1.
페이스북을 통해 어떤 분이 질문하였습니다.’universal portfolio’을 문의했습니다
“혹시 “유니버셜 포트폴리오” 알고리즘 트레이딩 매매에 대해서 혹시 알고 계십니까? 국내에서는 문병로 교수님(서울대)이 운영하고 계신다는데… 개인은 힘드니 “역발상 투자”라는 책을 소개시켜 주네요.”
사실 제 전공도 아닙니다. 수학기호가 많이 나오는 전략들을 풀어서 설명할 능력도 없습니다. 제가 할 수 있는 것은 학습을 위한 안내가 아닐까 합니다. 위키페디아를 보면 Universal Portfolio를 다음가 같이 정의하고 있습니다. “기계학습에 의해 얻은 정보로 포트폴리오를 구성하는 종목을 선택하는 방법”으로 보입니다.
The Universal portfolio algorithm is a portfolio selection algorithm from the field of machine learning and information theory. The algorithm learns adaptively from historical data and maximizes the log-optimal growth rate in the long run. It was firstly introduced by Thomas M. Cover. The algorithm rebalances the portfolio at the beginning of each trading period. At the beginning of the first trading period it starts with a naive diversification. In the following trading periods the portfolio composition depends on the historical total return of all possible constant-rebalanced portfolios.
universal portfolio를 다루는 한글 논문을 찾아보았지만 거의 없습니다. 유일한 것이 서울대 경제학부 소식지에 실린 최병선 교수의 글입니다.
저의 학문적 삶에 있어서 가장 큰 행운은 지도교수인 Thomas M. Cover교수를 만난 것입니다. Cover교수는 Stanford대학에서 통계학과, 재무수학과 그리고 전자 공학과의 겸임교수로 재직하는 능력이 출중한 학자로서, 정보통신분야에서 획기적인 논문들을 발표한 공로로 전자공학계의 가장 권위 있는 Shannon상과 Hamming 상을 수상하였습니다. Cover교수는 재무이론 분야에서도 꾸준한 연구를 해왔으며, 1991년에는‘Mathematical Finance’창간호에 universal portfolio라는 독창적인 재무이론을 제안하였고, 1999년 여름 실리콘밸리에 투자자문회사를 창업하여 자신의 이론을 실제 적용하고 있기도합니다. Cover교수가이렇게다양한학문분야들을 깊게 섭렵할 수 있는 힘의 근원은 다름 아닌 그의 뛰어난 직관력과 더불어 탁월한 수학적 능력이라고 생각합니다.
문병로 교수나 Cover 교수나 비슷한 경력을 가지고 있습니다. Cover교수가 처음 이론화한 universal portfolio 논문는 아주 쉽게 구할 수 있습니다. T. Cover Selected Papers on Portfolio Theory는 Cover교수가 직접 정리한 논문 목록입니다. 이중 Universal Portfolios이 1991년에 처음 작성한 논문입니다.
2.
경제학 논문이므로 수학기호가 넘쳐납니다. 깊은 이해가 아니라 얕은 이해를 위한 글을 소개합니다.
CASTrader Blog: Thomas Cover’s Universal Portfolio – Part I에서 소개한 ‘Universal portfolios’ take investors back to the future의 요약입니다. 강조한 부분을 눈여겨 보시길 바랍니다.
To create a universal portfolio, the investor buys very small amounts of every stock in a market — no small task in itself. The New York Stock Exchange, for example, lists 3,025 companies. In essence, the universal investor mimics the buy order of a sea of investors using all possible “constant rebalanced” strategies, in which the amount of money invested in each stock is adjusted each day to achieve a fixed proportion.
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“Imagine we have, for a simple example, two stocks,” Cover explains. “A good constant rebalanced portfolio might invest, say, one-fourth in one stock and three-fourths in the other. At the end of the day, the wealth you have in each stock would not be exactly one-fourth, three-fourths because the prices of the stocks change, so you would do the necessary buying and selling to restore it to one-fourth, three-fourths.”
Cover’s universal portfolio algorithm invests uniformly in all constant rebalanced portfolio strategies. The result is a strategy that is nearly optimal. Cover has shown, for any sequence of stock market outcomes, that this mixture of investments has as high a compound growth rate in the long run as the best constant rebalanced portfolio. Over time, the best strategy (that is, the best constant rebalanced portfolio) fights its way to the top of the fiscal food chain.
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One aspect of information theory is data compression. “The beauty of it is, the mathematics of growth-rate-optimal investment turns out to be parallel to the mathematics for optimal data compression,” Cover says. Thus universal investment algorithms are a counterpart to the universal data compression algorithms used to compress voice, fax and computer files.
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“This is an automatic investment algorithm in the stock market,” Cover says. “The portfolio rides the stocks and lives off the fluctuations. It essentially puts a little bit of money on every possible rebalanced investment algorithm, and the surviving algorithms — the ones that made most of the money — make enough so that your money grows at the same rate as if you had used the best algorithm to start with.”
The algorithm is “somewhat ponderous,” Cover says. “The performance of the algorithm, although good relative to the best portfolio in hindsight, is still slow in responding in an absolute sense. It sometimes requires hundreds of days before the initial conditions wash out, leaving the ‘fittest’ rebalanced portfolio dominating the performance. It’s guiding thinking, but no one’s making money off it yet.”
위에서 소개한 CASTrader의 필자는 CASTrader Blog: Thomas Cover’s Universal Portfolio – Part II에서 깊이 있는 공부를 자료를 풍부하게 소개합니다. William M. Beaver이 작성한 Universal Portfolios and Adaptive Trading라는 목록도 있습니다
Initial reading:
T. Cover and J. Thomas, Elements of Information Theory, Wiley & Sons, New York, 1991Universal portfolio:
Cover’s papers are on his website: http://www.stanford.edu/~cover/cover-papers.html
T. Cover. Universal Portfolios. Mathematical Finance, 1(1): 1-29, January 1991
T. Cover and E. Ordentlich. Universal Portfolios with Side Information. IEEE Transactions on Information Theory, 42(2):348-363, March 1996.
E. Ordentlich and T. Cover. The Cost of Achieving the Best Portfolio in Hindsight. Mathematics of Operations Research, 23(4):960-982, November 1998.
Online portfolio selection – Ordentlich, Cover – 1996Other authors and Universal Portfolios:
Universal Portfolios With and Without Transaction Costs – Blum, Kalai (1997)
Efficient Algorithms for Universal Portfolios – Kalai, Vempala (2002)
Universal portfolio selection, V. Vovk and C. Watkins.Online portfolio selection:
On the Competitive Theory and Practice of Portfolio Selection – Borodin, El-Yaniv, Gogan (2002)
On-Line Portfolio Selection Using Multiplicative Updates – Helmbold, Schapire, Singer, ..(1998)
Internal regret in on-line portfolio selection Gabor Lugosi and Gilles Stoltz 2005.Experts:
The weighted majority algorithm – Littlestone, Warmuth – 1994
Tracking the best expert – Herbster, Warmuth, Widmer and Kubat – 1995
Tracking a Small Set of Experts by Mixing Past Posteriors – Bousquet, Warmuth (2002)Games and experts:
Adaptive game playing using multiplicative weights Freund and Schapire 1999
Efficient algorithms for learning to play repeated games against computationally bounded adversaries. Freund, Kearns, Mansour, Ron, Rubinfeld, and Schapire. 1995
3.
이상이 이론적인 접근입니다. 금융공학에 대한 깊은 이해가 없으면 어렵습니다. 그래서 트레이더와 소프트웨어 개발자를 위한 글을 소개합니다. R을 사용하실 수 있으면 직접 시험까지 하실 수 있습니다. 혹 R과 금융수학을 잘 모르시면 ‘R과 함께 하는 금융통계과정‘을 등록해보시면?(^^)
Universal portfolio, part 1
Universal portfolio, part 2
Universal portfolio, part 3
Universal portfolio, part 4
Universal portfolio, part 5
Universal portfolio, part 6
Universal portfolio, part 7
Universal portfolio, part 8
Universal portfolio, part 9
Universal portfolio, part 10
Universal portfolio, part 11
