TSAA just announced that it will pay a dividend next year (t=1) of $1.00. The company will then increase its dividend by 10 percent per year for two years (t=2, t=3) after which it will maintain a constant 4 percent dividend growth rate. What is the value today of a share of TSAA at a required rate of return of 15 percent?

Question

BlackTom AI · Accepted Answer

Use the dividend-discount model (multi-stage DDM): dividends D1–D3 grow at 10% annually, then a perpetual 4% growth applies from D4 onward. Discount all cash flows at r = 15%.

1) Relevant knowledge points and concepts
- Dividend Discount Model (DDM): Price = present value of all future dividends.
- Multi-stage DDM: sum PVs of dividends during non-constant growth plus the PV of the terminal value (Gordon growth) at the start of the constant-growth period.
- Terminal value at t=3: P3 = D4 / (r − g), where D4 is the dividend one period after t=3 and g is the perpetual growth rate.

2) Complete reasoning process step by step
- Compute dividends:
  D1 = $1.00 (given)
  D2 = D1 × 1.10 = 1.00 × 1.10 = $1.10
  D3 = D2 × 1.10 = 1.10 × 1.10 = $1.21
- Compute D4 using perpetual growth g = 4%:
  D4 = D3 × 1.04 = 1.21 × 1.04 = $1.2584
- Compute terminal value at t = 3:
  P3 = D4 / (r − g) = 1.2584 / (0.15 − 0.04) = 1.2584 / 0.11 = $11.44
- Discount D1, D2, D3 and P3 back to t = 0 using r = 15%:
  PV(D1) = 1.00 / 1.15 = 0.869565
  PV(D2) = 1.10 / (1.15^2) = 1.10 / 1.3225 = 0.831601
  PV(D3) = 1.21 / (1.15^3) = 1.21 / 1.520875 = 0.795902
  PV(P3) = 11.44 / 1.520875 = 7.521820
- Sum PVs:
  PV_total = 0.869565 + 0.831601 + 0.795902 + 7.521820 = 10.018888 ≈ $10.02

3) Intermediate calculation summary (shown above) confirms numerical steps.

4) Why the correct answer is correct
The computed price includes the present value of the three explicitly modeled dividends plus the discounted terminal value that captures all dividends from year 4 onward under constant 4% growth. Discounting each cash flow at 15% yields approximately $10.02 today.

5) Why the other options are incorrect
- $11.44: This equals P3 (the value at t=3 of dividends from t=4 onward), not the present value today—it's not discounted to t=0.
- $10.45: Slightly above the correct price; likely a rounding or mis-discounting error (e.g., miscomputing one discount factor).
- $8.32: Too low—likely omits the terminal value or underestimates future dividends/discounting.

6) Conclusion
The detailed calculation gives a present value of approximately $10.02, matching the provided correct answer $10.02.

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