I'm trying to find optimal wights of stocks that maximises return per volatility. I'm able to solve it for two and three stocks within one second. But when I go for 4 stocks, the computer just never gives any result even in more than an our hour. Any idea on how to make it fast enough? I want to make a program for n stocks and I'm unable to go even for more than three stocks. I also tried Sympy but that is able to do for two stocks within a minute but hangs in three stocks for too long I don't have any hope with it.
I do have access to cloud computers and can make instances of 100s cores and 1000s GB RAM. I tried it on Windows server but it just gives runtime error soon enough. My own system runs indefinitely. Anyways, I suppose my code uses only one core for solving by default. All steps are pretty easy, it just hangs on the last solve() method. Is there a way I can utilise all other cores to solve it soon enough?
Here's the algorithm I'm using. r and v are the mean return and variance of the portfolio. r_i, sigma_i, sigma_ij, w_i are mean, standard deviation, covariance, and weights respectively. Equations eq1, eq2, eq3 have been derived by hand by minimising mean/standard_deviation.
r1, r2, r3, r4, s1, s2, s3, s4, s12, s13, s14, s23, s24, s34, w1, w2, w3, w4 = var('r_1, r_2, r_3, r_4, sigma_1, sigma_2, sigma_3, sigma_4, sigma_12, sigma_13, sigma_14, sigma_23, sigma_24, sigma_34, w_1, w_2, w_3, w_4')
r = r1*w1 + r2*w2 + r3*w3 + r4*(1-w1-w2-w3)
v = ((w1**2*s1**2 + w2**2*s2**2 + w3**2*s3**2 + (1-w1-w2-w3)**2*s4**2+2*s12*w1*w2 + 2*s13*w1*w3+
2*s14*(1-w2-w1-w3)*w1+s23*w2*w3 + 2*s24*w2*(1-w2-w1-w3) + 2*s34*w3*(1-w2-w1-w3)))
dr1 = derivative(r, w1)
dr2 = derivative(r, w2)
dr3 = derivative(r, w3)
dv1 = derivative(v, w1)
dv2 = derivative(v, w2)
dv3 = derivative(v, w3)
eq1 = (dv1/dr1 == 2*v/r)
eq2 = (dv2/dr2 == 2*v/r)
eq3 = (dv3/dr3 == 2*v/r)
eq4 = (w1+w2+w3+w4 == 1)
sol = solve((eq1, eq2, eq3, eq4), (w1, w2,w3, w4))
show(sol)
