Investment Analysis II & Advanced Investment Analysis ...people.exeter.ac.uk/wl203/BEAM015/Materials/Lecture 1/NewIA2 lect… · Investment Analysis 2 1 Investment Analysis II & Advanced

Alan Gregory 1/31/2005

Investment Analysis 2 1

Investment Analysis II &

Advanced Investment

Analysis – Lecture 1

Alan Gregory

Professor of Corporate Finance, Centre

for Finance and Investment, University of

Exeter

Introduction to Investment Analysis II

Objectives:

1. Revision of the theory of value and the

basic dividend valuation models

2. The discount rate and the market risk

premium – the issues in deciding the rate to

use in the models



The basic valuation model:

Fundamental principle:

Value = PV of future cash flows

Several points follow:

1. This determines fundamental value not relative value

2. Depending on how uncertainty is captured, can price

any security

3. All valuation models used in practice are attempts to

operationalise this

4. The most basic approach is the PV of dividends model

The Discounted Dividend Model:

• Assume a constant discount rate, r, and

suppress firm specific subscripts for clarity

• Define all cash flows (cash dividends, buy-

backs, rights issues [-ve dividends] etc)

to/from shareholders as “dividends”:

( )∑∞

= +=

1

01t

t

t

r

DP



Notes

• Note that this assumes the share has just gone ex

dividend and the next dividend is paid in one period’s

time

• This is the fundamental pricing model and ultimately all

models must reconcile to this (Barker, Ch 2)

• This is because the model focuses directly on cash flows

to shareholders

Operationalising the DDM

If we assume constant growth, we get the well-

known dividend growth model:

(2)

see next slide for proof…..

)()(

)1( 100

gr

D

gr

gDP

−=

−+

=

)()(

)1( 100

gr

D

gr

gDP

−=

−+

=



Proof of DDM (1)

• The generalised form of the present value

calculation used above can be written as:

• This carries on into perpetuity, and so n

becomes very large

n

n

r

gD

r

gD

r

gDP

)1(

)1(...

)1(

)1(

)1(

)1( 0

2

2

000 +

+++

++

+++

=

Proof of DDM (2)

• Note that the terms on the right hand side are

growing at a constant rate; (1 + g)/(1 + r)

• formula for the sum of such a progression, is:

• Sum = First term ÷ (1 - Common ratio)

• The common ratio is (1 + g)/(1 + r), and the first

term is D0(1 + g)/(1 + r)

• Substituting this into the expression and re-

arranging gives the DDM in (2)



Practical implementation

(2) can be combined with (1) to allow specific dividend forecasts for n years:

• Last term is the forecast share price using a constant growth from the horizon forecast

• This then must be discounted back to year 0 and added to PV dividends

( ) n

nnt

tt

t

rgr

gD

r

DP

)1)((

)1(

11

0 +−+

++

= ∑=

=

What does this mean?

• Simply that the share is worth the sum of:

• The present value of the dividends for the specific

forecast period, +

• The present value of the estimated share price at the

end of the forecast period

• In practice, most analysts using the DDM use

some sort of variation on this basic theme

• E.G. forecast dividends for 5 years and then a

general growth beyond that



Some examples:

• A high tech growth; B average firm; C regulated

utility

• All have 8% nominal r, real growth around

2.25%, inflation 2.5%

• Dividend info:

Co Yr0 Yr1 Yr2 Yr3 Yr4 Yr5 Growth

A 0 0 0 0 20 30 7.00%

B 10 11 12 13 14 15 4.25%

C 10 10.25 10.51 10.77 11.04 11.31 2.50%

Prices:

C o P ric e +

P r ic e - P r ic e c h a ng e a s %

A 2 9 5 4 .8 1 7 7 8 .8 3 3 .1 1% -1 9 .8 7%

B 3 5 6 .1 3 1 6 .7 6 .2 7% -5 .4 8%

C 1 9 3 .6 1 7 9 .8 3 .8 6% -3 .5 3%

Co PV Divs Yr 5 price CV = PV yr 5

price Price now Current DY CV as % P

A 35.1 3210.0 2184.7 2219.8 0.00% 98.42%

B 51.3 417.0 283.8 335.1 2.98% 84.69%

C 42.9 210.9 143.5 186.4 5.37% 77.00%

Sensitivity to a 0.25% change in growth:

The closer g is to r, the greater the impact of any change

in g



Problems of Dividend Valuation models

• Some firms do not pay dividends

• Increasing tendency in US to use share buy-backs – only

around 20% of US firms pay dividends (but 70% of

S&P)

• “Dividend Irrelevance” (MM 1961)

• Model implies all surplus cash flows not invested in

risky assets are paid out as dividends – any cash

retentions invested in risk-free deposits must change r

• Works best with steady and predictable payout policies

But note:

1. neither of the first two imply the model is

wrong. Remember dividends come in many

forms

2. dividend irrelevance comes about because

new issues of equity can be made and “home

made” dividends can be substituted for

corporate dividends – but price still depends on

future cash flows to shareholders



Uses of the model, besides valuation:

• Calculating the implied dividend growth rate on

a share

• Deriving a cost of capital (e.g. in regulation)

• Deriving the implied risk premium on the

market or required return on equity – a

particularly powerful application

The required return on equity:

Suppose the dividend yield on the FTASI is 3.5%

Long run real GDP growth is 2.25%

Expected inflation = 2.5% p.a.

Then by re-arranging the DGM, we have

gP

gDr +

+=

0

0 )1(



So applying this to our example:

since a yield is the dividend per £ invested,

r = [(3.5*1.0225)/100] + .0225 = .0583 or 5.83%

In nominal terms this is equivalent to:

(1.0583 x 1.025) – 1 = 8.47%

An Historical Perspective

How does an expected real return of 5.8%

compare with history?

Gregory, 2002: Table 4.1 Historical total real returns on equities (i.e. capital gains plus dividends) for the UK, 1900-2000. Compiled from: The Millennium Book – A Century of Investment Returns;

Dimson, Marsh and Staunton (2000), Tables 41 and 42

Period Real geometric mean return

Real arithmetic mean return

1900-2000

5.9% 7.8%

1900-1950

3.0% n.a.

1950-2000

8.9% n.a.



Geometric or arithmetic average returns?

Highly contentious issue as difference is enormous

– 1.9% p.a.!

Arithmetic averages:

• can be viewed as independent drawings from a

stationary independent probability distribution

• would then give the correct estimate of ex ante

expected returns if the representative investor’s

holding period is 1 year.

Geometric or arithmetic average returns, cont’d…

• Suppose share prices can move up by 10.52%

each year or down by 9.52%

• Each is equally likely ex ante

• The implied returns are lognormally distributed

• What will possible outcomes be after 2 years?

• What can we conclude about required returns?



Return Patterns and Prices

100

110.52

90.48

81.87

122.14

100

Start price=£100

Actual outcome in red

What were the averages?

• The geometric average = nth root of the end price/start

price -1 , where n = no. years

• In this case √(100/100) –1 = 0%• This is the actual compound return earned

• The arithmetic average was 10.52% - 9.52% ÷ 2 = +

0.5%

• Now think about a firm initially valued at £1000,

offering similar annual returns in perpetuity (each year’s

cash flows are independent of one another)



Firm cash flows

• At Year 1 either +105.2 (p = 0.5) or -95.2 (p =

0.5) so expected flow = +£5

• And so on, in perpetuity

• Discounting at 0% (the geometric average

return) gives an infinite PV!

• Discounting at 0.5% (the arithmetic average)

gives a PV of inflows of 5/0.005 = £1000 – the

correct value

But:

• there is evidence of negative autocorrelation in

long run returns

•does the representative investor really have such a

short time horizon?

•what happens if this is, say, 5 or even 10 years?



Ten year historical HPRs for the UK

To / From 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990

1910 1.8

1920 -1.4

1930 9.3

1940 2.6

1950 3.1

1960 13.7

1970 6.5

1980 -1.4

1990 15.4

2000 11.2

Average 6.08

Gregory 2001: Table 4.2 Historical annualised real returns on equities (i.e.

capital gains plus dividends) for successive 10 year holding periods for the

UK, 1900-2000. Compiled from: The Millennium Book – A Century of Investment Returns; Dimson, Marsh and Staunton (2000), Table 42

Barclays Capital Estimates of equity returns, 1899-2001

Nominal Rm Real Rm

1-year return 11.35% 7.17%

5-year rolling return 10.08% 5.82%

10 year rolling return 9.94% 5.51%



So what conclusion does this lead us to?

• These figures of roughly 5.5 to 6.1% are far

closer to the geometric average (5.9%) than the

annual arithmetic average (AA)

• Note that this does not refute the arguments for

the AA, but highlights how critically AA

estimates depend upon the holding period

assumed.

• Finally, note that a forward estimate from the

DGM is equivalent to a geometric average

So what is the expected cost of equity implied by the DDM?

• 5.83% is the geometric average – technically

should use arithmetic averages

• The difference between arithmetic and

geometric averages is approx half the variance

• Variance (SD squared) depends on holding

period

• From Barclays Capital data, 1 yr, 5 yr and 10 yr

SDs are:20.23%; 8.11%; 5.48%.



Converting geometric averages to arithmetic averages:

• The variance of the 1 year return is .20232



• So, if we believe the 5-year holding period is

representative, the adjustment is:

• +0.5 x .08112 = +0.3%

• This would imply an expected return of 5.8 +

0.3 = 6.1% for discounting/valuation purposes

The CAPM and the risk premium

• One approach is to use the CAPM to set r

• Frequently used by analysts and in investment appraisal

• CAPM assumes all assets earn a fair return in

equilibrium

• i.e. Excess return = ß x market risk premium

• But in this CAPM world all assets are fairly priced

(market efficiency)

• Same is true for any equilibrium based asset pricing

model (e.g. APT)



Estimating the Risk Free Rate and the Risk premium

• CAPM is a single period model, so use in valuation is

always a compromise involving assumptions

• One approach is sequential estimation of implied future

periodic returns from yield curve with a constant[?] risk

premium

• Alternatively, use a risk-free rate of equivalent duration

to asset being valued

• Problem is in low inflation, low return environments,

duration is long

• Closest match appears to be long index-linked gilt yield

The market risk premium

The historical risk premium for the last 100 years:

Period Total real return on equities

Total real return on gilts

Mean risk premium over gilts

Mean risk premium over Treasury Bills

1900-2000 5.9% 1.3% 4.6% 4.9% 1900-1950 3.0% 0.9% 2.1% 2.4% 1950-2000 8.9% 1.6% 7.3% 7.6% 1960-2000 7.7% 2.6% 5.1% 5.8% Gregory, 2001: Table 4.4 Geometric mean returns and premia for the UK Compiled from: The Millennium Book – A Century of Investment Returns;

Dimson, Marsh and Staunton (2000), Table 42

Note these are geometric premia



Or, applying the arguments on HPRs above and looking at gilts:

So taking Table 4.2 equity returns gives an historical premium of: 6.08 – 1.41 = 4.7%

To / From 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990

1910 -0.2

1920 -9.2

1930 8.3

1940 5.9

1950 0.7

1960 -2.3

1970 -1.5

1980 -4.4

1990 7.5

2000 9.3

Average 1.41

Gregory 2001: Table 4.5 Ten year holding period annualised real returns for UK gilts Compiled from: The Millennium Book – A Century of Investment

Returns; Dimson, Marsh and Staunton (2000), Table 42

The risk premium: how big is it?

• Recently, Fama and French (2000) run the ex antemodel historically for the US

• For 1872-1950, historical averages and DGM produce similar estimates

• These diverge from about 1950

• DGM premium estimate is 3.4% c.f. 8.28% realised

• FF conclude that the divergence is due to capital gains resulting from low expected future returns – (irrational exuberance [Shiller] or a realisation of the “equity risk premium” puzzle?) – their 1999 premium estimate is 1.32%!

• The then UK equivalent was about 2.5%



The problem

• Particularly acute for the US, where DGM estimates and

historical estimates diverge

• Following recent falls in the UK, divergence less of a

problem

• What seems certain is that future equity returns will be

lower than those experienced in the past few decades

• A problem in all valuation models since we always need

a discount rate

• In practice, investment banks seem to be using 4 to 5%

as the market risk premium

Multi-stage DGMs

• So far, we have seen one example of this – an

initial (or specific forecast period) set of growth

rates followed by a long run sustainable growth

rate

• But more formalised stage models are possible,

e.g.::

2 stage

“H” model

3 stage



Two-stage model

• This is less sophisticated than our specific

forecast period model

• It simply assumes an initial period compound

growth rate followed by a long run growth rate

• So, say we assume growth at gs in the short

term, followed by gl in the long term

• We then have a formulaic representation of the

dividend process from our earlier model:

The 2-stage model:

• As before, last term is the forecast share price using a

constant growth from the horizon forecast

• This then must be discounted back to year 0 and added

to PV dividends in initial period

• Difference is that Year n dividend is simply pre-

determined by short term growth rate

( ) n

l

l

n

snt

tt

t

s

rgr

ggD

r

gDP

)1)((

)1()1(

1

)1( 0

1

00 +−

+++

+

+=∑

=

=



Example

• Suppose we have:

General Mills div $1.10

Short run growth 11%

Long run growth 8%

Rf 6.7%, MRP 4%, Beta 1.0

• We can calculate:

PV divs in high growth period

Terminal value at end high growth period

What value would be with “normal” growth

Solution

• PV divs – either “longhand” or by recognising

we have a growth annuity

• Formula for growth annuity is:( )( )

s

n

n

s

gr

r

ggD

−

+

+−+

1

11)1(0

( )( )

s

n

n

s

gr

r

ggD

−

+

+−+

1

11)1(0



So PV dividends is:

• Discount rate = 6.7 + 1 x 4 = 10.7%

• PV = 1.1 x (1.11) x (1-(1.115/1.1075))

.1075 - .11

= $5.54

• Note it doesn’t matter that r – g is negative in the short

term – the formula still works

• Plus the PV of the year 5 terminal value

• =(1.1x1.115x1.08) / [(.1075 - .08) x 1.10755) = $44.60

• A total value of $5.54 + $44.60 = $50.14

In general

• We can break down value into that which is

found given “normal” growth in perpetuity, +

• The value of “abnormal growth”

• For General Mills example, with “normal

growth” share would have been worth:

• (1.1 x 1.08)/(.1075 - .08) = $44

• So “abnormal growth” is worth $50.14 - $44 =

$6.14



A problem with the 2 stage model:

• The growth pattern is unrealistic – dividend

growth “falls off a cliff” in year n:G from last yr

0.00%

2.00%

4.00%

6.00%

8.00%

10.00%

12.00%

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

Year

Growth

G from last yr

Enter the “H-model”

• Idea here is to have a gradual transition to the long run

growth position

• If we have a continuous rate of change (sometimes

misleading referred to as “linear”), then average growth

in “abnormal growth” period is (gs – gl)/2

• If H is half the abnormal growth duration, then extra

dividends (above “normal”) will be approximately 2 x H

x D0 x (gs – gl)/2

• = D0 x H x (gs – gl)

• This will add value to the firm with “normal” growth:



The H-model:

• So we get:

• This is an approximation, as neither the final dividend nor the PV of dividends are estimated with total accuracy

• The accurate solution can be estimated with a spreadsheet

l

sl

l

l

gr

ggHD

gr

gD

−−

+−+ )()1( 00

The growth assumption in the “H model”:

G from last yr

0.00%

5.00%

10.00%

15.00%

20.00%

25.00%

30.00%

35.00%

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

Year

Growth

G from last yr



Example

• Siemens AG has:

Dividend €1.00

Short term growth 29.28%, long term 7.26%, transition period 16

years (so H = 8 years)

Rf 5.34%, MRP 5.32%, beta 1.37, so r = 12.63%

• So value is:

• 1 x 1.0726 + 1 x 8 x (0.2928 - .0726)

(.1263-.0726) (.1263-.0726)

• = €19.97 (“normal” value) + €32.8 (“abnormal” value)

How inaccurate is this?

• With these assumptions, the true value is €60.77

• In general, the H-model will be less accurate when:• The abnormal growth period is long

• The difference in growth parameters is large

• The model was developed by Fuller & Hsia in 1984 –with modern spreadsheets, the need to rely on such approximations is reduced – though concept is useful

• Note that if the reduction in dividend growth is assumed to be linear (i.e. straight line as opposed to a constant rate of reduction) then the approximation is far less accurate (for example, true value is then €79.94)



3-stage models

• Continuing the principle, we could have a 3

stage model:

• Period of high growth (or specific dividend forecasts)

• Followed by a period of reversion to “normal”

growth rates – value could be determined by a “H-

model” or by spreadsheet

• Followed by a long-run rate of dividend growth

• For example, the Bloomberg model uses this

approach

Conclusions

• We have looked at various forms of the DGM which

measures cash flows to equity holders directly

• This implies the discount rate should be the return

required by equity holders – i.e. the cost of equity

capital, or r

• We have examined the likely ranges for r, and the issues

involved in calculating it

• Next week, we look at firm level (as opposed to equity

level) cash flows

• We also look at what determines growth

Documents

Investment Analysis II & Advanced Investment Analysis ...people.exeter.ac.uk/wl203/BEAM015/Materials/Lecture 1/NewIA2 lect… · Investment Analysis 2 1 Investment Analysis II & Advanced