4.6Lesson 4.6 · Module 4

Growth Opportunities, PVGO, and P/E

Stock value reflects both existing earnings and future investment opportunities. Decompose P₀ = EPS₁/r + PVGO, and connect it to P/E.

No-growth value = EPS₁/r
PVGO = P₀ − EPS₁/r
P/E = 1/r + PVGO/EPS₁
Safer earnings and profitable growth both raise P/E

Start

Learning objectives

By the end of this lesson, you should be able to:

1Define the no-growth value of a stock as EPS₁ / r.
2Decompose P₀ into EPS₁/r + PVGO.
3Determine the sign of PVGO from the ROE on new investment vs r.
4Derive P/E = 1/r + PVGO/EPS₁.
5Explain why safer companies tend to have higher P/E, all else equal.
6Explain why profitable growth can offset higher risk in P/E.
7Recognize that high or low P/E alone does not signal over- or undervaluation.

From Lesson 4.5

Lesson 4.5 showed where growth comes from: $g = b \times R O E$ . But that leaves a deeper question. If a firm can grow its earnings, does that growth automatically make shareholders better off? We saw the answer is no — growth only creates value when the return on new investment exceeds the cost of equity. This lesson makes that intuition precise by decomposing a stock's value into the part that comes from existing assets and the part that comes from future growth opportunities, and then connects the result to the P/E ratio.

6.1Section 1

The no-growth value

Imagine a company that distributes all of its earnings as dividends and reinvests nothing. Its earnings do not grow, so it is just a level perpetuity of $E P S_{1}$ per year.

No-growth value

P_{0}^{n o - g r o w t h} = \frac{E P S _{1}}{r}

If all earnings are paid out and nothing is reinvested, the stock is worth next year's earnings capitalized at the cost of equity.

E P S_{1} = $8.33, r = 15% \Rightarrow P_{0} = \frac{8.33}{0.15} \approx $55.56

P₀ ≈ $55.56

No-growth does not mean the business shuts down. Current operations continue exactly as they are; the firm simply pays out everything it earns rather than reinvesting.

6.2Section 2

ABC Software: value beyond no-growth

Now meet ABC Software. It has $E P S_{1} = $8.33$ , pays out $p = 60%$ , retains $b = 40%$ , earns $R O E_{n e w} = 25%$ on new investment, and faces $r = 15%$ . The dividend is $D_{1} = 0.60 \times 8.33 \approx $5$ and the growth rate is $g = 0.40 \times 25% = 10%$ .

ABC Software value (Gordon)

P_{0} = \frac{D _{1}}{r - g} = \frac{5}{0.15 - 0.10}

P_{0} = \frac{5}{0.05} = $100

P₀ = $100

Compare to the no-growth value of $55.56. The difference — $44.44 — is the value the market assigns to ABC's future investment opportunities. That is PVGO.

6.3Section 3

Decomposing P₀ = EPS₁/r + PVGO

Value decomposition

P_{0} = \frac{E P S _{1}}{r} + P V GO

Stock value = value of existing assets (no-growth) + present value of growth opportunities.

$E P S_{1} / r$: value of current operations if all earnings were paid out
$P V GO$: present value of all future value-creating investments

The first term is what the firm is worth just for the business it already has. PVGO is the extra value the market pays because it expects the firm to find profitable new investments.

6.4Section 4

NPV of the first new investment

Where does PVGO actually come from? Look at the very first project ABC undertakes with its retained earnings.

First project NPV

N P V_{1} = - Investment + \frac{R O E _{n e w} \times Investment}{r}

Retain b × EPS₁ as capital. It earns $R O E_{n} e w$ per year forever. The NPV is the present value of that perpetuity minus the capital invested.

Investment = 0.40 \times 8.33 = $3.33, return = 0.25 \times 3.33 = $0.83/ yr, P V = \frac{0.83}{0.15} = $5.55

NPV₁ = −3.33 + 5.55 = $2.22

The $3.33 of retained earnings is capital the shareholders already owned — it is not value created. Value created is the NPV: $2.22. The project is worth more than it costs because $R O E_{n} e w$ > r.

6.5Section 5

From one project to total PVGO

ABC does not just invest once. Each year it retains more earnings and finds more opportunities, and the size of those opportunities grows at $g = 10%$ . So the NPVs of successive projects form a growing stream.

Total PVGO

P V GO = \frac{N P V _{1}}{r - g}

The present values of all future investment NPVs, which themselves grow at g, form a growing perpetuity starting at NPV₁.

P V GO = \frac{2.22}{0.15 - 0.10} = \frac{2.22}{0.05} \approx $44.44

PVGO ≈ $44.44

Total PVGO is far larger than the first project's $2.22 because the firm keeps finding new, profitable opportunities year after year. This is why growth companies can trade well above their no-growth value.

6.6Section 6

Three PVGO cases

ROE_new > r

$P V GO > 0$ . New investment earns above its cost. Growth creates value.

ROE_new = r

$P V GO = 0$ . Investment earns exactly its cost. NPV-neutral.

ROE_new < r

$P V GO < 0$ . Investment earns below its cost. Growth destroys value.

6.7Section 7

Negative PVGO: growth that destroys value

Suppose $E P S_{1} = $10$ , $r = 10%$ , $b = 50%$ , and the firm earns only $R O E_{n e w} = 6%$ on new investment — below its cost of equity.

Value-destroying growth

P_{0} = \frac{D _{1}}{r - g}, g = b \times R O E_{n e w} = 0.50 \times 0.06 = 3%

D₁ = (1−b)·EPS₁ = 0.50 × 10 = $5. Plug into Gordon.

P_{0} = \frac{5}{0.10 - 0.03} = \frac{5}{0.07} \approx $71.43

PVGO = 71.43 − 100 = −$28.57

The no-growth value is EPS₁/r = 10/0.10 = $100. But because reinvestment earns only 6%, growth pushes the value DOWN to $71.43. Earnings rise — and value falls.

6.8Section 8

What really makes a 'growth stock'

Definition · A growth stock has PVGO > 0

A growth stock is not defined by fast revenue or EPS growth. It is defined by having $P V GO > 0$ — future investment opportunities whose returns exceed the cost of capital. A slow, boring company with one great new project can have positive PVGO. A flashy, fast-growing company that reinvests below its cost can have negative PVGO. The label "growth" belongs to the value created, not to the growth rate.

6.9Section 9

Defining P/E

Forward P/E

P / E = \frac{P _{0}}{E P S _{1}}

The price-earnings ratio using next year's expected earnings (the forward P/E).

P/E decomposition

P / E = \frac{1}{r} + \frac{P V GO}{E P S _{1}}

Divide P₀ = EPS₁/r + PVGO by EPS₁ to split P/E into a risk component and a growth component.

$1/ r$: the P/E a no-growth firm would have — purely a function of risk
$P V GO / E P S_{1}$: extra multiple the market pays for growth opportunities

The P/E is not one number with one cause. It is a baseline that reflects risk (1/r) plus a premium that reflects profitable growth (PVGO/EPS₁).

6.10Section 10

The no-growth P/E = 1/r

If $P V GO = 0$ , the growth component vanishes and $P / E = 1/ r$ . The multiple then depends only on the cost of equity. Lower risk means a higher baseline multiple.

r	1/r (no-growth P/E)
8%	12.5×
10%	10×
15%	6.67×
20%	5×

6.11Section 11

Risk and P/E

Higher risk means a higher $r$ , which lowers $1/ r$ and therefore lowers P/E, all else equal. Lower risk means a lower $r$ , a higher $1/ r$ , and a higher P/E. In other words, safer companies deserve higher P/E multiples, holding growth constant.

A caveat

This is an all-else-equal statement. It does not mean every safe company has a higher P/E than every risky one — growth opportunities (PVGO) vary too. It means: for the same growth, the safer firm commands a higher multiple.

6.12Section 12

When a riskier company has the higher P/E

Compare two firms. Company A is safe — $r = 8%$ — but has no growth, so $P / E = 1/0.08 = 12.5 \times$ . Company B is riskier — $r = 15%$ — but has strong profitable growth (large PVGO). B's growth component $P V GO / E P S_{1}$ can easily push its total P/E above 12.5×. Profitable growth can more than offset higher risk.

6.13Section 13

P/E under the Gordon model

Gordon P/E

P / E = \frac{1 - b}{r - b R O E}

Substitute D₁ = (1−b)·EPS and g = b·ROE into P₀ = D₁/(r−g), then divide by EPS₁.

P/E depends on retention, ROE, and r together. The same ROE can produce very different multiples depending on how much is retained and what the cost of equity is.

6.14Section 14

ABC's P/E, decomposed

Recall ABC: $P_{0} = $100$ , $E P S_{1} = $8.33$ . Its forward P/E is $100/8.33 \approx 12.0 \times$ . Where does that 12.0× come from?

Risk component 1/r

$1/0.15 = 6.67 \times$ . The multiple ABC would have with no growth.

Growth component PVGO/EPS₁

$44.44/8.33 \approx 5.33 \times$ . The extra multiple from profitable growth.

Total

$6.67 + 5.33 \approx 12.0 \times$ . Risk plus growth equals the observed P/E.

6.15Section 15

What high and low P/E really mean

A high P/E may mean

Lower risk (higher 1/r)
Durable, high-quality earnings
Strong, profitable growth opportunities (large PVGO)
Temporarily depressed earnings
Market optimism about the future

A low P/E may mean

Higher risk (lower 1/r)
Declining or cyclical business
Weak or no growth opportunities (negative PVGO)
Temporarily inflated earnings
Market pessimism

Key takeaway

High P/E does not mean overvalued, and low P/E does not mean undervalued. A multiple only means something once you know what drives it: risk, growth, or a temporary earnings distortion.

6.16Section 16

Forward vs trailing P/E

Trailing P/E

Price divided by the most recent actual (trailing) earnings. Backward-looking.

Forward P/E

Price divided by expected next-year earnings EPS₁. This is the version our model uses.

6.17Section 17

Normalized earnings and P/E distortions

Suppose a company's normal EPS is $10, but this year earnings are temporarily depressed to $2. The stock trades at $100. The trailing P/E looks enormous:

Measure	EPS	P/E
Trailing (depressed)	$2	50×
Normalized	$10	10×

Distortion warning

The 50× trailing P/E says nothing about overvaluation. It reflects temporarily weak earnings. Using normalized earnings gives a truer 10×. Always check whether earnings are normal before reading a multiple.

6.18Synthesis

Putting Module 4 together

Module 4 has built a single chain of reasoning, link by link:

Equity ownershipDistributionsDDMGordonMulti-stageEarnings/retentiong = b·ROEROE vs rPVGOP/E

The big picture

Fixed income is risk-specified: cash flows, timing, and priority are written into the contract. Equity is residual. Its value reflects both the existing assets — whose earnings are capitalized at $r$ — and the future opportunities — captured by PVGO. Everything in this module is a tool for quantifying that residual claim.

Try itPVGO decomposition lab

P₀ = EPS₁/r + PVGO

Decompose value into existing business and growth

Set next-year earnings EPS₁, the retention ratio b, the ROE on new investment, and the cost of equity r. The lab splits the stock price into the value of existing operations and the present value of growth opportunities, then decomposes the P/E the same way.

Next earnings EPS₁

Retention ratio b40%

ROE on new investment25.0%

Cost of equity r15.0%

No-growth value EPS₁/r

$55.53

$8.33 ÷ 15.0%

Implied growth g = b × ROE

10.00%

40% × 25.0%

Gordon growth value D₁/(r−g)

$99.96

$5.00 ÷ (15.0% − 10.0%)

PVGO

$44.43

$99.96 − $55.53

Growth-opportunity verdict

PVGO > 0

ROE on new investment (25.0%) exceeds r (15.0%). Retained capital earns above its cost, so future investment creates value.

Stock price = existing business + PVGO

$99.96 = $55.53 + $44.43

P/E = 1/r + PVGO/EPS₁ (forward)

Forward P/E

12.00×

$99.96 ÷ $8.33

Risk component 1/r

6.67×

1 ÷ 15.0%

Growth component PVGO/EPS₁

5.33×

$44.43 ÷ $8.33

12.00× ≈ 6.67× + 5.33×. The risk component is what a no-growth version of the firm would earn; the growth component is the extra multiple the market pays for future opportunities.

6.19Concept check

Decompose a P/E

Try itWorked check

EPS₁ = $6, r = 12%, price = $70

Find the no-growth value, PVGO, the forward P/E, and decompose the P/E.

6.20Common questions

PVGO, P/E, and what multiples mean

06Mastery

Summary and mastery check

Try itLesson 4.6 mastery check

Pass with 4 of 6 correct

Answer all questions, then check your work. You can retry any time — mastery is based on correctness, not speed.

01
EPS₁ = $8.33, r = 15%. No-growth value?
02
P₀ = $100, no-growth value = $55.56. PVGO?
03
If $R O E_{n} e w$ < r, PVGO is:
04
P₀ = $100, EPS₁ = $8.33. Forward P/E?
05
All else equal, a safer company (lower r) tends to have:
06
P/E decomposition: P/E = ?

Lesson summary

1No-growth value = EPS₁/r — the value of existing operations paying out all earnings.
2PVGO = P₀ − EPS₁/r — the value of future growth opportunities.
3PVGO is positive when ROE on new investment exceeds r, zero when equal, negative when below.
4P/E = 1/r + PVGO/EPS₁ — decomposes into risk and growth components.
5Safer companies tend to have higher P/E, all else equal.
6Profitable growth can offset higher risk in P/E.
7High or low P/E alone does not indicate over- or undervaluation.
8Equity is residual. Its value reflects both existing assets and future opportunities.

Replay this lesson Continue to Equity Valuation Case Lab