A Certainty Equivalent Based Sequential Three-Way Decision for Securities Investment

CAO Jiashuo,ZHOU Xianzhong,HUANG Bing,JIA Xiuyi,LI Huaxiong*

(1.School of Management and Engineering,Nanjing University,Nanjing 210093,China;2.School of Information Engineering,Nanjing Audit University,Nanjing 211815,China;3.School of Computer Science and Engineering,Nanjing University of Science and Technology,Nanjing 210094,China)

Abstract:In recent decades,the research on securities investment such as portfolio received attention,which is based on the current determination to invest,and research on uncertain investment decisions using artificial intelligence with uncertainty methods are be-ing studied more and more widely.The returns and risks of financial investment are full of uncertainty,we need to make decisions to judge whether and how investors should make investment decision.In three-way decision,the attitude of decision makers about risk is an important decision consideration.Based on the sequential three-way decision model,the problem of securities investment was studied by simulating the risk attitude of investors through certainty equivalent.A Certainty Equivalent based Sequential Three-Way Decision(CES3WD)model is introduced.The mean and variance of returns are obtained from historical data and used as the expected return and risk of investment.The expected total value is transformed into the risk attitude of decision makers by the expo-nential utility function,and the expected investment return of the decision makers are measured by the certainty equivalent method.Moreover,based on the principle of maximizing expected investment return,the decision rule is constructed.On this basis,the dy-namic decision rules are optimized according to the actual investment strategy,and the derivation algorithms of sequential three-way decision rules are constructed.Finally,experimental analysis validates the proposed methods.

Key words:sequential three-way decision;expected utility theory;Certainty Equivalent.

1 Introduction

As one of intensively investigated topics of arti-ficial intelligence,dealing with the complex and un-certain natural world phenomena has gained signifi-cant monument[1-3].Risk analysis is a widely con-cerned research field of uncertain decision-making at present.The returns and risks of financial invest-ment are full of uncertainty,it is required to use ar-tificial intelligence to control and make decisions,and use artificial intelligence to judge whether and how investors should invest.Financial data mining by artificial intelligence,calculating the expected re-turn rate and risks of investment,and then provid-ing investors with investment suggestions,has be-come a hot topic in current research.

Three-Way Decision(3WD)theory was intro-duced by Yao[4],which is a relatively new decision analysis theory and can be used to deal with incom-plete and inaccurate information.In the rough set theory model,Yao introduced the Bayesian risk deci-sion method and divided the partitioned object set into positive region,boundary region and negative region according to the lowest risk cost[5].On this basis,by obtaining more effective information,the delay region is further divided into three branches,thus forming a Sequential Three-Way Decision(S3WD)model,which is suitable for dealing with dynamic,complex and uncertain problems[6-11].The three-way decision is actually three-region decision.The positive region,boundary region and negative region respectively correspond to the decision of in-vestors to buy a certain security,the decision of in-vestors to maintain the status quo and the decision of investors to sell a certain security.

The determination of the boundary region of se-quential three-way decision namely the acquisition of threshold is a problem that is being attention.For the problem of investment,using expected utili-ty theory to describe the investment feeling of inves-tors is a common method.For investment decision problems to construct three-way decision by replac-ing the loss function with the satisfaction of expect-ed return,namely expected utility,of investors is suitable.In decision theory,the actual value of the decision result to the decision makers,that is,the order of preference of decision makers for the re-sult,is described by utility.Expected utility theory is a framework established by Von Neumann and Morgenstern on the basis of axiomatic hypothesis and using logical and mathematical tools to analyze the choice of rational actor under uncertain condi-tions[12].However,expected utility theory only con-siders the profitability of the scheme and evaluates the pros and cons of each scheme only from the per-spective of income,and it ignores the key role that the decision-maker plays in the decision result,which means that it does not reflect the attitude of decision makers towards risk.The certainty equiva-lent method makes up for the deficiency of the ba-sic risk assessment criteria.In certainty equivalent method,two factors of benefit and risk are consid-ered comprehensively,and the utility function is used to replace the return function[13-17].The goal of the decision-making strategy is to maximize the utility.In the field of decision making,many other expectations models were developed,among which the most representative ones were prospect theory[18],cumulative prospect theory[19],regret theory[20],weighted expected utility theory[21]and nonlinear ex-pected utility theory[22].

The research on the three-way decision mainly focuses on theoretical expansion[23-25],model im-provement[26-29]and practical application[30].The ba-sic idea of the three-way decision has been applied in different fields,such as shadow set[31],informa-tion system[32],cost sensitive learning[33],cognitive concept learning[34],recommendation system[35],medi-cal diagnosis[36]and cluster integration[37].In addi-tion,abundant current researches on securities in-vestment are based on the mean-variance model pro-posed by Markowitz.On the basis of mean-vari-ance model,many researches focus on how to im-prove the existing portfolio model and how to im-prove the current effective algorithm[38-40].In securi-ties investment,the three decisions of buying,wait-and-see and selling securities correspond to the three-way decision at a distance.However,there is almost no research on the application of three-way decision to securities investment.

Based on the directional consistency of three-way decision and investment decision,we firstly propose a Certainty Equivalent based Three-Way Decision(CE3WD)model by introducing the cer-tainty equivalent method from expected utility theo-ry,which can be applied to the securities investment problem.However,securities investment is a target-ed problem,and investment decision-making pro-cess is a dynamic decision-making process.There-fore,we redesigned the decision process for cyclical securities,and then established a novel Certainty Equivalent based Sequential Three-Way Decision(CES3WD)model,hoping to help a few investors with more professional knowledge reserves to simu-late the securities investment in the future to buy,sell or temporarily retain.

2 Preliminary

2.1 Sequential Three-Way Decision

The three-way decision model is described by two state sets and three action sets.State set Ω={X，XC}represents whether an object x belongs to X,which can be seen in this paper as whether the security has a positive return rate or a high risk.Action set A={aP，aB，aN}represents accepting ac-tion,delaying decision and rejecting action for an object,respectively,namely an object x is classified into positive region POS(X),boundary region BND(X)and negative region NEG(X)respectively.In this paper,it can be assumed that positive region POS(X)means investors buy securities,boundary re-gion BND(X)means investors keep existing assets unchanged,and negative region NEG(X)means in-vestors sell existing securities.In addition,the con-ditional probabilities of an object x belonging to X and not belonging to X are denoted by Pr(X|[x])and Pr(XC|[x]),respectively.Pr(X|[x])=|[x]∩X/[x]| and Pr(X|[x])+Pr(XC|[x])=1 hold.[x]is an equivalent class.

Similarly,we denote that λPP,λBP,λNPrespec-tively represents the loss function of object x taking three actions of aP,aBand aNwhen it belongs to X.λPN,λBN,λNNrespectively represents the loss function under three actions of aP,aBand aNwhen objectx does not belong to X.Therefore,the expected loss of x when the three actions aP,aBand aNare respec-tively taken is shown as follows:

According to the Bayesian decision criterion,the action set with the least expected loss should be selected in the specific decision process.Also a rea-sonable hypothesis can be obtained,namely0≤λPP≤λBP≤λNPand 0≤λNN≤λBN≤λPNhold.Therefore,a three-way decision rule(P1),(B1)and(N1)can be obtained by recalculating the above three rules in combination with this hypothesis.

(P1)If Pr(X|[x])≥ α1，decide x∈ POS(X)；

(B1)If β1

(N1)If Pr(X|[x])≤β1，decide x∈NEG(X)，

where

The three-way decision can be viewed as the middle part of the sequential three-way decision.Sequential three-way decision is a kind of dynamic three-way decision which progresses from coarse-grained to fine-grained.At each granularity level,delayed decision is adopted when there is not enough information available to support a definitive decision,and three-way decision is made at the next granularity level after more information is added.The proposed sequential three-way decision can fur-ther extend the application of three-way decision in real life.

Definition 1 Si=(Ui,ATi,Vi,fi)is an information system,where i=1,2,…,n,ATi=Ci∪ Di,Cirepresents the condition attribute set,and Direpresents the decision attribute set.Given a nested attribute set sequence C1⊂C2⊂…⊂Cn⊆C,an evoked equivalent relation sequence is En⊂ En-1⊂…⊂ E1.Eirepresents the equivalent relation arising from Ci,where1≤i≤n and Ei={(x,y)∈U ×U|∀a∈Ci,f(x,a)=f(y,a)}.The multi-grain structure constructed by nested attribute sets is denoted as GS,and GSirepresents the i-th granularity layer of GS.GS and GSiare respectively expressed as follows:

At the i-th granularity layer,Uirepresents the object to be processed,and ATirepresents a non-empty finite set of attributes.Virepresents the range of the attribute ATi,and fi:Ui×ATi→Virepre-sents an information function.A pair of threshold values(αi,βi)are given to satisfy αi>βi.Then the positive region,boundary region and negative re-gion on the i-th granularity layer are expressed as follows:

(P2)If Pr(Xi|[x]Ci)≥αi,decide x∈POS(Xi);

(B2)If βi

(N2)If Pr(Xi|[x]Ci)≤βi,decide x∈NEG(Xi).

At the i-th granularity layer,Xistands for the goal concept,Xi⊆ Ui.And POS(Xi)∪ BND(Xi)∪NEG(Xi)=Ui.From the first granularity layer GS1to the n-th granularity layer GSn,sequential three-way decision can be constructed.Sequential three-way decision mainly solves the dynamic decision problem under the granularity change,which is a multi-step decision.In sequential three-way deci-sion,when the available information is insufficient,the delay decision can be adopted to reduce the loss caused by the wrong decision.

2.2 Certainty Equivalent of Expected Utility Theory

The unique interests,feelings,and trade-off re-sponses to expected returns of decision makers are called utility.In decision theory,the actual value of an outcome to a decision maker,that is,preference ordering of the outcome,is described by utility.Utility is the quantification of preference.Prefer-ence ordering is as follows:

(1)x≥y:Weak preference for x,that is,xis at least as good as y;

(2)x≻y:Strong preference for x,where x≻y⇔x≥y is true but y≥xis not;

(3)x～y:There is no difference in preference for x and y,that is,x～y⇔x≥y and y≥x are true.

In the discussion of general economic equilibri-um without uncertainty,behaviour of economic ac-tors is to maximize the expectation of the utility function to make decisions.When considering the economic equilibrium discussion with uncertainty,it is assumed that the quantity of goods are random variables,and their values will depend on the uncer-tain state.When the utility function encounters un-certainty,it is difficult to determine its value,which leads to the concept of expected utility:the utility of investment is a random variableu,which may take two values(u1and u2),namely

The probability of u=uiare π1and π2,then the expected value of u is Eu=π1u1+π2u2,the expected utility of the investment.For the expected utility function,its mathematical properties can reflect the behavior of people.Since the expected utility is not certain to be realized,the real utility will be higher or lower than it,there is uncertainty,not reaching the expected utility,resulting in losses,and forming risks.Attitudes towards risk can be divided into three types:aversion,neutral and preference.These three attitudes are described in mathematical terms as follows:

(1)If the utility function is a concave function,as shown in Fig.1(a),u(Eω)>Eu(ω)means that the expected utility is better than the expected utili-ty,corresponding to the risk averse person;

Fig.1 Three types of utility function：（a）Risk averse.（b）Risk neutral.（c）Risk preference

(2)If the utility function satisfies u(Eω)=Eu(ω),as shown in Fig.1(b),that is,the expected utility is equal to the expected utility,then it corre-sponds to the risk neutral;

(3)If the utility function is convex,as shown in Fig.1(c),u(Eω)

Certainty equivalent refers to the payment de-sire of the economic activity subject for a certain in-vestment activity,that is,the mathematical expected value corresponds to the expected utility of an in-vestment activity.Risk premium is the return on an investment that a risk-averse person is willing to give up to avoid taking risks[41-43].

where ε is the random return of the investment and Wis the initial wealth.ρ is the return that investors are willing to give up in order to avoid risks.The greater the value,the higher the degree of risk aver-sion of investors.(W-ρ)refers to certainty equiva-lent.

If we take the Taylor series expansion of both sides of this equation in W,we get

Therefore,the risk premium ρ is

where the right side of the equal sign is composed of two parts:Var(ε)is the variance of the invest-ment random return,which means the uncertain risk.And the others are the factors reflecting the preference of investors.

If we strip out factor ε and leave only subjec-tive factors of investors behind,we can derive a more general measure of risk aversion RA(W)than risk premium,that is absolute risk aversion(ARA):

RA(W)can determine attitude of investors to-wards risk:

(1)If RA(W)>0,decide to risk aversion;

(2)If RA(W)=0,decide to risk neutral;

(3)If RA(W)<0,decide to risk preference.

This judgment is consistent with the concavity and convexity judgment of utility function.

The relative risk premium is

The relative risk aversion coefficient(RRA)is

Classical asset pricing theory introduces many kinds of utility functions,among which constant ab-solute risk aversion(CARA)and constant relative risk aversion(CRRA)can show some kind of risk aversions,which are widely used in the field of in-vestment decision making.Since CRRA utility func-tion is discussed in a more complex environment than CARA utility function in empirical test,CARA utility function is more commonly used and conve-nient in academic research in practice.Commonly used CARA utility function is negative exponential utility function:

2.3 Certainty equivalent based three-way deci-sion model

Similar to the classical three-way decision mod-el[4]and the three-way decision model based on utili-ty theory[26],we assume that:

The certainty equivalent of the utility function can be obtained by the following formula:

where CE represents Certainty Equivalent.Then,the certainty equivalent cij(i=P,B,N;j=P,N)in dif-ferent states is calculated.

According to formula(1),the certainty equiva-lent C(ai|[x])(i=P,B,N)corresponding to taking different actions aP,aBand aNcan be expressed as follows:

Since the decision maker may choose the op-tion that maximizes CE,the three-way decision model based on CE proposes the following optimi-zation problems to find the best action:

Therefore,the decision rule can be expressed as follows:

(P3)If Pr(X|[x])≥α2,decide x∈POS(X);

(B3)If β2

(N3)If Pr(X|[x])≤β2,decide x∈NEG(X),

where

3 Three-way decision for securities invest-ment

3.1 Certainty equivalent based three-way deci-sion model for securities investment

The returns of most securities are difficult to predict.For example,when investing in a stock,un-expected events,market conditions and other factors have different degrees of impact on the changes in stock prices.As a result,the future price,return and risk of a security are uncertain.In actual invest-ment activities,the average historical real rate of re-turn is usually used to replace the expected rate of return to measure the quality of investment portfo-lio[38].Assuming that there are M securities in the securities pool,the current price of security i is Pi,the actual rate of return of security i is calculated as follows:

where riis the actual return rate of security i,Pi(t)is the price of security i at the period t and the divi-dend during the holding period,and Pi(0)is the price of the i-th security at the beginning.

In investment activities,it is uncertain to use historical rate of return to estimate the expected rate of return,since the real return may be higher or lower than expected,which is a risk faced by many investors.At present,there is no consensus on the measurement of risk.Markowitz believes that risk can be quantified by the variance of expected return rate.For the i-th security,the following formula can be used to calculate its risk:

The above formula quantifies risks by using the volatility of securities returns,that is,the vari-ance of the expected return rate of a security.The greater the variance,the greater the deviation be-tween the actual return rate and the expected return rate,indicating that the return of the security has greater uncertainty and strong investment risk.

Similarly,in A={aP,aB,aN},aNindicates that the investor intends to sell the i-th security with a total value of b,then we proposed the expected total val-ue of the i-th security:

In A={aP,aB,aN},aBindicates that the investor intends to keep the i-th security unchanged,that is,neither buying nor selling,then we proposed the ex-pected total value of the i-th security:

Fig.2 Distribution of security return rate

Assuming that the risk attitude is CARA,the risk attitude of investors will be expressed in the form of exponential utility function[44],which can be expressed as:

where c is the risk aversion coefficient of investors,which is generally taken as 0.005.In addition,in order to eliminate the problems caused by too large or too small values,c Wishould be within 10 to 1 000.

The expected investment income of investors is measured by CE,and its calculation formula is as follows:

Definition 3 Denote A and B as two non-emp-ty sets.The following will compare the size of A and B.

Obviously,we need to select the action set with the greatest certainty equivalence.Thus,3WD rules(P4),(B4)and(N4)can be obtained:

decide to sell the securitie.

According to formula(26)-(27)and the deci-sion rule(P4)can be further written as:

Similarly,according to decision rules(B4)and(N4),the new 3WD rules(P5),(B5)and(N5)can be further written as:

According to formula(26)-(27),the decision rule(P4)also can be further written as:

Similarly,another decision rules(P6),(B6)and(N6)can be further written as:

3.2 Certainty equivalent based sequential three way decision model for securities investment

Although our proposed CE3WD model can be applied to securities investment problems.However,investment decision is a dynamic decision process,which cannot be handled by this model.In addition,for securities with different attributes,the CE3WD model cannot maximize the return on investment deci-sion of each security.Therefore,we decided to further improve the CE3WD model to make it a more specif-ic sequential three-way decision model for the invest-ment problem of cyclical securities.

Cyclical stocks are the most abundant stock type,which rise and fall with the ups and downs of the eco-nomic cycle,and are the most favored by short and medium term investors.The return of short and medi-um term investors is mainly through the price fluctua-tion law to obtain profits.They need to choose the volatile price,stable fluctuation range and stable busi-ness performance of the stock as the investment tar-get.The historical data of the stock can then be used to determine the selling and buying sub-points.

The selling sub-point should be selected as the time node when the stock price appears under a given probability and makes it reach the maximum price with the guarantee less than the high price of each his-torical wave peak.The price satisfies

where P is the current price,Psis the historically high price(HHP),and Pmis the highest historical price.

Similarly,the buying sub-point should be select-ed as the time node when the stock price appears un-der a given probability and makes it reach the mini-mum price with the guarantee greater than the low price of each historical wave peak.The price satisfies

where P is the current price,Pbis the historically low price(HLP),and Plis the lowest historical price.

Assume p=90%,q=10%.The former means that there is 90% probability that the current price P is less than the historical high price Ps,that is P

Similarly,we can set Pˉas the historical median price(HMP),that is,there is a 50% probability that current price P is greater than or less than Pˉ,respec-tively.According to formula(18),we proposed a new return calculation,which takes HMP as the bench-mark,as follows:

Assuming that all of the sale proceeds are used to buy the securities,the investors will sell all of the securities they holds at the high price and use all of the proceeds to buy the securities at the low price.Ob-viously,investors can not do any operations when prices are neither too high nor too low.We can call the duration of a low-price buying and high-price sell-ing as an operation round,which is divided into two decision rounds:a buying decision round and a sell-ing decision round.Each decision round is further di-vided into countless operational decision nodes,and each operational decision node will make an opera-tional decision.At each operational decision node,from the opening of the market to the closing of the market,decisions will be made continuously accord-ing to the changes in the stock price,which is a dy-namic decision-making process.In a buying decision round,the selling decision region is kept as small as possible to ensure that a buying decision can be made.The same is true for the selling decision round.Obvi-ously,both of these rounds are dynamic and can be viewed as sequential three-way decision.

According to formula(37),we know that t=0 means the period that the first data of the historical se-curity price data.However,we need to change this to mean the period at which the decision round begins,indicating the first operational decision node of the buying or selling decision round.When we enter an operation round,we start with a buying decision round and then a selling decision round.We let t=1 indicate the first operational decision node of the buy-ing decision round,rather than the first data of the his-torical data.If we can not make a buying decision in the first operation decision node of the buying deci-sion round,we will enter the second operation deci-sion node of the buying decision round,in which case we will sett=2.Therefore,we set up a database to store the dynamic prices of securities in each opera-tional decision node until the buying decision round ends and the buying decision is made.When we enter the first operation decision node of a selling decision round,reset t=1.

where if c gets smaller,β5will get larger,and the neg-ative region will get smaller.Prices need to be higher than before to make a selling decision.Decision rule(P5)-(N5)becomes:

where if c gets larger,α6will get smaller,and the posi-tive region will get smaller.Prices need to be lower than before to make a buying decision.Decision rule(P5)-(N5)also becomes:

When an investor wants to sell a security,he will choose to sell it at a high price.Therefore,if the cur-rent security price Siis lower than the lowest histori-cal price Pl,this price data should not be included in the dynamic database.If not,when entering the later operational decision node,the current security price data that is too low will have a bad impact on the en-tire historical data,which means that the decision re-sult will make the selling decision when the price is not high enough.Similarly,when an investor wants to buy a security,excessive price should not be included in the dynamic database.

From the above analysis,the detail of the selling decision round is shown by Algorithm 1(A1),the de-tail of the buying decision round is shown by Algo-rithm 2(A2),and the main algorithm,that is,the de-tail of the sequential three-way decision is shown by Algorithm 3(A3).And the historical data of M Securi-ties should include the opening price,low price,high price and closing price for each day.

4 Experiment and Analysis

Typical cyclical industries in China include basic bulk raw materials such as steel and non-ferrous met-als,construction materials such as cement,capital-in-tensive sectors such as construction machinery,ma-chine tools,heavy trucks and equipment manufactur-ing.Cyclical stocks are the largest class of stocks,which rise and fall with the ups and downs of the eco-nomic cycle.When the overall economy rises,so do the prices of such stocks;When the overall economy goes down,so do the prices of such stocks.

Collecting relevant historical data is crucial for making decisions on cyclical stock investments.His-torical data can reflect the cycles and amplitudes of stocks.Therefore,historical data and investment entry points need to be carefully selected before making in-vestment decisions.Historical data should not be too long or too short,and should reflect the cycles and amplitudes of recent cyclical stocks.The investment entry point should choose to start the operation when the current price is in the middle of the cycle fluctua-tion,and the current price should not be too high or too low.

We collect the data of four typical cyclical stock industries in China with similar periods of ferrous metals,non-ferrous metals,metal products and non-metal products from 2015 to 2021.Among them,there are 15 stocks of ferrous metals,17 stocks of non-ferrous metals,13 stocks of metal products,22 stocks of non-metal products.We collected the opening,low,high and closing prices of these stocks for each day in their history.This paper takes the former part of stock data as historical data and the latter part as future stock price data to simulate stock investment deci-sions.To facilitate batch processing,we divide stock data into four categories based on historical data days:100 days(17 stocks),140 days(15 stocks),170 days(15 stocks)and 250 days(20 stocks).

Take 100 days as an example,that is,the first 100 days of the stock data we collected are divided in-to historical data,the others are divided into future da-ta.And investment decisions will be simulated from the 100-th day.When dealing with historical data,since we cannot get the price of a stock at each point in time,we treat the opening,low,high and closing price as a range of historical prices.We start the in-vestment decision on 100-th day,that is when t=1,we simulate the price fluctuation of the stock over the course of the day by assuming that the price change for the day is from the opening price to the lowest price to the highest price to the closing price.Assum-ing that the price at that time of the day happens to be the highest price,the price is incorporated into the al-gorithm described in the previous section to make in-vestment decisions.After the decision is made,time continues to flow until the investment decision is made again when the price of the day happens to be the lowest.When the time flow comes to the end of the 100-th day,that is,the current price is the closing price,and if the results of the current decision and the previous three decision results are all buying decision or selling decision,the buying or selling operation will be carried out at the closing price,then the deci-sion-making process will be ended.Then,time enters 101-th day to restart the new investment decision,at which point we sett=1again.Otherwise,the invest-ment decision is continued at 101-th day,at which time we sett=2.

In addition to using the total rate of return(TRR)to reflect the result of the investment decision,this pa-per also introduces a new measure factor called rate of return of round(RRR),average total rate of return(ATRR)and average rate of return of round(ARRR):

As above described,an operation round is divid-ed into two decision rounds:a buying decision round and a selling decision round.Thus,the rate of return from buying at low price to selling at high price is called a rate of return of round.TRR gives the total re-turn over multiple operation rounds,whereas RRR gives the total return over a single operation round.It reflects the ability to buy at a lower price and sell at a higher price during investment decisions.

In addition to making investment decisions based on historical prices,short and medium term in-vestors will also make use of β coefficient to make in-vestment based on market fluctuations.Coefficient β comes from the Capital Asset Pricing Model(CAPM)[45],which is the cornerstone of modern finance.β re-flects the risk coefficient of the security to the market,presented as follows:

As above mentioned,there are few studies based three-way decision for securities investment at pres-ent.In order to facilitate the comparison,we need to transform the model that only focuses on portfolio in-to an investment decision model that solves the prob-lem of whether to invest or not.Therefore,the strate-gy of investing with HHP and HLP and investing with β coefficient can all be regarded as the 3WD models,which are 3WD and β3WD,respectively.If the cur-rent price is higher than HHP or the current β is less than most of the historical β,decide to sell;When the current price is lower than HLP or the current β is much higher than the historical β,decide to buy;When in between,decide to neither buy nor sell.

According to formula(24),we set aside 100 000 yuan for each stock as an investment starter.Then we recorded the return made at the end of each operation round and the total return made at the end of all rounds.Table 1 shows the ATRR values of the above four models.As you can see,the ATRR values of β 3WD and CE3WD are much lower than those of the other two models because they are not designed for cyclical securities.Therefore,in order to facilitate comparison,we will remove these two models and on-ly focus on comparing 3WD and CES3WD.

Table 1 ATRR of four models.

Table 2 Average numbers of operating rounds.

Fig.3 shows the calculated rate of return on in-vestment decisions.It can be seen from Fig.3(a)that the ARRR obtained by CES3WD model proposed in this paper is much higher than that obtained by the normal 3WD model.This means that our model can make a buying decision when the price is lower than the historically low price and a selling decision when the price is higher than the historically high price.From Fig.3(b),we can see that the CES3WD model can still achieve better results in ATRR.Although the number of operation rounds may not be as large as the 3WD model,the ATRR is still higher than the 3WD model since the model in this paper can achieve a higher ARRR.Then,Fig.4 shows the RRR of each stock in each pool.Obviously,the RRR of CES3WD model is greater than that of 3WD model.

Fig.3 Return on investment decisions.ARRR and ATRR obtained by selecting the days of different historical data and dividing it into four types of securities data to simulate decision-making.

Fig.4 RRR histogram.RRR of CES3WD and 3WD by using different historical data

Table 2 shows the average number of operation rounds obtained from the investment decisions of the four types of securities.Obviously,the number of op-eration rounds of the model decision in this paper is less than the number of operation rounds of the nor-mal model decision.Therefore,through the analysis of the above figures and tables,it can be concluded that the model in this paper achieves a higher total rate of return with fewer operation rounds by increas-ing the rate of return of each operation round.

5 Conclusion

How to describe the risk attitude of decision-makers is a focus topic in the three-way decision theo-ries.This paper is based on a specific problem,that is,investment decisions of decision makers on securities.We introduce the certainty equivalent method of utili-ty theory into three-way decision and construct a nov-el CES3WD model.However,expected return and risk are only regarded as interval numbers in this mod-el.In fact,return and risk can be described by intro-ducing normal fuzzy variables.Therefore,our future work will consider the improvement of the model and the construction of portfolio decision models for vari-ous securities to do a more in-depth study.