Problem 44 - Applied Calculus

Question

GENERAL: Permanent Endowments Show that the size of the permanent endowment needed to generate an annual \(C\) dollars forever at interest rate \(r\) compounded continuously is \(C / r\) dollars.

Step-by-Step Solution

Answer

The permanent endowment needed is \(C/r\) dollars.

Step 1: Understand the problem

The problem is to find out how much principal (endowment) is required to generate a specific amount of money annually (denoted as \(C\)) when the interest is compounded continuously at the rate \(r\).

Step 2: Recall continuous compounding formula

The formula for continuously compounded interest is \(A = Pe^{rt}\), where \(A\) is the amount after time \(t\), \(P\) is the principal amount, \(r\) is the interest rate, and \(e\) is the base of the natural logarithm.

Step 3: Apply the context

Since we want to generate a fixed annual amount \(C\) forever, the endowment earns \(C\) per year by interest alone. Thus, our effective formula simplifies to constantly earning \(C\) from interest \((P \cdot r = C)\).

Step 4: Solve the equation for principal \(P\)

Rearrange the formula \(P \cdot r = C\) to solve for \(P\):\[ P = \frac{C}{r} \]

Step 5: Interpret the result

The result \(P = \frac{C}{r}\) indicates that the initial principal must be \(\frac{C}{r}\) dollars to ensure that the interest earned annually is exactly \(C\), allowing for a perpetual endowment.

Key Concepts

Endowment

An endowment is a fund created to support a specific purpose indefinitely. It typically provides financial stability over the long term. When establishing an endowment, the principal amount is invested. Only the income generated from this investment is used for the intended purpose.

For instance, a university might have an endowment fund to support scholarships, research, or faculty positions. Such endowments help in preserving the primary sum while the generated income assists in covering annual expenses.

*Key Points:*

• The principal of an endowment remains intact.
• Income from the endowment is used to meet annual financial needs.
• Endowments support perpetual funding for specified causes, aligning with long-term goals.

Interest Rate

The interest rate is the percentage at which money grows when invested or decreases when borrowed. It represents the cost of borrowing or the gain on investments.

In the context of endowments and continuous compounding, the interest rate helps determine how much future income an investment will generate. A higher interest rate will result in more income being produced annually.

*Types of Interest Rates:*

• Simple Interest: Interest calculated on the principal amount alone.
• Compound Interest: Interest calculated on the principal and on accumulated interest.
• Continuous Compounding: Compound interest calculated continuously, leading to exponential growth.

Understanding the nature of the interest rate and its effects on investments is crucial for effectively managing endowments.

Compounding Formula

The compounding formula is a tool that determines the total amount of money accrued over time, taking into account interest. For continuously compounded interest, a particular form of compounding, the formula used is:\[ A = Pe^{rt} \]Where:

• \(A\) is the final amount after time \(t\).
• \(P\) is the principal or initial endowment.
• \(r\) is the annual interest rate (expressed in decimal form).
• \(e\) represents the base of the natural logarithm (approximately 2.71828).

This formula accounts for exponential growth as interest is calculated at every possible instant. This is particularly important for permanent endowments, as it helps in understanding how much principal is needed to generate a specified annual return via interest alone. For a perpetual endowment, the principal \(P\) needs to be \(\frac{C}{r}\), ensuring that the interest income equals the desired annual income \(C\).