Compounding under the microscope.

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*3!4!/5!
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Compounding under the microscope.

Post by *3!4!/5! »

This is how compounding works.

Suppose you have an investment that starts with value $1000 and the value fluctuates over time, for example
$1000
$1250
$1280
$640
$1600
$500
(Wild ride, but it's just an example to make a point while using round numbers.) So
$1000 was multiplied by 1.25 to get $1250
$1250 was multiplied by 1.024 to get $1280
$1280 was multiplied by 0.5 to get $640
$640 was multiplied by 2.5 to get $1600
$1600 was multiplied by 0.3125 to get $500
(Some people would express this in terms of percentage changes +25%, +2.4%, -50%, +150%, -68.75% but there is really never any reason to talk in terms of percentages except to communicate with people who talk in that obsolescent language.)

So here's where compounding comes in. We just multiply these factors
1.25 * 1.024 * 0.5 * 2.5 * 0.3125 = 0.5
But that's just the multiplication factor to go from the initial value to the final value!
$1000 * 0.5 = $500

That's it! That's compounding!

-------------------------------------------------------

Let's make it more general with arbitrary values, $a,$b,$c, ...

Suppose you have an investment that starts with value $a and the value fluctuates over time, for example
$a
$b
$c
$d
$e
$f
(Maybe a wild ride, maybe not. For the math here, it is irrelevant whether the value is increasing or decreasing or oscillating, nor whether the jumps are small or large.) So
$a was multiplied by (b/a) to get $b
$b was multiplied by (c/b) to get $c
$c was multiplied by (d/c) to get $d
$d was multiplied by (e/d) to get $e
$e was multiplied by (f/e) to get $f
(Some people would express this in terms of percentage changes [100(b-a)/a]%, [100(c-b)/b]%, etc, but that would be nuts.)

So here's where compounding comes in. We just multiply these factors
(b/a) * (c/b) * (d/c) * (e/d) * (f/e) = (f/a)
But that's just the multiplication factor to go from the initial value to the final value!
$a * (f/a) = $f

That's it! That's compounding!

-------------------------------------------------------

One more generalization, you can make the sequence any length.

Values:
v_0,v_1,v_2,v_3,...,v_{n-1},v_n

Multiplication factors from each values to the next:
(v_1/v_0), (v_2/v_1), (v_3/v_2), ... , (v_n/v_{n-1}),

So when you multiply these:
(v_1/v_0) * (v_2/v_1) * (v_3/v_2) * ... * (v_n/v_{n-1}) = (v_n/v_0)

and that's just the multiplication factor to go from the initial value to the final value!
v_0 * (v_n/v_0) = v_n

That's all there is to compounding! There's no "power" or "magic" to it. It's just plain simple math.

It basically comes down to the formula
(b/a) * (c/b) = (c/a)
Once you get that, you pretty much know everything there is to know about compounding. The rest is hype.
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Re: Compounding under the microscope.

Post by dbr »

I'm with you on the math and I'm with you on your rejection of hype.
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Re: Compounding under the microscope.

Post by qwertyjazz »

The magic is the belief that I hold that over a long enough time frame for the kind of things I 'invest' in that the final multiple is greater than one. That might be considered by an English major as 'Magical Realism'
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Re: Compounding under the microscope.

Post by nisiprius »

On the one hand... I do not like the "power and magic" stuff. It's overhyped. And it constantly leaves out the fact that inflation compounds, and the cumulative probability of avoiding societal or personal "black swans" compounds, too.

On the other hand... there is a chunk of mathematical and finance literacy that comes from understanding compounding, aka geometrical progression, aka exponential growth. A lot of people in STEM fields have been working with it for so long that it is second nature and we forget that it wasn't always obvious. (As witness the poster who thought I was doing something invalid in drawing a straight line on a semilog chart of a mutual fund to demonstrate that it had had a steady annual growth). (And don't get me started on Dave Ramsey).

On the third hand... (well, the "Car Talk" guys always have a "third half" of the program...) I assume everybody has noticed that in days of 0.5%, 1%, 2% returns... there's not too much magic in compounding. Twenty years of growth at 1% interest gives you a total cumulative gain of 22.019% over the whole period which, OK, it's more than 20% but "20 x 1% = 20%" isn't all that far off the mark.
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Re: Compounding under the microscope.

Post by dbr »

nisiprius wrote: A lot of people in STEM fields have been working with it for so long that it is second nature and we forget that it wasn't always obvious. (As witness the poster who thought I was doing something invalid in drawing a straight line on a semilog chart of a mutual fund to demonstrate that it had had a steady annual growth). (And don't get me started on Dave Ramsey).
You mean like around 300 BC or onwards of 2,300 years now: http://aleph0.clarku.edu/~djoyce/elemen ... pIX35.html not that there might not be an earlier analysis. Also just a little later this one: https://en.wikipedia.org/wiki/The_Quadr ... e_Parabola

Zeno, around 450 BC, apparently did not understand geometric series.
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Re: Compounding under the microscope.

Post by Phineas J. Whoopee »

dbr wrote:...
Zeno, around 450 BC, apparently did not understand geometric series.
Zeno of Elea's point was the reasoning they were using clearly was wrong, therefore they'd better think of something else. In particular it had to do with whether not-yet-named physics was continuous or discrete. He simply used his day's logic to prove things which obviously were incorrect.
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Re: Compounding under the microscope.

Post by dbr »

Phineas J. Whoopee wrote:
dbr wrote:...
Zeno, around 450 BC, apparently did not understand geometric series.
Zeno of Elea's point was the reasoning they were using clearly was wrong, therefore they'd better think of something else. In particular it had to do with whether not-yet-named physics was continuous or discrete. He simply used his day's logic to prove that which obviously was incorrect.
PJW
Yes, that is true. I was trying to ballpark when someone in history had looked at what we recognize today as geometric series and it's properties. I think we'll award that to Euclid.

The real point is why it is so difficult to provide mathematics education that extends to some of these ideas that come up with such frequency on phenomena of every day importance. A formal mathematical development is not needed, but a description and some foundation in mathematical principles would seem to me most valuable. Also helpful is some familiarity with writing things in mathematical notation or using graphic presentations such as nisi's always helpful postings. That would be preferable to a nearly willful glazing over of the eyes too often met with (not directed in particular to any person or posting on this forum).
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Re: Compounding under the microscope.

Post by Phineas J. Whoopee »

And on that, dbr, we completely agree.
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Re: Compounding under the microscope.

Post by sport »

dbr wrote:The real point is why it is so difficult to provide mathematics education that extends to some of these ideas that come up with such frequency on phenomena of every day importance.
Sometimes the teaching of this subject is provided. However, learning can be problematic.
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Re: Compounding under the microscope.

Post by nisiprius »

Speaking of "compounding under the microscope:"
https://www.youtube.com/watch?v=zrx7Xg0gkQ4

I hate to admit it but when I was kid, this taught me something--no, not Danny Kaye singing the words that are supposed to be the message of the song, but the children's chorus in the background behind him.

https://www.youtube.com/watch?v=fXi3bjKowJU
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Re: Compounding under the microscope.

Post by Frugal Al »

Wow, "the magic of compounding" is hype, who would have thought. Obviously, we all know there's no such thing as magic-right? Of course there's no magic in what is just math, but rather it's the power in the resultant, effortless, exponential growth over time within a positive return environment, which can seem somewhat magical, because of its simplicity. No, compounding does not lend itself well to sequences with negative returns. Still, even then, it should illustrate the importance of limiting losses, and ensuring the low risk side of a portfolio is indeed low risk. And if the overall long run environment is positive, rebalancing can help mitigate the bad sequences of returns.

And as far as inflation negating the effect of compounding, that only reinforces the need to get compounding working for you, rather than allowing inflation to work compounding against you.
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Re: Compounding under the microscope.

Post by gilgamesh »

We all know it's not magic and it's just math, but where does the 'magic' come from?

It comes from this. At 6% return it takes ~12 years to make $100,000 from your first $100,000, and with the same 6% return it only takes 6.77 years to make the next $100,000. When it reaches $1M it only takes 1.59 years to make the next $100,000 with the same 6% return. This 'ability to shrink time' is what's perceived/imagined to be 'magic'. Nothing wrong with that.
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Re: Compounding under the microscope.

Post by Sheepdog »

gilgamesh wrote:We all know it's not magic and it's just math, but where does the 'magic' come from?
This 'ability to shrink time' is what's perceived/imagined to be 'magic'. Nothing wrong with that.
No compounding is not magic, it's just math, but when you see the result over time, it's vision stands out.
When I purchased $10,000 worth of I Bonds (then 3.6% plus rate of inflation) in June 2000 (and more over the next couple of years) because Mel Lindauer, at M* Vanguard Diehard forum, hyped us to "back up the truck and buy I Bonds" and now I see that they are today worth $26,092 and will be worth $26,508 in May 2017 to conclude 17 years of compounding, that is the vision to show the magic (and beauty) of compounding.
We all can use a little 'magic' in our lives.
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Re: Compounding under the microscope.

Post by dbr »

nisiprius wrote:
I hate to admit it but when I was kid, this taught me something--no, not Danny Kaye singing the words that are supposed to be the message of the song, but the children's chorus in the background behind him.

https://www.youtube.com/watch?v=fXi3bjKowJU
I don't know if you mean this as a positive or a negative, but to me manipulating a beautiful mathematical concept into a tedious recitation may help explain why these concepts are not learned.

On the other had the example under the microscope shows why the idea is important.
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Re: Compounding under the microscope.

Post by Dottie57 »

gilgamesh wrote:We all know it's not magic and it's just math, but where does the 'magic' come from?

It comes from this. At 6% return it takes ~12 years to make $100,000 from your first $100,000, and with the same 6% return it only takes 6.77 years to make the next $100,000. When it reaches $1M it only takes 1.59 years to make the next $100,000 with the same 6% return. This 'ability to shrink time' is what's perceived/imagined to be 'magic'. Nothing wrong with that.
+1

I think saving / putting more money into the investment also spurs the "magical" aspect.
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Re: Compounding under the microscope.

Post by telemark »

Compare a fifteen year mortgage with a thirty year one and you'll really see the effects of compounding. Although there isn't usually a lot of sales talk about "magic" in that case.
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Re: Compounding under the microscope.

Post by Phineas J. Whoopee »

sport wrote:
dbr wrote:The real point is why it is so difficult to provide mathematics education that extends to some of these ideas that come up with such frequency on phenomena of every day importance.
Sometimes the teaching of this subject is provided. However, learning can be problematic.
If one student fails to learn, the student may be at fault. If many students fail to learn, the school is at fault.
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Re: Compounding under the microscope.

Post by *3!4!/5! »

The thing to understand is that the compounding math works just the same, regardless of the sequence of values. The sequence could be anything. There's nothing that says there should have exponential growth (or decay) (i.e. a geometric sequence), that's only the very special case when all the ratios between successive values are equal to one constant number.

There needs to be a good reason for there to be exponential growth/decay, or approximate exponential growth/decay. It's sometimes observed, but it is by no means guaranteed. Just look at the Nikkei. That's compounding at work.
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Re: Compounding under the microscope.

Post by dbr »

Phineas J. Whoopee wrote:
sport wrote:
dbr wrote:The real point is why it is so difficult to provide mathematics education that extends to some of these ideas that come up with such frequency on phenomena of every day importance.
Sometimes the teaching of this subject is provided. However, learning can be problematic.
If one student fails to learn, the student may be at fault. If many students fail to learn, the school is at fault.
PJW
I see a fair number of elementary age students on a daily basis. I think an informal impression would be that by third grade two-thirds of them have a definite antipathy to math. This is not necessarily related to not being very good at it. The other third seem to maintain an enthusiasm for it. I am not seeing bad teaching at the classroom level. The curriculum and even the concept of what math is seems to be coming from people who might be mathematically illiterate. Of course the grand experiment in having math education designed by real mathematicians played out its sad story with the "new math."

My theory is that a significant factor in this is that elementary school math is mainly tedious but also confusing drill in doing arithmetic computations starting with addition and subtraction problems (with regrouping if you have the modern nomenclature down) to long multiplication problems (you had better know the matrix method if you are a parent) to long division. I was a math major at one time and wouldn't ever have an enthusiasm for that stuff myself. The one's that are saved are the one's that get into something like Lego League, Math Olympiad, or Continental Math or generally anything that involves playing with and manipulating a computer.
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Re: Compounding under the microscope.

Post by dbr »

*3!4!/5! wrote:The thing to understand is that the compounding math works just the same, regardless of the sequence of values. The sequence could be anything. There's nothing that says there should have exponential growth (or decay) (i.e. a geometric sequence), that's only the very special case when all the ratios between successive values are equal to one constant number.

There needs to be a good reason for there to be exponential growth/decay, or approximate exponential growth/decay. It's sometimes observed, but it is by no means guaranteed. Just look at the Nikkei. That's compounding at work.
Yes, it is correct that compounding is a more general concept than geometric series and an exponential growth function, the point being that the ratios in general, as in stock market returns, are not constant every time period. We create a myth that they are when we compute CAGR.

It is also true that the underlying reason needs to be there and may change over time. One of the most interesting insights I got into a compound growth formula is radioactive decay (ie growth can be negative). In that case the result is a consequence of no more than the assumption based on observation that the decay of any given nucleus is random with a probability that is exactly the same for all nuclei in a collection and independent of how many nuclei are present. But physics is really simple.
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Re: Compounding under the microscope.

Post by gilgamesh »

*3!4!/5! wrote:The thing to understand is that the compounding math works just the same, regardless of the sequence of values. The sequence could be anything. There's nothing that says there should have exponential growth (or decay) (i.e. a geometric sequence), that's only the very special case when all the ratios between successive values are equal to one constant number.

There needs to be a good reason for there to be exponential growth/decay, or approximate exponential growth/decay. It's sometimes observed, but it is by no means guaranteed. Just look at the Nikkei. That's compounding at work.
Ah! Now I see your point. Thanks!
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Re: Compounding under the microscope.

Post by gilgamesh »

dbr wrote:
Phineas J. Whoopee wrote:
sport wrote:
dbr wrote:The real point is why it is so difficult to provide mathematics education that extends to some of these ideas that come up with such frequency on phenomena of every day importance.
Sometimes the teaching of this subject is provided. However, learning can be problematic.
If one student fails to learn, the student may be at fault. If many students fail to learn, the school is at fault.
PJW
I see a fair number of elementary age students on a daily basis. I think an informal impression would be that by third grade two-thirds of them have a definite antipathy to math. This is not necessarily related to not being very good at it. The other third seem to maintain an enthusiasm for it. I am not seeing bad teaching at the classroom level. The curriculum and even the concept of what math is seems to be coming from people who might be mathematically illiterate. Of course the grand experiment in having math education designed by real mathematicians played out its sad story with the "new math."

My theory is that a significant factor in this is that elementary school math is mainly tedious but also confusing drill in doing arithmetic computations starting with addition and subtraction problems (with regrouping if you have the modern nomenclature down) to long multiplication problems (you had better know the matrix method if you are a parent) to long division. I was a math major at one time and wouldn't ever have an enthusiasm for that stuff myself. The one's that are saved are the one's that get into something like Lego League, Math Olympiad, or Continental Math or generally anything that involves playing with and manipulating a computer.
Math??? ... 19% of high school graduates cannot read....high school graduates!...read!

http://www.statisticbrain.com/number-of ... cant-read/
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Re: Compounding under the microscope.

Post by daveydoo »

*3!4!/5! wrote: Once you get that, you pretty much know everything there is to know about compounding. The rest is hype.
So who is creating all this "hype" that you refer to? I think it's mostly us, right? I see every fund under the sun bragging about its returns and about its position relative to its peer group, but I can't recall much advertising hype about the miracle of compounding -- at least not in the past decade. And anyone who fails to grasp the concept of compounding will be stumped by your algebra -- I think that's why the graphs (silly or not) are generally used.

To me, the utility of a compounding illustration is not to figure out what something is worth today, as you imply (cuz that's pretty easy) -- it's to project what the value will be x years in the future. In the example you cite, with wildly varying annual returns, that is of course impossible owing to sequence-of-return issues. But as interest rates rise, there will be real-world examples again.
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Re: Compounding under the microscope.

Post by Valuethinker »

dbr wrote:
nisiprius wrote: A lot of people in STEM fields have been working with it for so long that it is second nature and we forget that it wasn't always obvious. (As witness the poster who thought I was doing something invalid in drawing a straight line on a semilog chart of a mutual fund to demonstrate that it had had a steady annual growth). (And don't get me started on Dave Ramsey).
You mean like around 300 BC or onwards of 2,300 years now: http://aleph0.clarku.edu/~djoyce/elemen ... pIX35.html not that there might not be an earlier analysis. Also just a little later this one: https://en.wikipedia.org/wiki/The_Quadr ... e_Parabola

Zeno, around 450 BC, apparently did not understand geometric series.
The underlying problem is that human beings think in a linear fashion, but not in an exponential or geometric one. Thus they tend to overestimate effects in the short run, and underestimate them in the long run. They also tend to believe trends can go on forever, beyond the physical limits of the environment in which they take place.

That's a blindness I have observed in very senior people in industry and politics, not just in badly educated high school kids.

Examples include understanding the effects of inflation on real incomes. Or the inevitability of slowing corporate growth with size.

Like a lot of "predictable irrationalities" it's very hard to educate out something that seems quite hard wired to human beings.

If you've ever been to a country where maps are not common, or where people are not taught to read maps (e.g. many modern kids, as far as I can figure out), then you will encounter this curious blindness to something that we probably think is obvious. Or ask for directions, some time, and note how contradictory and incorrect the answers you receive will be (this also worsens with some cultures).
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Re: Compounding under the microscope.

Post by jimb_fromATL »

*3!4!/5! wrote: (Some people would express this in terms of percentage changes +25%, +2.4%, -50%, +150%, -68.75% but there is really never any reason to talk in terms of percentages except to communicate with people who talk in that obsolescent language.)
:confused

I don't think percentages are obsolete.

The formula is I=PRT. Interest = Principal x Rate x TIME. Not just I = PR. And Rate is a percentage.

Whether you express the gain or loss as a number or a percentage, it's virtually meaningless without the factor of TIME. And you need a Time-weighted unit of measure such as APY or CAGR (Annual Percentage Rate or Compound Annual Growth Rate) to evaluate how one investment compares to another.

The product of your numbers is indeed 0.5. In other words if you started with $1,000 you will have 1000 x 0.5 = $500 at the end of the period. But without knowing how long it took, it means very little.

Here's where TIME as a factor in a unit of measure comes in:
  • If it was the yearly result for a period of 5 years, the geometric mean, or CAGR or average APY is -12.945%. Pretty bad for five years, but not really all that bad for a single year in an aggressive fund ... as long as it's in a good index fund and you leave it the heck alone.

    On the other hand, if it was over a period of 5 months the average annual rate was =RATE(5, 0, 1000, -500) * 12 … which is -155.339% for comparison. In this case, you might not want to plan on taking your loss and retiring early.
Speaking of wild rides, let's take a sample from the last big market crash.
  • Using monthly performance data for Vanguard's S&P500 index fund VFINX for 439 rolling 5 month reporting periods since 01/1980, the worst 5 month period for a lump sum investment had a -102.88% Annual Percentage Rate. In just the 5 months, an investment of $10,000 would have been worth $6,388 at the end of 02/2009. Not all that far off your example.
jimb
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Re: Compounding under the microscope.

Post by MIretired »

OP:
At first I thought, why does he have to bring "percentages or ratios" into it.
But 1st; ? on what + return you need to overcome a 3% inflation.
Answer? Not return/inf=1,
not (return - inf.) =0
It is: (1-inf.)(1+ret) = 1, or it is 1/inf. * ret/1 = 1 in ratios (or ret/inf = 1, which somebody had an equation for up-post.)
So, I say; in any one year, when the rates using are for an annual rate, it is the sequence of multipliers + and - about the number 1. (1 is identifying that we are multiplying our beginning portfolio by these terms.)
Port. * [(1-inf.)(1-WR)(1+ret)]
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Re: Compounding under the microscope.

Post by nisiprius »

dbr wrote:
nisiprius wrote:I hate to admit it but when I was kid, this taught me something--no, not Danny Kaye singing the words that are supposed to be the message of the song, but the children's chorus in the background behind him. https://www.youtube.com/watch?v=fXi3bjKowJU
I don't know if you mean this as a positive or a negative, but to me manipulating a beautiful mathematical concept into a tedious recitation may help explain why these concepts are not learned. On the other had the example under the microscope shows why the idea is important.
OK, one out of two ain't bad. No, I meant it seriously. It was one of those "teachable moment" things. I was just old enough and my mental math was just good enough to "get" it that sixteen was another doubling, and thirty-two was another. It was an example of repeated multiplication by two and probably one of the first I'd ever seen. It was exponentiation... generalized exponentiation, and they didn't teach that until junior high school IIRC. A year or two later I saw a digital computer with a row of light labelled 1-2-4-8-16-32-64 etc. and it was like hearing the song. But if it doesn't do a thing for you, that's fine.
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Re: Compounding under the microscope.

Post by nisiprius »

Does everyone know this one? There is a single lily pad growing in the middle of a huge lake. Every day, it doubles in size. Calculations show that if this continues, it will exactly cover the entire lake in 30 days. How many days does it take to cover half the lake?
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Re: Compounding under the microscope.

Post by MIretired »

nisiprius wrote:Does everyone know this one? There is a single lily pad growing in the middle of a huge lake. Every day, it doubles in size. Calculations show that if this continues, it will exactly cover the entire lake in 30 days. How many days does it take to cover half the lake?
Technically you'd have to subtract the are of the original lilypad from that of the lake. But, the geometric mean (or the geometric halfway point?) would be the sqrt(30). lol?

Yep. I see I missed that totally. lol.
Last edited by MIretired on Sun Feb 26, 2017 2:52 pm, edited 2 times in total.
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Re: Compounding under the microscope.

Post by bligh »

nisiprius wrote:Does everyone know this one? There is a single lily pad growing in the middle of a huge lake. Every day, it doubles in size. Calculations show that if this continues, it will exactly cover the entire lake in 30 days. How many days does it take to cover half the lake?
29 days.
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Re: Compounding under the microscope.

Post by *3!4!/5! »

nisiprius wrote:
dbr wrote:
nisiprius wrote:I hate to admit it but when I was kid, this taught me something--no, not Danny Kaye singing the words that are supposed to be the message of the song, but the children's chorus in the background behind him. https://www.youtube.com/watch?v=fXi3bjKowJU
I don't know if you mean this as a positive or a negative, but to me manipulating a beautiful mathematical concept into a tedious recitation may help explain why these concepts are not learned. On the other had the example under the microscope shows why the idea is important.
OK, one out of two ain't bad. No, I meant it seriously. It was one of those "teachable moment" things. I was just old enough and my mental math was just good enough to "get" it that sixteen was another doubling, and thirty-two was another. It was an example of repeated multiplication by two and probably one of the first I'd ever seen. It was exponentiation... generalized exponentiation, and they didn't teach that until junior high school IIRC. A year or two later I saw a digital computer with a row of light labelled 1-2-4-8-16-32-64 etc. and it was like hearing the song. But if it doesn't do a thing for you, that's fine.
What I got from that song, when they sing "two and two are four", is that even in the old days, the Math teaching was done by English majors! :happy
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Re: Compounding under the microscope.

Post by qwertyjazz »

nisiprius wrote:Does everyone know this one? There is a single lily pad growing in the middle of a huge lake. Every day, it doubles in size. Calculations show that if this continues, it will exactly cover the entire lake in 30 days. How many days does it take to cover half the lake?
Not 29 days as by then it will have been starved of all nutrients
My favorite version of the doubling chess board story includes the execution of the advisor who was granted the award
As Sagan noted, exponential can't go on forever or else they would gobble everything.
We tend to think our intuition is faulty in these cases, but it might be the math.
G.E. Box "All models are wrong, but some are useful."
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Re: Compounding under the microscope.

Post by mmcmonster »

*3!4!/5! wrote:This is how compounding works.
[...]
The rest is hype.
You can go through the physics of refraction and reflection, the speed of light in various materials, and the chemistry and evolutionary biology of the rods and cones in our eyes.

That doesn't make a rainbow any less miraculous.
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Re: Compounding under the microscope.

Post by *3!4!/5! »

mmcmonster wrote:
*3!4!/5! wrote:This is how compounding works.
[...]
The rest is hype.
You can go through the physics of refraction and reflection, the speed of light in various materials, and the chemistry and evolutionary biology of the rods and cones in our eyes.

That doesn't make a rainbow any less miraculous.
Aha! Now I see the light! :happy

COMPOUNDING
Dirghatamas
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Re: Compounding under the microscope.

Post by Dirghatamas »

dbr wrote: Of course the grand experiment in having math education designed by real mathematicians played out its sad story with the "new math."
dbr Probably OT from this thread but what do you mean? I don't have kids and am old (46) so I don't know how kids nowadays are being taught. Is this "new math" rote learning/remembering of tables of some kind because I don't recall mathematics somehow becoming "new" in the 80s when I went through school? Personally, I have been fascinated by math, science, engineering and nature all my life (leading to career choice) but I remember my childhood, being able to play (and destroy) stuff to figure out how things work, basically led to a lifelong interest. Rote learning is probably what causes people to lose interest in these noble fields.
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Re: Compounding under the microscope.

Post by *3!4!/5! »

Dirghatamas wrote:
dbr wrote: Of course the grand experiment in having math education designed by real mathematicians played out its sad story with the "new math."
dbr Probably OT from this thread but what do you mean? I don't have kids and am old (46) so I don't know how kids nowadays are being taught. Is this "new math" rote learning/remembering of tables of some kind because I don't recall mathematics somehow becoming "new" in the 80s when I went through school? Personally, I have been fascinated by math, science, engineering and nature all my life (leading to career choice) but I remember my childhood, being able to play (and destroy) stuff to figure out how things work, basically led to a lifelong interest. Rote learning is probably what causes people to lose interest in these noble fields.
https://en.wikipedia.org/wiki/New_Math
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Re: Compounding under the microscope.

Post by linenfort »

mmcmonster wrote:
*3!4!/5! wrote:This is how compounding works.
[...]
The rest is hype.
You can go through the physics of refraction and reflection, the speed of light in various materials, and the chemistry and evolutionary biology of the rods and cones in our eyes.

That doesn't make a rainbow any less miraculous.
I appreciate this post. And guess what, guys- even the people who don't know the math behind compounding know that there's math behind the compounding!
The Magic of Compounding Returns, The Tyranny of Compounding Costs
– John Bogle
http://acquirersmultiple.com/2016/07/th ... ohn-bogle/
My father explained compounding to me at an early age. Got me excited about investing and in saving.
Justifiable hype.

(edited in link)
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Re: Compounding under the microscope.

Post by dbr »

Dirghatamas wrote:
dbr wrote: Of course the grand experiment in having math education designed by real mathematicians played out its sad story with the "new math."
dbr Probably OT from this thread but what do you mean? I don't have kids and am old (46) so I don't know how kids nowadays are being taught. Is this "new math" rote learning/remembering of tables of some kind because I don't recall mathematics somehow becoming "new" in the 80s when I went through school? Personally, I have been fascinated by math, science, engineering and nature all my life (leading to career choice) but I remember my childhood, being able to play (and destroy) stuff to figure out how things work, basically led to a lifelong interest. Rote learning is probably what causes people to lose interest in these noble fields.
It was the 1960's. Reference in post above.
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Re: Compounding under the microscope.

Post by dbr »

Here is a good article on New Math: http://web.math.rochester.edu/people/fa ... /smsg.html
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Re: Compounding under the microscope.

Post by dbr »

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Re: Compounding under the microscope.

Post by nisiprius »

Here's the best:
Tom Lehrer, "The New Math"
(The song; not the animation, I just reviewed several on YouTube, didn't like the animations on any...)
Annual income twenty pounds, annual expenditure nineteen nineteen and six, result happiness; Annual income twenty pounds, annual expenditure twenty pounds ought and six, result misery.
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Re: Compounding under the microscope.

Post by nisiprius »

On the other hand, an opposing view... and the "New Math" came in while I was in school and I liked it...
...anyway, an illustration of the "back to basics" movement before there was a "back to basics" movement. This is from the novel, A Clergyman's Daughter, by George Orwell, set in a bad, cheap private school in England, probably in the early 1930s or thereabouts. The protagonist, desperate for work, has taken a job there as a teacher, and gets a letter from an angry parent:
Dear Miss,--Would you please give Mabel a bit more arithmetic? I feel that what your giving her is not practacle enough. All these maps and that. She wants practacle work, not all this fancy stuff. So more arithmetic, please. And remain,

Yours Faithfully,

Geo. Briggs

P.S. Mabel says your talking of starting her on something called decimals. I don't want her taught decimals, I want her taught arithmetic.
Annual income twenty pounds, annual expenditure nineteen nineteen and six, result happiness; Annual income twenty pounds, annual expenditure twenty pounds ought and six, result misery.
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Re: Compounding under the microscope.

Post by *3!4!/5! »

nisiprius wrote:Here's the best:
Tom Lehrer, "The New Math"
(The song; not the animation, I just reviewed several on YouTube, didn't like the animations on any...)
So maybe that's where my discomfort with percentages comes from. They're a bastion of base ten privilege. They are culturally insensitive to people with missing fingers. Let's do it in base eight. We need persixtyforthages!
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Re: Compounding under the microscope.

Post by nisiprius »

*3!4!/5! wrote:
nisiprius wrote:Here's the best:
Tom Lehrer, "The New Math"
(The song; not the animation, I just reviewed several on YouTube, didn't like the animations on any...)
So maybe that's where my discomfort with percentages comes from. They're a bastion of base ten privilege. They are culturally insensitive to people with missing fingers. Let's do it in base eight. We need persixtyforthages!
Believe it or not, my mother owned a copy of a book which fascinated me at a certain time, entitled New Numbers. It was a perfectly serious book advocating the use of a base-12 numbering system. The argument was that ten was an historical accident related to the number of fingers. It doesn't have any intrinsic merits as a number base, while 12 does: convenient divisibility by 2, 3, 4, and 6; already useful in commerce in the quantities "a dozen," "a gross," and "a great gross;" and, he claimed, was part of so many traditional weights and measures that in base 12 they were became just as mentally convenient as the metric system is in base 10. Why change the established units of weights and measures to fit the arbitrary base 10, why not changed the numbering base to fit the existing units?

It was actually pretty interesting, even somewhat convincing. Might have been more convincing if there had been twelve shillings to the pound.

Being a practical-minded guy, he didn't want to upset printers by adding a new character to the typecase (well, they were probably using linotypes even back then), so he suggested use a Greek "chi" for the digit 10, and a script "E" for the digit 11.

Searching... yep, Faber and Faber , London , 1936: ANDREWS, F. Emerson. New Numbers: How acceptance of a duodecimal (12) base would simplify arithmetic.

In one of the computer rooms at MIT, there was, honest to gosh, a mechanical Monroe-style "rotary calculator," the kind with something resembling a typewriter carriage, that had been built to calculate in octal (base 8). I'd love to know the story behind that, how many were built, what they cost, and what became of it... or them.
Annual income twenty pounds, annual expenditure nineteen nineteen and six, result happiness; Annual income twenty pounds, annual expenditure twenty pounds ought and six, result misery.
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Re: Compounding under the microscope.

Post by Dirghatamas »

dbr wrote:Here is an even better one: https://www.nytimes.com/2014/07/27/maga ... .html?_r=0
dbr *3!4!/5! and nisiprius

Thanks! I learned something new and feel kind of dumb. I thought "new math" would be some new thing they developed to teach younguns (2010s). Looks like it was something before I was born :oops: I will read up about it.
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Re: Compounding under the microscope.

Post by Oicuryy »

*3!4!/5! wrote:Multiplication factors from each values to the next:
(v_1/v_0), (v_2/v_1), (v_3/v_2), ... , (v_n/v_{n-1}),

So when you multiply these:
(v_1/v_0) * (v_2/v_1) * (v_3/v_2) * ... * (v_n/v_{n-1}) = (v_n/v_0)

and that's just the multiplication factor to go from the initial value to the final value!
v_0 * (v_n/v_0) = v_n

That's all there is to compounding!
I was taught to call that factoring.

This is the formula I was taught for compounding.
A = P * (1+r) ^ t

Ron
Money is fungible | Abbreviations and Acronyms
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Re: Compounding under the microscope.

Post by nisiprius »

Dirghatamas wrote:...I thought "new math" would be some new thing they developed to teach younguns (2010s). Looks like it was something before I was born :oops: I will read up about it.
Ha! It's easier to be knowledgable about the details of "history" if you experienced them as "news." The Russians put up the first artificial satellite, Sputnik, in 1957, a few months ahead of the US. At that time the US was having a lot of problems with its missile program, and unlike the Soviet Union, we had a free press and rocket failures made the new. It's a little reminiscent of the struggles SpaceX is having, and it seems as if every few weeks there would be dramatic news footage of US rockets exploding. (One of the most prominent was named Project Vanguard.) Concomitantly, there was a lot of reasonable concern about the state of science and math education (not then called STEM) in the US, which had been somewhat neglected during the war. There was also some appreciation of just how important to the war effort the Los Alamos scientists had been.

There were a number of well-intentioned efforts, with varying degrees of success, to revise curricula in accordance with what working scientists and mathematicians thought was really important. One of the better efforts was "PSSC Physics," which was great. "The New Math" was not successful, because it was intended to... how shall I put this... lay the foundations for learning mathematics, rather than finance. In quite clever ways, they were successfully teaching kids in the lower elementary grades the basic concepts of set theory. The problem was that--like George Orwell's angry parent--"I don't want her taught decimals, I want her taught arithmetic"--people didn't want their children taught set theory, they wanted them taught arithmetic. And they had a point, it wasn't likely ever to be directly useful to any but a very small percentage of students.
Annual income twenty pounds, annual expenditure nineteen nineteen and six, result happiness; Annual income twenty pounds, annual expenditure twenty pounds ought and six, result misery.
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Re: Compounding under the microscope.

Post by nisiprius »

An important sideline regarding the word "compounding," which I didn't realize until 2009, was that originally, compound interest meant the practice of allowing missed interest payments to be added to the loan principal, so that the creditor could then charge interest on the unpaid interest--and was often thought of as immoral and usurious.
Compound Interest as Bordering on Usury
According to something I quoted in that thread :happy , it may have been illegal for credit card companies to charge "compound interest on delinquent accounts" into the early 1970s.
Annual income twenty pounds, annual expenditure nineteen nineteen and six, result happiness; Annual income twenty pounds, annual expenditure twenty pounds ought and six, result misery.
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Re: Compounding under the microscope.

Post by Texanbybirth »

mmcmonster wrote:
*3!4!/5! wrote:This is how compounding works.
[...]
The rest is hype.
You can go through the physics of refraction and reflection, the speed of light in various materials, and the chemistry and evolutionary biology of the rods and cones in our eyes.

That doesn't make a rainbow any less miraculous.
Well said.
“The strong cannot be brave. Only the weak can be brave; and yet again, in practice, only those who can be brave can be trusted, in time of doubt, to be strong.“ - GK Chesterton
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Re: Compounding under the microscope.

Post by Phineas J. Whoopee »

My early math education, in a school district that rejected new math when it came out, was terrible. It was entirely based on memorizing tables. Even if we got the right answer we were penalized for, as they said and I well remember the admonishments, "counting on our fingers," also known as trying to figure out how addition and subtraction work.

Once we advanced beyond the right number answer, which we weren't allowed to understand, and into logic, that is to say algebra, things started improving for me, even before the move to a better district just in time for high school. I remember some of the higher-performing table memorizers being thrown for a loop.

The district's mission was to prepare us, really only the boys, for line jobs at factories. There's nothing inherently wrong with that, but they decided the more ignorant of math we were the better we'd suit the no-longer-existent occupations.

I'm one of the lucky individuals who made it out from there, and for that I'm grateful.

PJW
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