at the highest score in the yellow B category. He would also prefer the second histogram.

position in first histogram position in second histogram

9. Leann 73.5% on the test. In the first histogram she had the middle C score. What does the second histogram say about her scores?

9. In the new histogram she is grouped with a student who did slightly worse. Both are together in the category "73% to almost 78%". This puts her at the lowest score in the yellow B category. She would also prefer the second histogram.

position in first histogram position in second histogram

Mathematicians call histogram categories bins. It helps to picture the sections of the x-axis as physical bins that items are put into. That reminds us that designing the bins is a choice with consequences.

Someone who is careless, and makes a histogram without thinking carefully about the categories, is still making a choice! It is an unintentional and sloppy choice. Hopefully it does no harm.

Please remember to pay attention to histogram bins. Perhaps the person who made the histogram has an agenda. Perhaps that person is you!

videos

Khan Academy

How to create a histogram

OCLPhase2

Finding the mean of a data set

Finding the mean of a data list

Finding the mean from a frequency table

Finding the median from a frequency table

Finding the mode

Creating a histogram

Standard Deviation

When we want to talk about averages, not all groups of numbers are equally friendly. Here are four histograms that show test scores for a History class.

standard deviation example standard deviation example
standard deviation example standard deviation example

The class shown in histogram A has two subgroups of students. About half the students did poorly on the test, and half did well. We foolishly could calculate the mean of this group of numbers but doing so would be inappropriate and misleading. There are really two subgroups, each with their own typicality.

The class shown in histogram B has two subgroups of students, and test scores are even more extreme. In this class the students tended to do really terrible or really amazing. Very few were in between. As before, we foolishly could calculate the mean of this group of numbers but doing so would be inappropriate and misleading because there are really two subgroups with their own typicalities.

The class shown in histogram C does not have subgroups. This histogram has one big clump. For this classs we sensibly could calculate the mean of this group of numbers. Doing so is appropriate and helpful. The class has meaningful typicality.

The class shown in histogram D does has niether one big clump nor multiple subgroups. In this class the student scores look almost random. For this classs we hesitantly could calculate the mean of this group of numbers. Doing so is in theory appropriate because the students are in a single, meaningful group. But it is probably not be helpful. The class lacks typicality because no score is most common or most representative of the random mess. Yes, its single group has an average, but so what? For what purpose are we trying to condense this mess of test scores into a single summary number?

There is a name for how those four histograms difffer in obvious and important ways.

The name comes from considering a question. If the mean was representative of a standard value for the group, how much do all the numbers in the group deviate from that standard?

Standard Deviation

The measure of how poorly a group of numbers forms a single, meaningful clump is called standard deviation.

The formula for calculating standard deviation is not a part of this math class. But without a formula you can visually sort the four histograms.

Histogram C has the smallest standard deviation because in that group the numbers huddle in one big clump. They deviate only a little bit from the mean.

Histogram D has slightly bigger standard deviation because in that group the numbers look random. They are neither pulled toward nor pushed away from the mean.

Histogram A has a big standard deviation because in that group the numbers look pushed away from the mean. They are forming a pattern away from a single standard value.

Histogram B has the biggest standard deviation because in that group the numbers look almost allergic to the mean. They are hiddling against the far edges, as if avoiding being anywhere near a single standard value.

To repeat, the concept of standard deviation is important. Not all groups of numbers have a meaningful representative average. It is useful to be able to talk about this, to know there is a measure for this, and to realize you can visually sort using this.

If you ever actually wanted to know the value of the standard deviation for a group of numbers, you could always use an online tool.

10. Here are the four groups of numbers from the four History test histograms. Copy-and-paste each group into the online tool to find the value of its standard deviation.

10. The four standard deviations are:

Yes, or visual sorting was correct. Also notice how much lower the value for group C is than the others! It is the only group of test scores for which the average value is clearly meaningful.

Median

All four of this histograms were mostly symmetric. What happens if our group of numbers has some atypically low or high values that make its histogram unsymmetric?

crochet cthulu

Let's return to that preschool classroom in which the students' weights are 34, 36, 36, 36, 37, 37, 38, 39, 39, 40, 40, and 41 pounds.

Now three of the preschoolers insist their favorite stuffed animals, with weights 1, 1, and 2 pounds, also be included.

Those new weights atypically low. They make the histogram unsymmetric!

11. Draw the new histogram with all fifteen weights.

12. What is the new mean for the weights of all fifteen "friends" in the preschool classroom?

12. First we add up all the numbers.

34 + 36 + 36 + 36 + 37 + 37 + 38 + 39 + 40 + 40 + 41 + 1 + 1 + 2 = 457 pounds

Then we divide that total by 15, because there are 15 "friends".

457 pounds ÷ 15 ≈ 30 pounds

We calculated that answer correctly. But the answer feels very wrong. The number 30 does not describe what is most common, and is not representative of anything in the room.

It feels wrong that the stuffed animals have such a big influence. What can we do better?

We need a different kind of average. What if, instead of doing any calculation, we simply picked the middle number of a sorted list?

The Median

To find the median of a group of numbers, first sort the list of numbers in order and then pick the middle number in that sorted list.

If the list has an even number of values then there will be no middle value. Instead we find the mean of the two values most in the middle.

13. What is the median weight of all fifteen "friends" in the preschool classroom?

13. First we sort the list.: 1, 1, 2, 34, 36, 36, 36, 37, 37, 38, 39, 40, 40, 41

The two middle values are 36 and 37. Their mean is (36 + 37) ÷ 2 = 36.5 pounds

This answer feels right. There are indeed preschoolers who weigh around 36.5 pounds. The number 36.5 does an okay job showing what is most common, and is representative for that classroom.

That median is less than the mean when we did not include any stuffed animals. But it feels okay that the stuffed animals have a measurable but not dramatic influence.

Notice that process of finding the median throws out any atypical smallest or largest values. This is the type of average for those second category of people who felt most natural defining "typical" by throwing suspiciously extreme or rare amounts.

In other words, the median focuses on most representative numbers of the group. As the previous problem demonstrated, the median can also do a respectable job at estimating the most common.

mean versus median

You can think of the median as ignoring the most extreme values on a histogram and then finding where the rest of them balance. That is not quite how the median works, especially if the group of numbers is small. But in the real life situations where the median is used it will ideally behave that way.

The median is appropriate and often used for company salaries, state incomes, neighborhood house values, and other situations where the lowest and highest values really should not be thought of as representative of the group.

videos

Khan Academy

Histogram skew

Mode

Just for the sake of completeness, know there is a third kind of average called the mode.

The Mode

The mode of a group of numbers is the number that appears most ofen. If there is a tie, all ties are modes.

(On a histogram, the modes are the tallest bars.)

14. In a certain preschool classroom, the students' weights are 34, 36, 36, 36, 37, 37, 38, 39, 39, 40, 40, and 41 pounds. Which weight is the mode?

14. The only number to appear three times is 36. So the mode is 36 pounds.

The mode is so very rarely used in real life that it is only taught in math classes because of tradition. It will not appear in homework or a test.

The mode can count the "popular vote" in an election. But it focuses too much on most common and as a result does a terrible job of measuring most representative.

Ten Exercises

Try these ten exercises on scratch paper. Work in a study group if you can! Notice where your notes need improvement. After you are very happy with your answers, you can use this form to ask me to check your work. Can you get at least 8 out of 10 correct?

1. Find the mean of these six numbers: 135, 95, 11, 5, 33, 15.

2. Continuing the previous probem, find the median of those six numbers.

3. The histogram below (original source) shows the number of books read by different children. What is the mean number of books read?

the number of books read by twelve children

4. Continuing the previous problem, what is the median number of books read?

5. The home values on a certain street, in thousands of dollars, are: 384, 364, 342, 346, 360, 356, 265, 417, and 530. What is the mean of these home values?

6. Continuing the previous problem, what is the median of those home values? Why does this type of average better communicate the "typical" value of a home on that street?

7. A shipping company needs to transport seven freight containers. Their weights are 10, 16, 16, 18, 20, 60, and 77 tons. What is the mean and median weight of these freight containers?

8. Two company clerks receive a report that only contains the mean and median weights, and number of containers, from the previous problem. The first clerk tries to find the total weight by multiplying the mean by the number of containers. The second clerk tries to find the total weight by multiplying the median by the number of containers. Which clerk is correct? Why? How much error does the other clerk have?

9. Some news articles make a big deal when many countries have an average temperate increase well above global average (Alaska, Canada, Russia, Norway, Finland, Switzerland, China, Singapore, Australia, South Africa, etc.) How does a better understanding of averages explain that having many items above average is neither surprising nor sensationalism?

10. During the 2007 strike of the Writer's Guild of America, two different news reports painted very different pictures of these screen and television writers.

• According to CNBC, there were 4,434 guild writers who worked full-time in 2006, and their average salary was $204,000. (CNBC headline, October 11, 2007)

• According to the Los Angeles Times, the median income of the writers from their guild-covered employment is $5,000 a year. (Howard A. Rodman, October 17, 2007)

Were Hollywood's writers very wealthy and going to strike even though they earned much more than most Americans? Or were they poor and going on strike to defend the few thousand dollars they could earn from their writing? The headlines leave out two important facts. First, almost half of the guild's writers don't write anything in a given year (their salary that year is $0). Second, a very few writers earn millions of dollars. How does a better understanding of averages explain the situation more clearly?

Random Exercises

Try these exercises on scratch paper. Work in a study group if you can! Notice where your notes need improvement. Check your work when you are done.

Interlude from Factfulness

Factfulness book cover Conquering the Generalization Instinct

The sixth chapter of Factfulness includes many examples of when sorting information into the wrong "bins" causes incorrect conclusions. Hans Rosling describes several ways this happens.

Bins might hide differences. A bar graph showing the average income with a bin for each country has no practical value. Within each country are dramatic differences in income. Those bins attempt to group together people who are actually too different.

Bins might hide similarities. Everywhere in the globe, the main factor that affects how people live is their income level, not their country, culture, or religion. A bar graph showing literacy rates with a bin for each religion is promoting stereotypes. The meaningful comparison is literacy rates with income level. If bins about religion make any trend it would only be a side-effect of how religions and income levels match up.

Bins might measure averages. As we have seen, many groups of numbers have no meaningful average. No single number is "most common" or "most representative". If colllege students tend to equally drink no tea or a lot of tea, then a histogram showing the average number of cups of tea drunk by the students in different classrooms will have valid bins (each classroom) but meaningless heights (no students actually drink what looks like a common medium amount).

Bins might measure majorities. Many situations have no meaningful majorities. Most math students get a majority of questions correct on their math tests. But test scores of 51% and 91% are hugely different!

Bins might measure something meaningless. A high school could publish a histogram of its graduates' SAT scores. But those SAT scores do not reliably predict anything, so the histogram has no meaningful implications.

Bins might make generalizations. A bar graph might show what percentage of people in each country have annual dental cleanings. But different cultures place different emphasis on dental care. That statistic might meaningfully represent one country's overall health care, but say nothing meaningful about another country.

Only special cases might be included in the bins. A histogram showing the test scores of all a high school's math students would look very different from a histogram showing the test scores of the students in the AP math class.

chapter 6 logo

Bell Curves

Heritability describes something about a population of people, not an individual. It makes no more sense to talk about the heritability of an individual’s IQ than it does to talk about his birthrate.
    - Richard Hernstein

Making a Bell Curve

Some histogams form one very symmetric clump. These are called bell curves because they are shaped like a bell. They are sometimes instead called normal curves or a normal distribution.

three distribution curves

Bell curves can happen when an average result is the most common, being higher or lower than average is equally likely, and being way higher or lower could happen but is rare.

Repeatedly finding the sum of two dice will create a bell curve.

tallying the sum of two dice

In-Class Activity

15. Your turn! Get into groups of two or three students. Roll your pair of dice a bunch of times. Use a chart like the one above to keep track of how many times each sum happens.

Notice that making a bell curve reliably requires counting a very large number of things. The histogram for each student group might not look at all like one symmetrical clump. But after combining the entire classroom's counts we do get a bell curve.

Natural Bell Curves

We saw how a random event can make a bell curve. Where else do bell curves naturally happen?

A Few Physical Characteristics

Heights and weights are two natural physical characteristics of people that will form a bell curve if a large enough group of people are measured.

16. Other than people's height and weight, what natural, physical demographic characteristics form bell curves?

16. Not many! Natural characteristics that form bell curves can be difficult to find.

Vision is one. Most people have close to "normal" vision with near-sightedness and far-sightedness forming the sides of the curve.

Bicep strength is another. Perhaps our class should try dumbell curls?

Grip strength is another.

Within a homogenous population, blood pressure and life expectancy can be others.

Comparing Bell Curves

Consider a pair of mostly similar bell curves counting people.

Because the two bell curves have the same area, they both count the same number of people.

Because the two bell curves have the same middle value, the average is the same for both sets of people.

two bell curves

We already have the vocabulary to describe the difference. The blue curve is more spread out, so it has a higher standard deviation.

In the pink bell curve more people are average or nearly average, and fewer are notably higher or lower. Because of how bell curves work, this effect is exaggerated near the middle and extremes. The pink bell curve has a lot more average people, and a lot fewer extremely high or low people.

In the blue bell curve more people are funky. Most are still average or nearly average. But comparatively more are notably higher or lower. Because of how bell curves work, this effect is exaggerated near the middle and extremes. The blue bell curve has a lot fewer average people, and a lot more extremely high or low people.

Both bell curves have mostly average people. Both curves have the same meaningful average that represents both most common and most typical. Neither curve is a random mess.

Both bell curves have some very high or low people. The "tails" of both bell curves remain above zero as they go on and on. Neither curve has a monopoly on extremes.

Imagine these curves measured height. The pink bell curve is measuring a group of people that strongly tend to be about average height—not too many are much taller or shorter, and the group has very, very few giants and dwarves. The blue bell curve is measuring a group of people with less tendancy to be average height—many more are taller or shorter than average, and although the group still has very few giants and dwarves, if you happened to see a giant or dwarf it would probably be from the blue group.

Unnatural Bell Curves

Standardized Tests

Remember that few natural demographic characteristics formed bell curves. We had trouble brainstorming any besides height and weight.

Far more common is when people purposefully construct tests designed to sort people into a bell curve, such as IQ Tests or college entrance exams. Because these tests are designed to make the bell curve happen, they can do it very well. But that comes at the cost of sacrificing doing anything else well. IQ Tests are famous for measuring not intelligence but "whatever it is IQ Tests measure". College entrance exams do not reliably predict whether a student is actually ready for college and will successfully earn a college degree.

Socioeconomic Trends

Be especially wary if you read about socioeconomic characteristics that form a bell curve!

Very, very few places have bell curve distributions of wealth, income, education, etc. Those bell curves are rare!

But the field of sociology (especially in America) has an alarming track record of expecting these to form bell curves. So researchers use increasingly sloppy methods of data collection or analysis until their expectations are finally met.

Remember that histogram bin choices can change the shape of the histogram. There are other ways that sloppy choices (even if made unknowingly and accidentally) can misrepresent data.

Grading On a Curve

Part of math jargon is giving certain words a very precisely defined definition. The words "average", "function", and "parallel" have a meaning in math that resembles their standard use but is more specific. In other math classes you might learn about special math definitions for the words "commute", "set", "group", or "normal".

Sometimes the opposite happens, and phrases that were originally a precise math definition shift to casual English usage and acquire a broader and fuzzier meaning.

In math jargon, "grading on a curve" means something very specific. The test must be of a kind designed to have its scores produce a bell curve (such as the SAT or ACT college entrance exams). There must be enough scores to make sure that bell curve happens (the way it took many coin flips to make that situation have a bell curve).

Only then it would make sense to use the bell curve histogram to assign grades. Most scores are average and get a C. A smaller number of scores are slightly below or above average and get a D or B (and the number of D's and B's is nearly equal). A very few scores are exceptionally low or high and get a F or A.

When designed properly, and refined over time, tests like these are so predictable that grades can be assigned to scores before the test happens! The previous group of scores will have histograms almost identical to the next group of scores. The test designers know where the distinction between each letter grade will happen even before the next test happens.

All of this is what mathematicians mean by "grading on a curve". The test has such a careful design and well-established history that everyone knows in advance where the distinction between each letter grade will happen.

one bell curve broken into three parts

Needless to say, there are very, very few college tests that fit this model. Only large universities have classes with enough students so that there are enough scores to make a nicely full histogram. Even in those big classes, there are only a few tests are designed primarily to create a bell curve of scores.

And remember that tests designed to make a bell curve usually cannot also be designed to measure student readiness for future success—most instructors value the latter and try to write tests accordingly.

So when most college instructors says they "grade on a curve" they are probably not using those words as a mathematician would.

Ask those instructors what they do mean by that phrase!

Often an instructor who says that he or she "grades on a curve" is actually using that phrase as part of an attempt to explain that he or she knows the tests will not reliable and predictably form a bell curve, and that he or she knows better than to actually grade on a curve!

The Bell Shape

On another note, the following image from a book about mental health attempts to show that a normal amount of stimulus is healthy even though excessive stimulus is too stressful. The graph below is not a histogram, and thus not a bell curve! Why not?

is this a bell curve or not

How could we change one word to make the graph a bell curve describing a population's stress level?

How could we change one word to make the graph a bell curve describing a person's stress level in different days of their life?

Ten Exercises

Try these ten exercises on scratch paper. Work in a study group if you can! Notice where your notes need improvement. After you are very happy with your answers, you can use this form to ask me to check your work. Can you get at least 8 out of 10 correct?

The most interesting part of bell curves is how they are used to sort people and shape society. That is not a great source of small math problems! So please pardon a brief tangent into our local contributions towards climate change.

In Oregon, a household's energy usage per month forms a histogram that resembles a bell curve reasonably well for real-life data.

The histogram below shows the monthly electrical useage for a household that has electric heat.

electric usage histogram

1. In which month does this household use the most electricity?

2. Which month has the least deviation from year to year?

3. Someone asks, "What is this household's typical monthly electrical usage?" What would make a numeric answer to this question meaningful?

4. When did this family replace their old electric furnace with a modern and more efficient heat pump?

5. The total electrical usage for 2017 was 14,780 kilowatt hours. The total electrical usage for 2019 was 11,513 kilowatt hours. How much less electricity was used in 2019?

6. Electricity costs an average of 11.3 cents per kilowatt hour. How much less money was spent on electricity in 2019 than in 2017?

7. What was the percentage decrease of this household's total annual electricity usage when comparing 2017 to 2019?

8. In this city 80% percent of the electric power is from carbon-free hydroelectric energy, making an overall CO2 emission of 16.2 grams per kilowatt hour. How many kilograms of CO2 did this household's electric use create in 2019?

9. A typical gasoline automobile's emission is 8.8 kilograms of CO2 per gallon. This household's car gets 30 miles per gallon. How many miles would they need to drive to the equal CO2 emissions of their annual electricity use?

10. That household actually drives 10,000 miles per year, which is equivalent to about 333 gallons of gasoline. Find this household's total kg of CO2 emissions for house and car, and then divide by 1,000 to convert kilograms to metric tons. Purchasing a carbon offset costs about $14 per metric ton of CO2. What is the value of the carbon offset cost for this household's house and car?

For most households the energy used to grow and transport food is a larger carbon footprint than home heating or vehicle driving. You can use a website such as CarbonFootprint to estimate your own numbers.

The Central Lane Metropolitan Planning Organization has found that in the Eugene-Springfield area the mean household carbon footprint is 31.9 metric tons of CO2, and the mean carbon footprint per person is 13.8 metric tons of CO2.

Random Exercises

(no random exercises for this topic)

Interlude from Factfulness

Factfulness book cover Conquering the Blame Instinct

The ninth chapter of Factfulness talks about blames versus causes. Hans Rosling writes:

The blame instinct makes us exaggerate the importance of individuals or of particular groups. This instinct to find a guilty party derails our ability to develop a true, fact-based understanding of the world: it steals our focus as we obsess about someone to blame, then blocks our learning because once we have decided who to punch in the face we stop looking for explanations elsewhere.

The same instinct is triggered when things go well. "Claim" comes just as easily as "blame". When something goes well, we are very quick to give credit to an individual or simple cause, when again it is usually more complicated.

...It's almost always about multiple interacting causes—a system. If you really want to change the world, you have to understand how it actually works and forget about punching anyone in the face.

Statistics can be used to place blame. For example, Coleman Hughes explains the gap-lens and past-lens with fascinating examples.

Bell curves especially are misused by people to try to create villains or heroes. Bell curves had distract us from looking at root causes and systems.

"Did you know that adult chronic criminals, as a group, tend to have unusually low IQs? That must mean that IQ predicts something about misbehavior!" Actually there is no connection between IQ and villains. Instead there is a systemic cause. Men born with chromosonal disorders have a higher than normal chance of developing both behavior problems and lower IQs. Many of these men become adult chronic criminals. The actual issue is whether society can develop a system for dealing with chromosonal disorders, not how it blames low-IQ individuals.

chapter 9 logo

Mortgages

Normal is getting dressed in clothes that you buy for work and driving through traffic in a car that you are still paying for—in order to get to the job you need to pay for the clothes and the car, and the house you leave vacant all day so you can afford to live in it.
    - Ellen Goodman

How big a home can your certain household afford?

We can work step-by-step and answer this question.

Let's imagine that the Wahl family has an annual income of $60,000. That is close to the median income for both Oregon and the overall United States.

Monthly Payment Size

There is a general rule for the size of mortgages or apartment payments.

The Wahl family gets out their last few years' tax papers.

Their pre-tax income has indeed stayed near $60,000. $60,000 × 0.35 ÷ 12 = $1,750.

Their after-tax income has stayed near $52,000. $52,000 × 0.25 ÷ 12 = $1,083.

So that general rule advises them to spend between $1,083 and $1,750 as a monthly mortgage payment.

The Wahl family thinks about their priorities. They have only one child. The father suffers from some chronic health issues. They decide to aim at the lower end of that price range and look for a house with a $1,100 per month mortgage.

Loan Size

How big a house can they get for $1,100 per month?

That math problem is complicated. Over time the principal (the original home loan debt) lessens as payments are made. But at the same time the loan accumulates interest. Solving a problem that includes both growth and shrinkage would be hard.

amortization table

Fortunately, it has already been solved for us. Yay!

Amortization is the paying off of debt with repeating, scheduled repayments.

Accountants have solved the problem of loan payments and interest, and saved the answers as an amortization table.

In this class we will use the amortization table to the right.

In real life, the amortization table is huge. It has more rows (for interest rates with decimals to the hundredths place). It has more columns (for loans with other durations of months and/or years).

How do we use our amortization table?

The amortization table values show us the correct monthly payment for a $1,000 loan with a certain interest rate and duration.

Imagine that someone got a $1,000 loan with a 20 year duration. The amortization table tells us the proper monthly payment would be $6.60. If that person made monthly payments of $6.60 for all twenty years, then principal would slowly be paid off and interest would happen, and everything would work out perfect so just as the final year ended the loan would be completely paid off.

Of course, a house mortage will be much more than $1,000. So we scale up the table value. We multiply by how many thousands are in the actual mortgage amount.

17. Find the monthly payment for a $130,000 mortgage that is a twenty-year loan with an interest rate of 6%.

17. The amortization table value for 6% and 20 years is $7.16.

So the monthly payment is $7.16 for each $1,000 of loan.

There are 130 thousands in the loan size of $130,000.

So $7.16 per thousand × 130 thousands = $930.80 monthly payment.

That is easy to do!

Unfortunately, it is not practical. In real life, anyone who gets a loan is told their monthly payment amount. No bank expects customers to figure it out themselves!

We can also use the amortization table backwards to find how large a home someone can afford based on their desired monthly payment.

This is more practical for real life.

18. The Wahl family wants to spend $1,1000 per month on a mortgage. Mortgage interest rates are at 6%. How large a thirty-year loan can they afford?

18. Imagine that the Wahl family goes to the bank with that $1,100 of bills in a suitcase.

The amortization table value for 6% and 30 years is $6.00. Every time the Wahl family gives the bank $6 they can get $1,000 more loan.

They hand over $6. They are at 1,000 of loan.

They hand over another $6. They are at 2,000 of loan.

They hand over another $6. They are at 3,000 of loan.

How long can this go on before their suitcase runs out of money?

We need to divide. $1,100 ÷ $6 ≈ 183.

They can give the bank that $6 amortization table amount 183 times before they run out of money. They can get a 183,000 mortgage.

Notice that when we rounded the division step in the previous problem it forced the final answer to be rounded to the nearest thousands of dollars. That is fine. Mortgages are often rounded that way in real life.

From the point of view of the Wahl family, the problem seems finished. They know to look for a house with a value of about $183,000. Everyone can get excited and start looking at Zillow.

Minor Complications

No, silly Wahl family! Slow down. A bunch of minor issues make buying a home more complicated.

Let's finish helping the Wahl family.

They find a home priced at $183,000. Their down payment is 20%.

19. What is the actual mortgage amount?

19. The down payment is $183,000 × 0.2 = $36,600.

So the mortgage amount is $183,000 − $36,600 = $146,400

20. The loan is for 30 years with a 6% rate. What is the monthly payment?

20. The amortization table value for 6% and 30 years is $6.00.

So the monthly payment is:

$6.00 per thousand × 146.4 thousands of loan = $878.40 monthly payment

Their mortgage has a 3% fee. The first month's mortgage payment must be paid up front at closing. Inspections cost $200. The various up front costs for insurance and fees total about $900.

21. About how much cash must the Wahl family have available for all their up front costs?

21. The mortgage fee is $146,400 × 0.03 ≈ $4,392.

The total up front costs are:

$36,600 down payment + $4,392 mortgage fee + $878.40 first payment + $200 inspections + $900 other ≈ $43,000

To summarize, they did some math to learn that they could spend $1,100 per month and afford a $183,000 home. Then more math showed how they ended up paying $43,000 of up-front costs (mostly the downpayment) to get a $146,400 mortgage with monthly payments of $878.40.

That initial math over-estimated their monthly payment because it ignored the up-front costs.

Whew. Now we are done helping the Wahl family. Thanks!

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Khan Academy

Mortgage Interest Rates

Total Payments

But from the point of view of the bank, the story is just starting.

22. Continuing with the Wahl family, how much will be paid total over the thirty years?

22. The monthly payment of $1,100 happens twelve times per year for thirty years.

$874.40 × 12 × 30 = $314,784 paid total

23. Continuing with the Wahl family, how much of their total paid amount is interest?

23. Think about that $314,784 total. All the money is either loan or interest.

So there is $314,784 − $146,400 = $168,384 interest total

Over the entire thirty year time span, more is paid on the interest than on the principal!

This is actually normal for a thirty year loan. A longer time period allows a household to afford a larger mortgage for the same monthly payment. But eventually it costs more in interest.

But the bank is not greedy. Actually, it is afraid.

Bank Safety

Potential Profit

If the bank instead invested that $183,000 in the stock market, how much would it expect to have after thirty years? We will later learn how to do that problem. The answer is about $2,428,000.

By chosing to do business with the Wahl family the bank is giving up the opportunity to earn more than two million dollars! Why would it do that? What is it afraid of?

Here is a graph of how the stock market has behaved between 1950 and the 1997 recession.

stock market

Notice the areas with a red highlight. During those years the stock market was doing worse than its average trend. If bank customers wanted to withdraw money during those years, the bank would earn less than that amazing stock market prediction. And during a recession it might earn much less.

So banks do business with the Wahl family because mortgages are boring and reliable. The bank feels safe. Bank customers to withdraw money any year without causing problems. The expected income to the bank is much less than a stock market prediction, but the bank need not fear unpredictability.

The Great Depression

man with sign

Historically, the Great Depression was partly caused by banks investing in the stock market. Now regulations prohibit such reckless financial behavior. Banks now give mortgages not only because they have learned better, but because they are required to play it safe.

But banks did not cause the Great Depression. (Remember our interlude from Factfulness. Look deeply at systems instead of rushing to blame villains.)

During the 1920s consumer debt doubled. The Federal Reserve had made borrowing money too easy. Everyone was borrowing more than they safely could to buy those new-fangled household appliances and automobiles. The crisis began when everyone realized they could (borrow and) spend no more. Demand for more goods and construction plummeted, putting many people out of work.

Then businesses were stuck between a rock and a hard place. With consumer demand plummeting, they could not continue to expand. But there was nowhere to invest their money. It could not be invested in mortgages because consumer spending on homes was so low. It could not be invested in other companies because consumer spending on goods was so low. Business wealth could only be hoarded.

A dollar that is hoarded, whether in a business's safe or under a mattress, is only a dollar.

A dollar that is invested, whether in a mortgage or the stock of a growing business, is more than a dollar. That dollar helps pay for the home construction, and the construction workers use it again at the grocery store, who uses it again with the farmer, who uses it again...

How quickly a dollar moves around like that is called the velocity of money. Between 1929 and 1933 the velocity of money slowed to a crawl. The money supply dropped by almost one-third.

traffic merging You know what happens when a three-lane freeway loses a lane, and cars must merge. You can think of the Great Depression as an economic traffic jam.

It began with too much lending: too much money circulating, too many cars on the road. Then consumers realized they had overspent while businesses realized they needed to spend more. How to shift the money from businesses to consumers? There were too few lanes of traffic available to get the money where it needed to go.

The 2007 Recession

In 2007 a smaller recession happened. The causes were similar. It was again too easy to borrow money, and there was too much money circulating. Consumers had once again borrowed and spent too much on mortgages, appliances, and automobiles. Business were once again stuck between a rock and a hard place with a lot of savings but no where to invest the money.

Why did history repeat itself? Didn't economists learn their lessons?

They had been seduced by a math formula.

wrapped package The sloppy historical summary is that a famous economist named David Li invented a formula to measure how certain kinds of risks could be combined together. All kinds of businesses would love to be able to combine their risks into a simpler number. The formula was used more and more, to combine all kinds of risks for which it was never intended.

Economists treated the formula like magic gift wrap, able to neatly wrap up into tidy and pleasant packages the increasingly oversized consumer debt and circulation of money. Once mortgage lenders joined in then home loans became way too easy to get. History was doomed to repeat itself.

Ten Exercises

Try these ten exercises on scratch paper. Work in a study group if you can! Notice where your notes need improvement. After you are very happy with your answers, you can use this form to ask me to check your work. Can you get at least 8 out of 10 correct?

1. Brenda can afford to spend $900 per month on mortgage payments, with a 30-year loan. Currently mortgage rates are 5% per year. What price home can she afford?

2. Brenda gets a loan with a 25% downpayment. What will be the size of Brenda's actual mortgage and monthly payment?

3. Brenda's loan has a 2% mortgage fee, first month pre-paid, and $1,000 other up-front costs. What is the total of her downpayment and these other up-front costs?

4. Zane has an annual income of $65,000. He wants to spend 30% of his income on a mortgage, with a 15-year loan. The interest rate is 5%. What price home can he afford?

5. Zane gets a loan with a 15% downpayment. What will be the size of Zane's actual mortgage and monthly payment?

6. Zane's loan has a 4% mortgage fee, first month pre-paid, and $1,300 other up-front costs. What is the total of of his downpayment and these up-front costs?

Remember the Wahl family? Their numbers were:

7. Does doubling the monthly payment double the size of the other numbers? Consider the Moneybag family that has initially budgets $2,200 per month on a mortgage, with everything else the same as the Wahl family. Do the other numbers (except for the fixed "other up-front costs") double?

8. Does halving the length of time halve the size of the other numbers? Consider the Short family that uses a 15 year loan for their mortgage, with everything else the same as the Wahl family. Do the other numbers (except for the fixed "other up-front costs") halve?

9. Does halving the interest rate halve the size of the other numbers? Consider the Timing family that uses a 3% interest for their mortgage, with everything else the same as the Wahl family. (Their Amortization Table value is $4.22.) Do the other numbers (except for the fixed "other up-front costs") halve?

10. Does doubling the interest rate double the size of the other numbers? Consider the Calamity family that uses a 12% interest for their mortgage, with everything else the same as the Wahl family. (Their Amortization Table value is $10.29.) Do the other numbers (except for the fixed "other up-front costs") double?

Random Exercises

Try these exercises on scratch paper. Work in a study group if you can! Notice where your notes need improvement. Check your work when you are done.

Interlude from Factfulness

Factfulness book cover Conquering the Destiny Instinct

The seventh chapter of Factfulness includes many examples of trends that appear to be linear (a straight line increase or decrease) but actually are not. Long-term trends that form straight lines are actually very rare. If a trend appears to be a straight line that probably means we are zoomed in to an unhelpful short-term picture.

Don't assume straight lines. Many trends do not follow straight lines but are S-bends, slides, humps, or doubling lines. No child ever kept up the rate of growth it achieved in its first six months, and no parents would expect it to.

A wonderful website named Our World in Data has many interactive graphs. Here are a few that demonstrate long-term trends that are not straight lines but could appear linear if we zoomed in too much.

How much global population is increasing is an S-bend. It began flat, had a period of dramatic change, and now is flattening out again.

population acceleration

Unsurprisingly, the amount of fresh water used for agriculture has a similar S-bend shape.

fresh water used for agriculture

Global fertility a decreasing S-bend shaped like a slide. It dropped dramatically, and now is flattening out.

global fertility

The amount of people still illiiterate is also dropping like a S-bend slide. Fortunately it has barely begun to level out!

illiteracy

The relationship between income and life expectancy is called a logarithmic increase. It first begins by rocketing upward but then levels off.

life expectency compared to income

Some graphs show a general "magnetic" pull. The data is too messy to make a single curve. But clearly a trend is happening that pulls all the numbers together over time. The drop in fertility in each country has this trend.

fertility by country

So does the equity of boys and girls attending school in diffferent regions of the world.

educational equity

A few long-term are actually linear. Remember that the lesson is that these are rare, not mythical.

Global life expectancy has been increasing somewhat linearly.

global life expectancy

Child labor has been decreasing linearly.

global child labor

The relationship between increasing a maternal education and decreasing infant mortality is linear.

education compared to child mortality

Finally, some long-term trends are cyclical.

Another website, Engaging Data, has all sorts of fun graphs. Here is a cyclical graph of the water level in California reservoirs.

education compared to child mortality

So remember to be suspicious when you are told about long-term linear trends. They exist but are rare.

the back half of the chessboard Also, remember the Chessboard Story from our patterns homework. That story teaches that doubling growth cannot be sustained. The growth by doubling starts small, but soon races out of control.

In the business world this phenomenon is nicknamed the back half of the chessboard. Many new technologies indeed spread rapidly as inventions contine and consumers desire what their neighbors own. But no one expects that initially rapid spread to continue indefinitely.

chapter 7 logo

Interest

It has been said that the Fed's job is to take the punch bowl away just as the party gets going, raising interest rates when the economy is growing too fast and inflation threatens.
    - Alex Berenson

Interest is a fee the lender charges the borrower.

For your savings account or a government bond, you are loaning money to the bank or government, and you collect the interest. For your debts, a bank or credit card is loaning money to you, and they collect the interest.

The initial amount of a loan is called the principal. We already saw this term used with mortgage loans.

Please review percent change and the One Plus Trick before learning more about loans and interest.

Simple Interest §6.6

A One-Year Loan with One Payout

The loan with the least math would borrow money for one year, and at the end of that year the borrower pays back both the loan and interest.

78. A $500 loan lasts one year. It has a 2% annual simple interest rate. What is the total amount paid at the end of this annual loan?

$500 loan × 1.02 rate = $510 total paid

So far our "story" about interest has been the simplest case:

Let's modify that third issue. What if the loan lasts multiple years?

A Multiple-Year Loan with One Payout

For this type of loan it remains true that the fee is only paid once, at the very end of the loan. Then borrower finally pays back both the loan and several years' copies of than annual interest fee.

32. A $100 loan lasts 5 years. It has a 12% annual simple interest rate. What is the total amount paid at the end of this loan?

32. More than one year! We can do this with two steps.

First we find the total interest

5 years × 0.12 interest per year = 0.6 overall interest

Then we can use the One Plus Trick.

$100 prinicipal × 1.6 OPT = $160 total paid

For that loan the interest payments all stayed the same. That type of loan is called simple interest.

Many government bonds do work like that. Also, banks sometimes offer "certificates of deposit" that have simple interest. In all these situations, no interest payments are made until the end of the loan. Then the principal is returned along with a copy of the interest fee for each year of the loan.

A Part-of-a-Year Loan with One Payout

Let's modify that third issue again. What if the loan lasts less than a year?

What should happen if the borrower desired a loan that lasted only six months?

There are two sensible responses.

Perhaps the person or bank loaning the money would say, "Too bad! The fee is for a one-year loan. Sure, you can pay off the loan early. But you still owe me the fee." That makes perfect sense from the point of view of contract law.

But that would not make a happy customer, and would be bad for business. So the other sensible response is actually what happens in real life with simple interest. If the loan only lasts for half a year, then the borrower only pays half the annual fee.

33. A $100 loan lasts six months. It has a 12% annual interest rate. How much interest is owed?

33. First we find the annual interest, as we did before.

$100 prinicipal × 0.12 rate = $12 interest

Then we can divide by 2 since the loan only lasts half a year.

$12 interest ÷ 2 = $6 interest

Any Loan with One Payout

Actually, any time span can be converted into years.

34. A $2,000 loan lasts for 200 days. It has a 20% annual interest rate. How much interest will be due?

34. Notice that we can write 200 days as 200365 of a year.

$2,000 prinicipal × 0.2 rate = $400 interest per year

$400 interest per year × (200 ÷ 365) ≈ $219.18 interest

Notice that the parenthesis in the previous line are not needed for order of operations. They only help our eyes identify their contents as something that used to be written as a fraction.

The previous problem only asked for the interest, not for the total payment. When we are only asked to find the interest we can use a formula.

Simple Interest Formula

Simple Interest = Principal × Annual Interest Rate × Years

You can compare the previous problem's work to the formula. Do you see how they match up? The formula is just describing what we were already doing.

Unfortunately, this formula is traditionally written in a more confusing manner.

Traditional Simple Interest Formula

I = P × r × t


Pleae be careful with this version of the formula!

Remember that the interest it gives is simple interest.

Remember that the rate is an annual interest rate that needs RIP LOP.

Remember that the time is always measured in years and might need unit conversion if initially provided in days, weeks, or months.

35. A $3,500 loan lasts for 9 months. It has a 4.5% annual interest rate. How much interest will be due? Use the simple interest formula.

35. Notice that we can write 9 months as 0.75 of a year.

I = P × r × t = $3,500 prinicipal × 0.045 rate × 0.75 years = $118.13

Simple Interest Growth

The picture to the right shows a loan with simple interest for which the borrower never makes any payments. The loan starts on the left side with only the principal (the blue bar). As time goes on we move to the right. Each year a fee is added to the loan (each payout is a new green bar). All the green bars have the same height. The amount of the loan grows at a steady pace.

The most common real-life examples of simple interest are payday loans, bank certificates of deposits, and bond investments.

Simple interest used to be more popular. Before calculators and computers, everyone was happier with loans whose monthly payments never changed.

But now loans using simple interest are rare. Technology makes it easy to calculate a different monthly payment each month.

Let's conclude with a few more examples.

36. A $100 loan lasts 18 months and has 3% annual simple interest. What is the total amount of interest?

36. First we change the time span into years. 18 ÷ 12 = 1.5 years

Then we use the formula. We multiply the principal, annual interest rate, and number of years.

Simple Interest = P × r × t = $100 principal × 0.03 annual rate × 1.5 years = $4.50 total interest

37. An $750 bond pays a 2% annual simple interest rate, and matures in five years. How much will it be worth when it matures?

37. Be careful! This problem did not ask for the interest. It asked for the total!

So we need to find the principal with the formula, and then add back the principal.

Simple Interest = P × r × t = $750 principal × 0.02 annual rate × 5 years = $75

Then we add $750 principal + $75 interest = $825 total due

38. A $400 loan has a 5% annual simple interest rate and lasts 200 days. How much will the borrower need to repay at the end of that time?

Be careful again! This problem also did not ask for the interest. It asked for the total, so we need to find the principal with the formula, and then add back the principal.

Simple Interest = $400 principal × 0.05 annual rate × (200 ÷ 365) years ≈ $10.96

Then we add $400 principal + $10.96 interest = $410.96 total due

Appreciation and Depreciation

Similar to interest is a concept called appreciation. This describes when an investment increases in value just because the investment is worth more after time goes by (not because a fee is charged). People often see appreciation in the value of their home, in the value of old furniture or jewelry, and in the value of the stocks in which they invest.

The opposite of appreciation is depreciation, when something's value goes down.

In a slightly confusing manner, people use the simple interest formula for both interest and appreciation/depreciation. Just pretend that the change from appreciation/depreciation is a fee even though it really is not.

Scrooge McDuck

39. Scrooge McDuck puts one million dollars into an investment that appreciates 3% its first year. The second year the investment depreciates by 3%. Scrooge is sad, and thinks he is financially back where he started. When his account statement arrives in the mail he is surprised to see that he has less money than he started with! What happened?

39. The first year his principal was $1,000,000 and his gain from appreciation was:
$1,000,000 principal × 0.03 rate × 1 time = $30,000.

The second year his principal was $1,030,000 and his loss from depreciation was:
$1,030,000 principal × 0.03 rate × 1 time = $30,900.

Overall, he lost $900.

Remember, that same 3% of a larger amount is itself larger!

videos

InspireMath Tutorials

Simple Interest Formula

The Organic Chemistry Tutor

Simple Interest Formula

Mathispower4u

The Simple Interest Formula

Determine an Account Balance Using Simple Interest

Bittinger Chapter Tests, 11th Edition

Chapter 6 Test, Problem 14: What is the simple interest on a principal of $120 at the annual interest rate of 7.1% for one year?

Chapter 6 Test, Problem 15: A city orchestra invests $5,200 at 6% annual simple interest. How much is in the account after half a year?

Textbook cover

Textbook Exercises for Simple Interest

This list of recommended odd-numbered textbook problems is designed for a hypothetical student with a "typical" math background. If your math foundation is weak, do even more odd-numbered problems. If your math foundation is strong, do fewer.

Please read the advice on doing homework in the study skills page.

Section 6.6 (Page 356) # 1, 3, 7, 9, 11

Six Years of Rent Increases

Consider an apartment whose monthly rent starts at $500, and every year it increases by 5%.

40. Complete the table below.

rent increase table

40. Year 2: Starting amount $525.00, Increase $26.25, Ending amount $551.25

Year 3: Starting amount $551.25, Increase $27.56, Ending amount $578.81

Year 4: Starting amount $578.81, Increase $28.94, Ending amount $607.75

Year 5: Starting amount $607.75, Increase $30.39, Ending amount $638.14

Year 6: Starting amount $638.14, Increase $31.91, Ending amount $670.05

41. What is the rent at the end of the sixth year?

41. The ending amount for the sixth year was $670.05.

42. Why does the amount of increase keep changing?

42. Five percent of a bigger number is a bigger dollar amount. (Just like 5% of a millionaire's income is a lot more than 5% of my income!)

Each year the starting number grows, so 5% times that number grows too.

43. What was the overall percent change?

43. Recall the percent change formula.

Percent Change Formula

percent change = change ÷ original

After the division, use RIP LOP to change the decimal into percent format.

The dollar amount of the overall change was $670.05 − $500 = $170.05.

So percent change = $170.05 ÷ $500 ≈ 0.34 = 34%.

Notice that if the increase had remained at $25 per year then the overall change would have been 6 years × 5% = 30%. The growth of the annual increases added up to an extra 4% overall change.

We can simplify our table using the One Plus Trick.

44. Complete the table again, this time using the one plus trick in each row.

rent increase table

That was a lot nicer!

Don't stop now! We can make the table even simpler!

The year two starting amount of $525 was calculated using $500 × 1.05. What happens if instead of the dollar amount we put that calculation in the place of the year two starting amount? And keep doing that for every year's starting amount?

45. Complete the table below, using the initial $500, incremental exponents, and the one plus trick.

rent increase table

That was the nicest of all!

Now we have a table that shows us a pattern.

46. What is the formula for the ending amount y if we know the year number n?

rent increase table

46. The formula is y = $500 × 1.05n

Compound Interest

The English word "compound" means two different things. It can refer to an item made up of multiple and different parts (for example, a medicine that is a compound of ingredients). It can also refer to an item made greater by a certain type of thing (for example, strong winds compounding the difficulty of putting out a forest fire, or young siblings with a tendency to get into trouble that is compounded by their cousins visiting).

Compound interest uses the word "compound" in both ways. We just saw that happen with the six years of rent increases. There were lots of parts, and the interest amount for each step got bigger as time accumulated.

For a loan with compound interest the payouts do not stay the same size. At the moment of each payout, the annual interest rate is applied to the current loan size then.

Compound interest is also not a math decision. It is another type of loan.

If the agreement says that payments never change size (that accumulated interest does not earn interest) then we can use our tools for simple interest.

If the agreement says that accumulated interest also earns interest, then the payments will increase and our tools for simple interest no longer apply.

For simple interest we could smoosh a year's worth of payouts together because they were all the same size. With compound interest we cannot do that any more.

In real life most loans have a minimum monthly payment. For most loans, a smart borrower will try to pay off the loan quickly if possible. However, payments make the math more complicated. We are not ready for that yet. We will start by looking at a loan with compound interest that has no payments.

Compound Interest Growth

The picture to the left shows loan with compound interest for which the borrower never makes any payments. The loan again starts on the left side with only the principal (the blue bar). As time goes on we move to the right. The first year's fee is a percentage of the principal (the first and lowest green bar). Then things change. The second's year payout is based on both the principal (adding a second green bar) and the first year's interest (adding the first and lowest purple bar). In the third year there are four bars that contribute to the third payout. And so forth. The amount of the loan grows at an increasing pace, faster and faster.

For simple interest we could smoosh a year's worth of payouts together because they were all the same size. Now we cannot do that any more.

Here is a sample monthly compound interest spreadsheet. Notice that the annual interest rate of 22% is still divided by twelve to be shared among the twelve monthly payments.

compound interest table

That table should remind you of our in-class activity about six years of rent increases. You have already invented the compound interest formula!

The Compound Interest Formula

Final Amount = Principal × (1 + rate per payout)number of payouts

The end of the formula should be an exponent! If your display wraps it to the next line, please shrink the text so it properly looks like an exponent.

47. A $100 loan has a 3% annual interest rate, compounded monthly. (That means payouts happen monthly.) What is the total loan amount after the first 18 months?

47. First we find the interest rate per payout.

0.03 per year ÷ 12 payouts per year = 0.0025 per payout

Then we use the compound interest formula:
Final Amount = Principal × (1 + interest rate per payout)number of payouts
     = $100 × 1.002518
     ≈ $104.60

Notice that this formula gives us the total, final amount. If we were asked to only find the interest, we would need to subtract away the principal.

48. A $100 loan has a 3% annual interest rate, compounded monthly. What is the interest after the first 18 months?

48. As with the previous problem we find that the total loan amount is $104.60.

Then we subtract away the principal of $100.

$104.60 − $100 = $4.60

Compound interest is not especially significant with a small time span. Compare that problem to the earlier, nearly identical situation with simple interest. The simple interest was $4.50. The compound interest was $4.60. The short short time span did not allow the interest to build on itself much.

However, with a larger principal amount and (especially important) decades of time the effect of compounding can be huge. Let's do two more problems to see it happen.

49. If Ms. Caruso invests $2,000 today at 9% annual simple interest how much will she have after 30 years?

49. Simple Interest = $2,000 principal × 0.09 annual rate × 30 years = $5,400

Then add back the prinicipal. $2,000 + $5,400 = $7,400

50. If Ms. Caruso invests $2,000 today at 9% annual interest, compounded monthly, how much will she have after 30 years?

50. First we find the interest rate per payout.

0.09 per year ÷ 12 payouts per year = 0.0075 per payout

Next we find the number of payouts.

30 years ÷ 12 payouts per year = 360 payouts

Then we use the compound interest formula:
Final Amount = Principal × (1 + interest rate per payout)number of payouts
     = $2,000 × 1.0075360
     ≈ $29,461.15

With that larger principal and longer time span, switching from simple interest to compound interest made a huge difference!

Most real-life compound interest with bank accounts and credit cards happens monthly. But there are other options.

51. Complete this chart to explore various frequences if payouts.

Payout Frequencies Example

NameMeaningPayouts per Year
annuallyonce per year
monthlyonce per month
quarterlyonce per quarter
weeklyonce per week
semiannuallytwice per year
semimonthlytwice per month
biweeklyevery other week

51. annually = 1 per year
monthly = 12 per year
quarterly = 4 per year
weekly = 52 per year
semiannually = 2 per year
semimonthly = 24 per year
biweekly = 26 per year.

Some people prefer a less generic compound interest formula. They do not mind have different versions if the formula for different time spans of compounding.

As before, the end of these formulas should be an exponent! If your display wraps stuff to the next line, please shrink the text so it properly looks like an exponent.

The Compound Interest Formula for Annual Compounding

Final Amount = Principal × (1 + annual interest rate)years


The Compound Interest Formula for Monthly Compounding

Final Amount = Principal × (1 + annual interest rate ÷ 12)(years × 12)


The Compound Interest Formula for Quarterly Compounding

Final Amount = Principal × (1 + annual interest rate ÷ 4)(years × 4)


The Compound Interest Formula for Weekly Compounding

Final Amount = Principal × (1 + annual interest rate ÷ 52)(years × 52)

 

As a concluding comment, the compound interest formula often does not apply to real life. Even though most loans do have compound interest with monthly payouts, there are also deposits and withdrawls. In real life it is rare to makes a single initial deposit or charge, and then passively watch the interest grow! We will deal with that more practical but complicated situation later.

videos

Khan Academy

Introduction to Compound Interest

Mathispower4u

Compound Interest

Compounded Interest Formula - Quarterly

Compounded Interest Formula - Determine Deposit Needed (Present Value)

McClendon Math

How to compute compound interest

AcademicLeadersEd

Compound Interest: Easy example and practice

ProfessorMcComb

Doubling an investment compounded monthly at 3%

Annual Effective Interest

A special case of the compound interest formula happens when the principal is $1, the time for each payout is less than a year, and the number of payouts exactly fits a single year. This situation would measure how much compounding happens each year.

In other words, there really is interest compounding during that year, but we are going to "summarize" the increase as if it were simple interest. Accountants call this the annual effective interest rate or AEIR.

We can write the normal compound interest formula for this situation.

Final Amount = $1 × (1 + interest rate per payout)number of payouts

Then we make two changes. First, multiplying by $1 of principal does nothing, so we can leave that out. Second, we are really looking for the interest, not the final amount, so we subtract the $1 principal at the end.

The Annual Effective Interest Rate Formula

The number of payouts must fit a single year.

AEIR = (1 + interest rate per payout)number of payouts − 1

52. A credit card has a 24% annual interest rate. Payouts happen monthly. The loan uses compound interest. What is the annual effective interest rate?

52. First we find the interest rate per payout.

0.24 per year ÷ 12 payouts per year = 0.02 per payout

Then we use the annual effective interest rate formula:
AEIR = (1 + interest rate per payout)number of payouts − 1
     = 1.0212 − 1
     ≈ 0.268
     = 26.8%

So this credit card behaves like a simple interest rate of 26.8% regarding its effect on your wallet!

The Rule of 72

In real life we sometimes want a quick and easy estimate about how long it takes for a loan or investment to double. That might quickly help us plan for the future.

Let's use the compound interest formula to watch an investment grow. We can look for a pattern.

53. Imagine that you invest $1,000 in an account that earns 10% interest every year, compounded annually. How many years will it take to double your money?

53. We can use the compound interest formula to find the final amounts as years go by.

After 1 Year = $1,000 × 1.11 = $1,100

After 2 Years = $1,000 × 1.12 = $1,210

After 3 Years = $1,000 × 1.13 = $1,331

After 4 Years = $1,000 × 1.14 = $1,464.10

After 5 Years = $1,000 × 1.15 = $1,610.51

After 6 Years = $1,000 × 1.16 ≈ $1,771.56

After 7 Years = $1,000 × 1.17 ≈ $1,948.72

After 8 Years = $1,000 × 1.18 ≈ $2,143.59

The original investment had almost doubled after 7 years, and has more than doubled after 8 years.

54. Imagine that you invest $1,000 in an account that earns 7% interest every year, compounded annually. How many years will it take to double your money?

54. We can use the compound interest formula to find the final amounts as years go by.

After 1 Year = $1,000 × 1.071 = $1,070

After 2 Years = $1,000 × 1.072 = $1,144.90

After 3 Years = $1,000 × 1.073 ≈ $1,225.04

After 4 Years = $1,000 × 1.074 ≈ $1,310.80

After 5 Years = $1,000 × 1.075 ≈ $1,402.55

After 6 Years = $1,000 × 1.076 ≈ $1,500.73

After 7 Years = $1,000 × 1.077 ≈ $1,605.78

After 8 Years = $1,000 × 1.078 ≈ $1,718.19

After 9 Years = $1,000 × 1.079 ≈ $1,838.46

After 10 Years = $1,000 × 1.0710 ≈ $1,967.15

After 11 Years = $1,000 × 1.0711 ≈ $2,104.85

The original investment had almost doubled after 10 years, and has more than doubled after 11 years.

investment doubling chart Here is a chart with even more information.

Can you see a pattern?

In our first example 10 rate × 7 years = 70. That is a little below 72.

In our second example 7 rate × 10 years = 70. The is also a little below 72.

The chart show more examples when doubling happens when the rate (written as a whole number, not a decimal) times the years equals 72.

When the product is slightly below 72, as with our two examples, that means the doubling has not quite happened yet but will during the upcoming year.

The Rule of 72 is a way to estimate how long it takes for doubling to happen.

The Rule of 72 (for Doubling)

The Rule of 72 says

annual interest rate × years until doubling ≈ 72

Notice that we pretend the annual interest rate is a whole number. We do not turn it into a decimal.

55. Imagine that you invest $1,000 in an account that earns 2% interest every year, compounded annually. Use the Rule of 72 to estimate how many years will it take to double your money.

55. The Rule of 72 says 2 rate × years until doubling ≈ 72

We divide both sides by 2 to find an answer of 36 years.

A very long time! No one can save for retirement with only a 2% annual interest rate.

The mobile game State of Survival allows players to invest their biocaps (a zombie apocalypse currency) in four different ways.

rate comparison from State of Survival

Before doing the next four example problems, think about which of these four investment options is the best, and by how much.

  1. Use the Rule of 72 to estimate how much your investment would increase after 4 months a rate of 5% per day.

The Rule of 72 says your currency would double about every 72 ÷ 5 = 14.4 days

Four months is roughly 30 days × 4 = 120 days. So you would have time for your currency to double about 120 days ÷ 14.4 ≈ 8 times.

You would have a bit more than 28 = 256 times your original investment.

  1. Use the Rule of 72 to estimate how much your investment would increase after 4 months a rate of 5% per week.

The Rule of 72 says your currency would double about every 72 ÷ 5 = 14.4 weeks

Four months is roughly 16 weeks. So you would have time for your currency to double once, and bit of extra time.

You would have a bit more than double your original investment.

  1. Use the Rule of 72 to estimate how much your investment would increase after 4 months a rate of 20% every 15 days.

The Rule of 72 says your currency would double about every 72 ÷ 20 = 3.6 sets of 15 days

Four months is roughly 8 sets of 15 days. So you would have time for your currency to double about 8 sets ÷ 3.6 ≈ 2 times.

You would have a bit more than 4 times your original investment.

  1. Use the Rule of 72 to estimate how much your investment would increase after 4 months a rate of 50% every month.

The Rule of 72 says your currency would double about every 72 ÷ 50 = 1.44 months

So you would have time for your currency to double almost 4 months ÷ 1.44 ≈ 3 times.

You would have a bit less than 8 times your original investment.

The first option is incredibly better than any of the other options!

Does the Rule of 72 also work when an initial amount of money shrinks?

56. Imagine that you inherit $1,000. Each year you spend 10% of what is left. How many years will it take to halve your inheritance money?

56. We can use the compound interest formula to find the final amounts as years go by.

Instead of using the One Plus Trick to set (1 + rate) as 1.1, we need to use the One Minus Trick to set (1 + rate) as (1 − 0.1) = 0.9

After 1 Year = $1,000 × 0.91 = $900

After 2 Years = $1,000 × 0.92 = $810

After 3 Years = $1,000 × 0.93 = $729

After 4 Years = $1,000 × 0.94 = $656.10

After 5 Years = $1,000 × 0.95 = $590.49

After 6 Years = $1,000 × 0.96 ≈ $531.44

After 7 Years = $1,000 × 0.97 ≈ $478.30

The original investment had almost halved after 6 years, and has more than halved after 7 years.

Yes, the Rule of 72 again provided a close estimate. 10 rate × 7 years ≈ 72

Let's check again.

57. Imagine that you inherit $1,000. Each year you spend 7% of what is left. How many years will it take to halve your inheritance money?

57. We can use the compound interest formula to find the final amounts as years go by.

Instead of using the One Plus Trick to set (1 + rate) as 1.07, we need to use the One Minus Trick to set (1 + rate) as (1 − 0.07) = 0.93

After 1 Year = $1,000 × 0.931 = $930

After 2 Years = $1,000 × 0.932 = $864.90

After 3 Years = $1,000 × 0.933 ≈ $804.38

After 4 Years = $1,000 × 0.934 ≈ $748.05

After 5 Years = $1,000 × 0.935 ≈ $695.69

After 6 Years = $1,000 × 0.936 ≈ $646.99

After 7 Years = $1,000 × 0.937 ≈ $601.70

After 8 Years = $1,000 × 0.938 ≈ $559.58

After 9 Years = $1,000 × 0.939 ≈ $520.41

After 10 Years = $1,000 × 0.9310 ≈ $483.98

The original investment had almost halved after 9 years, and has more than halved after 10 years.

Yes, the Rule of 72 again provided a close estimate. 7 rate × 10 years ≈ 72

Hooray! The Rule of 72 works both ways.

The Rule of 72 (for Halving)

The Rule of 72 says

annual interest rate × years until halving ≈ 72

Notice that we pretend the annual interest rate is a whole number. We do not turn it into a decimal.

videos

Khan Academy

The Rule of 72

Ten Exercises

Try these ten exercises on scratch paper. Work in a study group if you can! Notice where your notes need improvement. After you are very happy with your answers, you can use this form to ask me to check your work. Can you get at least 8 out of 10 correct?

1. Tom buys a painting for $100. His friend, Eric, buys a painting for $200. Both paintings increasse in value by $30. What is the percent increase for each?

2. A business buys a copy machine for $2,500 by borrowing that $2,500 with a loan of 15% simple interest for three years and three months. What is the total cost (the copy machine plus the loan's interest)?

3. How much was borrowed at 120% annual simple interest for two weeks if the interest was $16.15?

4. An annual compound interest rate of 18% compounded monthly means that each monthly payout will have a monthly interest rate of

5. Find the number of compounding periods in 7 years and 6 months (ignore leap years).

6. If $3,865 is invested with 9% annual compound interest for 40 years, how much will it grow to be worth? (Hint: If no time period for payouts is mentioned, it has annual compounding.)

7. When Huey, Dewey, and Louie entered kindergarten their uncled started a college account for them. Each account had $5,000. That one deposit grew at 9% annual interest for 13 years. For Huey the interest was compounded weekly. For Dewey the interest was compounded monthly. For Louie the interest was compounded annually. How much was each account worth at the end of the 13 years?

8. Bluebeard has a balance of $1,834.90 on a credit card with an annual percentage rate of 22.4%. This month's minimum payment is $36.70. How much less than this minimum payment is the interest? (In other words, if he only pays the minimum payment, how much goes "past" the interest to pay off his principal?)

9. A credit card has an 18% annual interest rate. Payouts happen monthly. The loan uses compound interest. What is the annual effective interest rate?

10. Mr. Largo gives his 20-year-old sibling a $100 wedding present. The sibling invests it at 11% annual compound interest for 30 years. Then the sibling adds an extra $10,000, and moves the combined funds in a different investment that earns 4% annual compound interest for 10 years. What is the final value, when the sibling is 60 years old? (Hint: If no time period for payouts is mentioned, it has annual compounding.)

Random Exercises

Try these exercises on scratch paper. Work in a study group if you can! Notice where your notes need improvement. Check your work when you are done.

Percent Change (Review)

The One Plus Trick (Review)

Simple Interest


Saving for Retirement

For retirement brings repose, and repose allows a kindly judgment of all things.
    - John Sharp Williams

Often when you think you're at the end of something, you're at the beginning of something else.
    - Fred Rogers

Annual Annuity

Sum of Annuity Due

A glaring weakness of the simple and compound interest formulas was that the principal was the only deposit.

Saving for retirement never works like that! Most households deposit money each year in a retirement account.

We saw that patterns make formulas. In real life people save a different amount every year. They save less when they are young and have less income. They save more as they middle-aged and have more income. They save less when a year has unusually large expenses such as medical bills or children attending college. They save more when they inherit. Real life does not make nice patterns.

So we will simplify what actually happens in real life.

What if a household saves the same amount with every deposit, and does so once per year. That oversimplication can still help us understand more deeply what it means to save for retirement.

The result is called the Annual Increasing Annuity formula.

The Annual Increasing Annuity Formula (January 1st Version)

Final Amount = Deposit × (1 + rate) × [ (1 + rate)years    −    1] ÷ rate

Heads up: most books use the word "principal" for this annual, repeated deposit.

Notice that when using this formula on a calculator you do not actually need nested parenthesis. The bits that look like (1 + rate) can just become numbers. For example, with a 5% rate we can just write 1.05.

This formula might look big, but it is actually much easier to use than the compound interest formula. We no longer need to split up the annual interest rate into monthly or quarterly bits. With the sum of annuity due formula everything is annual.

58. Someone saves $1,200 at the start of each year for retirement. If this is invested with 8% annual interest over 45 years, how much will it grow to be worth?

Final Amount = Deposit × (1 + rate) × [ (1 + rate)years    −    1] ÷ rate

= $1,200 × 1.08 × ( 1.0845 − $1 ) ÷ 0.08

$500,911

Notice that the above formula was labeled the "January 1st Version" because whomever is saving started at the very beginning of the year.

Some people behave differently, and save at the end if the year. That means their first deposit earns no interest, since it only appears in their account on December 31st. So the "one plus trick" happens one fewer time. Fortunately, the previous formula has an instance of the "one plus trick" that we can easily remove, sitting to the right of the deposit amount.

The Annual Increasing Annuity Formula (December 31st Version)

Final Amount = Deposit × [ (1 + rate)years    −    1] ÷ rate

As before, keep in mind that most books use the word "principal" for this annual, repeated deposit.

59. Someone saves $1,200 at the end of each year for retirement. If this is invested with 8% annual interest over 45 years, how much will it grow to be worth?

Final Amount = Deposit × [ (1 + rate)years    −    1] ÷ rate

= $1,200 × ( 1.0845 − $1 ) ÷ 0.08

$463,806.74

The difference for delaying the first deposit until the end of the first year is quite noticeable!

Monthly Annuity

Some people, and most job-related retirement funds, make deposits monthly instead of annually.

Exactly parallel to when we used the compound interest formula, we can generalize the above formulas by changing "rate" to "rate per payout", and "years" to "payouts".

In our class we will ignore the extremely unrealistic case of a households that saves for retirement with identical weekly or quarterly deposits. Years or months are the only situations we will use. Remember that in real life almost everyone saves less when young, and more when older. These formulas are approximations, very helpful for getting a sense of how long-term saving works. But we need not go crazy trying to cover every situation.

For a similar reason, we will assume that any retirement system with monthly deposits does so at the end of the month.

Putting those decisions together, we create our monthly version of the formula by changing "rate" to "monthly rate", and "years" to "months" in the December 31st version of the annual increasing annuity formula.

The End-of-Monthly Increasing Annuity Formula

Final Amount = Deposit × [ (1 + monthly rate)number of months    −    1] ÷ monthly rate

Remember to divide the annual rate by 12 to find the monthly rate.

As before, keep in mind that most books use the word "principal" for this annual, repeated deposit.

Payout Annuity

Remember the amortization table we used when finding mortages? We defined amortization as the paying off of debt with repeating, scheduled repayments.

That sounds a lot like the opposite of these increasing annuity formulas.

You can think of amortization as "decreasing annuity". An account begins with a certain amount of money (the principal). The account does grow because of interest. But it also shrinks because of regularly repeated identical withdrawls at the end of every time period.

This situation is also called a Payout Annuity. This term also refers to a type of account that automates the process for a retiree. These accounts usually have a large annual fee that most investors should avoid. But a retiree with money to spare might sensibly decide that he or she has done enough of personally managing investments. For that person, moving his or her savings into a payout annuity account can be worthwhile. All done with thinking about asset allocation, capital gains tax, and everything else! Receiving guaranteed regular income during your retirement years without needing to buy or sell investments is indeed relaxing.

The formula for a decreasing annuity is pretty similar. Notice it has a negative exponent.

The Decreasing Annuity Formula (aka Amortization or Payout Annuity)

Principal = Withdrawl × [ 1  −  (1 + rate per payout) number of payouts ] ÷ rate per payout

Notice that retirees with payout annuity accounts do match this formula to real life. Although few people save using identical deposits, many people do take retirement income with identical withdrawls.

Moreover, these retirees can choose to receive their withdrawls weekly, monthly, quarterly, or annually.

So we keep this formula in its generic version. You might probably to divide the annual rate to find the rate per payout, and might need to multiply a given number of years to find the number of payouts.

This same formula works for mortgages, car loans, and any other situation in which increasing interest competes with regularly scheduled identical withdrawls.

videos

Khan Academy

Annuity Due and Future Value

Mathispower4u

Determining The Value of an Annuity

The Installment Loan Formula

Payout Annuity Formula - Part 1

Payout Annuity Formula - Part 2

How Much to Save for Retirement

Typical Retirement

market increase image

The general rule for retirement savings is to, at age 65, have saved $20 for each dollar that your retirement expenses will exceed your retirement income (from Social Security, pensions, etc.).

In other words, save as much as you need for the twenty years from age 65 to 84. Then the interest earned during those twenty years pays for the years of age 85 and beyond.

Typical American retirement expenses are greater than retirement income by about $20,000 per year. This means that for most Americans a good plan is to have $400,000 saved for retirement at age 60.

A second guideline guideline has more detail through the years. It recommends saving multiples of your salary by certain ages: ×1 at age 30, ×3 at age 40, ×6 at age 50, and ×8 at age 60.

The two guidelines match if your salary at age 60 is $50,000. $50,000 × 8 at age 60 = $400,000 saved at age 60.

The specific retirement goals for a household depend on many factors. For the example problems we do, we will use the goal of saving $400,000 by age 60.

Cindy, Clara, and Chloe

Cindy, Clara, and Chloe are three sisters. Each has plans to save for retirement, but their plans are somewhat different.

Despite their different lives, by the time each was 25 years old she had the ability to set aside $3,000 per year for retirement.

three sisters retirement problem chart

All three use a retirement account that earns 8% annual compound interest.

Which of them end up saving at least $400,000 for retirement?

Which of them saves the most?

59b. How much will Cindy have saved when she retires at age 65?

This is a two part problem. For the first part, we look at the first ten years. Regular deposits tell us to use the sum of annuity due formula.

Final Amount = Deposit × (1 + rate) × [ (1 + rate)years    −    1] ÷ rate

= $3,000 × 1.08 × (1.0810 − 1 ) ÷ 0.08

≈ $46,936.46

For the second part, we look at the last thirty years. Merely watching an initial amount grow tells us to use the compound interest formula.

Final Amount = Principal × (1 + interest rate per payout)number of payouts

= $46,936 × 1.0830

$472,306

Cindy does save at least $400,000 for retirement. She also only had to save for ten years. By starting early, she benefitted from the tremendous power of earning interest over a long time period.

60. How much will Clara have saved when she retires at age 65?

For the first ten years no savings happen!

The only math is for the last thirty years. Regular deposits tell us to use the sum of annuity due formula.

Final Amount = Deposit × (1 + rate) × [ (1 + rate)years    −    1] ÷ rate

= $3,000 × 1.08 × (1.0830 − 1 ) ÷ 0.08

$367,038

Clara almost saves at least $400,000 for retirement, but fails. That happened even though she put a lot of money into savings for thirty years! By starting late, she missed out on much of the benefit of earning interest over a long time period.

61. How much will Chloe have saved when she retires at age 65?

For all forty years regular deposits tell us to use the sum of annuity due formula.

Final Amount = Deposit × (1 + rate) × [ (1 + rate)years    −    1] ÷ rate

= $1,500 × 1.08 × (1.0840 − 1 ) ÷ 0.08

$419,672

Chloe does save at least $400,000 for retirement. She did start early enough to benefit from the power of earning interest over a long time period.

In real life, most households are a mix of Cindy and Chloe. They do not save anything in early years. Then they save some money but not enough.

For many households, the best way to save more is to kick a bad habit. The next problem uses cigarettes as an example of an expense that could be changed into savings. Perhaps for your household, the problem should instead discuss eating at restaurants too often, or buying fancy morning coffees?

cigarette money

62. Cliff is Clara's brother. He stops smoking at age 25, and decides to devote the money he used to spend on cigarettes to retirement. He used to smoke 1 pack per day, at $5.70 per pack. How much money per year was Cliff spending on cigarettes?

62. $5.70 × 365 = $2,080.50

63. If he instead puts that annual amount into a retirement account that earns 6% annual compound interest, how much will extra will he have for retirement after forty years?

63. For all forty years regular deposits tell us to use the sum of annuity due formula.

Final Amount = Deposit × (1 + rate) × [ (1 + rate)years    −    1] ÷ rate

= $2,080.50 × 1.06 × (1.0640 − 1 ) ÷ 0.06

$341,301

Cliff almost saves the $400,000 his household wants for retirement simply by changing his cigarette money into savings!

When you did the math for Clara, did you think to yourself, "Bah! Who has $3,000 per year to put into savings at age 25? This is unrealistic!" That complaint does make sense.

But Cliff shows that sometimes a lifestyle change is sufficient to claim the power of starting to save early. Every bit counts! Cliff and his household will barely need to do any other savings to be ready for retirement.

Ten Exercises

Try these ten exercises on scratch paper. Work in a study group if you can! Notice where your notes need improvement. After you are very happy with your answers, you can use this form to ask me to check your work. Can you get at least 8 out of 10 correct?

1. Tom buys a painting for $100. His friend, Eric, buys a painting for $200. Both paintings increasse in value by $30. What is the percent increase for each? How can the two items have different percent appreciation if they both increased by $30?

2. A business buys a copy machine for $2,500 by borrowing that $2,500 with a loan of 15% simple interest for three years and three months. What is the total cost (the copy machine plus the loan's interest)?

3.payday advance loan neon sign Oregon Senate Bill 1105 limits payday loan interest rates to 36% or less. Before that bill was passed, payday loans in Oregon often had interest rates of 120%. Nationally, payday loans can have interest rates as high as 7,000%. How much was borrowed at 120% annual simple interest for two weeks if the interest was $16.15?

(Note: Here is an article describing how check-cashing businesses can be helpful for their community. Nationally, the average payday loan is a two-week advance on $350.)

4.Huey, Dewey, and Louie When Huey, Dewey, and Louie entered kindergarten their uncled started a college account for them. Each account had $5,000. That one deposit grew at 9% annual interest for 13 years. For Huey the interest was compounded weekly. For Dewey the interest was compounded monthly. For Louie the interest was compounded annually. How much was each account worth at the end of the 13 years?

5.Einstein Mr. Largo gives his 20-year-old sibling a $100 wedding present. The sibling invests it at 11% annual compound interest for 30 years. Then the sibling adds an extra $10,000, and moves the combined funds in a different investment that earns 4% annual compound interest for 10 years. What is the final value, when the sibling is 60 years old?

6. A credit card has an 18% annual interest rate. Payouts happen monthly. The loan uses compound interest. What is the annual effective interest rate?

7. If $3,865 is invested with 9% annual compound interest for 40 years, how much will it grow to be worth? How much less than $176,696 is your answer? These questions are interesting because research by Vanguard shows that the average 25-year-old American has saved $3,865 for retirement, and the average 65-year-old American has saved $176,696. We are examining how much more the average American saves beyond the nest egg they have at age 25.

average savings chart

8. Fiddle around using the sum of annuity due formula to find what annual deposit would grow to $176,696 over 40 years at 9% annual interest. (Hint: an annual deposit of $400 is too small, but $550 is too big.)

The two previous problems show us that it is easy to save as much as an average American. Setting aside less than $500 per year is enough to be average. Do not be embarassed if you start small with retirement savings. About half of Americans have a below average nest egg at age 25. You can become above average by saving more later in life as your career progresses.

child care photo from the Michigan Health Lab

9. How does the long-term financial cost of infant care compare to saving for kids' college expenses? The "big three" child care centers in Eugene cost roughly $13,000 per year. (Vivian Olum at UO, Early Learning Childrens Community at LCC, and Child Development Center at EWEB. Oregon is an unusually expensive state for child care.) Use the sum of annuity due formula for two years (when the infant is age 1 and 2), and then the compound interest formula for sixteen years (until the child is 18). If the family instead saved the infant care money and earned a 5% annual interest rate, how much would be saved towards the child's college tuition?

Archer Queen from Clash of Clans

10. In a certain computer game the archer queen gets 10% stronger every time she gains a level. Two brothers play the game. The younger brother has a level 20 archer queen. The older brother has a level 27 archer queen. Roughly how much stronger is the older brother's archer queen?

Random Exercises

Try these exercises on scratch paper. Work in a study group if you can! Notice where your notes need improvement. Check your work when you are done.

Interlude from Factfulness

Factfulness book cover Conquering the Fear Instinct

The fourth chapter of Factfulness includes many examples of scary things, and a careful discussion about the different between risk and fear. Hans Rosling writes:

"The risk something poses to you depends not on how scared it makes you feel, but on a combination of two things. How dangerous is it? And how much are you exposed to it?"

He summarizes that as a formula: Risk = danger × exposure

Then he writes, "The world seems scarier than it is because what you hear about has been selected—by your own attention filter or by the media—precisely because it is scary."

Look at the data in the Fatality Rate Playground. That list has something surprising for almost everyone. What risks have you been worrying about too much? What risks should you pay more attention to?

Tangentially, investing can be scary. But it is no different from other risks. There are dangers, and there are exposures, and you can learn about those to minimize your risk.

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