Matlab Workshop MFE 2006 Lecture 4

Transcription

1 Matlab Workshop MFE 2006 Lecture 4 Stefano Corradin Peng Liu Haas School of Business, Berkeley, MFE 2006

2 Applications in Finance II 4.1 Optimum toolbox. 4.2 Mean-variance portfolio. 4.3 Risk management analysis. Haas School of Business, Berkeley, MFE

3 Optimum toolbox We are going to use the command linprog, quadprog and fmincon to solve constrained optimization problems. Let consider an easy problem min 2x 1 3x 2 s.t. x 1 + x 2 3 x 1 0. We are going to use the command linprog >> help linprog LINPROG Linear programming. X=LINPROG(f,A,b) attempts to solve the linear programming problem: min f *x subject to: A*x <= b x Haas School of Business, Berkeley, MFE

4 X=LINPROG(f,A,b,Aeq,beq) solves the problem above while additionally satisfying the equality constraints Aeq*x = beq. X=LINPROG(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. therefore >> s=linprog([2-3],[1 1],3,[],[],[0]) Optimization terminated. s = 1.205e where we use [] to indicate that we do not consider Aeq x = beq constraint. Haas School of Business, Berkeley, MFE

5 Let consider a quadratic problem min x 2 1 3x x 1 x 2 s.t. x 1 + 2x 2 = 4 x 1 0 x 2 0. We are going to use the command quadprog >> help quadprog QUADPROG Quadratic programming. X=QUADPROG(H,f,A,b) attempts to solve the quadratic programming problem: min 0.5*x *H*x + f *x subject to: A*x <= b x X=QUADPROG(H,f,A,b,Aeq,beq) solves the problem above while Haas School of Business, Berkeley, MFE

6 additionally satisfying the equality constraints Aeq*x = beq. 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. therefore >> H = [2 0;0 6]; c = [3-1]; >> Aeq = [1 2]; >> beq = 4; >> lb = [0 0]; >> x = quadprog(h,c,[],[],aeq,beq,lb) Optimization terminated. x = Haas School of Business, Berkeley, MFE

7 Mean-variance portfolio Let consider now a mean-variance portfolio problem with n stocks x: 1 n vector of weights; min x xωx s.t. x1 = 1 x R = R p 0 x i 1 Ω: n n variance-covariance matrix of returns; R: 1 n vector of expected returns; R p : 1 1 expected portfolio returns. Haas School of Business, Berkeley, MFE

8 We are going to analyze two codes: markowqp.m Let analyze the code step-by-step: Matrix of stocks prices. prices = [ ; ; ; ; ; ; ; ; ; Haas School of Business, Berkeley, MFE

9 ; ; ; ]; Create the matrix of returns and calculate the expected return and matrix of variance-covariance. [n,col] = size(prices); % Matrix of returns for i = 2:n ret(i,:) = (prices(i,:)-prices(i-1,:))./prices(i-1,:); end % Expected returns retatt = mean(ret); % Variance-covariance matrix mvarcov = cov(ret); Haas School of Business, Berkeley, MFE

10 Create a vector of portfolio expected returns. R p = [ R 1 p, R 2 p,..., R n p]. % Portfolio expected returns low = min(retatt); up = max(retatt); step =.001; retpor = (low:step:up) ; Solve the problem with quadprog function for every element of the vector R p. mm=length(retpor); varpor=zeros(1,mm); weights = zeros(col,mm); % Quadratic programming % min 0.5 x H x + c x tale che A x <= b % Aeq matrix Aeq = [retatt;ones(1,col)]; c = zeros(col,1); Haas School of Business, Berkeley, MFE

11 for i=1:mm beq = [retpor(i,1);1]; % lower bounds - weights > 0 vlb = zeros(n,1); % upper bounds - weights < 1 vub = ones(n,1); % starting values x0 = [.25;.25;.25;.25]; % quadratic programming function x = quadprog(mvarcov,c,[],[],aeq,beq,vlb,vub,x0); % calculate variance portfolio varpor(i) = x *mvarcov*x; % optimal portfolio weights weights(:,i)=x; end Haas School of Business, Berkeley, MFE

12 Plot the efficient frontier. Haas School of Business, Berkeley, MFE

13 portfolio.m We use the function fmincon to solve the problem and we have now two datasets: - a time series 762 daily observations of 20 stock prices (Italian stock market); - a vector of current portfolio weights. Let analyze the code step-by-step: Import the datasets. % Load dataset in txt format file1= mibtel.txt ; data =dlmread(file1, \t ); % Load portfolio weights in txt format in the workspace file2= weight.txt ; weight =dlmread(file2, \t ); % Size provides number rows and columns in the workspace [r,c] = size(data); Haas School of Business, Berkeley, MFE

14 % Select part of the data set start = 200; obs = 250; m = data(start:start+obs,1:c); Create the matrix of returns and calculate the expected return and matrix of variance-covariance. term = obs - 1; % Stocks returns rend = ((m(2:obs,:)-m(1:term,:))./m(1:term,:)); % Number of Stocks bb = c-1; r_equity = rend(:,1:bb); % Last columns: market index return r_index = rend(:,c); % Expected returns Haas School of Business, Berkeley, MFE

15 rendatt = mean(r_equity); % Var-Cov matrix mvarcov = cov(r_equity); Create a vector of portfolio expected returns. % Portfolio expected return: a vector is created rend_low = min(rendatt); rend_high = max(rendatt); nfac = 100; step = (rend_high - rend_low)/nfac; rendatt_por = rend_low:step:rend_high; cc = length(rendatt_por); n_equity = bb; Solve the problem with fmincon function for every element of the vector R p % Efficient portfolio xstar = zeros(cc,bb); for i = 1:cc x0 = 1/bb.*ones(1,bb); % starting values Haas School of Business, Berkeley, MFE

16 Aeq = [ones(1,bb); rendatt]; Beq = [1;rendatt_por(i)]; vlb = zeros(1,bb); % weights lower bounds vub = ones(1,bb); % weights upper bounds options = optimset( Display, off ); x = fmincon(@(x)funcov(x,mvarcov,n_equity),x0,... [],[],Aeq,Beq,vlb,vub,[],options); xstar(i,:) = x; end where funcov.m is a function used to define the objective function to minimize function [f] = funcov(x,mvarcov,n_equity) x = x(1:1:n_equity); varpor = x*mvarcov*x ; % function to minimize f = 0.5*varpor; Haas School of Business, Berkeley, MFE

17 Calculate the standard deviation of portfolio, σ p, given the optimal portfolio weights derived x. Plot the efficient frontier. % Efficient portfolio standard deviation for i = 1:cc devpor_eff(i) = sqrt(xstar(i,:)*mvarcov*xstar(i,:) ); end % Expected return and standard deviation of the current portfolio rend_attpor = rendatt*weight ; dev_attpor = sqrt(weight*mvarcov*weight ); Haas School of Business, Berkeley, MFE

18 Haas School of Business, Berkeley, MFE

19 Risk management analysis We can calculate the Value at Risk (VaR) to analyse the riskiness of our portfolio. The VaR is an estimate of the level of loss on a portfolio which is expected to be equaled or exceeded with a given small probability V ar α (X) = inf{x P [X x] α} where α is a confidence interval. The VaR of our portfolio is where V ar α = α T σ p Haas School of Business, Berkeley, MFE

20 α = norminv(p, 0, 1) and p can be.9,.95,.99,...; T is time horizon or unwinding period; σ p can be calculated according to two approaches: a) standard deviation of portfolio previously derived; b) standard deviation of portfolio derived using β of stocks r i t = γ + βr m t + e t where r i t is the stock return, r m t is the market return, γ is the intercept and σ p = xββ x σ m where σ m is the standard deviation of market returns. Haas School of Business, Berkeley, MFE

21 Calculate the VaR with the first approach (still we are working on portfolio.m). % Value at Risk - Var-Cov approach % VaR 1,5 e 10 days time = input( Decide unwinding period = ); time = sqrt(time); % Confidence Interval 95%,97.5% e 99% pp = input( Decide confidence interval between 0 and 1 = ); conf = norminv(pp,0,1); % VaR Var-Cov on optimal portfolio VaRcov = zeros(cc,1); for i = 1:cc VaRcov(i) = time*conf*devpor_eff(i); end Calculate the vector β (the last vector of our dataset is the stock market). Haas School of Business, Berkeley, MFE

22 % Stocks beta Beta_equity = zeros(bb,1); for j = 1:bb X = [ones(term,1),r_index]; coeff = inv(x *X)*X *r_equity(:,j); beta = coeff(2,1); Beta_equity(j,1) = beta; end disp([ Stocks Beta ]); Beta_equity = Haas School of Business, Berkeley, MFE

23 Haas School of Business, Berkeley, MFE

24 Calculate VaR according to the second approach. % VaR Beta OLS VaRbeta = zeros(cc,1); for i = 1:cc % Standard deviation portfolio % (weights*beta*beta *weights *varianze_market)^1/2 devst_por = sqrt(xstar(i,:)*beta_equity*... Beta_equity *xstar(i,:) *cov(r_index)); VaRbeta(i) = time*conf*devst_por; end Haas School of Business, Berkeley, MFE

25 Make a plot to compare the two approaches. Haas School of Business, Berkeley, MFE