Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 11, Number 2 (2015), pp. 941-948 © Research India Publications http://www.ripublication.com
Portfolio Optimization Theory in Computational Finance P.K. Parida1 and S.K. Sahoo2 1
Department of Mathematics, C.V. Raman College of Engineering, Mahura, Janla, Bhubaneswar-752054, Odisha, India. 2 Institute of Mathematics & Appl., Bhubaneswar-751003, Odisha, India. 1 e-mail:
[email protected] 2 e-mail:
[email protected]
Abstract In modern finance management Markowitz theory of portfolio management combines probability and optimization theories to model the behaviour of agents in economic change. The fundamental goal of portfolio theory is to optimally allocation our investment between different assets, are assumed to strike a balance between maximizing the return and minimizing the risk of investment. Return is quantified as mean and risk as the variance of portfolio of securities. In particular, we shall use the methods of constrained optimization to construct portfolios for a given collection of assets with desirable features as quantified by an appropriate utility function and constraints. To tackle the situation associated with vaugness and uncertainty, fuzzy optimization approach may be taken up. AMS Classification number (2010) – 90A09, 91B29, 91G10. Keywords: portfolio, risk, return, mean variance optimization, convex quadratic programming, fuzzy optimization.
1. Introduction We observe that modern finance has become increasingly technical requiring the use of sophisticated mathematical tools in both research and practice. Many computational finance problems of decision making in investment ranging from asset allocation to risk management, portfolio selection, option pricing etc. can be solved by optimization techniques. Dynamic programming technique is used for option
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pricing, stochastic programming method is used in asset/liability management. Robust optimization models are used for portfolio selection and option pricing. We find roots of this trend in the portfolio selection models and methods described by H. Markowitz [4] in 1950s and the option pricing formulas developed by F. Black and M. Scholes [1]; R.C. Merton [11] in late 1060’s and early 1970’s. In modern finance management Markowitz theory of portfolio management combines probability and optimization theories to model the behaviour of agents in economic change. The agents are assumed to strike a balance between maximizing the return and minimizing the risk of investment. Return is quantified as mean and risk as the variance of portfolio of securities. And, perhaps, for the enormous effect these works produced on modern financial practices, H. Markowitz [4] was awarded the Nobel Prize in Economics in 1990, while M. Scholes and R. Merton won the noble prize in Economics in 1997. The fundamental lesson of the Markowitz analysis is to show that investor must take care not only of the realized returns but also of the “risk” of their position, represented by the standard deviation of their portfolio return. So Markowitz proposes to measure the risk of return R by its standard deviation and to determine the minimal S.D. for any fixed expected return. The dual criterion can also be considered i.e. maximize the expected return for any fixed standard deviation optimization tools to be applied to studies of portfolio management. The twin objective of investors-profit maximization and risk minimization are thus quantified so as to maximize the expected value and to minimize the variance of the portfolio value has been seen that portfolio analysis problem in the mean-variance model can be solved in polynomial time. Sharpe has shown the estimation from O N 2 to O N . Most of the existing portfolio selection models are based on probability theory, which only partly capture the reality. Recently, Ramaswamy [12], Tamaka and Guo [5], Inuiguchi and Ramik [10] studied fuzzy portfolio selection to handle uncertainty, vauguess and impression in financial markets. Robust programming techniques also deal to certain extent these types of problems.
2. Nomenclature Q
A C
x xi
n n symmetric covariance matrix n n symmetric covariance matrix m n matrix p n matrix non empty polyhedral set flexible porfolio total funds invest
i
expected return or mean
i
standard deviation
Si
return of asset
Portfolio Optimization Theory In Computational Finance return of asset
Sj
correlation coefficient
i, j
E max
Vmin R
xR
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maximum return or mean minimum variance target value optimal solution
3. Preliminaries Policy of optimal selection of portfolios by Harry Markowitz, gives a mathematical model and solution method. This method is known as Mean Variance Optimization (MVO) and it provides a mechanisim for the selection of portfolios of securities (or asset classes) in a manner that trades off expected returns & risk. Portfolios are considered mean-variance efficient if they minimize the variance for a given mean return or if they maximize the expected mean return for a given variance. Meanvariance efficiency rests on firm theoretical grounds if either investors exhibit quadratic utility, in which case they ignore non normality in the data, or returns are multivariate normal, in which case the specific utility function is irrelevant, as the set of efficient portfolios is the same for all investors. Consider an investor who has certain amount of money to be invested in a number of different asserts/securities/stocks/bonds etc. named as s1 , s2 ,..., sn (n 2) with random returns. Let i and i denote the expected return and standard deviation respectively of the returns of asset s i , si , s j for i
j,
1
i, j
,
2
denote the correlation coefficient of the returns of assets
,...,
T n
, and Q
i, j
be the n n symmetric covariance
2 matrix with i , j j , and i , j j , xi be the proportion of the i if i i , j i j if i total funds invested in security i . So the expected return and the variance of the resulting portfolio x x1 , x2 ,..., xn
is given by E( x)
x
1 1
........
x
n n
T
x and Var x
i, j
i
j
xi x j
xT Q x ,
i, j
when
ii T
1.
So x Qx 0 for any x (as variance is non-negative) and Q is positive semi definite. Taking, no redundant assets in this collection s1 , s2 ,..., sn , we take Q is positive definite. We assume that the set of admissible portfolios is a nonempty polyhedral set and represent it as x : Ax b, Cx d
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where A is a m n matrix, b is a m -dimension vector, C is the p n matrix, d is a n
p -dimension vector. And also, one of the constraints in the set is
xi
1.
i 1
Remarks 1. Here the linear portfolio constraints such as short-sale restrictions or limits on asset/vector allocation etc. are not considered in the definition of for the polyhedral feasible set. 2. A feasible portfolio x is called efficient if it has the maximal expected return among all portfolio with the same variance or alternatively if it has the minimum variance among all portfolios that have at least a certain expected return. The collection of efficient portfolios forms the efficient frontier of the portfolios universe. The efficient frontier is often represented as a curve in a two-dimension graph, where the co-ordinates of a plotted point correspond to standard deviation and the expected return of an efficient portfolio. 3. When we assume Q is positive definite, the variance is strictly convex function of the portfolio variables and there exists a unique portfolio in that has the minimum variance. We denote this portfolio with xmin and its return T
xmin with Vmin . So, we note that xmin is an efficient portfolio. We let Emax denote the max return for an admissible portfolio.
The region of portfolios in E-V plane The minimum variance return Vmin and the maximum variance return Emax . Figure shows the region of portfolios in the plane of the axes of expected return and variance. The optimal solution which the decision makers satisfy can be obtained on the curve Vmin Emax in the region of portfolios.
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4. Mathematical Formulation MVO Problem formulation - Markowitz; Mean-Variance optimization problem can be formulated in three different but equivalent ways. One of the form is: Find the minimum variance portfolio of the securities 1 to n that yields at least a target value of excepted return (say b ). So a mathematical formulation gives a quadratic programming problem with the restriction that the expected return rate is made sure to be more than some amount.
1 T x Qx 2 subject to T x R min x
eT x 1, x 0, e 1, 1, ...,1 , Ax b, Cx d
(1)
Here the first constraint indicates that the expected return is no less than the target value R . Solving this problem for values of R ranging between Vmin and Emax , one obtains all efficient portfolios. Here the objective function corresponds to one half the total variance of the portfolio. The constant half is added for convenience in the optimality conditions and it obviously does not affect the optimal solution. This is a convex quadratic programming problem for which the first order conditions are both necessary and sufficient for optimality and are given below. So, if x R is an optimal solution of this problem iff there exists R R , E R m , I
R p satisfying the KKT conditions: xR T
xR
R
0,
I
0,
AT
R
R, Ax R T T I
E
CT
b, Cx R
I
0
d,
xR
R
0,
C xR
d
0.
(2)
In these formulations, it is assumed that almost all of decision makers have aversion to risk even if its return may be obtained less. With utility point of view, the behaviors of investors have propensity to risk which may be represented by a parabola shape graph in the plane of horizontal axis of risk and the vertical axis of expected return rate. The graph shows the region of admissible port folios in the plane of axes of expected return rate and risk. Of course investors have different utility functions and it is difficult to identify one. Broadly speaking there should be a convex utility function. The other two variations of MVO problems are:
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P.K. Parida and S.K. Sahoo T
max x
x
T
subject to x Q x Ax b Cx d
max x
T
x
and subject to Ax Cx d
2 b
2
(3)
xT Q x (4)
In “Eq. (3)”, 2 is a given upper limit on the variance of the portfolio. In “Eq. (4)”, the objective function is a risk adjusted return function where the constant serves as a risk aversion constant. “Eq. (4)” is a quadratic programming problem, “Eq. (3)” is a convex quadratic programming problem and is not a quadratic programming problem, so it can be solved using general non linear programming solution techniques. Also “Eq. (3)” can be reformulated as a second order cone programming. So, by this, opens the possibility of using specialized and efficient second order cone programming methods for it solutions. A flexible portfolio x is called efficient if it has maximal expected return among all portfolios with the same variance or alternatively if it has the minimum variance among all portfolios that have at least certain expected return. The collections of efficient portfolio form the efficient frontier of the portfolio universe.
5. Conclusion Usually investments are supposed to be influenced by local disturbances of social and economical situations and so the problems are ill structured. A satisfaction approach is preferred than usual max/min approach of optimization techniques. Sometimes aspiration level of decision maker is considered to solve a problem in context of satisfaction strategy which is vauge type. A variety of different asset characteristics can be taken into consideration, such as the amount of value, on average, an asset returns on over a period of time and the riskiness of reaping returns comparable to the average. The financial objectives of the investor and tolerance of risk determine what types of portfolios are to be considered desirable. Here we discussed some general approach of Portfolio Optimization. Later on discussed Mean Variance Optimization (MVO) problem formulated in three different but equivalent ways.
Future Research Uncertainties about future events make the behaviour of economic indicators unpredictable and at times bring about turbulence to financial markets. Resource
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allocation methods derived from modern mathematical model play an influential role in work practices of financial institutions and have become a significant tool used in financial markets. Sometimes aspiration level of decision maker is considered to solve a problem in context of satisfaction strategy which is vauge type. So a Fuzzy approach is a more suitable to tackle such situation with vauge aspiration level of investor.
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P.K. Parida and S.K. Sahoo [12] Ramaswamy, S., “Portfolio selection using Fuzzy decision theory”, BIS working paper, no. 59, 1998. [13] Wang, S.Y., and Xia, Y.S., “Portfolio selection and Asset pricing”, Springer, 2002. [14] Chopra, V., and Ziemba, W., The effect of errors in means, variances, and co-variances on optimal portfolio choice. Journal of Portfolio Management, 19, 6-11, 1993.
