You are evaluating Stock ABC, which has recently paid an annual dividend of $5.75 per share. The company is expected to experience significant growth, with dividends growing at a rate of 10% annually for the next 5 years. After this high-growth period, the dividend growth rate is expected to slow down to 6% indefinitely. Investors require a 10% return on this stock. Given this information, what is the estimated value of one share of Stock ABC?

Question

BlackTom AI · Accepted Answer

We start by laying out the problem details to clearly understand the cash flows and the valuation framework.
- The stock just paid a dividend D0 = 5.75. The dividends are expected to grow at 10% per year for the next 5 years, after which the growth rate slows to a perpetual 6%.
- Required return (discount rate) is r = 10%.
- We will value the stock using a two-stage dividend discount approach: compute the dividends for years 1 through 5 with the 10% growth, then compute the terminal value at the end of year 5 using a Gordon growth model with g = 6% for the perpetuity, and finally discount everything back to present value.

Step through each option:
Option A: 181.13
- First compute D1 through D5 under 10% growth:
  D1 = 5.75 × 1.10 = 6.325
  D2 = 6.325 × 1.10 = 6.9575
  D3 = 6.9575 × 1.10 = 7.65325
  D4 = 7.65325 × 1.10 = 8.418575
  D5 = 8.418575 × 1.10 = 9.2604325
- The present value of each of these dividends, at 10% required return, aligns with the geometric factor (1.10)^n. In fact, Dn / (1.10)^n equals 5.75 for n = 1,2,3,4,5 because Dn = D0 × (1.10)^n and dividing by (1.10)^n cancels the growth. Therefore the PV of each of D1 through D5 is 5.75, and the sum of PVs for D1–D5 is 5 × 5.75 = 28.75.
- Next, determine the terminal value at t = 5 using the perpetuity formula with g = 6% starting from D6 = D5 × 1.06 = 9.2604325 × 1.06 = 9.81605845. The value at t = 5 is P5 = D6 / (r − g) = 9.81605845 / (0.10 − 0.06) = 9.81605845 / 0.04 = 245.40146125.
- Discount P5 back to present: PV(P5) = 245.40146125 / (1.10)^5 = 245.40146125 / 1.61051 ≈ 152.36.
- Add the PVs of the first five dividends: 28.75 + 152.36 ≈ 181.11, which rounds to 181.13 in the options.
Option A yields a consistent total near 181.13, considering minor rounding differences in intermediate steps.

Option B: 275.17
- This would require a higher present value from either early dividends or a larger terminal value. Since D1–D5 contributions (each about 5.75 PV) sum to 28.75, and the terminal value is about 245.40 nominal at t=5 whose PV is about 152.36, the total around 181 is already close to the computed result. Achieving 275.17 would demand a significantly larger terminal value or earlier cash flows, which isn’t supported by the given growth assumptions.

Option C: 120.08
- This would indicate a much smaller present value, implying lower present value from dividends or a much smaller terminal value. Given that the five-year dividends contribute about 28.75 PV and the terminal value contributes about 152.36 PV, a total near 120 would require either inconsistent modeling or different discounting, contradicting the specified 10% required return and 6% perpetual growth after year 5.

Option D: 204.40
- This option sits above the precise calculated total. It would require a larger contribution somewhere in the cash-flow stream. Since the year-5 dividend and the perpetuity value have fixed relationships under the provided growth rates, the resulting present value is constrained around 181 with the given numbers; 204.40 would overstate the value unless a revising assumption (e.g., a higher terminal D6 or a lower discount rate) is introduced.

In summary, the calculations align with Option A by constructing a two-stage dividend model: a 10% growth phase for 5 years, followed by a perpetual 6% growth phase, discounted at 10%. The arithmetic steps show the close fit to 181.13, whereas the other choices diverge from the mathematically consistent result under the stated assumptions.

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