44 ATaleofTwoMarkets
LEs were created for use in connection with information theory,
to specify the likelihood that information conveyed in binary com-
puter language would be understood properly. LEs measured the
increase in uncertainty of a communication as additional bits of in-
formation were added to the system.
The notion of bits of information has been reconceptualized for
application to public capital markets as measures of our knowledge
of current conditions. For example, in a time series of stoc kprice
data (e.g., daily returns), a positive LE would indicate the amount
of information or predictive power lost each day.
An LE of .05 per day, for example, would mean that information
becomes useless after 20 days (i.e., 1/.05). Thus, the LE is a measure
of the reliability of information in making forecasts for specified pe-
riods.
Peters, the Boston money manager, also has calculated the LE
for the S&P 500 (1950–1989), using monthly data. His calculations
resulted in a stable LE equal to .0241 per month. An LE of .0241
per month means that information reliability decays at the rate of
.0241 bit of accuracy each month; thus, the average cycle length of
the system using this measure is approximately three and a half years
(1/.0241approximately 42 months). Note that this result substan-
tially matches the result that Peters got in hisHanalysis.
Peters also calculated the LE of 90-day trading data for the S&P
500 (1928–1990) and found an LE of .09883 per period. That result
substantially matches both the monthly LE and theHanalysis: The
average cycle length of the system was approximately four years
(1/.09883 approximately ten 90-trading-day periods). Based on
these calculations, the public capital markets do exhibit sensitive
dependence on initial conditions and chaotic behavior rather than
simple linear efficiency.
Fractals
Another way to test for chaos is to determine whether a system has
a fractal dimension. Systems with fractal dimensions do not follow
Euclidean laws. Euclidean geometry simplifies and organizes nature
dimensionally: There are points, which lac kdimension; lines, which
have one dimension; planes, which have two dimensions; and solids,
which have three dimensions. These simplifying images are heuris-
tics: Natural objects do not conform to these images. Until fractal