"Will Trice" <wwtrice[at]paragondynamics.com> wrote
- quote -

I have been reading Chapter V from his book _(Mis)behavior of Markets_,
downloaded from http://misbehaviorofmarkets.com/ .

It's not very mathematical at all. To me, it does read like a diatribe. It's
even gossipy.

And yet, yes, he makes some interesting points. His criticism of
contemporary modeling (of several flavors) has value.

Where I'm bothered so far is (1) how he offers nothing to take the place of
more traditional modeling, so he seems to be urging throwing out the baby
with the bathwater; and (2) how he fails to recognize that, at a minimum,
the typical allocation model compels investors to at least think about the
value of diversifying.

But I dunno. Maybe in this book's conclusion he says as much. Like you
noted, this seems to be something of a pulp book, not intended to treat the
math per se, but instead promote interest in his ideas and some valid
criticisms.

snip
- quote -

That's an awfully big "just" Will, IMO.

Also, I would bet that someone has found equally credible models correlating
star movement with human behavior.

The real test of both is indeed whether they can predict.

- quote -

I am happy to agree to disagree.

- quote -

No, because I don't feel all investing theory is based in patterns of the
type Mandelbrot seems to have in mind.

All modeling rests on certain assumptions, though, as we've all said a few
times here.

I am still stunned at how rare it is to meet anyone who can talk about the
uncertainty associated with a given, say, asset allocation model's output.

As I suggest above, increasingly I'm thinking asset allocation models main
value is to get people to diversify at least a little.

- quote -

That's a good point. So what are the assumptions of the typical free online
asset allocation tool (and probably the not-free ones, too)?

As I think I mentioned before, it's not easy turning up their assumptions. I
did find the following site to have some interesting commentary:

http://www.indexinvestor.com/Free/onlineCalc.html

I caution that this site is of course also trying to sell a product or two,
so a grain of salt is appropriate when reading it. It generalizes, so we
don't really know what any specific online asset allocator uses to come up
with an allocation.

I tend to think any allocator that uses at least several decades of actual,
historical, annual figures for returns, without averaging them, on different
assets is at least somewhat reasonable (it's also fairly simple, but so?),
as long as it has the caveat that "past performance is no guarantee... ",
and as long as it examines rolling periods such that when the investor
enters the markets is taken into some consideration. Like the Trinity Study.

- quote -

He's an accomplished mathematician, absolutely, but that's also quite
possibly part of the problem.

- quote -

Again, I do think his criticism has value.

snip
- quote -

I had myself in mind. I haven't applied statistics to the financial modeling
we're discussing here much in the past.

I don't care about post-os. We all make these.

Will Trice wrote:

- quote -

Taleb is saying that there is a higher pobability that the stock price
will collapse or skyrocket than would be predicted from the variance.
To protect against the stock prices doing that buy puts with a low
strike price and calls with a high strike price.

He claims that diversifying the portfolio doesn't protect against
disasters (e.g 9-11).

--
Ron

Elle wrote:

- quote -

This is a bit of a strong statement. From the Forbes article you quote
below:

'"One would like to predict prices, to predict economic development,"
allows Mandelbrot... "But a preliminary step is just to describe them.
It sounds down-to-earth and disappointing, and perhaps too modest, but
it is absolutely indispensable." In Mandelbrot's opinion, economists
have not described the stock market well--so how can they predict it?'

Description is not prediction. Look at weather forecasters. Further
research may someday allow us to predict the weather more accurately,
and it may allow us to predict stock prices more accurately. But not
today, and Mandelbrot acknowledges that.

- quote -

There's no credible evidence that star positions can be used to model
human behavior. There are many credible models of the price movements
of assets. They just may not be useful for prediction. I think you
have apples and oranges here.

- quote -

I assume that you are not applying this statement to Mandelbrot alone,
but anyone who attempts to model the economic world. Including the
folks who put together the asset allocation tools that you present
occasionally?

- quote -

I seem to have touched a nerve. Mandelbrot is an accomplished
mathematician who is continuing work he started in the 60's. Yes, he
has written a pulp book for the masses, but he has refereed papers you
can read on the topic as well.

- quote -

As both you and Beliavsky have pointed out, I need to work on my
vocabulary...

-Will

"Will Trice" <wwtrice[at]paragondynamics.com> wrote
- quote -

To me, that's akin to religious fervor: Mandelbrot (among others) think the
markets can be modeled and therefore predicted, but he has no more
scientific evidence for such a model existing than do the people who keep
tinkering with machines they think will one day achieve perpetual motion.

Mandelbrot is making a huge (and IMO grossly uninformed) leap in saying that
the sciences of biology and physics are akin to that in sociology and
economics.

In articles reasonably friendly to Mandelbrot, we have statements like the
following:
------------
Mandelbrot's first foray into fractal economics, when he was discovering
ways to use computers to predict fractal systems at IBM in the '60s, had
major impact on the field. But then he hit a wall: Either he could write
equations that looked like the stock market but didn't allow him to predict
stock price changes in any meaningful way, or he had a prediction system
that did not account for wild price swings.
http://www.forbes.com/2002/04/02/0402mandelbrot.html
--------

I do not see how this differs from correlating, say, the star's movements to
human behavior: The stars' patterns look like they can model human behavior
but don't allow predictions of human behavior in any meaningful way. Or they
do predict but only(!) somewhat.

Much of the mathematics in these discussions is interesting and does have
some application. But I can't help but think that what we have here is
simply a huge collection of numbers having no scientific basis but with
occasionally a "pattern," seducing what are actually not-so-brilliant
scientists into, well, outrageous hypotheses about what the numbers will do
in the future.

Or Mandelbrot has simply found a new vehicle for making money, and is
milking his notions for every buck possible. Charlatan comes to mind at this
moment.

I suppose there's some art (but little useful science) there, anyway.

I'll look for one of his more recent books on the subject and see if I can
be persuaded to feel otherwise. :-)

- quote -

I don't buy into titles, anyway. If a person reads enough, s/he can be more
expert in some areas than those with formal credentials. True "experts" will
be willing to take on challenges from the non-credentialed, assuming a
certain minimum vocabulary is shared by both.


======================================= MODERATOR'S COMMENT:
Please trim the post to which you are responding. "Trim" means that except for a few lines to add context, the previous post is deleted.

Elle wrote:

- quote -

Yeah, I blew that one.

- quote -

Nassim Taleb is an options trader in the Mandelbrot camp
(www.fooledbyrandomness.com). He suggests taking measures to keep
yourself out of the tails of the distribution, but I haven't read his
site enough to know how one does that. Perhaps the market circuit
breakers do just this?

- quote -

Well, this is a point that Mandelbrot addresses. Many natural patterns
are unpredictable yet follow simple mathematical principles (weather
for example). Mandelbrot believes that fractal analysis can be applied
to sociological behavior just as it has been successfully applied in
biology and physics.

- quote -

I'm not an economist, I just play one on the internet.

-Will

beliavsky[at]aol.com wrote:

- quote -

Sorry about that, you are correct of course.

- quote -

But if the actual distribution is an inverse power law, then when the
exponent alpha is less than 2 there is no variance. It appears that
there is some dispute over the value of alpha, but most references I've
seen put it close to 2. Mandelbrot and others claim it is less than 2,
based on empirical evidence, while others claim more than 2, also based
on empirical evidence. This reminds me a lot of the open vs. closed
universe arguments.

- quote -

Interestingly, these folks have assumed daily rebalancing of the
portfolio, and that the portfolio only uses futures contracts.

-Will

"Will Trice" <wwtrice[at]paragondynamics.com> wrote
- quote -

Oh okay; I figured this was just a post-O, but I wanted to double-check.

- quote -

You mean variance of market price _changes_, right? For the newbies...

- quote -

But, as you know, not all mathematical approaches use a Gaussian model. The
Trinity study, for example, argues for diversity using the simple facts of
actual bond and stock returns of the last 50 years or so. Its conclusion:
Have X% in stocks, Y% in bonds, and you'll do better than having all of one
or the other. That is, assuming the future somewhat resembles the past,
anyway. The study is not about explicit mathematical correlation (or lack
thereof), yet it nonetheless supports the notion that an investor should
seek a balance of seemingly uncorrelated vehicles for investing, to optimize
return.

I was trying to get a better grip on the underlying assumptions of some of
the free online portfolio allocators I have been exploring. Some definitely,
simply rely on actual historical returns, albeit sampled over different
periods. Some, like T Rowe Price's, appear to be using at least in part
Gaussian models.

- quote -

Two points occur to me that I think are important, both academically and
practically:

1.
Every site I've seen that discusses how a Gaussian distribution does not
reasonably foresee a crash like 1987's also insists that, therefore, insofar
as being able to plan financially, the sky is falling. They do not appear to
discuss the effect of the market circuit breakers put into place after and
because of 1987's crash. IMO, if the academic discussion is to have any real
value, then this is no small oversight. I presume the design of these market
"circuit breakers" was not undertaken lightly. Some serious financial and
mathematical (and probably psychological) thought must have went into them.
(Maybe I'm wrong and they were simple though; someone can google.) Other
steps are always being taken (e.g. laws have been passed, or lawsuits
brought) that have increased the pressure on markets and the companies who
make them up that go towards minimizing wild fluctuations. With some
exceptions, I would wager this is the general trend, anyway. The Enron
debacle, for example, has led to new measures. It seems to me that it's
really outrageous (and maybe even embarrassing) that Mandelbrot adherents
use the one-day 1987 crash to bolster their argument against ever using
(log-)normal distributions to model stock market price changes. I'd still be
interested in a measure of how likely the market weeks after 9/11/2001 were,
according to a Gaussian distribution analysis.

2.
The above realities remind me of what Skip posted (again?) recently about
the difficulty of predicting regulatory changes and their impact on
investing and so financial planning.

- quote -

I agree these are good questions.

I am a little bothered by the fact that, the longer the interval, the more
important from _where_ within each interval one starts measuring return. I'm
sure there's a simple answer to this. And it's model specific...

I am also leery about getting sucked into the numerology of some of this.
How valid is it to assume any particular pattern of stock price changes in
the past will continue into the future? Stock price changes are not a result
of, say, biological phenomenon, where AFAIC it is more reasonable to assume
certain patterns will re-occur. Stock price changes are a result of economic
"principles," which so often do not rely on science per se but rather on
human and sociological behavior.

- quote -

Likewise, though I realize you've been looking at these models longer than I
have.

Will Trice wrote:

<snip
- quote -

It is the variance of market RETURNS, not PRICES, that matter.
Empirically, when the Student t distribution is fit to stock index log
returns, the degrees-of-freedom parameter v is > = 3, indicating that
the variance does exist.

- quote -

<snip
- quote -

The primary reason to use higher-frequency data is that there are more
observations. Some research has found that one can better estimate
variances and covariances using high-frequency data, although one must
correct for bid-ask bounce and non-synchronous trading. You can find
research by doing a keyword search of "realized variance" and "realized
correlation" at http://papers.ssrn.com/sol3/DisplayAbstractSearch.cfm .
One paper claims that intraday data can be used to make asset
allocation decisions:

http://papers.ssrn.com/sol3/papers.c...ract_id=276921
The Economic Value of Volatility Timing Using 'Realized' Volatility
JEFF FLEMING
CHRIS KIRBY
BARBARA OSTDIEK
December 29, 2001
Rice University, Jones Graduate School Working Paper
Abstract:
Recent work suggests that intradaily returns can be used to construct
estimates of daily return volatility that are more precise than those
constructed using daily returns. We measure the economic value of this
"realized" volatility approach in the context of investment decisions.
Our results indicate that the value of switching from daily to
intradaily returns to estimate the conditional covariance matix can be
substantial. We estimate that a risk-averse investor would be willing
to pay 50 to 200 basis points per year to capture the observed gains in
portfolio performance. Moreover,these gains are robust to transaction
costs, estimation risk regarding expected returns, and the performance
measurement horizon.
Keywords: Realized volatility, volatility timing, tactical asset
allocation, portfolio optimization, mean-variance analysis
JEL Classifications: G11, C14

Elle wrote:

- quote -

You're right, I didn't mean to imply that Mandelbrot is the only one
saying this. He just seems to be jumping up and down about it at the
moment. The interesting thing for me is that if the market is
sufficiently fat-tailed, then the variance of market prices cannot be
calculated, and thus correlations cannot be calculated. Since
mathematical approaches to asset allocation revolve around the
correlation between assets, this has interesting ramifications for
financial planning. Indeed, even GARCH is sensitive to big events.
Robert Engle said that the inclusion (or not) of the 1987 crash makes a
huge difference in the choice of model parameters.

- quote -

An interesting question in and of itself is the choice of data one uses.
Other than availability of more data points, why would one choose
daily over monthly? If daily is better than monthly, is hourly better
than daily? Is minutely better than hourly?

Thanks to all for the discussion so far, it has been interesting (for me
at least)
-Will

"Will Trice" <wwtrice[at]paragondynamics.com> wrote
- quote -

I thought it wasn't just Mandelbrot who asserted that the Gaussian
distribution was inaccurate; I thought just about anyone who has tried to
model market changes has noted that the actual distribution was more
fat-tailed. See for example the disclaimer from T. Rowe Price I quoted
earlier.

- quote -

I suppose it depends on the specific application of the model.

When I first posted, daily made sense to me, because this would provide what
would probably be a statistically meaningful sample on which to base a claim
that the distribution of, say, the Dow's (or any other index's) percent
changes was roughly Gaussian.

I think I'd be more comfortable with a model that used daily data. Computing
power is cheap, as you or someone noticed here recently.

But that's just a first blush response. Maybe quarterly or even annual data
is fine, particularly given all the margin of error in all the assumptions.

- quote -

I think it was closer to 20%, and certainly not on a single day, like 1987,
for the most part, wasn't it?

I note this not to be a nit-picker but to point out that this particular
decline may have been within the realm of reasonable expectation for anyone
using a Gaussian distribution. (Unlike 1987's crash.) Though again, I think
a Gaussian model presumes no odd world events like 9/11.

So Gaussian models reject the likelihood of a Black Monday 1987, but may
reasonably embrace the likelihood of a market like that which occurred in
the weeks after Sept. 11, 2001.

Will Trice wrote:

<snip
- quote -

I think the book

The Econometrics of Financial Markets
by John Y. Campbell, Andrew W. Lo, A. Craig MacKinlay,

finds that returns over longer time periods, such as monthly, are
closer to normal than daily returns, although they may not be exactly
normal.

- quote -

Asset allocation models often use data on monthly returns from places
like Ibbotson.
- quote -

Note that the VIX index of IMPLIED volatility measures the market's
EXPECTATION of volatility over the next month. It is not a measure of
REALIZED volatility.

Elle wrote:

- quote -

No, but cicuit breakers many not have that effect on periods longer than
a day anyway. According to Mandelbrot, asset price movement is
self-similar on scales larger than a few minutes, so the fat-tailed
effect would apply to weekly, monthly, quarterly, and yearly volatility
as well. Aren't most models built on yearly or quarterly data (as
opposed to daily)?

- quote -

Volatility only increased by about 33% after the attack and settled down
to its pre 9/11 value by the end of September (judging from the CBOE
volatility index that Beliavsky showed in an earlier post in this thread).

-Will

I wish that fractal theory could help technical charting separate out
the short term noise from the more global trends. Something smarter or
more quickly responsive than, say, a 50 day moving average to look for
lows, highs, inflection points, or trade signals. But I take it no
such thing is known or at least admitted to.

- quote -

Funny that these posts brought up a smart-ad on my screen for a
magazine with a Mandelbrot contribution. Clicked on it and was exposed
to a cat fight between readers writing in and rejoinders by Mandelbrot.

Long ago I was briefly mentored by a (brilliant) collegue of
Mandelbrot, a few doors down from the sainted one's office (always
empty when I passed). What was their relationship... well, I was told
by others to never mention fractals or Mandelbrot's name in the
presence of my elderly mentor, because he would have such a fit of
physical outrage that there was a concern he may drop dead. So I never
learned much about fractals...

"Will Trice" <wwtrice[at]paragondynamics.com> wrote
- quote -

Do they really? I suspect the problem is that Usenet posts on complicated
topics are necessarily sound bites. Laziness and perhaps the goals or
requirements of a moderated newsgroup preclude full sharing of any one
participant's views on this subject. I think they know that the pure math is
black and white but the assumptions make any allocation model's output
rather grey or "fuzzy."

- quote -

You are recalling correctly. For newbies with a statistical background, see
http://www.lope.ca/markets/1987crash/economic.html , among others.

Have you considered how the market controls put in place subsequent to and
because of the 1987 crash may nudge the stock market's daily changes to
better conform to a Gaussian distribution? It seems to me the controls
should indeed nudge the market away from a fat-tailed distribution and
closer to an actual Gaussian distribution. For example, two of the post-1987
market controls are to halt trading for an hour if the Dow drops 10% before
2 pm; two hours if it drops 20%. (I think economists do not attempt to model
the effect of world events such as Sept. 11. 2001 on the markets. That is,
once a nuclear holocaust occurs, all models for the future are thrown out
the window. Still, it might be interesting to see how the market days
following 9/11 conformed to a Gaussian distribution. I can't remember if
controls kicked in or not. Someone can google.)

- quote -

Getting back to the practical side:
I'd like to see what Mandelbrot's model proposes for a given person's
situation and "risk tolerance" (good luck nailing this) and the uncertainty
it places on a recommended asset allocation.

As a related, practical aside, the only relevant assumption of publicly
available Monte Carlo asset allocation tools I could find was the following
disclaimer from T. Rowe Price for one of its asset allocator tools:
-----
The Calculator's limitations
Limitations include but are not restricted to the following:
The actual probability distributions of monthly returns may have a higher
concentration in the "tails" of the curve than the normal distribution. This
means the market extremes (and potential for loss) may occur more often than
we have projected.
-----

http://www.troweprice.com/tools/cic/...MD&legal=false

Will Trice wrote:
- quote -

In the literature, it is common to simulate from the Student t
distibution, whose tail thickness is controlled by a degrees-of-freedom
parameter. There are other continuous fat-tailed distributions . I use
the Fortran code at
http://users.bigpond.net.au/amiller/random/random.f90 for Student t
variates, but I can't advise you on Monte Carlo software packages .

- quote -

One can simulate from a GARCH model, which is designed to account for
autocorrelated volatility. The innovations of a GARCH model can have
fat tails.

beliavsky[at]aol.com wrote:

- quote -

<snip
- quote -

The Monte Carlo simulations I've read about just use a normal
distribution. Do you know of some that are publicly available that use
a fat-tailed distribution? I guess one could always roll their own.
Mandelbrot also makes the point that asset price volatility is
autocorrelated. Monte Carlo simulations used for engineering make use
of autocorrelated data, but I haven't seen a financial simulation that
does this. Still, if you're rolling your own...

- quote -

Mandelbrot discusses these models, but considers them to be kludgy
attempts to fix the problems associated with Gaussian models.


- quote -

So Mandelbrot was only partially correct. There are other models out
there that take into account non-normal distributions, but these models
are not typically used in the financial world. Doesn't this make the
asset allocations that are generated today suspect? Quoting from
Harvey's paper (p19):

"The multivariate normal distribution is an inappropriate probability
model for portfolio returns primarily because it fails to allow for
higher moments, in particular skewness and coskewness."

- quote -

Indeed, he has made statements to this effect. From the interviews I've
read, he seems to have an ego the size of a house.

-Will

Will Trice wrote:
- quote -

It's true that returns of stocks and other assets are non-normal, for
at least two reasons:

(1) Volatility changes over time. A graph of historical implied
volatility is at http://finance.yahoo.com/q/bc?s=%5EVIX .
(2) Jumps occur, so that the daily stock returns, standardized by
volatility, are still non-normal.

A good book covering these topics is

"The Econometric Modelling of Financial Time Series" (1999), by Terence
Mills.

- quote -

I don't agree. Researchers such as Markowitz, Black, and Scholes used
the assumption of normality for reasons of theoretical simplicity and
computational convenience, but their ideas CAN be extended to more
general settings. Nowadays analytic solutions are less important,
because cheap computing power makes feasible Monte Carlo simulations.

- quote -

That's true, so "jump diffusion" models of returns are used to get a
more realistic description of "tail events". One can do a keyword
search of http://papers.ssrn.com/sol3/DisplayAbstractSearch.cfm for
references.

- quote -

All models simplify reality, but some of them are still useful. I think
one can get useful results from portfolio optimization if reasonable
constraints, such as an upper bound on the proportion invested in any
stock, are imposed.

- quote -

There have been papers extending Markowitz portfolio optimization to
non-normal distributions. Here is one.

http://papers.ssrn.com/sol3/papers.c...ract_id=634141
Portfolio Selection With Higher Moments
CAMPBELL R. HARVEY
Duke University - Fuqua School of Business; National Bureau of Economic
Research (NBER)
JOHN LIECHTY
Pennsylvania State University, University Park
MERRILL W. LIECHTY
Drexel University - Department of Decision Sciences
PETER MUELLER
The University of Texas M. D. Anderson Cancer Center
December 13, 2004
Abstract:
We propose a method for optimal portfolio selection using a Bayesian
decision theoretic framework that addresses two major shortcomings of
the Markowitz approach: the ability to handle higher moments and
estimation error. We employ the skew normal distribution which has many
attractive features for modeling multivariate returns. Our results
suggest that it is important to incorporate higher order moments in
portfolio selection. Further, our comparison to other methods where
parameter uncertainty is either ignored or accommodated in an ad hoc
way, shows that our approach leads to higher expected utility than the
resampling methods that are common in the practice of finance.
Keywords: Bayesian decision problem, multivariate skewness, parameter
uncertainty, optimal portfolios, utility function maximization,
resampling, resampled portfolios, estimation error, mean-variance
portfolios, expected returns, Markowitz optimization
JEL Classifications: G11, G12, G10, C11

Wiley will soon publish the book that may answer some of your
questions:

Fat-Tailed and Skewed Asset Return Distributions: Implications for Risk
Management, Portfolio Selection, and Option Pricing
by Svetlozar T. Rachev, Frank J. Fabozzi, Christian Menn .

In general Mandelbrot writes as if financial theory has not advanced in
30 years. Researchers are aware of the problems he has identified with
Gassian theory and have proposed more realistic models.

Many participants in this newsgroup advocate the use of rigorous
risk-adjusted portfolios. Yet risk is typically calculated assuming
that asset returns conform to a Gaussian distribution. In his 2004
book, _The (Mis)Behaviour of Markets_, Benoit Mandelbrot (of fractal
fame) asserts that markets follow an inverse power law and that all
current calculations of risk (i.e. those based on simple measures of
volatility) are bunk. He further asserts that this means that efficient
market hypothesis and risk-adjusted asset allocations are bunk. Whether
he is correct or not, it is true (isn't it?) that events like the 1987
market crash are near impossibilities in a Gaussian market.
Nevertheless, the 1987 crash did occur.

Is Mandelbrot correct? Or are Gaussian models useful for financial
planning? Unfortunately, Mandelbrot does not suggest how multifractal
analysis can be applied to financial markets, so maybe Gaussian models
are better than nothing?

Thanks for your thoughts,
-Will