market efficiency in the long run
DESCRIPTION
Essay on Market Efficiency over the long run, analyzing the S&P 500 composite index historical performance to support aggregate markets Irrational Exuberance and behavioural finance hypoteses.BSc Thesis with honors in Economics and Finance from Scuola Superiore Sant'Anna - University of Pisa by Niccolò Ferragamo - 2012TRANSCRIPT
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Market Efficiency and Bubbles in the long run:
an empirical test on the S&P Composite
Niccolo Ferragamo
April 30, 2012
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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
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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
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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
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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.
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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
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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.
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1. Historical evidence from Stock Returns
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.
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1. Historical evidence from Stock Returns
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
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1. Historical evidence from Stock Returns
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.
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1. Historical evidence from Stock Returns
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.
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1. Historical evidence from Stock Returns
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.
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1. Historical evidence from Stock Returns
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
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1. Historical evidence from Stock Returns
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.
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1. Historical evidence from Stock Returns
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.)
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1. Historical evidence from Stock Returns
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
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1. Historical evidence from Stock Returns
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
-
1. Historical evidence from Stock Returns
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
-
1. Historical evidence from Stock Returns
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
-
1. Historical evidence from Stock Returns
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
-
1. Historical evidence from Stock Returns
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
-
1. Historical evidence from Stock Returns
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
-
2. Pricing and Market Efficiency
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
-
2. Pricing and Market Efficiency
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
-
2. Pricing and Market Efficiency
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
-
2. Pricing and Market Efficiency
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
-
2. Pricing and Market Efficiency
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
-
2. Pricing and Market Efficiency
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
-
2. Pricing and Market Efficiency
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
-
2. Pricing and Market Efficiency
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
-
2. Pricing and Market Efficiency
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
-
2. Pricing and Market Efficiency
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. Pricing and Market Efficiency
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
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2. Pricing and Market Efficiency
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
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2. Pricing and Market Efficiency
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
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2. Pricing and Market Efficiency
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
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2. Pricing and Market Efficiency
(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
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2. Pricing and Market Efficiency
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
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2. Pricing and Market Efficiency
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
-
2. Pricing and Market Efficiency
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
-
2. Pricing and Market Efficiency
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
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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
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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