If the expected ROE on reinvested earnings is equal to k, the multistage DDM reduces to

Question

BlackTom AI · Accepted Answer

If reinvested earnings earn exactly the required return k, the present value of retaining earnings is zero and the firm's value equals the capitalized expected dividend payout: P0 = (Expected dividend payout in year 1)/k.

1. Relevant knowledge points and concepts
- Dividend discount model (DDM) / Gordon growth model: P0 = D1 / (k − g) when dividends grow at constant rate g.
- Growth from retained earnings: g = b × ROE, where b is the retention ratio (fraction of earnings retained).
- Economic meaning: If reinvested earnings earn ROE equal to the discount rate k, reinvestments create no net present value (NPV = 0) for shareholders.

2. Complete reasoning process step by step
- Start with the DDM in its general form for a long-run stage: P0 = D1 / (k − g).
- Growth g arises from retained earnings: g = b × ROE.
- If ROE = k, then retained earnings generate returns at the same rate investors require. Those reinvestments neither add nor subtract net present value.
- Economically, reinvestment in projects that earn exactly k simply exchanges one asset paying k for another paying k; there is no incremental value beyond dividends actually distributed. Thus only the expected dividend payout in year 1 matters for value capitalization.

3. Intermediate algebra / calculation (interpretation)
- Although algebraically g = b × ROE = b × k, one can interpret the discounted value of the growth component as zero because retained earnings earn k and are discounted at k.
- Consequently the capitalization simplifies from P0 = D1/(k − g) to P0 = D1/k when the net NPV of retention is zero.

4. Why the correct answer is correct
- The expected dividend payout in year 1 is the cash actually returned to shareholders. If retained earnings produce no value beyond earning k, the firm’s equity value is simply the perpetuity value of that expected payout at discount rate k: P0 = D1/k.

5. Why the other options are incorrect
- P0 = (Expected dividend yield in year 1)/k: dividend yield is a rate, not an actual cash payout; dividing a rate by k gives wrong units.
- P0 = (Expected EPS in year 1)/k: this would be true only if all earnings are paid out (payout ratio = 100%); in general EPS includes retained earnings that may not be distributed.
- P0 = (Treasury bond yield in year 1)/k and P0 = (Market return in year 1)/k: these are unrelated macro yields/returns and do not represent the firm’s dividend cashflows to be capitalized.

6. Conclusion confirming the final answer
- Therefore the multistage DDM reduces to P0 = (Expected dividend payout in year 1)/k, which matches the provided correct answer.

If the expected ROE on reinvested earnings is equal to k, the multistage DDM reduces to单项选择题

类似问题

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The constant-growth dividend discount model (DDM) can be used only when the growth rate is __________ blank .

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