在线投资组合选择：一项综合调查

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"这篇在线投资组合选择的调查报告深入探讨了计算金融领域的一个基本问题——在线投资组合选择。报告由南洋理工大学的BIN LI和STEVEN C.H. HOI撰写，涵盖了金融、统计学、人工智能、机器学习和数据挖掘等多个研究领域的相关工作。" 在金融领域，"在线投资组合选择"（Online Portfolio Selection）是一个关键问题，它涉及到如何动态地调整投资组合以应对市场的不确定性。报告首先从在线机器学习的角度出发，将在线投资组合选择定义为一个序列决策问题。这表示投资者需要根据历史和实时的市场信息，不断更新投资策略。报告详细介绍了多种先进的方法，这些方法被分为几个主要类别： 1. **基准策略**：这些是最基础的方法，通常基于简单的规则，如均匀分配投资或跟随市场指数。它们提供了一种衡量其他复杂策略表现的标准。 2. **"Follow-the-Winner"方法**：这类策略倾向于增加最近表现良好的资产的权重，假设过去的赢家在未来仍有可能保持优势。然而，这种方法可能会导致过度集中在某些资产上，增加了投资风险。 3. **"Follow-the-Loser"方法**：与之相反，这类策略倾向于减少表现不佳的资产的权重，试图避免进一步的损失。这种方法可能更注重风险控制，但可能错过反弹的机会。 4. **基于模式匹配的方法**：这类方法试图识别市场模式，并据此调整投资组合。它们通常涉及复杂的数据分析和预测模型。 5. **元学习算法**：元学习策略通过学习过去的经验来适应未来的新情况，它们能够在不断变化的市场环境中快速适应。除了算法的介绍，报告还讨论了这些方法与资本增长理论的关系。资本增长理论，如马科维茨的现代投资组合理论，提供了理解投资组合优化的基本框架。通过对比这些算法与理论，可以更好地理解各种策略的相似性和差异性，以及它们在风险管理、收益最大化和市场适应性等方面的优劣。这份报告为研究者和从业者提供了一个全面的视角，帮助他们了解在线投资组合选择的最新进展和潜在的挑战，同时也为未来的理论和实证研究提供了指导。

A:6 Li and Hoi

both buying and selling. At the beginning of the t

period, the portfolio manager intends to rebal-

ance the portfolio from closing price adjusted portfolio

t−1

to a new portfolio b

. Here

t−1

calculated as,

t−1,i

t−1

·x

t−1

, i = 1, . . . , m. Assuming two transaction cost rates γ

∈ (0, 1 )

and γ

∈ (0, 1), where γ

denotes the transaction costs rate incurred during buying and γ

denotes

the transaction costs rate incurred during selling. After rebalancing, S

t−1

will be decomposed into

two parts, that is, the net wealth N

t−1

in the new portfolio b

and the transaction costs incurred

during the buying and selling. If the wealth on asset i before rebalancing is higher than that after

reblancing, that is,

t−1,i

t−1

·x

t−1

≥ b

t,i

t−1

, then there will be a selling rebalancing. Otherwise,

then a buying rebalancing is required. Formally,

t−1

= N

t−1

+γ

i=1



t−1,i

t−1

· x

t−1

− b

t,i

t−1



+γ

i=1



t,i

t−1

−

t−1,i

t−1

·x

t−1



Let use denote transaction costs factor [Gy¨orﬁ and Vajda 2008] as the ratio of net wealth after

rebalancing to wealth before rebalancing, that is, c

t−1

∈ (0, 1). Dividing above equation

by S

t−1

, we can get,

1 = c

t−1

+ γ

i=1



t−1,i

t−1

· x

t−1

− b

t,i

t−1



+ γ

i=1



t,i

t−1

−

t−1,i

t−1

·x

t−1



. (1)

Clearly, given b

t−1

, x

t−1

, and b

, there exists a unique transaction costs factor for each rebalancing.

Thus, we can denote c

t−1

as a function, c

t−1

= c (b

, b

t−1

, x

t−1

). Moreover, considering the

portfolio is in the simplex domain, then the factor ranges between

1−γ

1+γ

≤ c

t−1

≤ 1.

Finally, for each period t, the wealth grows by a factor as,

= S

t−1

× c

t−1

× (b

·x

) ,

and the ﬁnal cumulative wealth after n periods equals,

= S

t=1

t−1

× (b

·x

) ,

where c

t−1

is calculated as Eq. (1).

3. ONLINE PORTFOLIO SELECTION APPROACHES

In this section, we survey the area of online portfolio selection. Algorithms in this area formulate

the online portfolio selection task as in Section 2 and derive explicit portfolio update schemes for

each period. Basically, the routine is to implicitly assume various price relative predictions and learn

optimal portfolios.

In the subsequent sections, we mainly list the algorithms following Table I. In particular, we ﬁrst

introduce several benchmark algorithms in Section 3.1. Then, we introduce the algorithms with ex-

plicit update schemes in the subsequent three sections. We classiﬁes them based on the direction

of the weight transfer. The ﬁrst approach, Follow-the-Winner approach, tries to increase the rela-

tive weights of more successful experts/stocks, often based on their historical performance. On the

contrary, the second approach, Follow-the-Loser approach, tries to increase the relative weights of

less successful experts/stocks, or transfer the weights from winners to losers. The third approach,

Pattern-Matching based approach, tries to build a portfolio based on some sampled similar his-

torical patterns with no explicit weights transfer directions. After that, we survey Meta-Learning

Algorithms, which can be applied to higher level experts equipped with any existing algorithm.

3.1. Benchmarks

ACM Computing Surveys, Vol. V, No. N, Article A, Publication date: December YEAR.

Online Portfolio Selection: A Survey A:7

3.1.1. Buy And Hold Strategy. The most common baseline is Buy-And-Hold (BAH) strategy, that

is, one invests wealth among a pool of assets with an initial portfolio b

and holds the portfolio until

the end. The manager only buys the assets at the beginning of the 1

period and does not rebalance

in the following periods, while the portfolio holdings are implicitly changed following the market

ﬂuctuations. For example, at the end of the 1

period, the portfolio holding becomes

⊤

, where

denotes element-wise product. In a summary, the ﬁnal cumulative wealth achieved by a BAH

strategy is initial portfolio weighted average of individual stocks’ ﬁnal wealth,

(BAH (b

)) = b

t=1

The BAH strategy with initial uniform portfolio b



, . . . ,



is referred to as uniform BAH

strategy, which is often adopted as a market strategy to produce a market index.

3.1.2. Best Stock Strategy. Another widely adopted benchmark is the Best Stock (Best) strategy,

which is a special BAH strategy that puts all capital on the stock with best performance in hindsight.

Clearly, its initial portfolio b

◦

in hindsight can be calculated as,

◦

= ar g max

b∈∆

b ·

t=1

As a result, the ﬁnal cumulative wealth achieved by the Best strategy can be calculated as,

(Best) = max

b∈∆

b ·

t=1

= S

(BAH (b

◦

)) .

3.1.3. Constant Rebalanced Portfolios. Another more challenging benchmark strategy is the Con-

stant Rebalanced Portfolio (CRP) strategy, which rebalances the portfolio to a ﬁxed portfolio b ev-

ery period. In particular, the portfolio strategy can be represented as b

= {b, b, . . . }. Thus, the

cumulative portfolio wealth achieved by a CRP strategy after n periods is deﬁned as,

(CRP (b)) =

t=1

⊤

One special CRP strategy that rebalances to uniform portfolio b =



, . . . ,



each period is

named Uniform Constant Rebalanced Portfolios (UCRP). It is possible to calculate an optimal of-

ﬂine portfolio for the CRP strategy as,

⋆

= arg max

∈∆

log S

(CRP (b)) = a rg max

b∈∆

t=1

log



⊤



which is convexand can be efﬁciently solved. The CRP strategy with b

⋆

is denoted by Best Constant

Rebalanced Portfolio (BCRP). BCRP achieves a ﬁnal cumulative portfolio wealth and correspond-

ing exponential growth rate deﬁned as follows,

(BCRP ) = max

b∈∆

(CRP (b)) = S

(CRP (b

⋆

)) ,

(BCRP ) =

log S

(BCRP ) =

log S

(CRP (b

⋆

)) .

Note that BCRP strategy is a hindsight strategy, which can only be calculated with complete

market sequences. Cover [1991] proved the beneﬁts of BCRP as a target, that is, BCRP exceeds

the best stock, Value Line Index (geometric mean of component returns) and the Dow Jones Index

(arithmetic mean of component returns, or BAH). Moreover, BCRP is invariant under permutations

of the price relative sequences, i.e., it does not depend on the order in which x

, x

, . . . , x

occur.

ACM Computing Surveys, Vol. V, No. N, Article A, Publication date: December YEAR.

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