% Computer Lab 8 Solutions
%-----------------------------------------------------------------
% Q1(a)
load S_rio_m.txt
load asx200_m.txt
% Percentage monthly returns:
len=length(S_rio_m);
R_rio_m = 100*(S_rio_m(1:len-1)-S_rio_m(2:len))./S_rio_m(2:len);
len=length(asx200_m);
R_asx200_m = 100*(asx200_m(1:len-1)-asx200_m(2:len))./asx200_m(2:len);

% Q1(b)
mu_rio=mean(R_rio_m)
mu_asx200=mean(R_asx200_m)
cov_rio_asx200=cov(R_rio_m,R_asx200_m)
beta=cov_rio_asx200(1,2)/cov_rio_asx200(2,2)
alpha=mu_rio-beta*mu_asx200

% Q1(c)
figure(1)
set(gcf,'PaperType','A4')
plot(R_asx200_m,R_rio_m,'o')
xlabel('ASX200 % Monthly Return')
ylabel('Rio % Monthly Return')
% Plot rio's asx200 characteristic line:
x_lo=min(R_asx200_m);
x_hi=max(R_asx200_m)
x=linspace(x_lo,x_hi,101);
y=alpha+beta*x;
hold on
plot(x,y,'m')
title('RioTinto Characteristic Line')

% Q1(d)
load S_nab_m.txt
% Percentage monthly return:
len=length(S_nab_m);
R_nab_m = 100*(S_nab_m(1:len-1)-S_nab_m(2:len))./S_nab_m(2:len);

mu_nab=mean(R_nab_m)
cov_nab_asx200=cov(R_nab_m,R_asx200_m)
beta=cov_nab_asx200(1,2)/cov_nab_asx200(2,2)
alpha=mu_nab-beta*mu_asx200

figure(2)
set(gcf,'PaperType','A4')
plot(R_asx200_m,R_nab_m,'o')
xlabel('ASX200 % Monthly Return')
ylabel('NAB % Monthly Return')
% Plot nab's asx200 characteristic line:
y=alpha+beta*x;
hold on
plot(x,y,'m')
title('NAB Characteristic Line')

%-----------------------------------------------------------------
% Question 2
% We use the portfolio equations directly to answer this question.
r=[10 15]';
s=sqrt([8 12])';
C=[1 0.5 ; 0.5 1];
rho=0.5;
% Part (a):
S=diag(s)*C*diag(s)
% Part (b):
figure(3)
set(gcf,'PaperType','A4')
clf
x1=-1:.01:2;
x2=1-x1;
x=[x1;x2];
mu=10*x1+15*x2;
sigma=[];  % Clear or empty an array, in this case sigma, when constructing it element by element.
for i=1:length(x1)
sigma(i)=sqrt(x(:,i)'*S*x(:,i));
end
hold off
plot(sigma,mu)
title('Q3')
xlabel('\sigma')
ylabel('\mu')
hold on
% Part (c):
x1=0:.01:1;
x2=1-x1;
x=[x1;x2];
mu=10*x1+15*x2;
sigma=[];  % Clear or empty an array, in this case sigma, when constructing it element by element.
for i=1:length(x1)
sigma(i)=sqrt(x(:,i)'*S*x(:,i));
end
plot(sigma,mu,'r','linewidth',2)
plot(s(1),r(1),'o')
text(s(1),r(1)-2,'P_1')
plot(s(2),r(2),'o')
text(s(2),r(2)-3,'P_2')
% Part (d):
t=0.5;
for i=1:5 
Z=-2*i;
sigma=0:.1:7;
mu=(0.5*sigma.^2-Z)/t;
plot(sigma,mu,'g')
if Z==-2, text(sigma(11),mu(11),'Z=-2'),end
if Z==-10, text(sigma(11),mu(11),'Z=-10'),end
end
%-----------------------------------------------------------------
% Question 3
% We use the portfolio equations directly to answer this question.
% Part (a):
figure(4)
set(gcf,'PaperType','A4')
clf
x1=0:.01:1;
x2=1-x1;
x=[x1;x2];
mu=10*x1+15*x2;
sigma=[];  % Clear or empty an array, in this case sigma, when constructing it element by element.
for i=1:length(x1)
sigma(i)=sqrt(x(:,i)'*S*x(:,i));
end
hold on
plot(sigma,mu,'b')
title('Q4')
xlabel('\sigma')
ylabel('\mu')
hold on
% Part (b):
x_o=(s(2)^2-rho*s(1)*s(2))/(s(1)^2+s(2)^2-2*rho*s(1)*s(2)); % Minimum risk x.
x1=0:.01:x_o;
x2=1-x1;
x=[x1;x2];
mu=10*x1+15*x2;
sigma=[];  % Clear or empty an array, in this case sigma, when constructing it element by element.
for i=1:length(x1)
sigma(i)=sqrt(x(:,i)'*S*x(:,i));
end
plot(sigma,mu,'r','linewidth',2)
% Part (c) (no shorting):
% Minimum risk portfolio:
mu_o=10*x_o+15*(1-x_o);
sigma_o=sqrt([x_o,1-x_o]*S*[x_o,1-x_o]');
plot(sigma_o,mu_o,'o')
text(sigma_o+0.05,mu_o,'Minimum risk portfolio')
% Maximum risk portfolio:
x=0;
mu=10*x+15*(1-x);
sigma=sqrt([x,1-x]*S*[x,1-x]');
plot(sigma,mu,'o')
text(sigma-0.3,mu,'Maximum risk portfolio')
% Minimum return portfolio:
x=1;
mu=10*x+15*(1-x);
sigma=sqrt([x,1-x]*S*[x,1-x]');
plot(sigma,mu,'o')
text(sigma,mu+0.3,'Minimum return portfolio')
% Part (d):
t=0.5;
x=(t*(r(1)-r(2))+s(2)^2-rho*s(1)*s(2))/(s(1)^2+s(2)^2-2*rho*s(1)*s(2)); % Minimum risk x.
mu=10*x+15*(1-x);
sigma=sqrt([x,1-x]*S*[x,1-x]');
plot(sigma,mu,'o')
text(sigma+0.05,mu,'Optimum portfolio')
Z=-mu*t+sigma^2/2;
sigma=2.6:0.01:3.5;
mu=(0.5*sigma.^2-Z)/t;
plot(sigma,mu,'g')
text(3.1,16,[ 'Optimum level contour Z=' num2str(Z)])