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:

$Price=P+P\left(\frac{1+i}{1+r}\right)+P{\left(\frac{1+i}{1+r}\right)}^2+.....$

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

One should pay at most $72.7 .

Step-by-step solution

Given information

We are given$P=5,i=0.03,r=0.09$

Find the geometric sequence

We have

$$\begin{gathered} P_1=P=4 \\ {P}_n=\frac{1+i}{1+r}{a}_{n-1} \\ {P}_n=\frac{1+0.03}{1+0.09}{a}_{n-1} \\ {P}_n=0.945a_{n-1} \end{gathered}$$

Calculate the value of S

We use the formula

$$\begin{gathered} S=\frac{a_1}{1-0.945} \\ S=\frac{4}{1-0.945} \\ S=72.2 \end{gathered}$$

Conclusion

One should pay at most $72.7