exp_util_gm_portfolio_opt

Exponential utility and entropic value at risk (EVaR) portfolio construction under a Gaussian mixture model of returns in Python. This repository implements the exponential utility and EVaR objective portfolio optimization problems described in the paper Portfolio construction with Gaussian mixture returns and exponential utility via convex optimization.

Optimizing a portfolio with the exponential utility

To run a minimal example, simply load example problem data defined in problem_data.py via

from problem_data import mus, Sigmas, pi, n, k.

This defines mus, Sigmas, and pi as lists of the respective mixture component means, covariance matrices, and mixture weights of a Gaussian mixture return model with 4 components, for an 11 dimensional return distribution. The, the following code from section 3.2 in the paper solves the EGM portfolio construcion problem:

import cvxpy as cvx

gamma = 1

def K(w):
    u = cvx.vstack([cvx.log(pi[i])
      - gamma * mus[i] @ w
      + (gamma**2/2) * cvx.quad_form(w, Sigmas[i]) for i in range(k)])
    return cvx.log_sum_exp(u)

w = cvx.Variable(n)
objective = cvx.Minimize(K(w))
constraints = [ w >= 0, cvx.sum(w) == 1 ]
egm_prob = cvx.Problem(objective, constraints)
egm_prob.solve()
w.value

Optimizing a portfolio with EVaR

After loading the EVaR portfolio optimization code via from gm_evar_portfolio import min_EVaR_portfolio and with L the leverage limit and alpha the EVaR level, simply run

w,delta,evar = min_EVaR_portfolio(alpha,L,mus,Sigmas,pi)

to generate the minimum EVaR portfolio.

Citing

If you use gm_evar_portfolio in your research, please consider citing us by using the following bibtex:

@article{luxenberg2024portfolio,
  title={Portfolio Construction with Gaussian Mixture Returns and Exponential Utility via Convex Optimization},
  author={Luxenberg, Eric and Boyd, Stephen},
  journal={Optimization and Engineering},
  volume={25},
  number={1},
  pages={555--574},
  year={2024},
  publisher={Springer}
}