First of all we have to read all the data with an xlsread function and give names to the subsets of data that we are interested in:

Transcription

1 First of all we have to read all the data with an xlsread function and give names to the subsets of data that we are interested in: data=xlsread('c:\users\prado\desktop\master\investment\material alumnos\data.xlsx') rm=data(:,1); rf=data(:,2) size_portf=data(:,13:22); portfolio=data(:,3:end); Obtain the expression for the minimum volatility frontier in the case of the set of the 3 portfolios. Generate the points in the minimum volatility frontier for the correspondent expected return in a range of values between -.6 and. We have to take the mean E and the covariance matrix of the portfolio in order to obtain the expression for the minimum volatility frontier which is: Where: We obtain the covariance matrix and the mean of the 3 portfolios V_3=cov(portfolio); E_3=mean(portfolio)'; We compute the letters that we need for the formula: A_3=E_3'*inv(V_3)*E_3; B_3=E_3'*inv(V_3)*one_3; C_3=one_3'*inv(V_3)*one_3; D_3=A_3*C_3-B_3^2 At this point, where we have all this numbers we have to compute the volatility for every different value of mean. So that, we create a vector of consecutives values representing different points in the mean-variance space, as for example: Ep=linspace(-.6,,1); And we insert it in the formula: sigma_3=sqrt((1/(a_3*c_3-b_3^2))*(c_3*ep'.^2-2*b_3*ep'+a_3));

2 Finally we will get two different vectors, one for the values of the mean and the other one the respective variance for each of the mean values. We can plot both vectors, plotting the standard deviation in the x-label and the mean values in the y-label. Plot(sigma_3,Ep) Obtain the weights, the mean return and the standard deviation of the minimum variance portfolio in the investment opportunity set with 3 portfolios. Plot this portfolio in the minimum variance frontier. Portfolio with the lowest variance in the minimum variance frontier First we have to obtain the composition (weights) of the minimum variance portfolio with an optimization problem. Finally the weights are obtained through the formula placed below: Wmv=(inv(V_3)*one_3)/(one_3'*inv(V_3)*one_3) Emv=Wmv'*E_3; varmv=wmv'*v_3*wmv

3 Finally we plot it with the parabola obtained before: Plot(sigma_3,Ep) Hold on plot(sqrt(varmv),emv,'*') Using the mean of the T-Bill rate as a proxy for the risk free rate, obtain the composition, the mean return, and the standard deviation of the tangent portfolio in the case of 3 risky assets: 1 size-sorted portfolios, 1 BM-sorted portfolios, and 1 momentum-sorted portfolios. %%Composition, mean return, std of Tangent portfolio for 3 portfolio. RF_mean=mean(rf); X=inv(V_3)*(E_3-RF_mean*one_3) WT=(X/(one_3'*X)) %COMPOSITION TANGENT PORTFOLIO ET=WT'*E_3 %MEAN TANGENT PORTFOLIO To obtain the variance of the tangent portfolio we follow the same formula as before but in this case we use the obtained weights for the Tangent portfolio NOT the MV portfolio std_tangent=sqrt(wt'*v_3*wt) %STD TANGENT PORTFOLIO

4 Plot the minimum variance frontier with risky assets and the efficient frontier with riskless asset Now we want to plot the same parabola as before and the Efficient Frontier that represent the combination of weights between the risky portfolio and the Risk Free Rate. We use the same formula for the mean-variance frontier with a new range in this case: Ep=linspace(-.6,.2,1); %CREAMOS UN NUEVO RANGO sigma_3=sqrt((1/(a_3*c_3-b_3^2))*(c_3*ep'.^2-2*b_3*ep'+a_3)); %EFFICIENT FRONTIER %NEW RANGE SHARPE RATIO. sig=linspace(,.6,1) Eff_front=RF_mean+((ET-RF_mean)/std_tangent)*sig If a rational investor is willing to support a volatility investment of.7, what portfolio will he chose? %VOLATILITY=.7 Eff_front_1=RF_mean+((ET-RF_mean)/std_tangent)*.7 hold on plot(.7,eff_front_1,'*')

5 A second rational investor has very low risk aversion and he wants to borrow at the risk free rate for investing 12% in the risky assets. What portfolio will he chose? E_risky=-.2*RF_mean+1.2*ET VA=((1.2^2)*std_tangent^2) std_port=sqrt(va); hold on plot(std_port,e_risky,'*')