Legendre and Markowitz

A somewhat unorthodox approach to portfolio optimization

Let’s say we have a vector of forecast returns e = (e_1,…e_n) for the n assets we’re trading, and we want to compute the optimal portfolio p* that maximizes the expected return, net of transaction costs, and adjusted for quadratic risk. Assuming our initial position is p^0, and costs are proportional, and denoting by Ω the covariance matrix of the assets, we are led to the standard minimization problem

\((1)\ \ \ \ \ \ \sup_{p\in\mathbb{R}^n}\left\{ p^Te - \frac{\lambda}{2}p^T \Omega p - \alpha\sum_{k=1}^n|p_k - p^0_k| \right\},\)

where T is transposition, λ>0 is the risk-aversion coefficient, and α>0 is the proportional transaction cost (which could be made asset-dependent). The optimal portfolio is the value p = p* where the sup in (1) is achieved.

Let’s now adopt the following notations

\((2)\ \ \ \ \ \ \begin{equation*} \begin{split} &\phi_1(p) = \frac{\lambda}{2}p^T\Omega p\\ \\ &\phi_2(p) = \alpha\sum_{k=1}^n|p_k - p^0_k| \end{split} \end{equation*}\)

and note that φ—>φ* defined by

\((3)\ \ \ \ \ \ \phi^*(e) \equiv \sup_{p}\left\{ p^Te - \phi(p) \right\}\)

is the Legendre-Fenchel transform, we see that (1) can naturally be interpreted as the transform

\((4)\ \ \ \ \ \ \left(\phi_1 + \phi_2\right)^*(e) = \sup_{p}\left\{p^Te - \left(\phi_1(p) + \phi_2(p)\right)\right\}\)

Since the two functions in (2) are convex, we have the useful identity

\((5)\ \ \ \ \ \ \left(\phi_1 + \phi_2\right)^* = \phi_1^**_{inf}\phi_2^*,\)

where *_inf is the inf-convolution (see below). It is easy to compute the Legendre transform of φ_1 and φ_2

\((6)\ \ \ \ \ \ \begin{equation*} \begin{split} &\phi^*_1(\xi) = \frac{\lambda^{-1}}{2}\xi^T\Omega^{-1}\xi\\ \\ &\phi^*_2(\xi) = p_0^T\xi + \mathbb{1}^\infty_{|\xi|_\infty < \alpha}(\xi),\ \ \ \mathbb{1}^\infty_{|\xi|_\infty < \alpha}(\xi) = 0\ \ \hbox{if}\ \ |\xi|_\infty<\alpha,\ \ +\infty\ \ \hbox{otherwise} \end{split} \end{equation*}\)

Putting together (5) and (6), we get, for φ = φ_1 + φ_2:

\((7)\ \ \ \ \ \ \begin{equation*} \begin{split} \phi^*(e) = \phi_1^**_{inf}\phi_2^*(e) &= \inf_{\eta}\left\{ \frac{\lambda^{-1}}{2}(e-\eta)^T\Omega^{-1}(e-\eta) + \left(p^0\right)^T\eta + \mathbb{1}^\infty_{|\eta|_\infty < \alpha}(\eta) \right\}\\ \\ &= \boxed{\inf_{|\eta|_\infty < \alpha}\left\{ \left(p^0\right)^T\eta + \frac{\lambda^{-1}}{2}(e-\eta)^T\Omega^{-1}(e-\eta) \right\}} \end{split} \end{equation*}\)

We have therefore expressed (1) as another optimization problem involving the inverse of Ω instead of Ω. (Note that the computation above can also be interpreted via the dual problem of (1).) In practice, the formulation in (7) can be convenient if Ω is close to singular (large condition number), since we could choose a regularized pseudo-inverse to plug into (7), something that might not be as straightforward in (1).

Let’s see now how we can compute the optimal portfolio p* from (7). As is well-known, the gradient of a Legendre transform is the inverse function of the gradient of the original function (assuming convexity); therefore, taking account that p* is the point where the sup in (1) is achieved, we have

\((8)\ \ \ \ \ \ \begin{equation*} \begin{split} &D\phi(p^*(e)) = e\\ \\ &\boxed{p^*(e) = D\left(\phi^*\right)(e)}\\ \end{split} \end{equation*}\)

Practically speaking, all we need to do is therefore compute φ* in (7) using a standard quadratic optimizer with constraints; then the gradient of φ* at e is the optimal portfolio.

Let’s see finally how we can easily recover two trivial cases and thereby gain some intuition about (7). When α→0, clearly the inf in (7) is simply the value at η=0, and, computing the gradient, we get

\((9)\ \ \ \ \ \ p^* = \lambda^{-1}\Omega^{-1}e,\)

which is the usual optimal portfolio in the quadratic case without transaction costs. On the other hand, if α→∞, we find that the gradient, and therefore p*, is equal to p^0, as expected, since, in the presence of large transaction costs, it is optimal to not trade and keep our initial portfolio p^0.

Quantitatively Yours,