Problem 71 - Applied Calculus

Question

BUSINESS: Preferred Stock Since preferred stock can remain outstanding indefinitely, the present value per share is the limit of the present value of an annuity* paying that share's dividend \(D\) at interest rate \(r\) : $$ \left(\begin{array}{c} \text { Present } \\ \text { value } \end{array}\right)=\lim _{t \rightarrow \infty} D\left(\frac{1-(1+r)^{-t}}{r}\right) $$ Find this limit in terms of \(D\) and \(r\).

Step-by-Step Solution

Answer

The limit is \( \frac{D}{r} \).

Step 1: Understand the formula

The formula given is the present value of an annuity: \( D\left(\frac{1-(1+r)^{-t}}{r}\right) \). It represents the present value of receiving dividend \( D \) indefinitely with an interest rate \( r \).

Step 2: Analyze the limit expression

We need to find \( \lim_{t \to \infty} \left(D\left(\frac{1-(1+r)^{-t}}{r}\right)\right) \). As \( t \to \infty \), the term \((1 + r)^{-t}\) tends towards zero because the exponential decay makes it approach zero.

Step 3: Simplify the expression

Substitute \((1+r)^{-t} \to 0\) as \( t \to \infty \) into the expression: \( D\left(\frac{1-(1+r)^{-t}}{r}\right) \rightarrow D\left(\frac{1-0}{r}\right) = \frac{D}{r} \).

Step 4: Interpret the result

The limit as \( t \to \infty \) simplifies to \( \frac{D}{r} \), meaning the present value of the preferred stock that pays dividend \( D \) indefinitely at interest rate \( r \) is \( \frac{D}{r} \).

Key Concepts

Preferred Stock

Preferred stock is a type of equity security that may have characteristics of both stocks and bonds. It represents a share in a company, similar to common stock, but typically offers fixed dividends, which are paid out before dividends to common stockholders. This makes it an attractive option for investors looking for steady income.

Preferred stocks do not usually come with voting rights, unlike common stocks. One of the key features of preferred stocks is their fixed dividend, meaning the payment amount is set at issuance. This share type is called _"preferred"_ because it has preference over common stocks in terms of payment of dividends and asset liquidation.

Because preferred stock dividends can extend indefinitely, they are often analyzed through the lens of their present value, much like bonds. The present value of the preferred stock dividends is crucial for an investor to determine their return on investment. Understanding the present value helps in predicting the future value of an investment given a certain discount rate.

Annuity

An annuity is a sequence of equal payments made at regular intervals. Common examples include retirement fund payouts and monthly insurance installments. In the context of preferred stock, we can compare dividend payments to an annuity. With preferred stocks, the dividends serve as those regular payments that continue indefinitely.

A present value annuity formula is used to evaluate the current worth of a series of future cash flows given a constant interest rate. When involving preferred stock dividends, the formula helps in determining how much those future dividend payments are worth today.

• It calculates the sum of these future payments discounted back to their present day value.
• The discount rate reflects the time value of money, which acknowledges that a dollar today is worth more than a dollar in the future.

This allows investors to assess the fairness of the stock price relative to the dividend income it promises.

Limit Calculation

In the scenario of preferred stocks, where the dividends are equated to an annuity that lasts indefinitely, finding the limit for the present value formula is crucial. As noted, the formula used is:\[ \lim_{t \rightarrow \infty} D\left(\frac{1-(1+r)^{-t}}{r}\right) \]Here’s why calculating the limit is important: it provides the mathematical framework needed to understand how perpetual payments take shape when evaluated today, with continually decreasing present value factors.

• The analysis involves seeing how the struggle of multiplying decreasing terms at long periods simplifies the calculation by eliminating the dropped terms like \(1 + r\)^{-t}.
• As \(t\) approaches infinity, \( (1 + r)^{-t} \) vanishes, effectively leaving us with \( \frac{D}{r} \).

Thus, this calculation concludes that the present value of an indefinite dividend stream is determined simply by dividing the dividend amount \(D\) by the interest rate \(r\). This limit process shows how preferred stock behaves similarly to a perpetual bond in financial modeling.