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.

COMM_V 298 101 102 103 2025W1 2025W1 COMM 298 Midterm Exam October 17, 2025 - Requires Respondus LockDown Browser

View Explanation

Log in for full answers

Similar Questions

ABC, Inc. just paid a dividend of $2. ABC expects dividends to grow at 10% annually forever. The return on shares like ABC, Inc. is typically around 12%. What is the most you would pay for ABC’s share?

ALP’s most recent dividend was $2.00 per share and is selling today for $70. The dividend is expected to grow at 7% per year for the foreseeable future. Should you purchase the share if the market return is 10% on investments with comparable risk?

Using a dividend discount model, find the intrinsic value (price at time zero) for a firm whose expected dividend is $1.92 and growth in dividends for year 2, 3, and 4 is 23.2% and constant growth afterwards is 0.54%. The cost of equity is 9.21%. (Answer in 2 decimal places, 6.21 for $6.21).

Using a dividend discount model, find the intrinsic value (price at time zero) for a firm whose expected dividend is $1.37 and growth in dividends for year 2, 3, and 4 is 13.8% and constant growth afterwards is 1.42%. The cost of equity is 6.67%.

AWSM has just paid a dividend of $1.24 and is expected to increase the future dividends at a rate of 3% per year indefinitely. If you, as a shareholder, require 12% per year, what is the current price per share?

More Practical Tools for Students Powered by AI Study Helper

Homework AI Solver

Stylized AI Paper Writer

Plagiarism Checker Assistant

Citation AI Academic Writing Tool

In-Class Translation Assistant

AI Note Generator

AI Quiz Answers

Past Exam Questions from University Test Bank

Smart Practice Assistant

Adaptive Practice

Making Your Study Simpler