Lecture 3: Stocks and equity valuation

This lecture answers one question: what is a share worth? The answer is an application of lecture 2: a share is worth the present value of the money it will pay its owner, which is the dividends. From that idea come the dividend discount model, the Gordon growth model, the value of growth opportunities and the meaning of the price-earnings ratio.

Start here if this is new to you

Share (stock, equity). A public company is divided into millions of equal pieces called shares. Owning a share makes you a part-owner: you are entitled to your share of the profits and you vote at the shareholders' meeting. The company sells shares to raise money; afterwards they trade between investors on the stock exchange.
Common versus preferred shares. Common shares carry voting rights and a dividend that varies with profits. Preferred shares usually pay a fixed dividend, which must be paid before common shareholders get anything, but typically carry no vote. This course is about common shares.
Profit (earnings) and dividend. Earnings are what is left of revenue after all costs and taxes. The company can pay part of it to shareholders as a dividend and keep the rest (retained earnings) to invest in the business.
Book value. The value of the company according to its balance sheet: assets minus liabilities, at accounting values. The market value (market capitalisation) is the share price times the number of shares, what the stock market thinks the company is worth. The two are usually different.
Required rate of return (k). The return investors demand for holding a share, given its risk. It is the discount rate for stocks, the "i" of lecture 2. Riskier company, higher k.
Notation. P₀ is the price today, P₁ the expected price in one year, DIV₁ the dividend expected one year from now, EPS₁ the earnings per share next year, g a growth rate per year.

Ratios: a quick look at a company

Before valuing a share, analysts compute a few per-share numbers. They are simple divisions, but you must know the definitions by heart.

Earnings per share (EPS): total profit divided by the number of shares.
Dividend per share (DPS): total dividends paid divided by the number of shares.
Retained earnings per share (REPS): EPS − DPS, the part of the profit per share kept in the company.
Dividend yield (DY): DPS divided by the share price. The cash return you get for holding the share, like an interest rate.
Return on equity (ROE): profit divided by the value of equity. On the slides it is computed on the market value of equity (shares × price).
Price to book value (PTBV): market value of the company divided by its book value. Above 1 means the market values the company at more than its accounting value, usually because of expected growth or intangible assets.

Worked example (from the slides)

A public company has issued 1,000,000 shares with a current value of €40 each. Its last profit was €10,000,000, of which €4,000,000 was paid out as dividends. The book value of the company is €32,000,000.

EPS = 10,000,000 / 1,000,000 = €10
DPS = 4,000,000 / 1,000,000 = €4
REPS = (10,000,000 − 4,000,000) / 1,000,000 = €6 (also EPS − DPS = 10 − 4)
DY = 4 / 40 = 10%
ROE = 10,000,000 / (1,000,000 × 40) = 10,000,000 / 40,000,000 = 25%
PTBV = 40,000,000 / 32,000,000 = 1.25

The limits of ratios. Ratios give quick information at a glance, but they are static: they refer to past accounting information and say nothing about the future or about risk. A company with a dividend yield of 10% may be a bargain or a company about to cut its dividend. For a real valuation we need to look forward, which is what the rest of the lecture does.

The expected return of a stock

If you buy a share today at P₀, hold it one year, collect the dividend DIV₁ and sell it at P₁, your return k is what you gained divided by what you paid:

k = \frac{D I V _{1} + P _{1} - P _{0}}{P _{0}} = \frac{D I V _{1}}{P _{0}} + \frac{P _{1} - P _{0}}{P _{0}}

The first part is the dividend yield, the second the capital gain (the percentage price change). The sum is the total return.

Worked example (from the slides)

P₀ = €100, and you believe DIV₁ = €5 and P₁ = €110. The expected return is

k = \frac{5 + 110 - 100}{100} = \frac{15}{100} = 0.15 = 15%

5% comes from the dividend and 10% from the price rise.

From expected return to price: the dividend discount model

Now turn the formula around. If investors require a return k, the price they are willing to pay today is the present value of what they expect to receive in a year:

P_{0} = \frac{D I V _{1} + P _{1}}{1 + k}

But what determines P₁? The same logic one year later: P₁ = (DIV₂ + P₂) / (1 + k). Substitute this into the first equation:

P_{0} = \frac{1}{1 + k} (D I V_{1} + \frac{D I V _{2} + P _{2}}{1 + k}) = \frac{D I V _{1}}{1 + k} + \frac{D I V _{2}}{( 1 + k ) ^{2}} + \frac{P _{2}}{( 1 + k ) ^{2}}

Keep substituting for P₂, P₃, ... For t periods:

P_{0} = \frac{D I V _{1}}{1 + k} + \frac{D I V _{2}}{( 1 + k ) ^{2}} + \dots + \frac{D I V _{t} + P _{t}}{( 1 + k ) ^{t}}

As t goes to infinity, the last term vanishes (a price far in the future, discounted heavily, is worth nothing today) and we get the dividend discount model (DDM):

P_{0} = t = 1 \sum \infty \frac{D I V _{t}}{( 1 + k ) ^{t}}

In plain words

The price of a share today is the present value of all its future dividends, discounted at the return investors require. Even an investor who plans to sell next year is, in the end, paid out of dividends: the price they will get is itself the present value of later dividends. Everything else in this lecture is a special case of this formula with a simple assumption about how dividends evolve.

Case 1: a static company (no growth)

A static company does not reinvest its profits: it pays out everything, so DIV = EPS every year, forever. The dividends are then a perpetuity (lecture 2), and the price is

P_{0} = \frac{E P S}{k}

Worked example (from the slides): AZ

The price of AZ is €100, EPS = DIV = €10 and the company does not grow. Expected return = 10 / 100 = 10%. (Equivalently, P₀ = 10 / 0.10 = 100.)

Suppose AZ invests all of one period's profit (€10 per share) and as a result future dividends rise to €11 forever. Was this a good investment? Its net present value per share is the cost today plus the present value of the extra dividend of €1 forever:

N P V = - 10 + \frac{1}{0.10} = - 10 + 10 = 0

Zero: the investment earns exactly 10%, the same as the shareholders could earn themselves. It neither creates nor destroys value, and the share price does not change.

Case 2: growth through reinvestment

Most companies keep part of their profits and invest them. Two numbers describe this:

b, the plowback ratio (retention ratio): the fraction of profits reinvested. The fraction paid out, 1 − b, is the payout ratio.
r, the return on reinvested earnings: what the reinvested euros earn, per year, inside the company.

Worked example (from the slides): AZ reinvests 40% at 15%

AZ can invest b = 40% of its annual profits at r = 15%. Today: EPS₀ = €10.

DIV₀ = (1 − b) × EPS₀ = 0.60 × 10 = €6; reinvestment = b × EPS₀ = 0.40 × 10 = €4 per share.
In 1 year the €4 earns 15%, so profit rises by 0.15 × 4 = 0.6: EPS₁ = 10 + 0.6 = €10.60. DIV₁ = 0.60 × 10.60 = €6.36; reinvestment = 0.40 × 10.60 = €4.24.
In 2 years: EPS₂ = 10.60 + 0.15 × 4.24 = 10.60 + 0.636 = €11.236. DIV₂ = 0.60 × 11.236 = €6.7416; reinvestment = €4.4944.

Look at the pattern. Each year, EPS grows by the factor 1 + r × b = 1 + 0.15 × 0.40 = 1.06, and so does the dividend: 6 → 6.36 → 6.7416 is 6% growth each year. In formulas:

D I V_{1} = (1 - b) E P S_{1} = (1 - b) E P S_{0} [1 + r b] = D I V_{0} [1 + r b], D I V_{2} = D I V_{0} [1 + r b]^{2}

So the dividends grow at a constant rate:

g = r \times b

The growth rate of dividends is the return on reinvested earnings times the fraction reinvested. Reinvesting more (higher b) or more profitably (higher r) means faster growth, but reinvesting more also means a smaller dividend today.

The Gordon growth model

For a company whose dividends grow at a steady rate g forever, the DDM becomes a growing perpetuity (lecture 2) and the price is

P_{0} = \frac{D I V _{1}}{k - g}

This is the Gordon growth model. Note that it uses DIV₁, next year's dividend. If you are given today's dividend DIV₀, first compute DIV₁ = DIV₀ × (1 + g). The model only works when g < k.

Solving the same equation for k gives the required return implied by a price:

k = \frac{D I V _{1}}{P _{0}} + g

The expected return is the expected dividend yield DIV₁/P₀ plus the growth rate (which is also the expected capital gain, since price grows with dividends).

Worked example (from the slides): ABC

The price of ABC is €36 and DIV₁ = €1.87. Expected dividend yield = 1.87 / 36 = 0.052 = 5.2%. If g = 4.1%, the required rate of return is

k = 5.2% + 4.1% = 9.3%

Growth stocks versus income stocks. A growth stock has g > DIV₁/P₀: most of its return comes from growth (and so from capital gains). An income stock has g < DIV₁/P₀: most of its return is the dividend. ABC, with 4.1% growth and a 5.2% dividend yield, is an income stock.

The two-stage growth model

A young company may grow fast for some years and then settle to a normal rate. The Gordon model cannot handle that directly, but the DDM can: discount the dividends of the fast-growth years one by one, and apply the Gordon model to everything after that.

Worked example (from the slides): AB

The last dividend of AB was €2 (DIV₀). You expect g = 8% up to and including t+7 and g = 4% from t+8 on. k = 10%.

Step 1: the dividends of years 1 to 7 grow at 8%: DIV_t = 2 × 1.08^t, so DIV₁ = 2.16, DIV₂ = 2.3328, ..., DIV₇ = 3.4276.

Step 2: the price at the end of year 7 is the Gordon value of the dividends from year 8 on. DIV₈ = 3.4276 × 1.04 = 3.5648, so

P_{7} = \frac{D I V _{8}}{k - g} = \frac{3.5648}{0.10 - 0.04} = C = 59.41

Step 3: discount the seven dividends and P₇ to today:

P_{0} = \frac{2.16}{1.10} + \frac{2.3328}{1.1 0 ^{2}} + \dots + \frac{3.4276}{1.1 0 ^{7}} + \frac{59.41}{1.1 0 ^{7}} = C = 43.51

Most of the value (59.41 / 1.10⁷ = 30.49) comes from the terminal price, which is typical: the Gordon part dominates.

Present value of growth opportunities (PVGO)

Split the value of a growing company into two parts: what it would be worth if it stopped growing (paid out all earnings, a static company) plus the value added by its investments:

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

The present value of growth opportunities is the part of the price that reflects future profitable investments. It is positive only when the reinvested earnings earn more than k (r > k); if r = k, as in the first AZ example, PVGO = 0; if r < k, reinvesting destroys value and PVGO is negative.

Worked example (from the slides)

k = 15%, DIV₀ = €4.545, g = 10%, EPS₀ = €7.573 and r = 25%. Theoretical price and PVGO?

DIV₁ = 4.545 × 1.10 = €5; EPS₁ = 7.573 × 1.10 = €8.33.
Gordon: P₀ = 5 / (0.15 − 0.10) = €100.
Without growth from next year on: P₀ = EPS₁ / k = 8.33 / 0.15 = €55.56.
PVGO = 100 − 55.56 = €44.44: 44% of the price is the value of future investments.

Check with g = r × b: b = g / r = 0.10 / 0.25 = 0.40, and indeed DIV₁/EPS₁ = 5 / 8.33 = 0.60 = 1 − b. Because r = 25% is well above k = 15%, growth is valuable.

Using the models: theoretical price versus current price

Dividend discount models estimate the theoretical price (TP) of a stock, which we compare with the current price (CP) on the exchange:

TP > CP: the stock is undervalued: buy.
TP < CP: the stock is overvalued: sell.
TP = CP: the share is priced correctly.

Remember that the theoretical price depends entirely on your estimates of k and g. Small changes in g move the Gordon price a lot, because g sits in the denominator next to k.

The price-earnings ratio (P/E)

The P/E ratio is the current price divided by EPS: how many euros investors pay for one euro of annual profit. For a static company, P₀ = EPS/k, so

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

and k = EPS/P₀, the earnings yield. For a growth company, divide the PVGO equation by EPS₁:

\frac{P _{0}}{E P S _{1}} = \frac{1}{k} (1 + \frac{P V GO}{E P S _{1} / k})

The P/E is 1/k scaled up by the share of the no-growth value that growth adds. Interpretations:

A high P/E implies high growth prospects (large PVGO) or low past profits (a temporarily small EPS makes the ratio jump).
A high P/E also implies a low cost of capital k if the PVGO is zero.

But be careful: EPS may be too high or too low because of accounting tricks, and published P/E ratios use past profits, which do not account for future prospects. A P/E of 25 for a company whose profits will double is cheap; for a company in decline it is expensive.

Exercise from the slides: Commercial Bank, fully worked out

The price of Commercial Bank closed at €18.12. DIV₀ = €0.38 and the P/E ratio was 18.08.

a. Does the theoretical price match the current price if k = 8% and g = 5% forever?

DIV₁ = 0.38 × 1.05 = 0.399. Gordon:

T P = \frac{0.399}{0.08 - 0.05} = C = 13.30

TP = 13.30 < CP = 18.12: with these assumptions the stock is overvalued; the models says sell. Either the market expects more growth than 5%, or it requires less than 8%.

b. What steady growth rate is implied by the current price?

Set the Gordon price equal to 18.12 and solve for g:

18.12 = \frac{0.38 ( 1 + g )}{0.08 - g} ⟹ 1.4496 - 18.12 g = 0.38 + 0.38 g ⟹ g = \frac{1.0696}{18.50} = 5.78%

The market price is consistent with dividend growth of about 5.8% a year forever (at k = 8%).

c. With g = 5.8%, how much of the price is due to the PVGO?

EPS₀ = price / (P/E) = 18.12 / 18.08 = €1.0022. EPS₁ = 1.0022 × 1.058 = €1.0603. No-growth value: EPS₁ / k = 1.0603 / 0.08 = €13.25.

P V GO = 18.12 - 13.25 = C = 4.87

About 27% of the share price (4.87 / 18.12) is the value of growth opportunities.

d. With g = 5.8%, what is the implied return on reinvested earnings?

g = r × b, so r = g / b. The plowback ratio is b = 1 − DIV₀/EPS₀ = 1 − 0.38 / 1.0022 = 0.6208 (the bank retains 62% of profits).

r = \frac{0.058}{0.6208} = 9.34%

Reinvested earnings earn about 9.3%, above k = 8%, which is why the PVGO in (c) is positive.

e. If g = 5% for the next 4 years and 3% thereafter, what is the theoretical price?

Two-stage model. Dividends: DIV₁ = 0.399, DIV₂ = 0.419, DIV₃ = 0.4399, DIV₄ = 0.4619. From year 5, growth is 3%: DIV₅ = 0.4619 × 1.03 = 0.4757, so the price at the end of year 4 is

P_{4} = \frac{0.4757}{0.08 - 0.03} = C = 9.515

P_{0} = \frac{0.399}{1.08} + \frac{0.419}{1.0 8 ^{2}} + \frac{0.4399}{1.0 8 ^{3}} + \frac{0.4619 + 9.515}{1.0 8 ^{4}} = 0.369 + 0.359 + 0.349 + 7.334 = C = 8.41

Far below €18.12: at 3% long-run growth the stock looks heavily overvalued. Compare with (a): lowering the long-run growth from 5% to 3% cuts the theoretical price from 13.30 to 8.41. The models are very sensitive to g.

f. A €200 million investment with €28 million a year for 10 years: effect on the share price?

The investment's value is its NPV. The inflows are an annuity of 28 for 10 years at k = 8%:

P V = \frac{28}{0.08} (1 - \frac{1}{1.0 8 ^{10}}) = 28 \times 6.7101 = C = 187.88 million

N P V = 187.88 - 200 = - C = 12.12 million

The project destroys €12.12 million of value. Spread over 85.9 million shares, the share price should fall by about €0.14 (12.12 / 85.9 = 0.141), from 18.12 to roughly 17.98. A project with negative NPV lowers the PVGO.

In plain words

Lecture 3 in one breath: a share is worth the present value of its dividends. Constant dividends: P = EPS/k. Dividends growing at g = r × b: P = DIV₁/(k − g), and k = DIV₁/P + g. The part of the price beyond EPS₁/k is the PVGO. Compare the theoretical price with the market price to decide buy or sell, and remember that everything hinges on your estimate of g.

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