From: John Conover <john@email.johncon.com>
Subject: Re: forwarded message from John Conover
Date: Fri, 1 Nov 1996 16:42:53 -0800
John Conover writes:
> John Conover writes:
> >
> > Yesterday, Garzarelli reiterated her prediction that the U.S. stock
> > market will crash right after the elections. Today, the UBS reiterated
> > their prediction that it won't:
> >
> > SINGAPORE - Contrary to popular belief, the U.S. stock
> > market is not on the verge of a crash or even major correction,
> > Union Bank of Switzerland (UBS) said.
> > Based on price earnings (PE) ratios, the U.S. equity markets
> > were only slightly overvalued at the moment, a group of UBS
> > economists told a briefing in Singapore.
> >
> > John
> >
>
> So, we have two very respected institutions, Garzarelli and UBS, with
> a very impressive set of credentials and historical track records,
> making conflicting predictions. There is also substantial evidence
> that there is a 58%, or so, chance that Garzarelli will be correct,
> and a 42% chance that the UBS will be correct. Of what use is that?
> Well it appears to present a bit of dilemma, since if you sell your
> portfolio, there is a 48% chance that the market will moved up, and
> you would not have participated in it, (ie., you would have lost
> money.) Likewise if you keep your portfolio, there is a 52% chance
> that you will loose money when the market moves down. Sounds like a
> probabilistic loose-loose scenario, huh? Not really.
>
So, you are sceptical, (as well you should be,) that such a simple
formula as f = 2P - 1 can really be right. Well, here is a proof,
using only the law of large numbers, calculus to take the derivative
and find a local maxima, and algebra to reduce the equations. (Of
course now, it is always easy to come up with these simple proofs
after the information theorist tell you the answer.)
Following Reza[1] and Kelly[2]. Consider the case of a gambler with a
private wire into the future who places wagers on the outcomes of a
game of chance. We assume that the side information which he receives
has a probability, P, of being true, and of 1 - P, of being false.
Let the original capital of gambler be V(0), and V(n) his capital
after the n'th wager. Since the gambler is not certain that the side
information is entirely reliable, he places only a fraction, f, of his
capital on each wager. Thus, subsequent to n many wagers, assuming
the independence of successive tips from the future, his capital is:
w l
V(n) = (1 + f) (1 - f) V(0)
where w is the number of times he won, and l = n - w, the number of
times he lost. These numbers are, in general, values taken by two
random variables, denoted by W and L. According to the law of large
numbers:
1
lim - W = P
n -> infinity n
and:
1
lim - L = q = 1 - P
n - >infinity n
The problem with which the gambler is faced is the determination of f
leading to the maximum of the average exponential rate of growth of
his capital. That is, he wishes to maximize the value of:
1 V(n)
G = lim - log ---
n -> infinity n V(0)
with respect to f, assuming a fixed original capital and specified
P:
W L
G = lim - log (1 + f) + - log (1 - f)
n -> infinity n n
or:
G = P log (1 + f) + q log (1 - f)
which, by taking the derivative with respect to f, and equating to
zero, can be shown to have a maxima when:
dG P - 1 1 - P
-- = P(1 + f) (1 - f)
df
1 - P - 1 P
- (1 - P) (1 - f) (1 + f) = 0
combining terms:
P - 1 1 - P P P
P(1 + f) (1 - f) - (1 - P) (1 - f) (1 + f)
= 0
and splitting:
P - 1 1 - P P P
P(1 + f) (1 - f) = (1 - P) (1 - f) (1 + f)
taking the logarithm of both sides:
ln (P) + (P - 1) ln (1 + f) + (1 - P) ln (1 - f) =
ln (1 - P) - P ln (1 - f) + P ln (1 + f)
combining terms:
(P - 1) ln (1 + f) - P ln (1 + f) +
(1 - P) ln (1 - f) + P ln (1 - f) = ln (1 - P)
- ln (P)
or:
ln (1 - f) - ln (1 + f) =l n (1 - P) - ln (P)
and performing the logarithmic operations:
1 - f 1 - P
ln ----- = ln -----
1 + f P
and exponentiating:
1 - f 1 - P
----- = -----
1 + f P
which reduces to:
P(1 - f) = (1 - P) (1 + f)
and expanding:
P - Pf = 1 - Pf - P + f
or:
P = 1 - P + f
and, finally:
f = 2P - 1
Which is one of the most famous equations in finance and economics.
(When someone mentions the economic entropic optima function, now you
can tell them, "Oh, f = 2P - 1," you should say. "What is the Shannon
probability, P, in this case," you should say. It is always a good
ploy. They will say something to the effect, "we don't really know,
since we do not have a large enough data set." To which, you should
reply, "historically, something larger than 0.5 seems to be about
right-say 0.54." You too can impress people that you are a financial
wizard.)
BTW, as silly as I am being, I am very serious-you can throw a project
budget together that is astonishingly accurate knowing what f = 2P - 1
really means-in spite of having inadequate information on costs and
resources, (these are just other noise sources-note it is exactly the
same thing as optimizing stock investment returns.)
Why 0.54? Well, if it was 0.5, nothing financial would ever
increase-and we know that things like stock prices do. If it was less,
then everything financial would decay to zero-and we know they
don't. If it was, say, 0.6, for example, like Ascend stock was doing
mid last year, and that scenario was maintained for about 15 years,
then the value of one share of Ascend stock would be larger than the
US deficit-and we know that probably won't happen. So, something like
0.55 seems about right. Extensive measurements over the past decade by
economists all over the world seems to indicate that 0.52 to 0.54
seems a reasonable "magic" approximation for most things, (but not
all.) Best bet is to do like the programmed traders and measure it-but
this is not always practical, since it requires about 5000 data points
to get a 90% confidence level on the number. (BTW, why did I say that
Ascend would grow so rapidly? It is because f is directly related to
the exponential growth in a financial function, like the DJIA, or
stock's price. 0.6 corresponds to a compound interest rate of about a
half-a-percent per day! Financial functions are very sensitive to this
number. Extreme accuracy is recommended, but seldom practical. For
example, if you were analyzing the annual beer industry, you would
have to have data for 5000 years! Roughly back to the time of the
empire of Mesopotamia-which we have, oddly enough. 3002 was a good
year for barley, and a good year for beer.)
You have to admire the ancient Mosoptamians. They get together and
invent civilization-then wonder who brought the beer.
John
[1] "An Introduction to Information Theory," Fazlollah M. Reza, Dover
Publications, New York, New York, 1994.
[2] "A New Interpretation of Information Rate," Bell System Tech. J.,
vol. 35, pp. 917-926, 1956.
--
John Conover, john@email.johncon.com, http://www.johncon.com/