lecture_13-14 dividend discount model

8/17/2019 Lecture_13-14 Dividend Discount Model

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Professor Sang Byung [email protected]

The dividend-discount model



Last class

• Population statistics vs Sample statistics

• The Capital Asset Pricing Model

• CAPM formula

• How to compute beta• Two methods

• What determines beta?

• Correlation with business cycle

• Operating leverage



Today

•Finish the CAPM lecture• Application to capital budgeting

• NPV rule revisited

•Company beta vs project beta

• Dividend-discount model



PV methods to value equity

• Equity

• A claim on the assets of a corporation

• Also know as common stock.

•

Return for holding a share comes in two forms:• Dividend

• Capital gain (or price appreciation)

• Mathematical notation• Dividend/share at year 1

• Current price/share at year 0

•

Price/share at year 1




• The PV relation implies

1

1 1

where r is the firm’s cost of capital calculated using CAPM.

• We could apply the same formula to and find

1

1

• If we combine them together,

(1)

1

1




• If we continue this,

(1)

1 1

1

(1)

1

1 ⋯

1

1

• Under reasonable conditions for

(1) 1 1 ⋯ 1 ⋯

=

∞



Dividend-discount model

=

∞

• This is a formula for the price given the (possiblyinfinite) stream of (expected) future dividends!

• Some remarks:

• This is price per share. But we can also value the entirefirm if given total dividends.

• “Dividends” can be any cash flow from the corporationto investors.

• If the company liquidates, the liquidation value is treated as onebig dividend.

• If another company buys the shares for cash, that is also a

“dividend” in this analysis.



Dividend-discount model

• Bottom line

• We value equities the same way we value any otherasset!

• We discount the future stream of cash flows.

• In principal, this sounds simple.

• However, for equities, we need to forecast futuredividends, which is very difficult.

• Why is this approach important?

• As we will learn, the principle of using multiples (suchas the P/E ratio) to value equity all comes down to

thinking of equity as a stream of future cash flows.



Constant dividends (zero growth)

• Suppose that a stock pays the same dividend

every year forever:

(1)

1 1 ⋯

1 ⋯

• What would be the price in this case?



Constant dividends (zero growth)

• Let

denote the plowback (or retention) ratio.

• The portion of earnings kept in the firm.

• Thus, (1) is paid out as dividends.

(1 )• Note that

1

• The price-earning (P/E) ratio:

1



If dividends are not constant

•If dividends are not constant, it can be hard toforecast dividends of every period going

forward.

• So we make some simplifying assumptions:

• Constant dividend growth

• Differential growth



Constant dividend growth

• Dividend growth rate: g

• • 1 •

1

• and so on

(1) (1)

1 1

1 ⋯ 1 −

1 ⋯




• Applying the growing perpetuity formula:

• This equation only makes sense if > . Otherwise, the sumdoes not converge.

• Written in terms of earnings:

(1 )

•

The P/E ratio equals:

1

• We can rewrite this equation so it tells us what the market

expects the growth rate to be.



Examples

•

Example 1• $3• 10%•

15%• What is the price of this stock?

• Example 2

• / $18• 11.6%• Plowbackratio 1/3•

What is the growth expectation embedded in this P/E ratio?



Answers

•Example 1

$3

0.150.1 $60

• Example 2

1 ⇔

(1)

• Using this equation,

0.116 118 1 1

3 0.078 7.8%




1 ⇔ 1

• We can use this formula in one of two ways:

1. We can take a growth estimate, which will tell us a P/E.

• This may be different from the P/E we see when we look at the

actual price and earnings.

2. We can use the actual P/E from the data, and use theformula to tell us what the market expects growth to be.



Differential growth

• Constant dividend growth

• To make the PV converge, we need to assume that > .

• You may say

• Plenty of companies grow at a rate larger than 15%.

• Often, you hear forecasts of 50% growth!

• But, not of 50% growth rates that last forever!

• It may be unrealistic to assume a constant growthrate forever for some companies.

• Companies typically grow quickly when they are young andmore slowly when they get old.



Differential growth

• A company grows for N years at

then at

.

• Phase I:

• • 1 •

1 • …

• 1 −

• Phase II:

• + 1 1 − 1 • + 1 1 − 1

• + 1 1 − 1

•

…



Differential growth

•It is easiest to value each of the phaseseparately and then find:

ℎ (ℎ ) where

ℎ 1

1 1 ⋯ 1 −

1

ℎ (1)1 +

1 1 +

1 1 + ⋯



Differential growth

• We will start with phase I• We use the growing annuity formula:

• If ≠

ℎ 1 1 1

• If

ℎ 1



Differential growth

• Phase 2

• N-year delayed growing perpetuity formula!

ℎ

1

1

(1)

• Substituting in for 1 −

ℎ 1 −(1)1 ( )



Example

•Consider a company whose dividends areexpected to grow at 18% for the next five years,

and 10% after that.

• The dividend next year is expected to be $2.3

and the discount rate is 15%.

• What should the price be?



Answer

•Note that• 2.3, g 18%,N 5, g 10%, r 15%

ℎ 2.30.150.18 1 1.181.15 10.54

ℎ 2.3 1.18 (1.1)

1.15 (0.150.10) 48.77

ℎ ℎ 58.31



Multiple phases

• This is a flexible method to value stocks.

• You can see how it would change if, say, there

were three periods rather than two.

• Regardless of the number of phases, we always

have the same three steps:

1. Identify growth stages.

2. Calculate the PV of each stage.

3. Sum together.



Determining dividend growth

• Where does g comes from?

• We will relate g to be the firm’s profitability.

• Here, we will assume funds for investment are

generated internally.

• Rather than from issuing additional stock or bonds.

• We have the following




• Equivalently,

where b is the plowback ratio ( /).

• This implies that

• To understand dividends,

•We need to understand earnings. (They have the samegrowth rates.)

• How are they determined?

• By the amount invested in the company multiplied by a

profitability measure.




• Return on equity (ROE) is defined as:

• Let’s assume that the ROE is constant

• g = earnings (or dividends) growth rate

• Finding:

×




• By definition,

+ +

• Then multiply both sides by ROE:

()(+) ()() +

• Divide both sides by +

1

: Earnings at year t




+

1

• The equation implies that the earnings grow at

the rate of ().

• Note that this implies that the dividends also

grow at the rate of ().

• To estimate growth rate, we need to find

and .



Constant dividend growth (revisited)

•

Recall the formula under constant growth

• Now that we’ve decomposed growth, we can

substitute in to arrive at the following:

(1) ()

• The price-earning ratio:

1 ()



Cash cow

• In the constant growth model,

• If nothing is plowed back into the company ( 0)• If We have zero growth and we return to the perpetuity formula:

• This company is known as a cash cow

• When , as in this case, it doesn’t matter for price whether earnings are kept in the firm or not.

• ROE: rate of return on $ invested inside the firm

• r: rate of return on $ invested outside the firm.

• There’s no investment – the firm is just being milked for itscash.



Effects of plowback ratio

(1) ()• There are two effects of on price:

• ↑ ⟹ ↓ because the firm pays less cash out (numerator)

• ↑ ⟹ ↑ because of higher growth (denominator)

• Which effect wins?

()

• The denominator is always positive.

ff f



Effects of plowback ratio

()

• Three cases:

• > ⇒ value increases with increasing plowback

•

⇒increasing plowback has no effect on value

• < ⇒ value decreases with increasing plowback

• Intuition

• If > , the firm can use the money more productively.

• If < , investors can use the money more productively.

• Note again

• ROE: rate of return on $ invested inside the firm

• r: rate of return on $ invested outside the firm.

E l



Example

•

Suppose:• 12%, 10%, 0.6, $10

• Then the price is

10(10.6)0.120.6(0.1) $66.67

• As a CFO, how can you raise your stock price?

A



Answer

•

Set b=0 so nothing is retained!• Why? Because ROE is smaller than r.

10(10)

0.120(0.1) 10

0.12 $83.33

• Intuition?

• You are returning cash to shareholders so they can use it

more productively.

• This firm is also a takeover target:

• A raider can raise the price just by changing the amount

it pays out.

Wh t thi l i t h



What this analysis teaches us

•

This analysis teaches us not to confusecompany growth with higher value!

• At least in the constant growth framework, whether

earnings reinvestment raises value depends on whether

the discount rate r is below or above ROE.

• So growth can actually be bad and lower value!

• Only when growth comes in the form of positive NPV

investments does it increase value!

NPVGO



NPVGO

• Net present value of growth opportunities

• As discussed, growth does not necessarily lead tohigher value.

• Only when growth comes in the form of positive NPVinvestments does it increase value.

• There is a formula that makes this explicit:

where / is the cash cow value and is the NPV of growthopportunity.

• This equation is very general.

• It holds whether b and ROE are constant or not.

E l i l th t it



Example: single growth opportunity

• If a firm undertakes no investment, $10 (per share) in perpetuity.

• Now assume we have a single investment

opportunity (say, a marketing campaign) atyear 1.

• Cost: $10 per share at year 1

• Earnings will increase by $2.10 per share in all

subsequent periods.

• Assume that the discount rate is 10%.

St k i ith t G/O



Stock price without G/O

• Stock price if the firm does not take the growth

opportunity

$100.1 $100

0 1 2

……3

$10 $10 $100

4

$10

St k i ith G/O



Stock price with G/O

• NPVGO

101.1 2.11.1 2.11.1 2.11.1 ⋯

101.1 1

1.12.10.1 $10

• Stock price if the firm takes the growth opportunity

$100 $10 $110

0 1 2

……3

-$10 $2.1 $2.104

$2.1

L t’ if thi



Let’s verify this.

• Cash flows after taking the growth opportunity

• Year 1: invest $10 earnings in the growth opportunity. (so

there is no dividend at year 1.)

• From year 2, dividend amount = $12.1

• Stock price = delayed perpetuity!

1

(10.1)× 12.1

0.1 $110

0 1 2

……3

$0 $12.1 $12.10

4

$12.1

M ll



More generally

• More generally, suppose that

• At year 1, the firm pays $10(1) as dividends and

retain $10 for a new investment.

• Then, the firm’s earnings increase by $10 1 in

perpetuity.

10

1

1

1

10

0 1 2

…

…3

10()

4

10() 10()10

Wh NPVGO > 0



When NPVGO > 0

101

11

10 > 0

• Equivalently,

1

1

10

> 10

1 > • Positive NPV growth opportunities are those where >

• Note that

• It is possible to have growth and have it lower the firm value.

• This occurs when the firm takes a project with NPVGO<0.

NPVGO d t k i



NPVGO and stock price

• This equation is very general.

• For example, constant dividend growth model can be

viewed as taking the same growth opportunity every

year.

• This equation holds whether b and ROE are constant or

not.

NPVGO d P/E ti



NPVGO and P/E ratio

1

• P/E ratio tells us something about the firm’s

growth opportunities and discount rate:

•

If we have two firms of roughly the same risk level,• If one is priced more highly,

• P/E ratio tells us this pricing is due to the positive NPV

growth opportunities the firm will undertake.

lecture_13-14 dividend discount model

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