These are helper functions for this blog post.

buffettkelly.py

def integrand(r, f, rmin, rmax, mu, sigma):
    """defines the part inside the integral sign"""
    from scipy.stats import truncnorm
    a, b = (rmin - mu)/sigma, (rmax - mu) /sigma
    rv   = truncnorm(a, b, loc=mu, scale=sigma)
    return  rv.pdf(r) * np.log(1 + f*(r-1))

def logS(f, rmin, rmax, mu, sigma):
    """numerically solves the integral"""
    from scipy.integrate import quad
    lnS, err = quad(integrand, rmin, rmax, (f, rmin, rmax, mu, sigma))
    return lnS

def neglogS(f, rmin, rmax, mu, sigma):
    """scipy has a minimizer, not maximizer"""
    return -logS(f, rmin, rmax, mu, sigma)
    
def optimalAlloc(rmin, rmax, mu, sigma):
  """maximize logS = minimize -logS"""
  from scipy.optimize import fminbound
  fst = fminbound(neglogS, 0.0001, 0.9999, (rmin, rmax, mu, sigma))
  return fst