Project Report on
STUDY OF PORTFOLIO OPTIMZATION MODELS
Project Supervisor: Dr V.N.Sastry Associate Professor, IDRBT
Submitted by:
Chander Prakash Jumrani 2007MT50436 3rd Year Undergraduate Student IIT Delhi
JULY 2010
Institute for Development and Research in Banking Technology
Castle Hills, Masab Tank Hyderabad – 500 057 Phone: 90-40-23534981 (8 Lines); Fax: 90-40-23535157 Web: http://www.idrbt.ac.in
CERTIFICATE
This is to certify that the Summer Internship project work entitled “Study of Portfolio Optimization Models”, submitted for partial fulfilment of the requirements for the award of the Integrated Degree of Master of Technology in Mathematics and Computing to the Indian Institute of Technology Delhi is a record of bonafide project work carried out by Mr. Chander Prakash Jumrani (Entry No. 2007MT50436) of third year at IDRBT (Institute for Development and Research in Banking Technology), Hyderabad, for a period of 10 weeks under my guidance during May-July 2010.
The subject matter embodied in the project report has not been submitted for the award of any other degree or diploma.
V.N.Sastry Associate Professor, IDRBT, Castle Hills, Masab Tank, Hyderabad – 500 057. e-mail: [email protected]
Abstract:
Portfolio Optimization is the technique of adjusting the ratio of securities in the portfolio to meet ones needs in the best possible way. The objective of the project is to study the various single objective and multi-objective based portfolio optimization models which includes:
• Markowitz Mean Variance Model • Capital Asset Pricing Model • Goal Programming based Model
These models were tested on a basic two stock and three stock portfolios and solved using quadratic and goal programming techniques. The results were analysed and inference was made.
The study also included the use of tools such as Solver in MS EXCEL and the Optimization Toolbox in MATLAB to solve these optimization problems.
CONTENTS OF THE REPORT:
1. Introduction
2. Chapter 1 : Single objective based portfolio optimization models.
3. Chapter 2: Multi-objective based portfolio optimization models.
4. References INTRODUCTION
Optimization: It refers to choosing the best element from some set of available alternatives. The decisions are based on certain conditions that must be met.
Portfolio : It is a set of investments held by an individual or an organization. It includes risky (stocks, options and other derivative securities) as well as non -risky (bonds) assets.
Optimization of Portfolio : It is technique of changing the ratio and quantity of assets present in a portfolio in order to meet certain objective in the most optimal way. Objectives may include • Risk Minimization • Return Maximization • Other objective includes liquidity, market dynamics etc.
These objectives and constraints may be linear or non-linear; accordingly we may get LPP or Non-LPP. We have considered Quadratic Programming and Non-linear Multi-objective portfolio optimization problems.
Chapter 1: Single objective portfolio optimization models
1.1 The Markowitz Mean Variance Model
Markowitz Mean Variance Model also known as the Modern portfolio theory (MPT) is a theory of investment which tries to maximize portfolio expected return for a given amount of portfolio risk, or equivalently minimize risk for a given level of expected return, by carefully choosing the proportions of various assets. Markowitz portfolio shows that as we add assets to an investment portfolio the total risk of that portfolio - as measured by the variance (or standard deviation) of total return - declines continuously, but the expected return of the portfolio is a weighted average of the expected returns of the individual assets. In other words, by investing in portfolios rather than in individual assets, investors could lower the total risk of investing without sacrificing return. The fundamental concept behind MPT is that the assets in an investment portfolio cannot be selected individually, each on their own merits. Rather, it is important to consider how each asset changes in price relative to how every other asset in the portfolio changes in price. Investing is a tradeoff between risk and expected return. In general, assets with higher expected returns are riskier. For a given amount of risk, MPT describes how to select a portfolio with the highest possible expected return. Or, for a given expected return, MPT explains how to select a portfolio with the lowest possible risk MPT is therefore a form of diversification. Under certain assumptions and for specific quantitative definitions of risk and return, MPT explains how to find the best possible diversification strategy. The Markowitz model is based on several assumptions concerning the behaviour of investors and financial markets:
1. A probability distribution of possible returns over some holding period can be estimated by investors (here it is assumed to be normally distributed.).
2. Variability about the possible values of return is used by investors to measure risk.
3. Investors care only about the means and variance of the returns of their portfolios over a particular period.
4. Expected return and risk as used by investors are measured by the first two moments of the probability distribution of returns-expected value and variance.
5. The correlation factors are assumed to be fixed/ constant. MEASURMENT OF RETURN AND RISK
Throughout this chapter, investors are assumed to measure the level of return by computing the expected value of the distribution, using the probability distribution of expected returns for a portfolio. Risk is assumed to be measurable by the variability around the expected value of the probability distribution of returns. The most accepted measures of this variability are the variance and standard deviation.
Return
Given any set of risky assets and a set of weights that describe how the portfolio investment is split, the general formula of expected return for n assets is:
n
Er(P ) = ∑ wEr i() i (1) i=1 Where:
n
∑ wi = 1.0; i=1 n = the number of securities;
wi = the proportion of the funds invested in security i;
th ri, r P = the return on i security and portfolio p; and
E ( ) = the expectation of the variable in the parentheses.
The return computation is nothing more than finding the weighted average return of the securities included in the portfolio.
Risk
The variance of a single security is the expected value of the sum of the squared deviations from the mean, and the standard deviation is the square root of the variance. The variance of a portfolio combination of securities is equal to the weighted average covariance of the returns on its individual securities:
n n 2 Var()()rp=σ p = ∑∑ ww ij Cov, rr ij (2) i=1 j = 1 Covariance can also be expressed in terms of the correlation coefficient as follows:
Cov(rij , r ) =ρσσ ijij = σ ij (3)
where ρij = correlation coefficient between the rates of return on security i, ri , and the rates of return on security j, rj , and σ i , and σ j represent standard deviations of ri and rj respectively. Therefore:
n n
Var ()rp= ∑∑ w ijijij w ρ σ σ (4) i=1 j = 1
Overall, the estimate of the mean return for each security is its average value in the sample period; the estimate of variance is the average value of the squared deviations around the sample average; the estimate of the covariance is the average value of the cross-product of deviations.
1.1.2 TWO RISKY ASSET PORTFOLIOS
THEORY:
Two risky asset portfolio means that we are limiting our portfolio to just two stocks say
A(µ 1, σ1) and B(µ 2, σ2).Let the mean and variance of the portfolio be (µ, σ) respectively.
Let us take w 1 units of A and w 2 units of B such that w 1+ w 2 = 1.
The expected rate of return on the portfolio is
Er()P= wEr AA () + wEr BB () (5)
The variance of the rate of return on the two-asset portfolio is
2 22222 σP=+(ww AA σσ BB ) = w AA σ + w BB σ + 2 ww ABABAB ρσσ (6)
(A) The Correlation Factor ( ƥ)
We have the Correlation factor ( ƥ) between A and B defined as
ƥ = (Covariance A,B )/( σA. σB)
It is calculated using the past data and determines the relationship between the two stocks and its expected behaviors w.r.t the fluctuations in the market. We have • -1< ƥ < 1
• ƥ=0 => uncorrelated
• ƥ=1 => perfect positively correlated
• ƥ=-1 => perfectly negatively correlated
• ƥ< 0 => the stocks are negatively correlated i.e. the increase in the value of one causes a decrease in the value of the other.
• ƥ>0 => the stocks are positively correlated i.e. the increase in the value of one causes a increase in the value of the other.
Taking w 2=s and w 1=1- s and substituting in the equations of µ and σ, we get
µ = µ 1(1-s) + µ2s …….(A)
2 2 2 2 σ = { σ1 (1-s) + σ2 s + 2 ƥ σ1σ2s(1-s)} ^ 0.5 …….(B)
Using the above relation and taking s as the parameter a graph is plotted showing the relation between return and risk for two stocks and for different values of ƥ.
ρ = − 1 asset B ρ = 0 ρ =1
0<ρ < 1 Expected Return asset A ρ = − 1
0 Standard Deviation
Figure.1 Investment opportunity sets for asset A and asset B with various correlation coefficients
INFERENCE FROM THE GRAPH
1. In Figure.1 the opportunity set with perfect positive correlation – is a straight line through the component assets. No portfolio can be discarded as inefficient in this case, and the choice among portfolios depends only on risk preference. Diversification in the case of perfect positive correlation is not effective.
2. With any correlation coefficient less than 1.0( ρ <1), there will be a diversification effect, the portfolio standard deviation is less than the weighted average of the standard deviations of the component securities. Therefore, there are benefits to diversification whenever asset returns are less than perfectly correlated.
3. Where negative correlation is present, there will be even greater diversification benefits.
4. With perfect negative correlation, the benefits from diversification stretch to the limit. Graph points to the proportions that will reduce the portfolio standard deviation all the way to zero.
5. An investor can reduce portfolio risk simply by holding instruments which are not perfectly correlated. In other words, investors can reduce their exposure to individual asset risk by holding a diversified portfolio of assets. Diversification will allow for the same portfolio return with reduced risk.
The concept of Markowitz efficient frontier
Every possible asset combination can be plotted in risk-return space, and the collection of all such possible portfolios defines a region in this space. The line along the upper edge of this region is known as the efficient frontier. Combinations along this line represent portfolios (explicitly excluding the risk-free alternative) for which there is lowest risk for a given level of return. Conversely, for a given amount of risk, the portfolio lying on the efficient frontier represents the combination offering the best possible return. Mathematically the efficient frontier is the intersection of the set of portfolios with minimum variance and the set of portfolios with maximum return.
Figure2 shows investors the entire investment opportunity set, which is the set of all attainable combinations of risk and return offered by portfolios formed by asset A and asset B in differing proportions. The curve passing through A and B shows the risk-return combinations of all the portfolios that can be formed by combining those two assets. Investors desire portfolios that lie to the northwest in Figure X.2. These are portfolios with high expected returns (toward the north of the figure) and low volatility (to the west).
Figure2 Investment opportunity set for asset A and asset B
Calculating the Minimum variance portfolio
In Markowitz portfolio model, we assume investors choose portfolios based on both expected return, E( r p ) , and the standard deviation of return as a measure of its risk, σ p . So, the portfolio selection problem can be expressed as maximizing the return with respect to the risk of the investment (or, alternatively, minimizing the risk with respect to a given return, hold the return constant and solve for the weighting factors that minimize the variance).
Mathematically, the portfolio selection problem can be formulated as quadratic program.
For two risky assets A and B, the portfolio consists of wA, w B , the return of the portfolio is then, The weights should be chosen so that (for example) the risk is minimized, that is
2 22 22 Min σP=w AA σ + w BB σ + 2 ww ABABAB ρσσ w A
For each chosen return and subject to wwAB+=1, w A ≥ 0, w B ≥ 0 . The last two constraints simply imply that the assets cannot be in short positions.
Table 1 The mimimum variance portfolio weight of two-assets portfolio without short selling The correlation of two assets Weight of Asset A Weight of Asset B
σ B σA− 2 σ B ρ= 1 wA = wB = σA− σ B σA− σ B
σ B σ A ρ= -1 wA = wB = σA+ σ B σA+ σ B
σ 2 σ2− 2 σ 2 ρ= 0 w = B w = A B A σ2+ σ 2 B σ2+ σ 2 A B A B Above, we simply use two-risky-assets portfolio to calculate the minimum variance portfolio weights. If we generalization to portfolios containing N assets, the minimum portfolio weights can then be obtained by minimizing the Lagrange function C for portfolio variance.
SOLVING A REAL LIFE TWO STOCK EXAMPLE
AIM: To find the optimal portfolio for two given stocks using the data for the last 10 days.
The market data for Aptech Ltd and TCS for the 10 days period is considered.
(Reference: http://money.rediff.com/companies/aptech-ltd/13020130/bse/month http://money.rediff.com/companies/tata-consultancy-services-ltd/13020033/bse/month)
The data is shown in the table below.
TCS Aptech Ltd Day 1 10th May 769.7 146.65 Day 2 11th May 755.2 144.15 Day 3 12th May 758.4 142.45 Day 4 13th May 765.75 144.05 Day 5 14th May 768.45 141.25 Day 6 17th May 744.85 134.6 Day 7 18th May 737.85 135.35 Day 8 19th May 721.15 131.85 Day 9 20th May 730.1 128.95 Day 10 21st May 719.2 124.4
Mean( µ) 747.065 137.37 Using the above data graphs have been plotted as shown below:
TCS APTECH LTD
Calculation of the DRIFT( Mean µ) ,VOLATALITY (Variance σ^2) and STANDARD DEVIATION ( σ)
We have the following formula for calculating mean and variance. 2 2 Mean µ=E(x i) and σ =E[(x i- µ) ]
Using the above data mean and variance are calculated as shown in the tables below.
TCS µ=747.065 i X(i) X(i)- µ [X(i)- µ]^2
1 769.7 22.635 512.3432 2 755.2 8.135 66.17822 3 758.4 11.335 128.4822 4 765.75 18.685 349.1292 5 768.45 21.385 457.3182 6 744.85 -2.215 4.906225 7 737.85 -9.215 84.91623 8 721.15 -25.915 671.5872 9 730.1 -16.965 287.8112 10 719.2 -27.865 776.4582
Mean( µ) 747.065 Variance(σ^2) 333.913 standard Deviation(σ) 18.27329
From the above table we have µ1 =747.065 σ1=18.27329
Similarly the mean and variance are calculated for Aptech as shown in the table below:
APTECH LTD
µ=137.37 i X(i) X(i)- µ [X(i)- µ]^2
1 146.65 9.28 86.1184 2 144.15 6.78 45.9684 3 142.45 5.08 25.8064 4 144.05 6.68 44.6224 5 141.25 3.88 15.0544 6 134.6 -2.77 7.6729 7 135.35 -2.02 4.0804 8 131.85 -5.52 30.4704 9 128.95 -8.42 70.8964 10 124.4 -12.97 168.2209
Mean( µ) 137.37 Variance(σ2) 49.8911 standard Deviation(σ) 7.063363 Again, from the table we have µ2 =49.8911 σ2=7.063363
Calculation of the COVARIANCE and CORRELATION FACTOR between the two stocks
The covariance between the stocks is calculated in the table shown below: i X(i)- µ1 Y(i)-µ2 [X(i)-µ1]*[Y(i)-µ2]
1 22.635 9.28 210.0528 2 8.135 6.78 55.1553 3 11.335 5.08 57.5818 4 18.685 6.68 124.8158 5 21.385 3.88 82.9738 6 -2.215 -2.77 6.13555 7 -9.215 -2.02 18.6143 8 -25.915 -5.52 143.0508 9 -16.965 -8.42 142.8453 10 -27.865 -12.97 361.40905
Covariance 120.26345 Correlation factor( ƥ) 0.93176 From the table we have Covariance(X,Y) =120.26345
Using the formula of correlation ƥ=Cov(x,y)/ σx σy
Correlation factor( ƥ)= 0.93176
Let w2=s and w1= (1-s)
σ2 = σ12* (1-s) 2+ σ22* s2+2 * ƥ * σ1 * σ2*s*(1-s) dσ2/ds=0
On Solving we get s = ( σ12- ƥ σ1 σ2)/( σ12+ σ22-2p σ1 σ2)
Short selling is the practice by which we can borrow the stocks and sell them in the market. Thus in a way we hold a negative share of the stock.’
In certain markets the practice of short selling is prohibited .
CASE 1: Allowing short selling
Substituting we get s = 1.491 w1= - 0.491 w2= +1.491 µ = -292.421 σ 2=+15.327
CASE 2: Not allowing short selling
Minimum risk when s=1
Thus using this model we obtain the following result. w1=0 w2=1 µ=49.8911 σ 2=333.91312
1.1.3 THREE RISKY ASSET PORTFOLIOS
Stock A(µ 1, σ1,w 1) and B(µ 2, σ2, w2) Stock C(µ 3, σ3, w3)
Portfolio (µ, σ) and w 1+w2+w3 = 1
We obtain the variance covariance matrix C3x3 whose diagonal elements are the variances and the non-diagonal element (i , j) is the covariance between stock i and stock j. Thus we have the equation
• µ= ∑ µ iwi
• σ2= w T C w
It results in a quadratic programming problem min w T C w s.to ∑wi=1
TAKING A REAL LIFE 3 STOCK EXAMPLE
The market data for Aptech Ltd and TCS and Indian Oil for the past 10 days.
(Reference: http://money.rediff.com/companies/aptech-ltd/13020130/bse/month http://money.rediff.com/companies/tata-consultancy-services-ltd/13020033/bse/month) TCS Aptech Ltd Indian Oil Ltd Day 1 10th May 769.7 146.65 302.05 Day 2 11th May 755.2 144.15 300.65 Day 3 12th May 758.4 142.45 301.15 Day 4 13th May 765.75 144.05 301.9 Day 5 14th May 768.45 141.25 306.5 Day 6 17th May 744.85 134.6 317.65 Day 7 18th May 737.85 135.35 316.45 Day 8 19th May 721.15 131.85 312.75 Day 9 20th May 730.1 128.95 325.25 Day 10 21st May 719.2 124.4 330.25
Mean( µ) 747.065 137.37 311.46
The graphs of TCS and Aptech have already been in the two stock problems. The graph of Indian Oil is shown below
INDIAN OIL
Plotting on a single graph
Calculation of the DRIFT( Mean µ) ,VOLATALITY (Variance σ^2) and STANDARD DEVIATION ( σ)
TCS As calculated in the two stock model we have µ1=747.065 σ1=18.27329
APTECH LTD As calculated in the two stock model we have µ2=49.8911 σ2=7.063363
INDIAN OIL
The calculation of mean and variance are shown in the table below:
µ=137.37 i X(i) X(i)- µ [X(i)- µ]^2
1 302.05 -9.41 88.5481 2 300.65 -10.81 116.8561 3 301.15 -10.31 106.2961 4 301.9 -9.56 91.3936 5 306.5 -4.96 24.6016 6 317.65 6.19 38.3161 7 316.45 4.99 24.9001 8 312.75 1.29 1.6641 9 325.25 13.79 190.1641 10 330.25 18.79 353.0641
Mean( µ) 311.46 Variance(σ^2) 103.5804 standard Deviation(σ) 10.17745
Thus mean and variance for Indian Oil comes out to be
µ3=747.065 σ3=18.27329
Calculation of the COVARIANCE between the stocks
We have the formula of covariance as
Cov(x,y)=E[(x i- µ1)(y i- µ2)]
Using the formula the covariance has been calculated below:
X(i) X(i)- µ1 Y(i) Y(i)-µ2 Z(i) Z(i)-µ3 [X(i)-µ1]*[Y(i)-µ2][X(i)-µ1]*[Z(i)-µ3][Y(i)-µ2]*[Z(i)-µ3]
769.7 22.635 146.65 9.28 302.05 -9.41 210.0528 -212.995 -87.3248 755.2 8.135 144.15 6.78 300.65 -10.81 55.1553 -87.9394 -73.2918 758.4 11.335 142.45 5.08 301.15 -10.31 57.5818 -116.864 -52.3748 765.75 18.685 144.05 6.68 301.9 -9.56 124.8158 -178.629 -63.8608 768.45 21.385 141.25 3.88 306.5 -4.96 82.9738 -106.07 -19.2448 744.85 -2.215 134.6 -2.77 317.65 6.19 6.13555 -13.7109 -17.1463 737.85 -9.215 135.35 -2.02 316.45 4.99 18.6143 -45.9829 -10.0798 721.15 -25.915 131.85 -5.52 312.75 1.29 143.0508 -33.4304 -7.1208 730.1 -16.965 128.95 -8.42 325.25 13.79 142.8453 -233.947 -116.112 719.2 -27.865 124.4 -12.97 330.25 18.79 361.4091 -523.583 -243.706
Covariance 120.2635 -155.315 -69.0262 Correlation factor( ƥ) 0.93176 -0.83513 -0.9602
THE VARIANCE COVARIANCE MATRIX
From the above table we obtain the following variance covariance matrix:
333.913 120.2635 -155.315
120.2635 49.8911 -69.0262
-155.315 -69.0262 103.5804
The risk of the portfolio σ is σ = σ12 w1 2 + σ22 w2 2 + σ32 w3 2 +2w1w2 σ12 +2w3w2 σ32+2w1w3 σ13
This could be written as w tCw Case 1: Thus the problem of three stock portfolio optimization becomes a quadratic programming problem as min w tCw s.t w1+w2+w3 = 1
Case2: If we desire a fixed expected return say µ we have an additional constraint and the the programming problem becomes min w tCw s.t w1+w2+w3 = 1
µ= ∑ µ iwi
This quadratic programming problem could be solved using the methods for quadratic optimization such as the Lagrange’s method or the wolves or Bill’s method or by using the Optimization toolbox in MATLAB. We use the quadprog function to solve the given QPP.
QUADRATIC PROGRAMMING USING MATLAB
Equation
Finds a minimum for a problem specified by
H, A, and Aeq are matrices, and f, b, beq , lb , ub , and x are vectors.
Syntax x=quadprog(H,f,A,b,Aeq,beq,lb,ub) defines a set of lower and upper bounds on the design variables, x, so that the solution is in the range lb ≤ x ≤ ub. If no equalities exist, set Aeq = [ ] and beq = [ ].
Solving Case1: Matlab code: clc; clear all; C=[333.913 120.2635 -155.315;120.2635 49.8911 -69.0262;-155.315 -69.0262 103.5804] A1=[1 1 1]; b1=[1]; x=quadprog(C,[ ],[ ],[ ],A1,b1,[0 0 0],[1,1,1]); x
Output:
Thus we have w1=0 w2=0.5921 w3=0.4079
Risk σ = w tCw
Thus min risk is 1.3828. And expected return µ=w *m where m=[ µ1 µ2 µ3]
Thus µ = 156.5893
Conclusion: Thus we have using the above ratio of stock (no short selling) the expected return as 156.5893 with a min risk of 1.3828.
Solving Case 2: Assuming that we desire a fixed µ= 500 .It is solved using MATLAB and the solution comes out to be w 1= 0.4328,w 2=0,w 3=0.5672 and the minimum risk comes out to be σ =4.429 and µ= 500.
1.1.3 GENERAL N RISKY ASSET PORTFOLIOS
If we generalization to portfolios containing N assets, the minimum portfolio weights can then be obtained by minimizing the Lagrange function C for portfolio variance.
n n 2 Min σp= ∑∑ w ijijij w ρσσ i=1 j = 1
Subject to w1+ w 2 ++... w N = 1 n n n C=∑∑ wwijCov() rr ij +λ1 1 − ∑ W i i=1 j = 1 i = 1
in which λ1 are the Lagrange multipliers, respectively, ρij is the correlation coefficient between ri and rj , and other variables are as previously defined.
By using this approach the minimum variance can be computed for any given level of expected portfolio return (subject to the other constraint that the weights sum to one).
Calculating the weights of optimal risky portfolio
One of the goals of portfolio analysis is minimizing the risk or variance of the portfolio. Previous section introduce the calculation of minimum variance portfolio, we minimum the variance of portfolio subject to the portfolio weights’ summing to one. If we add a condition into the equation we can get the optimal risky portfolio.
n n 2 Min σp= ∑∑ WW ijijij ρσσ i=1 j = 1
n * * Subject to ∑WERi() i = E , where E is the target expected return and i=1
n
∑Wi =1.0 i=1
The first constraint simply says that the expected return on the portfolio should equal the target return determined by the portfolio manager. The second constraint says that the weights of the securities invested in the portfolio must sum to one.
The Lagrangian objective function can be written:
n n n n * C=∑∑ wwijijCov() rr +−λ1 E ∑ wEr ii() +− λ 2 1 ∑ w i i=1 j = 1 i=1 i = 1
Taking the partial derivatives of this equation with respect to each of the variables, w1, w 2 ,...., w N , λ1 , λ 2 and setting the resulting equations equal to zero yields the minimization of risk subject to the Lagrangian constraints. Then, we can solve the weights and these weights are represented optimal risky portfolio by using of matrix algebra. n
If there no short selling constraint in the portfolio analysis, second constraint, ∑ wi =1.0 , i=1
n should substitute to ∑ wi =1.0 , where the absolute value of the weights wi allows for a i=1 given wi to be negative (sold short) but maintains the requirement that all funds are invested or their sum equals one.
The Lagrangian function is
n n n n * C=∑∑ wwijijCov() rr +−λ1 E ∑ wEr ii() +− λ 2 1 ∑ w i i=1 j = 1 i=1 i = 1
If the restriction of no short selling is in minimization variance problem, it needs to add a third constraint:
K wi ≥0, i = 1, , N
The addition of this non-negativity constraint precludes negative values for the weights (that is, no short selling). The problem now is a quadratic programming problem similar to the ones solved so far, except that the optimal portfolio may fall in an unfeasible region. In this circumstance the next best optimal portfolio is elected that meets all of the constraints.
Using the above method we get an equations with n+1 variables which has been solved and we get the results
-1 T -1 wopt = (C e)/(e C e) and
2 T σ opt = w opt C w
1.1.3 GENERAL MARKET PORTFOLIOS
General market portfolios consist of risky as well as non-risky assets (also called risk-free asset). The introduction of a risk free security to the Markowitz model changes the efficient frontier from a curved line to a straight line called the Capital Market Line (CML). This CML represents the allocation of capital between risk free securities and risky securities for all investors combined.
The optimal portfolio for an investor is the point where the new CML is tangent to the old efficient frontier when only risky securities were graphed. This optimal portfolio is normally known as the market portfolio.
The points on the Capital market Line(CML) represents the optimal portfolio points. The points of the left are low risk and low return points and the points to the right are high risk and high return points. To choose between these points we use the concept of indifference curves.
INDIFFERENCE CURVES
One of the factors to consider when selecting the optimal portfolio for a particular investor is degree of risk aversion, investor’s willingness to trade off risk against expected return. This level of aversion to risk can be characterized by defining the investor’s indifference curve, consisting of the family of risk/return pairs defining the trade-off between the expected return and the risk. It establishes the increment in return that a particular investor will require in order to make an increment in risk worthwhile. The optimal portfolio along the efficient frontier is not unique with this model and depends upon the risk/return trade off utility function of each investor.
• All portfolios that lie on the same indifference curve are equally desirable to the investor (even though they have different expected returns and variance.).
• An investor will find any portfolio that is lying on an indifference curve that is "further northwest" to be more desirable than any portfolio lying on an indifference curve that is "not as far northwest."
• Thus using these indifference curves of different investors we could find the optimal portfolio as the point that lies on the capital market line and is tangent to the indifference curve
1.2 The Capital Asset Pricing Model
The asset return depends on the amount paid for the asset today. The price paid must ensure that the market portfolio's risk / return characteristics improve when the asset is added to it. The CAPM is a model which derives the theoretical required expected return (i.e., discount rate) for an asset in a market, given the risk-free rate available to investors and the risk of the market as a whole. The CAPM is usually expressed:
µk = µrf + βk * (µ m-µrf ) …….(c)
Equation (c) is also known as the security market line.
β, Beta, is the measure of asset sensitivity to a movement in the overall market. Beta is usually found via regression on historical data:
βk=Cov(r k,r m)/Var(r m) …….(d) where r k and r m are rate of return of the kth asset and the market respectively
By calculating beta we eliminate the need to calculate individual correlation coefficients between the new asset and each of the existing stocks in the market.
With the help of the security market line we analyze whether the addition of the certain stock to the portfolio will increase the expected return or will this reduces the total risk .Thus CAPM helps us to predict whether addition of a certain risky asset to the portfolio is beneficial or not.
MARKOWITZ MPT SHORTCOMINGS
• Markowitz Model has certain assumption which is not applicable to the practical world. Some of these assumptions are
• Asset returns are normally distributed.
• Correlations are assumed to be fixed/ constant.
• The Markowitz Model considers only single objective i.e. either risk minimization or profit maximization whereas in real world we need to satisfy both contradicting objectives simultaneously with certain priority levels.
• Also the market securities rates follow a stochastic behavior which cannot be captured by the deterministic Markowitz mean variance model.
Chapter 2: Multiple objective portfolio optimization
Multi-objective programming is the process of simultaneously optimizing two or more conflicting objectives subject to certain constraints. If a multi-objective problem is well formed, there should not be a single solution that simultaneously minimizes each objective to its fullest. In each case we are looking for a solution for which each objective has been optimized to the extent that if we try to optimize it any further, then the other objective(s) will suffer as a result. Finding such a solution, and quantifying how much better this solution is compared to other such solutions (there will generally be many) is the goal when setting up and solving a multi-objective optimization problem.
In such cases we come across solutions which may be optimal in one of the objectives but not necessarily the optimal solutions. For example in a maximization problem (two objectives case) we come across solutions as (2,3) and (3,2) which are optimal in one of the objectives .We cannot say which one is better solutions or there may exist another solution which is better than these two .These solutions which are optimal is some of the objectives are called pareto-optimal solutions . But our objective is to find the global optimal solution. Thus to solve such optimization problems we use techniques such as weighted objectives , goal programming, stochastic programming etc. In all the methods to solve the multi-objective optimization problem we first convert it into a single objective programming problem.
2.1 Multi-objective Programming in Portfolio Optimization (using weighted approach)
Multi-objective optimization, developed by French-Italian economist V. Pareto, is an alternative approach to the portfolio optimization problem. The multi-objective approach v v v K combines multiple objectives fxfx1( ), 2 ( ) , , fxn ( ) into one objective function by assigning a weighting coefficient to each objective. The standard solution technique is to minimize a positively weighted convex sum of the objectives using single-objective method, that is,
vn v K MinimizeFx ()()=∑ afxi i , a i > 0, i = 1,2, , n i=1
The concept of optimality in multi-objective optimization is characterized by Pareto v v optimality. Essentially, a vector x* is said to be Pareto optimal if and only if there is no x such v v v * K * that fxi( ) ≤ fx i ( ) for all i= 1,2, , n . In other words, x is the Pareto point if v F( x * ) achieves its minimal value. Since investors are interested in minimizing risk and maximizing expected return at the same time, the portfolio optimization problem can be treated as a multi-objective optimization problem (Model 5). One can attain Pareto optimality in this case because the formulation of Model 5 belongs to the category of convex vector optimization, which guarantees that any local optimum is a global optimum.
2.2 Goal Programming approach to Portfolio Optimization
Goal programming can be thought of as an extension or generalisation of linear programming to handle multiple, normally conflicting objective measures. Each of these measures is given a goal or target value to be achieved. Unwanted deviations from this set of target values are then minimised in an achievement function. This can be a vector or a weighted sum dependent on the goal programming variant used.
Objective Function: Minimizes the sum of the weighted deviations from the target values .
2.2.1 Steps to Solve Goal Programming 1. Rewrite information in a table format 2. State your (functional) constraints 3. Rewrite your goals as constraints if not given in that form 4. Decide decision variables 5. Rewrite your functional constraints using your decision variables 6. Rewrite your goal constraints using your decision variables 7. Add deviation variables to the goal constraints 8. Determine the variables need to be minimized in the objective function 9. Write the objective function with priorities 10. Write the Goal Programming Model
2.2.2 Goal Programming in Portfolio Selection Problem The GP model was proposed by Charnes et al. (1955) and Charnes and Cooper (1961) under its standard formulation as follows: - + Min (δi + δi ) - + s.to: fi(x) + δi + δi =f i for all i=1,2,…m
gk(x) <= b k for all k=1.2….k - - x ∈ X and δi and δi >=0 for all i=1,2,….m where the fi (i =1,2, ... ,m) represent the aspiration level (the goal) associated with the objective i, f i(x)= ∑jCij xij is the achievement level of the objective i, X represents the set of - + feasible solution where x >= 0 and x = x 1, x 2, ... , x n. The variables δi and δi indicate the negative and the positive deviations, respectively, between the achievement and the aspiration levels. The g k(x) = ∑jAkj xj <= b k (k=1,2, ... ,K) represent the constraints.
2.2.3 SLOVING THE TWO STOCK MODEL USING GOAL PROGRAMMING
Previously Two stock model of TCS and Indian Oil using the Markowitz risk minimization model had been solved to obtain the solution µ opt =348.934 and σopt =3.74. We solve the same example now using the goal programming approach
Problem: Suppose the investor is willing to take more risk in order to get higher return. So he sets the goal/objectives • the expected return to be at least 500 • the risk( σ) involved to be less than 5. Also he assigns priorities to these goals in the ratio 2:1.
Aim: To formulate the goal programming model and to solve it to find optimal expected return and risk.
SOLUTION:
Step1: Variables: w1= # of units of TCS
w2= # of units of Indian Oil Ltd
Step2: Constraints: w1+ w2=1 Step3: Goals:
1. µ 1w1+ µ 2w2 >= 500 Priority=2 2 2+ 2 2 2. σ1 w1 + σ2 w2 +2 ƥσ1 σ2w1w2 <= 25 Priority=1 Step4: To write the goal inequalities in the equation form with positive and negative deviations from the goal - + µ1w1+ µ 2w2 + z 1 - z1 = 500 2 2+ 2 2 - + σ1 w1 + σ2 w2 +2 ƥσ1 σ2w1w2 + z 2 - z2 = 25 Step5: Determine the variables that need to be minimized in the objective function. - + Looking at the goal we see that we want z 1 and z 2 to be as small as possible. Step6: Modeling the problem with priorities in the objective function - + Min 2z 1 + z 2
s.to w1+ w2=1 - + µ1w1+ µ 2w2 + z 1 - z1 = 500 2 2+ 2 2 - + σ1 w1 + σ2 w2 +2 ƥσ1 σ2w1w2 + z 2 - z2 = 25 - + - + w1, w2, z1 , z 1 , z 2 , z 2 >=0 This is now a single objective quadratic programming problem and has been solved using solver in excel sheet (see excelsheet3.1.)
The solution is µ opt =500 and σopt = 7.763. Thus we see here as compared to the Markowitz risk minimization model there is more expected return at the expense of greater risk.
Given µ 1= 747.065, σ1=18.273, µ 2=137.37, σ2=10.177 we solved Using Markowitz model µ opt =500 and σopt = 7.763. Now using Goal Programming we have the result summarized in the table
Desired µ Desired σ Calculated µ Calculated σ
500 5 500 7.763 600 9 600 11.892 700 2 700 16.212 800 3 747.065 18.273 300 2 348.32 3.74 300 5 340.78 3.75 200 3 348.35 3.74
CONCLUSIONS:
• µ > max(µ 1, µ 2) cannot be achieved . In this case µ>747.065 is not feasible.
• σ is always less than the min ( σ1, σ2) . It shows that diversification of portfolio minimizes risk involved. • If we desire a lower expected return we get values close to the optimal value as calculated in Markowitz model.
CONCLUSIONS AND FUTURE RESEARCH The Markowitz modern portfolio theory is a very basic model to the portfolio optimization problem .It has several assumptions and is restricted to the single objective optimization problem. Although modern portfolio theory has been used for long, it has now been replaced by the post-modern portfolio theory which extends MPT by adopting non-normally distributed, asymmetric measures of risk. This helps with some of these problems, but not others. Over the last 30 years, GP for Portfolio Selection problems have been deployed extensively. The project briefly reviews many of the highlights. GP models for PS allow incorporating multiple goals such as portfolio’s return, risk, liquidity, expense ratio, amongst other factors.
The analysis done in this project was restricted to prediction of the future through regression of the past data. But in real world scenario the stocks rates are stochastic and non- deterministic. It follows a certain probability distribution which is called a geometric Brownian motion. Thus we could extend portfolio optimization problem to stochastic programming .It is more realistic approach. In stochastic programming some or all of the variable or constraints are non-deterministic but their distributions are known. We convert this to a deterministic programming problem by taking a threshold value < 1 and converting the constraint into to their respected probabilistic value to be greater than the threshold values.
AKNOWLEDGEMENT
First and foremost, I would like to thank to our supervisor of this project Dr. V.N.Sastry for the valuable guidance and advice. He inspired me greatly to work in this project. I would also like to thank the authority of IDRBT for providing me this is opportunity and providing a good environment and facilities to complete this project. I would also like to thank the authority of IIT Delhi for allowing me to pursue a two month summer internship at IDRBT.
REFERENCES
Books: Operations Research Introduction 8 th Edition Hamdy.A.Taha Pearson Publication Numerical Optimization S.Chandra and Aparna Mehra 1st Edition Narosa Publication Roman, Steven, Introduction to the Mathematics of Finance: From Risk Management to Options Pricing. Springer 1 Edition, 2004 Benninga Simon, 1999, Financial Modeling, The MIT press, Combridge, Massachusetts, London, England 1999. Cerbone, Duong., and Noe, Tomayko. Multi-objective Optimum Design. Azarm, 1996. Bodie Z., A. Kane ,and A. J. Marcus, 2003, Essentials of Investment, Fifth edition, McGraw- Hill. Bodie Z., A. Kane ,and A. J. Marcus, 2005, Investment, Sixth edition, McGraw-Hill. Brandimarte Paolo, Numerical Methods in Finance: A MATLAB-Based Introduction, New York:John Wiley & Sons. Inc. Cheng F. Lee, Joseph E. Finnerty, Donals H. Wort ,1990, Security Analysis and Portfolio Management John Wiley & Sons. Inc.
Internet : http://www.wikepedia.org http://www.money.rediff.com
Research papers in Journals: A fuzzy Goal Programming Approach to Portfolio Selection European Journal of Operational Research , Volume 133,2001 , Pages 287-297 M.Arenas Parra, A. Bilbao terol, M.V. Rodriquez Uria
A MCDM approach to Portfolio Optimization European Journal of Operational Research , Volume 155,2004 , Pages 752-770 Matthias Ehrgott, kathrin Klamroth , Christian Schwehm