From: John Conover <john@email.johncon.com>
Subject: forwarded message from William F. Hummel
Date: Fri, 14 Aug 1998 21:55:06 -0700
John Conover writes:
>
> If I get time, I will write a theoretical tautology on how they arrive
> at the allocations-might be useful in managing personal finance.
>
Interestingly, a related issue is attached. The argument in the
attached is that the concepts of neo-classical, (game-theoretic) and
Keynesian supply and demand economics are incompatible. They are
compatible, but not in the traditional sense that an economy is a
deterministic system that can be manipulated through monetary policy.
To illustrate, I will build a very simplified model of an equity
market, that has a growth that is dependent on T-Bill rates-ie., can
be manipulated through monetary policy. An investment portfolio
strategy will be optimized, and verified through simulation. Finally,
the equity market model will be compared with the New York Stock
Exchange Composite. The idea being that if the optimizations work on
the model, and the NYSE behaves like the model, the optimizations
would be valid for the NYSE.
Fortunately, the data we need to construct the model is available on
the Internet. The monthly T-Bill rates are available from
http://www.stat-usa.gov/, and the NYSE Composite daily closes are
available from http://www.nyse.com/public/market/2c/2cix.htm.
The very simplified assumptions of the stock exchange model are:
1) Equity values can be represented as a Brownian Motion fractal
with independent increments, (ie., the values are determined by a
cumulative sum of a lottery.)
2) Equity values are determined by speculation, ie., there are no
dividends.
3) Individual equity values have optimal growth, and the
characteristics and statistics of this growth are identical for
all equities in the exchange.
4) Investors have only two investment choices, equities, or
T-Bills, and the T-Bill rate is fixed, and does not change.
5) There are 253 business trading days in a calendar year.
Astonishingly, these simplified assumptions construct an equity
exchange model that is relatively accurate, and represents the
characteristics of the NYSE to a considerable extent.
It will be shown that:
1) The optimal asset allocation for investors is 40% in equities,
and 60% in T-Bills.
2) The optimal number of equities invested in is at least 10.
3) The growth of the equity index will be twice the T-Bill rate,
ie., the equity market is dependent on monetary policy.
The programs used in the equity exchange simulations are tsinvest(1),
and tsinvestsim(1). The analytical utilities used to analyze the
simulations and the NYSE data are tsfraction(1), tsavg(1), tsrms(1),
tsshannoneffective(1), tsshannon(1), and tsunshannon(1). For most of
the data reduction, a common spread sheet would suffice. However, the
C sources to all programs are all available from
http://www.johncon.com/ndustrix/archive/fractal.tar.gz.
The formulas used in the model are all derived in the tsinvest(1)
manual page:
1) G = (1 + rms)^P * (1 - rms)^(1 - P), where G is the gain in
value of an equity, rms is the root mean square of an equity's
marginal increments, and P is the Shannon probability of an up
movement, ie., the likelyhood of an up movement in an equity's
value.
2) P = ((avg / rms) + 1) / 2, where avg is the average of an
equity's marginal increments.
3) Optimal growth of an equity, portfolio, or exchange, occurs
when f = 2P - 1, where f is the fraction of the portfolio invested
in equities. This implies that for optimal growth, avg = rms^2.
These three formulas are the basis of financial engineering. It is
important to understand that the avg of many equities in a portfolio
or exchange add linearly, and rms add root mean square. Singularly,
this is one of the most important concepts of hedging and
optimization, and are discussed in the tsinvest(1) manual page.
X^253 = 1.07000781
or:
X = 1.000267
which is the interest rate, compounded daily. I need to convert this
value to an effective Shannon probability iterating the programs
tsunshannon(1) and tsshannon(1). After some iterating:
tsunshannon 0.51155
gives:
2^C(0.511550) = 2^0.000385 = 1.000267
which is the interest of a T-Bill compounded daily. Checking, using
the tsshannon(1) program:
tsshannon 0.000385
gives:
C(0.511551) = 0.000385
which confirms the iteration, above. Calculating the rms from the
Shannon probability, P using the fact that an equity has optimal
growth where sqrt (avg) = rms:
rms + 1
P = ------- = 0.51155
2
or:
rms = 0.0231
and:
avg = rms^2 = 0.0005331
These values, P, rms, and avg, are remarkably representative of stocks
on the NYSE over the time period in question. If you think about, they
would have to be-they control the growth in value of a stock, and
equities and T-Bills, at least in our simple market, have to be in a
long term equilibrium. (In point of fact, it is a Nash equilibrium
in our case.)
Since, for n many identical stocks in an exchange or portfolio, the
rms add root mean square and the avg adds linearly:
P' = ((sqrt (n) * avg / rms) + 1) / 2
and:
G' = (1 + (rms / sqrt (n)))^P' * (1 - (rms / sqrt (n)))^(1 - P')
where P' is the Shannon probability of the entire exchange or
portfolio, and G' is its daily compounded growth.
These are two very important formulas, and form the basis of most
hedging strategies for investments. Note that the likelihood of an up
movement for the portfolio or exchange rises with the number of stocks
in the exchange or portfolio. Note, also, that the growth increases
with the number of stocks, ie., the probability is better, and the
growth is better than any stock in the portfolio or exchange. Quite a
significant double whammy. I'll spin through some numbers. For the
sake of simulation time, lets limit the number of stocks in the
exchange to 300-the NYSE has a little over 3,000, but it is a
reasonable expediency.
for n = 300, P = 0.51155, rms = 0.0231, avg = 0.00053361:
P' = ((sqrt (300) * 0.0005331 / 0.0231) + 1) / 2 = 0.7000519
and:
G' = (1 + (0.0231 / sqrt (300)))^0.7000519 *
(1 - (0.0231 / sqrt (300)))^(1 - 0.7000519) = 1.000533
Compare this with the value of any one of the 300 stocks:
P = 0.51155
and:
G = 1.000267
Note that the growth of the portfolio or exchange was DOUBLE any stock
in the portfolio or exchange!
Now you know why the index growth rate is twice what the T-Bill rate
is. And, you know why it is very difficult, (actually, it requires an
enormous amount of luck,) to beat the index by picking only a few
stocks in the long run.
If you look at the equation for P and G, as a function of n, you will
find that it has a shape like a square root function-there is little
advantage in holding more than 10 stocks, (although there is a grave
disadvantage in holding less.) The incremental gain in adding an
eleventh stock is modestly marginal.
Note that:
rms' = rms / sqrt (300) = 0.001333679
and:
avg' = avg =0.00053361
where avg' is the average of the incremental growth of the portfolio
or exchange and rms' is the root mean square of the incremental
growth, and is a quantitative expression of volatility.
Note, also:
G^253 = 1.000533^253 = 1.144323
or an average index growth of about 14% a year.
So, bottom line, by the simple procedure of adding or subtracting
stocks, we can control the growth, volatility, and probability of an
up movement in a portfolio. Optimally:
f = 2P - 1 = 2 * 0.7000519 - 1 = 0.4000104
or about 40% of our investment portfolio should be invested in
equities, the remainder in T-Bills. This will make the portfolio's
growth the maximum possible. (This is very close to what fund managers
and institutional investors run, by the way.)
Using the tsinvest(1) and tsinvestsim(1) programs to simulate the
exchange for 100,000 days, (and using tsrms(1) and tsavg(1) to analyze
the simulation):
avg = 0.000533
rms = 0.001438
or:
P = ((avg/rms) + 1) / 2 = 0.6853268
and the final value divided by the starting value of the index, Q:
Q = 1.26865767E23
where:
G^100000 = Q = 1.26865767E23
or:
G = 1.000532
which agree with the anticipated values for the model of the exchange
very well.
Why a 100,000 days? The reason is that there are two issues that have
to be addressed. 1) we need a excellent accuracy on avg, and avg is
"rattling" around in the volatility, rms, which is about twice as
large. 2) The stock's values move up and down, (the maximum / minimum
in a year = 2,) and we may make a serendipitous measurement that is
tsshannoneffective 0.0005331 0.0231 100000
where avg = 0.0005331 and rms = 0.0231 gives:
For P = (avg / rms + 1) / 2:
P = 0.511539
Peff = 0.507111
meaning that the effective Shannon probability, P = 0.5115, could only
be considered about 70% correct do to the consequences outlined above,
if measured for 100,000 days.
Note that the tsshannoneffective program is at odds with those that
think they can time the market, or pick stocks based on only a few
years of growth, and it has a better track record than any fund manager,
frequently beating the indices, substantially.
So, how well does it compare to the NYSE Composite. Doing the same
analysis on the real stock exchange data from 1 January, 1981 to 31
December, 1996:
avg = 0.000437
rms = 0.00874
Q = 5.012778
giving:
P = 0.525
G = 1.000399
and:
G^253 = 1.000399^253 = 1.106196
or about an 11% growth per year, which is reasonable in comparison
with the 14% predicted, above, (depending on one's point of view, of
course.)
John
------- start of forwarded message (RFC 934 encapsulation) -------
Message-ID: <35d812e2.7035657@news.concentric.net>
From: wfhummel@concentric.net (William F. Hummel)
Subject: Decay in Sweden
Date: 13 Aug 1998 09:39:27 PDT
Newsgroups: sci.econ
Another economics group that I subscribe to has been having a
lively discussion about what has been happening to Sweden under
the neo-classical economics school that has dominated that
country since about 1979. The discussion began in response to a
recent article in the New York Times. Two Swedish economists
have been participating in the discussion, and both tell pretty
much the same story. I have copied one of the posts for sci.econ
because it contains an important lesson for any other country,
including the U.S.
Start of quote:
First of all, let me say that as I read the NYT article I got a
feeling of hopelessness. Intellectually I have already left
Sweden (I've been working in Denmark for two years now) but since
I still have my family there I am still forced to face the
tragedy of Sweden every week. I see all the concrete effects of
two decades of anti-keynesianism escalating into pure fiscal
madness. One would hope that foreign media were sensible enough
to start asking the right questions at some point. But the NYT
article gave no reason to hope for this.
I have written before on this list about the intellectual climate
in Sweden. I worked for three years at the department of
economics at Stockholm University, and during those years I got a
pretty good inside view of how the effects of the 1979
intellectual coup d'etat were nursed by Assar Lindbeck and his
staff. Much of what Per [the other Swedish economist] says about
the libertarian over-representation at the IIES (Institute for
Economics Studies) is true, but I think one has to remember how
grad students at IIES end up being libertarian. This is actually
a key issue in understanding how Sweden became what it is today.
The problem is not that the very small minority of dedicated
libertarians outrun all competition for the salaried grad student
positions at IIES. The problem is that mainstream economics
unintendedly can open for libertarianism as a moral conclusion in
the minds of students. Irresponsible professors can easily give
students the impression that libertarian solutions are the only
reasonable solutions to economic problems. In a country of
pluralism, open public debate, intellectual diversity in academia
etc, the libertarian potential in mainstream economics is curbed
by the pressure from alternative ideologies and paradigms. But
in a country where the political opposition competes with the
incumbent government over who is the most fiscally prudent of the
bunch, things turn out to be a bit different.
Students of economics refer to the media debate over financial
markets, taxes, government spending, unemployment and exchange
rates when they try to understand the use and meaning of
economics. If that world outside the classrooms is unrestrictedly
conservative, if the media consumer is drenched with "Spending
money, like eating people, is wrong" (although this time they're
dead serious about it...) from the morning paper to the late
night TV news, seven days a week, then she has to have one hell
of an integrity not to become a libertarian while taking
economics. I chose to write my dissertation in Denmark. Per is
now going to the US to do it. Neither of us found any place in
the academic system in Sweden.
I've tried the questions I'm working with on former friends now
working high up in the Swedish government administration or at
universities in Sweden. They all dismiss them as unscientific,
uneconomical, or simply unintelligible. There is not one, I
daresay, not one graduate student of economics in Sweden who
would even think of working with the problems I address - or, not
to forget, the issues into which Per has been digging so well
over the past few years.
One has to remember that these students later become professors,
big-shots in the central bank or finance ministry or the private
banking industry. They get to set the intellectual agenda, to
define what's economics and what's whacked. They get to define
fiscal policy and the daily conditions of living for millions of
people. They're all deeply convinced that the only sort of
fiscal policy that can be tolerated is as far from keynesianism
as Big Bang is from us. At best these people regard keynesianism
as an interesting piece of the history of economic thought -
something one can chat about with trustworthy colleagues in the
lounge after a nice dinner.
Before I conclude, I must comment upon the ethnical remarks in
the NYT article. It is hinted that one of the problems with
Sweden today is ethnic diversity. As an answer I would like to
point at two things: a) out of one million (not 800 000)
immigrants of first and second generation in Sweden, half are
from neighboring Nordic countries, predominantly Finland, which
as I see it is not a contribution to ethnic diversity; b) Denmark
shows at least the same ethnic diversity as Sweden, but the
economic downturn here ended several years ago and the Danish
economy is in better shape than it has been over the last 25
years. So the ethnic remarks in the NYT article (though
disguised) are yet another sign of the insufficient research
behind the article that Per labeled "crap journalism".
Wrapping up, then: Together with Per I've been trying to predict
the economic and political development in Sweden throughout the
'90s. The accuracy of our predictions is remarkably high ('cause
we're keynesians!) but we've often been wrong about the timing.
It seems to be that when all the three power centers of a country
- - the legislative (government), the economic (private businesses,
in our case Wallenberg) and the intellectual (academic economics)
- - pull in the same direction, namely towards the abyss of
austerity, then the process tends to slide faster than one can
really imagine in advance. We should remember that never in
post-war history has there been such a full-scale experiment of
reversed keynesianism as in Sweden.
My current view is, sadly, that there is nothing we can do about
Sweden. The process of social and economic disintegration has
gone too far, the budget cutbacks have been too severe, and the
damage done to the labor market is so serious, that only an
international intervention can prevent that country from a
socio-economic meltdown within a few years. All it takes is
another round of 5+% of GDP in reduced demand via tax hikes and
plunges in government spending. When I tell politicians and
academics in Sweden that this is indeed what I think awaits them,
I'm dismissed as a funny oddball. Sometimes I wish I were...
End of quote.
I have intentionally withheld the names of the two economists,
but anyone wishing to learn them can subscribe to the group on
which they are writing. It is a listserv group whose name I will
provide by e-mail request. If there is further interest in their
views, I will be happy to copy other posts for sci.econ.
William F. Hummel
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--
John Conover, john@email.johncon.com, http://www.johncon.com/