three years of stock market decline 5

8/3/2019 Three Years of Stock Market Decline 5

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ON THE SIGNIFICANCE OF A THREE YEAR STOCK MARKET DECLINE

by

Charles J. Higgins, Ph.D.Dept. of Finance/C.I.S.

Loyola Marymount UniversityLos Angeles, CA 90045-8385

Revised No. 3March 25, 2003

Preliminary Draft, Not for Attribution


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ON THE SIGNIFICANCE OF A THREE YEAR STOCK MARKET DECLINE

At the end of 2002, the Dow Jones Industrial Average (DJIA) had

declined for a third year in a rowfor the first time in over sixty years. As

problematic as the DJIA is in terms of being a sample of only thirty

securities, of not being a price weighted nor a market capitalization weighted

index, and that a survivorship bias skews long term analyses given that de-

listed securities are generally not included throughout the performance

measure, the three year sequential negative performance was publicly

questioned in terms of its significance. Likewise the Standard & Poors 500

(S&P) index suffered a similar decline and while being generally free of

some of the deficiencies associated with the DJIAcan be criticized as also

having a survivorship bias. Had de-listed securities been included, then such

a corrected measure would likely have a lower mean return and a probably

larger standard deviation. The likelihood of a string of three consecutive

market losses would decrease with a higher mean average annual return and

would likely increase with a higher annual standard deviation.

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simulations are not generally de rigueur , it would seem that an examination

using a simulation approach would be capable of handling a once in a sixty

year period event. It should be noted here, that the Efficient Market

Hypothesis posits that subsequent security prices are a function of

subsequent information and is usually regarded as normal in distribution.

The price movements (P) and the derived returns (r) may in some measures

incorrectly omit the dividend portion (D) of the total realized return (as is the

case for the DJIA), and should be r t = (P t P t-1 + D t)/P t. The omission of the

dividend return component further understates market return if one uses the

DJIA.

A simulation was created with a random normal distribution, with a

mean return (r) and a standard deviation (s), or N(r,s), where the standard

deviation was created from an original normalized distributionN(0,1).

The standard deviation was calculated from [-2log (u 1)]sin(2piu 2) where

the sin is in radians, and the two different us are independently selected

uniformly distributed random numbers between 0 and 1. The simulation

was comprised of 40,000 trials of 1,000 sequential yearly returns each,

wherein each subsequent year was determined by P t = P t-1(1 + r). The

simulation was initially tested with a mean return of zero (r = 0) and a

standard deviation of s = .1. The initial test correctly resulted in a

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subsequent set of substantially unchanged mean prices equaling P o for each

time t and with subsequent standard deviations approximated and expected

with s t = .1t when computed from time zero.

When each of the aforementioned set of statistics was independently

simulated, the year (Y) which resulted in the first occurrence of a string of

three subsequent decreasing annual prices was stored, then the mean, median

(50 th percentile or .5 cumulative frequency), and .05 and .95 cumulative

frequency distributions (5 th and 95 th percentiles) for the years in each of the

series of simulated runs were:

Three Years in a Row of Negative Returnsr s Mean Y Y .05 Y .50 Y .95

.072 .204 31.36 4 23 90

.067 .202 29.66 3 21 84

.1439 .1712 152.46 10 108 457

.1503 .1673 187.54 13 136 578

.1583 .1371 378.29 33 416 620

.1331 .1671 130.26 9 92 385

Thus, while a three-year sequence of declining security prices can be

disturbing even after some sixty years, it is still difficult to reject a null

hypothesis that during the three years of 2000-2002 that the security return

performance was part of a series of random events. For the DJIA inflation

adjusted returns (the first two rows of the above table) a sixty year period

was well within the 5 th and 95 th percentiles, and that for the remaining

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nominal returns a sixty year period was well within the 5 th percentile and the

median (50 th percentile). Another way to approach a test for statistical

significance, although a bit problematic, is to simulate the sixty year period

and store the longest sequential periods of market declines, and then to note

the cumulative frequency distribution for the three year decline. This

approach becomes problematic in that the cumulative distributions are not

sufficiently differentiable in that there are few items in the distributionthat

is by the third year of sequential declines, the cumulative distribution was

already substantially complete:

Cumulative Distribution for the Occurrence of a MaximumYearly Sequence of Negative Returns at the Third Year

r s Maximum at Third Year .072 .204 51.3 percent.067 .202 48.0.1439 .1712 92.7.1503 .1673 94.6.1583 .1371 98.7.1331 .1671 90.8

Notice that generally the third year was the maximum sequence of years of

negative returns ranging from 48.0 to 98.7 percent. This means that in

approximately one-half to nearly all the cases (in the latter simulated runs)

that a substantial number of sequenced negative yearly returns were less than

three years. Moreover, the aforementioned problematic nature of this

approach becomes apparent now in that the other distinguishable results are

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so fewnamely just zero, one, and two years and already comprise a

substantial portion of the distribution. Thus there exists in this approach an

insufficient gradation of the results.

The means and median measures for a four year string of negative

realized returns increases the years of each occurrence:

Four Years in a Row of Negative Returnsr s Mean Y Y .05 Y .50 Y .95

.072 .204 89.60 7 63 264

.067 .202 82.83 7 58 243

.1439 .1712 468.33 41 544 757*

.1503 .1673 500.48 51 644 827*

.1583 .1371 594.85 241 632 871*

.1331 .1671 130.26 32 412 807

*These 95 th percentile cumulative years may be understated in thateach run was comprised of 1,000 years.

Should events unfold so as to create a fourth year of negative return

performance in 2003, then one could be justified in moving from a position

of being unreasonably disturbed to that of being quite concerned.

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three years of stock market decline 5

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