Finance Quiz: Multi Stage Dividend Models
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Multi Stage Dividend ModelsQuestion 1 of 20

An analyst is using a two-stage dividend discount model to value a stock. The first stage of high growth lasts for 3 years. The analyst correctly calculates the terminal value at the end of year 3 (V₃) to be $50.00. The required rate of return is 10%. When calculating the total intrinsic value of the stock today (V₀), which step must the analyst perform with this terminal value V₃?

Discount V₃ back three periods at the 10% required rate of return and add it to the present value of the first three dividends.
Discount V₃ back four periods at the 10% required rate of return, as it is based on the dividend in year 4.
Add the undiscounted V₃ to the present value of the first three dividends, as V₃ already represents a future value.
Calculate the terminal value using the dividend from year 3 (D₃) instead of year 4 (D₄) to maintain consistency in the timeline.
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Finance Quiz

Finance Quiz: Multi Stage Dividend Models

Practice Multi Stage Dividend Models in Finance with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

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This quiz focuses on Multi Stage Dividend Models, giving you a quick way to practice the rules, question types, and explanations that matter most for Finance.

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Question 1

An analyst is using a two-stage dividend discount model to value a stock. The first stage of high growth lasts for 3 years. The analyst correctly calculates the terminal value at the end of year 3 (V₃) to be $50.00. The required rate of return is 10%. When calculating the total intrinsic value of the stock today (V₀), which step must the analyst perform with this terminal value V₃?

  1. Discount V₃ back three periods at the 10% required rate of return and add it to the present value of the first three dividends. (correct answer)
  2. Discount V₃ back four periods at the 10% required rate of return, as it is based on the dividend in year 4.
  3. Add the undiscounted V₃ to the present value of the first three dividends, as V₃ already represents a future value.
  4. Calculate the terminal value using the dividend from year 3 (D₃) instead of year 4 (D₄) to maintain consistency in the timeline.
Explanation: The terminal value V₃ represents the value of all dividends from year 4 onwards, calculated as of the end of year 3. To find its value today (at time 0), it must be discounted back three periods. The formula for the stock's value is V₀ = PV(dividends in stage 1) + PV(terminal value). The present value of V₃ is V₃ / (1+r)³.

Question 2

An analyst is valuing a stock using a two-stage DDM. The current dividend (D₀) is $1.50. Dividends are expected to grow at 20% for the next two years. Afterwards, the growth rate will be a constant 5%. The required return is 13%. What is the present value of the dividends paid during the initial two-year high-growth stage?

  1. $3.28 (correct answer)
  2. $3.96
  3. $22.20
  4. $25.48
Explanation: The question asks only for the present value of the dividends in the first stage.
  1. Calculate D₁ = $1.50 * 1.20 = $1.80.
  2. Calculate D₂ = $1.80 * 1.20 = $2.16.
  3. Calculate the present value of each dividend: PV(D₁) = $1.80 / (1.13)¹ = $1.5929. PV(D₂) = $2.16 / (1.13)² = $1.6917.
  4. Sum the present values: $1.5929 + $1.6917 = $3.2846, which rounds to $3.28.

Question 3

A stock is trading at $50. An analyst using a two-stage DDM estimates the present value of the first stage (3 years) of dividends to be $5.00. The terminal value is calculated at the end of year 3 (V₃) and the required return (r) is 11%. What is the implied terminal value at the end of year 3 (V₃)?

  1. $61.54 (correct answer)
  2. $45.00
  3. $59.85
  4. $32.90
Explanation: The total stock value is the sum of the PV of stage 1 dividends and the PV of the terminal value.
  1. PV of terminal value = Total Value - PV of stage 1 dividends = $50.00 - $5.00 = $45.00.
  2. This $45.00 is the value of V₃ discounted back 3 years. So, PV(V₃) = V₃ / (1+r)³.
  3. Rearrange to solve for V₃: V₃ = PV(V₃) * (1+r)³ = $45.00 * (1.11)³ = $45.00 * 1.367631 = $61.54.

Question 4

An analyst uses a two-stage DDM and forecasts D₁ = $1.00, an initial growth rate of 20% for 3 years, a terminal growth rate of 5%, and a required return of 10%. After a review, the analyst revises the terminal growth rate to 4%. Which of the following describes the effect of this revision on the valuation?

  1. The value will decrease primarily because the denominator of the terminal value calculation, (r - g), will increase.
  2. The value will increase because a lower terminal growth rate implies lower risk and a lower discount rate.
  3. There will be no change to the present value of the dividends in the first three years.
  4. The terminal value will decrease due to both a lower numerator (D₄ = D₃ × 1.04 vs. D₃ × 1.05) and a larger denominator (r - g). (correct answer)
Explanation: The terminal value is calculated as D₄ / (r - g). Reducing the terminal growth rate from 5% to 4% has two effects that both decrease the terminal value: 1) The numerator decreases because D₄ = D₃(1+g) will be smaller with the lower growth rate, and 2) The denominator (r - g) increases from 0.05 to 0.06. Both effects reduce the terminal value and therefore the total intrinsic value. Choice D correctly identifies both impacts on the Gordon Growth formula.

Question 5

An analyst is valuing two companies, Firm A and Firm B, using a two-stage DDM. Both firms have identical required rates of return, initial high growth rates, and terminal stable growth rates. However, Firm A's high-growth stage is expected to last for 3 years, while Firm B's is expected to last for 5 years. All else equal, the valuation model will produce:

  1. a higher intrinsic value for Firm A because its terminal value is received sooner and discounted less.
  2. a higher intrinsic value for Firm B because it sustains a higher dividend growth rate for a longer period. (correct answer)
  3. the same intrinsic value for both firms, as the total number of future dividends is infinite in both cases.
  4. an ambiguous result, as the higher PV of Firm A's terminal value may be offset by Firm B's higher initial dividends.
Explanation: Holding all other factors constant, a longer period of high growth will result in a higher intrinsic value. Firm B will have two additional years of dividends growing at the high rate compared to Firm A. This directly increases the sum of the present values of the dividends during the explicit forecast period. Furthermore, Firm B's terminal value will be calculated off a higher dividend base (D₅ vs. D₃) at a later date, which also contributes to a higher overall valuation, even with additional discounting.

Question 6

A firm has a current dividend D₀ of $2.00, which is expected to grow at 15% for two years, and then at a constant 5% thereafter. The firm's required return is 12%. An analyst calculates the stock's value by discounting the first two dividends and the terminal value. Which of the following represents the correct calculation for the present value of the terminal value?

  1. D2(1+gL)(r−gL)(1+r)2\frac{D_2(1+g_L)}{(r-g_L)(1+r)^2} (correct answer)
  2. D2(1+gS)(r−gL)(1+r)2\frac{D_2(1+g_S)}{(r-g_L)(1+r)^2}
  3. D0(1+gL)(r−gL)(1+r)2\frac{D_0(1+g_L)}{(r-g_L)(1+r)^2}
  4. D2(1+gL)(r−gS)(1+r)2\frac{D_2(1+g_L)}{(r-g_S)(1+r)^2}
Explanation: The high-growth period lasts for two years (n=2). The terminal value (V₂) is calculated at the end of year 2 using the dividend for year 3 (D₃). D₃ can be expressed as D₂(1+gₗ), where gₗ is the long-term growth rate. The terminal value is then V₂ = D₃ / (r-gₗ). To find its present value, V₂ must be discounted back two years. Therefore, PV(V₂) = V₂ / (1+r)² = [D₂(1+gₗ) / (r-gₗ)] / (1+r)², which matches choice A.

Question 7

An analyst values a company using a two-stage DDM with a 5-year high-growth stage. The analyst finds that the present value of the terminal value accounts for over 80% of the total intrinsic value. This result is most likely to be caused by:

  1. a long initial high-growth period (e.g., 15 years) and a low terminal growth rate.
  2. a very high required rate of return relative to the dividend growth rates throughout the forecast period.
  3. a combination of a low dividend payout ratio in the initial stage and a low required rate of return.
  4. a combination of a high terminal growth rate (close to the required rate of return) and a relatively short initial high-growth stage. (correct answer)
Explanation: The terminal value's dominance is magnified by two main factors. First, a short initial stage means fewer explicit dividends are forecasted, placing more of the company's value in the terminal period. Second, a high terminal growth rate (g) that is close to the required rate of return (r) makes the denominator (r-g) very small, which greatly inflates the terminal value calculation. The other choices would tend to decrease the terminal value's relative importance.

Question 8

An analyst makes a common mistake when applying a two-stage DDM for a company with a 4-year supernormal growth period. The analyst calculates the terminal value at the end of year 4 (V₄) and then discounts it back four periods. However, in the numerator of the terminal value calculation, the analyst uses the dividend from year 4 (D₄) instead of the dividend from year 5 (D₅). How will this error affect the calculated intrinsic value?

  1. The intrinsic value will be overstated because an incorrect dividend is used in the terminal value calculation.
  2. The intrinsic value will be understated because the numerator of the terminal value calculation is too small. (correct answer)
  3. The intrinsic value will be unaffected because the error in the numerator is offset by the discounting period.
  4. The effect cannot be determined without knowing if the growth rate is positive or negative.
Explanation: The terminal value at time n, Vₙ, should be calculated as Dₙ₊₁ / (r - g). The analyst incorrectly used D₄ instead of D₅ in the numerator to calculate V₄. Since the company is growing, D₅ will be greater than D₄ (as D₅ = D₄ * (1+g)). By using the smaller D₄ in the numerator, the analyst calculates a terminal value that is too low, which in turn leads to an understated total intrinsic value for the stock.

Question 9

According to the sustainable growth rate framework (g = ROE × b), the decline in the dividend growth rate from the high-growth phase to the terminal phase in a multi-stage DDM is most likely driven by a(n):

  1. increasing retention ratio (b) as the firm finds more profitable projects to invest in.
  2. decreasing return on equity (ROE) from increased competition and a decreasing retention ratio (b) from fewer growth opportunities. (correct answer)
  3. increasing return on equity (ROE) due to stronger market position and an increasing retention ratio (b) to fund expansion.
  4. decreasing financial leverage that directly reduces the dividend growth rate without affecting ROE or the retention ratio.
Explanation: As a company matures, it typically faces more competition, which erodes its profitability and leads to a lower Return on Equity (ROE). Simultaneously, as its industry matures, the number of high-return investment opportunities diminishes, leading the company to retain less of its earnings (a lower retention ratio, b) and pay out more as dividends. Both a lower ROE and a lower b contribute to a lower sustainable growth rate (g).

Question 10

In a two-stage dividend discount model, an analyst decides to increase the assumed length of the initial high-growth period. Assuming the high growth rate is greater than the terminal growth rate and all other inputs remain constant, what will be the effect on the calculated intrinsic value of the stock?

  1. The intrinsic value will decrease because the higher terminal value is discounted over more years.
  2. The intrinsic value will increase because more dividends will grow at the higher rate for a longer duration. (correct answer)
  3. The effect cannot be determined without knowing the specific growth rates and the required rate of return.
  4. The intrinsic value will remain unchanged because the terminal value is pushed further into the future.
Explanation: Extending the high-growth period means that more dividends will be grown at the higher initial rate before the growth rate drops to the lower terminal rate. Each of these higher dividends replaces a lower dividend that would have been projected in the original model. This increases the sum of the present values of all future dividends, thus increasing the stock's calculated intrinsic value.

Question 11

An analyst uses the H-model to value a stock. The stock's current dividend (D₀) is $1.00. The initial dividend growth rate (gₛ) is 20%, which is expected to decline linearly over a 10-year period (2H) to a final constant growth rate (gₗ) of 4%. The required rate of return (r) is 10%. What is the approximate value of the stock using the H-model?

  1. $30.67 (correct answer)
  2. $17.33
  3. $33.33
  4. $44.00
Explanation: The H-model formula is V₀ = [D₀(1+gₗ)]/(r-gₗ) + [D₀H(gₛ-gₗ)]/(r-gₗ). Given: D₀=$1.00, gₛ=0.20, gₗ=0.04, r=0.10, 2H=10 so H=5. Term 1 (present value of perpetual growth): [$1.00(1.04)]/(0.10-0.04) = $1.04/0.06 = $17.33. Term 2 (present value of supernormal growth): [1.00∗5∗(0.20−0.04)]/(0.10−0.04)=(1.00 * 5 * (0.20-0.04)]/(0.10-0.04) = (1.00 * 5 * 0.16)/0.06 = $0.80/0.06 = $13.33. V₀ = $17.33 + $13.33 = $30.66, which rounds to $30.67.

Question 12

An analyst is forecasting dividends for a multi-stage DDM. The firm's ROE is expected to be 20% for the next 3 years before declining to 12% in the stable phase. The firm plans to maintain a retention ratio of 60% during the high-growth phase and reduce it to 30% in the stable phase. The required return is 11%. What is the firm's implied terminal dividend growth rate (g_L)?

  1. 12.0%
  2. 6.0%
  3. 3.6% (correct answer)
  4. 9.0%
Explanation: The sustainable growth rate is calculated as g = Return on Equity (ROE) × retention ratio (b). The question asks for the terminal growth rate, so we must use the parameters from the stable phase. Terminal ROE = 12% or 0.12. Terminal retention ratio (b) = 30% or 0.30. Terminal growth rate (g_L) = 0.12 * 0.30 = 0.036 or 3.6%. The information from the high-growth phase and the required return are distractors.

Question 13

An analyst is building a three-stage dividend discount model with a high-growth stage, a transitional stage, and a terminal stable-growth stage. Which of the following relationships between the model's parameters would render the valuation mathematically invalid?

  1. The terminal dividend growth rate is greater than the long-term forecast for nominal GDP growth.
  2. The high-growth rate in the first stage is greater than the required rate of return.
  3. The terminal dividend growth rate is greater than or equal to the required rate of return. (correct answer)
  4. The dividend payout ratio in the terminal stage is lower than the payout ratio in the high-growth stage.
Explanation: The terminal value in any multi-stage DDM is calculated using the constant-growth (Gordon) model formula: Vₙ = Dₙ₊₁ / (r - g). For this formula to be valid, the required rate of return (r) must be greater than the perpetual growth rate (g). If g ≥ r, the denominator becomes zero or negative, resulting in an infinite or meaningless valuation. This is a strict mathematical constraint.

Question 14

An analyst is valuing two companies, Firm A and Firm B, using a two-stage DDM. Both firms have identical required rates of return, initial high growth rates, and terminal stable growth rates. However, Firm A's high-growth stage is expected to last for 3 years, while Firm B's is expected to last for 5 years. All else equal, the valuation model will produce:

  1. a higher intrinsic value for Firm A because its terminal value is received sooner and discounted less.
  2. a higher intrinsic value for Firm B because it sustains a higher dividend growth rate for a longer period. (correct answer)
  3. the same intrinsic value for both firms, as the total number of future dividends is infinite in both cases.
  4. an ambiguous result, as the higher PV of Firm A's terminal value may be offset by Firm B's higher initial dividends.
Explanation: Holding all other factors constant, a longer period of high growth will result in a higher intrinsic value. Firm B will have two additional years of dividends growing at the high rate compared to Firm A. This directly increases the sum of the present values of the dividends during the explicit forecast period. Furthermore, Firm B's terminal value will be calculated off a higher dividend base (D₅ vs. D₃) at a later date, which also contributes to a higher overall valuation, even with additional discounting.

Question 15

A company is expected to pay a dividend of $2.00 per share one year from today (D₁). The dividend is expected to grow at 15% per year for two more years (for years 2 and 3). After year 3, the dividend growth rate will decline to a constant 5% per year indefinitely. The required rate of return is 12%. What is the terminal value of the stock at the end of year 3 (V₃)?

  1. $39.68 (correct answer)
  2. $37.79
  3. $28.24
  4. $34.50
Explanation: The terminal value at time n (Vn) is calculated using the Gordon Growth Model with the dividend from the next period (Dn+1). Here, we need V₃.
  1. Calculate D₃: D₁ = $2.00; D₂ = $2.00(1.15) = $2.30; D₃ = $2.30(1.15) = $2.645.
  2. Calculate D₄: D₄ = D₃(1 + g_L) = $2.645(1.05) = $2.77725.
  3. Calculate V₃: V₃ = D₄ / (r - g_L) = $2.77725 / (0.12 - 0.05) = $2.77725 / 0.07 = $39.675, which rounds to $39.68.

Question 16

An analyst is valuing a young technology company that recently initiated a dividend. The company's industry is highly competitive but is expected to consolidate and mature over the next decade. Which of the following is the most compelling reason for the analyst to use a multi-stage dividend discount model (DDM) instead of the constant-growth DDM?

  1. The company's dividend payout ratio is currently very low but is expected to increase as the firm matures.
  2. The company's dividend growth rate is currently high but is expected to decline over time to a sustainable, long-term rate. (correct answer)
  3. The required rate of return on the company's stock is significantly higher than the risk-free rate, reflecting its high risk.
  4. The constant-growth model cannot be used if the initial growth rate temporarily exceeds the required rate of return.
Explanation: The fundamental reason for using a multi-stage DDM is that firms often go through life cycles where their growth rates change. A multi-stage model is designed for companies whose growth is expected to change over time, typically starting with a high-growth period and eventually settling into a stable, mature growth rate. The constant-growth model's assumption of a single, perpetual growth rate is unrealistic for such firms.

Question 17

An analyst is valuing a young technology company that recently initiated a dividend. The company's industry is highly competitive but is expected to consolidate and mature over the next decade. Which of the following is the most compelling reason for the analyst to use a multi-stage dividend discount model (DDM) instead of the constant-growth DDM?

  1. The company's dividend payout ratio is currently very low but is expected to increase as the firm matures.
  2. The company's dividend growth rate is currently high but is expected to decline over time to a sustainable, long-term rate. (correct answer)
  3. The required rate of return on the company's stock is significantly higher than the risk-free rate, reflecting its high risk.
  4. The constant-growth model cannot be used if the initial growth rate temporarily exceeds the required rate of return.
Explanation: The fundamental reason for using a multi-stage DDM is that firms often go through life cycles where their growth rates change. A multi-stage model is designed for companies whose growth is expected to change over time, typically starting with a high-growth period and eventually settling into a stable, mature growth rate. The constant-growth model's assumption of a single, perpetual growth rate is unrealistic for such firms.

Question 18

A company is expected to pay a dividend of $2.00 per share one year from today (D₁). The dividend is expected to grow at 15% per year for two more years (for years 2 and 3). After year 3, the dividend growth rate will decline to a constant 5% per year indefinitely. The required rate of return is 12%. What is the terminal value of the stock at the end of year 3 (V₃)?

  1. $39.68 (correct answer)
  2. $37.79
  3. $28.24
  4. $34.50
Explanation: The terminal value at time n (Vn) is calculated using the Gordon Growth Model with the dividend from the next period (Dn+1). Here, we need V₃.
  1. Calculate D₃: D₁ = $2.00; D₂ = $2.00(1.15) = $2.30; D₃ = $2.30(1.15) = $2.645.
  2. Calculate D₄: D₄ = D₃(1 + g_L) = $2.645(1.05) = $2.77725.
  3. Calculate V₃: V₃ = D₄ / (r - g_L) = $2.77725 / (0.12 - 0.05) = $2.77725 / 0.07 = $39.675, which rounds to $39.68.

Question 19

In a two-stage dividend discount model, an analyst decides to increase the assumed length of the initial high-growth period. Assuming the high growth rate is greater than the terminal growth rate and all other inputs remain constant, what will be the effect on the calculated intrinsic value of the stock?

  1. The intrinsic value will decrease because the higher terminal value is discounted over more years.
  2. The intrinsic value will increase because more dividends will grow at the higher rate for a longer duration. (correct answer)
  3. The effect cannot be determined without knowing the specific growth rates and the required rate of return.
  4. The intrinsic value will remain unchanged because the terminal value is pushed further into the future.
Explanation: Extending the high-growth period means that more dividends will be grown at the higher initial rate before the growth rate drops to the lower terminal rate. Each of these higher dividends replaces a lower dividend that would have been projected in the original model. This increases the sum of the present values of all future dividends, thus increasing the stock's calculated intrinsic value.

Question 20

An analyst is choosing between a two-stage and a three-stage dividend discount model. The most appropriate reason to select a three-stage model is if the company is expected to experience:

  1. an abrupt drop in its growth rate from a high level to a sustainable level after a few years.
  2. a negative growth rate for the next several years before resuming stable positive growth.
  3. a period of supernormal growth, followed by a transitional period of declining growth, before reaching a stable mature phase. (correct answer)
  4. a constant growth rate in perpetuity that is slightly above the long-term economic growth rate.
Explanation: A three-stage model is specifically designed for the scenario where a company's transition from high growth to stable growth is not abrupt. It introduces a middle stage where the growth rate declines over time (e.g., linearly) to bridge the gap between the initial high-growth rate and the final stable growth rate, which is often considered more realistic.