A startup company's stock will pay its first dividend ever at Year 5. That is, there will be no dividends paid until Year 5. The stock's expected return is 10%, and the dividend growth rate is expected to be 5% forever after the first dividend is paid.

The stock price today is $100.

Calculate the value of the dividend at Year 5 (the date when the stock pays its first dividend).

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

Question

A startup company's stock will pay its first dividend ever at Year 5. That is, there will be no dividends paid until Year 5.  The stock's expected return is 10%, and the dividend growth rate is expected to be 5% forever after the first dividend is paid.

The stock price today is $100.

Calculate the value of the dividend at Year 5 (the date when the stock pays its first dividend).

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

BlackTom AI · Accepted Answer

The value one period before the first dividend (at t = 4) equals D5 / (r − g) by the Gordon growth model, because from Year 5 onward dividends grow at constant rate g forever. Since the stock price today (P0) is the present value of that price at t = 4, P0 = P4 / (1 + r)^4 = [D5 / (r − g)] / (1 + r)^4. Solve this for D5.

Relevant concepts:
- Gordon (constant-growth) model: P_t = D_{t+1} / (r − g) when dividends grow at constant rate g and required return is r.
- Discounting: to move a value at time 4 back to time 0, divide by (1 + r)^4.
- Algebraic rearrangement to solve for the unknown dividend.

Step-by-step reasoning and calculations:
1. Write P4 in terms of D5: P4 = D5 / (r − g).
2. Relate P0 to P4: P0 = P4 / (1 + r)^4 = [D5 / (r − g)] / (1 + r)^4.
3. Solve for D5: D5 = P0 * (r − g) * (1 + r)^4.

Insert given numbers: P0 = 100, r = 0.10, g = 0.05.
4. Compute r − g = 0.10 − 0.05 = 0.05.
5. Compute (1 + r)^4 = 1.10^4. Intermediate powers: 1.10^2 = 1.21; 1.10^3 = 1.331; 1.10^4 = 1.4641.
6. Multiply: D5 = 100 * 0.05 * 1.4641 = 5 * 1.4641 = 7.3205.

Round to nearest cent: $7.32.

Why this is correct:
- The first dividend occurs at Year 5, so the perpetual-growth price formula applies at t = 4 (one period before the first payment). Discounting that t = 4 value four years to today yields the relationship used. Common mistakes would be (a) applying the Gordon model at t = 0 (incorrect because no immediate dividend), or (b) discounting five years instead of four; both produce wrong answers.

No multiple-choice distractors here; typical incorrect formulas would omit the (1 + r)^4 term or use the wrong exponent.

Final confirmation: the calculated Year-5 dividend is $7.32, matching the provided answer.

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