Problem 74 - Applied Calculus

Question

73-74. GENERAL: Permanent Endowments The formula for integrating the exponential function \(a^{b x}\) is \(\int a^{b x} d x=\frac{1}{b \ln a} a^{b x}+C\) for constants \(a>0\) and \(b,\) as may be verified by using the differentiation formulas on page 289. Use the formula above to find the size of the permanent endowment needed to generate an annual \(\$ 12,000\) forever at \(6 \%\) interest compounded annually. [Hint: Find \(\left.\int_{0}^{\infty} 12,000 \cdot 1.06^{-x} d x .\right]\) Compare your answer with that found in Exercise 43 (page 413 ) for the same interest rate but compounded continuously.

Step-by-Step Solution

Answer

The permanent endowment needed is approximately $205,904.29.

Step 1: Understanding the Problem

We need to find the size of a permanent endowment that generates an annual payment of $12,000 forever with a 6% interest rate compounded annually.

Step 2: Set up the Integral

The hint suggests using an integral to find the present value of all future payments. We will calculate \[\int_{0}^{\infty} 12,000 \cdot 1.06^{-x} \, dx.\]

Step 3: Identify the Exponential Function

Recognize the exponential part of the integral, which is in the form of \(a^{bx}\). Here, \(a = 1.06\), \(b = -1\). The function becomes \((1.06)^{-x}.\) We will use the exponential integration formula provided earlier.

Step 4: Apply the Integration Formula

Substitute into the integration formula:\[\int a^{bx} \, dx = \frac{1}{b \ln a} a^{bx} + C.\]For our integral, substitute \(a = 1.06\) and \(b = -1\), then compute:\[\int 12,000 \cdot 1.06^{-x} \, dx = 12,000 \cdot \frac{1}{-1 \ln 1.06}(1.06)^{-x} + C.\]

Step 5: Evaluate the Integral at the Limits

We need to evaluate:\[\left[ -\frac{12,000}{\ln(1.06)} (1.06)^{-x} \right]_{0}^{\infty}.\]As \(x\) approaches infinity, \((1.06)^{-x}\) goes to 0, simplifying to 0. Evaluating at 0 gives:\[-\frac{12,000}{\ln(1.06)} (1.06)^{0} = -\frac{12,000}{\ln(1.06)}.\]

Step 6: Calculate the Endowment Size

The size of the endowment is the negative of the value found when \(x = 0\):\[\frac{12,000}{\ln(1.06)}.\]Calculate this value to find the endowment size needed.

Step 7: Final Calculation and Result

To get the final number, calculate:\[\frac{12,000}{\ln(1.06)} \approx \frac{12,000}{0.0582689} \approx 205,904.29.\]Thus, the permanent endowment needed is approximately \$205,904.29.

Key Concepts

Permanent Endowment

When you're considering a permanent endowment, it's about creating a sustainable financial resource that can generate income indefinitely. In this context, the idea is to maintain a principal amount that earns interest, which in turn provides a steady stream of funds like the $12,000 annual payment in our exercise. The key is that the principal, or endowment size, is calculated so that it remains intact while the interest pays out the desired income. Permanent endowments are often used by institutions like schools or charities, where they rely on these funds to cover ongoing expenses without touching the initial investment. Permanent endowments offer:

• A consistent inflow of funds to support specific needs or operations
• Financial stability without depleting the principal
• Potential growth if the interest generated exceeds the required payout

In solving for the endowment size, integration helps determine the present value of infinite future payments, making sure that the base capital remains for continuous yields.

Compounded Interest

Compounded interest is essentially interest earned on interest. This concept is powerful because it allows even small amounts of interest to accumulate significantly over time. In our exercise, interest is compounded annually at a 6% rate. This means that each year, the interest is added to the principal, and in the next compounding period, the interest calculation is based on this new total amount. When you think about compounded interest:

• It accelerates the growth of the principal since interest accumulates on reinvested earnings.
• The frequency of compounding matters; more frequent compounding results in more interest.
• It's a key factor in understanding the growth of investments and the cost of loans.

In contrast to simple interest, which is calculated only on the initial principal, compounded interest can significantly enhance the funds available in an endowment, thereby reducing the amount needed initially to meet your payout goals.

Present Value Calculation

Present value calculation is a financial concept that helps you understand what a future amount of money is worth today. This is crucial in endowment discussions because it determines how much you need now to meet future financial obligations.The idea is that money has a time value; a dollar today is worth more than a dollar tomorrow due to its potential earning capacity.In our exercise, the present value calculation is used to determine the size of the endowment necessary to provide infinite payments of $12,000 annually. Here's how it works:

• We calculate an integral to sum up an infinite series of future payments.
• Using the formula \( a^{bx} \), we decipher the present value of continuing payments, bringing the focus to the current value of those future earnings.
• This calculation helps in setting up the endowment such that the interest generated can sustain the desired outflow without depleting the principal.

By understanding the present value, you can effectively plan for future needs, ensuring your investments are capable of meeting long-term commitments.