Open access peer-reviewed chapter

Saving Time in Portfolio Optimization on Financial Markets

Written By

Todor Atanasov Stoilov, Krasimira Petrova Stoilova and Miroslav Dimitrov Vladimirov

Submitted: June 7th, 2019 Reviewed: August 2nd, 2019 Published: September 14th, 2019

DOI: 10.5772/intechopen.88985

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The time management is important part for tasks in real-time operation of systems, automation systems, optimization in complex system, taking explicit consideration in time constraints, scheduling of tasks and operations, making with incomplete data, and time management in different practical cases. The limit in time for taking appropriate decisions for management and control is a strong constraint for the implementation of autonomic functionalities as self-configuration, self-optimization, self-healing, self-protection in computer systems, transportation systems, and distributed systems. Time is an important and expensive resource. The time management in financial domain is a prerequisite for high competitiveness and an increase in the quality of the investment activities. It is the popular phrase that time is money, and particularly, the portfolio optimization targets its implementation in real cases. This research targets the identification of portfolio parameters, which are strongly influenced by time. We restrict our considerations only on portfolio optimization task, and we identify cases, which are strongly influenced by time constraints. Thus, the portfolio optimization problem is discussed on position how the time can influence the portfolio characteristics and solutions. This chapter starts with the description of the object portfolio management, which provides the cases where time in explicit way influences the portfolio problem.


  • data driven analysis
  • real-time portfolio optimization
  • decision making
  • automation in information systems

1. Introduction

The time management is important part for tasks in real time operation of systems, automation systems, optimization in complex system, taking explicit consideration in time constraints, scheduling of tasks and operations, making with incomplete data, time management in different practical cases. The limits in time for taking appropriate decisions for management and control is a strong constraints for the implementation of autonomic functionalities as self-configuration, self-optimization, self-healing, self-protection in computer systems, transportation systems, distributed systems. Time is an important and expensive resource.

The time management in financial domain is a prerequisite for high competitiveness and increase of the quality of the investment activities. It is the popular phrase that “time is money” and particularly the portfolio optimization targets its implementation in real cases. This research targets the identification of portfolio parameters, which are strongly influenced by time. We restrict our considerations only on portfolio optimization task and we identify cases, which are strongly influenced by time constraints. Thus, the portfolio optimization problem is discussed on position how the time can influence the portfolio characteristics and solutions. This chapter starts with description of the object “portfolio management” which provides the cases where time in explicit way influences the portfolio problem.


2. Portfolio optimization problem

The task, which is resolved by the portfolio optimization of financial resources, is related with maximization of the return and simultaneously minimization of the investment risk. The portfolio optimization can be applied also to assets, which belong to the stock markets, because the same valued characteristics are used for portfolio optimization. The goal of the portfolio problem is to share the amount of investments among a set of securities, which are chosen to enter into the portfolio. The portfolio goal is to allocate in optimal manner the parts of the investment for buying securities. The time management problem initially arises with its complexity on the stage of the portfolio definition. The investment procedure has to be implemented at time t0 (now). The assets’ characteristics can be evaluated for this time moment t0, Figure 1.

Figure 1.

Time schedule of the portfolio investment.

The portfolio management insists to make decision for buying (or selling) assets at the current time t0. Then after a period of time t>0at time moment T=t0+tthe investor has to sell (or buy) the assets from the portfolio and must receive positive return


The value of the Receiptis defined in the future time Tand the Expenditure—on the current time t0. The portfolio problem arises according to the difference of the time moment t0<T. The investment decisions are based on the assets’ characteristics for the moment t0, A(t0). But in time Tthese characteristics will be A(T) and in common case they will differ in values At0AT. These differences strongly influence the portfolio return at time T. In general, the assets’ characteristics are the return and risk, Ait0=AiReturnit0Riskit0,i=1,,N, Nis the types of assets in the portfolio which are evaluated for the current time t0. But the portfolio return is evaluated at the end of the investment period T. Respectively, the assets’ characteristics at time Tare different AiT=AiReturniTRiskiT,i=1,N. Hence, the final portfolio returns from (1) becomes


Following (2) for the implementation of the portfolio investment, the investor has:

  • to choose the types and number of assets N, which will participate in the portfolio;

  • to assess the assets’ characteristics Riski(t0) and Returni(t0), i = 1,…, Nat the current time t0;

  • to choose the duration ∆tof the investment, which defines the final investment time T;

  • to forecast the assets’ characteristics Riski(T) and Returni(T), i = 1,…, Nfor the end of the investment period T;

  • to define and solve the portfolio optimization problem which will give the relative weights wi, i = 1,…,Nof the investment, allocated for buying (selling) asset i. The relative values of weights introduce the analytical constraint


and wiare the solutions of the portfolio optimization problem. To move ahead about the time management problem and to recommend relations between t0, ∆tand Tthere is a need to analyze the character of the portfolio optimization problem.


3. Modern Portfolio Theory

The Modern Portfolio Theory (MPT) was quantitatively introduced from Markowitz, with his seminal work [1]. The problems, introduced for the portfolio optimization are defined with two formal descriptions:

  • maximization of portfolio Returnby finding optimal values of the assets’ weights wi, i = 1,…,N, satisfying constraints about portfolio Riskto stay below a predefined value


  • and/or minimization of portfolio Riskby finding optimal assets’ weights wi, i = 1,…,N, satisfying constraints about the portfolio Returnto stay over a predefined value


The notations used concern

Ei—the mean return of asset i = 1,…,N, ET = (E1, …, EN),

 − is the covariance matrix of the assets’ returns, square symmetrical N × Nmatrix,

σmax2—the maximal portfolio risk, which the investor can afford for problem (4),

Emin—the minimal portfolio return which the investor expects from the investment,

wT = (w1, …, wN)—a vector of relative weights of the investment, which will be allocated to asset i = 1,…,N, for buying or selling.

Particularly, additional nonnegative constraints are aided, wi  0, i = 1,…,N, which means that asset iwill be bought for the portfolio. The case with negative weights, wi < 0 means that the investor will sell asset iat time t0 and at the end of the investment period Tthe will buy these assets to recover his wealth. During these operations the investor has to achieve positive portfolio return. The case of portfolio optimization with negative weights is named “short sells” but it is allowed only for special types of investors [2]. That’s the reason that MPT mainly applies an additional constraint for nonnegative weights w  0to problems (4) and (5).

To be able to solve problems (4) and (5) the parameters Eand have to be numerically evaluated. These parameters are strongly influenced by time. The estimation of the mean assets’ returns Ei, i = 1,…,N, has to be made for historical period. The portfolio manager must use a time series of assets’ returns


where Rimis the return of asset iat time m, i = 1,…,N, m = 1,…,n; n-discrete points from the return history. These return values could be on daily, monthly, weekly basis for a past period of time. Because for that case the time is defined as integer number of days/months/weeks, the number ndescribes the length of the historical period, taken by the portfolio manager to estimate the mean assets’ returns Ei, 1,…,N. The value of nis a discrete time and it influences the values of the assets’ characteristics. For a discrete time diapason 1 ÷ nthe mean assets’ returns are


Having the values Ei, i = 1,…,Nfrom (7) the covariance matrix is calculated as


The covariance coefficient cijhas meaning, which defines how the time series of the assets’ returns iand jbehave. The case of positive correlation cij > 0 means that if the time series of returns Riof asset increase (or decrease) the same simultaneous change of increase (or decrease) takes place for the time series of returns Rj. For the case of negative correlation cij < 0, the time series Riand Rjmove in opposite directions. If the time series Riincrease (or decrease) the time series Rjdecrease (or increase). The negative correlation has advantage in usage by the portfolio managers to decrease the total risk of the portfolio. Because cij = cjifrom (8), the covariance matrix ∑ is symmetrical. For the case i = jthe value ciiis the variation of the row Ri, cii=σi2, σi—standard deviation of row Ri. Thus, the covariance matrix on its diagonal gives the variation of the assets’ returns. The components cijdefine the behavior of the time series of returns Riand Rm. The portfolio theory applies the variation σi2as quantitative values of the risk of asset i. The graphical interpretation of mean return and risk of asset iis given in Figure 2, where.

Figure 2.

Graphical interpretation of the risk and mean return of asset.

Riis the dynamically changed return of asset i,

Eithe mean value of return for the time period [t1, t2],

σi—standard deviation of Ritowards Eiand give value of the risk of asset i.

The risk of the asset graphically represents the diapason [+σi,σi] between which the real asset returns Rigenerally stay around the mean value Ei. After definition of the vector of mean assets’ returns ET = (E1, …, EN) and the covariance matrix COV(.) = , the portfolio return Epanalytically is evaluated as


The value of the portfolio risk is calculated by the quadratic term


The MPT uses and integration of the portfolio problems (4) and (5) by definition of a common optimization problem


The value of Ψis the “risk aversion” coefficient, which is normalized for the numerical diapason [0, 1].

  • For the case Ψ=0the investor doesn’t care about the portfolio return and his goal is to achieve minimal portfolio risk.

  • For the case Ψ=1the investor targets maximization of the portfolio return without considering the portfolio risk, becauseminΨETwmax+ΨETw.

By changing Ψ01different solutions woptΨare found from problem (11) which gives corresponding returns Ep=ETwoptΨand risk σp2=woptTΨwoptΨfor the portfolios. These set of solutions can be presented as a set of points [σp2,Ep] in this space which in continuous case origins the “efficient frontier” curve, Figure 3.

Figure 3.

The curve of “efficient frontier” and the market point.

The efficient frontier has quadratic character but it is not a smooth line [3]. This non-smooth character origins from the existence of non-negative constraints wi0, i = 1,…,Nin problem (11). Hence, the MPT recommends to be defined and solved portfolio problem (11). Because the investors have different ability to undertake risk, the portfolio manager has to estimate the correct value of the “risk aversion” parameter. Because such identification is strongly subjective influenced, the MPT recommends to be evaluated the “efficient frontier” of portfolios. The investor can choose appropriate point from the frontier, which corresponds to the relation Risk/Return, which the investor is willing to accept. The portfolio, applied in problem (11) is named also “mean-variance” (MV) portfolio model. From the time management considerations, the cases which are influenced by the time, for the portfolio problems are summarized as:

  • the portfolio manager has to choose the time for the portfolio implementation;

  • he has to decide the duration of the investment ∆t = T–t0; T—final investment time;

  • he has to choose the duration nof the historical period, which will be used for the evaluation of the mean assets’ returns Ei, i = 1, Nand the covariance matrix COV(.) = of the assets’ returns. The diagonal values of matrix gives assets’ risks σi2,i=1,,N.

Thus, the time is very important parameter, which influences the definition and implementation of the portfolio investment and optimization.


4. Capital Market Theory

The MPT originated by the works of Markowitz has its further developments. The next important stage of MPT is the definition of the Capital Market Theory, [2]. The Capital Market Theory introduces a new point on the “efficient frontier,” named “market portfolio.” It has analogical portfolio characteristics as market return EMand market risk σM2. This theory derives new analytical relations with the market characteristics, which are formal part of the Capital Asset Price Model (CAPM). This model added three additional linear relations named Capital Market Line (CML), Security Market Line (SML) and cHaracteristic Line (HL).

The graphical representation of the CML is given in Figure 3. It starts from the point (0,rf) which is a riskless asset with mean return rf. The market point (EM,σM2)is a tangent one over the “efficient frontier.” The CML gives relations between the portfolios returns and risks for a particular market, assessed by the characteristics rf,EM,σM2. Analytically, the CML is a linear relation between Epand σp,


By estimating the market parameters rf,EM,σM2the investor has information about the level of risk σp2, which has to be undertaken by means to obtain portfolio return Ep. This prevents the investor to have unrealistic expectation about the potential mean return, which has to be achieved by a portfolio. The values of the market parameters,EM,σM2are defined mainly according to the behavior of market index (S&P500, Dow Jones Industrial Average, NASDAQ Composite, NYSE Composite, FTSF100, Nikkei225, IPC Mexico, EURONEXT 100 and others). On each market the risk-free assets (deposits, long time bonds) has its own value rf.

The SML introduces linear relations between the mean return of a particular asset Eiand the market return EM


The coefficient “beta” (βi) is a value of the relation


The “beta” coefficient takes normalized values from the diapason [−1, 1]. Numerically, it defines how strong the mean return value Eiis related with the market return EM. If the return Eiis strongly related to the market behavior, the coefficient βihas high value, close to 1, if the covariation coefficient cov(i,M) is positive. The case of βi<0means that the covariance between the series of returns Riand RMare in opposite directions.

The HL line makes additional clarification between the current value of the asset return Riand market one RM


Relation (15) is timely influenced. If the market value RMis changed/predicted, the corresponding asset return Riof asset ican be estimated and/or predicted.

The CAPM does not apply explicit inclusion of time in its characteristics. Time explicitly influences only the values of the market return EMand market risk σM. Applying the same considerations which take place for the evaluation of the assets’ characteristics Ei,σii = 1,…,Nthe historical period for the evaluation of the market characteristics is recommended to be the same, with ndiscrete historical values of the market return RM=RM1RM2RMn.


5. Black-Litterman model for estimation of portfolio characteristics

The Black-Litterman (BL) model is based on both achievements of the MV portfolio model and CAPM. The idea behind the BL model is the ability to use additional information by means to estimate and to predict the assets’ characteristics Ei(T) and σiT, i = 1,…,N[4, 5, 6]. The difference and the added value Nfor the future time moment Twhen the portfolio investment will be capitalized e of the BL model is graphically interpreted in Figure 4.

Figure 4.

Additional modification of portfolio parameters by BL model.

The MV model estimates the assets characteristics Ei, σi, i = 1,…,Nusing historical data from ndiscrete points of the assets’ returns Rim,i=1,,N,m=1,,n. The BL model allows additional information to be used by means to modify the mean values of return ET = (E1, …, EN) as the assets’ risk characteristics, given by the covariation matrix . The modification of ETto a new vector EBLT=EBL1EBLNis made by two additional numerical matrices Pand Q. These matrices are evaluated from expert views, who make a subjective assessment about the future levels of assets’ returns at time T, when the portfolio investment should be capitalized.

Pis a k×Nmatrix, which contains kexpert views. The vector Qdefines quantitative information about the k-th expert view for increase or decrease the mean return Eiof i-th asset. The elements pkiof Pdefines the view of k-th expert about the change of the Eireturn of asset i. The component pkitakes value +1 for the case of increase, and respectively −1 for decrease.

The BL model added a new contribution to the MPT by introducing new characteristic of the portfolio asset: “implied return,” Пi,i=1,,N(“implied excess return,” when the return is evaluated according to the level of risk-free asset rf). These returns differ from the historically evaluated mean returns Ei, i=1,,N. The assumption behind these new “implied returns” is related with the existence of market point (EM,σM). For the case of market equilibrium, the CAPM asserts that all assets, which participate on this market should have appropriate mean returns ПТ=(П1, … ПN) and market weights wMT=(wM1,…, wMN). Hence from the market values (EM,σM) it follows exact values of Пand w. But the market is a stochastic system and it endues a lot of noises, which change the values of the “implied returns.” These returns Пi,i=1,,Nare values, which “should be.” But the noises make changes to Пiand the BL model evaluates the unknown mean values EBLwhich are the “best approximation to Пi.” These considerations origin the matrix linear relation in BL model


where the noise εis assumed to be with normal distribution, zero mean and volatility proportionally decreased from the historical covariance matrix, εN0τΣ, 0<τ<1. The subjective views formally are introduced by the linear stochastic relation


where Qis the quantitative assessment of the experts’ views about the value with which the historical returns will change; Pidentifies which assets’ returns will be changed. The expert views contain also noise η. Due to the independence of the expert views the noise ηis assumed with zero mean and volatility Ω, ηN0Ω. The matrix Ωis kxksquare one with only diagonal elements because of the independence of the expert’s views. The matrix Ωis presented mainly in the form [7].


The goal of the BL model is the evaluation of the returns EBLwhich have to approximate in maximal level the stochastic relations (16) and (17). The values of the vectors and matrices П, Q, P, ε, ηare assumed to be known and/or estimated. The definitions of these parameters are given in the next paragraph.


6. Definition of the “implied excess returns”

Using [8, 9] the assumption is made that the “implied excess return” Пmust satisfy the market portfolio. The goal function of the portfolio problem for that case is


where λ=1ΨΨis not normalized value of the risk aversion coefficient, λ0. Because the market point is used in (18) according to the CAPM the relation wMT.1=1is satisfied, 1T=11is a unity Nx1vector. The unconstrained minimization of (19) gives solution


By multiplication from left of the both sides of (20) with market capitalization weights wMTit follows


The right component of (21) contains the market “excess return” EMrf, according to (9). The left side gives the market volatility (risk) σM2, (10) or


Substituting (22) in (21) the “implied excess return” results as


The “implied return” П* is the value of Пto which the riskless return is added


This manner of definition of Пis known as “inverse optimization” because the market risk and return are known, but we need to calculate the asset returns.


7. Definition of P and Q from scientific views

Following [10] absolute and relative manner for the formalization of the expert views are applied. The explanation of these forms of formalization is given with a simple example. Let’s the portfolio contains N = 4 assets. Assuming that an expert expects that the first asset will increase its return with 2%; a second expert makes conclusion that the fourth asset will decrease its return with 3% the formalization of P, Qare


The relative form of views applies comparisons between the assets’ returns. Let’s the first expert expects that the first asset will outperform the third one with 2.5%; the second expert makes view that the second asset will outperform the fourth one with 3.5%. The formalization of matrices Pand Qare


The two types of formalization of expert views is widely mention in references dealing with the BL model [7, 10]. A new form of expert views has been developed in [11]. It has been applied a weighted form for the definition of matrix P, where its components can take values different from ±1. To provide this new formalization of the expert views the matrix Ωis analyzed. This matrix formalizes the variation of the expert views. Using relation (18) let’s assume that the portfolio contains three assets, N = 3 and two experts k = 2 make views in relative form formalized in the matrices Pand Q


Hence it follows


where is a symmetrical 4 × 4 matrix σ12σ12σ21σ31σ41σ22σ32σ42σ13σ14σ23σ32σ43σ24σ34σ42. The matrix multiplications results in 2 × 2 matrix Ω=ω100ω2where


Relations (29) have analytical structure with the risk relation of portfolio with two assets, N = 2, and negative correlation, [2] σp2=w12σ12+w22σ222w1w2σ12where σ12and σ22are the volatilities of the two assets, σp2is the volatility of the portfolio, σ12is the covariation between the two returns. Assuming equal weights in the portfolio, w1=w2, the portfolio risk is evaluated as


The comparisons between relations (29) and (30) can be interpreted that in (29) ω1and ω2are the values of risks of two virtual portfolios. The first one contains assets one and three. The second portfolio has the second and fourth assets. Thus, the values ωii=1,2, which formalize the risk of expert views are proportional to virtual portfolios with corresponding two assets, which have negative correlations and equal weights.

Now let’s assume that the matrix Pcontains weighted components αi, which differ from the values ±1. To simplify the formal notations we assume that the matrix Pis on the form.

P =α10α300α20α4. The weighted coefficients satisfy the equalities


For that case the corresponding values of the components ωii=1,2are


Relations (32) interpret that for the weighted form P(α)of the expert views the corresponding components ωii=1,2of the variation of the expert views are proportional to the risk of a portfolio with two assets and negative correlation, and the assets weights αare normalized because equalities (31) hold. The ability to define matrix Pwith components different to ±1allows the expert views to be generated not only by subjective assessments, but also with additional considerations, which are based on objective criteria, estimations and assessments.

This research makes several additions to the numerical definition of Pand Qmatrices.

  1. 1. Formalization P(α)based on the difference ПiEi, i = 1,…,N, normalized by the i-th volatility.

    Following [11] a row of matrix Pconcerning the view of an expert is defined in the form ps=0αi00αj0, 1xNvector. The values αiand αjmust satisfy the normalization equation αi+αj=1. The value αiis chosen from the maximal difference


Relation (33) presents that the mean history’ return of asset i, Ei, is lower from its “implied excess return” and the investor has to expect that the return of asset ihas to increase. The same considerations, but for decrease of the mean return Ejis made from the difference


Asset jis over performed and the investor has to expect decrease of the historical mean return Ejtowards the level of the “implied excess return” Пj.

The value of the component from matrix Qis


  1. 2. Formalization P(ПE)based on the difference ПiEi, i = 1,…,Nwithout normalization with volatilities.

    For that case relations (33) and (34) are slightly modified with lack of volatility normalization


  1. 3. Formalization of P(П)based only on the value of Пi, i = 1,,N.


  1. 4. A particular case can arise when all differences αi=ПiEi, i = 1, Nhave equal sign (+) or (−). Hence all assets’ returns have to be increased, when αi>0or decreased if αi<0.

For that case absolute views can be assign. The matrix Pwill be square NxNidentity matrix. 10101N×N. The Q, N×1 vector will have components equal to αi=ПiEi, i = 1,…,N.

Thus, for the formalization of p. 2 the matrices Pand Qare


for the case of p. 3. These four forms of weighted formalization of matrix P(α)allows to be overcome the need to have subjective expert views. With these formalizations the assets’ characteristics are evaluated not only by historical returns and covariances but by adding data, which in this case concerns differences from the “implied returns.” The |BL model incorporates such additional source of information, Figure 4. The formalism P(α)allows to be compared portfolio solutions, based on MV model and BL one because subjective influences in BL model now are missing. The BL model integrates different sources of information, concerning future assets’ characteristics, but this information is not subjectively generated and it origins from real and actual behavior of the market.


8. BL modification of the assets’ characteristics

Using relations (22) and (23) the BL returns EBLare found by means to approximate in best way these two linear stochastic relations. For simplicity additional notation are used in the next matrix relations




The general least square method with the minimization of the Mahalanobis distance


gives solution


and volatility VolEBL=BL=τΣ1+PTΩ1P1.

Taking into account the riskless return, the final BL assets’ returns and covariance matrix are

EBLfinal=EBL+rf and ΣBLfinal=Σ+BLE44

Using these modified assets’ characteristics, the portfolio problem (11) is solved and appropriate point from the efficient frontier is chosen. It is recommended the best portfolio to be taken with weights wiopt,i=1,,N, which belongs to portfolios with characteristics

Maximal Sharp excess ratio,maxwEprfσp2E45
or maximal information ratio,maxwEpσpE46

9. Numerical simulations and comparisons between MV and BL portfolios solutions

The numerical simulations are performed with real data from the Bulgarian Stock Exchange [12]. The riskless investment for several years gives very low or even negative return. That is, the reason for the investors to start to apply portfolio optimization with risky assets. Currently, the risky investments are performed with a set of about 125 mutual funds in Bulgaria nowadays. The mutual funds are operated by different business and economics entities. The goal of all mutual funds is to manage their portfolios by means to achieve positive return or to decrease the losses in nonfriendly behavior of the financial market. The success or not successful management of the mutual funds can be seen by their historical data about achieved returns and risks in their investments. Thus, our portfolio simulations will start with historical return data of a set of chosen Bulgarian mutual funds. It has been chosen seven mutual funds to participate in the portfolio: Concord Asset Management (CONCORD), Elana Asset Management (ELANA), Profit Asset Management (PROFIT), Texim (TEXIM), Central Cooperative Bank Lider (LIDER), Asset Management UBB Patrimonium (PATRIM), Asset Management DSK Growth (GROWTH). They invest both in currencies and shares. The Bulgarian Association of Asset Management Companies [13] and the Government Financial Supervision Commission [14] regularly record and update the activities of the Bulgarian mutual funds. For the simulation experiments it has been taken the mean monthly return of these 7 mutual funds for 2018-year, Figure 5.

Figure 5.

Monthly and annual returns, and the covariance matrix of the mutual funds for 2018.

The calculations in this research have been performed in MATLAB environment. The mean years returns and the covariance matrix are given also in Figure 5. The simulations apply multiperiod investment policy, described in Figure 6.

Figure 6.

Multi period investment with flowing historical window.

9.1 Initial evaluation of historical data

The monthly mean returns of the mutual funds for the first 8 months of 2018 were taken as historical period. It has been calculated the average return for each fund for this historical period, n = 8. The average returns and the corresponding covariance matrix are given in Figure 7.

Figure 7.

Mean returns and covariance matrix for the first 8 months of 2018.

The portfolio manager has to pay attention for the different values of mean returns and covariance, given in Figures 5 and 7. The first case is evaluated for n = 12, 12 time period. While the second evaluations are made for a shorter period, n = 8. That is, a case where the time management is important for the estimation of the assets’ characteristics.

9.2 Evaluation of the efficient frontier with MV model for the first 8 months

By changing the values of Ψ01the portfolio problem (11) is repeatedly solved. The interim values of the portfolio return, risk and portfolio weights are stored in working arrays in MATLAB environment. The evaluation step of changing Ψwas chosen Ψ=0.01resulting in 100 solutions of problem (11). The graphical presentation of the MV “efficient frontier” is given in Figure 8.

Figure 8.

Graphical presentation of the “efficient frontier” with historical data.

The Sharpe excess ratio (45) and the information ratio (46) are presented in Figure 9.

Figure 9.

Graphical presentation of Sharpe excess ratio and information ratio.

It is estimated the maximum Sharpe_excess_ratio = 4.321. This value corresponds to a portfolio with characteristics:

Return=0.0218,Risk=0.0143,woptT=0; 0.0304; 0.9696; 0; 0; 0; 0.E47

These results recommend that the portfolio manager has to allocate his investment only in two mutual funds: the second in the portfolio (ELANA) and the third one (PROFIT). This recommendation is valid for the investment month of September 2018.

9.3 Evaluation of the assets’ characteristics for the BL model

9.3.1 Definition of the risk-free return rf

In this research for the risk-free return rfhas been used an official index, evaluated and maintained by the National Bank of Bulgaria. The index is named LEONIA+ which is abbreviation of Lev (the name of the National currency) Over Night Index Average. This index is used by the mutual funds to take or giving loans for overnight activities on the financial market. This index is recommendation from the Bulgarian National Bank for all financial institution and authorities in Bulgaria dealing in overnight deposits with Bulgarian currency [15]. For this research the risk-free value is negative on monthly basis, rf = −0.4.

9.3.2 Evaluation of the market point

The characteristics of the market point are the mean return EMand the risk, numerically estimated by the standard deviation σM. The market point is found as a tangent one where the CML (Capital Market Line) makes over the “efficient frontier.” Additionally, the CML must pass through the riskless point (0, rf). The CML cannot be presented in analytical way because the “efficient frontier” is not analytically given. The last have been found numerically as a set of points in the plane (Risk/Return) from the multiple solutions of portfolio problem (11), given in p. 2. This research makes a quadratic approximation of the “efficient frontier” and finds analytical description of the “approximated efficient frontier.” Then with algebraic calculations using the linear equation of the CML and the approximated efficient frontier the tangent point is evaluated. The coordinates of the market point give the mean market return EMand the market risk σM. For these market values the market capitalization weights wMare found from the working arrays when problem (11) has been solved in p. 2. The “approximated efficient frontier” is a quadratic curve of the form


where a2 = −3980.6; a1 = 118.40; a0 = −0.9, x = Risk, y = Return.

The numerical values of the market point are:

EM = 0.0222, σM2= 0.0143, λ=4.3462(according to (22)).

The graphical presentation of the CML, the “efficient frontier” and its approximation and the market point are given in Figure 10.

Figure 10.

CML and approximated efficient frontier.

9.3.3 Evaluation of the implied excess returns Пi, i = 1,…,N.

Using relation (23) the “implied excess returns” Пi, i = 1,…,Nare:

ПT = [0.0523; −0.0126; 0.0235; 0.0635; 0.0375; 0.0433; 0.0427].

Respectively, the “implied returns” is П* = П + rfor.

П*T = [0.0923; 0.0274; 0.0635; 0.1035; 0.0775; 0.0833; 0.0827].

9.3.4 Definition of the characteristics of the expert views P and Q

The portfolio parameter, which is used for the estimation of matrices Pand Qis the difference between the implied returns Пand the mean assets’ historical returns E, (П–Е). These values are as follows:

  1. П*T = [0.0923; 0.0274; 0.0635; 0.1035; 0.0775; 0.0833; 0.0827];

  2. ET = [−0.0592; −0.0424; 0.0238; −0.0105; −0.1277; −0.1141; −0.1216];

  3. *-E)T = [0.1115; 0.0298; −0.0003; 0.0741; 0.1652; 0.1575; 0.1643].

Because the value of the third component of *-E)Tis less than 0.1% it is assumed to be zero. All differences *-E)have positive sign, which means that the assets are underestimated and their implied returns are higher. Hence, the portfolio manager has to expect an increase of the mean returns of the assets in the portfolio. This case of differences between implied and mean returns defines the usage of relation (39) for the definition of matrices Pand Q. The option (39) is also applied in this simulation work. The calculations have been performed with 7 × 7 identity matrix P, 101017 × 7 and two types of matrices Q:


9.3.5 Evaluation of the BL returns EBLand the BL covariance matrix ΣBL

The evaluations of the modified mean assets’ returns EBLaccording to the BL model are done according to relations (43) and (44). The value of the covariance matrix of the expert views is assumed to be as the historical covariance but the values of its components are decreased with equal value τ. Thus the covariance matrix of the expert views is τwhere the value of τmust be between 0 and 1. From practical recommendations [7, 16, 17], this research uses τ=0.5. The BL model evaluations are.


9.3.6 Solution of portfolio problem with EBLTand ΣBL

The portfolio problem (11) is repetitively solved by changing Ψ01with the BL evaluations of the assets’ characteristics EBLTand ΣBL. The new BL “efficient frontier” is found as a set of numerically evaluated points (100 points). For illustration purposes both “efficient frontiers” with historical data (MV model) and BL data (BL model) are given in Figure 11.

Figure 11.

Efficient frontiers with MV and BL models.

9.3.7 Evaluation of the BL weights wBLopt

The portfolio which has maximum Sharpe excess ratio is identified. This maximum is found from the numerically evaluated points of the BL “efficient frontier.” The needed portfolio parameters are stored in the arrays in MATLAB, during the sequential solutions of problem (11). The Sharpe excess ratio evaluated from (45) gives:


The difference between wBLoptand woptshows a bit increase of the weight for the second asset (PROFIT) for the BL portfolio.

9.4 Comparison of the MV solution wopt and the BL one wBLopt

The optimal weights wBLoptand woptare assumed to be implemented as portfolio solutions in the beginning of month of September 2018. At the end of this month we can estimate the actual mean returns of the assets for month of September Efand the modified actual covariation matrix fwhich is calculated again for 8 months history but from February to September 2018.

  • For the case when the MV weights woptare invested the investor results will be


  • For the case when wBLoptweights are applied the investor results will be


Then these portfolio results will be compared in the space Risk(Return). The portfolio point which is situated far on the Nord-West direction of the Risk(Return) space is the preferable portfolio. Such assessment will prove which portfolio model MV or BL gives more benefit and efficiency.

9.5 Multiperiod portfolio optimization

Following Figure 6 a next portfolio investment with MV and BL models is done by moving the history period 1 month ahead. The portfolio evaluations are done for a history period from February till September 2018. The evaluated weights wBLoptand woptare applied for the month of October. For this case of 8 months historical period and available data for all 12 months of 2018 such multiperiod investment policy will evaluate 4 portfolios using the two models MV and BL. This research did three modifications of the BL model, concerning the evaluation of the matrices Pand Q, related to the views for changing the assets characteristics:

For the cases when all components (ПE)or Пhave same sign, the procedures (32) or (33) are applied. The obtained results are given in Table 1.

MV modelBL model
Return (MV)fRisk (MV)fReturn (BL)fRisk (BL)fReturn (BL)fRisk (BL)fReturn (BL)fRisk (BL)f
Mean values

Table 1.

Results of multi-period portfolio management with MV and BL models.

The graphical presentation of the comparison of the multiperiod portfolio management between MV and BL with P(П)modification is given in Figure 12.

Figure 12.

Comparison of multiperiod MV and BL(P(П)) portfolio optimization.

The common results prove that the market situation in 2018 does not allow the mutual funds to achieve positive return. The results are negative but this negative value is less than the riskless return value rf = −0.4. Hence, the portfolio management allows reduction of the losses. Particularly, all three modifications of the BL model give better results in comparison with the classical MV portfolio model. The mean values of the returns with BL model are very close to the returns of the MV model. But the risk values are considerably lower, which means that the probability to be closer to the mean values of BL returns is higher than the case of MV model.


10. Time management considerations for the portfolio investments

This research illustrates that the task of portfolio investment is quite complicated. The meaning of portfolio optimization concerns the definition and solution of portfolio problem. In both these tasks the time is a prerequisite for successful portfolio investment.

10.1 Time requirements for the stage of definition of the portfolio problem

The content in the paragraph “Portfolio optimization problem” explicitly asserts that the investor has to choose the duration of the historical period. This duration, nis in discrete form. It has to be chosen in a way that can refer to the investment period (T-t0). Obviously, high number of nwill give influence for the slow changes in the market behavior. Respectively, the active portfolio management will not benefit with long duration of the historical period n.

The active management needs to follow the current dynamics of the market. The relations between nand (T-t0) cannot be derived on theoretical basis. Only practical considerations could be useful. The authors’ experience recommends duration of the historical period to be considered between 6 and 8 months. Such history period can be used for multiperiod portfolio management from 1 to 3 months ahead in the future.

An unexpected problem has been met by the authors, concerning the relation between the historical discrete points nand the number Nof the assets, included in the portfolio. The two parameters nand Nparticipate both for the evaluation of the covariance matrix ∑. This matrix should be in full rank by means that the portfolio problem (11) must generate regular solutions. If the rank of ∑ is less than Nproblem (11) gives unrealistic solutions. To keep ∑ with rank Nit is needed its components to be evaluated with historical data n > N. The practical minimal case is n + 1 = Nbut before solving the portfolio problem the investor has to check the rank of ∑. As practical consideration, if the portfolio contains many assets and Nis high, the data from the historical period nhave to be also high. For that case one can use not only monthly returns but also weekly average data. Thus, the value of ncan increase.

10.2 Time requirements for the solution of the portfolio problem

The solution of the portfolio problem (11) gives unique set of weights, which have to be implemented for the portfolio investment. Because the market behavior changes, reasonable policy is to perform repeatedly definition and solution of the portfolio problem. Potential beneficial strategy can be the multiperiod portfolio management, presented in Figure 6. It incorporates the multiperiod management and adopts the portfolio parameters with up to date market data. The relation between the duration of the historical period and the investment period is still an open question. But making additional simulations with 1, 2, 3 or more months (time) ahead the portfolio manager can change his decision on each investment step.

11. Conclusions

This research identifies in explicit way the influence of the time for the definition and solution of portfolio problems. These time requirements are considerably related with the estimation of the parameters of the portfolio problem. Respectively, the time requirements insist the portfolio management to be performed in multiperiod investment.

This research makes an analysis of the development of the portfolio theory. Starting with the Markowitz formalization, the MV portfolio problems are based only on historical data about mean returns and covariances between the returns. The development of CAPM gives new relations, originated from a new “market” point. The last gives additional information about the values of the parameters of the portfolio problem. Finally, the BL model introduces a new set of points, “implied excess returns,” which originate from the market point. As a result, new values for the parameters of the portfolio problem are found. Respectively, the portfolio problem gives weights of the assets, which are not sharp cut, which decreases the risk of the investment.

This research introduces new modifications of the BL model for the part of definition of expert views. Particularly the experts are substituted by additional data, which origins from the dynamical behavior of the assets’ returns. Thus, not only mean returns and covariances are taken into consideration, but also the difference between objective parameters as implied and historical mean returns. These modifications allow the portfolio model MV and these based on BL one to be compared on a common basis and to assess their performances. Such comparison cannot be made if subjective experts are used, because their mutual views will be different for the same historical data and with changes the members of the experts.

This research gives also a practical added value with the analysis of the behavior of the market with mutual funds in Bulgaria. This gives additional experience and bases for future comparisons and assessments of the different portfolio models.


This work has been partly supported by project H12/8, 14.07.2017 of the Bulgarian National Science fund: Integrated bi-level optimization in information service for portfolio optimization, contract ДH12/10, 20.12.2017.


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Written By

Todor Atanasov Stoilov, Krasimira Petrova Stoilova and Miroslav Dimitrov Vladimirov

Submitted: June 7th, 2019 Reviewed: August 2nd, 2019 Published: September 14th, 2019