is a 50 percent decline a random walk (4)

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Is a One-Year 50 Percent Decline a Random Walk?

Charles J. Higgins, PhDDept. Finance/CIS

Loyola Marymount Univ.

One LMU Drive

Los Angeles, CA 90045

310 338 7344

[email protected]

March 5, 2009

9th draft

In a previous working paper, “On the Significance of a Three Year

Stock Market Decline” March, 2003, I argued that the three year decline

then was not indicative of a deviation from a random walk , Burton

Malkiel’s early analogized market model (a drunk in an open field during a

new moon), where each subsequent step (or security price) is substantially

independent from the one prior. A notable demonstration of the market’s

random behavior was by Jensen (1967 and 1970). A pricing model of Pt+1

= Pt (1 + ř ) where ř is a random normal distribution may not necessarily be

mean zero; an 8.8 percent annual return as a market premium per Ibbotson

and Sinquefield is frequently cited (see Malkiel [1996] and Siegel [1998]).

In 1995, I noted in "A Distribution of Security Price Returns"

(with data supported by the LMU College of Business) that

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mailto:[email protected]



there was no autocorrelation between anterior and posterior

returns even when examined six days backward and

forward; moreover I spectrally sorted the returns by integer percentiles and

found nothing significant (save a moderate dead cat bounce which is

explainable by the omitted bankruptcy risk if the security was delisted).

However, the decline in the U.S. (and world) markets seems to present

evidence that we are now entering anomalous times. Whether using the

Dow Jones Industrials Average (DJIA), or Standard & Poor’s 500 (S & P

500), we are approaching a 50 percent yearly decline as we enter 2009:

Dow Jones Industrial Average (Bigcharts.com)



SPX, Standard & Poor’s 500 (Bigcharts.com)

Consider the annual rates of volatility measured in standard deviation:

Yearly

Mean Return Standard Deviation

DJIA less inflation 7.2 percent 20.4 percent

DJIA (Dimson) 6.7 20.2

1926-1997 nominal 14.39 17.12(Siegal) 15.03 16.73

S & P 500 15.83 13.71

1973-1999 13.31 16.71

The annual standard deviations are generally in their teens (see Ibbotson and

Sinquefield and Robert J. Shiller as detailed in Siegel [1998]); for a recent

examination outside the U.S., see Girmes and Benjamin (2006). One should

mention that the DJIA is a sample of only 30 non-price weighted securities

and that the simulation omitted dividends. Likewise, the annual

distributions become less symmetric as time increases (a Gamma



distribution in that security returns cannot be below -100 percent—security

prices are bounded by zero). I simulated daily returns, noting that security

return standard deviations are generally the square root of time, computed

with a mean zero normal distribution; the daily standard deviations ranged

from 1.0 to 1.9 percent in .1 increments. The frequency of declines greater

than 50 percent in a year’s time were counted (with some 257 trading days

simulated 1,600 times providing a .025 level of confidence). The normal

distribution was calculated from (-2log[u1])½•sin(2π•u2) wherein the sin is in

radians, and the two different u’s are independently selected uniformly

distributed random numbers between 0 and 1. This simulation did not

include a positive mean (which would have been .034 percent on a daily

basis). However, because the real market does, this test is more robust

(biased toward understatement). The results were:

Daily Yearly Frequency Once

Standard Standard Values Every

Deviation Deviation <.5 …Years

1.0 16.3 0 % …

1.1 17.3 0 …

1.2 19.5 0 …

1.3 21.0 .19 5261.4 22.5 0 …

1.5 25.0 .19 526

1.6 27.3 .56 178

1.7 27.6 .25 400

1.8 29.5 1.25 80

1.9 32.4 1.56 64



The above graphic of the frequency distribution was at a 1.5 percent daily

standard deviation (here there were 40,000 runs to provide more detail).

Note that the median is lower than the mean (here 100) as is the case for

distributions that have a lower boundary.

The conclusion is that it appears that we are living during an

anomalous event affecting the security markets wherein the annual standard

deviations in the teens produced no occurrence of an annual decline greater

than 50 percent. One thus must accept the alternative hypothesis that we are

outside the most extreme cases of a random walk. The inference is that we

have experienced a shock to the market that cannot be explained by only



random chance. Of course, given the stream of bad financial news during

this same period, then what is presented here is merely an obvious

confirmation.

References

Girmes, D. H. and Anne E. Benjamin, “Random Walk Hypothesis for 543

Stocks and Shares Registered on the London Stock Exchange” Journal of

Business Finance & Accounting Vol. 2, Issue 1, December 2006, pp. 135-

141.

Higgins, Charles J., “On the Significance of a Three Year Stock Market

Decline” working paper, 2003

Higgins, Charles J., "A Distribution of Security Price Returns"working paper, 1995

Jensen, Michael C., “Random Walks: Reality or Myth –

Comment” Financial Analysts Journal November-December,1967

Jensen, Michael C. and George A. Bennington, “RandomWalks and Technical Theories: Some Additional Evidence”

Journal of Finance Vol. 25, No. 2, May, 1970, pp. 469-482

Malkiel, Burton Gordon, A Random Walk Down Wall Street, W. W. Norton

& Company 1996

Siegel, Jeremy, Stocks for the Long Run, McGraw-Hill 1998

is a 50 percent decline a random walk (4)

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