% 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')