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It is now January \(1,2002 .\) Wayne-Martin Electric Inc. (WME) has just developed a solar panel capable of generating 200 percent more electricity than any solar panel currently on the market. As a result, WME is expected to experience a 15 percent annual growth rate for the next 5 years. By the end of 5 years, other firms will have developed comparable technology, and WME's growth rate will slow to 5 percent per year indefinitely. Stockholders require a return of 12 percent on WME's stock. The most recent annual dividend (D) , which was paid yesterday, was \(\$ 1.75\) per share. a. Calculate WME's expected dividends for \(2002,2003,2004,2005,\) and 2006 b. Calculate the value of the stock today, \(\hat{\mathrm{P}}_{0}\). Proceed by finding the present value of the dividends expected at the end of \(2002,2003,2004,2005,\) and 2006 plus the present value of the stock price that should exist at the end of \(2006 .\) The year-end 2006 stock price can be found by using the constant growth equation. Notice that to find the December \(31,2006,\) price, you use the dividend expected in \(2007,\) which is 5 percent greater than the 2006 dividend. c. Calculate the expected dividend yield, \(D_{1} / P_{0}\), the capital gains yield expected in 2002 , and the expected total return (dividend yield plus capital gains yield) for 2002 . (Assume that \(\mathrm{P}_{0}=\mathrm{P}_{0},\) and recognize that the capital gains yield is equal to the total return minus the dividend yield.) Also calculate these same three yields for 2007 . d. How might an investor's tax situation affect his or her decision to purchase stocks of companies in the early stages of their lives, when they are growing rapidly, versus stocks of older, more mature firms? When does WME's stock become "mature" in this example? e. Suppose your boss tells you she believes that WME's annual growth rate will be only 12 percent during the next 5 years and that the firm's normal growth rate will be only 4 percent. Without doing any calculations, what general effect would these growthrate changes have on the price of WME's stock? f. Suppose your boss also tells you that she regards WME as being quite risky and that she believes the required rate of return should be 14 percent, not 12 percent. Again, without doing any calculations, how would the higher required rate of return affect the price of the stock, its capital gains yield, and its dividend yield? Again, assume that the firm's normal growth rate will be 4 percent.

Step by step solution

01

Calculate Expected Dividends for 2002 to 2006

We start with the most recent dividend, \(D_0 = 1.75\). Since WME's dividends are expected to grow at a rate of 15% annually for the next five years, we calculate the expected dividends for the years 2002 to 2006 using the formula:\[D_n = D_0 \times (1 + g)^n\]where \(g = 0.15\) (15%). Hence, we calculate:- \(D_1 = 1.75 \times 1.15 = 2.0125\)- \(D_2 = 2.0125 \times 1.15 = 2.314375\)- \(D_3 = 2.314375 \times 1.15 = 2.66153125\)- \(D_4 = 2.66153125 \times 1.15 = 3.0607609375\)- \(D_5 = 3.0607609375 \times 1.15 = 3.519875078125\)

02

Determine the Year-End 2006 Stock Price

The stock price at the end of 2006 can be found using the Gordon Growth Model:\[P_6 = \frac{D_7}{r - g}\]where \(D_7 = D_6 \times 1.05\) (5% growth rate for 2007), \(r = 0.12\) (required return), and \(g = 0.05\).Calculate \(D_6\) first:\[D_6 = D_5 \times 1.15 = 3.519875078125 \times 1.15 = 4.04785633984375\]Then find \(D_7\):\[D_7 = 4.04785633984375 \times 1.05 = 4.25024915683593\]Now calculate \(P_6\):\[P_6 = \frac{4.25024915683593}{0.12 - 0.05} = 60.7178442405133\]

03

Calculate the Present Value of Future Dividends and Stock Price

To find \(\hat{P}_0\), the present value of expected dividends and stock price, we discount each dividend and the stock price back to the present value:\[\hat{P}_0 = \sum_{n=1}^{5} \frac{D_n}{(1 + r)^n} + \frac{P_6}{(1 + r)^6}\]Calculate the present value for each:- \(\frac{2.0125}{1.12} = 1.796875\)- \(\frac{2.314375}{1.12^2} = 1.84926339285714\)- \(\frac{2.66153125}{1.12^3} = 1.8908447506683\)- \(\frac{3.0607609375}{1.12^4} = 1.92201544936055\)- \(\frac{3.519875078125}{1.12^5} = 1.94456881120888\)- \(\frac{60.7178442405133}{1.12^6} = 30.7504521297942\)Sum these present values to find \(\hat{P}_0\):\[\hat{P}_0 = 1.796875 + 1.84926339285714 + 1.8908447506683 + 1.92201544936055 + 1.94456881120888 + 30.7504521297942 = 40.1540199338891\]

04

Calculate 2002 Expected Dividend Yield and Capital Gains Yield

The dividend yield for 2002 is given by:\[\text{Dividend Yield} = \frac{D_1}{\hat{P}_0} = \frac{2.0125}{40.1540199338891} = 0.050126703\]The total return expected is 12%. Thus, the capital gains yield is:\[\text{Capital Gains Yield} = r - \text{Dividend Yield} = 0.12 - 0.050126703 = 0.069873297\]

05

Calculate 2007 Expected Yields

By 2007, the growth rate has slowed to 5%, so the dividend yield involves the newly expected dividend:\[\text{Dividend Yield in 2006} = \frac{D_7}{P_6} = \frac{4.25024915683593}{60.7178442405133} = 0.0700000002647 \approx 0.07\]Given the constant growth rate of 5% thereafter:\[\text{Capital Gains Yield} = g = 0.05\]The expected total return will remain 12%, which is the sum of the dividend yield and capital gains yield in this scenario.

06

Comment on Investor's Tax Situation and Firm's Maturity

An investor's tax situation could influence their preference for high-growth firms with lower initial dividends but higher potential capital gains, versus mature firms with higher dividend payouts. WME's stock becomes 'mature' after 5 years when growth stabilizes at 5%.

07

Effect of Lower Growth Rates on Stock Price

If the growth rate is revised to 12% initially and then 4%, the stock price would likely decrease because reduced growth rates decrease expectations for future dividends and stock price appreciation, which ultimately lowers their present value.

08

Impact of a Higher Required Return

Increasing the required rate of return to 14% would also reduce the present value of expected future cash flows (dividends and stock price), leading to a lower stock value. This higher required return would increase the dividend yield but decrease the capital gains yield since the overall return is assumed to remain constant while adjusting the yield components.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Dividend Growth Model

The Dividend Growth Model is a valuable tool used for valuing the price of a stock by considering the future dividends it will bring. In this model, we assume dividends grow at a constant rate indefinitely. Let’s break down how it works:
- The formula to calculate the price of a stock today, known as the Gordon Growth Model, is: \( P_0 = \frac{D_1}{r - g} \ \) - Where \( P_0 \) is the current stock price, \( D_1 \) is the dividend expected next period, \( r \) is the required return, and \( g \) is the growth rate of dividends.
When utilizing the Dividend Growth Model, it’s important to note: - It works best for companies with stable and predictable growth rates.- It may not be ideal for firms undergoing rapid growth periods as their dividend growth rates are likely to change.- Margins for error can increase if estimates of future growth rates are inaccurate.
In the context of the exercise, the model helps find the year-end stock price based on expected dividends and a set growth rate.

Present Value

Present Value (PV) is a financial principle that helps us understand how valuable a future sum of money is in today's terms. We do this by discounting future amounts based on a rate of return, reflecting how much the money today is worth relative to receiving it in the future.
The formula to calculate the present value of a future cash flow is:- \[ PV = \frac{FV}{(1 + r)^n} \ \] - Where \( FV \) is the future value you want to calculate back to today's terms, \( r \) is the discount rate or required rate of return, and \( n \) is the number of periods until payment.
In the exercise, the stock's price today (\( \hat{P}_0 \)) is found by calculating the present value of all expected future dividends and the future stock price post-rapid growth. Calculating these present values gives investors a sense of what the stock is worth now based on expected future performance.
Understanding present value is key:- It helps us compare future cash flows on a consistent time-value basis.- Allows for better decision-making by comparing the present values of different investment alternatives.
In the realm of stock valuation, grasping present value principles ensures investors understand the today equivalent value of their future dividends.

Capital Gains Yield

Capital Gains Yield represents the portion of an investor's total return attributed to the change in the stock price. It reflects how much the value of the stock is expected to increase relative to its purchase price.
To calculate Capital Gains Yield, the formula is:- \[ \text{Capital Gains Yield} = \frac{P_1 - P_0}{P_0} \ \] - Here, \( P_1 \) is the price at the end of the period, and \( P_0 \) is the initial price.
In the exercise, the capital gains yield within the first rapid growth phase is part of the total expected return of 12% for 2002. The expected total return comprises both the capital gains yield and dividend yield, emphasizing the appreciation in stock value during the early high-growth period.
Key points about Capital Gains Yield:- It is largely influenced by market conditions and company prospects.- High growth firms provide potentially higher gains but come with increased risk.- Over time, as growth stabilizes, the focus might shift towards dividends, diminishing capital gains yield.
Understanding Capital Gains Yield enables investors to anticipate how much their stock value will grow, a critical factor when deciding to buy or sell.

Dividend Yield

Dividend Yield is a financial metric that indicates how much a company pays in dividends each year relative to its stock price. It's a vital measure for investors seeking income from their investments, as opposed to solely capital gains from stock value appreciation.
The formula for Dividend Yield is:- \[ \text{Dividend Yield} = \frac{D_1}{P_0} \ \] - \( D_1 \) is the expected dividend in the next period, while \( P_0 \) is the current stock price.
In the context of the exercise:- Initially, the dividend yield is relatively low because the high initial growth focuses on capital gains.- As the growth rate slows in 2007, the dividend yield increases, indicating a shift in investor expectations from growth to income.
Insights on Dividend Yield:- A higher yield may indicate that a company is returning a significant portion of earnings to shareholders through dividends.- It provides clues about how much an investor can expect to earn from holding a stock, aside from stock price changes.
Investors focused on secure cash flow will prioritize dividend yield, making it crucial to understand in any stock valuation analysis.