Ok, so now we are feeling pretty good about ourselves since we remember something about compound interest. But are simple calculator next to our check book does NOT do exponential equations. So we go back to simple interest. Shame on you. Are you planning on using hot oil to pop your next batch of popcorn ? Hell no, you are going to NUKE IT in the microwave. So get yourself a real calculator (solar power of course) so you can do some compound interest, and soon i will teach you logs.
It took me thirteen (yes, that is 13) to understand logs, so i am not going to try to explain to you the Law of Logs (LOL) in one blog.
My tutoree today called it "Lots of Luck". I told you he had a high social IQ.
So you will just have to do what i do !!
Lets say you want to know what 5^x = 2.
Algebra is NOT going to help you here. You can do trial and error like they did before the 1800's.
5^1=5, so the answer is less than 1.
5^.5 =5^(1/2)= square root of 5 = 2.24....getting closer.
How about 5^.4 = 1.9...a little low, but closer
Anyways the way we do it in math:
we say we "take the log of both sides"...(just like in algebra we take the square root of both sides of an equation sometimes)
So log 5^x = log 2.
Now we use the Law of Logs (LOL) and we are allowed to bring the "x" up front.
x log 5 = log 2
Finally we divide both sides by log 5
and our answer is x = (log 2) / (log 5) = .43....so 5^(.43) = 2 !!
So you might say WHO CARES ?? just give them the credit card...lol (laugh out loud)
Now how do we double our money ? (the Ben Franklin way !!)
Remember that that compounded continuously we use "Pert shampoo" or A=Pe^(rt)
If we buy a $100 US Bond, believe it or not, it only cost grandma $50 !! (but don't tell the grandkids)...the kids have to wait 20 or 30 years depending on the interest rate the govt is paying before the Bond "matures" or doubles in value up to $100.
So we have $100 = $50 e^(rt)...lets start with 6% interest rate (that was the rate back in the 1980's when things were going well with Ronald Reagan...yes, i had to put that plug in...remember 1989 is when the Russian Empire collapsed and i am a cuban exile...see my other blogs).
So we have $100 = $50 e^(.06 * t). How do we solve this ? with LOGS !!
First get rid of the 50 on both sides.
2 = e^(.06 * t)....(if you can not follow this step, ask you wife or your kids)
now we "take the logs of both sides", but here since the base is e, it will have to be log base e which we call natural log....or "ln"...you will find the button on your calculator (the new one you bought at Walmart for $5.95 that does more than just add and multiply or borrow the one your kids use in school)
ln 2 = ln e^(.06 * t)
using the LOL (Law of Logs) we are allowed to bring the (.06 * t) up front
ln 2 = (.06 * t) ln e
now take my word for it...the ln e = 1
so we have
ln 2 = (.06 * t)
and now we can divide both sides by .06
and we get our final answer.... (ln 2) / .06 = t = 11.5 years
It will take 11.5 years for your money to double at a rate of 6% compounded continuously.
Now we come to the "Rule of 72"
In the business world, if you want a good approximation for when your money will double at 6%, all you have to do is to divide 72 by 6 and you get about 12 years !!
At 3%, it would take 72/3 = about 24 years (this was the rate in the 1990's)
At 2%, it would take 72/2 = about 36 years (this is the rate in the 2000's)
in 2010, it is probably about 1%, or it would take 72 years to double...only the Chinese would buy such cheap stock or bonds....lol
(For the more advance math students, the Rule of 72 should really be the Rule of 70, since the ln 2 is closer to .70 then to .72, but the business world uses 72 ,because MORE numbers go into 72..2,3,4,6,8,9,12....and that keeps me awake at night.)
Money, Money
Money makes the world go around...
Saturday, January 22, 2011
Benjamin Franklin and Compound Interest
IN A CODICIL to his will, which we copied from here: The Last Will and Testament of Benjamin Franklin, Benjamin Franklin made an unusual pair of bequests. This is lengthy, but interesting, so bear with us. We’ve added some bold for emphasis:
… I wish to be useful even after my death, if possible, in forming and advancing other young men, that may be serviceable to their country in both these towns. To this end, I devote two thousand pounds sterling, of which I give one thousand thereof to the inhabitants of the town of Boston, in Massachusetts, and the other thousand to the inhabitants of the city of Philadelphia, in trust, to and for the uses, intents, and purposes herein after mentioned and declared.
The said sum of one thousand pounds sterling, if accepted by the inhabitants of the town of Boston, shall be managed under the direction of [various churches] who are to let out the sum upon interest, at five per cent, per annum, to such young married artificers, under the age of twenty-five years, as have served an apprenticeship in the said town … And in order to serve as many as possible in their turn, as well as to make the repayment of the principal borrowed more easy, each borrower shall be obliged to pay, with the yearly interest, one tenth part of the principal and interest, so paid in, shall be again let out to fresh borrowers.
If this plan is executed, and succeeds as projected without interruption for one hundred years, the sum will then be one hundred and thirty-one thousand pounds; of which I would have the managers of the donation to the town of Boston then lay out, at their discretion, one hundred thousand pounds in public works … .
The remaining thirty-one thousand pounds I would have continued to be let out on interest, in the manner above directed, for another hundred years, as I hope it will have been found that the institution has had a good effect on the conduct of youth, and been of service to many worthy characters and useful citizens.
At the end of this second term, if no unfortunate accident has prevented the operation, the sum will be four millions and sixty one thousand pounds sterling, of which I leave one million sixty one thousand pounds to the disposition of the inhabitants of the town of Boston, and three millions to the disposition of the government of the state, not presuming to carry my views farther.
All the directions herein given, respecting the disposition and management of the donation to the inhabitants of Boston, I would have observed respecting that to the inhabitants of Philadelphia …
Franklin died in 1790. Each city received £1,000, then worth about $4,400. What happened to the money? According to the Wikipedia article on Benjamin Franklin:
As of 1990, more than $2,000,000 had accumulated in Franklin’s Philadelphia trust, which had loaned the money to local residents. From 1940 to 1990, the money was used mostly for mortgage loans. When the trust came due, Philadelphia decided to spend it on scholarships for local high school students. Franklin’s Boston trust fund accumulated almost $5,000,000 during that same time, and was used to establish a trade school that became the Franklin Institute of Boston.
Why such different amounts?
… I wish to be useful even after my death, if possible, in forming and advancing other young men, that may be serviceable to their country in both these towns. To this end, I devote two thousand pounds sterling, of which I give one thousand thereof to the inhabitants of the town of Boston, in Massachusetts, and the other thousand to the inhabitants of the city of Philadelphia, in trust, to and for the uses, intents, and purposes herein after mentioned and declared.
The said sum of one thousand pounds sterling, if accepted by the inhabitants of the town of Boston, shall be managed under the direction of [various churches] who are to let out the sum upon interest, at five per cent, per annum, to such young married artificers, under the age of twenty-five years, as have served an apprenticeship in the said town … And in order to serve as many as possible in their turn, as well as to make the repayment of the principal borrowed more easy, each borrower shall be obliged to pay, with the yearly interest, one tenth part of the principal and interest, so paid in, shall be again let out to fresh borrowers.
If this plan is executed, and succeeds as projected without interruption for one hundred years, the sum will then be one hundred and thirty-one thousand pounds; of which I would have the managers of the donation to the town of Boston then lay out, at their discretion, one hundred thousand pounds in public works … .
The remaining thirty-one thousand pounds I would have continued to be let out on interest, in the manner above directed, for another hundred years, as I hope it will have been found that the institution has had a good effect on the conduct of youth, and been of service to many worthy characters and useful citizens.
At the end of this second term, if no unfortunate accident has prevented the operation, the sum will be four millions and sixty one thousand pounds sterling, of which I leave one million sixty one thousand pounds to the disposition of the inhabitants of the town of Boston, and three millions to the disposition of the government of the state, not presuming to carry my views farther.
All the directions herein given, respecting the disposition and management of the donation to the inhabitants of Boston, I would have observed respecting that to the inhabitants of Philadelphia …
Franklin died in 1790. Each city received £1,000, then worth about $4,400. What happened to the money? According to the Wikipedia article on Benjamin Franklin:
As of 1990, more than $2,000,000 had accumulated in Franklin’s Philadelphia trust, which had loaned the money to local residents. From 1940 to 1990, the money was used mostly for mortgage loans. When the trust came due, Philadelphia decided to spend it on scholarships for local high school students. Franklin’s Boston trust fund accumulated almost $5,000,000 during that same time, and was used to establish a trade school that became the Franklin Institute of Boston.
Why such different amounts?
Extra Help (LOL)
Gotcha ya !! You thought LOL stood for "Laugh Out Loud".
It does in the email world, but in the Math Classroom it stands for Law of Logs !!
Today we were studying Logs, no, NOT the kind you use for firewood or to build a log cabin house. These logs are the operation (function) we use to find exponents.
Ok, i just lost half of my audience. With only 4 people reading these, there goes 2 people. Please stay. Try to follow for fun. I guarantee you it is an important concept and it DOES have real world applications (NOT like Algebra).
Subtraction is the opposite of addition, Division the opposite of multiplication, square root the inverse of squaring...(are your palms getting sweating ? sorry)
3 square is 9...and the square root of 9 is 3...got it ?
In Math we first learn to go forwards and then we learn to go backwards.
In Algebra we learn to "FOIL" (ex: (x+1)(x+2) = x^2 + 3x + 2 )....then we learn to go backwards which we call Factoring...(ex: x^2 + 3x + 2 = (x+1)(x+2) )
Of course this has become obsolete because our graphing calculator have a program that solves the equation for you, so we really do NOT need to factor any more. (But please do not tell all those old ladies still teaching factoring after 30 years. Hey, even I still have to teach factoring after all these years...what a waste of time.)
Now we get into Algebra 2 and we start to study exponential functions.
The thing is that in the real world things do not grow in a nice line. (y = mx+b)
And only a few things follow the parabola (y = x^2 )
One of my favorite formula's for a baseball or football traveling through the air or a cannon ball ( y = -16x^2 + velocity x + height ). In Algebra 1, we find the maximum height of the ball and how long does it take to hit the ground ? (in the real world we call that a field goal or a home run !!)
But in the real world (nature) things grow exponentially. Like the growth of trees and the more importantly the growth of MONEY.
A little review is necessary. When we are in middle school, we learn about "simple interest". If you put $1,000 in the bank and it grows at 6% a year, in 5 years you should have $1,300 in the bank. The original $1,000 you put plus the extra $300 we call interest earned. Mathematically it looks something like this:
A = P ( 1 + rt )....therefore A = $1,000 ( 1 + .06(5) ) = $1,300.
Now our boy Benjamin Franklin loved something called "compound interest". This means we "earn interest on the interest". That is a beautiful thing if you are a banker or if you are saving money. It is an ugly thing if you are borrowing money...lol.
Compound Interest is an exponential function. We say it grows exponentially FAST.
(Population growth also grows exponentially until come force of nature slows it down. We have experience exponential growth in the 1900's but now that the age limit of humans as stabilized at 75 or so, the growth has slow down especially in "developed" continents like America and Europe. In Europe they are paying couples to have children and in the USA they give you tax incentives to have children. Africa population has stabilized for other reasons (hunger and Aids), but Asia continues to add to the overall World population especially India and China, two countries that have over half of the entire world's population. China has tried to control this by state law allowing only one child per family.)
but i have digress...one last word on world population. It was predicted we would double from 6 Billion in 2000 to 12 Billion by 2020, but it looks like that will NOT happen. The new date is now 2050. There is even fear among some people that the world's population will actually start to decline after 2050. We will be the only animals the purposely will bring our species population into decline. And the fear is that once that decline starts, it will be difficult to stop it.
Now getting back to money.
What if we were to compound our money "yearly" (every year). That $1,000 we started with would be $1,060 after the first year. Now if we earn 6% on that extra $60 we earned then after two years we would have $1,060 plus $ 63.60 for a total of $1,123.60.
But that is too much work so we create a formula...and it goes something like this
A = P ( 1 + r/k )^(kt)....A = $1,000 ( 1 + (.06/1))^(1 * 5) = $1,338.23
($1,000 compounded annually or once a year for 5 years at 6%)
But if we wanted to compound it monthly then k = 12
A = ( 1 + (.06/12))^(12*5) = $1,348.85
How about daily then k = 360 (in the business world there are 30 days per month)
A = ( 1 + (.06/360))^(360*5) = $1,349.83
Now we get real greedy and we want to compound it every hour, every second,
continuously !!! (aha, the thought of infinity...we call that Calculus...Archimedese in 250 BC thought about infinity but that if for another blog)
( 1 + (1/x))^x where x goes to infinity is 2.728....and we call that "e" (after the greatest mathematician of all time Euler 1700's)
and now we get "Pert shampoo" for compounded continuously
A = P * e^(rt)....yes, Pert shampoo
A = $1,000 e^(.06 * 5) = $1,349.59
Now we have to go backwards...and start talking about logs.
Let me tell you a story about "log diving" in the springs near Gainesville, Florida.......
Some believe it was Benjamin Franklin who said, "Compound interest is the eighth wonder of the world," while attributing "Compound interest is the world's greatest discovery," to Albert Einstein. Still others attribute the whole thing to John Maynard Keynes.
Historical note: the Indains were given $24 of trinkets in the 1500's for Long Island. At 6% interest rate over 500 years it would come out to
A = $24 e^(.06 * 500) = 2 followed by 14 zeros ($200,000,000,000,000 = 200 Trillion)
COMMENTS:
(OK, the Extra Help blog just makes me nervous and feel stupid as I can't figure any of that stuff out. I should know how to do all that stuff, but I don't and never will no matter how much I try! Just one of my many, many deficits.
The autobio blog looks great! I'm really glad you got to Cuba over the Christmas holidays; it must have been very emotional, and I really look forward to reading the chapters.
Thanks for passing the links on.
This writer is a computer nerd...and HE feels nervous...lol)
It does in the email world, but in the Math Classroom it stands for Law of Logs !!
Today we were studying Logs, no, NOT the kind you use for firewood or to build a log cabin house. These logs are the operation (function) we use to find exponents.
Ok, i just lost half of my audience. With only 4 people reading these, there goes 2 people. Please stay. Try to follow for fun. I guarantee you it is an important concept and it DOES have real world applications (NOT like Algebra).
Subtraction is the opposite of addition, Division the opposite of multiplication, square root the inverse of squaring...(are your palms getting sweating ? sorry)
3 square is 9...and the square root of 9 is 3...got it ?
In Math we first learn to go forwards and then we learn to go backwards.
In Algebra we learn to "FOIL" (ex: (x+1)(x+2) = x^2 + 3x + 2 )....then we learn to go backwards which we call Factoring...(ex: x^2 + 3x + 2 = (x+1)(x+2) )
Of course this has become obsolete because our graphing calculator have a program that solves the equation for you, so we really do NOT need to factor any more. (But please do not tell all those old ladies still teaching factoring after 30 years. Hey, even I still have to teach factoring after all these years...what a waste of time.)
Now we get into Algebra 2 and we start to study exponential functions.
The thing is that in the real world things do not grow in a nice line. (y = mx+b)
And only a few things follow the parabola (y = x^2 )
One of my favorite formula's for a baseball or football traveling through the air or a cannon ball ( y = -16x^2 + velocity x + height ). In Algebra 1, we find the maximum height of the ball and how long does it take to hit the ground ? (in the real world we call that a field goal or a home run !!)
But in the real world (nature) things grow exponentially. Like the growth of trees and the more importantly the growth of MONEY.
A little review is necessary. When we are in middle school, we learn about "simple interest". If you put $1,000 in the bank and it grows at 6% a year, in 5 years you should have $1,300 in the bank. The original $1,000 you put plus the extra $300 we call interest earned. Mathematically it looks something like this:
A = P ( 1 + rt )....therefore A = $1,000 ( 1 + .06(5) ) = $1,300.
Now our boy Benjamin Franklin loved something called "compound interest". This means we "earn interest on the interest". That is a beautiful thing if you are a banker or if you are saving money. It is an ugly thing if you are borrowing money...lol.
Compound Interest is an exponential function. We say it grows exponentially FAST.
(Population growth also grows exponentially until come force of nature slows it down. We have experience exponential growth in the 1900's but now that the age limit of humans as stabilized at 75 or so, the growth has slow down especially in "developed" continents like America and Europe. In Europe they are paying couples to have children and in the USA they give you tax incentives to have children. Africa population has stabilized for other reasons (hunger and Aids), but Asia continues to add to the overall World population especially India and China, two countries that have over half of the entire world's population. China has tried to control this by state law allowing only one child per family.)
but i have digress...one last word on world population. It was predicted we would double from 6 Billion in 2000 to 12 Billion by 2020, but it looks like that will NOT happen. The new date is now 2050. There is even fear among some people that the world's population will actually start to decline after 2050. We will be the only animals the purposely will bring our species population into decline. And the fear is that once that decline starts, it will be difficult to stop it.
Now getting back to money.
What if we were to compound our money "yearly" (every year). That $1,000 we started with would be $1,060 after the first year. Now if we earn 6% on that extra $60 we earned then after two years we would have $1,060 plus $ 63.60 for a total of $1,123.60.
But that is too much work so we create a formula...and it goes something like this
A = P ( 1 + r/k )^(kt)....A = $1,000 ( 1 + (.06/1))^(1 * 5) = $1,338.23
($1,000 compounded annually or once a year for 5 years at 6%)
But if we wanted to compound it monthly then k = 12
A = ( 1 + (.06/12))^(12*5) = $1,348.85
How about daily then k = 360 (in the business world there are 30 days per month)
A = ( 1 + (.06/360))^(360*5) = $1,349.83
Now we get real greedy and we want to compound it every hour, every second,
continuously !!! (aha, the thought of infinity...we call that Calculus...Archimedese in 250 BC thought about infinity but that if for another blog)
( 1 + (1/x))^x where x goes to infinity is 2.728....and we call that "e" (after the greatest mathematician of all time Euler 1700's)
and now we get "Pert shampoo" for compounded continuously
A = P * e^(rt)....yes, Pert shampoo
A = $1,000 e^(.06 * 5) = $1,349.59
Now we have to go backwards...and start talking about logs.
Let me tell you a story about "log diving" in the springs near Gainesville, Florida.......
Some believe it was Benjamin Franklin who said, "Compound interest is the eighth wonder of the world," while attributing "Compound interest is the world's greatest discovery," to Albert Einstein. Still others attribute the whole thing to John Maynard Keynes.
Historical note: the Indains were given $24 of trinkets in the 1500's for Long Island. At 6% interest rate over 500 years it would come out to
A = $24 e^(.06 * 500) = 2 followed by 14 zeros ($200,000,000,000,000 = 200 Trillion)
COMMENTS:
(OK, the Extra Help blog just makes me nervous and feel stupid as I can't figure any of that stuff out. I should know how to do all that stuff, but I don't and never will no matter how much I try! Just one of my many, many deficits.
The autobio blog looks great! I'm really glad you got to Cuba over the Christmas holidays; it must have been very emotional, and I really look forward to reading the chapters.
Thanks for passing the links on.
This writer is a computer nerd...and HE feels nervous...lol)
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