LECTURE 4 :LECTURE 4 :

EFFICIENT MARKETS AND EFFICIENT MARKETS AND PREDICTABILITY OF PREDICTABILITY OF STOCK RETURNSSTOCK RETURNS

(Asset Pricing and Portfolio (Asset Pricing and Portfolio Theory)Theory)

ContentsContents

EMHEMH– Different definitions Different definitions – Testing for market efficiencyTesting for market efficiency

Volatility tests and Regression based Volatility tests and Regression based modelsmodels

Event studies Event studies

Are stock returns predictable ? Are stock returns predictable ? Making money ? Making money ?

IntroductionIntroduction

Debate between academics and Debate between academics and practitioners whether financial practitioners whether financial markets are efficient markets are efficient

Are stock return predictable ? Are stock return predictable ? – Implications for active and passive Implications for active and passive

fund management. fund management. – Market timing : switching between Market timing : switching between

stocks and T-billsstocks and T-bills

Martingale and Fair Martingale and Fair Game PropertiesGame Properties Stochastic variable : E(XStochastic variable : E(Xt+1t+1||tt) = X) = Xtt

– XXtt is a martingale is a martingale– The best forecast of XThe best forecast of Xt+1t+1 is X is Xtt

Stochastic process : E(yStochastic process : E(yt+1t+1||tt) = 0 ) = 0 – yytt is a fair game is a fair game

If XIf Xtt is a martingale than y is a martingale than yt+1t+1 = X = Xt+1t+1-X-Xtt is a fair game is a fair game From EMH : for stock markets : yFrom EMH : for stock markets : yt+1t+1 = R = Rt+1t+1 – E – EttRRt+1t+1

implies that implies that unexpectedunexpected stock returns embodies a stock returns embodies a fair gamefair game

Constant equilib. required return : EConstant equilib. required return : Ett(R(Rt+1t+1 – k)| – k)|tt) = 0) = 0 Test : RTest : Rt+1t+1 = = + + ’’tt + + t+1t+1

Martingale and Martingale and Random WalkRandom Walk Stochastic variable : XStochastic variable : Xt+1t+1 = = + X + Xtt + + t+1t+1

where where t+1t+1 is iid random variable with E is iid random variable with Ettt+1t+1 = 0 = 0 and no serial correlation or heteroscedasticityand no serial correlation or heteroscedasticity

Random walk without drift : Random walk without drift : = 0 = 0 If XIf Xt+1t+1 is a martingale and is a martingale and XXt+1t+1 is a fair game is a fair game

(for (for = 0) = 0) Random walk is more restrictive than Random walk is more restrictive than

martingale martingale – Martingale process does not put any restrictions on Martingale process does not put any restrictions on

higher moments. higher moments.

Formal Definition of Formal Definition of the EMH the EMH ffpp(R(Rt+nt+n| | tt

pp) = f(R) = f(Rt+nt+n| | tt) )

ppt+1t+1 = R = Rt+1t+1 – E – Epp(R(Rt+1t+1 | | pp

tt))

Three types of efficiency Three types of efficiency – Weak form : Weak form :

Information set consists only of past prices (returns)Information set consists only of past prices (returns)– Semi-strong form : Semi-strong form :

Information set incorporates all publicly available Information set incorporates all publicly available informationinformation

– Strong form : Strong form : Prices reflect all information that are possible be Prices reflect all information that are possible be

known, including ‘inside information’. known, including ‘inside information’.

Empirical Tests of the Empirical Tests of the EMHEMH Tests are mainly based on the semi-Tests are mainly based on the semi-

strong form of efficiency strong form of efficiency Summary of basic ideas constitute the Summary of basic ideas constitute the

EMHEMH– All agents act as if they have an All agents act as if they have an

equilibrium model of returns equilibrium model of returns – Agents possess all relevant information, Agents possess all relevant information,

forecast errors are unpredictable from info forecast errors are unpredictable from info available at time t available at time t

– Agents cannot make abnormal profits over Agents cannot make abnormal profits over a series of ‘bets’. a series of ‘bets’.

Testing the EMHTesting the EMH

Different types of testsDifferent types of tests– Tests of whether excess (abnormal) Tests of whether excess (abnormal)

returns are independent of info set returns are independent of info set available at time t or earlieravailable at time t or earlier

– Tests of whether actual ‘trading Tests of whether actual ‘trading rules’ can earn abnormal profitsrules’ can earn abnormal profits

– Tests of whether market prices Tests of whether market prices always equals fundamental valuesalways equals fundamental values

Interpretation of Tests Interpretation of Tests of Market Efficiencyof Market Efficiency EMH assumes information is available at EMH assumes information is available at

zero costs zero costs Very strong assumption Very strong assumption Market moves to ‘efficiency’ as the Market moves to ‘efficiency’ as the well well

informedinformed make profits relative to the make profits relative to the less well informedless well informed – Smart money sells when actual price is Smart money sells when actual price is

above fundamental valueabove fundamental value– If noise traders (irrational behaviour) are If noise traders (irrational behaviour) are

present, the rational traders have to take present, the rational traders have to take their behaviour also into account. their behaviour also into account.

Implications of the Implications of the EMH For Investment EMH For Investment PolicyPolicy Technical analysis (chartists) Technical analysis (chartists)

– Without merit Without merit Fundamental analysis Fundamental analysis

– Only publicly available info not Only publicly available info not known to other analysis is useful known to other analysis is useful

– Active funds do not beat the market Active funds do not beat the market (passive) portfolio)(passive) portfolio)

Predictability of Predictability of ReturnsReturns

A Century of Returns A Century of Returns

Looking at a long history of data we find (Jan. Looking at a long history of data we find (Jan. 1915 – April 2004) : 1915 – April 2004) :

Price index only (excluding dividends). Price index only (excluding dividends). – S&P500 stock index is I(1) S&P500 stock index is I(1) – Return on the S&P500 index is I(0) Return on the S&P500 index is I(0) – Unconditional returns are non-normal with fat tails. Unconditional returns are non-normal with fat tails.

Number of observations (Jan 1915 – April 2004) : 1072 Number of observations (Jan 1915 – April 2004) : 1072 prices and 1071 returns prices and 1071 returns

Mean = 0.2123%Mean = 0.2123% SD = 5.54%SD = 5.54% From normal distribution would expect to find 26.76 From normal distribution would expect to find 26.76

months to have worse return than 2.5months to have worse return than 2.5thth percentile (- percentile (-10.64%)10.64%)

In the actual data however, we find 36 months !In the actual data however, we find 36 months !

US Real Stock Index : US Real Stock Index : S&P500 (Jan 1915 – April S&P500 (Jan 1915 – April 2004)2004)

0

10

20

30

40

50

60

70

80

Ja

n-1

5

Ja

n-2

3

Ja

n-3

1

Ja

n-3

9

Ja

n-4

7

Ja

n-5

5

Ja

n-6

3

Ja

n-7

1

Ja

n-7

9

Ja

n-8

7

Ja

n-9

5

Ja

n-0

3

US Real Stock Returns : US Real Stock Returns : S&P500 (Feb. 1915 – April S&P500 (Feb. 1915 – April 2004)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 : US Real Stock Returns : S&P500 (Feb. 1915 – April S&P500 (Feb. 1915 – April 2004)2004)

0

20

40

60

80

100

120

-0.15 -0.11 -0.07 -0.03 0.01 0.05 0.09 0.13

Fre

qu

en

cy

Volatility of S&P 500Volatility of S&P 500

GARCH Model : GARCH Model :

RRt+1t+1 = 0.00315 + = 0.00315 + t+1t+1

[2.09][2.09]

hht+1t+1 = 0.00071 + 0.8791 h = 0.00071 + 0.8791 htt + 0.0967 + 0.0967 tt22

[2.21] [33.0] [2.21] [33.0] [4.45] [4.45]

Mean (real) return is 0.315% per month (3.85% p.a.)Mean (real) return is 0.315% per month (3.85% p.a.)Unconditional volatility : Unconditional volatility :

22 = 0.00071/(1-0.8791-0.0967) = 0.0007276 = 0.00071/(1-0.8791-0.0967) = 0.0007276 SD = 2.697% (p.m.)SD = 2.697% (p.m.)

Conditional Var. : GARCH Conditional Var. : GARCH (1,1) Model (Feb. 1915 – April (1,1) Model (Feb. 1915 – April 2004) 2004)

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Feb-15 Feb-27 Feb-39 Feb-51 Feb-63 Feb-75 Feb-87 Feb-99

Return’s DataReturn’s Data

Stocks : Real Returns Stocks : Real Returns (1900 – 2000)(1900 – 2000)

InflationInflation Real ReturnsReal ReturnsArith.Arith. GeoGeo

mmArith.Arith.

MeanMeanSDSD s. e.s. e. GeoGeo

m m MeanMean

Min.Min. Max.Max.

UKUK 4.34.3 4.14.1 7.67.6 20.020.0 2.02.0 5.85.8 -57 -57 (1974(1974

))

+97 +97 (1975(1975

))

USAUSA 3.33.3 3.23.2 8.78.7 20.220.2 2.02.0 6.76.7 -38 -38 (1931(1931

))

+57 +57 (1933(1933

))

WorlWorldd

N.A.N.A. N.A.N.A. 7.27.2 17.017.0 1.71.7 6.86.8 N.A.N.A. N.A.N.A.

Dimson et al (2002)

Bonds : Real Returns Bonds : Real Returns (1900 – 2000)(1900 – 2000)

InflationInflation Real Return Real Return

Arith.Arith. Geom.Geom. Arith. Arith. MeanMean

SDSD s.e.s.e. Geom. Geom. MeanMean

UKUK 4.34.3 4.14.1 2.32.3 14.514.5 1.41.4 N.A. N.A.

USAUSA 3.33.3 3.23.2 2.12.1 10.010.0 1.01.0 1.61.6

WorldWorld N.A.N.A. N.A.N.A. 1.71.7 10.310.3 1.01.0 1.21.2

Dimson et al (2002)

Bills : Real Return Bills : Real Return (1900 – 2000) (1900 – 2000)

InflationInflation Real ReturnReal Return

Arith. Arith. GeomGeom..

Arith. Arith. MeanMean

SDSD s.e.s.e.

UKUK 4.34.3 4.14.1 1.21.2 6.66.6 0.70.7

USAUSA 3.33.3 3.2 3.2 1.01.0 4.74.7 0.50.5

Dimson et al (2002)

US Real Returns (Post 1947) US Real Returns (Post 1947) : Mean and SD (annual : Mean and SD (annual averages)averages)

Standard deviation of returns (percent)

Avera

ge R

etu

rn (

perc

en

t)

0 4 8 12 16 20 24 28 32

4

8

12

16

Government Bonds

Corporate Bonds T-Bills

S&P500 Value weighted, NYSE

Equally weighted, NYSE

NYSE decile size sorted portfolios

Simple ModelsSimple Models

EEttRRt+1t+1 r rtt + rp + rptt Assuming that k and rp are constant Assuming that k and rp are constant

than : than : RRt+1t+1 = k + = k + ’’tt + + t+1t+1

or or

RRt+1t+1–r–rtt = k + = k + ’’tt + + t+1t+1

Tests : Tests : ’ = 0 ’ = 0 tt can contain : past returns, D-P ratio, can contain : past returns, D-P ratio,

E-P ratio, interest ratesE-P ratio, interest rates

Long Horizon ReturnsLong Horizon Returns

Evidence of mean reversion in stock Evidence of mean reversion in stock returnsreturns

RRt,t+kt,t+k = = kk + + kk R Rt-k,tt-k,t + + t+kt+k

Fama and French (1988) estimated Fama and French (1988) estimated models for k = 1 to 10 yearsmodels for k = 1 to 10 years

Findings : Findings : – Little or no predictability, except for k = 2 and 7 years Little or no predictability, except for k = 2 and 7 years

is less than 0. is less than 0. – k = 5 years k = 5 years -0.5; -10% return over previous 5 years, -0.5; -10% return over previous 5 years,

on aver., is followed by a +5% over next 5 yearson aver., is followed by a +5% over next 5 years

US Long Horizon US Long Horizon Returns Returns

Dimson et al (2002)

Poterba and Summers Poterba and Summers (1988) : Mean (1988) : Mean Reversion Reversion hht,t+kt,t+k = (p = (pt+kt+k – p – ptt) = k) = k + ( + (t+1t+1 + + t+2t+2 + … + + … + t+kt+k)) Under RE, the forecast errors Under RE, the forecast errors tt are iid with zero are iid with zero

meanmeanEEtthht,t+kt,t+k = k = k and Var(h and Var(ht,t+kt,t+k) = k) = k22

If log-returns are iid, then If log-returns are iid, then Var(hVar(ht,t+kt,t+k) = Var(h) = Var(ht+1t+1 + h + ht+2t+2 + … + h + … + ht+kt+k) = kVar(h) = kVar(ht+1t+1))

Variance ratio statistic Variance ratio statistic VRVRkk = (1/k) [Var(h = (1/k) [Var(ht,t+kt,t+k)/Var(h)/Var(ht+1t+1)] ≈ 1 + 2/k )] ≈ 1 + 2/k (k-j)(k-j)jj

Findings : Findings : VR > 1 for lags of less than 1 yearVR > 1 for lags of less than 1 yearVR < 1 for lags greater than 1 year (mean reversion)VR < 1 for lags greater than 1 year (mean reversion)

VR of Equity Returns VR of Equity Returns

CountryCountry 1 Year1 Year 3 Year3 Year 5 Year5 Year 10 Year10 Year

Monthly Data, Jan 1921 – Dec 1996Monthly Data, Jan 1921 – Dec 1996

USUS 1.01.0 0.9940.994 0.9900.990 0.8280.828

UKUK 1.01.0 1.0081.008 0.9640.964 0.8170.817

GlobalGlobal 1.01.0 1.2111.211 1.3091.309 1.2381.238

Test stats, Test stats, 5%, 1-sided5%, 1-sided

-- 0.7120.712 0.5710.571 0.3140.314

MCS (Normality)MCS (Normality)

Median VRMedian VR -- 0.9600.960 0.9160.916 0.8100.810

55thth percent percent -- 0.7310.731 0.5980.598 0.3980.398

Long-Horizon Risk and Long-Horizon Risk and Return : 1920 – 1996Return : 1920 – 1996

Probability of LossProbability of Loss

1 year1 year 5 years5 years 10 years10 yearsUS (Price US (Price change)change)

36.636.6 34.334.3 33.733.7

US (total Return)US (total Return) 30.830.8 20.720.7 15.515.5

UK (Price UK (Price change)change)

40.340.3 32.532.5 45.245.2

UK (total Return)UK (total Return) 30.130.1 22.122.1 30.830.8

Median (P. Median (P. change) – 30 change) – 30 countriescountries

48.248.2 46.846.8 48.248.2

Median (total Median (total Ret.) – 15 Ret.) – 15 countriescountries

36.136.1 26.926.9 19.919.9

Global index (P. Global index (P. c.)c.)

37.837.8 35.435.4 35.235.2

Global index (t. Global index (t. R.)R.)

30.230.2 18.218.2 12.012.0

Predictability and Predictability and Market TimingMarket Timing Cochrane (2001) estimates Cochrane (2001) estimates RRt,t+kt,t+k = a + b(D/P) = a + b(D/P)tt + + t+kt+k

US data, 1947-1996 US data, 1947-1996 – for one-year horizons : b ≈ 5 (s.e. = for one-year horizons : b ≈ 5 (s.e. =

2), R2), R22 = 0.15 = 0.15– for 5 year horizons : b ≈ 33 (s.e. = for 5 year horizons : b ≈ 33 (s.e. =

5.8), R5.8), R22 = 0.6 = 0.6

1 - Year Excess 1 - Year Excess Returns Returns

US : 1 Year returns : 1947 - 2002 (actual, fitted)

-40

-30

-20

-10

0

10

20

30

40

50

60

1940 1950 1960 1970 1980 1990 2000 2010

5 – Years Excess 5 – Years Excess Returns Returns

US : 5 year returns : 1947 - 2002 (actual, fitted)

-80

-60

-40

-20

0

20

40

60

80

100

120

1940 1950 1960 1970 1980 1990 2000 2010

Price-Dividend Ratio : Price-Dividend Ratio : USA (1872-2002)USA (1872-2002)

0

10

20

30

40

50

60

70

80

90

100

1860 1880 1900 1920 1940 1960 1980 2000 2020

Predictability and Predictability and Market Timing (Cont.)Market Timing (Cont.) Cochrane (1997) – estimation up to 1996Cochrane (1997) – estimation up to 1996 RRt+1t+1 = a + b(P/D) = a + b(P/D)tt + + t+1t+1 (1.)(1.)

(P/D)(P/D)t+1t+1 = = + + (P/D)(P/D)tt + v + vt+1t+1 (2.)(2.)

Predict P/DPredict P/D19971997 using equation (2.) and using equation (2.) and than Rthan R19981998 using (1.), etc. using (1.), etc.

Findings : Findings : Equation predicts excess return for 1997 to be Equation predicts excess return for 1997 to be -8% p.a. and for 2007 -5% p.a. -8% p.a. and for 2007 -5% p.a.

1-Year Excess Return and PD 1-Year Excess Return and PD Ratio : Annual US Data, 1947-Ratio : Annual US Data, 1947-0202

-40

-30

-20

-10

0

10

20

30

40

50

60

0 20 40 60 80

P-D ratio

Exc

ess

Retu

rn

Cointegration and ECM Cointegration and ECM

Suppose in the ‘long-run’ the dividend-price Suppose in the ‘long-run’ the dividend-price ratio is constant (k) ratio is constant (k)

d - p = k d - p = k or p – d = 1/kor p – d = 1/k

where p = ln(P) and d = ln(D) where p = ln(P) and d = ln(D)

Regression model : Regression model :

pptt = = 00 + + 11’(L)’(L)ddt-1t-1 + + 22’(L)’(L)ppt-1t-1 – – (z-k)(z-k)t-1t-1 + + tt where z = p-dwhere z = p-d

MacDonald and Power (1995)MacDonald and Power (1995)

Annual US data 1871-1976(1987) Annual US data 1871-1976(1987) RR22 ≈ 0.5 ≈ 0.5

Profitable Trading Profitable Trading Strategies ? Strategies ? Pesaran and Timmermann (1994) ‘Forecasting Pesaran and Timmermann (1994) ‘Forecasting

Stock Returns : …’, Journal of Forecasting, 13(4), Stock Returns : …’, Journal of Forecasting, 13(4), 335-67335-67– Excess returns on S&P500 and Dow Jones indices over Excess returns on S&P500 and Dow Jones indices over

one year, one quarter and one month. one year, one quarter and one month. – SMPL 1960 – 1990 (monthly data)SMPL 1960 – 1990 (monthly data)– 3 Portfolios : 3 Portfolios :

Market portfolio (passive) Market portfolio (passive) Switching portfolio (active)Switching portfolio (active) T-billsT-bills

– If predicted excess return (model based on If predicted excess return (model based on fundamentals) is positive then hold the market portfolio fundamentals) is positive then hold the market portfolio of stocks, otherwise bills/bond. of stocks, otherwise bills/bond.

– Switching strategy dominates the passive portfolioSwitching strategy dominates the passive portfolio

Predicting Returns and Predicting Returns and Abnormal Profits : Abnormal Profits : S&P500S&P500

Market Port.Market Port. Switching Switching Port.Port.

T-BillsT-Bills

Transaction Costs Transaction Costs StockStockss

0.00.0 0.50.5 1.01.0 0.00.0 0.50.5 1.01.0 -- --

BillsBills -- -- -- 0.00.0 0.10.1 0.10.1 0.00.0 0.10.1

Sharpe RatioSharpe Ratio0.310.31 0.300.30 0.300.30 0.820.82 0.790.79 0.760.76

Wealth at end of period ($ 100 invested in Jan. Wealth at end of period ($ 100 invested in Jan. 1960)1960)

1,911,9133

1,881,8844

1,851,8555

3,833,8333

3,553,5599

3,343,3466

749749 726726

Risk Adjusted Rate of Risk Adjusted Rate of ReturnReturn Can ‘predictability’ be used to Can ‘predictability’ be used to

make profits adjusted for risk and make profits adjusted for risk and transaction costs ? transaction costs ? – Transaction costs : bid – ask spread Transaction costs : bid – ask spread

(and other commission)(and other commission)– Risk adjusted rate of return measuresRisk adjusted rate of return measures

Sharpe ratio : Sharpe ratio : SR = (ERSR = (ERpp – r – rff)/)/pp

Treynor ratio : Treynor ratio : TR = (ERTR = (ERpp – r – rff)/)/pp

Jensen’s alpha : Jensen’s alpha : (R(Rpp – r – rff))tt = = + + (R(Rmm-r-rff))tt

SummarySummary

Different forms of market efficiency Different forms of market efficiency Important implications if market are Important implications if market are

efficient, opportunities if markets are efficient, opportunities if markets are inefficient inefficient

Hong horizon returns are less risky Hong horizon returns are less risky than returns over short horizonsthan returns over short horizons

Predictability of returns – difficultPredictability of returns – difficult Some variable have been identified Some variable have been identified

which help to predict stock returns which help to predict stock returns

References References

Cuthbertson, K. and Nitzsche, D. Cuthbertson, K. and Nitzsche, D. (2004) ‘Quantitative Financial (2004) ‘Quantitative Financial Economics’, Chapters 3 and 4 Economics’, Chapters 3 and 4

Cuthbertson, K. and Nitzsche, D. Cuthbertson, K. and Nitzsche, D. (2001) ‘Investments : Spot and (2001) ‘Investments : Spot and Derivatives Markets’, Chapter 13 Derivatives Markets’, Chapter 13

References References

Jorion, P. (2003) ‘The Long-Term Risk of Jorion, P. (2003) ‘The Long-Term Risk of Global Stock Markets’, University of Global Stock Markets’, University of California-Irvine Discussion PaperCalifornia-Irvine Discussion Paper

Dimson, E., Marsh, P. and Staunton, M. Dimson, E., Marsh, P. and Staunton, M. (2002) Triumph of the Optimists : 101 (2002) Triumph of the Optimists : 101 Years of Global Investment Returns, Years of Global Investment Returns, Princeton University PressPrinceton University Press

Cochrane, J.H. (2001) ‘Asset Pricing’, Cochrane, J.H. (2001) ‘Asset Pricing’, Princeton University PressPrinceton University Press

ReferencesReferences

MacDonald, R. and Power, D. (1995) ‘Stock Prices, MacDonald, R. and Power, D. (1995) ‘Stock Prices, Dividends and Retention : Long Run Relationship and Dividends and Retention : Long Run Relationship and Short-run Dynamics’, Journal of Empirical Finance, Short-run Dynamics’, Journal of Empirical Finance, Vol. 2, No. 2, pp. 135-151Vol. 2, No. 2, pp. 135-151

Pesaran, M.H. and Timmermann, A. (1994) Pesaran, M.H. and Timmermann, A. (1994) ‘Forecasting Stock Returns : An Examination of Stock ‘Forecasting Stock Returns : An Examination of Stock Market Trading in the Presence of Transaction Costs, Market Trading in the Presence of Transaction Costs, Journal of Forecasting, Vol. 13, No. 4, pp. 335-367. Journal of Forecasting, Vol. 13, No. 4, pp. 335-367.

Cochrane, J.H. (1997) ‘Where is the Market Going?’, Cochrane, J.H. (1997) ‘Where is the Market Going?’, Economic Perspectives (Federal Reserve Bank of Economic Perspectives (Federal Reserve Bank of Chicago), Vol. 21, No. 6. Chicago), Vol. 21, No. 6.

END OF LECTUREEND OF LECTURE