Problem 39 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

An investment guarantees annual payments of \(\$ 5,000\) in perpetuity, with the payments beginning immediately. Find the present value of this investment if the prevailing annual interest rate remains fixed at \(5 \%\) compounded continuously.

Step-by-Step Solution

Answer

The present value of the investment is \text{\textdollar}100,000.

Step 1: Understand the Concept of Perpetuity

A perpetuity is a type of annuity that provides continuous payments indefinitely. The present value of a perpetuity can be calculated using a formula that considers the payment amount and the interest rate.

Step 2: Identify the Formula for Present Value of a Perpetuity

For continuous compounding, the present value of a perpetuity can be calculated using the formula \[ PV = \frac{P}{r} \] where \(P\) is the payment amount and \(r\) is the annual interest rate.

Step 3: Plug in the Given Values

Given that the annual payment \(P\) is \(\text{\textdollar} 5,000\) and the annual interest rate \(r\) is \(5 \text{\text{\bfseries\textpercent}} = 0.05\), substitute these values into the formula: \[ PV = \frac{5000}{0.05} \]

Step 4: Calculate the Present Value

Perform the division to find the present value: \[ PV = \frac{5000}{0.05} = 100,000 \]

Key Concepts

perpetuity

A perpetuity is a type of financial instrument that provides a series of infinite payments. Unlike other investments, which have a fixed end date, a perpetuity continues indefinitely. This makes it a unique form of annuity. The key aspect of a perpetuity is its endless nature. It guarantees regular payments without an end. This is often seen with certain types of bonds and dividends. Understanding this continuous component is central to mastering the concept of perpetuity.

present value

The present value (PV) of an investment represents the current worth of a future stream of payments. The idea is to figure out how much those future payments are worth in today's dollars. This requires discounting future payments based on an interest rate. For perpetuities, the formula is especially simple: \[PV = \frac{P}{r}\] where \(P\) is the payment amount and \(r\) is the annual interest rate. This formula assumes payments begin immediately. If the payments start after a delay, an adjustment would be needed.

continuous compounding

Continuous compounding refers to the process where interest is calculated and added to the account balance constantly. Unlike annual compounding, which happens once per year, continuous compounding happens every infinitesimally small moment. This effectively grows the investment at an ever-accelerating rate. For the present value of a perpetuity with continuous compounding, the standard formula still holds true: \[PV = \frac{P}{r} \]. Although the compounding is continuous, the fact that payments are consistent means the formula remains straightforward.

annuity

An annuity is a financial product that pays out a fixed stream of payments over time. When discussing investments like perpetuities, it's important to differentiate between different types of annuities. Regular annuities have a defined end, with a clear number of payments. Perpetuities, however, do not end. They are a subtype of annuity with infinite payments. Understanding this distinction is critical to applying the correct formulas and concepts to solve problems involving these financial instruments.