Dividend Discount Model (DDM) - Finance

1. A company with stock trading at $90 per share announces a 3-for-1 stock split. Before the split, the company's most recent annual dividend was $1.80 per share, and it was expected to grow at 5% annually. Assuming the split has no impact on the company's total market value or dividend policy, what is the expected dividend per share one year from now, after the split takes effect?
   Answer: $0.63 A 3-for-1 stock split means that for every one share an investor owned, they will now have three. To keep the total value constant, per-share metrics must be divided by 3. The most recent dividend on a post-split basis would be $\$1.80 / 3 = \$0.60$. This is the new $D_0$. The question asks for the expected dividend one year from now (the new $D_1$), which is found by growing the new $D_0$ at the constant growth rate: New $D_1 = (\text{New } D_0)(1+g) = \$0.60(1.05) = \$0.63$.
2. Apogee Inc. has a required rate of return of 12%. It is expected to earn $5.00 per share in the coming year and plans to maintain a 60% dividend payout ratio. The company's stock currently trades at $75 per share. Based on this information, what is the market's assessment of the present value of Apogee's growth opportunities (PVGO)?
   Answer: $33.33 The value of a stock can be partitioned into its no-growth value and the present value of its growth opportunities (PVGO). The no-growth value is the value the company would have if it did not reinvest any earnings, which is calculated as next year's earnings discounted by the required return: No-growth value = $E_1 / r = \$5.00 / 0.12 = \$41.67$. The PVGO is the difference between the market price and the no-growth value: PVGO = $P_0 - (E_1 / r) = \$75.00 - \$41.67 = \$33.33$.
3. Phoenix Corp. is a mature company in a declining industry. Its most recent dividend was $5.00 per share. Due to declining sales, the company is expected to decrease its dividend by 2% per year indefinitely. If the required rate of return for Phoenix is 8%, what is its estimated stock price?
   Answer: $49.00 The constant growth DDM can accommodate negative growth. The growth rate $g$ is -2.0% or -0.02. First, calculate the dividend for next year: $D_1 = D_0(1+g) = \$5.00(1 - 0.02) = \$4.90$. Then, apply the DDM formula, being careful with the signs: $P_0 = D_1 / (r - g) = \$4.90 / (0.08 - (-0.02)) = \$4.90 / (0.08 + 0.02) = \$4.90 / 0.10 = \$49.00$.
4. A stock is correctly priced at $40.00. It is expected to pay a dividend of $1.20 one year from now, and its required rate of return is 10%. Based on the constant growth DDM, what is the expected capital gain in dollar terms on this stock over the next year?
   Answer: $2.80 First, find the capital gains yield, which is equivalent to the constant growth rate ($g$). We know that $r = \text{Dividend Yield} + \text{Capital Gains Yield}$. The dividend yield is $D_1 / P_0 = \$1.20 / \$40.00 = 0.03$ or 3%. So, $g = r - (D_1/P_0) = 10\% - 3\% = 7\%$. The expected dollar capital gain is the current price multiplied by the capital gains yield: Dollar Gain = $P_0 \times g = \$40.00 \times 0.07 = \$2.80$.
5. A company has a required rate of return of 11% and a return on equity of 15%. The company plans to maintain a dividend payout ratio of 40%. Based on the constant growth dividend discount model, what is the company's justified leading price-to-earnings (P/E) ratio?
   Answer: 20.0 The justified leading P/E ratio is derived from the DDM as $P_0/E_1 = (D_1/E_1) / (r-g)$. The term $D_1/E_1$ is the dividend payout ratio, which is given as 40% or 0.40. The growth rate $g$ is calculated as $g = ROE \times b$, where $b$ is the retention ratio. The retention ratio is $1 - \text{payout ratio} = 1 - 0.40 = 0.60$. So, $g = 0.15 \times 0.60 = 0.09$ or 9%. Plugging these values into the P/E formula: $P_0/E_1 = 0.40 / (0.11 - 0.09) = 0.40 / 0.02 = 20.0$.
6. A company's stock has an intrinsic value of $80.00 today based on a constant growth DDM valuation. The valuation assumes a required rate of return of 12% and a perpetual dividend growth rate of 7%. What is the expected price of the stock three years from today?
   Answer: $98.00 In the constant growth Dividend Discount Model, the stock price is expected to grow at the same rate as the dividends, which is $g$. Therefore, the expected price at time $t$, $P_t$, can be calculated as $P_t = P_0(1+g)^t$. For this problem, $P_3 = P_0(1+g)^3 = \$80.00(1.07)^3 = \$80.00(1.225043) = \$98.00$.
7. An analyst has the following dividend per share forecasts for a company: Year 1: $2.00 Year 2: $2.10 Year 3: $2.205 The analyst believes that after Year 3, dividends will grow at a constant rate equal to the Year 2 to Year 3 growth rate. If the required return is 12%, what is the stock's estimated value at the end of Year 2 ($P_2$)?
   Answer: $31.50 First, determine the constant growth rate (g) from the forecasted dividends. The growth from Year 2 to Year 3 is $(\$2.205 - \$2.10) / \$2.10 = 0.05$, or 5%. This is the long-term growth rate. The value of a stock at time $t$ using the constant growth model is the dividend at time $t+1$ divided by $(r-g)$. To find the value at the end of Year 2 ($P_2$), we use the dividend from Year 3 ($D_3$): $P_2 = D_3 / (r-g) = \$2.205 / (0.12 - 0.05) = \$2.205 / 0.07 = \$31.50$.
8. An analyst is valuing a high-growth technology company that just paid a $1.00 dividend. The analyst forecasts a perpetual dividend growth rate of 15%. The company's beta is 2.0, the risk-free rate is 4%, and the expected market return is 9%. Which of the following is the most appropriate conclusion?
   Answer: The constant growth model is inappropriate for valuing this company. When you encounter dividend discount model problems, always check whether the model's assumptions are met before attempting any calculations. The constant growth dividend discount model (Gordon Growth Model) requires that the required return exceed the growth rate for the model to produce meaningful results. Let's first calculate the required return using CAPM: $r = r_f + \beta(r_m - r_f) = 4\% + 2.0(9\% - 4\%) = 14\%$. Now we have a growth rate of 15% and a required return of 14%. Since the growth rate (15%) exceeds the required return (14%), the constant growth model breaks down mathematically and conceptually. The formula $P = \frac{D_1}{r-g}$ becomes $P = \frac{1.15}{0.14-0.15} = \frac{1.15}{-0.01}$, yielding a negative denominator. Answer A ($115.00) likely comes from incorrectly calculating $\frac{1.15}{0.15-0.14} = 115$, which reverses the denominator terms. Answer B (negative intrinsic value) reflects the mathematical result but misses that this signals model failure, not actual negative value. Answer C incorrectly assumes high growth automatically means attractive investment without considering the required return relationship. Answer D correctly recognizes that when growth rates exceed required returns, the constant growth model is inappropriate. This scenario suggests either unrealistic assumptions or the need for a multi-stage growth model that eventually assumes growth rates decline to sustainable levels. Study tip: Always verify that $r > g$ before applying the Gordon Growth Model. When this condition fails, consider multi-stage models or question whether the growth assumptions are realistic.
9. A company's stock is trading at $50.00 per share. It is expected to pay a dividend of $2.00 per share one year from now, and the dividend is expected to grow at a constant rate thereafter. If the stock's required rate of return is 10%, what is the implied constant dividend growth rate according to the Dividend Discount Model?
   Answer: 6.0% The constant growth Dividend Discount Model (DDM) formula is $P_0 = D_1 / (r - g)$, where $P_0$ is the current price, $D_1$ is the dividend in one year, $r$ is the required rate of return, and $g$ is the constant growth rate. We can rearrange the formula to solve for $g$: $g = r - (D_1 / P_0)$. Plugging in the given values: $g = 0.10 - (2.00 / 50.00) = 0.10 - 0.04 = 0.06$, or 6.0%.
10. A stock has a required rate of return of 13%. An analyst using the constant growth DDM estimates the stock's intrinsic value to be $75.00. The stock is expected to pay a dividend of $3.00 next year. What is the implied capital gains yield for this stock?
    Answer: 9.0% In the context of the DDM, the total required rate of return ($r$) is the sum of the dividend yield and the capital gains yield. The dividend yield is $D_1 / P_0 = \$3.00 / \$75.00 = 0.04$ or 4.0%. Therefore, Capital Gains Yield = Total Return - Dividend Yield = $13\% - 4\% = 9\%$. In the constant growth model, the capital gains yield is equal to the dividend growth rate, $g$.
11. An analyst is valuing Titan Corp. using the constant growth DDM. Titan just paid a dividend of $2.20 per share. The analyst assumes Titan's dividend growth rate will be the same as that of its closest competitor, Atlas Inc. Atlas stock trades at $63 per share, its required return is estimated at 9%, and it is expected to pay a dividend of $2.52 next year. If Titan's required return is 10%, what is the estimated value of Titan's stock?
    Answer: $46.20 First, calculate the implied growth rate ($g$) for the competitor, Atlas Inc., using the DDM formula $P_0 = D_1 / (r-g)$. Rearranging gives $g = r - (D_1/P_0)$. For Atlas, $g = 0.09 - (\$2.52 / \$63.00) = 0.09 - 0.04 = 0.05$, or 5%. Second, use this growth rate to value Titan. Calculate Titan's expected dividend next year: $D_1 = D_0(1+g) = \$2.20(1.05) = \$2.31$. Finally, calculate Titan's stock price: $P_0 = D_1 / (r-g) = \$2.31 / (0.10 - 0.05) = \$2.31 / 0.05 = \$46.20$.
12. An analyst values a stock at $60.00 using a constant growth model with a required return of 10% and a perpetual growth rate of 5%. The analyst then revises the forecast for the constant dividend growth rate from 5% to 6%, with the revision affecting all future dividends starting with $D_1$. Assuming the most recent dividend ($D_0$) has already been paid and the required return remains 10%, what is the estimated percentage change in the stock's value?
    Answer: 26.2% The price is given by $P_0 = D_0(1+g)/(r-g)$. Let $P_{old}$ be the price with $g=5\%$ and $P_{new}$ be the price with $g=6\%$. The ratio of the new price to the old price is $P_{new}/P_{old} = [\frac{D_0(1+g_{new})}{r-g_{new}}] / [\frac{D_0(1+g_{old})}{r-g_{old}}] = \frac{(1+g_{new})}{(1+g_{old})} \times \frac{(r-g_{old})}{(r-g_{new})}$. Plugging in the values: $P_{new}/P_{old} = \frac{1.06}{1.05} \times \frac{0.10-0.05}{0.10-0.06} = \frac{1.06}{1.05} \times \frac{0.05}{0.04} \approx 1.00952 \times 1.25 = 1.2619$. The percentage change is $1.2619 - 1 = 0.2619$, or approximately 26.2%.
13. An investor plans to buy a stock today and hold it for two years. The stock just paid a dividend of $1.50. The dividend is expected to grow at a constant rate of 6% per year. The investor's required return is 11%. Assuming the constant growth model is appropriate, what is the maximum price the investor should be willing to pay for the stock today?
    Answer: $31.80 The intrinsic value of a stock according to the DDM represents the present value of all future dividends. An investor's planned holding period does not change this intrinsic value, because the price they receive upon selling the stock (the terminal value) will itself reflect the present value of all subsequent dividends. Therefore, we value the stock as if it were held forever using the standard constant growth DDM. First, $D_1 = D_0(1+g) = \$1.50(1.06) = \$1.59$. Then, $P_0 = D_1 / (r-g) = \$1.59 / (0.11 - 0.06) = \$1.59 / 0.05 = \$31.80$.
14. The expected total return on a company's stock is 12%, and its dividend yield is 4.5%. The company adheres to a constant dividend growth policy. If the company's most recently paid dividend ($D_0$) was $2.18, what is the current price of the stock?
    Answer: $52.08 First, determine the constant growth rate ($g$), which equals the capital gains yield. Total Return = Dividend Yield + $g$, so $g = 12\% - 4.5\% = 7.5\%$. Next, calculate the expected dividend in one year ($D_1$): $D_1 = D_0(1+g) = \$2.18(1.075) = \$2.3435$. Finally, use the dividend yield formula (Dividend Yield = $D_1 / P_0$) to solve for the price ($P_0$): $P_0 = D_1 / \text{Dividend Yield} = \$2.3435 / 0.045 = \$52.08$.
15. A stock is valued at $50.00 using the constant growth DDM with a required rate of return ($r$) of 10% and a constant growth rate ($g$) of 6%. The government then announces a new tax policy that lowers the effective tax rate on dividends for all investors. This policy change leads to a reduction in the required rate of return on the stock from 10% to 9.5%. Assuming no change to the company's expected dividends or growth rate, what is the new estimated value of the stock?
    Answer: $57.14 This is a two-step problem. First, use the initial data to find the implied dividend for next year ($D_1$). $P_0 = D_1 / (r-g) \Rightarrow \$50.00 = D_1 / (0.10 - 0.06) \Rightarrow D_1 = \$50.00 \times 0.04 = \$2.00$. Second, use this $D_1$ with the new required rate of return ($r' = 9.5\%$) to calculate the new price ($P_0'$). $P_0' = D_1 / (r' - g) = \$2.00 / (0.095 - 0.06) = \$2.00 / 0.035 \approx \$57.14$.
16. A firm maintains a return on equity (ROE) of 15% and follows a policy of retaining 40% of its earnings. The most recent annual dividend was $2.50 per share. If the required rate of return for the stock is 11%, what is its estimated value per share?
    Answer: $53.00 This is a multi-step problem. First, calculate the dividend growth rate (g) using the sustainable growth formula: $g = ROE \times \text{retention ratio} = 0.15 \times 0.40 = 0.06$ or 6%. Second, calculate the dividend expected in one year ($D_1$): $D_1 = D_0(1+g) = \$2.50(1.06) = \$2.65$. Finally, use the constant growth DDM to find the price ($P_0$): $P_0 = D_1 / (r-g) = \$2.65 / (0.11 - 0.06) = \$2.65 / 0.05 = \$53.00$.
17. Sterling Industries is expected to have earnings per share (EPS) of $6.00 next year. The company's return on new investments (ROE) is 12%, and its cost of equity is 10%. The company maintains a constant dividend growth rate, and its stock is currently trading at a price of $120, which is assumed to be its fair value. What is the company's implied dividend payout ratio?
    Answer: 28.6% Let $p$ be the payout ratio. Then $D_1 = E_1 \times p = 6p$. The retention ratio $b = 1-p$, so the growth rate $g = ROE \times b = 0.12(1-p)$. Substitute these into the DDM formula $P_0 = D_1 / (r-g)$: $\$120 = 6p / (0.10 - 0.12(1-p))$. Solving for $p$: $\$120 = 6p / (0.10 - 0.12 + 0.12p) = 6p / (0.12p - 0.02)$. $\$120(0.12p - 0.02) = 6p \Rightarrow 14.4p - 2.4 = 6p \Rightarrow 8.4p = 2.4 \Rightarrow p = 2.4 / 8.4 \approx 0.2857$, or 28.6%.
18. A company has a return on equity (ROE) that is consistently higher than its required rate of return (r). According to the constant growth DDM, which of the following actions would most likely lead to a decrease in its stock's intrinsic value, all else equal?
    Answer: An increase in its dividend payout ratio. This question tests your understanding of the constant growth Dividend Discount Model (DDM) and how different variables affect stock valuation. The DDM formula is: $P_0 = \frac{D_1}{r - g}$, where P₀ is stock price, D₁ is next year's dividend, r is the required return, and g is the growth rate. The key insight here involves the relationship between dividend payout ratio and growth. When ROE consistently exceeds the required return, the company creates value by retaining earnings and reinvesting them. Growth rate equals: $g = ROE \times (1 - \text{payout ratio})$. So when payout ratio increases, growth rate decreases. Answer D is correct because increasing the dividend payout ratio reduces the retention ratio, which lowers the growth rate g. Since this company's ROE exceeds its required return, reducing reinvestment destroys value. The decrease in g in the denominator (r - g) increases the denominator, reducing the stock price. Answer A is wrong because a decrease in beta reduces systematic risk, lowering the required return r, which decreases the denominator and increases stock value. Answer B is incorrect because a lower risk-free rate also reduces r through the CAPM formula, again increasing stock value. Answer C is wrong because an increase in g directly decreases the denominator (r - g), increasing stock value. Remember this pattern: when a company's ROE exceeds its cost of equity, higher retention (lower payout) creates more value than paying dividends. Always consider how payout ratio affects both current dividends and future growth.
19. An analyst is valuing a company using the constant growth DDM. The risk-free rate is 3%, the market risk premium is 6%, and the company's beta is 1.5. The company just paid an annual dividend of $2.00, which is expected to grow at 4% per year. What is the stock's intrinsic value?
    Answer: $26.00 First, calculate the required rate of return ($r$) using the Capital Asset Pricing Model (CAPM): $r = R_f + \beta(E[R_m] - R_f) = 3\% + 1.5(6\%) = 3\% + 9\% = 12\%$. Next, calculate the dividend expected in one year ($D_1$): $D_1 = D_0(1+g) = \$2.00(1.04) = \$2.08$. Finally, apply the DDM formula: $P_0 = D_1 / (r-g) = \$2.08 / (0.12 - 0.04) = \$2.08 / 0.08 = \$26.00$.
20. A stock just paid an annual dividend of $2.00 per share. The dividend is expected to grow at a constant rate of 6% per year. If the stock is currently trading at $53.00 per share, what is the implied market cost of equity capital?
    Answer: 10.0% When you encounter a dividend growth model question, you're dealing with one of the fundamental ways to estimate a company's cost of equity capital. The Gordon Growth Model (also called the Dividend Discount Model) relates a stock's current price to its expected future dividends and required return. The formula is: $P_0 = \frac{D_1}{r - g}$, where $P_0$ is current price, $D_1$ is next year's expected dividend, $r$ is the required return (cost of equity), and $g$ is the growth rate. Rearranging to solve for the cost of equity: $r = \frac{D_1}{P_0} + g$ First, calculate next year's dividend: $D_1 = \$2.00 \times 1.06 = \$2.12$ Then: $r = \frac{\$2.12}{\$53.00} + 0.06 = 0.04 + 0.06 = 0.10 = 10.0\%$ This confirms answer D is correct. Answer A (9.8%) likely comes from using the current dividend ($2.00) instead of next year's dividend in the numerator. Answer B (3.8%) represents just the dividend yield component ($2.00/$53.00) without adding the growth rate. Answer C (4.0%) uses next year's dividend yield ($2.12/$53.00) but forgets to add the growth rate. Key tip: Always remember that the dividend growth model requires next period's expected dividend, and the cost of equity has two components: the dividend yield plus the growth rate. Both pieces are essential for capturing the total return investors expect.

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