stock market inefficiencies through the decades market... · stock price movements are subjected to...

Stock Market Inefficiencies Over

the Last Six Decades

C. Michael Carty, Principal

New Millennium Advisors, LLC

QWAFAFEW

New York City Chapter

July 30, 2013


Why is Stock Market Efficiency So Important?

To provide an effective means for surplus economic units to

purchase previously issued securities

To permit investors to convert their holdings into cash at the

smallest differential from the last price (and vice versa)

To facilitate flow of investment funds through the economy,

encouraging sustainable growth

It has major implications for adopting appropriate strategies (passive

vs. active) and long-term portfolio performance


Letting the Efficient Market Work for You


Efficient Market Investment Strategies

Use index funds

Diversify

Dollar-cost average

Rebalance periodically, e.g., annually

Recognize the time-varying nature of stock/bond correlations

Do not try to time the market

Costs matter, minimize expense ratios, brokerage fees, etc


Or, You Can Shake the Money Tree


Non-Efficient Market Investment Strategies

Use enhanced strategies

Identify relevant investment factors

Pursue creative analytical tools and approaches

Analyze economic and market trends

Apply reversion-to-the-mean tactics

Make opportunistic portfolio adjustments

Consider alternative financial instruments


Characteristics of an Efficient Stock Market

All investors seek maximum possible returns at specific risk levels

Unlimited funds can be borrowed or lent at the risk-free rate

All investors have homogeneous expectations

All investors have a one-period time horizon

Investments are indivisible

Frictionless transactions; no taxes or trading costs

All changes in interest rates and inflation, if any, are fully anticipated

Capital markets are in equilibrium


How Do Stock Prices Behave?

Stock prices instantaneously reflect all relevant information

Stocks trade at their “fair value” in a centralized market

Investors cannot buy undervalued stocks or sell overvalued stocks

It is impossible to outperform the overall market through superior

stock selection or market-timing

Higher-than-market returns are only achieved with riskier

investments

“Properly anticipated prices fluctuate randomly,” Paul Samuelson

(1965)


“What the Hell is a Random Walk?”*

Random Walk Theory states, “Successive security price

changes can be treated as identically distributed independent

random variables.”

* Adam Smith (The Money Game, 1967)


The Early Random Walk Model (Bachelier, 1900)

(1) p(t) – p(t-1) = Є(t) t=1,2,…

where,

p(t) = the security price at time t

Є(t) = a Gaussian (Normal) process of independent variables


An Alternative Specification (Moore, 1962)

(2) lnep(t) – lnep(t-1) = Є(t) t=1,2,…

where,

lne is used to compensate for disparate price levels over time


A Generalized Model (Mandelbrot, 1962)

(2) lnep(t) – lnep(t-1) = Є*(t) t=1,2,…

where,

Є*(t) is a non-Gaussian stable process with a finite mean

The Gaussian distribution is only one of a family of stable Paretian

distributions

Other family members better approximate historic empirical

distributions which are more peaked and with fatter tails than a

Gaussian distribution


A Typical Empirical Distribution


Alcoa, Inc. (NYSE:AA)

Percent Changes in Daily Closing Prices*

January 2, 1962 to June 28, 2013

0

1,000

2,000

3,000

4,000

5,000

6,000

-0.24 -0.22 -0.20 -0.18 -0.16 -0.14 -0.12 -0.10 -0.08 -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24

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* Daily closing prices adjusted for stock splits and dividends

Adding Reflective Barriers (Cootner, 1964)

One group of investors will face surprises coming this week that

have no relation to those that came last week, i.e., the standard

random walk

Another more knowledgeable group of investors have an idea of

what will happen in the future

They cannot benefit from it unless the market price wanders

sufficiently from the expected price to provide an adequate profit

When that happens they commit sufficient funds to prevent prices

from wandering further

Prices therefore follow a random walk between those reflective

barriers

However, the nature of the distribution is altered by the

existence of reflective barriers


Compound Poisson Model (Press, 1967)

Stock price movements are subjected to two influences:

1) The normal institutional arrangements implicit in the traditional

random walk model, and

2) Unanticipated exogenous economic shocks that move trading to

new levels about which prices fluctuate randomly

The institutional element of the process emulates a Gaussian

distribution

The exogenous element emulates a Poisson distribution

Being more peaked with heavier tail density it should provide a

superior fit to historical price changes

It has theoretical appeal in recognizing exogenous influences


The Poisson Distribution*


* Named after Siméon Denis Poisson, French mathematician

Expresses the probability of a given number of events occurring in a fixed

interval of time independent of the time of the last event

Testing for Skewness and Kurtosis

30 stocks in the Dow Jones Industrial Average on June 28, 2013

were selected

All available daily price data were collected from January 2, 1962

to June 28, 2013 and intervening periods

Prices were adjusted for stock dividends and splits

The first four moments were calculated for each stock over its full

range, each decade and year, i.e., the mean, standard deviation,

skewness and kurtosis

Results are tabulated and ranked

Histograms are constructed for stocks highly ranked in positive

and negative skewness


Annual Positive Skewness

Ticker Rank Maximum Skewness Year Ticker Rank Maximum Skewness Year

CSCO 1 3.2617 2013 AXP 16 1.1401 2005

DIS 2 2.9609 2004 JPM 17 1.1156 1995

HPQ 3 2.6667 2005 MSFT 18 1.1099 2007

WMT 4 2.0649 1977 T 19 1.0989 2008

UNH 5 1.9283 2008 AA 20 1.0930 1999

HD 6 1.8386 1986 GE 21 0.9764 1985

BA 7 1.8311 1970 TRV 22 0.9376 2008

IBM 8 1.7623 1994 MMM 23 0.8921 2000

BAC 9 1.6651 1988 VZ 24 0.8520 1993

MRK 10 1.3450 2007 MCD 25 0.8267 1999

PG 11 1.2579 1970 CVX 26 0.8015 2008

UTX 12 1.2529 1972 DD 27 0.7865 1967

CAT 13 1.2050 1993 PFE 28 0.7005 2005

JNJ 14 1.2029 2008 XOM 29 0.6393 2008

KO 15 1.1673 2008 INTC 30 0.6348 2002


Annual Negative Skewness

Ticker Rank Minimum Skewness Year Ticker Rank Minimum Skewness Year

DD 1 -9.2447 1964 KO 16 -1.7614 2004

MRK 2 -7.4071 2004 JNJ 17 -1.7182 2002

UNH 3 -4.0675 1996 AXP 18 -1.7111 1987

JPM 4 -3.9051 1987 PFE 19 -1.6304 2004

UTX 5 -6950 2001 HD 20 -1.6017 2000

IBM 6 -3.4663 1987 CVX 21 -1.4777 1987

PG 7 -3.2201 2000 INTC 22 -1.4532 2006

DIS 8 -3.0898 1983 XOM 23 -1.4417 1987

CSCO 9 -2.7488 2009 TRV 24 -1.4292 1987

MMM 10 -2.6273 1987 BA 25 -1.3529 2001

AA 11 -2.4888 1987 VZ 26 -1.2421 1987

BAC 12 -2.2636 2003 MCD 27 -1.2198 1987

MSFT 13 -2.2507 2006 GE 28 -0.9266 1987

HPQ 14 -2.1658 2011 T 29 -0.9253 1986

CAT 15 -2.0706 2006 WMT 30 -0.5358 2012


Annual Test Results

Two-thirds or 20 of the 30 DJIA stocks exhibit positive skewness in at

least one year

27 of the 30 stocks exhibit negative skewness in at least one year

10 of the 27 stocks exhibiting maximum negative skewness did so in

1987

All 30 stocks are severely peaked (leptokurtotic) in at least one year


A Positively Skewed Distribution - DIS



Mean 0.0009

Minimum -0.0386

Maximum 0.1462

Std. Dev. 0.0161

Skewness 2.9609

Kurtosis 25.9547

# Obs. 263

# Positive 126

# Negative 123

Disney (The Walt) Company (NYSE:DIS)


January 2, 2004 to December 31, 2004

0

10

20

30

40

50

60

70

80

90

100

-0.039 -0.026 -0.014 -0.002 0.011 0.023 0.035 0.048 0.060 0.072 0.085 0.097 0.109 0.122 0.134 0.146

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A Negatively Skewed Distribution - MRK



Mean -0.0010

Minimum -0.2678

Maximum 0.0430

Std. Dev. 0.0221

Skewness -7.4071

Kurtosis 86.0376

# Obs. 253

# Positive 136

# Negative 116

Merck & Company (NYSE:MRK)



0

10

20

30

40

50

60

-0.033 -0.028 -0.023 -0.018 -0.013 -0.009 -0.004 0.001 0.006 0.011 0.015 0.020 0.025 0.030 0.035 0.039 0.044

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Decade Positive Skewness

Ticker Rank Maximum Skewness Decade Ticker Rank Maximum Skewness Decade

UNH 1 1.0976 00’s T 16 0.3434 00’s

BAC 2 1.0272 00’s KO 17 0.3394 10’s

JPM 3 0.8752 00’s WMT 18 0.3373 70’s

TRV 4 0.8327 10’s MMM 19 0.3331 00’s

BA 5 0.7980 70’s PG 20 0.3075 70’s

CSCO 6 0.6338 00’s HPQ 21 0.2967 00’s

AA 7 0.5968 90’s MCD 22 0.2738 90’s

DIS 8 0.5923 70’s JNJ 23 0.2736 70’s

AXP 9 0.5362 00’s MSFT 24 0.2724 90’s

CVX 10 0.5165 00’s DD 25 0.2722 60’s

IBM 11 0.4563 70’s MRK 26 0.2273 70’s

VZ 12 0.3994 00’s CAT 27 0.1781 00’s

GE 13 0.3805 00’s HD 28 1.1069 10’s

XOM 14 0.3700 90’s PFE 29 0.1013 10’s

UTX 15 0.3479 70’s INTC 30 0.0578 10’s


Decade Negative Skewness

Ticker Rank Minimum Skewness Year Ticker Rank Minimum Skewness Year

DD 1 -2.7148 70’s CAT 16 -0.6765 80’s

PG 2 -2.7022 00’s T 17 -0.6764 80’s

JPM 3 -2.5615 00’s UTX 18 -0.6376 00’s

HD 4 -1.5173 90’s AA 19 -0.6100 80’s

DIS 5 -1.4684 80’s BAC 20 -0.5537 80’s

TRV 6 -1.3575 80’s AXP 21 -0.5174 70’s

MSFT 7 -1.2466 80’s CSCO 22 -0.4157 10’s

IBM 8 -1.1814 80’s KO 23 -0.4021 80’s

UNH 9 -1.1575 90’s JNJ 24 -0.3405 80’s

MMM 10 -0.9783 80’s CVX 25 -0.2844 10’s

MRK 11 -0.8747 00’s MCD 26 -0.2817 80’s

VZ 12 -0.8326 80’s GE 27 -0.2506 80’s

XOM 13 -0.7914 80’s PFE 28 -0.2489 80’s

HPQ 14 -0.7713 10’s BA 29 -0.1655 90’s

INTC 15 -0.6886 90’s WMT 30 -0.0887 10’s


Decade Test Results

Only 2 of the 30 stocks exhibit marginal positive skewness over at

least one decade

Positive skewness is therefore most likely to be found over shorter

periods

9 stocks exhibit negative skewness, 5 of which occurred in the

80’s suggesting the statistical results may have been distorted by

the crash of 1987

All 30 stocks in the sample are severely peaked (leptokurtotic) in

at least one year which appears to confirm previous research

findings


A Positively Skewed Distribution - UNH



United Healthcare (NYSE:UNH)



0

100

200

300

400

500

600

700

800

900

1000

-0.20 -0.18 -0.16 -0.14 -0.12 -0.10 -0.08 -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34 0.36

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Mean 0.0009

Minimum -0.1864

Maximum 0.3476

Std. Dev. 0.0237

Skewness 1.0976

Kurtosis 25.2419

# Obs. 2,515

# Positive 1,285

# Negative 1214

A Negatively Skewed Distribution - DD



DuPont & Company (NYSE:DD)



0

100

200

300

400

500

600

700

800

900

-0.16 -0.15 -0.14 -0.13 -0.12 -0.11 -0.1 -0.09 -0.08 -0.07 -0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 More

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Mean -0.0003

Minimum -0.1884

Maximum 0.0676

Std. Dev. 0.0122

Skewness -2.7148

Kurtosis 41.5556

# Obs. 1987

# Positive 899

# Negative 973

Full Range Statistical Results – Part I


Ticker Start Mean Std. Dev. Median Skewness Kurtosis # Obs.

Date

MMM 1/2/70 0.0004 0.0147 0.0000 -0.1141 6.8738 10,974

T 1/2/85 0.0005 0.0164 0.0000 0.1654 5.6737 7,183

AA 1/2/62 0.0003 0.0209 0.0000 0.1124 9.4197 12,960

AXP 1/4/78 0.0006 0.0213 0.0000 0.1354 9.6284 10,974

BAC 1/2/87 0.0006 0.0275 0.0000 0.7685 0.0006 6,678

BA 1/2/62 0.0006 0.0207 0.0000 0.3385 5.1284 12,960

CAT 1/2/62 0.0005 0.0187 0.0000 -0.0249 5.7555 12,960

CVX 1/2/70 0.0005 0.0164 0.0000 0.1679 7.2720 10,974

CSCO 1/2/91 0.0013 0.0282 0.0009 0.2617 0.0013 5,668

KO 1/2/62 0.0005 0.0151 0.0000 -0.0103 12.3876 12,960

DIS 1/2/62 0.0007 0.0204 0.0000 -0.0507 8.8560 12,960

DD 1/2/62 0.0003 0.0162 0.0000 -0.1832 6.8593 12,960

XOM 1/2/70 0.0005 0.0145 0.0000 -0.0252 14.9295 10,974

GE 1/2/62 0.0004 0.0164 0.0000 0.1767 8.2954 12,960

HPQ 1/2/62 0.0006 0.0229 0.0000 0.0408 5.7030 12,960

Full Range Statistical Results – Part II


Ticker Start Mean Std. Dev. Median Skewness Kurtosis # Obs.

Date

HD 1/2/85 0.0010 0.0224 0.0000 -0.6035 16.2259 7,183

IBM 1/2/62 0.0004 0.0161 0.0000 0.0309 9.9901 12,960

INTC 1/2/87 0.0010 0.0260 0.0003 -0.2043 0.0010 6,678

JNJ 1/2/70 0.0005 0.0150 0.0000 -0.0358 6.2369 10,974

MCD 1/2/70 0.0007 0.0177 0.0000 0.0162 7.1284 10,974

MRK 1/2/70 0.0005 0.0165 0.0000 -0.3853 11.2977 10,974

MSFT 1/2/87 0.0011 0.0225 0.0000 -0.1604 0.0011 6,678

JPM 1/2/85 0.0006 0.0248 0.0000 0.4466 13.0665 7,183

PFE 1/4/82 0.0006 0.0179 0.0000 -0.0835 4.0721 7,953

PG 1/2/70 0.0005 0.0145 0.0000 -1.0652 37.8531 10,974

TRV 1/2/87 0.0006 0.0182 0.0000 0.5603 0.0006 6,678

UTX 1/2/70 0.0006 0.0179 0.0000 -0.1001 7.8516 10,974

UNH 1/2/91 0.0011 0.0246 0.0009 -0.2433 0.0011 5,668

VZ 1/2/85 0.0005 0.0161 0.0000 0.1892 7.5317 7,183

WMT 1/3/73 0.0009 0.0198 0.0000 0.2480 5.2130 10,216

Conclusions Regarding Full Range Testing

All 30 DJIA stocks had positive returns over the full range of

available price data for each stock

There was no evidence of positive skewness and only one stock

exhibited marginally negative skewness

24 of the 30 stocks experienced positive kurtosis


Non-Random Walk Model (Lo and MacKinlay, 1999)

(3) lnep(t) = α + β*lnep(t-1) + Є(t)

where,

for a random process, the expected values are α=0, β=1and Є(t)=0


Annual Test Results for the N-R Model

Adjusted R-Square = 0.908

Standard Error = 0.015

Observations = 253


lnep(t) = 0.074 + 0.977*lnep(t-1) + Є(t)

(1.546) (65.665)

Positively skewed distribution (Disney, 2004):

Negatively skewed distribution (DuPont, 1964)

lnep(t) = 0.126 + 0.953*lnep(t-1) + Є(t)

(2.484) (49.861)

Adjusted R Square = 0.945


Observations = 252

Decade Test Results for the N-R Model

Adjusted R-Square = 0.995


Observations = 2515


lnep(t) = 0.008 + 0.998*lnep(t-1) + Є(t)

(2.878) (1148.833)

Positively skewed distribution (UnitedHeath, 2000-09):

Negatively skewed distribution (P&G, 2000-09)

lnep(t) = 0.009 + 0.998*lnep(t-1) + Є(t)

(1.674) (720.143)

Adjusted R Square = 0.998


Observations = 2515

Conclusions Regarding the N-R Testing

Overall, the Non-Random Walk Model produced mixed results

The independent variable is statistically significant in all tests at

the 95% confidence level.

The constant term is statistically significant in 2 of the 4

regression; the negatively skewed annual test, DuPont, and the

positively skewed decade test, UNH.


Summary and Conclusions

While the market has provided an effective secondary market,

permitting transactions to occur close to previous prices, thereby

promoting economic activity, it is not perfectly efficient, nor does it

have to be perfect.

The few statistical and regression tests conducted demonstrate

that the Gaussian distribution is not an accurate representation of

the process.

It does not explain the historically verified presence of peaked

central tendencies, heavy tail densities, skewness and evidence of

statistically important omitted variables.


Thank you!

C. Michael Carty

Principal and CIO


[email protected]

917-697-9464


Bibliography

[1] Bachelier, L, Théorie de la spéculation. Paris: Gautier Villars, 1900. Reprinted in Cootner [ 2], pp 17-78.

[2] Cootner, P. H., The Random Character of Stock Prices. Cambridge: M.I.T. Press, 1964.

[3] ______, “Stock Prices: Random vs. Systematic Changes,” Industrial Management Review, vol. III (Spring 1962), pp. 25-45.

[4] ______, “Comments on the Variation of Certain Speculative Prices,” in Cootner [2].

[5] Fama, E.F., “The Behavior of Stock prices,” Journal of Business, XXXVIII, No. 1 (1965), pp. 34-105.

[6] ______, “Mandelbrot and the Stable Paretian Hypothesis,” Journal of Business, xxxvi, (October, 1965), pp. 297-307.

[7] ______, “Portfolio Analysis in a Stable Paretian Market,” Management Science, Vol. 111, No. 3 (January 1965), pp. 404-19.

[8] King, B. F., “Market and Industry Factors in Stock Price Behavior,” Journal of Business, XXXIX, No. 1, Part II (1966), pp. 138-89.

[9] Lo, A. W. and A. C. MacKinlay, A Non-Random Walk Down Wall Street. Princeton: Princeton University Press, 1999.

[10] Malkiel, B. G., A Random Walk Down Wall Street. New York: W.W. Norton & Company, 2011.

[11] Mandelbrot, B., “The Variation of Certain Speculative Prices,” Journal of Business, xxxvi, No. 4 (1963), pp. 394-419.

[12] Moore, A. B., “Some Characteristics of Changes in Common Stock Prices,” Unpublished Ph.D. Thesis, Graduate School of Business, University of Chicago, 1962. Reprinted in Cootner [x], pp. 139-61.

[13] Niederhoffer, V. and M. F. M. Osborne, “Market-Making and Reversal on the Stock Exchange,” Journal of the American Statistical Association, Vol. 61, No. 316 (December, 1966), pp. 897-916.

[14] Osborne, M. F. M., “Brownian Motion in the Stock Market,” Operations Research, vii (March-April, 1959), pp. 145-73.

[15] Press, S. J., “A Compound Events Model for Security Prices,” Journal of Business, vol. 40, No. 3 (July 1967), pp. 317-35.

[16] Samuelson, P. A., “Proof that Properly Anticipated Prices Fluctuate Randomly,” Industrial Management Review, Vol. VI (Spring 1965), pp. 41-49.

[17] Smith, A., The Money Game. New York: Random House, Inc.,


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