Re: Bayes inference for comparing stock portfolios


Nick Radov wrote:
Reef Fish wrote:
Nick Radov wrote:
I am working on an optimization problem to figure out which stock
portfolio out of a large set of possible portfolios has the highest
Sharpe ratio (risk-adjusted return) based on historical data.

In the study of the historical data of stock performance and groups
of stock performance (portolio), it's known as HIND SIGHT, and
the University of Chicago has been doing it for DECADES and
they have the complete data of stocks on CRSP (Center for
Research in Security Prices) tapes and other storage media, used
to DISCLAIM all "Winning Strategy in Stock Investing" type of books
by proving that they are only HIND SIGHT and none of them
(NONE, ZERO, ZILCH) works on future data.


Yes, I am aware of that research and I have used CRSP data for several
projects before. I know that trailing data is of very limited use in
predicting future returns. My objective on this particular project is
not to "beat the market". It is simply to select the most efficient
asset allocation from a limited set of 20 mutual funds available in my
own retirement account (I probably should have explained that from the
beginning rather than mentioning stocks). So I would like to use
diversification to maximize the expected Sharpe ratio of the overall
portfolio.

Now I understand your problem better and you seem to know quite a
bit about the market and investment strategies. You want to
diversify your risk, but you seem to prefer to choose among a set of
mutual funds which are KNOWN to be not competitive against
individual stocks (load and management fees and transaction fees).
Why? Why not just diversify your risk among a fixed set of stocks
among industries you like by the dart throwing method?


That is part of the support for the proven Hypothesis about the
Efficient Market that the only useful information for predicting the
FUTURE performance on past performance is the ILLEGAL
Insider information, as Martha Stewart went to jail for. The
Efficient Hypothesis is the theory developed at the Graduate
School of Business by Fama and his colleagues, including the
recent Nobel price winner of Economics who Mert Miller who
was in the area of Finance at GSB rather than Economics.

The efficient market hypothesis has never been proven.

While that is true, but there has not been a winning strategy against
choosing stocks at random; and no one has succeeded in beating
the random walk model. So, if the market is NOT efficient, someone
should have beaten the socks of it by a winning non-random
strategy. The "proof" is in the absence of a counterexample. It
is not a mathematical proof, but is a good enough proof for the
investment market.

In general, any
scientific hypothesis can never be proven. One can only find evidence
that either falsifies or does not falsify the hypothesis. In the
specific case of the efficient market hypothesis, it has been falsified
for some limited cases but it does provide a fairly good approximation
most of the time.


As part
of that larger problem, I need an efficient way to determine which of
two portfolios has the highest Sharpe ratio. The obvious brute-force
solution is to randomly pick a large sample of historical days, run
each portfolio through all of those days to find what the returns would
have been, and then calculate the Sharpe ratio for each.

You call THAT the brute force method? You would only introduce
errors from sampling. The brute force method is the CRSP method
of running it through the entire HISTORY of stocks (without missing
a single day) on debunking the "How to WIn" Quackery that sold
books, which promoted the well-known Dart-Throwing porfolio by
randomly throwing darts at the newspaper lists of stocks and showed
that OUTPERFORM the Mutual Funds and other investment funds
which cost stock investers millions and billions of dollars by paying
the cost of transaction (on the Fund Manager's "window dressing"
on trading) as well as the salary and bonuses of the Fund
Managers themselves.


I do have the entire history of closing price data (adjusted for
dividends) for the relevant securities. The particular data set I am
using isn't from CRSP, but it's the same numbers. And I am fully aware
that most mutual funds underperform the market indexes. But since I
only have a set of 20 mutual funds to pick from in this portfolio, that
point is moot.

My only question would be why pick from a suboptimal set?

Quarter after Quarter, the Funds as a whole could not out perform
any of the "averages" that require no investor selection of stocks
-- the DOW, NASDAQ, SP500, SP300, etc. because these
averages don't have transaction cost and management cost.

I suspect I
could get faster and more accurate results by using Bayes inference and
a smaller number of days, but I don't understand how to apply it to
this case. Can anyone give me a hint, or point me toward a web page
with some relevant examples?

The best way to do it WITHOUT sampling, Intro to CRSP:

http://www.library.hbs.edu/helpsheets/wrdscrspstock.html

The URL below, "This paper is the Stanford University Graduate
School of Business ..." tells how you can access the CRSP data with
SAS, FORTRAN, and other methods:

http://web.mit.edu/doncram/www/crsp.html

Mert Miller went from Chicago to Stanford after his Nobel in
Economics. Nice fella but not a particularly good poker player. :-)
We used to teach the Chicago downtown MBA program on some
evenings and go dinner in a group afterwards.

The URL gives the licence agreement on access:
http://ssdc.ucsd.edu/crsp/doc/stationlicense.txt

"CRSP Data License This is a legal agreement between you (either
an ...Permission to Use CRSP Data CRSP permits the use of its
data in scholarly papers written ... "

So, rather than telling you how to use a Bayesian (or other statistical
method) on a sample, your problem of evaluating PAST performance
on Sharpe's ratio (or whatever performance criterion) is best suited
on using the CRSP brute-force method of using the ENTIRE history.


Perhaps I didn't state the problem clearly enought. For this project I
don't care about the Sharpe ratios of individual securities.

You are repeating yourself that you are not interested in selecting
a portolio of stocks but a portfolio of mutual funds without giving
any compeling reason WHY, given that we already know the
mutual funds perform worse than the market averages. Why not
go with a index such as SP500?

What I
want to maximize is the Sharpe ratio of the overall portfolio. For any
given pair of portfolios, each with different asset weights, I can
calculate the expected Sharpe ratio of each portfolio after simulated
day 1, 2, 3, ..., n. The simulation of each day is run by randomly
picking a historical day, then updating the value of each item in each
portfolio based on the excess return earned on the chosen historical
day. After running through at least three simulated days I can
calculate the Sharpe ratio for each of the two portfolios, and one will
be higher than the other. So my real question is: how can I know when
we have reached a particular level of confidence (let's say p=0.9) that
one portfolio really does have a higher Sharpe ratio than the other? I
thought Bayesian inference might be useful here since it can reevalute
the probability that a hypothesis is true after each new data point is
received. The hypothesis being tested in this case is that portfolio 1
has a higher Sharpe ratio than portfolio 2, and we can assume that the
initial probability of the hypothesis being true is p=0.5.

Could anyone perhaps tell me the appropriate equation to use for
evaluating the new probability? Or if Bayesian inference isn't
appropriate for this case, is there a better statistical method? I just
want to know when I can stop the simulation since it takes a lot of
processing time to check each portfolio.

I hope someone else can help you the way you sought to be helped.
I understand what you are trying to do, but I don't understand WHY
you insists on choosing mutual funds or finding the sharpe's ratio
through sampling. Therefore, whatever else I say will not be of any
help to you.

-- Reef Fish Bob.

Now the future distribution of returns isn't necessarily going to look
like the past distribution of returns, especially since the sample size
for some of the securities is quite small. But from what I have
observed, the correlation relationships between various pairs of these
funds do tend to hold fairly constant over time. Therefore, think the
general method I have chosen should be good enough for my purposes
since it is really looking for the best way to diversify rather than
seeking the highest absolute return.

Forget about Statistics for this PERFORMANCE HISTORY problem.

-- Reef Fish Bob.

To state the problem more clearly, let's assume we have two stock
portfolios S1 and S2. After each simulated day, I would like to
calculate the probability p that the Sharpe ratio for S2 really is
higher than the Sharpe ratio for S1. At the start time t=0 we can
assume that p=0.5 since it is equally likely that either portfolio will
have the higher ratio. After we run the simulation for one day we are
at time t=1. And at that time we can calculate updated Sharpe ratios
for both S1 and S2, giving us an additional data point for which one
has the higher ratio. Now, how can I calculate the new probability p'
based on the values of p, t and that new data point? Once p' rises
above 0.95 or falls below 0.05 we can be confident that we have
identified the better portfolio.

Hopefully that made sense. Any assistance would be appreciated.

.

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