A stock will pay its first dividend at Year 5 (no dividends between today and 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 (first dividend paid by the stock).

Round your final answer to the nearest cent. Enter two decimals, without punctuation e.g. enter 2500.35 for $2,500.3531

Question

A stock will pay its first dividend at Year 5 (no dividends between today and 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 (first dividend paid by the stock).

Round your final answer to the nearest cent. Enter two decimals, without punctuation e.g. enter 2500.35 for $2,500.3531

BlackTom AI · Accepted Answer

We start by restating the problem in our own terms to ensure understanding: a stock pays its first dividend at Year 5, with no dividends before that, an expected return of 10%, and a dividend growth rate of 5% forever after the first dividend. The current price is 100, and we’re asked to find the value of the dividend at Year 5 (D5).

Key idea: the present value of all future dividends equals the current price. Since the first dividend is at Year 5 and thereafter grows at 5% per year, the stream of dividends from year 5 onward is a perpetuity growing at 5% starting at D5.

We can express P0 as the present value of the dividend stream starting at t = 5:
P0 = sum_{t=5 to ∞} [ D5 * (1.05)^{t-5} ] / (1.10)^t
    = D5 * sum_{k=0 to ∞} (1.05)^k / (1.10)^{k+5}, where k = t-5
    = D5 / (1.10)^5 * sum_{k=0 to ∞} (1.05/1.10)^k.

The ratio 1.05/1.10 equals 0.954545..., so the infinite geometric series sums to 1 / (1 - 0.954545...) = 1 / 0.045454... = 22.0 exactly (since 0.045454... is 1/22).
Thus, P0 = D5 * [22 / (1.10)^5].

Now plug in P0 = 100 and solve for D5:
100 = D5 * [22 / (1.10)^5]
=> D5 = 100 * (1.10)^5 / 22.
Compute (1.10)^5: 1.1^5 = 1.61051.
Hence D5 = 100 * 1.61051 / 22 = 161.051 / 22 ≈ 7.3205.

Rounding to two decimals as requested, D5 ≈ 7.32.

To check the intuition: because the first dividend is several years away, its amount must be smaller than the current price might naïvely suggest, yet grows at a modest 5% thereafter, with a 10% discount rate; the math confirms a D5 around 7.32. If you were to recompute using a different framing (e.g., discounting a growing perpetuity starting at year 5), you should arrive at the same value for D5 given the same P0, r, and g.

Common pitfalls to avoid:
- Assuming dividends start immediately or that there is a dividend in year 1, which would change the PV calculation.
- Confusing the growth rate with the discount rate when summing the perpetuity; the ratio 1.05/1.10 must be used for the growth-adjusted stream.
- Forgetting to account for the year 5 timing in the discounting (the (1.10)^5 term is essential).

In summary, the math ties the current price to the present value of a growing perpetuity that begins at year 5, yielding D5 ≈ 7.32 when rounded to two decimals.

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