LECTURE 4 :
EFFICIENT MARKETS AND
PREDICTABILITY OF STOCK
RETURNS
(Asset Pricing and Portfolio Theory)
Contents

EMH
– Different definitions
– Testing for market efficiency
Volatility tests and Regression based models
 Event studies



Are stock returns predictable ?
Making money ?
Introduction


Debate between academics and
practitioners whether financial markets
are efficient
Are stock return predictable ?
– Implications for active and passive fund
management.
– Market timing : switching between stocks
and T-bills
Martingale and Fair Game
Properties

Stochastic variable : E(Xt+1|Wt) = Xt
– Xt is a martingale
– The best forecast of Xt+1 is Xt

Stochastic process : E(yt+1|Wt) = 0
– yt is a fair game




If Xt is a martingale than yt+1 = Xt+1-Xt is a fair game
From EMH : for stock markets : yt+1 = Rt+1 – EtRt+1
implies that unexpected stock returns embodies a fair
game
Constant equilib. required return : Et(Rt+1 – k)|Wt) = 0
Test : Rt+1 = a + b’Wt + et+1
Martingale and Random
Walk




Stochastic variable : Xt+1 = d + Xt + et+1
where et+1 is iid random variable with Etet+1 = 0 and
no serial correlation or heteroscedasticity
Random walk without drift : d = 0
If Xt+1 is a martingale and DXt+1 is a fair game (for d =
0)
Random walk is more restrictive than martingale
– Martingale process does not put any restrictions on higher
moments.
Formal Definition of the
EMH


fp(Rt+n| Wtp) = f(Rt+n| Wt)
hpt+1 = Rt+1 – Ep(Rt+1 | Wpt)
Three types of efficiency
– Weak form :

Information set consists only of past prices (returns)
– Semi-strong form :

Information set incorporates all publicly available information
– Strong form :

Prices reflect all information that are possible be known,
including ‘inside information’.
Empirical Tests of the
EMH


Tests are mainly based on the semi-strong
form of efficiency
Summary of basic ideas constitute the EMH
– All agents act as if they have an equilibrium
model of returns
– Agents possess all relevant information, forecast
errors are unpredictable from info available at
time t
– Agents cannot make abnormal profits over a
series of ‘bets’.
Testing the EMH

Different types of tests
– Tests of whether excess (abnormal)
returns are independent of info set
available at time t or earlier
– Tests of whether actual ‘trading rules’ can
earn abnormal profits
– Tests of whether market prices always
equals fundamental values
Interpretation of Tests of
Market Efficiency


EMH assumes information is available at zero
costs  Very strong assumption
Market moves to ‘efficiency’ as the well
informed make profits relative to the less well
informed
– Smart money sells when actual price is above
fundamental value
– If noise traders (irrational behaviour) are present,
the rational traders have to take their behaviour
also into account.
Implications of the EMH
For Investment Policy

Technical analysis (chartists)
– Without merit

Fundamental analysis
– Only publicly available info not known to
other analysis is useful
– Active funds do not beat the market
(passive) portfolio)
Predictability of Returns
A Century of Returns


Looking at a long history of data we find (Jan. 1915 –
April 2004) :
Price index only (excluding dividends).
– S&P500 stock index is I(1)
– Return on the S&P500 index is I(0)
– Unconditional returns are non-normal with fat tails.





Number of observations (Jan 1915 – April 2004) : 1072 prices and
1071 returns
Mean = 0.2123%
SD = 5.54%
From normal distribution would expect to find 26.76 months to
have worse return than 2.5th percentile (-10.64%)
In the actual data however, we find 36 months !
Jan-03
Jan-95
Jan-87
Jan-79
Jan-71
Jan-63
Jan-55
Jan-47
Jan-39
Jan-31
Jan-23
Jan-15
US Real Stock Index : S&P500
(Jan 1915 – April 2004)
80
70
60
50
40
30
20
10
0
US Real Stock Returns : S&P500
(Feb. 1915 – April 2004)
0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
Feb-15
Feb-27
Feb-39
Feb-51
Feb-63
Feb-75
Feb-87
Feb-99
US Real Stock Returns : S&P500
(Feb. 1915 – April 2004)
120
100
Frequency
80
60
40
20
0
-0.15
-0.11
-0.07
-0.03
0.01
0.05
0.09
0.13
Volatility of S&P 500

GARCH Model :
Rt+1 = 0.00315 + et+1
[2.09]
ht+1 = 0.00071 + 0.8791 ht + 0.0967 et2
[2.21]
[33.0]
[4.45]
Mean (real) return is 0.315% per month (3.85% p.a.)
Unconditional volatility :
s2 = 0.00071/(1-0.8791-0.0967) = 0.0007276
SD = 2.697% (p.m.)
Conditional Var. : GARCH (1,1)
Model (Feb. 1915 – April 2004)
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
Feb-15
Feb-27
Feb-39
Feb-51
Feb-63
Feb-75
Feb-87
Feb-99
Return’s Data
Stocks : Real Returns
(1900 – 2000)
Inflation
Real Returns
Arith.
Geom
Arith.
Mean
SD
s. e.
Geom
Mean
Min.
Max.
UK
4.3
4.1
7.6
20.0
2.0
5.8
-57
+97
USA
3.3
3.2
8.7
20.2
2.0
6.7
-38
+57
World
N.A.
N.A.
7.2
17.0
1.7
6.8
N.A.
N.A.
Dimson et al (2002)
(1974) (1975)
(1931) (1933)
Bonds : Real Returns
(1900 – 2000)
Inflation
Real Return
Arith.
Geom.
Arith.
Mean
SD
s.e.
Geom.
Mean
UK
4.3
4.1
2.3
14.5
1.4
N.A.
USA
3.3
3.2
2.1
10.0
1.0
1.6
World
N.A.
N.A.
1.7
10.3
1.0
1.2
Dimson et al (2002)
Bills : Real Return
(1900 – 2000)
Inflation
Real Return
Arith. Geom. Arith.
Mean
SD
s.e.
UK
4.3
4.1
1.2
6.6
0.7
USA
3.3
3.2
1.0
4.7
0.5
Dimson et al (2002)
Average Return (percent)
US Real Returns (Post 1947) :
Mean and SD (annual averages)
NYSE decile size sorted portfolios
16
Equally weighted, NYSE
12
S&P500
Value weighted, NYSE
8
4
Corporate Bonds
T-Bills
Government Bonds
0
4
8
12
16
20
24
28
Standard deviation of returns (percent)
32
Simple Models


EtRt+1  rt + rpt
Assuming that k and rp are constant than :
Rt+1 = k + g’Wt + et+1
or
Rt+1–rt = k + g’Wt + et+1


Tests : g’ = 0
Wt can contain : past returns, D-P ratio, E-P
ratio, interest rates
Long Horizon Returns
Evidence of mean reversion in stock returns
Rt,t+k = ak + bk Rt-k,t + et+k
Fama and French (1988) estimated
models for k = 1 to 10 years

Findings :
– Little or no predictability, except for k = 2 and 7 years
 b is less than 0.
– k = 5 years  b -0.5; -10% return over previous 5
years, on aver., is followed by a +5% over next 5 years
US Long Horizon Returns
Dimson et al (2002)
Poterba and Summers
(1988) : Mean Reversion


ht,t+k = (pt+k – pt) = km + (et+1 + et+2 + … + et+k)
Under RE, the forecast errors et are iid with zero mean
Etht,t+k = km and Var(ht,t+k) = ks2

If log-returns are iid, then
Var(ht,t+k) = Var(ht+1 + ht+2 + … + ht+k) = kVar(ht+1)

Variance ratio statistic
VRk = (1/k) [Var(ht,t+k)/Var(ht+1)] ≈ 1 + 2/k S(k-j)rj

Findings :
VR > 1 for lags of less than 1 year
VR < 1 for lags greater than 1 year (mean reversion)
VR of Equity Returns
Country
1 Year
3 Year
5 Year
10 Year
Monthly Data, Jan 1921 – Dec 1996
US
1.0
0.994
0.990
0.828
UK
1.0
1.008
0.964
0.817
Global
1.0
1.211
1.309
1.238
-
0.712
0.571
0.314
Median VR
-
0.960
0.916
0.810
5th percent
-
0.731
0.598
0.398
Test stats,
5%, 1-sided
MCS (Normality)
Long-Horizon Risk and
Return : 1920 – 1996
Probability of Loss
1 year
5 years
10 years
US (Price change)
36.6
34.3
33.7
US (total Return)
30.8
20.7
15.5
UK (Price change)
40.3
32.5
45.2
UK (total Return)
30.1
22.1
30.8
Median (P. change)
– 30 countries
48.2
46.8
48.2
Median (total Ret.)
– 15 countries
36.1
26.9
19.9
Global index (P. c.)
37.8
35.4
35.2
Global index (t. R.)
30.2
18.2
12.0
Predictability and Market
Timing



Cochrane (2001) estimates
Rt,t+k = a + b(D/P)t + et+k
US data, 1947-1996
– for one-year horizons : b ≈ 5 (s.e. = 2),
R2 = 0.15
– for 5 year horizons : b ≈ 33 (s.e. = 5.8),
R2 = 0.6
1 - Year Excess Returns
US : 1 Year returns : 1947 - 2002 (actual, fitted)
60
50
40
30
20
10
0
1940
-10
-20
-30
-40
1950
1960
1970
1980
1990
2000
2010
5 – Years Excess Returns
US : 5 year returns : 1947 - 2002 (actual, fitted)
120
100
80
60
40
20
0
-201940
-40
-60
-80
1950
1960
1970
1980
1990
2000
2010
Price-Dividend Ratio :
USA (1872-2002)
100
90
80
70
60
50
40
30
20
10
0
1860
1880
1900
1920
1940
1960
1980
2000
2020
Predictability and Market
Timing (Cont.)





Cochrane (1997) – estimation up to 1996
Rt+1 = a + b(P/D)t + et+1
(1.)
(P/D)t+1 = m + r(P/D)t + vt+1
(2.)
Predict P/D1997 using equation (2.) and than
R1998 using (1.), etc.
Findings :
Equation predicts excess return for 1997 to be -8%
p.a. and for 2007 -5% p.a.
1-Year Excess Return and PD
Ratio : Annual US Data, 1947-02
60
Excess Return
50
40
30
20
10
0
-10 0
-20
20
40
-30
-40
P-D ratio
60
80
Cointegration and ECM

Suppose in the ‘long-run’ the dividend-price ratio is
constant (k)
d-p=k
or p – d = 1/k
where p = ln(P) and d = ln(D)

Regression model :
Dpt = a0 + b1’(L)Ddt-1 + b2’(L)Dpt-1 – g(z-k)t-1 + et
where z = p-d
MacDonald and Power (1995)
Annual US data 1871-1976(1987)
R2 ≈ 0.5
Profitable Trading
Strategies ?

Pesaran and Timmermann (1994) ‘Forecasting Stock
Returns : …’, Journal of Forecasting, 13(4), 335-67
– Excess returns on S&P500 and Dow Jones indices over one
year, one quarter and one month.
– SMPL 1960 – 1990 (monthly data)
– 3 Portfolios :



Market portfolio (passive)
Switching portfolio (active)
T-bills
– If predicted excess return (model based on fundamentals) is
positive then hold the market portfolio of stocks, otherwise
bills/bond.
– Switching strategy dominates the passive portfolio
Predicting Returns and
Abnormal Profits : S&P500
Market Port.
Switching Port.
T-Bills
Transaction Costs
Stocks
Bills
0.0
0.5
1.0
0.0
0.5
1.0
-
-
-
-
-
0.0
0.1
0.1
0.0
0.1
Sharpe Ratio
0.31
0.30
0.30
0.82
0.79
0.76
Wealth at end of period ($ 100 invested in Jan. 1960)
1,913 1,884 1,855 3,833 3,559 3,346 749
726
Risk Adjusted Rate of
Return

Can ‘predictability’ be used to make
profits adjusted for risk and transaction
costs ?
– Transaction costs : bid – ask spread (and
other commission)
– Risk adjusted rate of return measures
Sharpe ratio :
 Treynor ratio :
 Jensen’s alpha :

SR = (ERp – rf)/sp
TR = (ERp – rf)/bp
(Rp – rf)t = a + b(Rm-rf)t
Summary





Different forms of market efficiency
Important implications if market are
efficient, opportunities if markets are
inefficient
Hong horizon returns are less risky than
returns over short horizons
Predictability of returns – difficult
Some variable have been identified which
help to predict stock returns
References


Cuthbertson, K. and Nitzsche, D.
(2004) ‘Quantitative Financial
Economics’, Chapters 3 and 4
Cuthbertson, K. and Nitzsche, D.
(2001) ‘Investments : Spot and
Derivatives Markets’, Chapter 13
References



Jorion, P. (2003) ‘The Long-Term Risk of
Global Stock Markets’, University of CaliforniaIrvine Discussion Paper
Dimson, E., Marsh, P. and Staunton, M. (2002)
Triumph of the Optimists : 101 Years of Global
Investment Returns, Princeton University
Press
Cochrane, J.H. (2001) ‘Asset Pricing’,
Princeton University Press
References



MacDonald, R. and Power, D. (1995) ‘Stock Prices,
Dividends and Retention : Long Run Relationship and
Short-run Dynamics’, Journal of Empirical Finance, Vol.
2, No. 2, pp. 135-151
Pesaran, M.H. and Timmermann, A. (1994) ‘Forecasting
Stock Returns : An Examination of Stock Market Trading
in the Presence of Transaction Costs, Journal of
Forecasting, Vol. 13, No. 4, pp. 335-367.
Cochrane, J.H. (1997) ‘Where is the Market Going?’,
Economic Perspectives (Federal Reserve Bank of
Chicago), Vol. 21, No. 6.
END OF LECTURE