PORTFOLIO ESTIMATION FOR COMPOUNDING WEALTH
Stat 676/678a Fall 2003
Instructor: Andrew Barron ,
Phone: office 2-0634, dept 2-0666, home 248-5386
Office Hours: Monday and Wednesday, 2:15-3:15.
Course Time: CHANGED TO WEDNESDAY 4:00-6:00pm (beginning September 17).
[It previously was Tuesday, Thursday, 1:00-2:15].
Location: 24 Hillhouse (room 107)
PREFACE:
Looking at the stock market in recent years, there are selections of stocks for
which if the portfolio is rebalanced monthly to a constant proportion then over
about a decade, e.g. from Jan 1992 through Dec 2002, one's wealth would have
multipled 5462 fold! Moreover, investing in only 4 particular stocks in this way
would have had a wealth factor of over 4000. That is, $1000 invested in that
selection of stocks would have become over $4 million. Other strategies with non-
constant portfolios would have had even higher return. This leads us to ask basic
questions. How does one characterize the behavior of the market with hindsight?
By what factor less than the maximum wealth are realizable strategies that don't
have the advantage of hindsight. Can one make guarantees (concerning this factor
of drop from the maximum) that hold for all possible price sequences?
COURSE DESCRIPTION:
This course is a study in growth rate optimal investment. We explore properties
of compounded wealth in repeated gambling and in stock market investment. Wealth
concentration properties are explored and the strategies of highest concentrated
wealth are derived. Normal theory is developed for log-wealth. Relationships are
given to the principle of maximum likelihood in statistics. Universal portfolios
and their relationships to Bayes methodology are also examined. The ratio of
idealized wealth (best with hindsight) to actual wealth is studied both for
stochastic stock price sequences and its minimax behavior for arbitrary price
sequences. Algorithms are developed for computation of universal portfolios.
We explore both the development of the theory, through mathematical analysis,
and the practice, through empirical study of its implications in stock markets.
This course is available to any students who have completed introductory
probability. Some introductory statistics exposure is desirable but not
required.
RESPONSIBILITIES:
Students are expected to participate through discussion in class and by
giving a mid-October report and a late-November/early-December report.
The reports may cover aspects of either theory or practice related to the
material of the class. At least one of the reports should have an oral
component of about 15-45 minutes presented to the class and there should
be at least one written report. The mid-term report may present a preliminary
investigation expanded on in more depth in the final report.
Some of the core material is similar to what was covered last year in Statistics
676a ``Some Topics in Portfolio Selection.'' Students who took Stat 676a last
year are also permitted in 678a this year under the restriction that they report
on topics not addressed in the previous year.
SOME REFERENCES which you may look up. Some of these may be distributed in class:
BOOK CHAPTERS:
Thomas M Cover and Joy A Thomas (1991), Elements of Information Theory, Wiley.
Chapter 6 and Chapter 15.
David Luenberger (1998), Investment Science, Oxford University Press, Chapter 9
and Chapter 15.
Robert C Merton (1990), Continuous-Time Finance, Blackwell Publishers,
Chapters 4 and 5.
Jonathan E Ingersoll (1987), Theory of Financial Decision Making,
Rowman and Littlefield Publishers, Chapter 11.
JOURNAL ARTICLES:
Paul Algoet and Thomas M Cover (1988), Asymptotic optimality and asymptotic
equipartition property of log-optimal investment. Annals of Probability, vol 16,
pp 876-898.
Andrew R Barron and Thomas M Cover (1988), A bound on the financial value of
information. IEEE Transactions on Information Theory. vol 34, pp 1097-1100.
Robert Bell and Thomas M Cover (1980), Competitive optimality of logarithmic
investment. Mathematics of Operations Research, vol 5, pp 161-166.
Robert Bell and Thomas M Cover (1988), Game-theoretic optimal portfolios.
Management Science, vol 34, pp 724-733.
Daniel Bernoulli (1954), Exposition of a new theory on the measurement of risk.
Econometrica vol 22, pp 23-36 (Translation of D. Bernoulli, 1738, Specimen
theoriae novae de mensura sortis, Papers of the Imperial Acadamy of Science of
Saint Peterburg, vol 5, pp 175-192).
Leo Brieman (1960), Investment policies for expanding business optimal in a
long-term sense. Naval Research Logisitics Quarterly, vol 7, pp 647-651.
Leo Brieman (1961), Optimal gambling systems for favourable games. In the Fourth
Berkeley Symposium on Mathematical Statistics and Probability, University of
California Press, pp 65-78.
A. Blum and A Kalai (1999). Universal portfolios with and without transaction
costs. Machine Learning. vol 35, pp 193-205.
Thomas M Cover (1984), An algorithm for maximizing expected log investment return.
IEEE Transactions on Information Theory. vol 30, pp 369-373.
Thomas M Cover (1991), Universal portfolios. Mathematical Finance, vol 1, pp 1-29.
Thomas M Cover and Eric Ordentlich (1996). Universal portfolios with side
information. IEEE Transactions on Information Theory, vol 42, pp 348-363.
Jason E Cross and Andrew R Barron (2003). Efficient universal portfolios for past
dependent target classes. Mathematical Finance. Vol.13, No.2, pp.245-276.
M. Finkelstein and R. Whitley (1981). Optimal strategies for repeated games.
Advances in Applied Probability, vol 13, pp 415-428.
Dean P Foster and R V Vohra (1999). Regret in the on-line decision problem.
Games and Economic Behavior, vol 29, pp 7-35.
N. Hakansson (1979). A characterization of optimal multiperiod portfolio
policies. In Portfolio Theory, 25 Years After: Essays in Honor of Harry
Markowitz, editors Elton and Gruber. TIMS Studies in Management Sciences,
North-Holland Publishers, vol 11, pp 169-177.
F. Jamshidian (1992). Asymptotically optimal portfolios. Mathematical Finance,
vol 2, pp 131-150.
J. L. Kelly (1956), A new interpretation of information rate. Bell System
Technical Journal, vol 35, pp 917-926.
H. A. Latane (1959). Criteria for choice among risky ventures. Journal of
Political Economy, vol 38, pp 145-155.
H. A. Latane and D. L. Tuttle (1967). Criteria for portfolio building. Journal of
Finance, vol 22, pp 359-373.
David C. Larson (1986). Growth optimal trading strategies. Ph.D. Thesis, Stanford
University.
J. B. Long (1990) The numeriare portfolio. Journal of Financial Economics, vol 26,
pp 29-69.
D. G. Luenberger (1993). A preference foundation for log mean-variance criteria
in portfolio choice problems. Journal of Economic Dynamics and Control, vol 17,
pp 887-906.
H. Markowitz (1952). Portfolio selection. Journal of Finance, vol 8, pp 77-91.
H. Markowitz (1976) Investment for the long run: New evidence for an old rule.
Journal of Finance, vol 31, pp 1273-1286.
Robert C Merton and Paul A Samuelson (1974). Fallacy of the log-normal
approximation to optimal portfolio decision-making over many periods.
Journal of Financial Economics, vol 1, 67-94.
J. Mossin (1968), Optimal multiperiod portfolio policies. Journal of Business,
vol 41, pp 215-229.
Eric Ordentlich and Thomas M Cover (1996). Online portfolio selection.
Proceedings of the 9th Annual Conference on Computational Learning Theory.
Paul A Samuelson (1969), Lifetime portfolio selection by dynamic stochastic
programming. Review of Economics and Statistics, vol 1, pp 236-239.
Paul A Samuelson (1971), The ``fallacy'' of maximizing the geometric mean in long
sequences of investing or gambling, Proceeding of the National Academy of Science
USA, vol 68, pp 2493-2496.
Paule A Samuelson (1979), Why we should not make mean of log of wealth big
though years to act are long. Journal of Banking and Finance, vol 3, pp 305-307.
Y. Singer, D. P. Helmbold, R. E. Schapire, and M. K. Warmuth (1998), On-line
portfolio selection using multiplicative updates. Mathematical Finance. vol 8,
pp 325-347.
Stephen Stearns (2000), Daniel Bernoulli 1738: evolution and economics under risk.
Journal of Biosciences, vol 25, pp 221-228.
E. Thorp (1963), Portfolio choice and the Kelly criterion, Business and Economic
Statistics Section in the Proceedings of the American Association, pp 215-224.
Vladimir Vovk and C.J.H.C. Watkins (1998). Universal portfolio selection.
In Proceedings of the 11th Annual Conference on Computational Learning Theory,
pp 12-23.
W E Young and R H Trent (1969), Geometric mean approximation of individual
security and portfolio performance, Journal of Financial and Quantitative
Analysis, vol 4, 179-199.
Qun Xie and Andrew R Barron (2000), Asymptotic minimax regret for data
compression, gambling, and prediction. IEEE Transactions on Information Theory,
vol 46, 431-445.