Learn dividend discount model (DDM)

Derivation: From First Principles

Start with the fundamental question: What am I buying when I buy a stock?

Year 1: I receive dividend $D_1$ in one year. What's that worth today?
- Worth $\frac{D_1}{1 + r}$ because I could invest $\frac{D_1}{1 + r}$ today at rate $r$ and have $D_1$ next year.
Year 2: I receive $D_2$ in two years. Present value?
- $\frac{D_2}{(1 + r)^2}$ — discounted twice because it's two years away.
Pattern: Each future dividend $D_t$ has present value $\frac{D_t}{(1 + r)^t}$ .
Sum them all: $P_0 = \frac{D_1}{(1 + r)} + \frac{D_2}{(1 + r)^2} + \frac{D_3}{(1 + r)^3} + \cdots = \sum_{t=1}^{\infty} \frac{D_t}{(1 + r)^t}$

Why the discount rate $r$ ? It's your required rate of return—the minimum you demand to compensate for:

Time value of money: You could invest elsewhere
Risk: Dividends aren't guaranteed
Opportunity cost: What you give up by buying this stock

Figure — Learn dividend discount model (DDM)

The Gordon Growth Model (Constant Growth DDM)

Derivation from infinite sum:

Start with $P_0 = \sum_{t=1}^{\infty} \frac{D_t}{(1 + r)^t}$ .

If $D_1$ is next year's dividend and dividends grow at $g$ :

$D_1 = D_1$
$D_2 = D_1(1 + g)$
$D_3 = D_1(1 + g)^2$
$D_t = D_1(1 + g)^{t-1}$

Substitute: $P_0 = \sum_{t=1}^{\infty} \frac{D_1(1 + g)^{t-1}}{(1 + r)^t} = D_1 \sum_{t=1}^{\infty} \frac{(1 + g)^{t-1}}{(1 + r)^t}$

Simplify the sum: $= D_1 \sum_{t=1}^{\infty} \left(\frac{1 + g}{1 + r}\right)^{t-1} \cdot \frac{1}{1 + r}$

Let $x = \frac{1 + g}{1 + r}$ , so we need $\sum_{t=1}^{\infty} x^{t-1} = \sum_{k=0}^{\infty} x^k$ .

Geometric series: $\sum_{k=0}^{\infty} x^k = \frac{1}{1 - x}$ when $|x| < 1$ (i.e., $g < r$ ).

So: $P_0 = \frac{D_1}{1 + r} \cdot \frac{1}{1 - \frac{1+g}{1+r}} = \frac{D_1}{1 + r} \cdot \frac{1 + r}{(1 + r) - (+ g)} = \frac{D_1}{r - g}$

Why this matters: The Gordon model is the workhorse for valuing mature, dividend-paying companies. Simple, elegant, deadly accurate when assumptions hold.

Worked Examples

Worked example Example 1: Stable Utility Stock

Given: A utility company pays $D_0 =$ 3.00 $per share this year. Dividends grow at$ g = 4% $per year. Your required return is$ r = 9%$.

Find: Intrinsic value $P_0$ .

Step 1: Calculate $D_1$ (next year's dividend). $D_1 = D_0(1 + g) = 3.00 \times 1.04 = \$3.12$ Why this step? Gordon model uses $D_1$ (next year's dividend), not $D_0$ .

Step 2: Apply Gordon formula. $P_0 = \frac{D_1}{r - g} = \frac{3.12}{0.09 - 0.04} = \frac{3.12}{0.05} = \$62.40$

Interpretation: If the stock trades below $62.40, it's undervalued. Above? Overvalued. At$ 62.40? Fairly priced.

Worked example Example 2: Multi-Stage Growth

Given: A tech company currently pays $D_0 = \emptyset$ 1.00$. Expected growth:

Years 1-3: $g_1 = 20\%$ (high growth phase)
Year 4 onward: $g_2 = 5\%$ (mature growth)
Required return: $r = 12\%$

Find: $P_0$ .

Why multi-stage? Young companies grow fast initially, then slow down. Single-stage Gordon doesn't capture this.

Step 1: Calculate dividends for high-growth years. $D_1 = 1.00 \times 1.20 = \$1.20$ $D_2 = 1.20 \times 1.20 = \$1.44$ $D_3 = 1.44 \times 1.20 = \$1.728$ Step 2: Calculate $D_4$ (first year of stable growth). $D_4 = 1.728 \times 1.05 = \$1.8144$

Step 3: Find terminal value at end of year 3(value of all dividends from year 4 onward, as of year 3). $P_3 = \frac{D_4}{r - g_2} = \frac{1.8144}{0.12 - 0.05} = \frac{1.8144}{0.07} = \$25.92$ Why this step? At year 3, the company enters perpetual growth. We use Gordon to value that infinite stream as of year 3.

Step 4: Discount everything back to today. $P_0 = \frac{D_1}{(1.12)^1} + \frac{D_2}{(1.12)^2} + \frac{D_3}{(1.12)^3} + \frac{P_3}{(1.12)^3}$ $= \frac{1.20}{1.12} + \frac{1.44}{1.2544} + \frac{1.728}{1.4049} + \frac{25.92}{1.4049}$ $= 1.071 + 1.148 + 1.230 + 18.45 = \$21.90$

Why discount $P_3$ ? It's the value at year 3, not today. We need present value.

Common Mistakes

Common mistake Mistake 2: Using

$g \geq r$ Wrong: Company growing at $g = 15\%$ , required return $r = 12\%$ . Try Gordon: $P_0 = \frac{D_1}{0.12 - 0.15} = \frac{D_1}{-0.03} = \text{negative price!?}$

Why it feels right: High-growth companies seem more valuable, so high $g$ should give high price.

The problem: The math literally breaks. When $g \geq r$ , the infinite sum diverges (doesn't converge to a finite value). It means: "This company will grow so fast forever that it's worth infinite money." Impossible.

The fix: Use multi-stage DDM. High growth can't last forever. Model the high-growth phase explicitly (years 1-5), then assume $g$ drops below $r$ for the terminal value.

Active Recall Flashcards

#flashcards/stock-market

What does the Dividend Discount Model (DDM) value a stock as? :: The present value of all future dividends: $P_0 = \sum_{t=1}^{\infty} \frac{D_t}{(1 + r)^t}$

What is the Gordon Growth Model formula?

P_0 = \frac{D_1}{r - g}

where

D_1

is next year's dividend,

r

is required return, and

g

is constant growth rate

Why must

g < r

in the Gordon Growth Model?

Because if

g \geq r

, the geometric series diverges—mathematically it means the stock has infinite value, which is impossible. Also economically: if dividends grow faster than required return forever, no investor would ever sell.

In DDM, what is the "discount rate"

r

and why do we use it?

r

is your required rate of return—it compensates for time value of money, risk, and opportunity cost. We discount because

\

today is worth more than

$1$ in the future.

What's the difference between

D_0

and

D_1

in Gordon Growth Model?

D_0

is the dividend just paid (historical).

D_1 = D_0(1+g)

is next year's expected dividend. Gordon formula requires

D_1

because it values future cash flows.

How do you value a stock with multiple growth stages?

1) Calculate dividends during high-growth years, 2) Find terminal value at end of high-growth using Gordon model, 3) Discount all cash flows (dividends + terminal value) back to present.

Why does a higher required return

r

decrease stock price in DDM?

Because higher

r

means you're discounting future dividends more heavily. Each future dollar is worth less today when your required return is higher. Mathematically:

P_0 = \frac{D_1}{r - g}

, so

r \uparrow

means

P_0 \downarrow

What type of stock is Gordon Growth Model best suited for?

Mature, stable companies with predictable dividend growth (utilities, consumer staples). Not suitable for high-growth companies, non-dividend payers, or cyclical businesses.

Feynman Technique

Recall Explain to a 12-year-old

Imagine your friend has a lemonade stand. You're thinking of buying it from them. How much should you pay?

Well, the stand will give you money every summer—let's say $\$ 10 $this year,$ $11 $next year (because they raise prices a bit),$ $12$ the year after, and so on forever.

But here's the thing: $\$ 11 $next year isn't as good as$ $11today, right? Because if you had $\$ 11$ today, you could put it in a savings account and have even more next year. So we need to "discount" future money to make it fair to compare.

DDM is just fancy math that says: "Add up all the money you'll get in the future, but make each future dollar worth a little less because it's far away." That total is what the lemonade stand (or stock) is worth.

The Gordon Growth Model is a shortcut: if the stand's earnings grow by the same percent every year forever, you can use one simple formula instead of adding up infinite numbers. It's like a cheat code for the calculation!

Mnemonic

Connections

2.6.01-Understand-intrinsic-value-vs-market-price: DDM calculates intrinsic value
2.6.02-Time-value-of-money-inuation: DDM applies present value discounting
2.6.03-Learn-price-to-earnings-(PE)-ratio: PE is relative valuation; DDM is absolute valuation
2.6.08-Discounted-cash-flow-(DCF)-analysis: DDM is a special case of DCF (dividends are a type of cash flow)
2.5.03-Understanding-dividends-and-payout-ratios: Dividend policy determines $D_t$ and $g inputs
3.2.04-Required-rate-of-return-andCAPM: CAPM calculates $r$ for DDM
2.6.10-Compare-valuation-methods: DDM vs. PE vs. DCF vs. asset-based

Study this note with active recall: Cover the answers, try to derive formulas from scratch, work examples without peeking. Steel-man the mistakes—understand why they're tempting.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dividend Discount Model (DDM) ka basic concept bahut simple hai: ek stock ki asli value kya hai? Woh hai total present value of sare future dividends jo apko milne wale hain. Socho ki ap ek stock khareedte ho—apko paise kab milte hain? Dividends ke roop mein, har saal. Toh agar main har saal $5,$ 6, $7... aur age tak dividends lu, un sabka aj ka value kya hai? Yahi DDM calculate karta hai.

Gordon Growth Model isse aur simple banata hai. Agar dividends har saal constant rate se badhte hain (jaise 4% har saal), toh ek shortcut formula hai: $P_0 = D_1 / (r - g)$ . Yahan $D_1$ next year ka dividend hai, $r$ apka required return hai (kitna profit chahiye apko), aur $g$ growth rate hai. Par dhyan rakhna— $g$ hamesha $r$ se chhota hona chahiye, nahi toh formula toot jayega aur infinite value aa jayega jo real life mein impossible hai.

Yeh model utilities aur mature companies ke liye best hai jinke dividends stable hain. High-growth startups ke liye multi-stage model use karo jisme pehle fast growth phase ko alag se model karo, phir stable growth assume karo. DDM valuation ka foundation hai—agar yeh samajh gaya toh DCF aur dusre advanced methods bhi clear ho jayenge. Yeh basically time value of money ko apply karte hue stock ki true worth nikalna hai, not just market ka speculation.

Core Intuition

Derivation: From First Principles

The Gordon Growth Model (Constant Growth DDM)

Worked Examples

Common Mistakes

Active Recall Flashcards

Feynman Technique

Mnemonic

Connections

Concept Map

Hinglish (regional understanding)

Test yourself — Valuation Methods

Connections