Comparison of Portfolio Management Strategies

Despite the great potential of the developed models, most of them have not been applied to real data. To prove the usefulness of portfolio management models for practitioners, we apply some of the contemporary results in this field to real MICEX trading data and give recommendations for their usage. Our database consists of the complete tick-by-tick limit order book for MICEX shares from January 2006 through June 2007. We consider only liquid shares, such as LKOH, RTKM and GAZP, because only during sufficiently intensive trading does it become possible to calibrate models for the real market.

We consider the problem of optimal purchase of a single-asset portfolio over a given period and compare the performance of the following strategies:

1. Immediate strategy—portfolio is obtained via a single trade at the moment of decision-making. This strategy must lead to the largest costs but eliminates market risk completely. It is recommended for high-volatility markets or in case of information about unfavourable future price movements.

2. Fruth et al.’s (2011) strategy—this has the same goal as uniform strategy, i. e. minimization of expected transaction costs but not market risk. The main advantage of the model is its flexibility and consideration of several main microstructure effects, such as time-varying immediate price impact, dynamic model of the order book and time-varying resilience rate. Authors define price impact for buy and sell sides (Et and Dt) as the difference between best price in the book and unaffected price. Permanent impact is proportional to volume of the order and constant over time while immediate response function K(t, v) = Ktv changes over time. Temporary impact decays exponentially with a fixed time – dependent, deterministic recovery rate pt, so that temporary impact of trade v,

t

—f Pudu

occurred at time s, at time t equals Kse s v. General framework considers

both continuous and discrete time market models. It generalizes Obizhaeva and Wang’s approach and postulates the following strategy: when price impact is low and the agent still has much to buy, she buys until the ratio of impact to remaining position is high enough, otherwise she waits for the impact to lower. After that, the agent can make another deal or wait, etc. So, for each moment of time, the agent has a barrier dividing her “Buy” and “Wait” regions.

3. Andreev et al. (2011) approach—a generalization of the Almgren and Chriss framework. Optimality is considered as minimization of both transaction costs and risk. This model has been obtained specifically for the MICEX market and incorporates a parametric dynamic model of cost function, which provides more accurate results: market model uses fundamental price instead of best bid-ask

image009

Fig. 4 Immediate response coefficient Kt for the whole trading day (7 February 2006) and dynamics during decline period (LKOH shares)

prices, which follows arithmetic Brownian motion. Transaction costs function has polynomial form (third degree polynom) with stochastic coefficients, which follow simple AR(1) model. No price impact is assumed. The strategy, unlike the previous three, considered agent risk aversion, which is characterized by the weighted sum of two criteria of optimality in minimization of functionality. Thus, problem formulates as minimization of — E(WT) + XVar(WT), where WT is terminal wealth and X is a priori risk aversion parameter.

For example, consider a 100,000 LKOH-share portfolio, liquidated via six consequent trades with 60-s wait periods. Consider also linear immediate response function with coefficient Kt. Rough estimate of Kt is obtained via least-squares

M

method: Kt = arg min ^ (@vC (t, v,) — Ktv,) , where C(t, v) is cost of trade with

І = 1

volume v, reconstructed from order book shape, and 0 < v1 < ••• < vM = V is a priori volume grid, for V we take half of available trading volume at the moment. Figure 4 shows dynamics of immediate response coefficient Kt. Liquidation begins when decline in response has been observed for some time (selected region in Fig. 4).

Strategies 2 and 3 are presented in Fig. 5 and have quite different behaviours. The form of the first strategy is obvious from the description. For Strategy 3, we use the simplest calibration assumptions, considering resilience rate a constant and immediate response as linear in time and volume. Assumptions are appropriate for medium periods of time.

We ascertain that the performance of Fruth et al.’s approach is the best of the three, while immediate buy is the worst. This result was expected because Strategy 2 is better adjusted to a specific form of response and can often show better performance if the form was guessed right. The strategy of Almgren and Chriss shows inferior performance and higher aggressiveness (see Fig. 6) due to

image010

Fig. 5 Trading strategies for immediate strategy, approach by Fruth et al. (2011) and approach by Andreev et al. (2011) with X = 0.01 for purchase of portfolio of 100,000 Lukoil shares via six trades with 1-min intervals. Date: February 7, 2006

image011

Fig. 6 In Almgren and Chriss framework aggressiveness of the strategy increases with risk – aversion parameter X. The figure demonstrates how volume left for execution depends on the number of trade for different values of X. X = 0 leads to equal size of trades. Initial volume is 10,000 shares, strategy allows the maximum of 20 trades

minimization of market risk if risk-aversion is sufficiently high.[1] The choice of risk-aversion parameter heavily influences resulting strategy but cannot be chosen automatically. Unfortunately some practitioners interpret this as a misspecification and excessive difficulty of the model and therefore favor simpler strategies. It is also not surprising that the Fruth et al. approach leads to lower costs than immediate
strategy: the strategies in the model contains immediate buy, and dynamics in the parameters of the market are taken into account. Immediate strategy doesn’t consider specifics or the current situation on the market, so it can be frequently outperformed by more elaborate methods.

Подпись: Conclusion Due to development of microstructure models and availability of high- frequency historic data, mathematical portfolio selection strategies have been extensively researched since the early 1990s. Nevertheless, very few frame-works were applied by practitioners because underlying models of the market were too unrealistic at the time. The aim of this research is to provide a review of modern accomplishments in the field, including the ongoing work, and demonstrate more realistic market models used in contemporary frameworks. To illustrate the effect of using automatic algorithms of portfolio selection and, in particular, optimal purchase/liquidation, we apply several approaches to real MICEX shares-related trading data and compare the results.

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