Finance-Theory · Unit 9 · Video 6 · Interactive Practice

Why Stock Prices Explode Near r = g: The Gordon Growth Model and Bursting Bubbles

IKey Formulas

FormulaNameWhat it needs
Pt=∑k=1∞D(1+r)k=DrP_t = \displaystyle\sum_{k=1}^{\infty} \frac{D}{(1+r)^k} = \frac{D}{r}Constant-dividend perpetuityThe same expected dividend DD every period and one discount rate rr for every date. A $5 dividend at 5% gives $100; $6 gives $120; doubling the rate to 10% halves the price to $50
Pt=∑k=1∞D(1+g)k−1(1+r)k=Dr−g,r>gP_t = \displaystyle\sum_{k=1}^{\infty} \frac{D(1+g)^{k-1}}{(1+r)^k} = \frac{D}{r-g}, \quad r > gGordon growth modelThe first dividend DD one period from now, growing at gg every period forever, discounted at rr; the growing perpetuity with DD in place of CC
D12(r−g)=2⋅Dr−g\dfrac{D}{\tfrac{1}{2}(r-g)} = 2 \cdot \dfrac{D}{r-g}Halving the gap doubles the priceWith a $1 dividend and r=10%r = 10\%: growth of 8%, 9% and 9.5% gives $50, $100 and $200

Key Insight: The growth rate gg is a forecast. With a $1 dividend and r=10%r = 10\%, raising expected growth from 0% to 1% adds about 11% to the price, while cutting it from 9% to 8% takes away half.

IIConstant Dividends: P=D/rP = D/r

The price rises in proportion to the expected dividend and is inversely proportional to the discount rate.

IIIWhere the Gordon Price Comes From

D/(r−g)D/(r-g) is an infinite sum; near r=gr = g most of it comes from dividends decades away.

💡 Splitting the price by year is our own illustration, computed from the growing-perpetuity sum; the video's point is the same one: the formula assumes its rates hold forever, not for ten or twenty years.

IVOne Point of Growth, Far from r and Near It

With D=1D = 1 and r=10%r = 10\%, the same one-point revision of gg barely moves a no-growth price but halves a price near rr.

VQuiz Questions

Problem 1 · Flat and Growing Dividends

Given: A stock is expected to pay a dividend of $3 one year from now and the same $3 every year after that, forever. Its risk-adjusted discount rate is 8%. Find its price, and then its price if the dividend, still $3 next year, is instead expected to grow 2% a year forever.

Price with a constant $3 dividend?

Price with 2% growth?

✅ Correct! 3/0.08=37.53 / 0.08 = 37.5, so the flat stock is worth $37.50; growth shrinks the denominator to 0.08−0.02=0.060.08 - 0.02 = 0.06, and 3/0.06=503 / 0.06 = 50, so the growing stock is worth $50.
❌ Check the flat price. A constant dividend forever is a perpetuity: divide the dividend by the discount rate.
❌ Multiplied instead of divided. 3×0.08=0.243 \times 0.08 = 0.24 is one year's return on $3. The perpetuity value is D/r=3/0.08=37.5D / r = 3 / 0.08 = 37.5.
❌ Check the decimal. 8% is 0.080.08, not 0.0080.008: 3/0.08=37.53 / 0.08 = 37.5, not 375.
❌ That values only the first dividend. 3/1.08≈2.783 / 1.08 \approx 2.78 is the present value of next year's payment alone. The share pays $3 every year forever: D/r=37.5D / r = 37.5.
❌ Check the growing price. Use the Gordon growth model, D/(r−g)D / (r - g), with DD the first dividend.
❌ Wrong sign on g. 3/(0.08+0.02)=303 / (0.08 + 0.02) = 30 would make growth lower the price. Growth is subtracted: 3/(0.08−0.02)=503 / (0.08 - 0.02) = 50.
❌ Divided by g alone. 3/0.02=1503 / 0.02 = 150 drops the discount rate. The denominator is the gap r−g=0.06r - g = 0.06.
❌ That grows the flat price by one year. 37.50×1.02=38.2537.50 \times 1.02 = 38.25 applies growth once. Growth continues every year forever, and the formula captures all of it: 3/(0.08−0.02)=503 / (0.08 - 0.02) = 50.
Show solution

Constant dividend. With D=3D = 3 every year and r=0.08r = 0.08:

P=Dr=30.08=37.5P = \frac{D}{r} = \frac{3}{0.08} = 37.5

The share is worth $37.50.

Growing dividend. The first dividend is still D=3D = 3, one year from now, and g=0.02g = 0.02:

P=Dr−g=30.08−0.02=30.06=50P = \frac{D}{r - g} = \frac{3}{0.08 - 0.02} = \frac{3}{0.06} = 50

The share is worth $50. Growth shrinks the denominator, so the growing dividend is worth more than the flat one.

Problem 2 · Which Dividend Is D?

Given: A company has just paid a dividend of $2.00 per share. Dividends are expected to grow 4% a year forever, and the discount rate is 9%. Find the price of a share today.

✅ Correct! The $2.00 has already been paid, so the first dividend the buyer receives is 2×1.04=2.082 \times 1.04 = 2.08, and 2.08/(0.09−0.04)=41.62.08 / (0.09 - 0.04) = 41.6.
❌ Close, but that uses the dividend already paid. In D/(r−g)D/(r-g), DD is the first dividend, one period from now: 2×1.04=2.082 \times 1.04 = 2.08, so P=2.08/0.05=41.6P = 2.08 / 0.05 = 41.6.
❌ Growth is missing. 2/0.09≈22.222 / 0.09 \approx 22.22 treats the dividend as constant. With 4% growth the denominator is 0.09−0.04=0.050.09 - 0.04 = 0.05.
❌ Wrong sign on g. 2.08/(0.09+0.04)=162.08 / (0.09 + 0.04) = 16 adds the growth rate. The denominator is r−g=0.05r - g = 0.05.
❌ Not quite. Find next year's dividend first, then divide by r−gr - g.
Show solution

The Gordon growth model's DD is the first dividend, arriving one period from now. The dividend just paid was $2.00, so the next one is expected to be

D=2×(1+0.04)=2.08D = 2 \times (1 + 0.04) = 2.08 P=Dr−g=2.080.09−0.04=2.080.05=41.6P = \frac{D}{r - g} = \frac{2.08}{0.09 - 0.04} = \frac{2.08}{0.05} = 41.6

The share is worth $41.60. Plugging in the $2.00 already paid gives $40, which undervalues the share by one year's growth.

Problem 3 · Run-Up, Then Crash

Given: A stock's next dividend is expected to be $4 and its discount rate is 11%. The market expects dividends to grow 7% a year forever, so the price is $100. Good news raises expected growth to 9%; later, bad news cuts it to 3%. Find the price after the good news, and the percentage change in price caused by the bad news.

Price after expected growth rises to 9%?

Percentage change when expected growth falls from 9% to 3%?

✅ Correct! The gap r−gr - g halves from 4% to 2%, so the price doubles to $200; the gap then quadruples to 8%, so the price falls to one quarter, 200/4=50200 / 4 = 50, a change of 14−1=−75%\tfrac{1}{4} - 1 = -75\%.
❌ Check the new price. Recompute D/(r−g)D / (r - g) with g=0.09g = 0.09.
❌ The price is not proportional to g. Growth rising from 7% to 9% is a 2/7≈28.6%2/7 \approx 28.6\% rise in gg, but the price depends on the gap: 4/(0.11−0.09)=4/0.02=2004 / (0.11 - 0.09) = 4 / 0.02 = 200.
❌ Divided by g alone. 4/0.09≈44.444 / 0.09 \approx 44.44 drops the discount rate. The denominator is r−g=0.02r - g = 0.02.
❌ The $4 is already next year's dividend. 4×1.09/0.02=2184 \times 1.09 / 0.02 = 218 grows it once more. DD in D/(r−g)D/(r-g) is the first dividend, so P=4/0.02=200P = 4 / 0.02 = 200.
❌ Check the change. Find the price at g=0.03g = 0.03, then compare it with the $200 price, not the original $100.
❌ The price does not move in proportion to g. Growth falling from 9% to 3% is a 6/9≈66.7%6/9 \approx 66.7\% cut in gg, but the gap r−gr - g goes from 2% to 8%, so the price falls to one quarter of $200.
❌ Wrong starting price. 50/100−1=−50%50 / 100 - 1 = -50\% measures the fall from the original $100. The bad news arrived when the price was $200: 50/200−1=−75%50 / 200 - 1 = -75\%.
❌ That is the change in g, not in the price. gg fell by 6 percentage points; the price fell from $200 to $50.
Show solution

Check the starting price: P=4/(0.11−0.07)=4/0.04=100P = 4 / (0.11 - 0.07) = 4 / 0.04 = 100, the $100 given.

Good news, g=0.09g = 0.09:

P=40.11−0.09=40.02=200P = \frac{4}{0.11 - 0.09} = \frac{4}{0.02} = 200

The price is $200: the gap halved and the price doubled.

Bad news, g=0.03g = 0.03:

P=40.11−0.03=40.08=50P = \frac{4}{0.11 - 0.03} = \frac{4}{0.08} = 50 50200−1=−0.75\frac{50}{200} - 1 = -0.75

The gap went from 0.020.02 to 0.080.08, so the price is multiplied by 14\tfrac{1}{4} and the change is −75%-75\%. Two revisions of gg, and no change in dividends or in rr, produced a doubling followed by a loss of three quarters of the value.

Problem 4 · When Rates Rise

Given: Interest rates rise across the market, lifting the discount rate on two stocks from 10% to 11%; nothing else changes. Both are expected to pay $1 next year. Stock A's dividend is expected to stay flat forever; stock B's is expected to grow 9% a year forever. Find each stock's percentage price change.

Stock A (no growth)?

Stock B (9% growth)?

✅ Correct! A's denominator rises from 0.100.10 to 0.110.11, so its price falls from $10 to about $9.09, about 9.1%. B's gap r−gr - g doubles from 1% to 2%, so its price halves from $100 to $50. The same rise in rates hits the stock priced near r=gr = g more than five times as hard.
❌ Check stock A. Compare 1/0.111 / 0.11 with 1/0.101 / 0.10.
❌ Close, but the price is inversely proportional to r. rr rising 10% (from 0.10 to 0.11) multiplies the price by 1/1.11/1.1, not by 0.90.9: 10/11−1≈−9.1%10/11 - 1 \approx -9.1\%.
❌ That is the change in the rate. The rate rose 1 percentage point; the price went from $10 to about $9.09.
❌ That is stock B's change. Stock A has no growth: its price goes from 1/0.10=101/0.10 = 10 to 1/0.11≈9.091/0.11 \approx 9.09.
❌ Check stock B. Compare 1/(0.11−0.09)1 / (0.11 - 0.09) with 1/(0.10−0.09)1 / (0.10 - 0.09).
❌ Growth changes the sensitivity. B's denominator is the gap r−gr - g, which goes from 0.010.01 to 0.020.02: it doubles, so the price halves.
❌ That is the change in the rate. A 1-point rise in rr doubled B's gap r−gr - g from 1% to 2%, so the price went from $100 to $50.
❌ Wrong direction. The gap doubled, but it is in the denominator: a doubled gap halves the price.
Show solution

Stock A, P=D/rP = D / r:

10.10=10→10.11≈9.09,1011−1=−111≈−9.1%\frac{1}{0.10} = 10 \quad\to\quad \frac{1}{0.11} \approx 9.09, \qquad \frac{10}{11} - 1 = -\frac{1}{11} \approx -9.1\%

Stock B, P=D/(r−g)P = D / (r - g):

10.10−0.09=100→10.11−0.09=50,50100−1=−50%\frac{1}{0.10 - 0.09} = 100 \quad\to\quad \frac{1}{0.11 - 0.09} = 50, \qquad \frac{50}{100} - 1 = -50\%

Stock A falls from $10 to about $9.09 and stock B from $100 to $50. A rise in rates lowers every stock price, as the video says, but a stock whose expected growth sits close to its discount rate falls much further.

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