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Chapter 5Portfolio

O. Afonso, P. B. Vasconcelos

Computational Economics: a concise introduction

O. Afonso, P. B. Vasconcelos Computational Economics 1 / 22

Overview

1 Introduction

2 Economic model

3 Numerical solution

4 Computational implementation

5 Numerical results and simulation

6 Highlights

7 Main references


Introduction

The portfolio optimisation model, originally proposed by Markowitz (1952),selects proportions of assets to be included in a portfolio.

To have an efficient portfolio:the expected return should be maximised contingent on any given number ofrisks; orthe risk should be minimised for a given expected return.

Thus, investors are confronted with a trade-off between expected returnand risk.The expected return-risk relationship of efficient portfolios is representedby an efficient frontier curve.

Optimisation knowledge is required to solve this problem. Focus is on a MonteCarlo optimisation and on advanced numerical solutions provided byMATLAB/Octave.


Economic model

The aim is to maximise the expected return constrained to a given risk

maxx

cT x , s.t. xT Hx = σ2,

n∑i=1

xi = 1 and xi ≥ 0 , (1)

wheren is the number of assets,x , n × 1, is the vector of the shares invested in each asset i ,c, n × 1, is the vector of the average benefit per asset,H, n × n, is the covariance matrix, andσ2 is the expected risk goal.

Problem (1) is know as a quadratic programming problem.Alternatively, minimise the risk subject to an expected return, c,

minx

xT Hx , s.t. cT x = c,n∑

i=1

xi = 1 and xi ≥ 0. (2)


Economic model

The global minimum variance portfolio is the one satisfying

minx

xT Hx , s.t.n∑

i=1

xi = 1 and xi ≥ 0. (3)

The efficient frontier is the set of pairs (risk, return) for which the returnsare greater than the return provided by the minimum variance portfolio.

The aim is to find the values of variables that optimise an objective,conditional or not to constraints.Numerical methods overcome limitations of size, but there is no universalalgorithm to solve optimisation problems.The topic is addressed only in a cursory manner exploiting theMATLAB/Octave optimisation potentialities.


Numerical solution

Consider the minimisation problem

minx∈Rn

f (x) (4)

s.t. ci(x) = 0, i ∈ Eci(x) ≥ 0, i ∈ I

where f : Rn → R, cE : Rn → RnE and cI : Rn → RnI , respectively, the equalityand inequality constraints.

A feasible region is the set points satisfying the constraintsS = {x : ci(x) = 0, i ∈ I and ci(x) ≥ 0, i ∈ D}.Problems without restrictions I = D = ∅ emerge in many applications andas a recast of constraint problems where restrictions are replaced bypenality terms added to the objective function.


Numerical solution

Optimisation problems can be classified in various ways, according to, forexample: (i) functions involved; (ii) type of variables used; (iii) type ofrestrictions considered; (iv) type of solution to be obtained; and (v)differentiability of the functions involved.Among the countless optimisation problems, linear, quadratic andnonlinear programming are the most usual.Many algorithms for nonlinear programming problems only seek localsolutions; in particular, for convex linear programming, local solutions areglobal.


Numerical solution

Unconstrained optimisation in practice

Unconstrained optimisation problemsMethods such as steepest descent, Newton and quasi-Newton are the mostused.MATLAB/Octave:

fminunc(f,x0) attempts to find a local minimum of function f, starting at pointx0;similarly with fminsearch(f,x0) but using a derivative-free method;x0 can be a scalar, vector, or matrix.


Numerical solution

Constrained optimisation in practice: linearprogramming

Linear programming problem: both the objective and constraints arelinear

minx

cT x

s.t. Ax ≤ b, Aeqx = beq , lb ≤ x ≤ ub

where c and x are vectors.MATLAB/Octave: linprog(c,A,b,Aeq,beq,lb,ub).


Numerical solution

Constrained optimisation in practice: quadraticprogramming

Quadratic programming problem (portfolio problem): this involves aquadratic objective function and linear constraints

minx

12

xT Hx + xT c

s.t. Ax ≤ b, Aeqx = beq , lb ≤ x ≤ ub

where c, x and ai are vetors, and H is a symmetric (Hessian) matrix.MATLAB: quadprog(H,c,A,b,Aeq,beq,lb,ub)Octave: qp([],H,c,Aeq,beq,lb,ub)


Numerical solution

Constrained optimisation in practice: nonlinearprogramming

Nonlinear programming: f and/or constraints are nonlinear

minx

f (x)

s.t. c(x) ≤ 0, ceq(x) = 0,Ax ≤ b, Aeqx = beq , lb ≤ x ≤ ub .

MATLAB: fmincon(f,x0,A,b,Aeq,beq,lb,ub,nonlcon)Octave: minimize(f,args) (where args is a list or arguments to f)


Numerical solution

Monte Carlo approach

Monte Carlo: experiments anchored on repeated random sampling toobtain numerical approximations of the solution.A Monte Carlo procedure can be schematised as follows.

1 Set a possible solution, and consider it to be the best for the moment2 For a certain number of times, do:

1 generate (randomly) a set of feasable solutions from the best one available;2 select (possibilly) a better one;3 repeat the process.


Computational implementation

Consider the following data, respectively, for the returns vector andcovariance matrix

c =

0.1000.2000.150

and H =

0.005 −0.010 0.004−0.010 0.040 −0.002

0.004 −0.002 0.023

.



Monte Carlo approach: portfolio with minimumvariance

function [ x , x _ h i s t ] = p o r t f o l i o _ m c a r l o _ f u n (H, nruns , const )% Monte Carlo s o l u t i o n approach% Implemented by : P .B . Vasconcelos and O. Afonso% based on : Computat ional Economics ,% D. A . Kendrick , P . R. Mercado and H. M. Amman% Pr inceton U n i v e r s i t y Press , 2006% inpu t :% H, covar iance mat r i x% nruns , number o f Monte Car lo runs% const , constant to increase / reduce the magnitude of the random% numbers generated% output :% x , best found p o r t f o l i o% x_h is t , search h i s t o r y f o r best p o r t f o l i o



% i n i t i a l i z a t i o n parameters and weights ;popsize = 10;n = size (H, 1 ) ;pwm = ( 1 / n ) ∗ones ( n , popsize ) ;c r i t = zeros (1 , popsize ) ;x _ h i s t = zeros ( n , 1 ) ;

% compute nruns x popsize p o r t f o l i o sfor k = 1: nruns

for j = 1 : popsize ;c r i t ( j ) = pwm( : , j ) ’∗H∗pwm( : , j ) ;

end

% s e l e c t i o n o f the best p o r t f o l i o[ ~ , top_index ] = min ( c r i t ) ;x = pwm( : , top_index ) ;

% sto re the best p o r t f o l i ox _ h i s t ( : , k ) = x ;



i f k == nruns , break , endpwm( : , 1 ) = x ;for i = 2 : popsize ;

x = x+randn ( n , 1 ) ∗const ;pwm( : , i ) = abs ( x /sum( abs ( x ) ) ) ;

endend

To solve the problem just do:nruns = 40;const = 0.1;[x,x_hist] = portfolio_mcarlo_fun(H,nruns,const);disp(’best portfolio:’);for i=1:length(x)

fprintf(’Asset %d \t %5.4f \n’,i,x(i));endfprintf(’expected return: %g \n’,c’*x);fprintf(’risk : %g \n’,sqrt(x’*H*x));


Numerical results and simulation

---------------------------------------------------------Portfolio optimization: global minimum variance

Monte Carlo solution approach---------------------------------------------------------best portfolio:Asset 1 0.7649Asset 2 0.2355Asset 3 0.0004expected return: 0.123644risk : 0.0392812



0 5 10 15 20 25 30 35 400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

share

of each a

sset

number of Monte Carlo runs

Asset 1

Asset 2

Asset 3

Monte Carlo convergence path for the portfolio with minimum variance



Quadratic programming approach: portfolio withminimum variance

Aeq = ones(1,length(c)); beq = 1; lb = zeros(1,length(c));x = quadprog(2*H,[],[],[],Aeq,beq,lb);disp(’best portfolio:’);for i=1:length(x)fprintf(’Asset %d \t %5.4f \n’,i,x(i));

endfprintf(’expected return: %g \n’,c’*x);fprintf(’risk : %g \n’,sqrt(x’*H*x));



---------------------------------------------------------Portfolio optimization: global minimum variance

quadratic programming approach---------------------------------------------------------best portfolio:Asset 1 0.7692Asset 2 0.2308Asset 3 0.0000expected return: 0.123077risk : 0.0392232


Highlights

The portfolio optimisation model selects the optimal proportions ofvarious assets to be included in a portfolio, according to certain criteria.A rational investor aims at choosing a set of assets (diversification)delivering collectively the lowest risk for a target expected return.A portfolio is considered efficient if it is not possible to obtain a higherreturn without increasing the risk.The expected return-risk relationship of efficient portfolios is representedby an efficient frontier curve.The model is a quadratic programming problem. It is solved by using asimple Monte Carlo approach that only requires the notion of a minimumconditioned to a set of restrictions, and by a more sophisticated methoddeployed by MATLAB/Octave.


Main references

References

R. A. Haugen and N. L. BakerThe efficient market inefficiency of capitalization-weighted stock portfoliosThe Journal of Portfolio Management, 17(3): 35–40, 1991

H. MarkowitzPortfolio selectionThe Journal of Finance, 7(1): 77–91, 1952

R. C. MertonAn analytic derivation of the efficient portfolio frontierThe Journal of Financial and Quantitative Analysis, 7(4): 1851–1872, 1972

J. Nocedal and S. J. WrightNumerical OptimizationSpringer (2006)

D. Pachamanova and F. J. FabozziSimulation and optimization in finance: modeling with MATLAB, @RISK, or VBAvol. 173, John Wiley & Sons (2010)


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