Summary of Ch4, Online Algorithms for the Portfolio Selection Problem
Portfolio Selection Problem
The portfolio selection problem (PSP) aims to maximise the wealth of the investor at the conclusion of the investment horizon.
The online algorithms attempting to solve PSP assume:
- There is no short-selling, the investor owns only positive number of shares at each time instant
- The investor does not take out or add wealth during the investment horizon
Consider an investor with an investment horizon of t = 1, …, T trading periods. The investor can allocate his wealth to A _1,…., A_m assets.
The algorithm seeks to solve how the investor should choose each b_t in the sequence of decisions noted as b = b_1, … , b_t to maximise the terminal wealth W_t.
Virtual Market
In evaluating portfolio selection algorithms, we benchmark our performance against the the strategies: Buy-and-Hold Portfolio, Constant Rebalancing Portfolio, Switching Portfolio
We assume that in a virtual market, the investor seeks to earn wealth, µ > 0 by selecting one of the following strategies:
(i) Buy-and-hold: The portfolio allocation b is realised once at the beginning of the trading period; No adjustment of portfolio allocation
(ii) Constant Rebalancing Portfolio: The portfolio allocation b is held constant for all trading periods, t = 2, …, T.
(iii) Switching Portfolio: Portfolio allocation switches during odd and evening trading periods. During odd trading periods the portfolio is allocated by b_t = (b_1, b_2). During even trading periods, the portfolio is allocated by b_t = (1-b1, 1-b2)
The investor’s decision problem consists of two parts:
(1) Choosing one of the three strategies
(2) Committing to a single portfolio allocation b to the strategy
As depicted, there exists a limit to the exponential growth rate for μ for BH, CR, and SW strategies. The BH strategy does not yield μ > 0 at any portfolio allocation. However, at 0 < b< 1, the CR strategy yields μ > 0.
Preliminaries for FTW and FTL Algorithms
The BH, CR, and SW strategies build a foundation for follow-the-winner and follow-the-loser algorithms.
Assume that for for each strategy, there exists an optimal b_* which yields the highest terminal wealth W_t. Generally, follow-the-winner algorithms track the b_* of constant rebalancing portfolios. Follow-the-loser algorithms track the b_* of switching portfolios.
Other Assumptions and Definitions for online algorithms:
- All online algorithms require at least one or more trading periods to produce the portfolio allocation b_(t+1)
- Until an online ALG is able to produce b(t+1), we can use assume the algorithms start with naive diversification
- An algorithm with window size requires at least w trading periods before a calculation is possible
- Some algorithms exploit first and second order information, which are related to holding period return, to determine the best portfolio allocation
- First order describes price change of A_i in relation to the holding period return of the current portfolio allocation
- Second order information describes combined price change of A_i and A_j for i, j = 1, …, m in relation to the quadratic holding period return
- Algorithms can be built using external information regarding the market into account. As an example, an algorithm can consider the three forecasting indications: “Buy”, “Hold” and “Sell”
Follow-the-Winner Algorithms
The objective of FTW is to track the best constant rebalancing portfolio or the best CR expert. FTW algorithms are generally implemented by transferring the portfolio weights from the underperforming assets to the outer-performing assets.
Successive constant rebalanced algorithm
SCR chooses the best CR-expert for a trading period t for a set of Ω CR-experts. The portfolio allocation is calculated by:
Universal portfolio algorithm
Similarly, consider a scenario with ω = 1, …, Ω CR-experts. Each CR-expert has a unique portfolio allocation, b_ω.The algorithm determines the proportion of wealth for A_i by considering current period wealth and the fraction of A_i in the portfolio 𝔟_ω.
With a continuous simplex 𝔅_m, the algorithm is defined as:
Exponential gradient algorithm
The EG algorithm uses first order information to determine the best portfolio allocation. The proportion of A_i for t+1 is increased if the return factor of A_i during trading period t is larger than the current period wealth change.
Online Newton Step Algorithm
ONS takes both first order and second order information into account. First, a matrix A_t is calculate for each trading period. Then, a vector o_t is calculated combining second and first order information. The algorithm is defined as:
Follow-the-Loser Algorithms
The FTL algorithms assume that underperforming assets will revert and outperform other assets in subsequent periods. These algorithms generally move portfolio weights from outperforming assets to the underperforming ones.
Anti Correlation Algorithm
The AC algorithm assumes that all stocks perform similarly in their long-term exponential growth. If a stock in a past time window shows different performance, then the stock will likely exhibit a counter movement in future performance. The AC algorithm is defined as follows:
Where Claim_{i,j} is the claim from transferring wealth from A_i to A_j.
Passive Aggressive Mean Reversion Algorithm
The PAMR algorithm punishes assets which exhibit larger return factors than the market average. The PAMR alternates between aggressive and passive investment strategies through a loss function.
The algorithm is defined as:
Where v_t is the sum of the square deviation of the average market change. x̄_t is the mean reversion level. Loss_t is a loss function which becomes negative if the holding period return is larger than the mean reversion level. λt is a multiplier function. λt can be differentiated into three varying degrees of a multiplier value, PAMR1, PAMR2, and PAMR3.
Confidence Weighted Mean Reversion Algorithm
CWMR adapts upon the PAMR algorithm by applying statistical learning techniques. Two versions of CWMR algorithm exist. One is based upon variance and the other is based on standard deviation.
The algorithm is defined as:
Online Moving Average Mean Reversion Algorithm
OLMAR is an extension of the PAMR algorithm. Unlike PAMR, it evaluates more than one past trading periods.
The algorithm is defined as:
Where the variable x̄_{it}^w signifies if the current price of A_i is greater or lower than the current moving average. x̄_{t}^w is the market average of all x̄_{it}^w.
Robust Median Reversion Algorithm
RMR is an extension of OLMAR. However, instead of using the current moving average, RMR leverages a vector of median asset prices as a predictor.
The algorithm is defined as:
Conclusion
Generally, the algorithms have the following time complexities:
Moreover, several analyses and investigations have indicated that FTL algorithms outperform FTW algorithms when maximising terminal wealth.
Works Cited
Dochow, Robert. “4.” Online Algorithms for the Portfolio Selection Problem, Springer Fachmedien Wiesbaden, Imprint: Springer Gabler, Wiesbaden, 2016.
