Legendre and Markowitz (part 2)

A somewhat unorthodox approach to portfolio optimization

Last week we talked about the primal/dual equivalence of a typical portfolio optimization problem with proportional costs:

\((1)\ \ \ \ \ \ \begin{equation*} \begin{split} \phi^*(e) &= \sup_{p\in\mathbb{R}^n}\left\{ p^Te - \frac{\lambda}{2}p^T\Omega p - \alpha |p - p^0|_1 \right\}\\ \\ &=\inf_{|\eta|_\infty < \alpha}\left\{ (p^0)^T\eta + \frac{\lambda^{-1}}{2}(e-\eta)^T\Omega^{-1}(e-\eta)\right\},\\ \end{split} \end{equation*}\)

where λ is the risk aversion coefficient, Ω is the covariance matrix of the n assets we’re trading, α > 0 the proportional cost, p^0 = (p^0_1,…,p^0_n) our current position, e = (e_1,…,e_n) is the vector of expected returns, and | . |_1, | . |_∞ are the usual 1- and infinity- norms. In the former formulation, we look for the optimal p which achieves the maximum of the net risk-adjusted expected return (inside the brackets), and that’ll be our optimal portfolio, whereas in the latter, the optimal portfolio will be obtained as the gradient of φ* at e.

Let’s now explore some advantages of the infimum formulation in (1). It lends itself well to robust optimization. For instance, we could assume there’s noise around the expected return vector e, with size ρ > 0 measured in the appropriate metric of the inverse covariance matrix (where assets with different volatilities have different scales, as they should). In this context the new function to minimize reads

\((2) \ \ \ \ \ \ \phi^*_\rho(e) = \inf_{|\eta|_\infty < \alpha}\left\{ (p^0)^T\eta + \frac{\lambda^{-1}}{2}\inf_{h^T\Omega^{-1}h < \rho^2}(e+h-\eta)^T\Omega^{-1}(e+h-\eta) \right\}\)

Expanding (2), noting that the quadratic term in h is constant at the minimum, and that the minimum of the linear term in h is achieved when h is proportional to Ω^{-1}(e-η), we get

\((3)\ \ \ \ \ \ \phi^*_\rho(e) = \inf_{|\eta|_\infty < \alpha}\left\{ (p^0)^Te + \frac{\lambda^{-1}}{2}(e-\eta)^T\Omega^{-1}(e-\eta) - \rho\lambda^{-1}\frac{||\Omega^{-1}(e-\eta)||^2}{\sqrt{(e-\eta)^T\Omega^{-3}(e-\eta)}} +\frac{\lambda^{-1}}{2}\rho^2 \right\},\)

which can then be fed into a nonlinear optimizer (there are ways we could regularize the nonlinearity, which won’t be developed here).

Another robustness element we can easily incorporate is with respect to the covariance matrix. For instance, if we assume that

\((4)\ \ \ \ \ \ \omega_1 \le \Omega^{-1}\le\omega_2,\)

in the sense of quadratic forms, then we can simply replace Ω^{-1} by ω_1 in (1) or in (3).

Another nice feature of (1) is that the constraint in η can be regularized, for instance by

\((5)\ \ \ \ \ \ \phi^*_\gamma(e) = \inf_{\eta\in\mathbb{R}^n}\left\{ (p^0)^Te + \frac{\lambda^{-1}}{2}(e-\eta)^T\Omega^{-1}(e-\eta) + \sum_{k=1}^n\left| \frac{\eta_k}{\alpha} \right|^\gamma \right\},\ \ \ \ \gamma \gg 1\)

which is palatable to some nonlinear optimizer that perform well with separate-variable nonlinearities, like Mosek.

Quantitatively Yours,