% Lab 7 Solutions
%-------------------------------------------------------------------------
% Question 1
% Q1(a)
load S_rio.txt
% Plot adjusted share price:
figure(1)
len=length(S_rio);
plot(len:-1:1,S_rio)
xlabel('Trading Day')
ylabel('Rio')

% Q1(b)
% Percentage daily returns:
R_rio = 100*(S_rio(1:len-1)-S_rio(2:len))./S_rio(2:len);
% Plot histogram:
figure(2)
hist(R_rio,20)
xlabel('Rio % Daily Return')
ylabel('Frequency')
% Mean, standard deviation, variance:
mu_rio=mean(R_rio)
sigma_rio=std(R_rio)
var_rio=sigma_rio^2
% Plot normal distribution:
x=linspace(mu_rio-3*sigma_rio,mu_rio+3*sigma_rio,101);
y=exp(-(x-mu_rio).^2/(2*var_rio))/(sqrt(2*pi)*sigma_rio);
hold on
plot(x,(len-1)*y,'m')
% Annualised values:
mu_rio_yr=365*mu_rio
sigma_rio_yr=365*sigma_rio
var_rio_yr=sigma_rio_yr^2

% Q1(c)
load S_nab.txt
figure(3)
len=length(S_nab);
plot(len:-1:1,S_nab)
xlabel('Trading Day')
ylabel('NAB')

% Percentage daily returns:
R_nab = 100*(S_nab(1:len-1)-S_nab(2:len))./S_nab(2:len);
% Plot histogram:
figure(4)
hist(R_nab,20)
xlabel('NAB % Daily Return')
ylabel('Frequency')
% Mean, standard deviation, variance:
mu_nab=mean(R_nab)
sigma_nab=std(R_nab)
var_nab=sigma_nab^2
% Plot normal distribution:
x=linspace(mu_nab-3*sigma_nab,mu_nab+3*sigma_nab,101);
y=exp(-(x-mu_nab).^2/(2*var_nab))/(sqrt(2*pi)*sigma_nab);
hold on
plot(x,(len-1)*y,'m')
% Annualised values:
mu_nab_yr=365*mu_nab
sigma_nab_yr=365*sigma_nab
var_nab_yr=sigma_nab_yr^2

% Q1(d)
%Covariance matrix of Rio and NAB:
cov_rio_nab=cov(R_rio,R_nab)
%Correlation matrix of Rio and NAB:
rho_rio_nab=corrcoef(R_rio,R_nab)
%-------------------------------------------------------------------------
% Question 2
% Run the following commands in Matlab:
% (a)
n=50;
p=0.3;
k=0:n;
for i=0:n P(i+1)=nchoosek(n,i)*p^i*(1-p)^(n-i);end
figure(5)
subplot(3,2,1)
plot(k,P,'.')
hold on
p=0.5;
k=0:n;
for i=0:n P(i+1)=nchoosek(n,i)*p^i*(1-p)^(n-i);end
plot(k,P,'r.')
title('Q2(a)')
hold off
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% (b)
m=3;
r=[0:10];
p=exp(m)*m.^r./gamma(r+1);
subplot(3,2,2)
plot(r,p,'.')
hold on
m=5;
p=exp(m)*m.^r./gamma(r+1);
plot(r,p,'r.')
title('Q2(b)')
hold off
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% (c)
k=[1:10];
p=0.3;
q=1-p;
P=q.^(k-1)*p;
subplot(3,2,3)
plot(k,P,'.')
hold on
p=0.7;
q=1-p;
P=q.^(k-1)*p;
plot(k,P,'r.')
title('Q2(c)')
hold off
%-------------------------------------------------------------------------
% Question 3
% Run the following commands in Matlab:
% (a)
a=2;
b=5;
% Make f in three pieces:
x1=[2*a-b:0.01:a];
y1=zeros(size(x1));
x2=[a:0.01:b];
y2=ones(size(x2));
x3=[b:0.01:2*b-a];
y3=zeros(size(x3));
x=[x1 x2 x3];
f=[y1 y2 y3]/(b-a);
subplot(3,2,4)
plot(x,f)
title('Q3(a)')
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% (b)
lambda=2;
% Make f in two pieces:
x1=[-lambda:0.01:0];
y1=zeros(size(x1));
x2=[0:0.01:4*lambda];
y2=exp(-x2/lambda)/lambda;
x=[x1 x2];
f=[y1 y2];
subplot(3,2,5)
plot(x,f)
title('Q3(b)')
axis([-lambda 4*lambda 0 1])
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% (c)
mu=0;
sigma=1;
x=-3*sigma:0.01:3*sigma;
f=exp(-(x-mu).^2/(2*sigma^2))/(sqrt(2*pi)*sigma);
subplot(3,2,6)
plot(x,f)
title('Q3 Normal Distribution')