Problem 43 - Applied Calculus

Question

GENERAL: Permanent Endowments Find the size of the permanent endowment needed to generate an annual \(\$ 12,000\) forever at a continuous interest rate of \(6 \%\).

Step-by-Step Solution

Answer

A permanent endowment of \$200,000 is needed.

Step 1: Understanding the Concept of Perpetuity

A permanent endowment provides funds indefinitely, which means it is a perpetuity. In this case, the annual payment from the endowment is $12,000.

Step 2: Identifying the Formula for Perpetuity

The formula to calculate the present value of a perpetuity is \(PV = \frac{C}{r}\), where \(PV\) is the present value (endowment size), \(C\) is the annual cash flow (\$12,000 in this case), and \(r\) is the interest rate.

Step 3: Substitute Known Values into the Formula

Substitute \(C = 12,000\) and \(r = 0.06\) (since 6% in decimal form is 0.06) into the perpetuity formula: \(PV = \frac{12,000}{0.06}\).

Step 4: Calculate the Present Value (Endowment Size)

Calculate the division: \(PV = \frac{12,000}{0.06} = 200,000\). This means \\(200,000 is needed to generate \\)12,000 annually at a 6% continuous interest rate.

Key Concepts

Perpetuity

A perpetuity is a type of financial arrangement that provides continuous payments over an indefinite period. This concept is central to understanding how certain financial products and endowments work. The idea is that a fixed amount of money, known as the cash flow, is paid out forever. As long as the conditions remain the same, the payments never cease. This makes perpetuities a fascinating subject in financial mathematics and a popular tool for planning long-term financial commitments. In the case of our exercise, the perpetuity is represented by a permanent endowment that pays \( \$12,000 \) annually. The term 'permanent' here indicates that the payments are aimed to go on forever, a distinct characteristic of perpetuities.

Present Value

Present value is a core concept in financial mathematics that refers to the current value of a future series of cash flows, given a specific rate of return or interest rate. In our scenario, the present value refers to the amount of money required today to ensure a perpetuity of \( \\(12,000 \) annual payments. By applying the formula for perpetuity, \( PV = \frac{C}{r} \), we can find the present value needed. This formula essentially tells us how much money we'd need to invest today at a certain interest rate to achieve our desired future cash flow. Here, \( C \) represents the annual cash flow of \( \\)12,000 \), and \( r \) is the interest rate of \( 6\% \) (expressed as a decimal, or \( 0.06 \)). By substituting these values into the formula, we determine that \( \$200,000 \) is the required endowment size.

Endowment Calculation

Endowment calculation involves determining the size of an initial investment or fund required to achieve a specific financial goal. An endowment is a fund established to provide ongoing financial support, with its earnings used for a designated purpose. In this context, the endowment is intended to generate a perpetual cash flow of \( \\(12,000 \). The working formula \( PV = \frac{C}{r} \) allows us to swiftly compute the needed fund size. By inputting \( 12,000 \) as the annual payout and \( 0.06 \) as the continuous interest rate, the calculation results in \( \\)200,000 \). This sum forms the principal amount that needs to be invested to support the specified perpetual distribution. Understanding how endowment calculations are structured helps in planning robust long-term financial strategies.