IN THE NAME Of GOD

Fuzzy Grey Cognitive Maps Approach for portfolio Optimization

Tayebeh ZanganehPh.D. student in Finance, Islamic Azad University, Research and Science University,

Tehran branch,T_zangene@yahoo.com

Fraydoon Rahnamay Roodposhtia, Department of Financial Management, Science and Research Branch, Islamic Azad

University, Tehran, Iran; f-rahnamayroudposhti@srbiau.ac.ir , rahnama.roodposhti@gmail.com

Amir Hosein ZanganehMaster in financial engineering, Amirkabir University of technology,

hosein.zanganh@gmail.com

Introduction

• Portfolio optimization is allocating a limited capital(fund) to the different financial asset in order to having an acceptable trade-off between risk and return during last years, researches tried to find and develop the proper measures for estimating risk.

• Risk measures are categorized in three groups:Volatility measures: standard deviation, variance Sensitivity measures: Beta, DurationDown-side measures: semi-variance, semi-standard deviation,

Introduction

• Between these measures VaR is the most commonly usedmetric because of it’s good performance in measuring risk.

• Investment is one of the most important issues in worldwide economic issues in which risk has great importance for investors. So that , in the last two centuries new financial instruments has been developed.

Introduction

• Financial decisions in financial environment includes risk-return trade-off. And measuring the portfolio’ risk is going to be essential nowadays.

• VaR is a measure of the risk of investment. It estimates how much a set of investments might lose, given normal market conditions, in a set time period such as a day.

• VaR is introduced first time by Bamol at 1963. however, at the last 1990 and after recession on economy in this decade it was becoming more interesting. It’s attractness is because of simplicity and better performance in compared to other metrics.

VaR Definition

• VaR is a measure of the risk of investment. It estimates how much a set of investments might lose, given normal market conditions, in a set time period such as a day.

• VaR metric has three main components:

• Time horizon, confidence level, capital(portfolio value)• Portfolio value at time zero• Current portfolio value• Statistical confidence level,

Pr 𝑝$ − 𝑝& ≥ 𝑉𝑎𝑅 ≤ 𝛼𝑜𝑟

Pr{𝑝& − 𝑝$ ≤ −𝑉𝑎𝑅} ≤ 𝛼

Portfolio optimization (portfolio selection)

Two main theories:• Modern portfolio optimization(MPT) theory• Down-side risk methodIn first one, allocation of asset is based on mean and variance of return.

In second one, finding optimal portfolio is based on expected return and down-side risk relationship.

Markowitz Portfolio optimization (portfolio selection)• The main goal of the portfolio optimization is allocating limited capital to the different financial assets so that we have good trade-off between risk-return.

• According to modern portfolio theory of Markowitz(1952) portfolio selection is based on :

• having minimum risk at the specific level of return• having maximum expected return at the specific level of risk. • If the covariance between assets are zero or near

zero(independent) or negative the risk of the portfolio will be reduced.

These portfolio is located on the efficient market line.

• shortcomings of Markowitz theory:1.Numerous estimations2.High cost3.Taking lots of timeFor selecting efficient portfolio we solve linear programming equation.

Markowitz based portfolio optimization

VaR-based portfolio optimization

In spite of the Markowitz model we want to minimize VaR of the portfolio with considering limitation(Campel et al. 2001).

Research methodology

Our contribution in this study is introducing the new method of Fuzzy Grey Cognitive Map for calculating VaR of portfolio. The reason for using this method is simplicity in calculation and desirable performance in the implicated situations(Kosko, 1986).Since the Tehran stock exchange is an inefficient market and because of the volatility of the stock price is high, calculating VaR using such a data isn’t reliable. Therefore, for estimating VaR we use the last month data. Also using the little data also deals to unreliability. So, These are the acceptable reasons for using Grey systems theory according to cognitive maps in order to omit above problems. In this way, we have the new method includes two mentioned before so called : Fuzzy Grey Cognitive Map(FGCM)

Grey systems theory

The most commonly application of Grey systems is when there are many high uncertainty and the data sample is small and incomplete. Base on the accessible data we have three systems:1.Black systems: when there isn’t any data2.White systems: when there is enough data3.Grey system: when there is incomplete and little data

Grey systems theory

• If U is complete set , grey set is subordinate of U as follow equation.

• is down limitation membership • Is up limitation membership and we have • The process of converting grey number to white number is

called “whitening” as follow:

• If α=0.5, whitening is called “whitening by equal weights”.

Proposed model(FGCM

• Fuzzy Grey Cognitive Maps• Fuzzy Grey Cognitive Maps represent the un-structured knowledge which is defined based

on Grey relations between concepts in Fuzzy Cognitive Maps, respectively. Fuzzy cognitive map are dynamic systems which has feedback, they change in a node might leads to a change in other nodes [12]. These models can be used in conditions of uncertainty and doubt in the information available to calculate the map concepts state vector. With this feature, we can get a good and reliable model in terms of price fluctuations in the market in Tehran, as well as with low data. In the fuzzy cognitive map, instead of using the weights with discrete values, in such a way in which weights are considered by fuzzy cognitive map, gray weights are in form of interval, and this gives a better view on the impact of concepts on each other. Also, in the Fuzzy Grey Cognitive Maps, because the concepts are gray mode, you can also specify the degree of uncertainty of concept. In addition, the maps convert decision making model to verbal relations, decision-makers without any technical background will also understand all the implications of the current situation on the map [13].

• In these maps, nodes are shown in gray variables, severity of impact of each node on each other node is specified by its gray degree severity (weight) [12, 13].

Proposed model(FGCM)

• In these maps, nods are shown as grey variables. Also the effects of each node on each others is shown by the فلش . The amount of effects between nodes is defined by the degree of grayness according to the below equation:

⊗𝑤34 ∈ 𝑤34, 𝑤34 |∀𝑖, 𝑗 → 𝑤34 ≤ 𝑤34, 𝑤34, 𝑤34 ∈ [−1,+1]

• In this formula i is the impressive node and j is impressible node. The steps of developing fuzzy grey cognitive map are as follow:

Proposed model(FGCM

Proposed model(FGCM

• The first step is defining inputs and outputs. Since there isn’t any proved theory for defining inputs, we should use literature. Then we considered first auto-regressive of prices as input data.

• The out coming of the model is daily volatility, so we can extract VaR by using following formula:

𝑉𝑎𝑅AB&C = 𝜎AB&F𝐶C

• Cq is the normal standatrd value in the conficence level of q.

Proposed model(FGCM

• The second step is developing initial map. For eliminating the probability of human mistake and accelerating calculation time, all process of developing map, from initial map, training map by using historical data, we use the learning algorithm of data-base Habiyan algorithm(DD-NHL).

• The reasons of selection this learning algorithm are the high speed for reaching response and the simplicity of calculations. Also this algorithm doesn’t need to any expert point of view and is capable to develop maps employing historical data.

Proposed model(FGCM

• After learning final map using historical data and DD-NHL algorithm, we estimate daily VaR.

• Since calculating VaR for all stocks, among optimizing method which is mentioned earlier, we solve non-linear programming in order to find optimal porfolio.

Findings:

Data sample: weakly data price of the 8 diversified active companies form Tehran Stock Exchange between 22 December 2016-23 February 2017 including 8 weeks. For estimating VaR and selecting optimal portfolio we need to calculate stock returns using following formula:

𝑅3 =𝑃3A

𝑃3AI&J − 1

Using series of index price, we calculate return series.

Findings

• By employing KPSS test, we can show the data has static station. Therefore we employ this data for employing model.

• For finding the number of delays we use PACF diagram which shows the effective delays.

Findings

• Using this result, the input of FGCM is considered first 6 delays of auto regressive data.

• After defining input and output we must convert usual number to grey number. Then we employ the group of grey numbers which is volatile during basic value.

⊗𝐶3 = 𝐶3 − 𝜃. 𝐶3 + 𝜃

Findings

• VaR is shown In the below table:

• For calculating total return of the portfolio , the weight of the stocks is considered equal.

Findings

No.Company stockVaR at the

considered time period

531کپارس1518دلقما2153شیران3221وصنا4523خموتور5191شکلر6157کبافق7400آسیا8

Company iMean

return)𝑹N𝒊(

Weight of ithstock in

portfolio)𝒘𝒊(𝒘𝒊𝑹N𝒊

0.02430.110.0026کپارس0.02670.110.0029دلقما0.03380.110.0037شیران0.01360.110.0015وصنا0.01850.110.002خموتور0.02960.110.0032شکلر0.03050.110.0033کبافق0.01020.110.0011آسیا

𝑹∗(0.0205(بازده کل سبد

Calculated VaR(95%

confidence level)

Total return of Portfolio

Findings

• For estimating VaR of the portfolio, Covariance matrix between stock is necessary.

آسیاکبافقشکلرخموتوروصناشیراندلقماکپارس

1کپارس

0.02761دلقما

1-0.14080.0137شیران

0.0410.13420.10171وصنا

0.10430.33320.03250.1681خموتور

0.10331-0.01140.04760.02490.0404شکلر

1-0.020.0074-0.0174-0.16640.10220.0108کبافق

1-0.07520.06120.00680.01550.1760.11540.0591آسیا

Weakly return covariance matrix

• As it is shown in the table the covariance between stocks is little and near to zero. By considering the assumption that the covariance between stocks is zero we can calculate the portfolio VaR by following formula.

• 𝑉𝑎𝑅S = ∑ 𝑤3U𝑉𝑎𝑅3UV3W&

�= 𝑤&U𝑉𝑎𝑅&U + ⋯+𝑤VU𝑉𝑎𝑅VU

�

• For finding the amount of each stock in the portfolio we should solve the following model:

Portfolio optimization model

Solving the model

• By solving the model using Lingo Software the optimal weights of the stocks are as follow:

In continue, for doing stress test we re-estimate our calculation in the 99% confidence level.

Findings

• Calculated VaR(95% confidence level)

Company stockشمارهVaR at the considered period

of time752کپارس1733دلقما2217شیران3313وصنا4741خموتور5271شکلر6222کبافق7567آسیا8

optimal weights of the stocks in the selected portfolio:

Company

stockآسیاکبافقشکلرخموتوروصناشیراندلقماکپارس

Optimal

weight0.0560.070.1970.1480.0580.1920.1940.082

Conclusion

• The results show that by increasing confidence level the amount of VaR is going to be increased. and the new weights in the optimal portfolio are as follow:

• The results show that the optimal portfolios in the both 95% and 99% are the same and the results aren’t sensitive to confidence level .

Comparison between different methods of VaR

For examining the performance of the proposed FGCM model in camprision to other so-called model for calculating VaR we estimate VaR by other models, the results are as follow:

As you see, the proposed model has the dominate performance other models.

Thank for your attention