Lecture_13-14 Dividend Discount Model
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Professor Sang Byung [email protected]
The dividend-discount model
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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
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Today
•Finish the CAPM lecture• Application to capital budgeting
• NPV rule revisited
•Company beta vs project beta
• Dividend-discount model
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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
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PV methods to value equity
• 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
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PV methods to value equity
• If we continue this,
(1)
1 1
1
(1)
1
1 ⋯
1
1
• Under reasonable conditions for
(1) 1 1 ⋯ 1 ⋯
=
∞
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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.
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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.
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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?
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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
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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
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Constant dividend growth
• Dividend growth rate: g
• • 1 •
1
• and so on
(1) (1)
1 1
1 ⋯ 1 −
1 ⋯
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Constant dividend growth
• 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.
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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?
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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%
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Constant dividend growth
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.
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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.
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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
•
…
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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 + ⋯
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Differential growth
• We will start with phase I• We use the growing annuity formula:
• If ≠
ℎ 1 1 1
• If
ℎ 1
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Differential growth
• Phase 2
• N-year delayed growing perpetuity formula!
ℎ
1
1
(1)
• Substituting in for 1 −
ℎ 1 −(1)1 ( )
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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?
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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
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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.
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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
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Determining dividend growth
• 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.
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Determining dividend growth
• Return on equity (ROE) is defined as:
• Let’s assume that the ROE is constant
• g = earnings (or dividends) growth rate
• Finding:
×
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Determining dividend growth
• By definition,
+ +
• Then multiply both sides by ROE:
()(+) ()() +
• Divide both sides by +
1
: Earnings at year t
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Determining dividend growth
+
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 .
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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 ()
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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.
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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.