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 parody 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 years:

"...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 have 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 your responses.

NAME____________________________________________________________

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
    =1000*(1+ben1)^100
    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.

    Tables

    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.
    For example, C2 should read
    =1000*(1+benone)^100
    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).
    A B C D
    1 Info Time Money Formula Defined Rate Names
    2 Ben's Plan 100 years    
    3 Ben's Plan 200 years    
    4 Boston 200 years    
    5 Philly 200 years    


    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.
    A B C D
    1 Info Time Money in Pounds or Dollars Average Earned Interest Rate as a Percentage - don't round
    2 Ben's Plan 100 years    
    3 Ben's Plan 200 years    
    4 Boston 200 years    
    5 Philly 200 years    

    Questions to Answer in Complete Sentences

    1. 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).







    2. How does the fund earn the interest that is compounded annually?







    3. 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?







    4. 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).







    5. 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.