Ben Franklin's Will - Part 1
While this was adapted by Dr. Sarah from
Middle Atlantic Consortium for Mathematics and its
Applications Throughout the Curriculum, this all actually happened!
It is a great project since it combines
financial mathematics with the
disciplines of history, economics,
political science, ethics and philosophy.
In addition, it gives you a chance to see how interest is actually earned
by a fund.
This will eventually become your first major writing project
once I give you Part 2.
The Franklin Technical Institute of Boston owes its existence in
part to a codicil in
Benjamin Franklin's will
(skim through the codicil by clicking on the underlined link
from the web page).
In 1785 a French mathematician named Charles-Joseph Mathon de la Cour wrote a
mocking the spirit of American optimism represented by Franklin.
The Frenchman wrote a piece about Fortunate Richard leaving a small sum
of money in his will to be used only after it had collected interest
for 500 years.
Franklin, who was 79 years old at the time, wrote back to the Frenchman,
thanking him for a great idea and telling him that he had decided to leave a
bequest to his
native Boston and his adopted Philadelphia of 1,000 pounds to each on the
condition that it be placed in a fund that would gather interest over
a period of 200
"...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 both in Boston and Philadelphia. To this end I devote Two thousand Pounds
Sterling, 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 and for the Uses, Interests and
Purposes herinafter mentioned and declared....."
Franklin's plan was to lend money to young apprentices.
The fund would charge a certain interest rate to the borrowers,
and would earn annual interest since
each borrower would pay back principal plus interest each year.
In addition, Franklin had plans for how the money would be used:
If this plan is executed and succeeds as projected without interruption for
one hundred Years,
the Sum will be one hundred and thirty-one thousand Pounds of which I would
the Managers of the Donation to the Inhabitants of 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.....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....
of which I leave one million sixty-one thousand pounds to
the disposition of the inhabitants of the town...,
and three millions to the disposition of the government of the State,
not presuming to carry my views farther."
Make sure you answer all the questions since your lab grade
today will be based on the depth, clarity and correctness of
Ben Franklin's plans
1. What interest was Franklin planning on for the first 100 years?
If 1000 pounds were invested for 100 years, which interest rate
compounding yearly would give a balance of 131000 pounds?
We could guess by trying
different rates, but instead we'll use Excel.
Click on this excel file
from Netscape Communicator
You will see some symbols that don't look like they make sense. If so, then
you are in the right place!
Under File, release on Save As...
and make sure that you go to the pubic folder/save files here folder,
which is located on the desktop.
Then click Save to save the file as benf.xls and replace if that comes up.
Under the apple, under Math 1010 Apps, release on Microsoft Excel.
Under File, release on Open, look in the public folder and then click on
benf.xls and Open it. You will see a chart that is partly filled in.
In the C2 box, type EXACTLY
and then hit return.
You will see "#NAME?" since excel doesn't know what ben1 means.
We'll define it.
Click on D2, and you will see that the D2 box is outlined.
Under Insert, scroll down to Name and then over to Define
and release the mouse button.
Type in ben1 (in the Names in workbook slot). Notice that
the Refers to line will already say =Sheet1!$D$2. Now click on OK.
Click on the C2 box (which now reads 1000). We want excel
to solve for the interest rate that will result in 131000.
Under Tools, scroll down to Goal Seek... and release.
Set cell: should already read C2
Type 131000 in the To value: slot
Type D2 in the By changing cell: slot
Goal Seek will find a solution, so then click on OK. Notice that
the interest rate solution is in box D2.
Save your excel sheet into the public folder and then show me your work.
Once I have approved your work,
fill in the first row of both tables in the Tables section below
and answer questions 1 - 3 in the Questions to Answer section below.
2. Franklin planned that reinvesting 31000 pounds for the second
hundred years would result in 4061000 pounds. If
31000 pounds were invested for 100 years, which interest
rate compounding yearly would give a balance of
4061000 pounds? Answer this in your excel sheet in boxes
C3 and D3 using a similar
process to the one described
above. Think about what your new lump sum formula will be.
Be sure to use a different
name for the rate. For example,
you could use ben2. In addition, be careful that you insert/name
into the correct box (D3), and that you use goal seek on the
correct box (C3). Once I have checked your work, save your
Excel file and then fill in the second row of the tables.
The Actual First Hundred Years
Even though the fund charged borrowers 5% interest,
it was not always possible to find as many borrowers as
Franklin had planned and there were other problems as well since
some of the borrowers did not pay back their loans.
In January 1894, at the end of 100 years from the
inception of the Franklin gift, because of these problems, the Boston
fund had grown from 1000 pounds to almost
90,000 pounds, instead of the 131,000 pounds
that Franklin had imagined.
After legal battles 3/4 of this money (about $300,000 US)
was used to fund the
Franklin Institute of Boston, and the remainder,
about $100,000 US was maintained for loans for the 2nd century.
The Actual Second Hundred Years
3. In real life,
the $100,000 reinvested at the end of the first hundred years
(which was lent to borrowers at 5% interest)
grew to 5 million (5000000.00)
dollars in Boston at the end of the second hundred years.
What average earned interest rate was responsible for this Boston growth
(the answer is not 5% since the fund faced similar problems as were faced
during the first hundred years).
Answer this in your excel sheet in boxes
C4 and D4 and then fill in row 3 of the tables below.
4. Assume that Philadelphia had also reinvested
$100,000 at the end of the first hundred years
(I haven't been able to find the actual amount that was reinvested
by Philadelphia). In real-life, the fund grew
to $2256952.05. What average earned interest rate was responsible for
this Philadelphia growth?
Answer this in your excel sheet in boxes
C5 and D5.
Note that a small difference in the average earned
interest rate resulted in a
substantial difference in the fund's earned income!
Fill in row 4 of the tables. Wait to answer questions 4 and
5 until the end of lab or outside of class.
Fill in the following 2 tables.
For the first table, in Column C, you should
put the formula that you used and in Column D,
you should put the names that you defined.
C2 should read
and D2 should read ben1. To remind yourself of the
formulas that you used, click on the corresponding box in
Excel, and look at the top next to the equal sign (under view,
release on Formula Bar if you do not see this already).
For the second table, you should fill in the numbers that you see
in Excel. Be sure to write down whether the money is in dollars
or pounds. Turn the interest rates into percentages, but don't round them.
||Money in Pounds or Dollars
||Average Earned Interest Rate
as a Percentage - don't round
Questions to Answer in Complete Sentences
- Write down the lump sum formula, filling in for the variables
that we already know and highlighting the unknown variable.
(No need for complete sentences for this question).
- How does the fund earn the interest that is compounded annually?
- Why is the lump sum formula appropriate to use here - if
the borrowers are paying back money each year, shouldn't we
use the periodic payment formula?
- After 200 years, Boston and Philly ended up with a lot of money using
Ben's loan method, but they ended up with a lot less money than he had
planned (note the difference between pounds and dollars at the time).
Notice that D2 says "Average Rate". Explain in your own words
why this isn't the actual lent interest rate of approximately 5% that
was charged to the lenders.
(Hint - think about the lending process, the problems that occurred
and how this would affect the average earned rate that the fund yielded).
- Worth More
What would you do with these funds for the cities of Philadelphia and Boston?
Think about this question carefully, especially in relationship to
Ben Franklin's goals, and then answer the question here.