Equity Risk Premium

The Relationship Between the Equity Risk Premium, Duration

and Dividend Yield 1,2

Ruben D. CohenCorporate Finance - Structured Products

Citigroup, London E14 5LBUnited Kingdom

1 I express these views as an individual, not as a representative of the company with which I am connected.

2 http://rdcohen.50megs.com/papers.html. Also, in the December 2002 issue of Wilmott Magazine, pp.84-97.

2

Ockham's Razor:

"All other things being equal, the simplest solution tends to be the right one."

William of Ockham(14th century English philosopher and theologian)

3

A Snapshot of the S&P Composite

Source of the underlying data: Shiller’s website.

1.E-01

1.E+00

1.E+01

1.E+02

1.E+03

1.E+04

1870

1880

1890

1900

1910

1920

1930

1940

1950

1960

1970

1980

1990

2000

IndexEarningsDividends

The S&P Composite Data

The Great Depression

The 1970s bear market

The 1987 crash

4

Relevant Questions

1. Has the behaviour of the S&P market changed markedly and fundamentallyover the last 130 years ?

2. If yes, when and how did it happen?

5

Other Outstanding Issues

1. The dividend puzzle: why investor’s demand dividends and why firms pay it?

2. Has the S&P market ever followed, and/or is it now following, a certain type of dividend policy? If so, what is it?

3. Why has the dividend yield been declining over the past 50 years?

4. How could one describe the behaviour of the market’s risk premium – past, present and going forward?

5. Is there a relationship between the equity risk premium and dividends?

* The equity premium puzzle will not be addressed here.

*

6. Are all the above points related to each other?

6

A Short Introduction to Equity ValuationCalculating the “Expected” Return on Equity

1. Market’s total “expected” rate of return

)(

)(

)(

)()(

tS

t

tS

tStSR ff

M

δ+

−≡

RM(2000)

Forecast for δ(2003)

Forecast for S(2003)

S(2002)

0.28

4

60

50

Example

2. Fundamentals – Using “projected” IS & BSi. At the firm’s level, using forecasts of equity earningsii. At the investor’s level, using forecasts of dividends

7

A Short Introduction to Equity ValuationFundamental Calculation

Expected EBIT 20Interest (at 5%) 5EBT 15Tax (at 40%) 6Expected profit 9Dividends 4Retained earnings 5

ROE 18%

Assets 150Debt 100Equity 50Total Debt & Equity 150

D/E 2.00

Income Statement

Balance Sheet

Relevant Ratios

Earnings, Ef, at the firm’s level

Dividends, δf, are earnings at the investor’s level

8

A Short Introduction to Equity Valuation Summary

Three distinctive ways for computing the “expected” rate of return on equity (ROE):

[28%]

[18%]

RI(2000)

S(2002)

Forecast for δ(2003)

δ(2002)

22%

50

4

3.5

Example

)2(:modelgrowthsGordon' Iff R

S≡+

− δ

δ

δδ

)3(:earningsequity ofDCF Ff R

S

E≡

)1(:ROEexpectedsMarket' Mff R

SS

SS≡+

− δ

The Notion of Equilibrium* & Its Historical Significance

* This is not economic equilibrium, where supply equals demand.

10

Define: “Equilibrium”

FIM RRR == (4)

(6) leads to constant dividend yield.

(6)MI RR =

Thr

ee c

ompo

nent

sFM RR = (5)

(5) leads to the relationship: re-investment = growth.

IF RR = (7)

(7) leads to efficient transmission of information from the firm to the equity investor.

11

Historical Significance of Eq. 5

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

1880 1900 1920 1940 1960 1980

RM

RF

FM RR =

12

Historical Significance of Eq. 6

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

1880 1900 1920 1940 1960 1980

RM

RI

MI RR =

13

Historical Significance of Eq. 6Translated into Price & Dividends

0.4

0.6

0.8

1

1.2

1.4

1.6

1870 1880 1890 1900 1910 1920 1930 1940

δ(t+1)/δ(t)S(t)/S(t-1)

• Statistically significant 1:1 relationship, with a correlation coefficient of 0.73.

14

Historical Significance of Eq. 7

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

1880 1900 1920 1940 1960 1980

RFRI

IF RR =

15

Historical SignificanceSummary

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

1880 1900 1920 1940 1960 1980

RM

RF

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

1880 1900 1920 1940 1960 1980

RF

RI

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

1880 1900 1920 1940 1960 1980

RM

RI

Results:1. RM ≠ RF

2. Close to RM = RI from 1870 to 1940

3. RF = RI from 1950 to the present

16

Historical SignificanceConclusions

• The notion of re-investment = growth has never held!!

• From 1870 to 1940:

i.e. dividend payments have followed a policy of constant dividend-yield.

• From 1950 to the present:

i.e. dividends have provided information on future earnings.

)1()(

)()1(

−=

+tS

ttS

t δδ

)(/)(1)(/)1(1

)()1(

tSttStE

tt

δδδ

+++

=+

How is it typically done and how to overcome the common problems?

Approach to Equity Valuation

18

The Approach to Valuation

)3()()(

)(tR

tS

tEF

f ≡

• Recall: )2()()(

)(

)(

)()(tR

tS

t

t

ttI

ff ≡+− δ

δ

δδExpected

dividend

growth rate

)(

)()(

tkR

ttS

I

f

δ

δ

−=• which gives:

• And:

• which gives:)(

)()(

tR

tEtS

F

f=

“Gordon’s growth model”

“DCF model”

[ ])(/)(ln ttf δδ≈

19

The DCF & GGM Main Underlying Assumption

F

f

R

tEtS

)()(:ValuationDCF =

ON THE SIDE

( ) ( )...

111)(:from Originates 32 +

++

++

+=

F

f

F

f

F

f

R

E

R

E

R

EtS

where Ef & RF are assumed to remain constant through time.

Iff R

S≡+

− δ

δ

δδExactly the same applies also to GGM:

which can be written as:δ

δkR

ttS

I

f

−=

)()(

20

An Alternative Valuation Relation• Problems with using relations of the type give by the DCF & Gordon’s

growth model arise from an unknown discount rate and the form of the denominator in Gordon’s growth model.

• How to overcome these problems?

Use the dividend policy relation, which has been valid since 1950:

Written here in terms of expected dividends and earnings forecast.Now, solve for S(t) in terms of other parameters:

)(/)(1

)(/)(1

)(

)(

tSt

tStE

t

t ff

δδδ

++

=

[ ])(/)(ln

)()()(

tt

ttEtS

f

ff

δδ

δ−= Note: This is a

combination of GGM & DCF.

21

Alternative Valuation RelationValidity

-2

0

2

4

6

8

0 2 4 6 8

1870-19401950-1998

ln S (actual)

ln S

(im

plie

d)

1 9 5 0 - 1 9 9 8 C o e f . S t d E r r o r t S t a tαo - 0 . 1 6 0 0 . 4 1 7 - 0 . 3 8 4α1 0 . 9 7 4 0 . 0 8 4 1 1 . 6 6 0

1870-1940 C o e f . S t d E r r o r t S t a t

αo - 1 . 2 5 0 0 . 8 0 7 - 1 . 5 4 9

α1 0 . 9 9 6 0 . 3 7 4 2 . 6 6 4

[ ])(/)(ln

)()()(

tt

ttEtS

f

ff

δδ

δ−=

22

The Steady-State Equilibrium

23

... and ask the question:

What would happen if all uncertainties disappeared?*

* A strongly hypothetical situation, as unforeseen changes in virtually anything (weather, disease outbreaks, wars, famines, etc.) make this highly unlikely.

Consider the Loop ...

Economic & financial volatility

Interest-rate volatility

Natural and man-induced uncertainties

and volatility

24

Outcome

Volatility, economic, financial and otherwise, would all disappear.

Economic & financial growth at

constant rate

Constant interest rate

25

Other Consequences

– Yield curve will remain straight, horizontal and frozen in time.

– All arbitrage opportunities between different investment instruments are eliminated - i.e. stocks and bonds will yield at the same constant rate, equal to the interest rate.

– Also, GDP’s growth rate will remain constant, equal to the interest rate - i.e. the “Golden Rule of Economics.” The GDP, therefore, will grow exponentially with time.

26

PIFM RbRRR +===

Proposition 1: At b = b* = constant, RP = 0.

Two Propositions for Steady-state Equilibrium

Sb

S

SS ff δ−=

−*

Sb ff δ

δ

δδ−=

−*

By ourdefinition:

27

Proposition 2: Investor seeks to maximise both, the rates of growth in price and dividends.

Two Propositions for Steady-state Equilibrium - cont’d

0→∴S

fδ

i.e. Zero dividend yield satisfies both.

Note:Combination of the two is analogous to maximising the market’s rate of return, Equation 1.

28

End Result

*bS

SS f =−

*bf =−

δ

δδ

*bS

E f =

29

A Couple of Questions??

• The outcome so far is that in this hypothetical scenario of no changes, expected and unexpected, the investor demands no dividend yield.

• However, in the real world full of volatility and changes, markets do provide a finite dividend yield.

• Therefore,– (1) could the dividend yield reflect the premium paid to the

investor to compensate for risks inherent in uncertain markets? - Which leads to:

– (2) could the dividend yield be the risk premium?

30

The Falling Dividend Yield

31

The Falling Dividend Yield• Well-known fact that the dividend yield has been on a declining trend

since 1950, whereas prior to that it was relatively constant.

0%

1%

2%

3%

4%

5%

6%

7%

8%

9%

10%

1870

1880

1890

1900

1910

1920

1930

1940

1950

1960

1970

1980

1990

2000

TR

AN

SIT

ION

Historical dividend yieldTheoretical dividend yield

32

The Falling Dividend Yield• What is the cause? Common explanations:

– (i) cash back via share buybacks & – (ii) orientation to growth.

• It is proposed here that the current dividend policy is the predominant cause for this decline!

Take the dividend policy from1950 and write it in terms of forecasts:

– Juggle it and re-write it as:

)(/)(1

)(/)(1

)(

)(

tSt

tStE

t

t ff

δδδ

++

=

[ ] [ ] [ ][ ] [ ])(/)1()1(/)(1

)(/)1()1(/)()(/)(1

)(

)(

tStStSt

tStStSttStE

tS

t ff

−×−+−×−×+

=δ

δδ

33

The Falling Dividend YieldCont’d

[ ] [ ] [ ][ ] [ ])(/)1()1(/)(1

)(/)1()1(/)()(/)(1

)(

)(

tStStSt

tStStSttStE

tS

t ff

−×−+−×−×+

=δ

δδ

Proposition 3: The forward-looking investor values financial securities according to constant parameters, all going forward.

Recall:

*bS

E f =

*...)1(

)1()()(

)()1()(

)()(b

tStStS

tStStS

tS

tStS f ==−

−−=−+=−

34

0%

1%

2%

3%

4%

5%

6%

7%

8%

9%

10%

1870

1880

1890

1900

1910

1920

1930

1940

1950

1960

1970

1980

1990

2000

TR

AN

SIT

ION

Historical dividend yieldTheoretical dividend yield

Let b* equal 5%, ∆(0) be about 5.5% and solve the “difference equation”:

The Falling Dividend YieldCont’d

[ ])(1)(1

)1()( *

*

tbtb

ttf ∆++∆+

=+∆≡∆∴

35

Is the Equity Risk Premium

Related to the Dividend Yield?

36

Recall Again…

*)(

)()1(b

tStStS

=−+

In a forward-looking world:

ConstantWritten in terms of real time:

*)(

)()(b

tS

tStS f =−

37

Outcome & Extension

In differential calculus form:

which implies: ln S = b*t

or: ln S = ln S(b*,t)

Extend to variable b, b = b(t),

Which gives: ln S = ln S(b,t)

*

*constant

lnb

tS

bb

=

∂∂

==

38

Outcome & ExtensionSome Mathematical Manipulation

Total differential:

Therefore:

In difference form:

×

∂∂

+

∂∂

=dtdb

bS

tS

dtSd

tb

lnlnln

b Equityduration )(tb&

bDbdt

SdE

&−=ln

[ ])()1()()(

)()1(tbtbDtb

tStStS

E −+×−=−+

39

Outcome & ExtensionBehaviour (1950 – 2000)

[ ])()1()()(

)()1(tbtbDtb

tStStS

E −+×−=−−+

y = 2.8575x + 0.0255

-0.5

-0.3

-0.1

0.1

0.3

-0.07 -0.05 -0.03 -0.01 0.01 0.03 0.05

-[b(t+1)-b(t)]

ln[S

(t+1

)/S

(t)]

-b(t

)

1950 - 2000

40

Outcome & ExtensionFurther Manipulation

bDbdt

SdE

&−=lnStart with:

Recall:

Combine:

pM RbRtS

tdt

Sd+==

++

)()1(ln δ

[ ])()1()(

)1(tbtbD

tSt

R Ep −+−+

=δ

41

Outcome & ExtensionFrom a Forward-looking Point of View

Proposition 3: The forward-looking investor values financial securities according to constant parameters, all going forward.

)()(constant

tbtbb

f =⇒=∴

Recall:

)(

)(

tS

tR f

p

δ=

The forward-looking equity risk premium is the expected dividend yield!!

)]()([)(

)(tbtbD

tS

tR fE

fp −−=

δ

42

Summary & ConclusionsI. Related to “Equilibrium”

1. The combination of the three fundamental equations of valuation leads to our definition of equilibrium.

2. It consists of three components:A. Constant dividend yield policy.B. Efficient transfer of information from firm to investor.C. Re-investment equals growth.

3. In this context, the S&P’s performance over the past 130 years could be divided into two parts:

A. A period of constant dividend yield policy between 1870 & 1940.B. A period of information efficiency from 1950 to the present, where

firms convey information on expected profits via dividends.

43

Summary & ConclusionsII. Alternative Valuation Relation

1. The existing dividend policy leads to an alternative valuation relation.

2. The relation is long-term, but, nonetheless, it captures the basic features of price, given forecasts in earnings and dividends and, at the same time, it

A. bypasses the need for any type of discount rate and

B. avoids the problem with the denominator in Gordon’s Growth Model.

44

Summary & ConclusionsIII. The Dividend Puzzle

1. Investors demand a dividend yield because of uncertainties.

2. The market provides dividends to convey information on expected profits.

3. The decline in the dividend yield over the last few decades is aconsequence of the current, effective dividend yield. Therefore, as long as this policy is in place, the dividend yield should continue to fall.

45

Summary & ConclusionsIV. Relationship between the ERP & DY

1. The ex-post or historical equity risk premium is related to the dividend yield, duration and time-wise changes in the interest rate.

2. The ex-ante or forward-looking dividend yield is exactly equal to the expected dividend yield. This arises from our Proposition 3and is consistent with the outcome of Propositions 1 & 2, being that investors demand a dividend yield because of uncertainties.

3. Therefore, as long as the dividend yield is expected to decline,owing to the current dividend policy, the ex-ante equity risk premium should also continue to fall.