Chapter 2 Discounted Dividend Valuation. Challenges Defining and forecasting CF’s Estimating appropriate discount rate.

Chapter 2 Discounted Dividend

Valuation

Challenges

Defining and forecasting CF’s Estimating appropriate discount rate

Basic DCF model

An asset’s value is the present value of its (expected) future cash flows

10 )1(t

tt

r

CFV

Comments on basic DCF model

Flat term structure of discount rates versus differing discount rates for different time horizons

Value of an asset at any point in time is always the PV of subsequent cash flows discounted back to that point in time.

Three alternative definitions of cash flow

Dividend discount model Free cash flow model Residual income model

Dividend discount model

The DDM defines cash flows as dividends.

Why? An investor who buys and holds a share of stock receives cash flows only in the form of dividends

Problems: Companies that do not pay dividends. No clear relationship between dividends

and profitability

DDM (continued)

The DDM is most suitable when: the company is dividend-paying the board of directors has a dividend policy

that has an understandable relationship to profitability

the investor has a non-control perspective.

Free cash flow

Free cash flow to the firm (FCFF) is cash flow from operations minus capital expenditures

Free cash flow to equity (FCFE) is cash flow from operations minus capital expenditures minus net payments to debtholders (interest and principal)

Free cash flow

FCFF is a pre-debt cash flow concept FCFE is a post-debt cash flow concept FCFE can be viewed as measuring what

a company can afford to pay out in dividends

FCF valuation is appropriate for investors who want to take a control perspective

FCF valuation

PV of FCFF is the total value of the company. Value of equity is PV of FCFF minus the market value of outstanding debt.

PV of FCFE is the value of equity. Discount rate for FCFF is the WACC.

Discount rate for FCFE is the cost of equity (required rate of return for equity).

FCF (continued)

FCF valuation is most suitable when: the company is not dividend-paying. the company is dividend paying but

dividends significantly differ from FCFE. The company’s FCF’s align with company’s

profitability within a reasonable time horizon. the investor has a control perspective.

FCF valuation is very popular with analysts.

Residual income

RI for a given period is the earnings for that period in excess of the investors’ required return on beginning-of-period investment.

RI focuses on profitability in relation to opportunity costs.

A stock’s value is the book value per share plus the present value of expected future residual earnings

Residual income (continued)

RI valuation is most suitable when: the company is not dividend-paying, or as an

alternative to the FCF model. the company’s FCF is negative within a comfortable

time horizon. the investor has a control perspective.

RI valuation is also popular. The quality of accounting disclosure can make the use of RI valuation error-prone.

Which is best, DDM, FCF, or RI?

One model may be more suitable for a particular application.

Analyst may have more expertise with one model.

Availability of information. In practice, skill in application, including

the quality of forecasts, is decisive for the usefulness of an analyst’s work.

Discount rate determination

Jargon Discount rate: any rate used in finding the

present value of a future cash flow Risk premium: compensation for risk,

measured relative to the risk-free rate Required rate of return: minimum return

required by investor to invest in an asset Cost of equity: required rate of return on

common stock

Discount rate determination

Weighted average cost of capital (WACC): the weighted average of the cost of equity, after-tax cost of debt, and cost of preferred stock

Two major approaches for cost of equity

Equilibrium models: Capital asset pricing model (CAPM) Arbitrage pricing theory (APT)

Bond yield plus risk premium method (BYPRP)

CAPM

Expected return is the risk-free rate plus a risk premium related to the asset’s beta:

E(Ri) = RF + i[E(RM) – RF]

The beta is i = Cov(Ri,RM)/Var(RM)

[E(RM) – RF] is the market risk premium or the equity risk premium

CAPM

What do we use for the risk-free rate of return? Choice is often a short-term rate such as the 30-day T-bill

rate or a long-term government bond rate. We usually match the duration of the bond rate with the

investment period, so we use the long-term government bond rate.

Risk-free rate must be coordinated with how the equity risk premium is calculated (i.e., both based on same bond maturity).

Equity risk premium

Historical estimates: Average difference between equity market returns and government debt returns. Choice between arithmetic mean return or geometric

mean return (see Table 2-2 p. 50)

Survivorship bias

ERP varies over time

ERP differs in different markets (see Table 2-3 p. 51)

Equity risk premium

Expectational method is forward looking instead of historical

One common estimate of this type: GGM equity risk premium estimate

= dividend yield on index based on year-ahead dividends

+ consensus long-term earnings growth rate

- current long-term government bond yield

Arbitrage Pricing Theory (APT)

CAPM adds a single risk premium to the risk-free rate. APT models add a set of risk premiums to the risk-free rate:

E(Ri) = RF + (Risk premium)1

+ (Risk premium)2 + … + (Risk premium)K

(Risk premium)i = (Factor sensitivity)i × (Factor risk premium)i

Arbitrage Pricing Theory (APT)

Factor sensitivity is asset’s sensitivity to a particular factor (holding all other factors constant)

Factor risk premium is the factor’s expected return in excess of the risk-free rate.

APT models

One popular model is the Fama-French three factor model using company-specific attributes: RMRF – return on equity index minus 30

day T-bills SMB (small minus big) – return on small

cap portfolio minus return on large cap portfolio

HML (high minus low) – return on high book-to-market portfolio minus return on low book-to-market portfolio

APT models

The Burmeister, Roll, and Ross (BIRR) model uses five macroeconomic factors Confidence risk Time-horizon risk Inflation risk Business-cycle risk Market timing risk

Using BIRR model

Use BIRR model to calculate required return on the S&P 500 (data in example 2-4, p 53)

The required return is:

r = 5.00% + (0.27×2.59%) – (0.56×0.66%) – (0.37×4.32%) + (1.71×1.40%) +(1.00×3.61%)

r = 9.74%

Sources of error in using models

Three sources of error in using CAPM or APT models: Model uncertainty – Is the model correct? Input uncertainty – Are the equity risk

premium or factor risk premiums and risk-free rate correct?

Uncertainty about current values of stock beta or factor sensitivities

BYPRP method

The bond yield plus risk premium method finds the cost of equity as:

BYPRP cost of equity= YTM on the company’s long-term debt

+ Risk premium

The typical risk premium added is 3-4 percent.

Build-up method

Cost of equity is the risk-free rate plus one or more risk premiums, one or more of which is usually subjective rather than theoretically sound.

For example, cost of equity is risk-free rate + equity risk premium +/- company-specific risk premium

BYPRP is an example of this. Buildup method sometimes used for stocks that are

not publicly traded.

Dividend discount models (DDMs)

Single-period DDM:

Rate of return for single-period DDM111

11

11

0)1()1()1( r

PD

r

P

r

DV

0

01

0

1

0

11 1P

PP

P

D

P

PDr

More DDMs

Two-period DDM:

Multiple-period DDM:222

11

22

22

11

0)1()1()1()1()1( r

PD

r

D

r

P

r

D

r

DV

nn

n

tt

t

r

P

r

DV

)1()1(10

nn

nn

r

P

r

D

r

DV

)1()1()1( 11

0

Indefinite HP DDM

For an indefinite holding period, the PV of future dividends is:

n

n

r

D

r

DV

)1()1( 11

0

.)1(1

0

tt

t

r

DV

Forecasting future dividends

Using stylized growth patterns Constant growth forever (the Gordon

growth model) Two-distinct stages of growth (the two-

stage growth model and the H model) Three distinct stages of growth (the three-

stage growth model)

Forecasting future dividends

Forecast dividends for a visible time horizon, and then handle the value of the remaining future dividends either by Assigning a stylized growth pattern to

dividends after the terminal point Estimate a stock price at the terminal point

using some method such as a multiple of forecasted book value or earnings per share

Gordon Growth Model

Assumes a stylized pattern of growth, specifically constant growth:

Dt = Dt-1(1+g)

Or

Dt = D0(1 + g)t

Gordon Growth Model

PV of dividend stream is:

Which can be simplified to:

n

n

r

gD

r

gD

r

gDV

)1(

)1(

)1(

)1(

)1(

)1( 02

200

0

gr

D

gr

gDV

100

)1(

Gordon growth model

Valuations are very sensitive to inputs. Assuming D1 = 0.83, the value of a stock is:

g = 3.45% g = 3.70% g = 3.95%

r = 5.95% $33.20 $36.89 $41.50

r = 6.20% $30.18 $33.20 $36.89

r = 6.45% $27.67 $30.18 $33.20

Other Gordon Growth issues

Generally, it is illogical to have a perpetual dividend growth rate that exceeds the growth rate of GDP

Perpetuity value (g = 0):

Negative growth rates are also acceptable in the model.

r

DV 1

0

Expected rate of return

The expected rate of return in the Gordon growth model is:

Implied growth rates can also be derived in the model.

gP

Dg

P

gDr

0

1

0

0 )1(

PV of growth opportunities

If a firm has growing earnings and dividends, it can be worth more than a non-growing firm:

Value of growth = Value of growing firm – Value of assets in place (no growth)

OR

PVGOr

EV 0

Gordon Model & P/E ratios

If E is next year’s earnings (leading P/E):

If E is this year’s earnings (trailing P/E):gr

b

gr

ED

E

P

)1(/ 11

1

0

gr

gb

gr

EgD

E

P

)1)(1(/)1( 00

0

0

Strengths of Gordon growth model

Good for valuing stable-growth, dividend-paying companies

Good for valuing indexes Simplicity and clarity, also helps

understanding of relationships between V, r, g, and D

Can be used as a component in more complex models

Weaknesses of Gordon growth model

Calculated values are very sensitive to assumed values of g and r

Is not applicable to non-dividend-paying stocks

Is not applicable to unstable-growth, dividend paying stocks

Two-stage DDM

The two-stage DDM is based on the multiple-period model:

Assume the first n dividends grow at gS and dividends then grow at gL. The first n dividends are:

nn

n

tt

t

r

P

r

DV

)1()1(10

tSt gDD )1(0

Two-stage DDM (cont)

Using Dn+1, the value of the stock at t=n is

The value at t = 0 is

0 (1 ) (1 )nS L

nL

D g gP

r g

0 00

1

(1 ) (1 ) (1 )

(1 ) (1 ) ( )

t nnS S Lt n

t L

D g D g gP

r r r g

Two-stage DDM example

Assume the following values D0 is $1.00

gS is 30% Supernormal growth continues for 6 years gL is 6% The required rate of return is 12%

Two-stage DDM example

Time

Value

Calculation

Dt or Vt

Present Values Dt/(1.12)t or Vt/(1.12)t

1 D1 1.00(1.30) 1.30 1.161 2 D2 1.00(1.30)2 1.69 1.347 3 D3 1.00(1.30)3 2.197 1.564 4 D4 1.00(1.30)4 2.856 1.815 5 D5 1.00(1.30)5 3.713 2.107 6 D6 1.00(1.30)6 4.827 2.445 6 V6 1.00(1.30)6(1.06) / (0.12 – 0.06) 85.273 43.202 Total 53.641

“Shortcut” two-stage DDM (not in the book)

If gS is constant during stage 1, this works:

For gS=30%, gL=6%, D0=1.00 and r=12%

0 00

(1 ) (1 ) (1 ) (1 )1

(1 ) (1 ) ( )

n nS S S L

n nS L

D g g D g gP

r g r r r g

)06.012.0()12.1(

)06.1()30.1(00.1

)12.1(

)30.1(1

30.012.0

)30.1(00.16

6

6

6

0

V

64.53202.42439.10)12.1(

274.854454.1222.7

60 V

Using a P/E for terminal value

The terminal value at the beginning of the second stage was found above with a Gordon growth model, assuming a long-term sustainable growth rate.

The terminal value can also be found using another method to estimate the terminal value at t = n. You can also use a P/E ratio, applied to estimated earnings at t = n.

Using a P/E for terminal value

For DuPont, assume D0 = 1.40

gS = 9.3% for four years

Payout ratio = 40% r = 11.5% Trailing P/E for t = 4 is 11.0

Forecasted EPS for year 4 is E4 = 1.40(1.093)4 / 0.40 = 1.9981 = 4.9952

Using a P/E for terminal value

Time Value Calculation Dt or Vt Present Values

Dt/(1.115)t or Vt/(1.115)t 1 D1 1.40(1.093)1 1.5302 1.3724 2 D2 1.40(1.093)2 1.6725 1.3453 3 D3 1.40(1.093)3 1.8281 1.3188 4 D4 1.40(1.093)4 1.9981 1.2927 4 V4 11 [1.40(1.093)4 / 0.40]

= 11 [1.9981 / 0.40] = 11 4.9952 54.9472 35.5505

Total 40.88

Valuing a non-dividend paying stock

This can be viewed as a special case of the two-stage DDM where the dividend in stage one is zero:

Forecasting the length of stage one and the dividend pattern in stage two are the challenges.n

nn

tt

t

r

P

r

DV

)1()1(10

The H model

The basic two-stage model assumes a constant, extraordinary rate for the super-normal growth period that is followed by a constant, normal growth rate thereafter.

The H model

Fuller and Hsia (1984) developed a variant of the two-stage model where the growth rate begins at a high rate and declines linearly throughout the super-normal growth period until it reaches the normal growth rate at the end. The normal growth rate continues thereafter.

The H model

The value of the dividend stream in the H model is:

V0 = value per share at time zero D0 = current dividend r = required rate of return on equity H = half-life of the high growth period (i.e., high growth period = 2H

years) gS = initial short-term dividend growth rate gL = normal long-term dividend growth rate after year 2H

0 00

(1 ) ( )L S L

L L

D g D H g gP

r g r g

H model example

For Siemans AG, the inputs are: Current dividend is €1.00. The dividend growth rate is 29.28%, declining linearly over a

sixteen year period to a final and perpetual growth rate of 7.26%.

The risk-free rate is 5.34%, the market risk premium is 5.32%, and the Siemens beta, estimated against the DAX index, is 1.37.

The required rate of return for Siemens is:

r = rf + bi(rm – rf) = 5.34% + 1.37(5.32%) = 12.63%.

H model example

Using the H model, the value of the company is:

V0 = 19.97 + 32.80 = €52.77. If Siemens experienced normal growth

starting now, its value would be €19.97. The extraordinary growth adds €32.80 to its value, which results in Siemens being worth a total of €52.77.

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0726.01263.0

)0726.1(00.1)()1( 000

L

LS

L

L

gr

ggHD

gr

gDV

Three-stage DDM

There are two popular version of the three-stage DDM The first version is like the two-stage model, only

the firm is assumed to have a constant dividend growth rate in each of the three stages.

A second version of the three-stage DDM combines the two-stage DDM and the H model. In the first stage, dividends grow at a high, constant (supernormal) rate for the whole period. In the second stage, dividends decline linearly as they do in the H model. Finally, in stage three, dividends grow at a sustainable, constant rate.

Three-stage DDM with three distinct stages

Assume the following for IBM: Required rate of return is 12% Current dividend is $0.55 Growth rate and duration for phase one

are 7.5% for two years Growth rate and duration for phase two are

13.5% for the next four years Growth rate in phase four is 11.25%

forever

Three-stage DDM with three distinct stages

Time

Value

Calculation

Dt or Vt

Present values Dt/(1.12)t or

Vt/(1.12)t 1 D1 0.55(1.075) 0.5913 0.5279 2 D2 0.55(1.075)2 0.6356 0.5067 3 D3 0.55(1.075)2(1.135) 0.7214 0.5135 4 D4 0.55(1.075)2(1.135)2 0.8188 0.5204 5 D5 0.55(1.075)2(1.135)3 0.9293 0.5273 6 D6 0.55(1.075)2(1.135)4 1.0548 0.5344 6 V6 0.55(1.075)2(1.135)4(1.1125)/(.12 – .1125) 156.4620 79.2685 Total 82.3897

Spreadsheet modeling

Spreadsheets allow the analyst to build very complicated models that would be very cumbersome to describe using algebra.

Built-in functions such as those to find rates of return use algorithms to get a numerical answer when a mathematical solution would be

impossible or extremely complicated.

Spreadsheet modeling

Because of their widespread use, several analysts can work together or exchange information through the sharing of their spreadsheet models.

Finding rates of return for any DDM

For a one-period DDM

For the Gordon model

For the H-model

0

01

0

1

0

11 1P

PP

P

D

P

PDr

gP

Dg

P

gDr

0

1

0

0 )1(

0

0

((1 ) ( ))L S L L

Dr g H g g g

P

Finding rates of return for any DDM

For multi-stage models and spreadsheet models it can be more difficult to find a single equation for the rate of return. Trial and error is used instead of an

equation. Using a computer or trial and error, the

analyst finds a discount rate such that the present value of future expected dividends equals the current stock price.

Finding r with trial & error

Johnson & Johnson’s current dividend of $.70 to grow by 14.5 percent for six years and then grow by 8 percent into perpetuity. J&J’s current price is $53.28. What is the expected return on an investment in J&J’s stock?

Finding r with trial & error

For a good initial guess, we can use the expected rate of return formula from the Gordon model as a first approximation: r = ($0.70 1.145)/$53.28 + 8% = 9.50%. Since we know that the growth rate in the first six years is more than 8 percent, the estimated rate of return must be above 9.5 percent.

Let’s use 9.5 percent and 10.0 percent to calculate the implied price.

Finding r with trial & error

The present value of the terminal value

= V6 / (1+r)6 = [D7/(r-g)]/(1+r)6 The calculations for 9.5% and 10.0% are shown in the

table. Actual r is 9.988%.

Time t Dt Present Value of Dt and V6

at r = 9.5% Present Value of Dt and V6

at r = 10.0% 1 $0.8015 $0.7320 $0.7286 2 $0.9177 $0.7654 $0.7584 3 $1.0508 $0.8003 $0.7895 4 $1.2032 $0.8369 $0.8218 5 $1.3776 $0.8751 $0.8554 6 $1.5774 $0.9151 $0.8904 7 $1.7035 6 $65.8838 $48.0805 Total $70.8085 $52.9245

Strengths of multistage DDMs

Can accommodate a variety of patterns of future dividend streams.

Even though they may not replicate the future dividends exactly, they can be a useful approximation.

The expected rates of return can be imputed by finding the discount rate that equates the present value of the dividend stream to the current stock price.

Strengths of multistage DDMs

Because of the variety of DDMs available, the analyst is both enabled and compelled to evaluate carefully the assumptions about the stock under examination.

Spreadsheets are widely available, allowing the analyst to construct and solve an almost limitless number of models.

Strengths of multistage DDMs

Using a model forces the analyst to specify assumptions (rather than simply using subjective assessments). This allows analysts to use common assumptions, to understand the reasons for differing valuations when they occur, and to react to changing market conditions in a systematic manner.

Weaknesses of multistage DDMs

Garbage in, garbage out. If the inputs are not economically meaningful, the outputs from the model will be of questionable value.

Analysts sometimes employ models that they do not understand fully.

Valuations are very sensitive to the inputs to the models.

Weaknesses of multistage DDMs

Subjective assessments may be better than systematic, quantitative assessments in some cases.

Programming and data errors in spreadsheet models are very common. These models must be checked very thoroughly.

Weaknesses of multistage DDMs

The choice of model should be made very carefully. There is a tendency to grab a model, put in the data, get the results, and use them without carefully justifying the logic of the underlying model and the appropriateness and realism of the values inserted into the model.

Equity durations(not in the book)

Duration is a measure of the interest rate risk of fixed income securities. The concept of duration can also be adapted to equities.

The mathematical definition of duration is

The percentage change in price is dr

PdP

dr

dP

PD

/1

drDPdP /

Gordon model duration(not in the book)

The stock price is

The derivative with respect to r is

The duration is

gr

cP

1

21

)( gr

c

dr

dP

grD

1

Forecasting growth rates

There are three basic methods for forecasting growth rates: Using analyst forecasts Using historical rates (use historical

dividend growth rate or use a statistical forecasting model based on historical data)

Using company and industry fundamentals

Finding g

The simplest model of the dividend growth rate is: g = b x ROE where g = Dividend growth rate b = Earnings retention rate (1 – payout ratio) ROE = Return on equity.

Finding g

The ROE, found with the duPont model is:

The growth rate can also be expressed as:

'

Net Income Sales AverageTotal AssetsROE

Sales AverageTotal Assets Average Stockholders Equity

Net income Dividends Net income Sales Assets

Net income Sales Assets Shareholders' equityg

-

DDMs and portfolio selection

Investment managers have used DCF models, including dividend discount models as part of a systematic approach to security selection and portfolio formation.

If a manager just chooses the most undervalued securities without any risk discipline, his selections might concentrate on a particular (or a few) risk factors. He might often fail to meet his risk objective. A risk control discipline must be used.

DDMs and portfolio selection

Sort stocks into groups according to the risk control methodology. For example, put stocks into groups of similar beta risk.

Rank stocks by expected return within each group using a DCF methodology. Rank stocks from highest to lowest expected return within each sector grouping.

DDMs and portfolio selection

Select portfolio from the highest expected return stocks consistent with the risk control methodology. All selected securities are equal weighted, but more important sectors have a larger number of securities; the result is approximate sector neutrality.

DDMs and portfolio selection

Six analysts follow a universe of 250 stocks. Company uses a three-stage, H-model. For each, an analyst estimates 1) the initial growth

rate, 2) the length of the initial phase, and 3) the length of the transitional phase.

Initial growth rate estimated with duPont model.

DDMs and portfolio selection

Length of initial phase and transitional phase based on fundamental analysis.

Growth rate for maturity phase assumed to be the same for all stocks.

Stocks assigned to five beta quintiles.

DDMs and portfolio selection

Company invests in the top return quintile in each beta quintile (10 stocks in each beta quintile).

Method had superior returns for several years.