Chapter 9: Problem 11
A stock is expected to pay a dividend of \(\$ 0.50\) at the end of the year (that is, \(D_{1}=0.50\) ), and it should continue to grow at a constant rate of \(7 \%\) a year. If its required return is \(12 \%,\) what is the stock's expected price 4 years from today?
Short Answer
Expert verified
The expected stock price 4 years from today is approximately \(\$13.20\).
Step by step solution
01
Identify the Given Variables
We start by identifying the parameters given in the problem:- Dividend expected at the end of the year, \( D_1 = 0.50 \).- Constant growth rate of the dividend, \( g = 7\% = 0.07 \).- Required rate of return, \( r = 12\% = 0.12 \). We need to find the stock's expected price 4 years from today.
02
Calculate Future Dividends
Determine the expected dividend in year 4 using the dividend growth formula:\[ D_n = D_1 \times (1+g)^{n-1} \]Where \(D_n\) is the dividend in year \(n\). Here, we want the dividend at year 4 (\(D_4\)):\[ D_4 = 0.50 \times (1+0.07)^{4-1} \]Calculate this to find \(D_4\).
03
Compute Dividend at Year 4
Calculate \(D_4\) using the formula from Step 2:\[ D_4 = 0.50 \times 1.07^3 = 0.50 \times 1.225043 = 0.6125 \]Thus, the dividend in year 4 is approximately \(\$0.61\).
04
Determine Stock Price 4 Years from Today Using Gordon Growth Model
Using the Gordon Growth Model (a type of dividend discount model), the price of the stock in year 4 can be calculated as:\[ P_4 = \frac{D_5}{r-g} \]Where \(D_5\) is the dividend at year 5:First, calculate \(D_5\) as:\[ D_5 = D_4 \times (1+g) = 0.6125 \times 1.07 \]Then, substitute \(D_5\) into the formula to find \(P_4\).
05
Calculate Dividend at Year 5
Calculate \(D_5\):\[ D_5 = 0.6125 \times 1.07 = 0.655375 \approx 0.66 \]The dividend at year 5 is approximately \(\$0.66\).
06
Calculate Stock Price at Year 4
Use the calculated \(D_5\) to find the stock's price in year 4:\[ P_4 = \frac{0.66}{0.12 - 0.07} = \frac{0.66}{0.05} = 13.20 \]The expected stock price 4 years from today is approximately \(\$13.20\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Gordon Growth Model
The Gordon Growth Model, named after economist Myron Gordon, is a popular method for valuing a stock's price based on expected future dividends and growth rates. It is a type of dividend discount model which assumes dividends will continue to grow at a constant rate indefinitely. This method is particularly useful for companies with stable growth rates.
To use the Gordon Growth Model, you need a few key pieces of information:
In our original exercise, we applied this model not just at the present time but to predict a future price by estimating dividends in subsequent years and accounting for growth. This forward-looking adaptation helps assess long-term investments.
To use the Gordon Growth Model, you need a few key pieces of information:
- Expected dividend for the next year (D_1).
- Dividend growth rate (g).
- Required rate of return (r).
In our original exercise, we applied this model not just at the present time but to predict a future price by estimating dividends in subsequent years and accounting for growth. This forward-looking adaptation helps assess long-term investments.
Stock Valuation
Stock valuation is the method used to determine the intrinsic value of a company's shares. Assessing this value involves evaluating key financial indicators and future potential, rather than just the current market price. The core goal is to identify whether a stock is over or undervalued compared to its calculated intrinsic value.
Different methods for stock valuation include:
Different methods for stock valuation include:
- Past Performance Analysis: Looks at historical data and performance trends.
- Relative Valuation: Compares measures such as P/E ratios to peers or industry averages.
- Discounted Cash Flow: Projects future cash flows and discounts them back to present value.
Dividend Growth Rate
The dividend growth rate is the annualized percentage rate of growth that a company's dividend is expected to undergo. It plays a crucial role in valuing stocks using models like the Gordon Growth Model by indicating how much dividends are likely to increase each year.
A stable dividend growth rate implies predictable income streams for investors, making these stocks attractive for long-term portfolio growth. Common factors influencing dividend growth rates include:
A stable dividend growth rate implies predictable income streams for investors, making these stocks attractive for long-term portfolio growth. Common factors influencing dividend growth rates include:
- Company Earnings: Increased earnings can lead to higher dividends.
- Business Expansion: Growth in operations often supports higher dividend payouts.
- Market Conditions: Economic factors can accelerate or hamper growth rates.