What Is a Stock?
Follow Dolphin AI Agents to explore stock ownership, company leadership, dividends, voting, public-company reports, and the stock-price equation.
0) Overview
Forward-looking valuation asks: what is a fair price today given the cash we expect tomorrow and the return we require? In the Dividend Discount Model (DDM), the relevant cash to shareholders is the dividend stream. Under constant growth forever, we get the Gordon Growth Model.
📽️ Chapter 8 Slides (PPT)
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1) What is stock valuation?
- Goal: convert future cash flows into a present price using a required return (r).
- Price ↑ when expected cash flows ↑ or risk ↓; return ↑ when price ↓ relative to future cash flows.
- Cash definitions: different models use different cash flows:
- FCFF (firm): cash available to debt + equity; discount at WACC → enterprise value → subtract net debt for equity.
- FCFE (equity): cash available to common shareholders after net borrowing; discount at cost of equity → equity value.
- Here we focus on dividends (DDM/Gordon) for simple, stable-payout cases.
Quick formulas (reference): FCFF ≈ EBIT·(1−T) + Dep/Am − CAPEX − ΔNWC; FCFE ≈ CFO − CAPEX + (Debt Issued − Debt Repaid).
2) Dividends as cash flow
Dividends are the actual cash paid to equity holders. For firms with stable payout policies, valuing the dividend stream is intuitive and tractable.
Reminder: Buybacks also return cash. For heavy repurchasers, consider total shareholder yield or an FCFE model.
Quick links — Official dividend / IR sources
- Coca-Cola (KO) — official dividend page.
- Walmart (WMT) — established quarterly payer.
- Apple (AAPL) — modest, growing quarterly dividend.
- NVIDIA (NVDA) — very small quarterly dividend.
- Amazon (AMZN) — official FAQ: no cash dividend.
- Tesla (TSLA) — official FAQ: no cash dividend.
- Ford (F) — official stock information / dividend history page.
2.1) Dividend profiles & DDM suitability (teaching heuristics)
| Ticker | Company | Dividend status | Payout profile | Style | DDM suitability | Notes |
|---|
Heuristics for classroom use (not investment advice). “DDM suitability” is about whether a dividend-discount model is a reasonable anchor (stable, meaningful dividend stream).
2.2) Why dividends can forecast price
Bottom line: A stock’s fair price equals the present value of future cash distributions. Dividends are the most direct, observable cash flow to shareholders, so a view on the path of dividends and the required return lets us forecast price.
Valuation identity
P0 = Σt=1∞ Dt / (1+r)t
If dividends grow at constant g (r > g): P0 = D1 / (r − g)
Expected return ≈ dividend yield + growth: r ≈ D1/P0 + g
Show how to derive P0 = D1 / (r − g)
Start from the present-value add-up equation and assume dividends grow at a constant rate:
P0 = D1/(1+r) + D1(1+g)/(1+r)2 + D1(1+g)2/(1+r)3 + ...
Now multiply both sides by (1+g)/(1+r):
P0(1+g)/(1+r) = D1(1+g)/(1+r)2 + D1(1+g)2/(1+r)3 + ...
Subtract the second line from the first line. Most terms cancel out:
P0 − P0(1+g)/(1+r) = D1/(1+r)
Factor out P0 on the left:
P0[1 − (1+g)/(1+r)] = D1/(1+r)
Simplify the bracket:
P0[(1+r − 1 − g)/(1+r)] = D1/(1+r)
P0(r − g)/(1+r) = D1/(1+r)
Multiply both sides by (1+r):
P0(r − g) = D1
So:
P0 = D1 / (r − g)
Key condition: this only works when r > g. Otherwise the infinite series does not converge.
Why dividends work as a forecasting anchor
- Cash, not accounting: dividends are real cash; earnings/book can be noisy.
- Policy is sticky: boards smooth dividends, so paths are more predictable.
- Signal of capacity: raises reveal confidence in future cash generation.
- Ties straight to r: the DDM pins down price once you choose r and g.
- Anchors terminal value: Gordon logic underlies long-run DCF terminal value.
- Clean long-run series: split-adjusted dividend history helps estimate g.
Quick recipe to forecast price
- Get D₀: use a public source (Company IR → Dividend History, or Nasdaq Dividend History). Use split-adjusted values.
- Estimate g: 5-yr dividend CAGR, or sustainable g ≈ ROE × (1 − payout).
- Set r: from CAPM or your course’s required return.
- Compute D1: D1 = D0(1+g).
- Price: P0 = D1 / (r − g). Check r > g.
- Cross-check: r ≈ D1/P0 + g.
3) Gordon model — infinite horizon derivation
Assume dividends grow at a constant rate g forever and the investor lives forever.
Dt = D1(1+g)t−1, t = 1,2,3,…
P0 = ∑t=1∞ Dt/(1+r)t = D1/(r − g), r > g
This is the PV of a growing perpetuity. Not constant? Use a multi-stage model (below).
4) Key equations
Price (given r, g, D1)
P0 = D1 / (r − g)
Often we’re given D0 (the most recent dividend). Then D1 = D0(1+g).
Required return (given P0, g, D1)
r = D1 / P0 + g = dividend yield + growth
4.1) Equations — Dividend Growth (Gordon)
D0 = most recent dividend; D1 = next year’s dividend; require r > g.
Use these as a quick formula sheet. For stable dividend-paying firms, the Gordon model links price, required return, dividend yield, and growth in one simple setup.
Price (Gordon / constant g)
P0 = D1 / (r − g)
P0 = D0(1 + g) / (r − g)
Required return r (yield + growth)
r = D1 / P0 + g
r = D0(1 + g) / P0 + g
Components: dividend yield D1/P0 and capital-gain yield g.
Growth rate g
g = r − D1 / P0
Dividends D1 and D0 from price
D1 = P0(r − g)
D0 = P0(r − g) / (1 + g)
Future dividends (constant g)
Dt = D0(1 + g)t
Related calculators
Use the dividend calculator for Gordon-growth pricing and the DCF calculator when a firm does not pay a stable dividend or when you want to value cash flows directly.
5) The growth rate g — what it is & how to estimate
g is the long-run growth rate of dividends per share (not revenue). It should be plausible and sustainable (for mature firms, usually ≤ nominal GDP).
Common approaches
- Historical dividend CAGR (5–10y), adjusted for cyclicality/payout changes.
- Sustainable growth: g ≈ ROE × (1 − payout)
- Analyst long-term EPS growth as a proxy.
- Macro anchor for mature names: inflation + real growth.
Why it’s hard
- Payout policy shifts (buybacks vs. dividends).
- Leverage/ROE changes alter sustainable g.
- Industry/regulatory/structural changes.
6) Interactive DDM calculator
Set any two and solve the third. All numbers are annualized. Constraint: r must be greater than g.
Single-stage (Gordon)
Gordon model
Behavioral tweak
Two-stage
Years 1…N at g₁; N+1→∞ at g₂
Stage 1: years 1…N at g₁. Terminal is at year N using DN+1 = DN(1+g₂).
Excel hint (one-line NPV): Put dividends in a column for years 1…N. In year N, add the terminal to DN.
Then use a single NPV, e.g. if N=5 and r=8%:
=NPV(8%, D1, D2, D3, D4, ... DN+PN)
7) Worked examples
Example A — Price from r and g
Suppose D₀ = $1.80, g = 5%, r = 8%. Then D₁ = 1.80×1.05 = 1.89 and
P₀ = D₁/(r−g) = 1.89 / 0.03 = $63.00 (illustrative).
Example B — r from P₀ and g
P₀ = $50, D₀ = $2.00, g = 4% ⇒ D₁ = 2.08.
r = D₁/P₀ + g = 2.08/50 + 0.04 = 8.16%.
7.1) Non-Constant Dividend Growth (method only)
- Project early dividends (or FCF) year by year at the near-term rate g₁.
- When growth becomes constant at year N, compute the terminal price: PN = DN+1 / (r − g₂) with DN+1 = DN(1+g₂).
- Discount everything back to today and add: P0 = Σ(Dt/(1+r)t, t=1…N) + PN/(1+r)N.
Excel one-liner:
=NPV(r, D1, D2, …, D{N-1}, D{N}+P{N})
— e.g., =NPV(8%, D1, D2, D3+P3) (your D3+P3 pattern).
Need numbers? Use the Two-Stage calculator in Section 6, then do the full guided practice below.
Go to ICE Problems Open Two-Stage Calculator Open DCF Calculator
8) Common pitfalls
- Using g ≥ r (model breaks). Keep r > g.
- Single-stage on firms with non-constant growth or changing payout policy.
- Ignoring repurchases (dividends aren’t the only cash returned).
- Mixing nominal/real rates (match units).
ICE — Non-Constant Dividend Growth (NPV method)
For non-constant growth: (1) forecast early cash flows, (2) compute a terminal value at the first year of constant growth, (3) discount everything to today and add. You can also verify with the DCF tool: jufinance.com/dcf.
Nonconstant Growth Valuation Explained | 3 Examples with Timelines, Excel NPV & Calculators
Watch the three worked examples with cash-flow timelines, terminal values, discounting, Excel NPV, and the JUFinance calculators.
ICE 1 — Enterprise & Equity Value from Free Cash Flows
Given (AAA): WACC r = 15%, long-run g = 6% from year 6; FCF (millions) for years 1–5: 75, 84, 96, 111, 120. Debt = $500m; Shares = 14m.
Show steps (NPV)
Idea: forecast the first 5 yearly FCFs, place the continuing-value piece at year 5, then discount the full timeline back to today (t=0).
Cash-flow timeline
| Year | 0 | 1 | 2 | 3 | 4 | 5 | 6 → ∞ |
|---|---|---|---|---|---|---|---|
| FCF | Today | 75 | 84 | 96 | 111 | 120 | Grow at 6% |
| Terminal | — | — | — | — | — | P5 = FCF6/(r−g) | All year-6-and-beyond FCFs are collapsed into P5 |
| Cash flow used in NPV | — | 75 | 84 | 96 | 111 | 120 + P5 | Already included in P5 |
One graph — why use NPV, and why is the terminal at year 5?
Why P5? Because constant growth starts in year 6. So first compute FCF6, then value the growing perpetuity as of year 5:
P5 = FCF6 / (r − g)
Then put P5 next to FCF5 and discount that combined year-5 amount back to today with the other yearly FCFs.
- Compute the year-5 terminal value (constant growth begins in year 6):
FCF6 = 120×(1+0.06) = 127.20
P5 = FCF6 / (r − g) = 127.20 / (0.15 − 0.06) = 1,413.33 - Put the terminal at year 5 and add it to the year-5 free cash flow:
Year-5 cash flow used in NPV = FCF5 + P5 = 120 + 1,413.33 = 1,533.33
- Discount the whole timeline back to today:
EV = ∑t=1..5 CFt/(1+r)t = NPV(15%; 75, 84, 96, 111, 1,533.33) = 1,017.66
- Move from enterprise value to equity value and price:
Equity = EV − Debt = 1,017.66 − 500 = 517.66
P0 = Equity / Shares = 517.66 / 14 = $36.98
Excel layout (illustrative):
| t | FCF | Terminal | Cash flow |
|---|---|---|---|
| 1 | 75 | — | =B2 |
| 2 | 84 | — | =B3 |
| 3 | 96 | — | =B4 |
| 4 | 111 | — | =B5 |
| 5 | 120 | =120*(1+0.06)/(0.15-0.06) | =B6 + C6 |
EV: =NPV(0.15, D2:D6)
Equity: =EV - 500
Price: =Equity / 14
Equity: =EV - 500
Price: =Equity / 14
Check: in steady state, Firm value ≈ FCF1/(WACC−g) = FCF0·(1+g)/(WACC−g).
ICE 2 — Zero Dividends Until Year 2
Given (AAA): D0=0, D2=0.56, g=4% thereafter, r=12%. Find P0.
Show steps (NPV)
Cash-flow timeline
| Year | 0 | 1 | 2 | 3 → ∞ |
|---|---|---|---|---|
| Dividend stream | Today | D1=0 | D2=0.56 | Grow at 4% |
| Terminal | — | — | P2 = D3/(r−g) | All dividends from year 3 onward are collapsed into P2 |
| Cash flow used in NPV | — | 0 | 0.56 + P2 | Already included in P2 |
Why the terminal is at P2 and why we use NPV
t=0
→
t=1
→
t=2
P0
discount back here
discount back here
D1=0
first future cash flow
first future cash flow
D2 + P2
last explicit cash flow plus terminal value
last explicit cash flow plus terminal value
Build P2 at year 2 because the constant-growth stream begins after D2.
Then discount the future cash flows 0 and (0.56 + P2) back to today with NPV.
Then discount the future cash flows 0 and (0.56 + P2) back to today with NPV.
- Find the first dividend in the constant-growth period:
D3 = 0.56×(1+0.04) = 0.5824
- Get the terminal price at year 2:
P2 = D3/(r−g) = 0.5824/(0.12−0.04) = 0.5824/0.08 = 7.28
- Now discount all future cash flows back to today:
P0 = NPV(12%; D1, D2 + P2)
= NPV(12%; 0, 0.56 + 7.28) = NPV(12%; 0, 7.84) = 6.25
Excel layout:
| t | Dividend | Terminal | Cash flow |
|---|---|---|---|
| 1 | 0 | — | =B2 |
| 2 | 0.56 | =0.56*(1+0.04)/(0.12-0.04) | =B3 + C3 |
P0: =NPV(0.12, D2:D3)
ICE 3 — High Growth for 4 Years, Then Stable Forever
Given: r = 12%, D0 = 1.00; dividends grow 30% for t=1..4, then g = 6.34% thereafter. Find P0 (≈ $40).
Show steps (NPV)
Cash-flow timeline
| Year | 0 | 1 | 2 | 3 | 4 | 5 → ∞ |
|---|---|---|---|---|---|---|
| Dividend stream | Today | 1.30 | 1.69 | 2.197 | 2.8561 | Grow at 6.34% |
| Terminal | — | — | — | — | P4 = D5/(r−g) | All dividends from year 5 onward are collapsed into P4 |
| Cash flow used in NPV | — | 1.30 | 1.69 | 2.197 | 2.8561 + P4 | Already included in P4 |
Why the terminal is at P4 and why we use NPV
t=0→
t=1→
t=2→
t=3→
t=4
P0
discount back here
discount back here
D1
1.30
1.30
D2
1.69
1.69
D3
2.197
2.197
D4 + P4
last explicit dividend plus terminal value
last explicit dividend plus terminal value
Build P4 at year 4 because stable growth begins at t=5.
Then discount D1, D2, D3, and (D4+P4) back to today with NPV.
Then discount D1, D2, D3, and (D4+P4) back to today with NPV.
- Near-term dividends during high growth:
D1=1.00×1.30 = 1.30
D2=1.30×1.30 = 1.69
D3=1.69×1.30 = 2.197
D4=2.197×1.30 = 2.8561 - Get the first stable-growth dividend and the year-4 terminal value:
D5 = 2.8561×(1+0.0634) = 3.0372
P4 = D5/(r−g) = 3.0372/(0.12−0.0634) = 3.0372/0.0566 = 53.6604 - Discount everything back to today:
P0 = NPV(12%; D1, D2, D3, D4 + P4)
= NPV(12%; 1.30, 1.69, 2.197, 2.8561 + 53.6604) ≈ 39.99 → about $40.00
Excel layout:
| t | Dividend | Terminal | Cash flow |
|---|---|---|---|
| 1 | =1*(1+0.30) | — | =B2 |
| 2 | =B2*(1+0.30) | — | =B3 |
| 3 | =B3*(1+0.30) | — | =B4 |
| 4 | =B4*(1+0.30) | =(B5*(1+0.0634))/(0.12-0.0634) | =B5 + C5 |
P0: =NPV(0.12, D2:D5)
Tip: In Excel,
One-cell pattern (explicit):
NPV(rate, range) discounts a series of future end-of-period cash flows. Add any t=0 cash separately if present.One-cell pattern (explicit):
=NPV(r, D1, D2, …, D{N-1}, D{N}+P{N}) ← put the terminal into the last period’s cash flow (your “D3+P3” fix).
HOMEWORK (Due with Final) — Dividend Growth Model
Try each problem first. Open the hint only after you set up the equation yourself.
- Northern Gas: D0=2.80, g=3.8%, P0=26.91. Find r. (Northern Gas recently paid a $2.80 annual dividend on its common stock. This dividend increases at an average rate of 3.8 percent per year. The stock is currently selling for $26.91 a share. What is the market rate of return? (14.60 percent))
Show hint / formula / Excel
D1=2.80×(1+0.038)
r = D1/P0 + gExcel hint:=(2.80*(1+0.038))/26.91 + 0.038 - Douglass Gardens: g=4.1%, r=12.6%, P0=24.90. Find D1 (Douglass Gardens pays an annual dividend that is expected to increase by 4.1 percent per year. The stock commands a market rate of return of 12.6 percent and sells for $24.90 a share. What is the expected amount of the next dividend? ($2.12)).
Show hint / formula / Excel
D1 = P0(r − g)Excel hint:=24.90*(0.126 - 0.041) - IBM: D0=3.00, g=10%. Find D1 (IBM just paid $3.00 dividend per share to investors. The dividend growth rate is 10%. What is the expected dividend of the next year? ($3.3)).
Show hint / formula / Excel
D1 = D0(1+g)Excel hint:=3.00*(1+0.10) - Given: P0=50, D1=2, g=6%. Find r. (The current market price of stock is $50 and the stock is expected to pay dividend of $2 with a growth rate of 6%. How much is the expected return to stockholders? (10%))
Show hint / formula / Excel
r = D1/P0 + gExcel hint:=2/50 + 0.06 - Creamy Custard: r=15%, D0=6.00, g=6% forever. Find P0 (Investors of Creamy Custard common stock earns 15% of return. It just paid a dividend of $6.00 and dividends are expected to grow at a rate of 6% indefinitely. What is expected price of Creamy Custard's stock? ($70.67)).
Show hint / formula / Excel
P0 = D1/(r−g) = D0(1+g)/(r−g)Excel hint:=6*(1+0.06)/(0.15-0.06)
9) Instructor notes
Walmart (WMT) stock split — what it means
- What happened: Walmart executed a 3-for-1 stock split (our dividend table flags 2024 as a split year).
- Mechanics: Shares outstanding ↑×3; price per share ↓÷3; total market value unchanged by the split itself.
- Dividends & EPS: Per-share dividend and EPS are adjusted ÷3 so totals stay the same.
- Why splits: Liquidity, “friendlier” price range for employees/retail, optics, and some index/plan constraints.
- Price outcome (general): The quote adjusts mechanically on the ex-split date (e.g., $180 → ~$60 in a 3-for-1). No free value created.
- Valuation reality: DDM/DCF don’t change if you use split-adjusted inputs. This page is split-aware and suppresses YoY/CAGR around the split.
- Missed class? Recording on Blackboard → Course Content → Recordings.
- UI tip: Use the Theme picker (Light/Dark/Warm). The left TOC scrolls if long.
Quiz 3 — Help Videos (4 parts)
10) Optional — FYI Only: Python Coding, Mutual Funds & Retirement
Optional — FYI only. These resources are not required coursework. FIN509 MBA students may explore these FIN310 activities to connect valuation and risk to personal investment decisions and try a practical coding application.
10.1) Session 10 — Investing toward retirement
Explore investing from age 22 to 62: compare stocks, bonds, diversified mixes, gold, and crypto. The classroom example starts with $1,000 per month and increases deposits 5% annually. A parametric Monte Carlo simulation uses 3,000 paths per choice to compare lower, median, and higher outcomes, purchasing power, fees, and Roth versus pre-tax treatment.
MBA application: Compare two portfolios, change an inflation assumption, and defend your choice using risk and purchasing-power evidence. Explain which assumptions could change the recommendation.
These are illustrative projections, not promised retirement balances; the model does not enforce account limits or capture every risk.
10.2) Session 11 — Choose a mutual fund
Use Yahoo Finance to research funds for different goals: long-term growth, retirement income, or a nearer-term purchase. Compare the fund’s category, holdings, share class, expenses, restrictions, and risks. The same person may need different investments for different deadlines.
MBA application: Recommend a fund for a stated goal, compare one alternative, and explain the fit and a material risk.
10.3) PYTHON CODING — Beginner Google Colab Lab
Optional — FYI only. Try running Python code to analyze a mutual fund.
- Download the notebook: open the Python coding lab below and choose “Download Google Colab notebook.”
- Open Google Colab: select File → Upload notebook, then upload the downloaded notebook.
- Run the code: enter an eligible fund ticker, select Runtime → Run all, and review the results and charts.
Download the notebook, enter an eligible fund ticker, and run the code in Colab. The lab uses historical adjusted prices and six-month block resampling to explore 1-, 3-, and 5-year outcomes. Interpret the charts, model ranges, and limitations; past returns do not establish future performance.