# Empirical Analysis of an Online Algorithm for Multiple Trading Problems

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5 Figure 1: Holding period for MT RANDOM (Rand) Rand will buy and sell at randomly chosen prices p j (t) within the holding period of MT (cf. Figure 2). Figure 2: Trading Possibilities for Rand BUY AND HOLD (B+H) B+H will buy at the first day t of the period and sell at the last day T of the period. MARKET (MA) To evaluate the performance of these three policies empirically we use as a benchmark the optimal offline policy. It is assumed that MA knows all prices p j (t) of a period including also these which were not presented to MT if there were any. In each period i MA will buy at the minimum price p min > m(i) and sell at the maximum possible price p max < M(i) within the holding period of MT (cf. Figure 3). Figure 3: Trading possibilities for MA The performance of the investment policies is evaluated empirically. Clearly, all policies cannot beat the benchmark policy MA. The policies are tested using historical 5

8 ticker prices and buys (sells) if slopes of long (short) and short (long) match (both negative / positive). The fourth policy implemented is the reverse strategy discussed in [FRY04]. All four policies were tested over a 15-day period from to with NASDAQ order book information. Three mixed policies which combine 2, 3 or all of the four policies were considered: SOBI + VWAP + Reverse + TF, SOBI + Reverse and SOBI + Reverse + TF. Results compare returns and Sharpe ratio. For a period length of 15 days the best combined policy is SOBI + Reverse + TF in terms of return, the reverse policy is the overall winner in terms of Sharpe ratio. 5 Experimental Results We want to investigate the performance of the trading policies discussed in Section 3 using experimental analysis. Tests are run for all p = 1, 2,, 6 period types with the number of periods n(p) from {52, 26, 13, 4, 2, 1} and period length h from {7, 14, 28, 91, 182, 364} days. The following assumptions apply for all tested policies: 1. There is an initial portfolio value greater zero. 2. Buying and selling prices p j (t) of an asset j are the closing prices of day t. 3. At each point of time all money is invested either in assets or in 3% overnight deposit. 4. Transaction costs are % of the market value but between 0.60 and Euro. 5. When selling and buying is on different days the money is invested in overnight deposit. 6. At each point of time t there is at most one asset in the portfolio. 7. In each period i at least one buying and one selling transaction must be executed. At the latest on the last but one day of each period asset j has to be bought and on the last day it has to be sold. 8. In period i = 1 all policies buy the same asset j on the same day t at the same price p j (t); the asset chosen is the one MT will chose (MTasset). 9. In periods i = 2,,n(p)-1 trades are carried out according to the different policies. 10. In the last period i = n(p) the asset has to be sold at the last day of that period. No further transactions are carried out from there on. 11. If the reservation price is calculated over h days, the period length is (also) h days. We simulate all policies using historical XETRA DAX data from the interval until This interval we divide into n(p) periods where n(p) is from {52, 8

9 26, 13, 4, 2, 1} and p is from {7, 14, 28, 91, 182, 364}. With this arrangement we get 52 periods of length 7 days, 26 periods of length 14 days, etc. We carried out simulation runs in order to find out (1) if MT shows a superior behaviour to buy-and-hold policies (2) the influence of m and M on the performance of MT (3) the average competitive ratio for policies for MA and MT Two types of buy-and-hold policies are used for simulation; one holds the MTasset within each period (MT_B+H) and the other holds the index over all periods (Index_B+H) of a simulation run. Thus, MT_B+H is synchronized with the MT policy, i.e, MT_B+H buys on the first day of each period the same asset which MT buys first in this period (possibly not on the first day) and sells this asset on the last day (note that this asset may differ from the one MT is selling on the last day) of the period. Using this setting we compare both policies related to the same period. Index_B+H is a common policy applied by ETF investment funds and it is also often used as a benchmark although it is not synchronized with the MT policy. In addition to these policies also the random policy Rand is simulated. Rand buys the same asset which MT buys on a randomly chosen day within a holding period. We first concentrate on question (1) if MT shows a superior behaviour to the policies MT_B+H and Index_B+H. For calculating the reservation prices we use estimates from the past, i.e. in case of a period length of h days m and M are taken from the prices of these h days which are preceding the actual day t* of the reservation price calculation, i.e. m = min {p(t) t = t*-1, t*-2,..., t*-h} and M = max {p(t) t = t*-1, t*-2,..., t*-h}. In Table 5-1 the trading results are displayed considering also transaction costs. The return rates are calculated covering a time horizon of one year. For the three active policies (MA, MT, Rand) the transaction costs are the same because all follow the holding period of MT; in all these cases there is a flat minimum transaction fee. 9

10 Historic Annualized Returns Including Transaction Costs Policy 1 Week n(7) = 52 2 Weeks n(14) = 26 4 Weeks n(28) = 13 3 Months n(91) = 4 6 Months n(182) = 2 12 Months n(364) = 1 MA % % % 47.93% 72.95% 61.95% MT 41.08% 1.37% 54.86% 6.08% 32.39% 31.35% MT_ B+H 9.70% 0.50% 17.18% 15.80% 45.30% 35.29% Index_B+H 20.78% 20.78% 20.78% 20.78% 20.78% 20.78% Rand % % 17.18% % 6.20% 15.42% Table 5-1: Annualized Return Rates for different period lengths MT dominates MT_B+H and Index_B+H in two cases (1 and 4 weeks). MT_B+H dominates MT and Index_B+H in two cases (6 and 12 months). Index_B+H dominates MT and MT_B+H in two cases (2 weeks and 3 months). MT generates the best overall annual return rate when applied to 4 weeks. MT_B+H generates the worst overall annual return rate when applied to 2 weeks. MT_B+H policy improves its performance in comparison to Index _B+H and MT policy proportional to the length of the periods. We might conclude the longer the period the better the relative performance of MT_B+H. MT outperforms Index B+H in four of six cases and it outperforms MT_B+H in three of six cases; MT and MT_B+H have the same relative performance. If the period length is not greater than 4 weeks MT outperforms MT_B+H in all cases. If the period length is greater than 4 weeks MT_B+H outperforms MT in all cases. Index_B+H outperforms MT_B+H in three of six cases. If we consider the average performance we have 27.86% for MT, 20.78% for Index_B+H, and 20.63% for MT_B+H. MT is not always the best but it is on average the best. From this we conclude that MT shows on average a superior behaviour to buy-and-hold policies under the assumption that m and M are calculated by historical data. In general we would assume that the better the estimates of m and M the better the performance of MT. Results in Table 5-1 show, that the longer the periods the worse the relative performance of MT. This might be due to the fact that for longer periods historical m and M are worse estimates in comparison to those for shorter periods. In order to analyse the influence of estimates of m and M we run all simulations also with the observed m and M of the actual periods, i.e. we have optimal estimates. Results for optimal estimates are shown in Table 5-2 and have to be considered in comparison to the results for historic estimates shown in Table

11 Now we can answer question (2) discussing the influence of m and M on the performance of MT. The results are displayed in Table 5-2. It turns out that in all cases the return rate of policy MT improves significantly when estimates of m and M are improved. For all period lengths now MT is always better than MT_B+H and Index_B+H. From this we conclude that the estimates of m and M are obviously of major importance for the performance of the MT policy. Clairvoryant Annualized Returns Including Transaction Costs Policy 1 Week n(7) = 52 2 Weeks n(14) = 26 4 Weeks n(28) = 13 3 Months n(91) = 4 6 Months n(182) = 2 12 Months n(364) = 1 MA % % % % 86.07% 70.94% MT % 87.90% 76.10% 81.38% 55.11% 54.75% MT_ B+H 9.70% -4.40% 22.31% 19.79% 45.30% 35.29% Index_B+H 20.78% 20.78% 20.78% 20.78% 20.78% 20.78% Rand % % % 47.37% 46.08% 15.42% Table 5-2: Annualized returns for optimal historic estimates Now we concentrate on question (3) discussing the average competitive ratio for policies MA and MT. We now compare the experimental competitive ratio c ec to the analytical competitive ratio c wc. To do this we have to calculate OPT and ON for the experimental case and the worst case. We base our discussion on the return rate R = sp / bp as the performance measure where sp is the selling price and bp is the buying price. We assume that we have precise forecasts for m and M. A detailed example for the evaluation of the competitive ratio is presented in Table 5-3 considering a period length of 12 months. In this period six trades were executed using reservation prices based on the clairvoyant test set. The analytical results are based on the consideration that MA achieves the best possible return and MT achieves a return of zero. E.g. for the first trade MA achieves a return rate of % and MT achieves a return rate of 0 % i.e. MT achieves absolutely % less than MA and relatively a multiple of The experimental result are also based on the consideration that MA achieves the best possible return and MT now achieves the return rate generated during the experiment. E.g. for the first trade MA achieves a return rate of % and MT achieves a return rate of 13.22%, i.e. MT achieves absolutely 0.82 % less than MA and relatively a multiple of

12 Clairvoyant Analytical Results Experimental Results # Trades n(364)=1 Holding Period Buy at Sell at Periodic Return Ratio MA/MT Buy at Sell at Periodic Return Ratio MA/MT 1 st trade Week , MA 37,91 43, ,91 43,23 1,1403 MT p* p* 1 37,91 42,92 1, nd Week , MA ,25 38,15 1,1139 MT p* p* 1 34,25 37,89 1, rd Week , MA ,54 13,69 1,0111 MT p* p* 1 13,54 13,69 1, th Week , MA ,57 35,73 1,0643 MT p* p* 1 34,13 35,73 1, th Week , MA ,23 58,86 1,1489 MT p* p* 1 52,37 56,52 1, th Week , MA ,16 89,4 1,0881 MT p* p* 1 82,66 89,4 1,0815 Table 5-3: Periodic results for period length one year We compared the analytical results with the experimental results based on annualized returns for the period lengths 1, 2, 4 weeks, 3, 6, and 12 months. The overall annual return rates for all period lengths are presented in Table 5-4. Transaction costs are not taken into account in order not to bias results. As the policies are always invested there is no overnight deposit. 12

13 Clairvoyant Ratio Annualized Return Period # Trades OPT/1 OPT/ON 1/ON MA MT MA-MT MT/MA 12 Month % 71.08% 54.89% 16.19% 77.22% n(364)=1 6 Months 7 1, % 86.24% 55.28% 30.96% 64.10% n(182)=2 3 Months % % 81.82% % 44.50% n(91)=4 4 Weeks % % 77.02% % 27.33% n(28)=13 2 Weeks % % 89.05% % 28.10% n(14)=26 1 Week % % % % 32.75% n(7)=52 Average 57,07% % 76.98% % 45.67% Table 5-4: Competitive ratio and annualized return rates For the period of 12 months the analytical worst case ratio (OPT/1) is and the average experimental ratio (OPT/ON) is The percentage the experimental ratio reaches of the worst case ratio which can be calculated by (1/ON) is 64.56%. The values for the other period lengths are also given in Table 5-4. It turns out that the average experimental ratio reaches at least 49.06%, at most 64.56% and on average 57.07% of the analytical worst case ratio; the longer the period the closer is the experimental ratio to the analytical ratio. In terms of return the experimental return reached by MT reaches at least 27.33%, at most 77.22% and on average 45.67% of the analytic and experimental return of MA. 5.Conclusions We carried out several experiments to answer three questions. The first is whether MT shows a superior behaviour to buy-and-hold policies or not. The second discusses the influence of estimates for upper and lower bounds, m and M, for asset prices on the performance of MT. The third question asks for the average competitive ratio for policies MA and MT. In order to answer these questions six clairvoyant simulations with optimal estimates for m and M as well as six simulations with historical estimates for m and M were performed. To answer the first question MT outperforms buy-and-hold in all cases even when transaction costs are incorporated in the clairvoyant test set. Simulations based on historical estimates of m and M show that MT outperforms buy-and-hold in one third of the cases and on average. We conclude that if the period length is small enough MT outperforms B+H. 13

14 From this we can answer the second question discussing the influence of m and M on the performance of MT. It is obvious that the better the estimates of m and M the better the performance of MT. Results show that the longer the periods, the worse are estimates by historical m and M in comparison to those for shorter periods. As a result, the performance of MT gets worse the longer the periods become. The third question asks for the average competitive ratio for policies MA and MT. Fortunately, these results show that it is very difficult to get close to the (analytical) worst cases under (simulated) real-life considerations. It turns out that the shorter the periods are the less MT achieves in comparison to MA. The longer the periods, the closer is the experimental ratio to the analytical ratio. A MT trading policy which is applied to short periods leads to small intervals for estimating historical m and M. In this case there is a tendency to buy too late (early) in increasing (decreasing) markets and to sell too late (early) in decreasing (increasing) markets due to unknown overall trend directions, e.g. weekly volatility leads to wrong selling decisions during an upward trend. The paper leaves also some open questions for future research. One is that of better forecasts of future upper and lower bounds of asset prices to improve the performance of MT. The suitable period length for estimating m and M is an important factor to provide an optimal trading signal, e.g. if the period length is h days estimates for historical m and M were also be calculated over h days. Simulations with other period lengths for estimating m and M could be of interest. Moreover, the data set of one year is very small. Future research should consider intervals of 5, 10, and 15 years. References Yan98 YFKT01 YFKT92 El-Yaniv,R., Competitive solutions for online financial problems, ACM Computing Surveys 30 (1), 28-69, 1998 El-Yaniv, R., Fiat,A., Karp,R., Turpin,G., Optimal search and one-way trading algorithm, Algorithmica 30, , 2001 El-Yaniv,R., Fiat,A., Karp,R., Turpin,G., Competitive analysis of financial games, IEEE Symposium on Foundations of Computer Science , 1992 She02 Shen,P. Market Timing Policies that Worked, Working Paper RWP 02-01, Research Division, Federal Reserve Bank of Kansas City, May 2002 CE08 Chavarnakul,T., Enke,D., Intelligent technical analysis based equivolume charting for stock trading using neural networks, Expert Systems and Applications 34, ,

15 RL99 FRY04 RS03 SR05 Ratner,M., Leal,R.p.C., Tests of technical trading policies in the emerging equity markets of Latin America and Asia, Journal of Banking and Finance 23, , 1999 Feng,Y., Ronggang,Y., Stone,P., Two Stock Trading Agents: Market Making and Technical Analysis, In: Peyman Faratin, David C. Parkes, Juan A. Rodriguez-Aguilar and William E. Walsh (eds.), Agent Mediated Electronic Commerce V: Designing Mechanisms and Systems, Lecture Notes in Artificial Intelligence, 18 36, Springer Verlag, 2004 Ronggang,Y., Stone,P., Performance Analysis of a Counter-intuitive Automated Stock Trading Stategy, ACM International Conference Proceeding Series Vol. 50, In: Proceedings of the 5th International Conference on Electronic Commerce, Pittsburgh, Pennsylvania, 40-46, 2003 Silaghi,G.C., Robu,V., An Agent Policy for Automated Stock Market Trading Combining Price and Order Book Information, ICSC Congress on Computational Intelligence Methods and Applications, 4-7,

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