The Sisyphean Company's common stock is currently trading for $25.50 per share. The stock is expected to pay a $2.80 dividend at the end of the year and the Sisyphean Company's equity cost of capital is 10%. If the dividend payout rate is expected to remain constant, then the expected growth rate in the Sisyphean Company's earnings is closest to ________.

Question

BlackTom AI · Accepted Answer

Using the Gordon growth model (constant-growth dividend discount model), the stock price equals the next dividend divided by the difference between the required return and the growth rate: P0 = D1 / (r - g).

Relevant concepts:
- Gordon Growth Model: P0 = D1 / (r - g) for a perpetuity of dividends growing at constant rate g, where r is the equity cost of capital and D1 is the dividend next period.
- With a constant dividend payout ratio, dividend growth equals earnings growth, so g in the model is the expected earnings growth rate.

Step-by-step algebra and arithmetic:
1. Start with P0 = D1 / (r - g).
2. Rearrange to solve for g: r - g = D1 / P0  => g = r - (D1 / P0).
3. Plug in given numbers: r = 0.10, D1 = 2.80, P0 = 25.50.
4. Compute D1 / P0: 2.80 / 25.50 = 0.1098039216 (≈ 10.98039216%).
5. Compute g: g = 0.10 - 0.1098039216 = -0.0098039216 = -0.98039216%.

Interpretation: The negative growth rate (about −0.98%) means dividends and earnings are expected to decline slightly each year under the constant payout assumption.

Why the correct answer is correct:
- Substituting g = −0.00980392 back into the Gordon formula gives P0 = 2.80 / (0.10 − (−0.00980392)) = 2.80 / 0.10980392 = 25.50, which matches the given price exactly.

Why the other choices are incorrect:
- g = −1.96% → r − g = 0.1196 → implied price = 2.80 / 0.1196 ≈ 23.41 (not 25.50).
- g = −1.47% → r − g = 0.1147 → implied price ≈ 24.42 (not 25.50).
- g = −0.49% → r − g = 0.1049 → implied price ≈ 26.69 (not 25.50).

Final confirmation: The correct expected earnings growth rate is −0.98%, matching the provided answer.

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