Problem 15 - Thomas Calculus

Question

Benjamin Franklin's will The Franklin Technical Institute of Boston owes its existence to a provision in a codicil to Benjamin Franklin's will. In part the codicil reads: 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 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 hereinafter mentioned and declared. Franklin's plan was to lend money to young apprentices at 5\(\%\) interest with the provision that each borrower should pay each year along \cdots with the yearly Interest, one tenth part of the Principal, which sums of Principal and Interest shall be again let 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 Inhabitants of the Town of Boston, then lay out at their discretion one hundred thousand Pounds in Public Works. \(\ldots\) 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.\ldots 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. It was not always possible to find as many borrowers as Franklin had planned, but the managers of the trust did the best they could. At the end of 100 years from the reception of the Franklin gift, in January 1894 , the fund had grown from 1000 pounds to almost exactly \(90,000\) pounds. In 100 years the original capital had multiplied about 90 times instead of the 131 times Franklin had imagined. What rate of interest, compounded continuously for 100 years, would have multiplied Benjamin Franklin's original capital by 90\(?\)

Step-by-Step Solution

Answer

The continuous interest rate is approximately 4.5\( \% \).

Step 1: Understanding the Problem

The problem seeks to find the continuous compound interest rate such that the original capital grows by a factor of 90 over 100 years.

Step 2: Using the Compound Interest Formula for Continuous Compounding

The formula for continuous compounding is given by \( A = Pe^{rt} \), where \( A \) is the final amount, \( P \) is the principal amount, \( r \) is the rate of interest, and \( t \) is the time in years.

Step 3: Setting Up the Equation

Let \( P = 1000 \) pounds and \( A = 90000 \) pounds. The time \( t \) is 100 years. Substitute these values into the formula: \( 90000 = 1000e^{100r} \).

Step 4: Simplifying the Equation

Divide both sides of the equation by 1000 to isolate \( e^{100r} \): \( 90 = e^{100r} \).

Step 5: Solving for \( r \)

Take the natural logarithm of both sides to solve for \( r \): \( \ln(90) = 100r \).

Step 6: Calculating \( r \)

Solve for \( r \) by dividing \( \ln(90) \) by 100: \( r = \frac{\ln(90)}{100} \).

Step 7: Compute the Final Interest Rate

Calculate \( r \) using the expression \( r = \frac{\ln(90)}{100} \). This gives us the continuous compounding interest rate as approximately \( 0.045 \) or 4.5\( \% \).

Key Concepts

Compound Interest Formula

The compound interest formula for continuous compounding is a fundamental concept in finance. This formula helps calculate the future value of an investment or loan based on continuous interest compounding. Continuous compounding means that interest is added an infinite number of times per period, resulting in a smooth, exponential growth of the investment.

The formula is expressed as:

• \[ A = Pe^{rt} \]

Here, \( A \) represents the final amount, \( P \) is the principal amount, \( r \) is the rate of interest per year, and \( t \) is the time in years.

The interesting aspect of continuously compounded interest is its relationship to the exponential function, denoted by \( e \). This relationship introduces a concept of exponential growth, which is essential for understanding how investments grow rapidly over time. One might think of it as the most aggressive form of compounding because it assumes the highest possible number of compounding intervals.

Natural Logarithm

The natural logarithm is crucial when dealing with continuous compound interest. It is particularly useful when one needs to solve equations involving exponential functions.

The natural logarithm, often represented as \( \ln \), is the logarithm to the base \( e \), where \( e \) is an irrational constant approximately equal to 2.71828. This logarithm is the inverse of the exponential function. So, when you have an exponential equation like \( e^{100r} = 90 \), the natural logarithm allows you to solve for \( r \) by taking the logarithm of both sides:

• \( \ln(e^{100r}) = \ln(90) \)
• Using the property that \( \ln(e^x) = x \), it simplifies to \( 100r = \ln(90) \)
• From this, you can isolate \( r \) as \( r = \frac{\ln(90)}{100} \)

The natural logarithm can appear intimidating, but it is a reliable tool that simplifies the process of solving problems involving exponential growth.

Interest Rate Calculation

Calculating the interest rate in the context of continuous compounding may seem challenging at first, but it can be manageable with the right steps. The problem statement describes how Franklin's initial fund of 1,000 pounds reached nearly 90,000 pounds over 100 years under continuous compounding. To find the rate of interest, the expression \( 90 = e^{100r} \) simplifies this relationship.

• The next step involves applying the natural logarithm to both sides, turning the exponential equation into a linear form: \( \ln(90) = 100r \)
• It is then straightforward to isolate \( r \) by dividing both sides by 100: \( r = \frac{\ln(90)}{100} \)
• This computation yields an interest rate of approximately 4.5\( \% \)

This rate signals how Franklin's fund, utilizing continuous compound interest, grew significantly despite not reaching the projected 131-fold increase. Understanding this process gives insight into financial growth and the power of compound interest, even when results vary from original projections.

Benjamin Franklin's Will

Benjamin Franklin was not only a renowned polymath but also a visionary with financial foresight, as exemplified by his will. His bequest aimed to foster the development of young apprentices and, in the long term, benefit the public. Franklin allocated substantial funds to the cities of Boston and Philadelphia, with an ambitious plan to grow the funds through lending at interest.

Franklin specified the use of the interest earned to grant loans to new borrowers, structured in a way that promised exponential growth over a century. His method is a prime example of using compound interest in a practical legacy.

• He imagined a compound interest rate of 5\( \% \) would elevate his bequest to 131,000 pounds over 100 years.
• The actual growth, managing to reach just 90,000 pounds, reflected challenges in implementing his plan, but demonstrated immense value over time.

Understanding Franklin's vision in terms of modern financial concepts helps emphasize the power of long-term financial planning and continuous compound interest, preserving wealth and encouraging societal benefit years beyond an individual's lifetime.