Construction of Optimal Portfolio Using Efficient Portfolio and Zero Opportunity Cost

In this manuscript, we introduce asset allocation and portfolio selection techniques based on efficiency condition, Sharpe ratio error condition and order three zero opportunity condition. Investors expect the same level of risk and return from alternative investment options unless they want the advantage of diversification for risk. There are two fundamental investment portfolios. The first one is risk free fundamental portfolio, and the second one is risky fundamental portfolio. Investors use zero opportunity cost to select portfolio objective. In this research, mathematical derivation of portfolio construction approach is described in advance. Historical data of this research show that there is a positive linear relationship between natural logarithm of standard deviation of securities’ return and square of co-variance between securities and market return. Furthermore, it shows that there is a positive linear relationship between Treynor ratio of securities’ return and Sharpe ratio of securities’ return. We use global search optimization tool in MatLab and R software to solve empirical portfolio selection and asset allocation problem. Moreover, we apply direct and indirect mathematical proof methods to prove mathematical facts of this study.


Introduction
Investors optimize investment asset allocation to reduce unnecessary losses based on their investment objectives and constraints. Traditionally, investors use 40-60 asset allocation strategies. This 40-60 classical asset allocation strategy need to be optimal. Thus, investors should set their investment objectives and constraints to optimize asset allocation. Practitioners use mean variance model to select portfolio even if the model assumes weights of securities are independent of time parameter. Another problem of mean variance model is that the model does not consider zero opportunity cost between fundamental portfolios. Single index model is developed because mean variance model needs estimations of expected return, risk and covariance of securities. This indicates that mean variance model is not easy for portfolio with large number of securities. In this research, we propose new portfolio construction method using order three zero opportunity condition, efficiency condition and Sharpe ratio error condition. Clearly, one can express variance of portfolio in terms of expected return by using efficiency condition. Similarly, the variance of portfolio can be expressed in terms of expected return due to Sharpe ratio error condition.
Let w * f , w * p and w * m be weight of risk free security, weight of risky portfolio and weight of market portfolio, respectively. Suppose that r f , r tp and r tm represents risk free rate of return, risky portfolio rate of return and market portfolio rate of return at time t, respectively. Then we define rate of return of complete portfolio r tcp by r tcp = w * f r f + w * p r tp + w * m r tm . In this research, we determine r tp , w * f , w * p and w * m using order three zero opportunity condition, efficiency condition and Sharpe ratio error condition. Moreover, we use weight-variance regres-

Motivation
Since investors have different objectives, their portfolio selection need not be the same using mean variance model. So which investor is right? This question motivates researchers to study portfolio selection and asset allocation techniques using zero opportunity cost.

Statement of the problem
The statement of the problem is constructing optimal complete portfolio using order three zero opportunity condition, efficiency condition and Sharpe ratio error condition. Mathematically, we obtain the optimal risk free security weight w * f , market portfolio weight w * m and risky portfolio weight w * p to construct complete portfolio P c such that r tcp = w * f r f + w * p r tp + w * m r tm using order three zero opportunity condition, efficiency condition and Sharpe ratio error condition.

Objectives of the study
The general objective of this study is to construct complete portfolio using order three zero opportunity condition, efficiency condition and Sharpe ratio error condition. Specific objectives are: 1. to show that there is positive linear relation ship between Treynor ratio and Sharpe ratio of securities, 2. to estimate parameters of weight-variance regression model. 3. to build General order three opportunity model (GOOM), 4. to construct Order three asset allocation model (OAAM).

Research gap
Since efficient frontier contains many efficient portfolios, investors should choose efficient portfolio using logical constraints. Mean variance model does not explain how to choose efficient portfolio using logical constraints. In this research, we use (GOOM) and (OAAM) to construct optimal complete portfolio.

Literature review
Investment is an allocation of resources for medium or long term and the expected effect is to recover the investment costs and have a high profit, [1]. Risk is the possibility that desired objective would not be attained, [2]. One cannot definitely determine whether the risk is only good or bad, [3].

Financial risk preference and tolerance
Risk preference is the individual's orientation towards risktaking, [4]. Financial risk tolerance is the level of risk that clients believe they are willing to accept, [5]. Financial risk tolerance is the maximum amount of volatility that investor is willing to accept when making a financial decision, [5]. Risk aversion means an attitude of reluctance to take risky decisions, [6]. Financial Risk Tolerance is the willingness of individual investors to receive negative changes of investment value or the result opposite or different from the expected results, [7]. Willingness to take financial risk depends on portfolio structure, gender, age, educational attainment, income, financial stability, financial literacy, marital status, and family size, [8].

Markowitz
Note that return maximization and risk minimization objectives are conflicting objectives. If investors attempt to increase return, they will be penalized by risk. The solution of M M depends on investors' risk preference. Consider the following special cases of M M .

Risk neutral investors
Risk neutral investors are investors who are interested to maximize return without paying attention on risk, [9]. These investors do not care about risk minimization. Risk neutral Markowitz's model (RIN M M ) is given as follows: (2)

Return neural investors
Some investors prefer risk minimization rather than return maximization. These investors are called return neutral investors. Return neutral Markowitz's model is given as follows: (3)

Maximum expected return for a given level of risk
Investors who know their risk tolerance potential would like to maximize return for their risk level. These investors would like to solve the following optimization problem: Readers are referred to [9] for further detail about maximum expected return for a given level of risk.

Minimum risk for a given level of expected return
Those investors who are interested to achieve targeted expected return by minimizing risk use the following model: The detail of this model is given in [9].

Risk adjusted model (RAM)
Investors may have different weight levels for risk and return. The investors who think both risk and return are equally important use 50-50 weights for risk and return. In general, risk adjusted portfolio model has the following form as given in [9,10]: where ρ is a parameter.

Sharpe ratio maximization model (SRM M )
Portfolio comparison is relative to some reference. For instance, managers may compare two portfolios relative to Sharpe ratio. This does not mean portfolio with high Sharpe ratio is absolutely better than portfolios with low Sharpe ratio. However, portfolio with high Sharpe ratio is more desirable than the one with low Sharpe ratio provided an investor cares about Sharpe ratio rather than other criteria. See reference [11,12] for more details. Sharpe ratio maximization model is given by: where r f risk free rate of return.

Diversification maximization model (DM M )
Diversification is a key technique to minimize investment risk. Investors who think diversification as a good tool to minimize risk maximize diversification ratio. Diversification ratio maximization model is given in, [13]. The following model is called diversification maximization model.
The above investment portfolio construction models presented above are well known fundamental portfolio selection models. However, there are some issues which also affect portfolio selection decision. For instance, managers should consider opportunity cost, variability (instability) of security weight over time, inefficiency, return errors and market factors when they construct portfolio. In this research, portfolio errors such as return error, frontier error, opportunity error, Sharpe error, Treynor error, Sharpe-Treynor error and weight error are considered.

Global minimum variance and expected return
Suppose that investors want to maximize expected return for a given level of risk, see problem (4)- (5). Apply Lagrange multiplier method to solve expected return maximization problem for a given risk level. Let λ 1 and λ 2 be two scalars. Then define Solve ∇L(w, λ 1 , λ 2 ) = 0 to find critical points of L. Thus, the following equations follow from ∇L(w, λ 1 , λ 2 ) = 0.
This implies that It follows from equation (14) that This implies that Since w T l = 1, we get Thus, the following equations are valid.

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Construction of Optimal Portfolio Using Efficient Portfolio and Zero Opportunity Cost It follows from equation (17) that Substitute the value of λ 2 in to the equation (22) to get the following equation (23).
Substitute equation (23) in to the equation (21) to get constant weight efficient frontier of portfolio p. Therefore, constant weight efficient frontier of portfolio p is given by: Minimize σ 2 p with respect to µ p to derive global minimum variance of portfolio p.
The global minimum variance of portfolio p is given by: Clearly, the global minimum variance of portfolio p is obtained at µ p = b c , [14,15,16,17].

Complete portfolio
A complete portfolio of risk free asset and risky asset portfolio is defined as a combination of a risky asset portfolio and the risk-free asset, [18]. The return r c of complete portfolio is defined by: where r p is risky asset portfolio return, r f is risk free asset return and w p is wight of risky asset portfolio.

Capital market line
Capital Market Line (CML) contains all optimal portfolios that are combination partners of the risk-free asset and the market portfolio, [19]. All of the portfolios on the CML have the same Sharpe ratio as that of the market portfolio, [19]. The tangency portfolio is the intersection point of the CML and the efficient frontier, [17,20]. Let r m be return of market portfolio. Then the return of tangency portfolio r p is defined by: where r f is risk free asset return and w p is wight of market portfolio. In this research, we relax the tangency portfolio condition r p = w p r m + (1 − w p )r f to order two opportunity cost condition 1]. Note that tangency portfolio condition is a special case of order two opportunity cost condition, choose ρ = 1.
We observe from literature review that portfolio selection models depend on investors' decisions and objectives. Tangency portfolio condition helps investors to choose efficient portfolio when investors would like to construct a complete portfolio of risk free asset and market index. However, there exist order two and three zero opportunity condition which are general conditions of Tangency portfolio condition. In summary, researchers study portfolio selection techniques based on zero opportunity cost condition as investors' decision.

Portfolio selection using zero opportunity cost
This research presents the mathematical modeling for portfolio selection based on zero opportunity cost. Hence there are many mathematical expressions used throughout this research. We have used a number symbols in this mathematical expressions which are given in Table 1.

Frontier error minimization model (F EM M )
The following model is called frontier error minimization model: A portfolio that corresponds to F EM M is called frontier error (F E) portfolio.
Definition 3.2 (Efficiency condition). The equation is called efficiency condition.

Zero opportunity portfolio
Let r f be the rate of return of risk free asset. Let r m be return of market portfolio (index). Let µ m be expected return of market portfolio. Let r p be return of risky portfolio. Scenario one portfolio (risk free fundamental portfolio) SP 1 is a portfolio with return r SP1 = w p r f + (1 − w p )r m , w p ∈ [0, 1] . Scenario two portfolio (risky fundamental portfolio) SP 2 is a portfolio with return r SP2 = ρr p + (1 − ρ)r m , ρ ∈ [0, 1]. Scenario three portfolio (free-risky fundamental portfolio) SP 3 is a portfolio with return r SP3 = δr f + (1 − δ)r p , δ ∈ [0, 1]. SP 1 and SP 2 are said to be zero opportunity portfolios if µ m ̸ = r f , w p ̸ = 0, ρ = w p and r SP1 = r SP2 . SP 1 and SP 2 are said to be equivalent portfolios of order two if r SP1 = r SP2 . SP 1 ,SP 2 and SP 3 are said to be equivalent portfolios of order three if r SP1 = r SP2 = r SP3 . Consider equivalent portfolios of order two. There are two cases for equivalent portfolios of order two. The first case ρ = w p . The second case for equivalent portfolios of order two is ρ ̸ = w p . Let us begin with ρ = w p case. The following results are obtained by the help of ρ = w p case.
Theorem 3.4. Let α i and β i be two real numbers. Define If SP 1 and SP 2 are zero opportunity portfolios, then Suppose that SP 1 and SP 2 are zero opportunity portfolios. Then and It follows from equation (32) that and It follows from equation (30) that Solve for w p from equation (35) to get equation (36).
Clearly the following equations hold.
It follows from equation (40) that It follows from equations (39) and (40) that Note that market portfolio expected return and standard deviation are constant numbers, whereas weight w p and σ p are variables. Clearly, Substitute equation (42) in to the equation (45) to find equation (46).
Not that (47) with respect to w p to get equation (48).
It follows from equation (49) that We also note the following equations: . It follows from equations (52) and (53) that Solve for h from equation (51). Substitute h in to equation (54) and solve for g from equation (54). Hence proved.
Theorem 3.5. If SP 1 and SP 2 are zero opportunity portfolios, then Proof. Clearly, α p result follows from equation (48). Now solve for σ SP1 from equation (41). Thus, the following result follows: It follows from theorem 3.4 that The value of β is zero from equations (55) and (56). Hence the proof followed.
Lemma 3.6. If risky and risk free fundamental portfolios have equal Sharpe ratio, then Proof. Suppose that risky and risk free fundamental portfolios have equal Sharpe ratio. Clearly, Since both risky and risk free fundamental portfolios have equal Sharpe ratio, we get Solve for w p from equation (59). Thus, the following equation follows.
It follows from equation (61) that Hence the proof followed.
Corollary 3.7. If SP 1 and SP 2 are zero opportunity portfolios, then Proof. The result of this theorem follows from Theorem 3.5.
Theorem 3.8. If SP 1 and SP 2 are zero opportunity portfolios, then Proof. Suppose that SP 1 and SP 2 are zero opportunity portfolios. Then This implies that Clearly, It follows from equations (65) and (66) that Consider the following equation.
Let us consider the following equation.
It follows from corollary 3.7 that Substitute equations (68) and (69) in to the equation (70) to get (71).
Consider the following equation.
(76) This implies that Hence the proof followed.

Sharpe square error model (SSEM )
We define zero Sharpe square error ZSSE by: Investors who prefer to construct efficient portfolio using zero Sharpe square error would like to solve Sharpe square error model. Sharpe square error model is given by: A portfolio that corresponds to SSEM is called Sharpe square error (SSE) portfolio.
Definition 3.10 (Sharpe ratio error condition). The equation is called Sharpe ratio error condition.

Return neutral portfolio
Use return neutral model to construct return neutral portfolio. Clearly, ⃗ w = Σ −1 l c .

Sharpe ratio efficient opportunity portfolio
Sharpe ratio efficient opportunity model (SREOM ) is an optimization model with Sharpe ratio objective subject to efficient-opportunity constraint and unit weight sum constraint. Sharpe ratio efficient opportunity model (SREOM ) is given by: Sharpe ratio efficient opportunity (SREO) portfolio is a portfolio with optimal weight vector of SREOM .

Equally weighted portfolio
Investors select equally weighted portfolio to keep diversification. However, equally weight portfolio may or may not be optimal portfolio. Let P ew be equally weighted portfolio. Then expected return and risk of P ew are given by: µ ew = l T µ n and σ ew = √ l T Σl n . Thus, Sharpe ratio of equally weighted portfolio is defined by:

Order three complete portfolio
The return of order three complete portfolio P 3c is defined by r p . Without loss of generality, let us assume that w p is given. Then one can find δ using weight-variance regression model and ρ using order three weight equation. Therefore, for a given w p there exist δ and ρ which satisfy order three weight equation.

Weight-variance regression model
The regression model ln(σ p ) = α + βσ 2 pm + ϵ p is called weight-variance regression model. We . Suppose that there is no opportunity cost between the two fundamental portfolios. That is, fundamental portfolios have the same level of risk and expected return. This assumption is correct because in finance there is no free lunch or arbitrage opportunity. Define expected return of risky fundamental portfolio by µ c = w p µ p + (1 − w p )µ m . Similarly, define risk of risky fundamental portfolio by σ 2 c = w 2 p σ 2 p +(1−w p ) 2 σ 2 m +2w p (1−w p )σ pm . Notation:µ p = µ T w and σ 2 p = w T Σw. Now one can build a model based on efficiency condition, Sharpe ratio error condition and weight-variance regression model. Define σ pm = (σ 1m , σ 2m , . . . , σ nm ) T . Therefore, we have the following Sharpe variance model (SV M ).

SV M : min
SV is called Sharpe variance portfolio.
is called Treynor-Sharpe regression model. The line tr i =â + bsr i called Treynor-Sharpe regression line. We have seen the case ρ = w p for equivalent portfolios of order two. Let us consider the opposite case of ρ = w p . Suppose that w p r f + (1 − w p )r m = ρr p + (1 − ρ)r m and ρ ̸ = w p . Let us consider three investment portfolio construction scenarios. Let P f r , P mr and P f mr denote free-risky portfolio, market-risky portfolio and free-market-risky portfolio, respectively. Consider the following return condition of these portfolios. Suppose that n = w p r f +(1−w p )r p = ρr m +(1−ρ)r p = δr f + (1 − δ)r m . The condition w p r f + (1 − w p )r p = ρr m + (1 − ρ)r p = δr f + (1 − δ)r m is called order three opportunity condition.  Proof. Suppose that order three opportunity condition is valid.
Hence the proof followed. Proof. Suppose that order three opportunity condition holds.
σm . Hence the proof followed. Proof. Suppose that order three opportunity condition holds. Let t p and t m be Treynor ratio of portfolio p and market portfolio m, respectively. Clearly, t m = 0. Claim: To show that Hence the proof followed.
Theorem 5.6. If order three opportunity condition holds, then both order three complete portfolio P 3c and market portfolio m have equal Treynor ratio.
Proof. Suppose that order three opportunity condition holds. Then r 3c = ρr m + (1 − ρ)r p . Define Treynor ratio t 3c of order three opportunity complete portfolio by: Similarly, Treynor ratio t p of risky portfolio is defined by: It follows from equations (84) and (85) that Consider the following result.
This implies that (87) Substitute the equation (87) in to equation (86) to get the equation Clearly, Since t p = 0, we get t 3c = 0. Hence the proof followed.
Theorem 5.7. Let s 3c be Sharpe ratio of order three complete portfolio P 3c . Suppose that 0 < w p < 1. If order three opportunity condition holds, then s 3c = s p .
Theorem 6.1. The following equations are necessary conditions for order three opportunity condition.

Sharpe ratio process
Sharpe ratio process (vector) is defined by ⃗ sr tp = ( µtp−r f σtp ) τ , where τ is an investment horizon.

Results
Secondary data on stocks return is used for demonstration purpose. In this research, we use 14 oct 2019-7 oct 2020 daily return historical data of USA 22 stock companies from yahoo finance. We also use one year bond rate r f = 0.11%(daily bond rate r f = 0.000003) for empirical study. All empirical results of this study is for long selling option. In this study, we have not included short selling empirical results. We have used global search in MatLab to solve optimization problems of the study. Regression analysis is done by the help of R. In this research, we have constructed Sharpe variance portfolio (SV ), order three portfolio (OT H) and complete portfolio (CP ). We solve Sharpe variance model (SV M ) to find optimal weights of Sharpe variance portfolio (SV ). We use general order three opportunity model (GOOM ) to solve for optimal weights of order three portfolio (OT H). Return of complete portfolio is given by r tcp = w * f r f + w * p r tOT H + w * m r tm , where r tOT H is rate of return of order three portfolio (OT H) at time t, w * f = . We apply order three asset allocation model (OAAM ) to solve for w p , δ and ρ. The empirical study of this research gives w p = 0, δ = 0.3661 and ρ = 0. Thus, we have w * f = 0.122, w * p = 0.6667 and w * m = 0.2113. Let us see weights of Sharpe variance portfolio (SV ), order three portfolio (OT H) and complete portfolio (CP ) as given in Table 2. Observe from Table 2 that SV is very close to market portfolio S&P 500. Efficiency condition is not considered for SV construction. Note that order three opportunity condition is considered for OT H construction. We assume that there is no opportunity cost between two investment options. Thus, OT H is more realistic model than SV . Since OT H is not complete portfolio, we want to construct complete portfolio based on order three zero opportunity condition and efficiency condition. Therefore, CP is realistic portfolio for investors who consider order three zero opportunity condition and efficiency condition. Consider table 3 to observe the following results. The minimum standard deviation corresponds to COST . AAP L has maximum expected return. The company with maximum Sharpe ratio is AM ZN . CAG has smallest beta where as AAP L has maximum alpha. Clearly, the market index has zero Sharpe error. Finally, AM ZN has maximum alpha to beta ratio(Treynor ratio). Opportunity models require estimation of weight-variance model parameters. We assume that portfolio follows weight-variance regression line. That is, natural logarithm of standard deviation of portfolio is assumed as linear function of square of co-variance between portfolio return and market index return. Opportunity models take weightvariance regression line as a constraint. Before applying ordinary least (OLS) square parameters estimation method, one should check the validity of (OLS) assumption. If all assumptions of (OLS) hold, then one can apply (OLS) to estimate parameters of linear regression. We want to test weight-variance model (OLS) assumptions validity using R program. Ordinary least square method has some basic assumption. Some of these assumptions are linear relation ship between explanatory and response variable, constant variance of residual square, error is normally distributed with expected value zero and constant variance, explanatory variables should be independent to each other. Let us consider hetroscedasticity problem testing methods. In this research, we apply white test and alternative white test to check the hetroscedasticity problem. Once weight-variance regression line parameters are estimated, use the estimated parameters in opportunity models to select portfolio based on zero opportunity cost.
Consider the following scatter plot. As shown in figure 1, there is linear relation ship between natural logarithm of risky portfolio standard deviation and square of co-variance between risky and market portfolio. Both white and alternative white test hetroscedasticity problem testing methods show that there is no statistically significant hetroscedasticity problem. Therefore, we can apply OLS to estimate weight-variance model parameters. The required weightvariance model equation is given by: The next step is estimating the value of 1−δ . Usê β = 2.121(10 6 ) and σ m = 0.02145974 to find 1−δ 1−w . Hence 1−δ 1−w = 1.054398. Let us consider basic portfolios and their optimal weight vector. In practice, investors use long selling assuming that short selling is not allowed. Thus, we assume weights of securities as non-negative. Treynor ratio maximization model (T RM M ) is given by: Now we want to test the assumption ρ = w P for second order zero opportunity portfolio. It is proved that tr p =â + bsr p is sufficient condition for ρ = w p . Thus, we should test the linear relation between Treynor ratio and Sharpe ratio using linear regression. One can observe from figure 2 that there is positive linear relation ship between Treynor ratio and Sharpe ratio.  Figure 2 shows that there is positive linear relation ship between Sharpe ratio of companies and Treynor ratio of companies. Thus, we assume portfolio follows Treynor-Sharpe regression line. As a result, we assume ρ = w P . Let consider the performance of OT H and CP . Note that CP is more diversified portfolio than OT H. We observe from Figure 3 that CP outperforms OT H based on Sharpe ratio process.

Discussion
Mean variance model is applicable only either expected return of portfolio or variance of portfolio return is given. But in reality both expected return and variance of portfolio are not known. Therefore, mean variance model has no supportive evidence unless either expected return of portfolio or variance of portfolio return is given. Another problem of mean variance model is it does not consider some logical constraints like opportunity cost and arbitrage opportunity constraints. By market efficiency hypothesis, information about securities prices is available in public. Thus, there is no arbitrage opportunity and opportunity cost. In this research, we have investigated two portfolio selection and asset allocation strategies based on the evidence that there is positive linear relation ship between Treynor ratio of securities and Sharpe ratio of securities and positive linear relation ship between natural logarithm of standard deviation of securities' return and square of co-variance between securities' return and market return. These two strategies are portfolio selection and asset allocation strategies using order three asset allocation model and general order three opportunity model.
We would like to test the quality or (validity) of order two zero opportunity cost condition using historical data. We observe from results of this study that securities follow Treynor-Sharpe regression line. Thus, it is reasonable to construct a portfolio that follows Treynor-Sharpe regression line. It follows from fundamental theorem of order two investment opportunity that risk free fundamental portfolio and risky fundamental portfolio are zero opportunity portfolios.
Order three zero opportunity condition suggests that there is linear relation ship between variance of security return and covariance of security return with market return. However, we realized that there is positive linear relation ship between natural logarithm of standard deviation of securities' return and square of co-variance between securities' return and market return using historical data of securities' return. We have introduced portfolio weight-variance regression line constraint in GOOM . We have included portfolio market line constraint and portfolio beta estimation constraint in GOOM . Note that GOOM needs optimization tool for its optimal solution.
The tangency portfolio is constructed by the help of efficiency condition and tangency portfolio condition. Let consider efficiency condition and order three zero opportunity condition to construct OAAM . Order three Sharpe error condition and order three efficiency condition follow from efficiency condition and order three zero opportunity condition. We use OAAM to determine weight of risky portfolio, risk free portfolio and market portfolio in order three complete portfolio.
The two models OAAM and GOOM are favorable for investors who do not want opportunity gain or loss from the three investment scenarios. Tangency portfolio does not con-sider portfolio beta estimation constraint, portfolio market line constraint and weight-variance regression line constraint. Let us discuss on mean variance portfolio, tangency portfolio and order three complete portfolio. Tangency portfolio is a special case of mean variance portfolio with capital market line constraint. Thus, mean variance portfolio may outperform tangency portfolio because tangency portfolio is constrained by capital market line constraint. However, mean variance portfolio has opportunity cost risk. Since tangency portfolio is a special portfolio with special case of order three opportunity cost condition, it has order three opportunity cost risk. Order three complete portfolio has no order three opportunity cost risk. Therefore, investors who take zero order three opportunity cost would choose order three complete portfolio for their investment portfolio selection and asset allocation decision. In summary, we have relaxed tangency portfolio condition to order three zero opportunity cost condition to construct order three complete portfolio. In summary, the two models OAAM and GOOM are realistic models than mean variance portfolio and tangency portfolio for investors who takes order three zero opportunity cost risk.

Conclusion and recommendation
In this research, we have introduced basic portfolio conditions, namely, efficiency condition, Sharpe ratio error condition and order three zero opportunity condition. We have proved some mathematical facts related efficiency condition, Sharpe ratio error condition and order three zero opportunity condition. We have shown that there is positive linear correlation between Treynor ratio and Sharpe ratio of companies using empirical study. Furthermore, we observed that logarithm of standard deviation of companies and square of co-variance of companies with market index are positively correlated. We constructed GOOM using portfolio Treynor-Sharpe line and order three opportunity condition. Moreover, we constructed OAAM using efficiency condition, Sharpe ratio error condition and order three zero opportunity condition. Finally, we applied GOOM and OAAM to construct complete portfolio CP .
Let us consider practical application of GOOM and OAAM . First, we checked Treynor-Sharpe regression model using historical data of securities' return. Second, we checked the validity of weight-variance regression model using historical data of securities' return. Third, we solved for optimal solution of GOOM . We constructed OT H from GOOM . Fourth, we applied OAAM to determine weights of assets in order three complete portfolio. Finally, we evaluated the performance of OT H and CP using Sharpe ratio process. The result of this study shows that CP is outperforming OT H with respect to Sharpe ratio process. Moreover, we constructed CP using proposed models and historical data. The result of this study suggests that investors who want to take order three zero opportunity cost risk should allocate 2 3 of the investment capital for risky portfolio.
In mean variance model, weight of each company is assumed as independent of time. In this research, we assumed