Robust risk budgeting

Abstract

Risk based portfolio construction and particular risk parity or equally weighted risk contribution became popular among practitioners. These approaches focus only on risk and are agnostic with respect to the expected returns. In this paper, we consider risk budgeting; a generalization of risk parity. We propose an alternative formulation that is more efficient computationally. We introduce the robust risk budgeting, a robust variant of the standard risk budgeting that deals with the uncertainty in the input parameters. We show that the problem remains tractable under different types of uncertainty. We evaluate the proposed framework on real data and we observe a positive premium associated with the robust variant.

A general case: NLP-SDP formulation

Recall, the robust optimization problem under the general case for the uncertainty definition (8–10)

$$\begin{aligned} \min _{y \ge 0} \max _{Q \in {\mathbb {Q}}} ~&~ \varvec{y} ^\top Q \varvec{y} - \sum _{i=1}^n b_i \ln y_i\end{aligned}$$

(18)

$$\begin{aligned} \text {s.t. } ~&~ Q^l \le Q \le Q^u \end{aligned}$$

(19)

$$\begin{aligned}&~ Q \succcurlyeq 0. \end{aligned}$$

(20)

For a fixed $\varvec{y}$, since $\varvec{y} ^\top Q \varvec{y} = \Big < Q,\varvec{y} \varvec{y} ^\top \Big>$, the inner problem can be written as

$$\begin{aligned} \max _{{\mathbb {Q}}}~&~ \Big \{ \Big < Q,\varvec{y} \varvec{y} ^\top \Big >:~ Q - Q^l \ge 0,~ -Q+Q^u \ge 0,~ Q\succcurlyeq 0 \Big \}, \end{aligned}$$

(21)

where $\Big < A,B \Big > = Trace(AB).$ The Lagrangian of the above is

$$\begin{aligned} {\mathcal {L}}(.) = \Big< Q,\varvec{y} \varvec{y} ^\top \Big> + \Big< Q-Q^l,L \Big> + \Big< -Q+Q^u,U \Big>+ \Big <Q,Z \Big >, \end{aligned}$$

(22)

where $U\ge 0$, $L\ge 0$ and $Z\succeq 0$. Therefore, the dual of the above is

$$\begin{aligned}&~\min _{U \ge 0, L \ge 0} \Big \{ \Big< U,Q^u \Big> - \Big < L,Q^l \Big > : \varvec{y} \varvec{y} ^\top - U - L + Z = 0, Z\succeq 0 \Big \} \end{aligned}$$

(23)

$$\begin{aligned} \Rightarrow ~&~ \min _{U \ge 0, L \ge 0} \Big \{ \Big< U,Q^u \Big> - \Big < L,Q^l \Big > : \varvec{y} \varvec{y} ^\top - U - L \preceq 0 \Big \}. \end{aligned}$$

(24)

Therefore, the optimization problem (18–20) becomes

$$\begin{aligned} \min _{\varvec{y} \ge 0, U \ge 0, L \ge 0} ~&~ \Big<U, Q^u \Big> - \Big < L, Q^l \Big > - \sum _{i=1}^n b_i \ln y_i \end{aligned}$$

(25)

$$\begin{aligned} \text {s.t. } ~&~ \varvec{y} \varvec{y}^\top - U + L \preceq 0. \qquad \qquad \qquad \end{aligned}$$

(26)

$\square$