Question

One method of pricing a stock is to discount the stream of future dividends of the stock. Suppose that a stock pays \(\$ P\) per year in dividends, and historically, the dividend has been increased \(i \%\) per year. If you desire an annual rate of return of \(r \%,\) this method of pricing a stock states that the price that you should pay is the present value of an infinite stream of payments: $$\text { Price }=P+P \cdot \frac{1+i}{1+r}+P \cdot\left(\frac{1+i}{1+r}\right)^{2}+P\cdot\left(\frac{1+i}{1+r}\right)^{3}+\cdots$$ The price of the stock is the sum of an infinite geometric series. Suppose that a stock pays an annual dividend of \(\$ 4.00\), and historically, the dividend has been increased \(3 \%\) per year. You desire an annual rate of return of \(9 \%\). What is the most you should pay for the stock?

Short answer

The most you should pay for the stock is approximately 72.73 dollars.

Step-by-step solution

- Identify Constants

First, we need to identify the constants from the problem: The annual dividend, P, is 4.00 dollars. The historical increase in dividend, i, is 3%, which is 0.03 in decimal form. The desired annual rate of return, r, is 9%, which is 0.09 in decimal form.

- General Formula for Geometric Series

The price of the stock can be found by using the formula for the sum of an infinite geometric series. The general formula for the sum S of an infinite geometric series with the first term a and common ratio r is:equation\[ S = \frac{a}{1 - r} \]

- Apply Constants To Formula

In this problem, a represents the annual dividend P, and r is the ratio \( \frac{1+i}{1+r} \). Substitute these values into the formula:equation\[ \text{Price} = \frac{P}{1 - \frac{1+i}{1+r}} \]

- Plug In the Values

Substitute P = 4.00 dollars, i = 0.03, and r = 0.09 into the formula:\[ \text{Price} = \frac{4.00}{1 - \left(\frac{1+0.03}{1+0.09}\right)} \]

- Simplify the Ratio

Calculate the ratio \( \frac{1+0.03}{1+0.09} \) which gives:\[ \frac{1.03}{1.09} \approx 0.945 \]So, the formula now becomes:\[ \text{Price} = \frac{4.00}{1 - 0.945} \]

- Simplify Denominator

Calculate the denominator:\[ 1 - 0.945 = 0.055 \]So, the formula now becomes:\[ \text{Price} = \frac{4.00}{0.055} \]

- Final Calculation

Perform the final division to get the price:\[ \text{Price} \approx 72.73 \]

Key concepts

stock pricing

Stock pricing can often be a complex task involving various metrics and calculations. One common method of stock pricing is to discount the stream of future dividends. Essentially, you are calculating the present value of all likely future dividend payments the stock will make, taking into account growth and a desired rate of return. This yields a theoretical fair price for the stock.
In simpler terms, it’s like asking: 'How much is it worth today, if I expect to get regular payments from this stock in the future?' This method requires the use of mathematical concepts such as a geometric series.
In the referenced exercise, we are given specific numbers: a historical increase in dividends of 3% per year and a desired rate of return of 9%. Using these numbers, we can formulaically determine the price one should pay for the stock now, to achieve the targeted returns.

infinite series

An infinite series is a sum of infinitely many terms. In the context of stock pricing, we often deal with an infinite geometric series, which has a common ratio between successive terms.
Geometric series have the property that under certain conditions, they converge to a finite sum. For a series to converge, its common ratio must have an absolute value less than one. The formula for the sum of an infinite geometric series with first term \( a \) and common ratio \( r \) is given by:
\[ S = \frac{a}{1 - r} \]
This concept becomes useful when pricing stocks because we model the price of the stock as the sum of an infinite geometric series of discounted future dividends. This approach allows investors to find a single price representing the value of all those future dividends considered together. The exercise illustrates using this series formula to price a stock given specific dividend growth and desired return rates.

present value

Present value is a key financial concept that represents the current worth of a sum of money or a series of future cash flows, given a specified rate of return. The idea is to understand how much future cash flows are worth in today’s dollars.
In simple terms, would you rather have \( \$100 \) today or \( \$100 \) a year from now? Most people would prefer \( \$100 \) today due to the time value of money. Present value calculations adjust future amounts so they can be compared to present amounts.
In the exercise, the present value of future dividends is calculated in the context of stock pricing. The sum of the series of dividends includes a growth factor (historical increase in dividends) and a discount factor (desired rate of return). The exercise guides through the steps to determine the most one should pay for the stock today, assuming future dividends grow consistently and an investor has a specific target rate-of-return in mind. This is where the infinite geometric series comes in handy to sum up all those discounted future dividends into a single present value calculation.