ON TRANSACTION COSTS IN STOCK TRADING

QUANTITATIVE METHODS IN ECONOMICS Volume XVIII, No., 07, pp. 58 67 ON TRANSACTION COSTS IN STOCK TRADING Marek Andrzej Koiński Faulty of Applied Informatis and Mathematis Warsaw University of Life Sienes SGGW, Poland e-mail: marek_koinski@sggw.pl Abstrat: Liquidity is an important harateristi of a stok traded on the stok exhange. The expeted value of transation osts, whih takes into aount the transation's volume and duration, may be a onsidered as an important measure of a liquidity of a traded stok. In this paper the formulas for expeted transation ost, aused by bid-ask spread and market impat are presented. Moreover, in this artile, the problem of determining a duration of a transation of a stok sale whih minimizes the transation ost and takes into aount the foreast of the expeted stok prie on the stok exhange, is onsidered. Keywords: liquidity, transation ost, bid-ask spread, market impat, trade duration INTRODUCTION Liquidity is an important issue in stok markets. In fat, a liquidity of a stok traded on the stok exhange is measured by the ost of its trading. For the purposes of market partiipants, the orret way to view liquidity should imply the possibility of suffiiently aurate foreasting the stok prie hange aused by the trade initiator and estimating the transation ost. Transation osts (trading osts) are widely reognized as an important fator whih determines the finanial investment performane. Their understanding and assessment is important for eonomi theory and partiipants of finanial markets. Transation osts are substantial omponent of realisti models of the stok market mirostruture. In ase of the stok transation between two parties: buyer and seller, transation osts refer to osts paid by one party of the transation and not reeived by the seond transation party. The trading osts an be expliit or impliit. The major soures of transation osts usually onsidered in finanial investment are: DOI: 0.630/MIBE.07.8..06

On transation osts in stok trading 59 ommissions (and similar payments), bid-ask spreads and market impat [Elton et al. 999]. In this artile the analysis of the dependene of the transation ost on the bid-ask spread and the market impat is presented. Moreover, the formula for the average transation ost is applied to determine the strategy whih maximizes the expeted amount of money reeived for selling the stok shares. One of the reason of planning the sale of the shares of the stok may be the foreast of a derease in the future stok prie. In this artile, the model of the stok prie proess is proposed, with the possibility of the shorter duration of the negative drift in the stok prie proess, than the time for the investor s stok sale. The expliit method for determination of the strategy minimizing the expeted transation ost of selling X shares of the stok, is applied to the numerial omputation of the duration of the stok sale whih minimizes the ost of trading. ON SOURCES OF TRANSACTION COSTS Broker ommissions are expliit osts of trading. They are usually easy to evaluate (as perentage of the transation value) before the start of the trade and therefore they are not the soure of the finanial risk. In this artile, the ommissions and similar expliit transation osts paid by the investor are not taken into aount in alulating the average transation ost of purhase or sale of the shares of the stok. The bid-ask spread is defined as the differene between the stok's highest bid ask bid S and the lowest S ask pries of one share of the stok in the stok bid ask exhange. The average of S and S may be onsidered as the market prie of the stok. The half of the bid-ask spread is the ost of trading one share of the stok. Market impat (also alled prie impat) an be defined as a hange in the stok prie with respet to a referene prie, aused by the transation. This hange is disadvantageous to the initiator of this transation and therefore the market impat is a soure of the trading ost. In theory of finane some distinguish between temporary and permanent market impat. The temporary prie impat is onsidered as the ost of providing laking liquidity to exeute the trade in short time. It affets only a single trade and is assumed not to hange the market value of the traded stok. Suh impat is aused by supply and demand imbalane. Permanent prie impat is onsidered as the hange in the market value of the stok due to the transation, whih remains at least to the ompletion of this transation. In ase of the stok, a buy transation signals that the stok may be undervalued and a sell transation is a signal to the market that the stok may be overvalued. Therefore, the permanent market impat is pereived as a result of an adjustment of the market to the information ontent of the trade.

60 Marek Andrzej Koiński MARKET IMPACT FORMULAS AND TRADING COST In theoretial finane prie impat formulas an be used to build more realisti market models and explain the empirial phenomena whih seem to ontradit the market effiieny [Czekaj et. al. 00]. In reent years, there has been observed a trend toward the applying of eletroni trading algorithms based on a market impat model [Shied and Slynko 0]. The well aommodated to the real finanial market prie impat model ould help the market partiipants in pre-trade assessment of the performane of their trading strategies. In pratie of finanial management it is important to hek whether the oeffiients and even the funtional form of the prie impat formula reflet the reent stok market data. The transation of the stok purhase or sale, whih is the soure of market impat, may have a omplex struture: it an be fragmented and exeuted inrementally by the sequenes of single finanial orders. A popular formula for market impat, defined as the expeted average prie return between the beginning and the end of the stok transation is given as follows: X MI () V where is the stok s daily volatility, X is the volume of the exeuted transation, V denotes the average daily number of the traded shares of the stok, and are the numerial onstants that an be estimated from the sample of historial transations. The onstant is usually desribed to be of order unity and it seems that it just means that is approximately equal to. The onstant does not exeed, is typially found to be approximately / [Donier J., Bonart J. 04] and is usually singly estimated for an entire stok market. It is not obvious that does not vary stok by stok. The positive and negative sign of MI respetively orresponds to the stok purhase and selling transation. The variant of the equation () with equal to, whih means that market impat is linear in the traded volume, is used, for example, in [DeMiguel et al. 04]. The linear dependene of market impat on the transation volume an be partly justified in finanial market mirostruture theory by the Kyle model [Bouhaud 009]. The formula () measures, in fat, the differene in the pries of the stok transation. However, the equation () an be used to obtain the average ost of the transation. Consider the transation of purhase of X shares of the stok, whih starts at time 0 and ends at time T. Let S 0 denote the market prie of one share of the stok just before the transation and let TC denote the expeted transation ost,

On transation osts in stok trading 6 per unit of the stok, paid by the initiator of this purhase transation. The amount S X TC of money paid for the stok purhased is It an be alulated that: 0. s X TC () V In ase of a transation of stok selling the amount of money reeived by the stok seller is S0 X TC and the transation osts TC is also given by the equality (). In formula () the algorithm of the transation exeution, applied by a market partiipant, is not taken expliitly into aount. In pratie there are many types of stati and dynami trading strategies. However, for a given transation volume, the trading strategy seems to be roughly haraterized by the transation's duration whih measures how long the transation lasts. The duration of the transation is determined by the speed of exeution (trading rate) and the transation volume. The omitting in () of expliit dependene of market impat on the stok trade duration may mean that the influene of the stok trading speed on prie impat exists but is signifiantly smaller than the dependene of the prie impat on the number of traded shares of the stok. In [Almgren et al. 005] the volume time is defined as the fration of a stok average day s volume that has been traded up to lok time t. Under assumption that the stok's daily volume is independent of a trading day and the speed of trading of the stok's daily volume in the market is onstant during a trading day the volume time oinides with physial time. In [Almgren et al. 005] the two variables I and J are defined. I denotes the permanent market impat and J is the transation ost per unit of the stok. Let I and J respetively denote the average values of I and J. The formulas for I and J an be alulated as follows: and 4 I X V V (3) 3 5 I X J sgnx (4) VT where and are numerial onstants, X denotes the number of traded shares of the stok, V is the average daily volume of the stok, denotes the number of outstanding stok s shares, T is a trade duration and is the daily volatility of the stok.

6 Marek Andrzej Koiński If J is negative, it means that the it was alulated for the stok sale transation and then J is the ost of selling per unit of the stok. The values of and were determined in [Almgren et al. 005] by linear regression: 0,34 0,04 and 0,4 0, 006. The prie impat model desribed in [Almgren et al. 005], refers to the transations ompleted by uniform rate of trading over their volume time interval and takes into aount the trade duration, whih means the investor an apply this model in analysing the effet of the speed of his trading on the prie impat. In model presented in [Almgren et al. 005], the bid-ask spread ost is a part of the transation ost. THE STRATEGY OF SELLING STOCK WITH TRANSACTION COSTS Consider the following formula for the expeted transation ost implied by the bid-ask spread and the market impat, whih inludes the stok s transation duration and volume (as fration of the average daily number of the traded shares of the stok): a b (5) T where is the stok s daily volatility, denotes the volume of the investor s stok and T is the trade duration. Moreover, a, b and denote the numerial oeffiients and 0. Consider an investor who, at time 0, holds X units of the stok and expets that the trading ativity of other market partiipants in time interval 0, T will ause a linear, negative drift in the proess of the stok s market prie. He wants to sell his stok s shares up to time T where T T. The drift an be interpreted as the reation of the market on the information announed at time 0, onerning the finanial foreast of the ompany whih issued the stok. For example, the negative trend might be generated by the information that the stok dividend would be less than the market partiipants expeted. It an be assumed that the investor assesses the trend duration on the basis of his past experienes with the market reations on the announements onerning the finanial situation of this ompany or other ompanies whih issued stoks traded on the stok exhange. Sine the investor does not foreast a diretion of the stok prie movements, whih are independent of his trading, after time T therefore he assumes a drift equal to 0 in the market prie of the stok, in the interval T,T.

On transation osts in stok trading 63 Let S t denote the expeted prie of the stok at time t. For 0, T, S t is determined by the equality: S 0 if t T T S 0 if t T, T t S t Figure. The example of investor s views of the expeted market prie of the stok s share up to time T, due to the stok trading of other market partiipants (6) Soure: own preparation Under assumption of the onstant speed of selling the investor s stok in the interval 0,T, it an be alulated that the average transation ost aused by the drift is given by the formula: T if T T TC (7) T T if t T,T T The objetive of the investor is to maximize the amount of money reeived from the sale of his X shares of the stok. In the hoie of the duration T whih maximizes the amount of money obtained for selling the stok s volume X, the investor takes into aount that dereasing the speed of the exeution of selling the stok redues the trading ost of market impat but also, in time interval 0, T, inreases the ost aused by the drift. By (5) and (7) the expeted ost of the investor s transation is

64 Marek Andrzej Koiński TC T a b T for T T T T a b T for T T T T T For T T the derivative of (8) with respet to T equals 0 for b T and T minimum of (8) on the set 0, T is obtained for TC is onvex on the interval, b b if T, T b T if For 0, T minimum of (8) on the set T,T is obtained for 0 T T TC is onave on the interval T T TCT T TC T if TC T, T if TC T,T (8) 0 T. Thus, the (9). Therefore, the T T (0) In onsequene, by (9) and (0) it follows that Numerial example T TCT 0, T 0, T T, T T () TT, T if TCT 0, T TCT T, T if TC T Consider an investor who intends to sell 5% of the stok s average daily number of the traded shares of the stok. In this numerial example of the formula (5) appliation, the values of a is set to, b is set to 0.4 and is set to 0.6. For these values of the oeffiients a, b and, the dependene of the average transation ost on the transation volume in (5) is the same as the dependene of the expeted ost of trading on the number of stok s shares in the linear version of () and the dependene of the average transation ost on the trading rate in (5) is suh as the dependene of the average transation ost on the speed of trading in (4). Moreover, the values of T and T are set to 0.5 and, respetively.

On transation osts in stok trading 65 Table presents the values of the trade duration T as the funtion of the stok s prie drift and the average daily volatility of the stok prie. Table. μ T as funtion of and σ.50% 7.50%.50% 7.50%.50% 7.50% 3.50%.00% 0.6 0.449.000.000.000.000.000 4.00% 0.47 0.9 0.40.000.000.000.000 6.00% 0.4 0.6 0.3 0.384 0.449.000.000 8.00% 0.095 0.89 0.60 0.3 0.375 0.46.000 0.00% 0.083 0.64 0.6 0.79 0.37 0.370 0.4.00% 0.074 0.47 0.0 0.49 0.9 0.330 0.367 4.00% 0.067 0.33 0.83 0.6 0.65 0.300 0.333 6.00% 0.06 0.3 0.69 0.08 0.43 0.76 0.306 8.00% 0.057 0.4 0.57 0.93 0.6 0.56 0.85 0.00% 0.054 0.07 0.47 0.8 0. 0.40 0.66 Soure: own omputation Table show dependene of the value of the expeted transation ost on the drift of the stok and the stok s prie. Table. The value of average transation ost depending on the on and μ σ.50% 7.50%.50% 7.50%.50% 7.50% 3.50%.00%.3% 3.07% 4.65% 6.% 7.77% 9.3% 0.88% 4.00%.4% 3.43% 5.6% 6.96% 8.5% 0.07%.63% 6.00%.54% 3.68% 5.6% 7.45% 9.% 0.8%.38% 8.00%.64% 3.89% 5.90% 7.80% 9.63%.4% 3.3% 0.00%.73% 4.07% 6.4% 8.0% 9.98%.8% 3.60%.00%.8% 4.% 6.35% 8.36% 0.9%.6% 3.99% 4.00%.88% 4.36% 6.55% 8.60% 0.57%.48% 4.34% 6.00%.94% 4.49% 6.7% 8.8% 0.8%.76% 4.66% 8.00%.00% 4.6% 6.88% 9.0%.05% 3.03% 4.96% 0.00%.06% 4.7% 7.04% 9.0%.7% 3.8% 5.3% Soure: own omputation Figure shows graphially how the expeted transation ost depends on the stok s prie drift and the average daily volatility of the stok.

66 Marek Andrzej Koiński Figure. The example of the average ost of selling the stok as the funtion of and Soure: own preparation CONCLUSION In this artile the problem of determining the trading rate whih minimizes the average ost of selling the investor s stok is expliitly solved in a framework of the model,whih an be applied in the ase when the negative drift in the stok prie lasts less than the investor s stok selling. The numerial example inluded in the paper shows that the solution of the problem of the expeted transation osts minimization in ase of selling the stok, may be signifiantly affeted by the volatility of the stok and the drift in the stok prie proess. REFERENCES Almgren R., Thum C., Hauptmann E., Li H. (005) Diret Estimation of Equity Market Impat Risk. 8, 58 6. Bouhaud J. P. (009) Prie Impat, https://arxiv.org/pdf/0903.48.pdf. Czekaj J., Woś M., Żarnowski, J. (00) Efektywność giełdowego rynku akji w Polse. Z perspektywy dziesięioleia, Warszawa: Wydawnitwo Naukowe PWN. DeMiguel V., Martín-Utrera A., Nogales F. J. (04) Parameter unertainty in multiperiod portfolio optimization with transation osts. http://faulty.london.edu/avmiguel/dmn- 04-0-09.pdf. Donier J., Bonart J. (04) A million metaorder analysis of market impat on the Bitoin. https://arxiv.org/pdf/4.4503.pdf.

On transation osts in stok trading 67 Elton R. J., Gruber M. J., Brown S. J., Goetzmann W. N. (00) Modern Portfolio Theory and Investment Analysis, John Wiley & Sons, Hoboken. Koiński M. (05) Trade Duration and Market Impat. Quantitative Methods in Eonomis XVI/, 37 46. Shied A., Slynko A. (0) [in:] Blath J., Imkeller P., Roelly S. (eds.) Surveys in Stohasti Proesses, European Mathematial Soiety Publishing House, Zürih. Tóth B., Lempérière Y., Deremble C., de Lataillade J., Kokelkoren J., Bouhaud J.-P. (0) Anomalous Prie Impat and the Critial Nature of Liquidity in Finanial Markets. http://journals.aps.org/prx/pdf/0.03/physrevx..0006.