This paper describes a community of investors forming their expectations with heterogeneous strategies in order to optimize their portfolios by means of a Sharpe ratio maximization. Traders are distinguished according to their methodology used in forecasting. Twelve acknowledged algorithms of technical analysis have been implemented to compare portfolios performances and assess profitability of each technique.
of opinions (views) of analysts about the market. Obtained results lead to more stable and diversified optimal portfolio into the Markowitz model. This model was studied by [ 9], [10], [11] and [12], among many others.
Another way to find the optimal portfolio is based on the maximization of a portfolio performance index, as the Sharpe ratio ([13],[14]), which is defined as a ratio of the expected portfolio return relative to his standard deviation. Actually, it was Roy [15] who first suggested a portfolio valuation strategy based on a risk-return index; later, Sharpe applied his idea to the Markowitz approach by creating his famous index, which is one of the most used one, despite it is based on unrealistic hypotheses (among which, for instance, the assumption of Gaussian-distributed returns).
Since standard deviation is not always an appropriate measure of risk, many alternatives to the Sharpe ratio have been advanced. Among others, the Treynor ratio [16] substitutes the standard deviation with the portfolio beta, while the Sortino ratio [17] uses the downside risk as a risk measure, and the ratio of Bodnar and Zabolotskyy [18] replaces standard deviation with var. Other portfolio performance indices have also been developed, such as Jensen’s alpha [19] and many measures based on portfolios weights, as in [20], [21], [22] and [23].
Financial markets can be seen as complex systems ([24], [25]) in which agents cannot be described as copies of the representative agent [26] endowed with rational expectations ([27], [28] and [29]), but have heterogeneous features and base their behaviour on a bounded rationality ([30], [31]) and interact with each other ([32], [33]). Agent-based models (abm) allow treatment of complex systems by putting the evidence on the relevance of interactions and the unpredictable sequentiality governing the system dynamics. The application of such a methodology to Economics is called Agent-based Computational Economics (ace), defined by Tesfatsion [ 34] as “the computational study of economic processes modelled as dynamic systems of interacting agents”. This methodology has been fruitfully applied to the study of financial markets, as in [35], [36], [37], [38], [39], [40], [41], [42], [43], [44] and [45], among others, as surveyed by LeBaron [46]. It is important to mention the work of [47] in which they used an agent-based model for a portfolio optimization. In this field, [ 48] made an experimental study to assess the efect of complexity on asset trading.
Heterogeneous expectations and behaviours are often studied in related literature, by referring to the endogeneity of the signal followed to form expectations (i.e., between fundamentalists -who maintain a view on an exogenous signal- and chartists -who define the expected dynamics by inferring the trend of past prices). A vast body of references can be reported, among which: Alfarano et al. [49] created an agent based model in which agents can switch between fundamentalist and chartist strategies; Alfi et al. [ 50] analysed price dynamics considering the presence of forces due to trend adverse and trend following strategies; Flaschel et al. [51] showed how chartists tend to destabilize the economy; Lux and Marchesi [52] developed a model in which fundamentalists, optimistic and pessimistic trend followers can switch between strategies, thus giving rise to more instability independently of expected news; Lux and Marchesi [53] studied the volatility clustering; Westerhof [ 54] and Chiarella et al. [55] considered multi-asset markets, in which the interaction between them causes clustered volatility and fat tail of asset returns; Westerhof and Reitz [ 56] estimated an heterogeneous agent model underlying the impact of fundamentalists.
The present paper proposes a model based on the Sharpe ratio maximization, in which agents form their expectations by following heterogeneous strategies, in order to optimize their portfolio. More precisely, we propose a set of behavioural rules of prediction widely acknowledged in technical analysis (see [57], [58], [59], [60], [61], [62], [63], [64], [65]): three variants of the Moving Average Convergence Divergence model - macd; the Relative Strength Index - rsi; Rate of Change - roc; Crossover; Stochastica; Linear Forecasting; and Compound Forecasting. Finally, we consider also fundamentals, chartists and the completely random approach, proposed in [66], [67], [68], [69], as detailed below. All algorithms have been implemented to create portfolios whose values have been compared in order to assess the profitability of each technique. In literature, there are some papers that compare the performance of diferent strategies, among others: [70],[71],[72],[73],[74],[75]. The paper is organized as follows: section two contains the model, section three shows results of simulations; section four presents concluding remarks.
The model simulates diferent processes of expectations formation, each using a diferent technique. Such expected values will be used in Sharpe-ratio optimization procedures by traders in order to determine the quantities of two assets in the optimal portfolio. All common caveats of Markowitzian optimization hold: first of all, we assume that each investor allocates all her wealth in both assets - thus assuming that any possible decision inherent to saving has been already taken and the investor has decided the amount of resources for financial trading; secondly, we admit that the whole informative set consists in the available data on market dynamics - this will have an exception for one category of investors, as it will be detailed below; ifnally, we are not considering any interaction or imitation among traders - thus, no exchange of information or influence is playing here.
The evaluation of portfolio performances is aimed to check whether given return and risk requirements have been met by fund management. This assessment is usually operated with reference to asset classes, i.e., major categories of assets (stocks, bonds and money market instruments), industrial sectors for a portfolio of national equities, or countries for portfolios of international equities. The measure of portfolio performance should be increasing with respect to the expected return and decreasing with respect to the estimated risk. The Sharpe ratio (sr) is one of the most discussed metrics within the class of reward-to-variability measures, as related to the classical mean-variance model of portfolio selection. In fact, under mild assumptions, the optimization problem of finding the maximum expected portfolio return, given a target level of risk as measured by the variance of portfolio return, is equivalent to the maximization of sr.
In the current paper we build a dynamic discrete time model where asset prices vary within a fixed time horizon [0, ], > 0. In term of performance evaluation, financial institutions usually split the strategic horizon [0, ] into sub-periods of the same length, and try to predict the optimal sequence of portfolio weights maximizing its performance at the final date . Then, let us choose sr as the relevant measure of performance and use subscripts to denote any integer sub-intervals [ − the rate of return is = ( −
1, ] for = 1, 2, . . . , . time ∈ [0, ]. The market price of the th asset from time − 1 to time is represented by , − 1)/− 1 and the investment horizon [0, ] is a sequence of to optimal weights , with ∑︀ = 1.
An sr-optimizer trader maximizes the performance, measured by means of the sr, of her portfolio, composed by allocating, at time , the personal endowment to two assets, according
The multi-period optimization problem solved by each trader at each time step can be stated as: argmaxx subject to
E(r x ) sd(r x ) (1 + rx)− 1 xe = = 1 = 1, . . . , = 1, . . . , .
The objective function is the sr, i.e., the ratio of the expected return of the portfolio to its
standard deviation. The control variable is x = (x)∈ , which represents the result of diferent
trading strategies to be set up for both assets at each time and e is a vector of ones. Eq.(
√xxCmx
xe min w s.t.
12 wCw we wm traders, at each time step ∈ [0, ], aimed at maximizing their sr.
The numerical implementation of problem (
) for each and . mmCC−−11e ,
(
for = 1, . . . , and = 1, 2.
Since the model designs the existence of heterogeneous agents, we assume diferent, subjective, probability spaces (Ω, , P), for = 1, 2, ..., . More precisely, each probability measure P pertains to trader -th and it is defined through what he expects from diferent hypotheses on market movements. Thus, diferent traders have the diverse expectations about future prices and fundamental values. Expectations are then used to define the probabilities P themselves, see [77] and [78]. Ω can be considered as individual conception of the ideal experiment associated to forecasting future prices, while actual experiments only reveal the level of traders observation. Therefore, for each trader (based on his subjective judgment) there is a diferent expectation
m at time . In order to compute it, each trader adopts a given strategy and make a forecast for the price for the next time step, thus obtaining the expected return.
The basic setting is built upon nine diferent trader types, adopting the following strategies: 1) fundamentalists; 2) chartists; 3) macd traders (in turn divided in three groups, i.e., 3.1) macdbasic, 3.2) macd-plus, and 3.3) macd-divergence); 4) crossover traders; 5) rsi traders; 6) random traders; 7) stochastic traders; 8) roc traders; 9) two econometric forecasting approaches, i.e., 9.1) linear traders and 9.2) compound traders. Strategies 2), 3), 4), 5), 7) and 8) are based on descriptions given in relevant related literature (see [57] for further details). In this paper, for each strategy, is considered a single trader who will decide to buy if the expected price is higher than the current market price and sell in the opposite case.
Fundamentalists consider the fundamental value of the asset, , as the relevant indicator to
infer the dynamics of the price time series. Such a fundamental value is, according to their view,
the true value of the asset, which embeds the “correct” evaluation of all relevant information and
also includes the efects of the dividend yields. It is defined as an exogenous random variable
whose dynamics may be assumed as a martingale-dividend-driven. Thus, the expected price,
for a fundamentalist can be written as:
= − 1 + D
(
= + Θ
where Θ is randomly chosen in the interval (− , ), in order to account for the heterogeneity
of investors. This implies that when the expected price is greater (resp., smaller) than the
observed current market price she decides to buy (resp., sell), because she is expecting that the
asset always converges to its fundamental.
(
The second type of investos has been designed as momentum trader. Momenutm is the name of a very broad category of investment strategies, based on the observation of past values of ifnancial time series. In particular, a very general definition of a momentum oscillator is given by:
= − − momentum strategy is based on the sign changes of : where the comparison between the current market price and the past price registered a given number of periods ago, − is used to infer the future dynamics. In particular, the basic ︂{ − → + → − + buy sell Related literature calls "chartists” one of possible expressions of such a kind of strategy: a group of technical analysts who found their trading decision on the observation of trends and past dynamics of financial prices. More precisely, each of them refers to a time window, whose length is individually heterogeneous, and compute a reference value , computed at any by averaging prices included in a time window of length , diferent for each chartist and randomly chosen in the interval (2, ): Thus, the expectation determined by the chartist is =
1 ∑︁ .
=−
= ± ( − ) less than . where the choice of setting ± in the formula depends on whether is respectively greater or The set of financial oscillators is very wide and a complete survey of them goes far beyond the scope of the present paper. Nonetheless, the model includes also other three kinds of traders based on technical analysis adopting the so-called Moving-Average-Convergence-Divergence (aka macd) strategy, developed by Gerald Appel. Such an oscillator combines exponential moving averages, , referred to diferent time intervals, .
The basic computation is considered as the diference between the averages over = 12 and where = − 1 + (︀ − − 1︀) , with 0 = ∑︀ /, and = +21 .
Oscillations of the two moving averages are diferent: the wider the days interval over which it is calculated, the slower it moves. Thus the slower is used to observe the trend, whereas the faster provides the trigger signal: when the latter crosses the former from the bottom (top), it is a bullish (bearish) signal. Such an oscillator is used according to three diferent versions, basically responding to the same rationale with diferent levels of refinement and data usage. The first, henceforth named macd-basic, is: =− ︂{ > 0 < 0 → → buy sell ︂{ > 9 < 9 → → buy sell
(
The Crossover method, inspired by the triple crossover method mentioned for the first time by R.C. Allen ([79], [80]), requires exponential moving averages for diferent time periods (d), Finally, the third version, henceforth named macd-divergence, is based on a more detailed data observation, which is based on divergences. In particular, in a windowed period of past days, the original time series of the asset price and the correspondent macd time series are compared. If either the two maximum points or the two minimum points in the interval of both series are oriented in the same direction of the trend in the same interval, then there is not a divergence. If, instead, a divergence exists, then it serves as a trigger. Let us indicate by the slope of the trend inferred by the time series and by the slope of the line joining the two local extrema included in the considered time window. The signalling role on the trading strategy played by the divergence can be described as: ︂{ < 0 ∧ > 0 > 0 ∧ < 0 → → buy sell , as for the case of macd strategies. More precisely, this method considers three moving averages, with = 12, = 20 and = 26, calculated as above described. A buy signal will occur in a downtrend, in which the shorter average (12 day) is bigger than both the medium (20 day) and the longer ones (26 day). A sell signal, instead, will occur in the opposite situation, in which the longer average is the biggest one.
︂{ 26 < ( 20; 12) 26 > ( 20; 12) → → buy sell
Inspired by the Relative Strength Index, introduced by Wilder [62], our corresponding trading
strategy considers once a time window of the recent past = 30 values of the time series to
infer possible forecasts on future dynamics. In particular, the index is computed as
(
The role of random traders has been investigated in several contributions under the names “zerointelligence”, “noise”, etc. They are defined as traders who decide when/how to invest at random. For such a reason, they have been used as the representation of the lack on influence/imitation in order to challenge the normal financial dynamics and show interesting insights to dampen market fluctuations, i.e., reducing volatility. In the present model, random traders are the sole group that will not use a portfolio optimization: they will simply decide whether to buy or to sell by tossing a fair coin. The quantity of both assets to have is extracted randomly by the algorithm governing the simulations, in such a way that both weights, 1 ∈ [0, 1] and The stochastic oscillator, popularized by George Lane, is based on the idea that when prices rise, closing prices tend to approach the upper end of the price range. The opposite occurs where prices fall. We present a simplified computational version of this method, that we implement as follow. Recent past values of prices are selected in order to compute two indexes: = 100 · 5 − 5 − 5 and days of 5 − 5. The strategy can therefore be described as: where: is the last price; 5 is the minimum value in the last 5 days; 5 is the maximum value in the last 5 days; 3 is the sum of the last 3 days of − 5; 3 is the sum of the last 3 ⎨
The rate of change (roc) is a ratio which assumes increasing values in cases of an upward trend and decreasing values in cases of downward trend. It is computed as follows: where: is the latest close price and is the former closing price. Once defined a given threshold = 0.05, roc traders’ strategy has been simply modelled as follows: = ︂{ ≥ 1 + < 1 + → → buy sell = + · = · (24) (25) (26) (27) (28) (29)
Two last techniques adopt ols estimation in order to predict future prices, by means of a linear and a compound model. In the first case, the prediction is obtained by: where and indicate a constant and the growth proportion, which could be interpreted as = 1+ growth-rate, and indicates time. where and are, respectively, the constant and the slope of the trend-line, and indicates time, thus implying a constant growth.
The second approach assumes, instead, that the variable grows by a constant proportion each period. In this case, the prediction is obtained by:
The model has been engineered as a sort of back-testing framework in which a fictitious reality is simulated. In Table 1 time series used for simulations are listed, each reporting the initial date of the time series. The simulation considers all possible couples of assets and defines a ifnancial market where traders try to get the highest gain in terms of portfolio value. For each couple, the length of the simulation is set to the length of the shortest time-series, in such a way that at every simulation step, traders will always have newly disclosed prices for both assets. In such a configuration, the simulated market will operate as if investors were living in the past, thus making the measurement of performance possible, while a sort of financial race is run among traders, in order to discover which strategy accumulates the highest wealth. Each trader adopts a strategy and cannot change it. At the beginning, all traders are given the same amount of money, = ¯ = 250. By means of their trading decision they will trade assets, thus computing their wealth, , possibly greater or smaller than the initial endowment. Further, traders are assumed to be marginal with respect to the market, which means that they will always find the possibility to negotiate assets as they want. In other words, they will not be constrained to buy and sell selected quantities. If a trader goes bankruptcy, he will not be replaced and this will correspond to the worst outcome (i.e., Δ = − 100%).
Unlike fundamentalists, all other types of traders considered observe past values of assets to take decisions. In particular, most of these traders are momentum traders and we assume that they must have at least 30 data available for each asset. Instead, econometrician forecasters operate by observing a wider time window of at least 180 data.
Results reported in Table 2 show the number of times that each strategy scored the first, the second and the third in the ranking and the amount of times they fail. Fig.1 compares how many times the strategies have placed first and fail while Fig.2 shows how many times they arrived in the first three positions of the 1225 rankings analysed.
More in detail, stochastica strategy have been place in the top three position of the ranking 696 times, followed by the crossover strategy with 581 placements, fundamentalists with 511 and the rsi with 532. Then we find the rnd with 283 placements and macd-divergence with 258, continuing with the chartists that have been in the top three positions 237 times and the macd-basis strategy with 231 placements. The roc with 111 times follows the macd-plus strategy with 133 placements. At the end there are the two econometric forecasting approaches, linear (59) and compound (37).
Data show that, overall, the stochastica strategy performed the best and that the worst are, instead, the econometric forecasting approaches.
For each couple of assets, simulations give twelve vectors containing portfolio values obtained by each strategy. The volatility of the best performing strategy has then been computed as the standard deviation of the distribution of portfolio values associated to that strategy for all occurrences in which it has ranked first. Fig.3 compares the volatility of all winning strategies, thus showing that even best performing ones manifest wide variability across the distribution of results.
We carried out the same analysis considering also the intraday data, more specifically the hourly data and data with a frequency of five minutes and one minute. All data were downloaded using Refinitiv’s Eikon ©, and the intraday series were considered with their maximum available length, corresponding to the last year for hourly data and the last three months for five minutes and one minute data. For data with a frequency of five minutes and one minute, the given threshold for the roc strategy was reduced ( = 0.01).
For hourly data we found the following results,reported in Table 3, Fig. 4 and Fig. 5.
Observing these results, we note that chartists reached the first position more times than the others, but by observing the number of times in which strategies have reached the top three and last positions, it is noted that the strategy that has performed better overall is the crossover, which it finished in the top three 468 times and fail 397 times. The worst are confirmed to be macd-plus and the econometrician forecasters, with compound strategy that failed 1118 times out of 1225.
For data with a frequency of 5 minutes we found the following results, reported in Table 4,
The best performances are those of fundamentalists (in the top three positions 387 times and, again, with the least number of failures 211) and once again the macd-basic strategy which reaches the top three positions 480 times . Chartists have reached the first position several times (206) but have failed many times (631). The worst is the compound strategy, that failed 1210 times.
In this paper we presented a model testing diferent strategies, based on technical analysis, of expectations formation and show how they afect performances of investors.
The proposed approach lies on back-testing simulations over true data. Data is used with reference to each day, as agents experience each time step ignoring the future. Thus, they choose on the basis of their expectations only, thus making it possible to measure performance of their portfolio investments, as in a financial race. Portfolios are obtained by considering all possible couples of the 50 assets.
Simulations have been devoted to investigate two sets of aspects. First of all, the performance of diferent strategies has been analysed. For each couple of assets, a ranking of strategies was identified on the basis of portfolio values. Overall, results show that there are no concordant rankings of various strategies at diferent time lengths resolution. Thus, for instance, a winning strategy in simulations with daily data reveals to loose if used with infra-day data. This provides evidence that optimal strategies are more a desire of investors than an actual results. The reason why financial success can be reached in speculative investments must depend on occasional reasons and not on the chosen strategy. In other words, the complexity of the system prevents any forecast and past trends, elaborated as much as algorithm and experienced traders want, need not be replicated.
Secondly, we analysed the volatility of the best performing strategy and the instability of the market. Consequently to our first finding, we found that even best performing strategies manifest wide variability. One can conclude that the variability of results is not simply linked to the goodness of the adopted strategy. Something lying between the lines makes all strategies uncertain and potentially harmful.
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