% Exercise 4.60: Log-optimal investment strategy
% Boyd & Vandenberghe "Convex Optimization"
% JoĆ«lle Skaf - 04/24/08
% (a figure is generated)
%
% The investment strategy x that maximizes the long term growth rate
% R = sum_{j=1}^m pi_j*log(p_j^Tx)
% is called the log-optimal investment strategy, and can be found by
% solving the optimization problem
% maximize sum_{j=1}^m pi_j log(p_j^Tx)
% subject to x >= 0, sum(x) = 1,
% where p_ij is the return of asset i over one period in scenario j and
% pi_j is the probability of scenario j. There are n assets and m scenarios.
% We consider the case of equiprobable scenarios.
%
% The log-optimal long term growth rate is found and compared to the one
% obtained with a uniform allocation strategy, i.e., x_i=(1/n).
% Sample trajectories ofthe accumulated wealth for the optimal strategy and
% the uniform one are plotted.
% Input data
P = [3.5000 1.1100 1.1100 1.0400 1.0100;
0.5000 0.9700 0.9800 1.0500 1.0100;
0.5000 0.9900 0.9900 0.9900 1.0100;
0.5000 1.0500 1.0600 0.9900 1.0100;
0.5000 1.1600 0.9900 1.0700 1.0100;
0.5000 0.9900 0.9900 1.0600 1.0100;
0.5000 0.9200 1.0800 0.9900 1.0100;
0.5000 1.1300 1.1000 0.9900 1.0100;
0.5000 0.9300 0.9500 1.0400 1.0100;
3.5000 0.9900 0.9700 0.9800 1.0100];
[m,n] = size(P);
Pi = ones(m,1)/m;
x_unif = ones(n,1)/n; % uniform resource allocation
% Find the log-optimal investment policy
cvx_begin
variable x_opt(n)
maximize sum(Pi.*log(P*x_opt))
sum(x_opt) == 1
x_opt >= 0
cvx_end
% Long-term growth rates
R_opt = sum(Pi.*log(P*x_opt));
R_unif = sum(Pi.*log(P*x_unif));
display('The long term growth rate of the log-optimal strategy is: ');
disp(R_opt);
display('The long term growth rate of the uniform strategy is: ');
disp(R_unif);
% Generate random event sequences
rand('state',10);
N = 10; % number of random trajectories
T = 200; % time horizon
w_opt = []; w_unif = [];
for i = 1:N
events = ceil(rand(1,T)*m);
P_event = P(events,:);
w_opt = [w_opt [1; cumprod(P_event*x_opt)]];
w_unif = [w_unif [1; cumprod(P_event*x_unif)]];
end
% Plot wealth versus time
figure
semilogy(w_opt,'g')
hold on
semilogy(w_unif,'r--')
grid
axis tight
xlabel('time')
ylabel('wealth')