You buy 100 shares in Company X. The shares are entitled to an annual dividend.

You will receive your first dividend 3 years from today. Dividends are expected to grow by 2% each year thereafter. Your required return is 6% annual.

If the first dividend payment per share (at Year 3) is expected to be $10, how much are your 100 shares worth today (altogether)?

Round to the nearest cent. Enter two decimals, without punctuation e.g. enter 2500.35 for $2,500.353

Question

You buy 100 shares in Company X. The shares are entitled to an annual dividend.

You will receive your first dividend 3 years from today. Dividends are expected to grow by 2% each year thereafter. Your required return is 6% annual.

If the first dividend payment per share (at Year 3) is expected to be $10, how much are your 100 shares worth today (altogether)?

Round to the nearest cent. Enter two decimals, without punctuation e.g. enter 2500.35 for $2,500.353

BlackTom AI · Accepted Answer

The Gordon (constant-growth) dividend-discount model values a growing perpetuity as P = Dnext / (r − g). Because the first dividend of $10 occurs at Year 3, the appropriate “next” dividend for the perpetuity formula is D3 = $10, and the model gives the price one period before that payment (i.e., at t = 2). Then discount that price back two years to today.

1) Knowledge points:
- Gordon growth formula for a perpetuity: Pt−1 = Dt / (r − g) when Dt is the next dividend at time t.
- Time-value of money: discount future value to present by (1 + r)^n.

2) Step-by-step reasoning and calculations:
- Given: D3 = $10, growth g = 2% = 0.02, required return r = 6% = 0.06.
- Compute price at t = 2 (immediately before the Year 3 dividend) using the Gordon formula:
  P2 = D3 / (r − g) = 10 / (0.06 − 0.02) = 10 / 0.04 = 250.00 (per share).
- Discount P2 back two years to today (t = 0) at r = 6%:
  PV0 per share = P2 / (1 + r)^2 = 250.00 / (1.06)^2.
- (1.06)^2 = 1.1236, so PV0 per share = 250.00 / 1.1236 = 222.49911... ≈ $222.49911 per share.
- For 100 shares: total value = 222.49911 × 100 = 22,249.911... ≈ $22,249.91 (rounded to cents).

3) Intermediate arithmetic shown:
- P2 = 10 / 0.04 = 250.00
- Discount factor = (1.06)^2 = 1.1236
- PV per share = 250.00 / 1.1236 = 222.49911...
- Total for 100 shares = 222.49911... × 100 = 22,249.911... → $22,249.91

4) Why this is correct:
- The payout stream is a growing perpetuity starting at Year 3, so valuing the perpetuity at t=2 with Gordon’s formula and then discounting back exactly matches the cash-flow timing.

5) Common incorrect approaches:
- Treating the $10 as D1 and using D1/(r−g) would give 10/0.04 = 250 as if that were the present value at t=0 (wrong timing).
- Discounting the single $10 only (ignoring the perpetuity after Year 3) undervalues the asset.
- Discounting 10 two years directly (10/(1.06)^2) ignores the ongoing growth and perpetuity nature.

Conclusion: The correct present value for 100 shares is $22,249.91, which matches the provided answer.

类似问题

Amanda would receive an annual cash flow of $100,000 at the end of each year that will continue indefinitely. This cash flow is expected to grow at 5% per annum. Calculate the present value of these cash flows if the effective annual interest rate is 10%.

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