Market Efficiency in the Long Run

Market Efficiency and Bubbles in the long run:

an empirical test on the S&P Composite

Niccolo Ferragamo

April 30, 2012

Contents

1 Historical evidence from Stock Returns 5

1.1 Defining returns in the equity market . . . . . . . . . . . . . . . . 5

1.2 Returns and the S&P Composite . . . . . . . . . . . . . . . . . . 7

1.2.1 S&P Composite as proxy for the world equity market. . . . 8

1.2.2 Total Return Index . . . . . . . . . . . . . . . . . . . . . . 11

1.2.3 Dividends and Capital Gains components in the S&P Com-

posite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.2.4 Equity, risk and volatility . . . . . . . . . . . . . . . . . . 16

1.2.5 Future returns . . . . . . . . . . . . . . . . . . . . . . . . . 20

2 Pricing and Market Efficiency 22

2.0.6 Why should the equity market be efficient? . . . . . . . . . 24

2.0.7 Testing market efficiency and the joint stock hypothesis . . 25

2.1 The Net present Value . . . . . . . . . . . . . . . . . . . . . . . . 26

2.1.1 The long run matters . . . . . . . . . . . . . . . . . . . . . 32

2.1.2 Dividend policy . . . . . . . . . . . . . . . . . . . . . . . . 34

2.2 The market equity premium . . . . . . . . . . . . . . . . . . . . . 35

3 Abnormal returns from fundamental analysis 43

3.1 The P/E10 ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.1.1 Price-Earnings as fundamental market-timing indicators . 46

3.1.2 Test on Total Real Returns . . . . . . . . . . . . . . . . . 48

3.1.3 Test on Extra Returns . . . . . . . . . . . . . . . . . . . . 52

3.1.4 Test on Real Dividend Returns . . . . . . . . . . . . . . . 56

3.2 Statistical Significance issues with the P/E10 test . . . . . . . . . 60

3.3 An approach to solve the overlapping problem . . . . . . . . . . . 61

3.4 Where could the PE ratio go? . . . . . . . . . . . . . . . . . . . . 67

1

CONTENTS

4 EMH and Behavioural Finance 72

4.1 Cross-section abnormal Returns on single value stocks . . . . . . . 73

4.2 Speculative Bubbles . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.2.1 Overconfidence, Overreaction and trend extrapulation . . . 79

4.3 Shillers Variance Test . . . . . . . . . . . . . . . . . . . . . . . . 80

4.3.1 Reconciling excessive volatility with EMH . . . . . . . . . 84

4.3.2 How can mispricing persist? . . . . . . . . . . . . . . . . . 85

2

Abstract

One of the fundamental assumptions of modern finance in that investors have

risk averse preferences and that, therefore, higher returns from an investment

should be the result of an higher systematic risk. The aim of this dissertation

is to analyse the equity market performance in aggregate from an historical

perspective and to evaluate if its movements can be totally explained by the

efficient market hypothesis. In particular we want to evaluate how reasonable it

is to explain some of the market movements as a consequence of noisy traders

who over-react and generate speculative bubbles.

In order to do so, we use as proxy for the global equity market the U.S. S&P

Composite, representing the world biggest equity financial market, and attempt

to find abnormal returns from long term investments. In particular, we use the

Price-Smoothed Earnings Ratio, as introduced by Shiller, as a fundamental value

indicator. If extreme Price Earnings ratios represent a valid indicator of market

over-valuation, we suggest that both policy-makers and investors can use this

indicator as an early warning system for bubble formation.

This dissertation is organized as follows: chapter 1 formally defines different

returns in the equity market and analyses the S&P Composite performances in

terms of total real returns, dividend returns, extra returns on U.S. t-bonds and

volatility. In order to do so, we compute different indexes and statistics from

Shillers data on the S&P Composite and show how this index has offered, in the

long run, returns that highly outperformed fixed income securities.

Chapter 2 defines the efficient market hypothesis and introduces different

Net Present Value methodologies, showing the impact of expectations on cur-

rent prices and which variables influence the most risk premiums required by

operators.

In Chapter 3 we test the historical correlation between Shillers PE10 ratio

and returns from 20-years buy and hold strategies on the S&P Composite. We

perform linear regressions on the S&P Composite historical data and show that

this correlation has been persistent in time. In order to check if this relation has

3

CONTENTS

to be attributed only to variations of real interest rates, we repeat the test using

extra returns instead of total real returns and show that the negative correlation

is less strong than before, but still existing. We also show that the PE10 ratio has

mainly had an impact on the capital gain component of total returns, and this

supports the irrational exuberance and bubble formation theories. These tests,

however, violate the linear regression model assumption of independence and

identical distribution of the dependent variables. To overcome this problem and

evaluate the impact of the PE10 on subsequent returns, we decompose the extra

returns over a generic -years period into different i.i.d. yearly k-forward extra

returns. We then perform linear regressions on the latter and recombine their k

coefficient estimates to get the cumulative correlation coefficient between PE10

ratios and the average extra return over the whole -years period. This approach

strengthens the previous conclusions and shows that the negative correlation is

statistically significant.

Chapter 4 offers a review of theories in favour of the irrational exuberance

and bubble formation thesis and analyses the concepts of over-reaction and trend

extrapolation from a behavioural perspective. We conclude repeating Shillers

1981 variance-bound exercise with the newly available data in order to show what

price levels should have ex-post been had operators forecast exactly the future

level of dividends. The excessive volatility that emerges as result of this analysis

supports, despite its limitations, the hypothesis in which the equity market as a

whole over-reacts.

4

Chapter1Historical evidence from Stock Returns

1.1 Defining returns in the equity market

Stocks are financial instruments that represent an ownership position in a certain

company, do not expire over time and give the right to their holder to claim on

a proportional share in a corporations assets and profits. Ordinary stocks own-

ers are provided with voting rights which can be used in shareholders meetings

to decide important issues of the company such as the election of the board of

directors. Even though only incorporated companies have stocks, these are par-

ticularly important in modern economies as their ownership rights can be more

easily traded on both regulated and over the counter financial markets. The

direct consequence of these peculiarity is the possibility for investors to easily

allocate part of their portfolios in these instruments in order to get a return on

their investment. The stock of a business is divided into multiple shares, the

return on which can be divided into two components:

dividends;

capital gains.

Dividends Dt represent the part of corporate profits which is distributed from

a corporation to its shareholder at time t. The ratio DtEt

of earnings paid out as

dividends in a fiscal year is called dividend payout ratio. This ratio can change

over time and from company to company at the discretion of management, de-

pending on the amount of earning that is retained and re-invested in the business

every year. Differently from other instruments such as bonds, the cash flows that

a shareholder has the right to receive are not certain in their amount and, as we

will discuss later, present a certain level of risk. In particular, common stocks

should be riskier than debt issued by the same corporation as their holders re-

ceive only companys residual cash flows and cannot be paid dividends until all

5

1. Historical evidence from Stock Returns

the interests on the companys debt and the preferred stock dividends are paid

in full. In other words, earnings are subject to a leverage effect, which results

in a shares price volatility which is, at least in the short run, higher than fixed

income instruments.

Capital gains or losses are represented by the variation of the market price

P of the share over time. The market price of the share is the amount of money

at which operators exchange at a certain time a given share on a financial market

and, determined by the interaction of supply and demand, is different from the

share face value. We will see later that, in efficient markets, a share should be

priced according to its net present value.

In a stock investment, the total nominal return rn,t from time t 1 to time tcan be defined as the sum of nominal capital gains and nominal dividends returns

over (t-1,t). Formally:

rn,t =(Dn,t + Pn,t Pn,t1)

Pn,t1=

Dn,tPn,t1

+Pn,t Pn,t1

Pn,t1.

Where Dn,t and Pn,t represent nominal dividend and share price at time t.

When a certain investments is evaluated, it is possible to consider either nominal

or real returns. Real variables take into account the variation usually negative

of purchasing power in time caused by inflation. If the utility of an operator is

function only of its level of consumption, we should expect a rational investor to

care only about the variation of its purchasing power and about real variables

while evaluating a given investment. It is interesting to notice that the Media and

the financial press almost never present data and stock charts only in nominal

terms, putting not much emphasis on real returns and, perhaps, stimulating a

certain money illusion effect1.

To compute the real return on a stock investment we have to deflate cash flows

by an appropriate price index, such as the European Harmonised Consumer Price

Index or the United States Consumer Price Index published by the U.S. Bureau

of Labour Statistics. If CPIt is the consumer price index at time t, we can

compute real prices Pt and real dividends Dt as:

1The term money illusion was introduced by John Maynard Keynes to indicate the tendencyof people to mistake the nominal value of money for its purchasing power. If money illusionexists, then investors may irrationally consider nominal variables when they evaluate returnson investments.

6


Pt =Pn,tCPIt

Dt =Dn,tCPIt

and we can then define total real returns as:

rt =D,tPt1

+Pt Pt1Pt1

.

Where the first part of the equation on the right represents the real dividends

return and the second part the real capital gain return.

Note that, given an inflation t =CPItCPIt1

CPIt1in (t, t+1), the relation between

total real returns rt and total nominal returns rn,t is equal to:

rt =(1 + rn,t)

1 + t 1 (1.1)

1.2 Returns and the S&P Composite

The aim of this dissertation is to consider returns and abnormal returns from

an aggregate market perspective. What we need to do, hence, is to consider an

highly diversified index which can approximate as well as possible the risk and

return profiles of the equity market.

The evolution of financial markets in developed countries and the consequent

fall in transaction costs has made it easy and realistic for both private and in-

stitutional operators to diversify their investments into portfolios with hundreds

of securities, held in proportions that reflect their market capitalization. Diver-

sification enables investors to get rid of idiosyncratic risks2 of single assets by

exploiting lower than unitary covariances among returns from different assets. A

fast way for an investor to do so is to buy a share of a mutual fund or, in recent

times, of a large Exchange traded fund.

An ETF is an investment fund which passively replicates the returns of a

certain stock index, such as the American S&P500 or the Italian FTSE Mib.

ETFs differ from traditional mutual funds in the fact that their shares can be

bought and sold throughout the day like stocks on a securities exchange through

a broker-dealer. Thanks to their low costs, tax efficiency and stock-like features,

ETFs have proliferated significantly in the last ten years, reaching in the United

2Idiosyncratic risk is often called specific risk as well.

7


States at the end of February 2012 a total amount of assets under management

of 1,18 trillion dollars3. The concept of total real returns and its combination

of dividend and real returns can be applied to these instruments as well as to

corporation shares.

In order to evaluate the return opportunities from stocks, it is important for

an investor to analyse how equities have performed in aggregate in the past.

The ideal analysis would involve considering data on the equity market portfolio,

which consists of a weighted sum of every stock security in the market, with

weights in the proportions that they exist in the market and assuming that these

assets are infinitely divisible. This would theoretically include all existing stocks,

from both listed and unlisted corporations and from every country in the world,

and would represent the equity portfolio with the highest possible degree of

diversification. Note that this would still be only a subset of the even bigger true

market portfolio, as introduced by Markowitz, which includes virtually anything

with marketable value not only equities in every market.

Roll4 criticised Markovitzs model considering the fact that the true market

portfolio is unobservable as not only it is not possible to physically invest in all

these assets, but also it is impossible to observe the returns of many of these. A

similar critique can be addressed to the equity market portfolio, as it is practically

impossible to observe all existing corporations and to invest on all of them. What

is possible to do is, at most, to combine ETFs from different countries into a

portfolio with a degree of diversification which is comparable to the theoretical

equity market portfolio.

The problem that emerges at this time is where to find data on a global

equity index going sufficiently back in time to analyse its return and volatility

in an historical perspective. As we lack such a data set, in this dissertation we

will use the S&P Composite index, with information available on a monthly base

since 1871, as a proxy for the equity market portfolio. Before we go further, it

is important to discuss on how reasonable this assumption is.

1.2.1 S&P Composite as proxy for the world equity mar-

ket.

The correlation presented in figure 1.1 partly supports the use of the S&P Com-

posite as a proxy for the global equity market.

3Data from ETF Industry Association, March 2012.4See A critique of the asset pricing theorys tests, Journal of Financial Economics, Vol.

4, March 1977

8


Figure 1.1: Historical correlation between real GDP growth rates in the U.S. andin the World. Personal elaboration from World Bank Data

Nevertheless, it is important to take into account the limitations of this ap-

proach. First of all, the coefficient of determination R2 shows that only 68.8% of

the variance of World GDP real growth is related to the variance of U.S. GDP.

Secondly, there are several reasons why different equity markets could move dif-

ferently in countries with the same GDP growth. Equity returns are related to

corporate earnings and, thus, to the share of national GDPs represented by cor-

porate profits. This share changes from country to country and could be, at a

global level, different from the U.S. Secondly, even though the corporate profits

share was exactly the same in the U.S. and in the World, equity returns could be

functions of different risk premiums required by local investors to cover different

risks. An investor in a less developed and less liquid market could, for example,

request an higher risk premium for its equity investment.

Despite these factors, the increasing level of market openness and globaliza-

tion are making local markets and probably, as investors diversify their portfolio

internationals and as multinationals listed on a certain stock exchange develop

their businesses abroad, the equity market will tend to become more and more

correlated.

To show how global equity markets and the U.S. S&P 500 have been correlated

in recent years, we take the Global S&P 1200 index as proxy for global returns

and show how the two price indexes have moved from January 2007 to April

2012.

9


The S&P Global 1200 Index is a free-float weighted stock market index of

global equities from Standard & Poors which covers 31 countries and approxi-

mately 70 percent of global stock market capitalization5. The S&P 500 is a subset

of the S&P Global 1200, and it account, as on April 2012 , for approximately

52% of the capitalization of the latter.

Figure 1.2: S&P 500 (Gray line) and S&P Global 1200 (Black line) price indexesfrom January 2007 to April 2012

Figure 1.2 shows how similar the paths of the two indexes, computed on a

yearly base, have been in the last 5 years. For simplicity, we have set the initial

price levels of both the indexes as on January 2007 at 100. This means that

the part of the S&P Global 1200 which is not made by the S&P 500 is highly

correlated with the movements of the latter. Unfortunately, data on the S&P

Global 1200 index is not available for past decades and, consequently, it is not

possible to see whether this correlation did hold in past years or not.

Even if this approach is susceptible to critiques, and even though this does

not take into account the potential survivorship bias6of the U.S. stock market, if

the mentioned limitations are clear, then we conclude that the S&P can be used

as a proxy for global equity returns.

Even in the case in which the S&P Composite did not represent a good proxy

for the market as a whole, an analysis of the returns of the world biggest stock

5The S&P Global 1200 combines six different regional indexes: the S&P 500, S&P TSX 60(Canada), S&P Latin America 40 Index (Mexico, Brazil, Peru, Chile), S&P TOPIX 150 Index(Japan), S&P Asia 50 Index (Hong Kong, Korea, Singapore, Taiwan), S&P/ASX 50 Index(Australia), S&P Europe 350 Index. Source: Standard and Poors.

6We will discuss on this bias, as reported in Jorion and Goetzmann (1999), in chapter 4.

10


market is still crucial to make capital allocation decisions and understand how

equities have behaved historically.

An operator can easily invest on the S&P Composite index, now named S&P

500, by buying one of its replicating ETFs7.

The S&P 500 is a free-float capitalization-weighted index published since 1957

of the prices of 500 large-cap common stocks actively traded in the United States.

The stocks included in the S&P 500 are those of large publicly held companies

traded on both the New York Stock Exchange NYSE and the NASDAQ. These

securities vary in time in order to keep the index reflective of American stocks8,

and are included with weights that are proportional to the capitalization of their

public float9. It is possible to extend the data set back in time by using data on

previous versions of the index, created since 1871. Even though the composition

of the index has slowly changed through time, the S&P Composite has always

been a good and objective representation of the U.S. equity market. With its

actual capitalization of 12,38 trillion dollars, the S&P 500 represents about 70%

of the total capitalization of listed companies in the U.S.10

In order to proceed with our analysis, in the following sections we will build a

few useful return indexes on the S&P Composite. The data set used in this dis-

sertation consists of monthly observations for the Standard and Poor Composite

Stock Price Index, extended back to 1871 by using the data in Cowles as made

available by Shiller. Figure 1.3 shows the S&P Composite Total Return Index

from 1871 to 1945 and from 1946 to 2012 with an initial value of the index set

to 1$ in January 1871.

1.2.2 Total Return Index

The total return index can be computed from an underlying price index, such

as the S&P500 by assuming that all dividends and distributions are reinvested

immediately on the underlying index11.

TRIt = TRIt1(1 + rt)

where TRIt is the level of the total return index at time t and (1 + rt) is the

7The Standard & Poors 500 Index Depository Receipts ETF SPY.N with its 99.6billion dollars net assets as in March 2012, is currently the biggest U.S. EFT by capitalization

8Siegel (2009) shows that, out of the 500 firms in 1957, only 125 remained in the samecorporate form in 2003.

9Public float refers to the number of outstanding shares in the hands of public investorswhich are publicly traded. The capitalization of a company is usually higher than the capi-talization of its public float because of the existence of shares holed by insiders which are nottraded.

10Data on the U.S. market capitalization from Worldbank, 2011.11Index Mathematics Methodology - S&P index official data.

11


Figure 1.3: Real S&P Composite Stock. Total real return from 1871 to the end ofWWII and from 1946 to March 2012. Personal calculations from Shillers Data.

total real return on the index over (t-1,t), representing the amount of money one

receives in t after investing one monetary unit in t-1:

1 + rt =Pt +DtPt1

(1.2)

Where Pt and Dt represent respectively the real price and the real dividend

paid in t of the underlying price-index. The initial value of TRI0 is set to 1

dollar.

As we would expect, the U.S. equity market has produced returns in the long

term not only in nominal terms, but also when real variables are considered. An

investor who had invested one dollar in 1871 would have multiplied that amount

over 141 years, in real terms, by over 7300 times thanks to the effect of compound

12


capitalization. It is essential to note that this total return index does not take

into account the important impact of individual taxes on both dividends and

capital gains, which would noticeably decrease the final value of the investment

in 2012.

Despite the volatility of returns, it is significant that stocks in the U.S., as

noted by Siegel, have never delivered to investors a negative real return over

periods of 17 years or more12. Even if an operator had invested his savings in

the S&P Composite in September 1929, at the peak of the U.S. stock market

and just before the biggest drop ever in the total real return index13, he would

have recovered all the losses by October 1936. After October 1947, moreover,

the total real return index has never reached a value lower than the 1929 peak.

Did the same recovery happen after the other two most impressive shocks in

recent history? Regarding the dot come bubble, real prices had reached a peak

in August 2000 while the Total Return Index TRI was at its highest level in

December 1999. After the collapse of the bubble, by September 2002 the TRI

had dropped 50% of its value: that means that, in real terms, any investor who

had put his money in the U.S. aggregate equity market at its peak would have

lost 50% of the value in less than 3 years. By March 2012, had the investor

held his investment, he would have recovered 60% of this loss, achieving a minus

20% in real term from the 1999 peak. Such a negative performance is pretty

impressive if we consider that almost 13 years have passed since 1999. This

empirical evidence shows us that, hence, in the short to medium run the stock

market can make investors lose a significant part of their capital.

If we consider that in 2007-2009 the world experienced the second most serious

financial crises after 1929, and if look at how high some would say irrational14

P/E ratios15 had become in 1999, however, even this minus 20% could appear

to be not so impressive. Despite the superficiality and abstractness of this con-

sideration, it is interesting to note that an investor who had bought the S&P

Composite at any time before December 1998, when the dot com bubble was

already swelling, by March 2012 would have received a positive real return.

From an investor perspective anyway, it is not only important to consider

absolute performances, but also returns compared to substitute investments. It

is essential, in particular, to understand how risk-free investments did perform

12The sample used by Siegel for the U.S. stock market is actually even larger than the oneused by Shiller, going back to 1802. Siegel 2008 (op cit)

13According to our calculations, the S&P 500 total return index lost 75% of its value fromSeptember 1929 to February 1932

14See Shillers Irrational Exuberance (op. cit).15We will discuss about this ratio and about ways to use it as a fundamental value indicator

in chapters 3 and 4.

13


historically. The rating agency Standard & Poors, with its recent downgrade,

has removed the United States government from its list of risk-free borrowers.

Despite this decision and despite the tightness of the concept of risk free asset,

for our purposes we will consider the 10-years U.S. treasury bonds as a safe asset.

If we compare the returns from the 1999 peak to March 2012 of both the

S&P 500 index and 10-years government U.S. bonds, for example, we see that

the minus 20% loss becomes a minus 33% with respect to the risk free asset. In

the last 10 years, moreover, the geometric average of real returns on t-bonds has

slightly outperformed by an yearly 0.18% the S&P Composite.

If we consider longer periods, anyway, the situation becomes much more

favourable for the equity market. Lets compare, for example, returns of Portfolio

E and Portfolio B on different 20 years periods. Portfolio E is made by the only

S&P Total Return Index and Portfolio B by a 10-years zero coupon U.S. t-bond

bought at time t, payed back after 10 years, and reinvested into another 10-years

t-bond.

The geometric average of real yearly returns on B on a twenty years period

are equal to:

rB, =20

(1 + y )10(1 + y+10)10 1

Where y is equal to the real annual yield of a 10-years zero coupon U.S.

treasury bond at year 16.

Table 1.1 shows the geometric and arithmetic averages or total real returns

of the S&P Composite portfolio and the Risk Free Portfolio bonds from 1872

to 2011 and on twenty-years sub-periods. Even though there have been others

20-years periods when t-bonds did perform better than the S&P Composite, in

the considered sample the extra return of equity on t-bonds has always been

positive.

From what we see, the S&P Composite performed well in the long run not

only in absolute terms, but also when compared to risk-free investments.

If we take U.S. t-bonds with an even higher maturity and yearly yield, it

is interesting to analyse how many times an equity investment did outperform

treasuries depending on the length of the holding period. Taken the 30-years

U.S. government bonds as the benchmark for long term risk free rate, and taken

a period going from 1871 to 2006, Siegel17 shows that the percentage of times

16In order to compute these yearly real ratios, we take into account the average inflationrate over the maturity periods of both the t-bonds

17See Siegels Stock for the long Run (op. cit.)

14


S&P Composite S&P Composite Risk-Free PortfolioPeriod Geometric Mean Arithmetic Mean Geometric Mean

1872-2011 6,47% 8,22% 2,5%1872-1891 8,40% 9,35% 7,38%1892-1911 6,86% 8,18% 1,79%1912-1931 4,91% 7,07% 1,37%1932-1951 5,37% 9,01% -0,32%1952-1971 9,21% 10,35% 0,98%1972-1991 4,76% 6,15% 3,71%1992-2011 5,68% 7,26% 3,47%

Table 1.1: Geometric and aritmetic averages of total real returns on the S&PComposite Index and on 10-years U.S. treasury bonds.

that stock returns outperformed bond increase dramatically as the holding period

increases. For 10, 20 and 30 years horizons stocks outperformed 30-years bonds

respectively 82.4, 95.6 and 100 times out of 100.

According to Siegel (2009):

The high probability that bonds and even bank accounts will out-

perform stocks in the short run is the primary reason why it is so

hard for many investors to stay in stocks.

If one accepts the fragile assumption that past evidence is a predictor of

future events, hence, the presented data shows that the equity market should

be considered a good buy-and-hold investment for an operator with long-term

perspectives and who is willing to accept the potential short-term losses. Equi-

ties should definitely be, thus, a relevant part of the diversified portfolio of any

operator.

1.2.3 Dividends and Capital Gains components in the

S&P Composite

Total returns have a dividend and a capital gain component. To analyse the

historical size of these components in the S&P, we decompose the Total Return

index in the product of a Dividend Only Index and the Price Index.

TRIt = TRIt1

(Pt +DtPt1

)= TRIt1

(1 +

DtPt

)PtPt1

(1.3)

=ti=1

(1 +

DiPi

) ti=1

PiPi1

= DRItPIt (1.4)

Where Pt =PtP0

, where P0 is set to 1, is the normalized price index and DRIt

is the Dividend Only Index. The first is the index which is commonly used by the

15


Media and that we have used to build the Total Return Index. The normalized

Price Index shows how much 1$ would have become, in real terms, by investing

in the S&P, if dividend payouts are not taken into account.

On opposite, the dividend real return index shows how much 1$ invested in

1871 would have become in 2012 if the real price of the S&P had stayed at the

same level i.e., if no real capital gains occurred and dividends were continuously

reinvested in the index.

Figure 1.4 shows the real normalized price, real dividend Only and total real

return indexes from 1871 to 2012. In order to linearise the exponential trends,

we present the data on a logarithmic y-axis.

From 1871, out of a geometric average of yearly total real returns of 6,46% and

with a geometric average of yearly real capital gain return of 1,94%, dividends

have been not only the less volatile, but also the most relevant component of the

total return index, with an average of 4,43% real return on a yearly base18. This

evidence makes it difficult to understand why it is so rare to find total return

charts for stocks and indexes in the Media and in the Financial Press. From a

behavioural perspective, it could be argued that this emphasis on capital gains

and stock prices could lead several non-professional investors to overestimate

the importance of capital gains when they evaluate their investments and their

expected future returns.

The weight of the dividend component, however, has become less important

in time due to variations in the dividend payout policy of U.S. firms. If we repeat

the same analysis from 1946 to 2012, infact, out of an average yearly total return

of 6,49%, the dividend component has accounted for an yearly 3,48% and the

capital gain for an yearly 2,9%. In more recents year, moreover, capital gains

have surpassed dividend returns. From 1985 to 2012, infact, out of an average

total real return of 7,33%, yearly capital gains have averaged 4,89% on a yearly

base, while dividends have returned an yearly 2,38%. The second graph in figure

1.4 shows exactly this weight variation.

1.2.4 Equity, risk and volatility

Volatility is a measure for variation of price or of the total return of a financial

instrument over time. Two different kinds of volatility can be defined:

Historic Volatility, derived from time series of past market prices or pastequity total returns;

18Data from personal calculations using the previously built real total return, real dividendand real price indexes.

16


Figure 1.4: Real S&P Composite Stock. Comparison between Total Return,dividend Return and price indexes on the 1871-2012 and the 1985-2012 periods.Personal calculations from Shillers Data.

17


Implied volatility, derived from the market price of options having as un-derlying variable a stock or another financial instrument.

Both entities are used to quantify the risk of financial instruments over a

specified time period. Hereby we focus on the first concept and define historical

volatility from t k to t to as the standard deviation of total real yearly returnsover that period. We estimate the standard deviation of the population using a

sample of monthly observations of yearly rt. Formally:

r,t,tk =

i=ti=tk (ri rt,t+k)

2

n 1(1.5)

Where rt,t+k is equal to the arithmetic average of returns over (t, t + k) and

n represents the number of monthly observations in the same period. We then

put k equal to 5, 10 and 20 years and plot the charts in figure 1.5 by computing

r,t,t+k for every monthly t from 1891 to 2011. We keep using the database on

S&P Composite made available by Shiller.

We repeat the same exercise with the 5-years standard deviations of dividend

returns, by simply substituting ri in equation 1.5 with the yearly real dividend

returns di.

Figure 1.5 and figure 1.6 highlight five different evidences.

The higher k, the lower the volatility of r,t,tk i.e. the volatility of theyearly returns volatility. This evidence is strictly related to the way in

which r,t1,t2 in built in equation 1.5. The higher k, the more overlapping

periods exist and the smoother we expect the charts to be.

The standard deviations of ri are less volatile than yearly ri themselves. Ifwe consider chart A, we notice that 5-years standard deviations have been

fluctuating between 8% and 20% in 100 out of the 120 considered years.

Volatility seems to be positively autocorrelated. Once returns becomehighly volatile, thus, they tend to stay volatile for some time. Some of this

autocorrelation derives of course from the way in which r,t1,t2 is computed:

r,1882,1887 will structurally be highly correlated with r,1881,1886, as the two

statistics are computed over a sample where 80% of the observations are

identical.

Despite the low variance of volatility over most of the period, there havebeen periods with a much higher than average uncertainty about returns.

Between 1929 and 1946, when both the most serious economic crises and

18


Figure 1.5: 5-years, 10-years and 20 years standard deviations of total real returnson the S&P Composite from 1881 to 2012. Personal calculations from ShillersData.

19


Figure 1.6: 5-years standard deviations of dividends real returns on the S&PComposite from 1881 to 2012. Personal calculations from Shillers Data.

the most dramatic war in history occurred, r,t,t600 5 years standard

deviation, with monthly observations reached a peak of 45,7%19.

As we could expect, dividend smoothing policies make dividend returnsmuch less volatile than total real returns. The fact that 5-years volatility on

dividend returns has never been higher than 1,4%, and that its coefficient

of variation over the same period never exceeded 80%, tells us that almost

all the volatility is due to changes in prices and capital gain returns.

1.2.5 Future returns

Despite the quantity of historical data, one can never be certain that the under-

lying factors that generate asset prices have remained unchanged20. As Nobel

laureate Paul Samuelson said, we have but one sample of history. In the im-

possibility to repeat controlled experiments, holding some factors constant while

estimating the value of the target parameters, past economical events are not

a guarantee of future events. Nothing guarantees, hence, the equity market

will continue to outperform in such a relevant way the bond market21, or even

that the S&P Composite will never present a period longer than 17 years with

non-positive total returns. Even when past relations are statistically significant,

moreover, valuation benchmarks are valid only as long as underlying economic

and financial conditions do not change. Structural changes in the economy and

19This peak was reached, in particular, in 193720Siegel 2008, op. cit.21The relevance of this impressive performance is at the center of the debate on the Equity

Market Premium Puzzle. This term was coined in 1985 by Mehra and Prescott to show that,to reconcile the much higher returns of stocks compared to bonds, individuals should haveimplausibly high risk aversion according to standard economics models.

20


in financial markets should be, therefore, carefully analysed when attempting to

use the past to predict future returns.

21

Chapter2Pricing and Market Efficiency

The concept of market efficiency has become widely known through the work

of professor Eugene Fama and his colleagues in the late 1960s. The concept,

however, has been less formally known for decades1, supported by the intuitive

evidence that it is extremely difficult to obtain abnormal returns by buying low

and selling high in the stock market. According to Jensens definition (1978):

A market is efficient with respect to information set t if it is

impossible to make economic profits by trading on the basis of infor-

mation set t.

An economic profit is defined as the net adjusted rate of return, net of all

costs. It is, in other words, a net return that its higher than returns that are

required for a certain level of risk. In literature this is also called abnormal

return.

According to efficiency market hypothesis from now on EMH the price of

securities and, in aggregate, of markets should reflect perfectly all the available

information at any time. Whenever a new information is available, through

variations of demand and supply, markets are supposed to adjust in a short time

instantly, in theory and to eliminate every possible source of economic profit.

The efficient markets theory claims that no asset in the market can be either

overpriced or under priced and that, thus, the smartest investor will not be able

to outperform a casual investor in terms of final return for a taken risk. In

other words, financial markets are efficient when they do not allow investors to

earn above-average returns without accepting above-average risks2. Expected

1In 1889 in a book by George Gibson entitled The stock Markets of London, Paris and NewYork, the author wrote that When shares become publicly known in an open market, the valuewhich they acquire may be regarded as the judgement of the best intelligence concerning them.(Shiller, op.cit)

2Malkiel 2003 (op. cit.)

22

2. Pricing and Market Efficiency

performance indicators such as the Sharpe Ratio3, thus, should be equal among

all investors.

Depending on how wide the information set t is, three different market effi-

ciency forms have been introduced in financial literature.

1. The Weak Form of the Efficient Market Hypothesis, in which the informa-

tion set t is taken to be solely the information contained in the past price

history of the market as of time t. If markets are efficient in the Weak

Form, it should be impossible to make economic profits by exploiting re-

current price patterns and technical analysis cannot work. Its interesting

to note that is weak form efficiency holds, the terms bull and bear market,

which are often used among Media, professional and casual investors to

describe expected upward and downward market trends, completely lack

of sense4.

2. The Semi-Strong Form, in which t represents all the information that

is publicly available at time t. This includes not only past prices, but

also information about fundamental indicators, companies balance sheets,

financial reports, macroeconomics public research etc. In this form funda-

mental analysis, which is the evaluation of securities, firms and markets

mispricing based on the analysis of economic and financial factors, should

not provide abnormal returns.

3. The Strong Form of the Efficient Market Hypothesis, in which t is taken

to be all information known to anyone at time t, including thus all informa-

tion which has not been published but that is available to companies and

insiders. Price gains due to a takeover or a merger, for instance, should be

priced well before the official announcement is made.

Under the various forms of EMH, price variations can be accounted only to

the availability of purely new information. If it is impossible to forecast future

information and events happen randomly, a consequence of EMH is that the

underlying stochastic process for price formation, after adjustments for required

returns, is a martingale. In probability theory, a martingale is a model of a fair

game where no knowledge of past events can help to predict future winnings. In

3The Sharpe ratio is the amount of expected extra return an investor receives for everyunit of standard deviation of its portfolio. Formally: S = E( rtrft ), where is the standarddeviation of the portfolio, rt its return and rft the risk free rate of return.

4The same applies to the famous Wall Street phrase The trend is your friend or torelative-strenght and momentum strategies.

23


particular, a martingale is a sequence of random variables for which the expec-

tation of the next value in the sequence is equal to the present observed value

even given knowledge of all prior observed values at a current time. Formally:

E(Pt+1|(t)) = Pt(1 + cgt).

Where pt and pt+1 are the prices of a security at time t and t+1 and cgt is

the required capital gain return on the asset for period (t-1,t). The rate cgt, in

particular, is independent from Pt. We will turn back to cgt and required returns

in the following section.

2.0.6 Why should the equity market be efficient?

The efficient market hypothesis relies on the assumption that, whenever an ex-

isting abnormal profit opportunity is discovered, investors will take advantage of

it and adjust the demand and the supply on the security and bring its price at its

rational level. Investment strategies intended to take advantage of inefficiencies

are actually the fuel that keeps a market efficient.5.

In order to make this happen, the following conditions should be respected:

the market has to be liquid;

information has to be available in terms of accessibility and cost and shouldbe released to investors at the same time. In other words, information

efficiency has to hold;

rational investors must have enough funds to take advantage of inefficiencyuntil it disappears.

It is important to note that, if markets become efficient through the contin-

uous exploitation of abnormal returns, EMH does not rule out small abnormal

returns, before fees and expenses. Grossman and Stiglitz (1980) claim that an-

alysts should still have an incentive to acquire and analyse valuable information

as, without any incentive to do so, no one would spend time to acquire the infor-

mation needed for markets to be efficient. The profits derived from speculation,

hence, are the result of being faster in the acquisition and correct interpretation

of existing and new information6.

5To make sense, the concept of market efficiency has to admit the possibility of minor marketinefficiencies. The evidence accumulated during the 1960s and 1970s appeared to be broadlyconsistent with this view. Dimson and Mussavian (op. cit.).

6Cuthbertson and Nitzsche 2004 (op. cit).

24


Shleifer (2000) claims that as soon as investors begin to understand the exis-

tence of an anomaly and learn something about fundamental values of securities,

they quiclky respond to the new information and eliminate the anomaly7.

Another important aspect regards the presence of irrational traders into the

market. EMH does not require tat all participants in the market are efficient

and well informed.

The EMH only requires that there are sufficient smart money traders who recog-

nise mispricing and, by either buying or short selling the asset, will arbitrage the

opportunity an bring back prices to their fundamental values.

Arbitrage is one of the fundamental concepts of finance and has been defined

by Sharpe and Alexander as the simultaneous purchase and sale of the same, or

essentially similar, security in two different markets for advantageously prices8.

Theoretically, investors could even try to pursue an inter-temporal arbitrage by

taking simultaneously short and long positions in the same market on securities

that have the same risk profile but different current prices.

An arbitrage is such if it requires zero initial outlay of capital to be exploited

and if it generates a positive return with probability one regardless of future

events. In chapter 4, however, we will analyse in which situations arbitrage

opportunities could either be risky or difficult to exploit due to the presence of

various frictions.

2.0.7 Testing market efficiency and the joint stock hy-

pothesis

Financial Literature has tested the Market Efficiency hypothesis in its various

forms9.

Weak form efficiency tests have been performed by evaluating how distant

price patterns have historically been from the null random walk hypothesis.

Studies on the semi-strong form of the efficient markets hypothesis, differ-

ently, are tests of the speed of adjustment of prices to new information, in

the form of event studies. An event study computes the cumulative abnormal-

performance of stocks from a given number of time periods before an event to

a given number of periods afterwards. Semi-strong form tests include looking

for trading strategies (such as the value and growth strategies) that, after taking

account of their transaction costs and their systematic risk, could outperform the

rest of the market. In chapter 3 we will focus on the semi-strong form efficiency

7An example of anomaly which has diminished over time, for example, is the January effect.8Shleifer and Vishny 1997 (op. cit.)9For a comprehensive review on these tests, see Dimson and Mussavian 2000 (op cit).

25


and test if market timing through the fundamental PE10 ratio can generate ab-

normal returns.

Another kind of efficiency test checks if market prices always equal fundamen-

tal value. These tests use past data and dividend discount models to compute,

ex-post, the perfect forecast fundamental value of a stock or of an index. After

doing this, these tests compare the ex-post fundamental value volatility with

variations of the actual prices: in chapter 4 we will show such a test by repeating

Shillers variance-bound test.

As it is difficult to observe information that are not publicly available, tests of

the strong form efficiency consider the performances of operators who are consid-

ered more capable of obtaining this kind of information: investment professionals

and mutual funds.

Jensen stated in 1978 that testing market efficiency in all its three forms has

an intrinsic problem. In most cases, tests of market efficiency are tests of joint

hypotheses : market efficiency and the pricing models chosen to predict returns.

The magnitude of abnormal returns of a stock or an index depends critically

on the choice of benchmark and this makes it difficult to interpret the results.

The tests can fail either because one or both the hypotheses are false or because

both parts of the joint hypothesis are false. In other words, as we cannot be

sure about which kind of risk is priced by markets and about the effectiveness of

pricing models, a market efficiency test could give negative results simply because

our pricing model is wrong. On the one hand, anomalous behaviour may be an

indication of market inefficiencies. On the other hand, even if there is no bias

in computed abnormal returns, the regularity in returns may be indicative of

shortcomings in the underlying asset pricing model.

It is important to note that prices, even when market accurately reflect the

available information, could be not representative of fundamental values simply

because the information is not reliable or not sufficient. Even if markets are

efficient, then, some EMH empirical tests could give negative results because of

some kind of information inefficiency.

2.1 The Net present Value

EMH states that demand and supply should adjust at any time to give the correct

price to any traded stock. But what is the rationale behind the formation of a

certain price level?

The fundamental sources of stock valuation are earnings and dividends. Stocks,

in other words, have value only because of the cash flows that current investors

26


receive or the price appreciation caused by cash flows that future investors expect

to receive. In order to derive a share present value, future cash flows should be

discounted because cash received in the future is worth less than cash received

in the present. This fundamental assumption is based upon four reasons:

Time preferences of consumers. Consumers are supposed to prefer con-suming today rather than wait for tomorrow.

Productivity. One amount of money today can be invested in productivebusiness activities which can create value and turn into an higher amount

of money tomorrow.

Inflation, which usually reduces the purchasing power of money throughtime. Only nominal cash flows have to be discounted with a rate that takes

into account this variable.

Risk, for all cash flows which are uncertain in either their amount of theirpayment date. Modern finance assumes that individuals are risk adverse

and, thus, are willing to take risk only if this risk is rewarded with higher

expected returns10..

The price of an asset should be equal to its Net Present Value NPV, which

represents the sum of all the future cash flows that the owner will receive in the

future, discounted using a yield that compensates the investor for the factors

mentioned above. This yield kt is the rate of return that is just sufficient to con-

vince an investor, according to his preferences, to invest his money in an asset

from time t 1 to t. The term investor, in this case, refers to a representativeoperator whose actions reflect the beliefs of those people who are currently trad-

ing a stock. This is also called the marginal investor, who is the operation with

the higher probability in a given moment to trade the considered asset and who

determines its price11. Return kt should be such that, given an information set

0 at time 0, the stochastic behaviour of rt kt, where rt represent returns thatare really made on the market, assures that, on average, no abnormal returns

are made.

Given a cost of equity capital of kt over a (t-1, t) period, and given an infor-

mation set t, expected return ret+1 should be equal to kt. At the same time, the

martingale price-generating process requires that, over the same periods, prices

should increase by an yearly amount (1+cget ) equal to expected capital gains due

10Siegel 2008, op. cit.11Damodaran 2009, op. cit.

27


to re-investments of retained earnings in corporations. Under these assumptions,

we can compute the expected yearly capital gain cget :

P et +Det = (1 + kt)Pt1

Pet

Pt1= (1 + kt)

DetPt1

cget = kt det

Which is equal to the difference between the required cost of capital and

the expected return from dividends, due to re-investments of retained earning in

firms. The resulting process should than generate prices following an exponential

trend which grows faster as the dividend yield dt is reduced.

NPV of expected future cash flows.

Before introducing a model to determine the required return k, we show how

to price assets with a generic discount rate. If both the payment date and

the amount of the cash flows from an asset are certain, under the no arbitrage

condition we can price the risk free asset in this way:

Prf,0 =CF1

(1 + krf,1)+

CF2(1 + krf,1)(1 + krf,2)

+ + CFnni=1(1 + krf,i)

=

=ni=1

CFiij=1(1 + krf,i)

(2.1)

Where CFt represents the asset cash flow at time t; kfr,t the required return

of a risk free asset at time t and n is the period in which the last cash flow is

paid.

As weve seen in chapter 1, an investment in equity presents a given level of

variance and does not guarantee neither the amount nor the payment date of

its future cash flows. In order to price such a risky asset, then, equation (2.1)

should be changed into:

P0 =CF e1

(1 + k1)+ + CF

enn

i=1(1 + ki)=

ni=1

CF eiij=1(1 + ki)

(2.2)

Where e are the expectations of the operator based on the information set 0

28


at time 0, E(CFt|0) = CF et and ki are the yearly returns required from operator,which reflect the risk of the investment.

In the case of a share of a stock or of an ETF, we know that its cash flows

derive either from dividends or from selling of the share at a future time. Consider

the case of an operator investing in a share at time t 1 and willing to sell it attime t. From his point of view, the correct price should be equal to:

Pt1 =Det + P

et

1 + kt

If the operator is consistent, anyway, he will expect P et to be derived exactly

in the same way that it does for Pt1. By repeated forward induction, each

investor with a one-period horizon should believe the same. For a generic equity

share, hence, EMH asserts that its price in each period should be equal to the

net present value of its expected dividends:

P0 =i=1

Deiij=1(1 + kj)

= NPV (De) (2.3)

NVP with constant cost of capital or constant growth expectations.

If at time 0 dividends and real total returns are expected to remain constant in

the future, then Det = D0 t and ket = k t. We can use simple algebraic passagesto simplify this equation:

P0 =D0

(1 + k)+ . . .+

D0(1 + k)n

=

= D0

(1

1 + k+

1

(1 + k)2+ . . .+

1

(1 + k)n

) P0(1 + k) = D0

(1 +

1

1 + k+ . . .+

1

(1 + k)n1

)By subtracting P0 from P0(1+k), all but two of the elements of the geometric

progressions are eliminated:

29


P0(1 + k) P0 =

= D0(11

1 + k+

1

1 + k . . .+ 1

(1 + k)n1 1

(1 + k)n

P0 = D0

(1 1

(1+k)n

k

)

If we set n =, we get the perpetual rent formula:

P0 =D0k

(2.4)

As we are considering real dividends and real rates of return, note that the

above formula is actually considering nominal dividends which grow at the same

rate of the inflation.

Another particular case of NPV formula is determined by the assumption

that future dividends will grow, in real terms, at a certain constant rate g. If

information about future changes in the economy that will affect earnings, such

as changes in the tax rates or in the share of GDP of corporate profit, are

not available, it could be reasonable for example to assume for the S&P 500

dividends a future growth equal to the expected growth of the national GDP12.

We introduce hereby the Gordon Constant Growth model. Formally:

P0 =D0(1 + g

e)

1 + k+ . . .+

D0(1 + ge)n

(1 + k)n

P01 + k

1 + ge= De

(1 +

1

1 + k+ . . .+

1

(1 + k)n1

)With a procedure identical to the one used to get (2.4), we get the sum of

the geometric series from t = 1 to t = n, which is equal to:

P0 = D0(1 + ge)

(1 1

(1+k)n

k ge

)(2.5)

With n = we get the Gordon model :12As U.S. corporation open more and more international branches around the world, we

could also consider the expected global GDP growth in order to get the expected growth rateof dividends in the S&P 500

30


Figure 2.1: Price of a share at time 0, function of the future expected returnsat time 0 -Figure A- and of the future expected growth rate of real dividends.Dividend level at time 0: 100. Future expected growth rate in Figura A: 1.5%.Future expected returns in Figure B: 6.5%. Personal calculations.

P0 =D0(1 + g

e)

k ge(2.6)

Where:

P0 = f(+

D0,+

ge,k)

In this model the actual price of a share becomes function of three factors:

the current level of dividends, the expected growth rate and the required rate

of return. In the following paragraphs we will analyse all these three elements.

Before doing that it is useful to do a simple sensitivity analysis of the three

factors in order to understand how much the price should fluctuate when these

parameters change. In particular, the elasticities of P to D0, ge, k are equal to:

P0D0 =P0D0

D0

P0=

(1 + ge

k ge

)D0

P0

P0ge =P0ge

ge

P0=

(D0(1 + k)

(k ge)2

)ge

P0

P0k =P0k

ge

P0=

(D0(1 + k)

(k ge)2

)ge

P0

In the first case, P is directly proportional to the level of dividends at time

0. Regarding the expected steady state growth rate and the cost of capital,

differently, their impact of rational prices are represented in 2.1. Note how very

small variations in both the variables can justify important fluctuations of the

31


Figure 2.2: Ratio of the current price of a share due to actualized future cashflows from time 0 to time x. A constant expected real growth rate of dividendsof 1.5% and a 6.5% real cost of capital are used. Personal calculations.

value of P0. In particular, as the expected future growth rate gets near to the

cost of capital, the denominator of the Gordon model tends to zero and prices

strongly rise.

2.1.1 The long run matters

The return rate k used to discount future cash flows makes payments made later

in time less important than near ones. While pricing an asset, anyway, one must

not undervalue the importance of cash flows which are very distant in time.

To highlight this importance, we compute how much of the current price of

a share is due to cash flows paid before a future date, displayed of the x-axis

of figure 2.2. From the Gordon Growth Model we know that a share paying a

real dividend of 100$ every year, with a constant expected growth rate of real

dividends of 1,5% and with a real cost of capital of 6.5% should be priced today

2000$. Figure 2.2 shows that the cumulated actualized cash flows from year 1

to year 10 account for just 38% of this value. Actualized cash flows from the

first 20 years account for about 61% of the total and, after 50 years of actualized

flows, 10% of the value is still due to the following periods.

This inherent property of the NPV model has important consequences for

investors when pricing a share or evaluating the impact of new information on

market value. In particular, investors should carefully try to understand which

news will have a structural and long lasting impact on corporations and which

are only contingencies.

32


Example: positive shock on the dividend level

Suppose that the previously considered share, with a current dividend level of

100, is expected to experience a positive shock that will make earnings and

dividends deviate from their natural trend for the first 10 years. Lets suppose a

shock that adds 10 to D1, 9 to D2. . . 1 to D10. Then:

Pi =(100)(1, 015) + 10

1.065+

(100)(1, 015)2 + 9

1.0652+ . . .+

(100)(1, 015)11

1.06511+ . . .

. . .+(100)(1, 015)n

1.065n= 2043, 2

.

This 10% initial shock on the dividend level should impact price of asset i by

43, which is about 2% of the pre-shock value. This simulation shows that relevant

but temporary shocks on earnings and dividends level should not influence an

asset price considerably. Any variation greater than 2%, in the given example,

would be a sign of investors over-reaction to positive news.

Example: negative monetary shock

We now consider the effect of a negative monetary shock. Lets suppose that

the market premium on risk free assets required from investors is 4.5% and that

the central bank suddenly rise real interest rates from what investors think is

the structural interest rate, for example 2%, to 7%. Lets than suppose that it

is unreasonable to think that this rate will be kept this high forever and that

reversion will start after 5 years by 1% per year. Interest rates, thus, are expected

to reach 2% again at t=10. In this case, the shock affects not only discounts rate

for dividends paid before time 10, but also the way in which all future dividends

are discounted. Formally:

Pi =(100)(1, 015)

1.115+

(100)(1, 015)2

1.1152+ . . .+

(100)(1, 015)6

(1, 115)5(1.095)+

+(100)(1, 015)7

(1, 115)5(1.095)(1.085)+ . . .+

(100)(1, 015)n

1.065n= 1519, 5

.

The shock produced a significant 25% variation on the rational price levelaccording di NPV. This example clearly shows how relevant monetary policy is

in determining financial markets valuations and how large its impact on current

stock prices can be.

33


2.1.2 Dividend policy

It could be argued that several corporations do not distribute dividends and that,

consequently, it is not always possible to use the Gordon growth model to price

shares. Even if this claim is correct, anyway, one must not be confused and try

to use future earnings instead of future dividends when pricing an asset. Firstly,

as long as firms earn the same return on its retained earnings as shareholders

demand on its stock, then future dividend policy should not impact market

value of the firm13. Retained earnings should generate future higher dividends

that have the same actualized value at t=0 as dividends that would have been

otherwise paid.

Evaluating stocks as the present discounted value of future earnings, then,

greatly overstates the value of a firm. As Miller and Modigliani argued14, this

method would counts earnings benefits twice: earnings are discounted as if they

were distributed to shareholders and ready to be used for other investments

when they actually are retained, generating an additional growth in future

earning which is priced as well. In order to show this, we price two different

firms A and B that, at time 0, are identical, have in equilibrium a cost of equity

of 10% and have a real earning per share of 100. Suppose now that firm A

chooses to distribute all its earnings while firm B decides to re-invest all of them

and obtain the 10% yearly return. If everything else remains stable, firm A will

continue to earn 100$ per year and distribute everything, while firm B earnings

will grow in a geometric progression. Pricing A and B by using their future

earnings instead of their future dividends would bring to the following paradox:

PA =100

1.1+

100

1.12+ . . .+

100

1.1n=

100

0.1= 1000

PB =100

1.1+

110

1.12+ . . .+

(100)(1.1)n

1.1n=

(100)(1.1)

0.1 0.1=

Thus, not only two identical firms would have different valuations, but also

the valuation of B would not lack of sense. This example confirm the fact that

only future dividends and cash flows for the investor should be considered when

pricing a share.

13We will see in chapter 4 that this, if taxation is taken into account, could be untrue dueto a deferall benefit.

14Shiller 1981, op.cit

34


2.2 The market equity premium

In the introduced NPV pricing model, weve seen how important the required

cost of capital k is in determining the current price of an asset or of a share of

equity, as very small movements of k can determine huge fluctuations in these

value. It is essential, then, to understand how to compute this rate in order to

price any kind of asset.

The notion that riskier investments, under risk aversion should have higher

expected returns than safer investments, is central to modern finance theory.

Considering this assumption we can expect the return on any investment to be

equal to the required risk free rate of return rrf plus a premium rate i for the

risk taken. Then:

ki = krf + i

The same applies to the market in aggregate and for the S&P Composite

proxy. The difference in any particular period between the actual rate of return

on a risky asset and the risk-free rate is called excess return15. The disagreement

among both academics and practitioners remains on both:

how to measure risk of an investment and how to forecast future risk;

how to price this risk and convert it into a risk premium i that compensatesinvestors for the variance of future returns.

Regarding the first point, finance theories agree on the fact that the risk of

any investment has an idiosyncratic component, due to specific characteristics

of the asset itself, and a systematic part, which affects all the assets in the

market and which cannot be further diversified. Risk should be measured from

the perspective of a well-diversified investor who, consequently, will measure and

price only the latter component.

We can define the risk that cannot be further diversified in terms of variance

in actual returns around an expected return16. Excess return on the market

portfolio, in particular, can be seen as proportional to the expected standard

deviation of its returns.

t = rem,t+1 krf = (em,t+1) (2.7)

15Bodie, Kane, Marcus 2009 (op.cit.).16An investment is riskless if rt = k

erf,tt.

35


Where rem,t+1 is the expected return of the market. If EMH holds, rem,t+1 =

kt+1 and:

kt = krf,t + (em,t+1) (2.8)

While we have models, such as the CAPM 17, to estimate the return of a

single asset as a function of its correlation with the market portfolio and the

market risk premium, it is difficult to create models that predict the future risk

of the market in aggregate and its future excess return. Systematic risk depends

upon different sources that impact the whole economy, such as fiscal, monetary,

and regulatory policy, natural disasters, international conditions or technological

innovations. As there is no natural level of risk in the market, what we can do

is to look again at figure 1.5 and estimate future risk as a function of past

standard deviations. We need to keep in mind that, even though the 5-years

standard deviation on returns has been fluctuating for most of the past century

between 8 and 10%, there have been past periods when m of total real returns

have diverged significantly from its historical average. The evidence from the

1929-1946 period, in this sense, is emblematic. Nothing guarantees that future

standard deviation will remain almost stationary18. When we determine em,t+1,

so, we should also take into account the eventuality of this future periods of high

uncertainty.

If we manage to compute e realistic em,t+1, we still need to convert this

measure into a risk premium. From equation (2.8) we see that, given a em,t+1

and a krf,m, i is ultimately determined by the parameter . Note that this

parameter is equal to the definition of Sharpe Ratio, which indicates the amount

of extra return on the risk free asset required from investors for every unit of

standard deviation taken.

Damodaran suggests three approaches to simplify this approach and quantify

the t that investors will be likely to consider during their investments decisions:

survey investors or managers with the intent of finding out what they re-quire as a premium for investing in equity as a class, relative to the risk-free

rate;

back out an equity risk premium from market prices today;

look at the premiums earned historically by investing in stocks, as opposedto risk-free investments.

17Note that the CAPM, despite its wide diffusion among academics and financial operators,is capable of predicting only a fraction of the total variation in asset returns.

18This evidence is better analysed in chapter 1

36


In the following sections we will focus on the latter methodology. Using the

categorization suggested by Damodaran, we examine some of the factors that

influence and em,t+1.

(1) Risk aversion

The first and most important influencing factor of the market premium is in-

vestors aversion to standard deviations of future returns. Risk aversion varies

among investors depending on elements such as their age, their different prefer-

ences for consumption and their education.

Regarding age, for example, people have distinct financial needs at differ-

ent periods of their life, typically borrowing when young, investing in stocks

and bonds for retirement when middle-aged, and disinvesting during retirement.

Early literature on the subject claimed that individuals become less risk prone

as they get older: according to this view, the younger the average investor, the

lower the aggregate market risk premium should be19. Even though this point

is controversial20, demographic trends are still considered to have a huge impact

on market premiums. According to Geanakoplos and Quinzii (2000):

It seems plausible that a large middle-aged cohort seeking to save

for retirement will push up the prices of securities, and that prices

will be depressed in periods when the middle-aged cohort is small.

Bakshi, Chan and others argued that price-earnings (PE) ratios have moved

over the last century proportionally to the ratio of middle-aged to young adults

in the U.S. Note that if markets were not myopic and did forecast and consider

the impact of demographic trends in advance21 such a relation should not exist.

As far as consumption preferences are concerned, differently, we expect equity

risk premiums to be lower in markets where individuals prefer to consume tomor-

row rather than today and are net savers than in markets where individuals

are net consumers. Equity risk premiums, hence, should be a positive function

of investors marginal propensity to consume today.

In determining the market risk premium and, in particular, factor , we need

to consider how consumption preferences vary in aggregate over time in order to

understand which are the preferences of the theoretical marginal investor.

19Bakshi and Chen examined risk premiums in the United States and noted an increase inrisk premiums as investors aged. See Bakshi and Chan 1994, op. cit.)

20The observation, exploited by Bakshi and Chen, that young people are more risk-tolerantthan old people, suggests that the equity premium should be smallest when the proportion ofyoung people is highest. But this is exactly contrary to the historical record. Geanakoplos andQuinzii (2000)

21Demography is the future that already happened. - Peter Drucker.

37


(2) Inflation

The equity risk premium should be lower in an economy with lower expected

em,t+1. Predictable inflation, stable interest rates and stable economic growth

are amongst the factors that reduce this variable.

Literature examined the relationship between equity risk premium and in-

flation but did not reach any definitive conclusion. Brandt and Wang (2003)

present evidence that risk premiums in equity markets tend to increase when

inflation is higher than expected and decrease when it is lower than expected.

According to Damodaran (2011), it seems reasonable to conclude that it is not

so much the current level of inflation that influences equity risk premiums but

uncertainty about that level.

(3) Information asymmetries

If all existing information was available with high transparency to every oper-

ator in the economy, investors could evaluate their investments by only consid-

ering expected fundamental values and their expected volatility. This volatility,

anyway, does increase as information asymmetries arise and as we move from

markets where the quantity and quality of information available is good to stock

exchanges where information cannot be considered reliable. Differences in the

level of information efficiency may be one reason why investors demand larger

risk premiums in some emerging markets.

Changes in both the quantity and quality of information available to investors

can occur also in time, as regulators improve their controls and ITC makes

information transmission easier. According to Damodaran, during the market

boom in the late 1990s, there were some operators who argued that the higher

market prices observed in that period reflected the fact that investors had access

to more information about their investments and were thus requiring lower risk

premiums than before.

(4) Liquidity

Liquidity represents the degree to which a security can be rapidly traded in the

market without affecting its price. Market liquidity is strictly related to the

concept of market deepness: the deeper a market is, the higher the volume of

investors willing to trade in both directions and the larger is the number of shares

that can be bought and sold without changes in current market price.

When volumes or the presence of market markets in a stock exchange are not

sufficient to provide liquidity, investors may have to accept large discounts on the

current market value to promptly liquidate equity positions and, consequently,

38


they will be willing to pay less for equities. It should not be a surprise that higher

risk premiums and average higher returns are observed on stocks which are not

publicly traded or in the private equity industry. The same can be said about

young stock exchanges in developing economies or small capitalization stocks,

where prices may be determined by the actions of a relatively small amount of

investors.

The notion that developed markets for publicly traded stocks are wide and

deep has led to the argument that illiquidity should not have an high impact

on the prices of an aggregate portfolio, such as the S&P Composite. However

according to Damodaran (2011), we need to be sceptical about this argument.

The cost of illiquidity in aggregate, in fact, can vary over time as a consequence

of economic cycles. During a period of financial crisis, for example, credit crunch

phenomenons can lead financial operators to sell their equity positions all at the

same time to avoid internal illiquidity and, if there are not enough counterparts

willing to buy at current market prices, this can highly increase equity market

risk premium.

(5) Black Swan Risk

When investing in equities, there is always the potential for events that occur

infrequently but can cause dramatic consequences. The rise of Nazism, the great

29-32 depression in the U.S., WWII, the Japanese collapse of stock markets in

the late 1980s and the 11 September attack are amongst this kind of events.

Such once-in-a-lifetime circumstances may be so difficult to forecast and so rare

that the standard deviations of ex-post past returns could not be representative

of ex-ante expected standard deviations. Talebs Black Swanw are a general

extension of this kind of events.

A Black Swan is an event with the following three attributes.

First, it is an outlier, as it lies outside the realm of regular expecta-

tions, because nothing in the past can convincingly point to its possi-

bility. Second, it carries an extreme impact. Third, in spite of its out-

lier status, human nature makes us concoct explanations for its occur-

rence after the fact, making it explainable and predictable. . .Rarity,

extreme impact, and retrospective predictability. A small number of

Black Swans explains almost everything in our world, from the success

of ideas and religions, to the dynamics of historical events. . . 22

22Quotation from The Black Swan: The Impact of the Highly Improbable. Nassim Taleb,The New York Times Press, 2007.

39


Despite their low probability, this kind of events must be taken into account

when computing expected volatility and required risk premiums. Some argue

that it is due to the eventuality of this catastrophic shocks that risk premiums

and stock prices have been fluctuating more than they should have according to

what really happened in the past. We will come back on this point in Chapter

4.

Determine the Market premium from past history

Under adaptive expectations, in economics, people form their beliefs about future

events based on what has happened in the past, considering thus past values

as the best estimate of future values. In this section we analyse how, under

this assumption, future market premiums and future returns should expected

depending on how much in the past we go.

Figure 2.3: Geometric average of the S&P Composite Total Return, computedfrom time x, shown on the x-axis, to March 2012. Personal calculations fromR.J. Shillers Data.

Figure 2.3 shows the geometric moving average of past total real yearly re-

turns computed from time periods on the x-axis to March 2012. This average

can be computed either by compounding return rates from a year t to 2012, or

by more simply using the previously Total Real Return index TRI. Formally:

40


rt,2012 =

(2012i=t

(1 + ri)

) 12012t

1 =(TRI2012TRIt

) 12012t

1

Where ri is equal to the total real return from year i1 to year i. As we couldexpect, the longer the period, the less the geometric moving averages change and

the nearer they get to the historical geometric average rate of total real returns

of 6.46%.

In order to derive the historical equity risk premium t,2012 from t to January

2012 we compute, according to equation (2.8), the geometric moving average of

past risk free rates an subtract it from rt,2012 for every t in the sample. Formally:

t,2012 = rt,2012 rf t,2012 = rt,2012

(2012i=t

(1 + rfi)

) 12012t

1

Where rf t,2012 is the geometric moving average of past risk free rates from

year t to 2012 and rfi is the one year yield of 10-years U.S. t-bonds at time i.

Figure 2.4 shows how the market premium varies depending on how much we

go back in time. Regarding risk premiums, if we dont consider the negative

values from the last 15 years, they have been fluctuating between a low 2,7%

for t = 1973 and an high of 5,8% for t = 1982, with an historical geometric

average from 1872 of 4.2%.

If we want to use past market performance to predict the future, we must

be aware of two main issues. Firstly, the nearer we go in the past, the more the

exact date in the past from which we start considering returns is relevant for the

determination of the geometric average. This is evident from the volatility of

figure 2.3, which gets higher as we get nearer to the current date. We get rid of

this problem if we consider a geometric average going sufficiently back in time.

What we have to consider, on the other hand, is that the more we go back in

time, the more is probable that preferences, rules and the structure of the econ-

omy have significantly changed. Risk premiums computed at the end of the XIX

century could be no longer reflective of premiums of a developed economy with

mature and regulated financial markets. As we mentioned before, for instance,

the lower transaction costs and the higher availability of information as made

possible by the ICT revolution could have lowered the required market premiums

over the last 20 years.

41


Figure 2.4: Average historical risk premium on the S&P Composite, computedfrom time x, shown on the x-axis, to January 2012. Personal calculations fromR.J. Shillers Data.

With notions on market efficiency, pricing models and methods to estimate

future risk premiums the base of past values, in the following chapters we will

analyse how much the real behaviour of the S&P Composite can be considered

a reflection of the fundamental value of the equity market in aggregate. If this

is not the case, than abnormal returns opportunities could be identified ex-post

and interpreted in the light of the behavioural finance literature.

42

Chapter3Abnormal returns from fundamental

analysis

A number of tests on the semi-strong form of market efficiency measure the

reactivity of prices in response to new useful information. Even when price

adjustments occur rapidly, it still remains possible that securities remain persis-

tently over- or under-valued over long periods of time. It is more difficult to test

whether prices conform to fundamental values, than it is to test whether prices

respond appropriately to information. Financial literature, nonetheless, has also

evolved in this direction.

Considerable empirical tests have been conducted to determine the predictabil-

ity of market returns on the basis of different initial valuation parameters. Fama

and French (1988) and Campbell and Shiller (1988) analysed the relation be-

tween future returns and initial dividend yields, concluding that as much as 40

percent of the variance of future returns for the stock market in aggregate can be

explained by different levels of initial dividend yields. As dividend payout ratios

have been continuously declining over the last half century, however, Malkiel

(2003) claims dividend yield may not be as predictive of future returns as in the

past.

In his book Irrational Exuberance, Shiller (2009) analysed the historical nega-

tive relation between the Price-Smoothed Earnings ratio and subsequent returns

from a long-term buy and hold investment His hypothesis is that extreme Price-

Earning values could be an indicator of market mispricing and that, hence, reflect

the tendency of operators to over-react to information and to be subject to ir-

rational behaviours. Hereby we reproduce Shillers analysis and extend it to

consider also extra returns and dividend returns.

43

3. Abnormal returns from fundamental analysis

3.1 The P/E10 ratio

Price earnings are amongst the most used indicators in fundamental analysis and

represent the ratio between the market price of a share and the earning per share

level at in a certain period. Given the previously defined net present value of a

share, we can compute the Price-Earnings ratio from the Price-Dividends ratio

which, over an infinite period and under the assumption of stable future growth

of dividends is equal to the Gordon growth model:

PtDt

=

(1 + ge

k ge

)Where ke is the

Documents

Market Efficiency in the Long Run