Stocks & Valuation

Over the last two posts we’ve priced bonds and measured their sensitivity to interest rates. The coupon amounts were written in the contract, the payment dates were fixed, and the face value at maturity was guaranteed (credit risk aside). Pricing them was mechanical: we discounted known cash flows at a known yield.

Stocks offer none of that. A stockholder is a residual claimant, entitled to whatever is left after everyone else (bondholders, employees, the tax collector) has been paid. Nobody promises a specific dividend. Nobody promises any dividend. The amounts are uncertain, the timing is at management’s discretion, and the stream of payments theoretically extends forever.

The logic does not change, though. A stock is still a claim on future cash flows, and its price is still the present value of those cash flows. What we have to work out is what the cash flows are.

Dividends and the Resale Chain

If a stock’s price is the present value of its dividends, an objection arises right away. Most investors sell their shares long before collecting dividends forever, which makes it hard to see why dividends should determine the price at all.

We buy the stock today at $P_{0}$ , expect a dividend $D_{1}$ next year, and plan to sell at $P_{1}$ , so the stock is worth $P_{0} = (D_{1} + P_{1}) / (1 + r)$ to us. But what sets $P_{1}$ ? The next buyer’s expected dividend $D_{2}$ and resale price $P_{2}$ . And $P_{2}$ rests on $D_{3}$ and $P_{3}$ .

Keep unrolling this chain, and every future price resolves into the dividend paid that period plus the price after it. All the future prices cancel, and we are left with an infinite stream of dividends discounted back to today.

Nobody needs to hold the stock forever for this to work. Each price in the chain already includes the value of every dividend after it, so the conclusion does not depend on any investor planning to stick around.

One clarification: “dividends” here means all cash returned to shareholders, including buybacks. A company that returns cash through repurchases rather than dividends is not an exception to the model, only a user of a different channel.

The Constant-Growth DDM

An infinite stream of dividends growing at a steady rate already has a closed form, and we have one. Post 2 introduced the growing perpetuity and named its price the Gordon growth formula.

If dividends start at $D_{1}$ and grow at rate $g$ forever, and investors require a return of $r$ , the dividend discount model (DDM) prices the stock at:

P_{0} = \frac{D _{1}}{r - g}

Three inputs drive the whole thing:

$D_{1}$ , next year's dividend, usually estimated as $D_{0} \times (1 + g)$ where $D_{0}$ is the last dividend paid. Price is proportional to it.
$r$ , the required return, meaning what investors demand for bearing the stock's risk. Because it appears in the denominator, a higher $r$ means a lower price. We estimate it from data later in the series, and until then we take it as given.
$g$ , the rate at which dividends grow forever. A higher $g$ raises the price twice over, since it increases $D_{1}$ and shrinks the denominator.

Suppose a utility’s last dividend was 2.40 per share. Analysts expect dividends to grow at 3.5% indefinitely, and investors in similar utilities demand a 9% return. We grow the dividend one period and divide by $r - g$ :

D_{1} = 2.40 \times 1.035 = 2.484

P_{0} = \frac{2.484}{0.09 - 0.035} = \frac{2.484}{0.055} = 45.16

The model says this stock is worth about 45.16 per share.

Drag the three sliders below to move the last dividend, the growth rate, and the required return.

Model Parameters Last Dividend (D₀): $2.40 Growth Rate: 3.5% Required Return: 9% Constant Growth Multi-Stage Stock Price $45.16 D₁ (Next Dividend) $2.48 Dividend Yield 5.50% Price / Dividend 18.2x

Push $g$ toward $r$ and the denominator $(r - g)$ shrinks while the price shoots up. At $g = r$ the price goes to infinity, which is the model reporting its own breakdown rather than a valuation. No stock is worth infinity. A growth estimate close to the discount rate leaves the model too sensitive to be useful: in that zone a quarter-point change in either assumption doubles or halves the price.

Implied Return

The DDM works in both directions. With a price already quoted by the market, we can solve for the return the market implicitly expects:

r = \frac{D _{1}}{P _{0}} + g

The first term is the dividend yield and the second is the capital gain from growth. Together they give the total expected return.

A REIT trades at 72.00 per share, its next annual dividend will be 4.32, and its cash flows are expected to grow at 2% per year. We read the implied return straight off the price:

r = \frac{4.32}{72.00} + 0.02 = 0.06 + 0.02 = 0.08 = 8%

Whether 8% is reasonable for a REIT is a question of comparison: against similar REITs, against a hurdle rate, against the returns available on comparable investments. An implied return too low for the risk marks the stock as overpriced, and one that looks too high may mean it is cheap. This is how analysts use the DDM as a sanity check on market prices.

Multi-Stage Growth

The constant-growth model assumes one growth rate forever. That is fine for a mature utility and not for a company whose dividends are growing fast now and will slow down later. The fix is a two-stage model: high growth for $N$ years, then a stable rate forever.

Price equals the present value of the high-growth dividends plus the present value of the terminal value, which is the Gordon growth formula applied at year $N$ , when growth stabilizes:

P_{0} = t = 1 \sum N \frac{D _{t}}{( 1 + r ) ^{t}} + \frac{P _{N}}{( 1 + r ) ^{N}}

where $P_{N} = D_{N + 1} / (r - g_{s})$ and $g_{s}$ is the stable growth rate.

Consider a tech company whose last dividend was 1.20 per share. Dividends will grow at 15% for the next 5 years, then slow to a 4% stable growth rate, and investors require 11%.

The high-growth dividends run 1.38, 1.59, 1.83, 2.10, and 2.41 in years 1 through 5. The year-6 dividend would be $2.41 \times 1.04 = 2.51$ , so we price the terminal value at year 5 at $2.51/ (0.11 - 0.04) = 35.86$ . Discounting the five dividends and the terminal value back to today, we get a stock worth roughly 28.

Switch the calculator above to multi-stage mode, set the last dividend to 1.20 and the required return to 11%, and its schedule reproduces that arithmetic row by row. The breakdown bar underneath splits the price between the two stages, and the terminal value dominates. Even though the high-growth dividends compound at 15%, their present value comes to under a quarter of the total. The perpetuity beyond year 5 accounts for the rest, as it does in almost every multi-stage valuation.

Sensitivity to the Inputs

The DDM has a well-known practical weakness: it is extremely sensitive to its inputs. A two-stage terminal value runs through the same denominator as a constant-growth price, so both versions inherit it.

The denominator $(r - g)$ is usually a small number, maybe 5 or 6 percentage points. Shift the growth rate by half a point and we move the denominator by something close to 10%, which moves the price by as much. Change both $r$ and $g$ in opposite directions and the swings get larger still.

This is not a bug in the model. It reports something real: stock prices are sensitive to assumptions about growth and risk. But it does mean that two analysts with slightly different views on a company’s long-run growth rate can arrive at prices 20% or 30% apart, both using the same model competently.

That is why analysts build sensitivity tables, varying $r$ and $g$ systematically to see how the price moves across a range of assumptions. That range of prices, not a number pulled out of it, is what the analyst reports.

The grid below is one such table. Read across a row to hold the required return fixed and vary growth, and down a column to do the reverse.

Last Dividend (D₀): $2.40 1% 2% 3% 4% 5% 6% 7% 6% 48.48 61.20 82.40 124.80 252.00 -- -- 7% 40.40 48.96 61.80 83.20 126.00 254.40 -- 8% 34.63 40.80 49.44 62.40 84.00 127.20 256.80 9% 30.30 34.97 41.20 49.92 63.00 84.80 128.40 10% 26.93 30.60 35.31 41.60 50.40 63.60 85.60 11% 24.24 27.20 30.90 35.66 42.00 50.88 64.20 12% 22.04 24.48 27.47 31.20 36.00 42.40 51.36 13% 20.20 22.25 24.72 27.73 31.50 36.34 42.80 14% 18.65 20.40 22.47 24.96 28.00 31.80 36.69 Rows: Required Return Columns: Growth Rate

Prices climb steeply toward the top right of the grid, where a high growth rate meets a low required return, and in the top-right corner itself, where growth has caught the discount rate, the grid stops reporting a price at all.

The P/E Shortcut

All of this presumes estimates of dividends, growth, and the discount rate are already in hand. In practice, analysts often reach for a quicker tool, the price-to-earnings ratio (P/E).

P / E = \frac{Price}{Earnings per Share}

A stock trading at 80 with EPS of 4 has a P/E of 20, which is 20 dollars of price for every dollar of current earnings.

The ratio is not unrelated to the DDM. We start from the Gordon growth formula and note that $D_{1} = p \times E_{1}$ , where $p$ is the payout ratio (the fraction of earnings paid out as dividends):

P_{0} = \frac{D _{1}}{r - g} = \frac{p \times E _{1}}{r - g}

Dividing both sides by $E_{1}$ gives us:

\frac{P _{0}}{E _{1}} = \frac{p}{r - g}

This is the fundamentally justified P/E ratio, and it connects the multiple directly to growth and risk. The formula gives a high P/E to a company with high growth and a low required return, and a low one to a slow-growing utility with a fat payout. The market is not irrational for giving tech stocks higher P/Es than utilities. It is pricing in different growth expectations.

Two caveats come attached. P/E is a relative metric: a P/E of 25 says nothing about whether a stock is cheap or expensive in isolation, only relative to the growth and risk we assume behind it.

The other caveat is the distinction between trailing P/E (using last year’s earnings) and forward P/E (using next year’s forecast). Trailing P/E is backward-looking and can be distorted by one-time charges or windfalls. Forward P/E is more relevant but depends on the accuracy of the earnings forecast.

Drag the required return slider below and watch the P/Es of three archetypes respond: a steady utility, a mature industrial, and a growth tech name.

Required Return: 10% Steady Utility 10.0x P/E Ratio Payout 75% Growth 2.5% r - g 7.5% Mature Industrial 8.0x P/E Ratio Payout 40% Growth 5% r - g 5.0% Growth Tech 15.0x P/E Ratio Payout 15% Growth 9% r - g 1.0% All three earn $4.00 per share.

When $r$ drops toward the growth rate, P/E explodes, and at $r = g$ the card stops quoting a ratio and flags $g \geq r$ instead. The growth tech stock is the most sensitive, since its growth rate is closest to the discount rate. The utility barely moves, because its $(r - g)$ spread is wide and comfortable.

Now push $r$ the other direction, up to 16%. The ranking flips, and the utility takes the highest P/E while the tech stock takes the lowest. At high discount rates the $(r - g)$ spreads widen for everyone, which neutralizes the growth advantage. The numerator takes over, and a 75% payout beats a 15% payout. So growth stocks get hit harder than value stocks when interest rates rise: their P/E premium depends on a tight spread, and rising rates blow it open.

The Limits of These Models

That rate sensitivity is the model working correctly. Other limits are not: the DDM and its P/E shortcut break down in four ways worth knowing.

Companies that pay no dividend. Amazon does not pay one, and the DDM prices such a company at zero, which is wrong. Modeling future dividends or free cash flows instead is possible, at the cost of forecasting far into an uncertain future.
Cyclical earnings. A company whose earnings swing wildly from year to year does not fit neatly into a constant-growth assumption.
Growth that outruns the economy. A $g$ of 12% in perpetuity asserts that the company eventually grows larger than the entire economy, which is not going to happen. Stable growth rates belong at or below long-run GDP growth plus inflation, somewhere around 2-4%.
Garbage in, garbage out. The model is only as good as the estimates of $r$ and $g$ behind it, and small errors in those produce large errors in the price, as the grid above makes plain.

None of this makes the models useless. The discipline of thinking in terms of cash flows, growth, and risk holds its value even when the numbers are imprecise. A DDM output is still best read as a range rather than a point estimate, and it belongs beside other evidence (comparable companies, market multiples, qualitative judgment) rather than standing alone.

What’s Next

The required return has done a lot of work in this post. We assumed 9% for the utility and 11% for the tech company, and we read 8% back out of the REIT’s price, without ever saying where such a number comes from. What makes one stock’s required return higher than another’s is a question about measuring risk. In the next post we build that toolkit: expected return, variance and standard deviation, the normal model, Value at Risk, and the Sharpe ratio.