BSP Research Academy for the invitation

The authors for the chance to read theoretical work

• Very gratifying to see a paper like this being written in the Philippines
• A cool result which is easy to deploy in practice

Goal: Explain the findings of Paskaramoorthy and Woolway (2022) by refining the bounds of Best and Grauer (1991)

Purpose:

• Theoretical: Convex analysis proof, refined bounds
• Practical: Use as signal to consider stabilizing optimization routine

Presumed audience:

• Portfolio manager: Should I still use Excel Solver?
• Your inner mathematician: Connections, geometry
• Future teachers of portfolio theory: I could actually teach this.

(New-ish) Where is the source of instability of portfolio weights?

(New-ish) How seriously should we take Markowitz?

Best and Grauer (1991) bounds vs the authors \[\begin{eqnarray*} && \Bigl\Vert\mathsf{ideal\;weights}-\mathsf{perturbed\;weights}\Bigr\Vert_{2}\\ &\leq & \dfrac{\left(\mathsf{estimation\;error}\right)*\left(1+\mathsf{\color{red}{condition\;number}}\right)}{\left(\mathsf{risk\; aversion\;parameter}\right)*\left(\mathsf{minimum\;eigenvalue}\right)}\end{eqnarray*}\]

• Sharpened bounds
• Refined lower bound if feasible set were affine

Provided that feasible set is affine, orthogonality condition for the optimal portfolio weights

• Difference between optimal weights and any other valid weights is orthogonal to the gradient of the objective function
• Without the affine restriction, you lose orthogonality.

The bounds are based on perturbing the vector of returns by \(\tau \mathbf{q}\) with \(\mathbf{q}\) having length 1.

• Consider perturbing the covariance matrix of returns too.
• Perhaps use the opportunity to figure out how shrinkage improves the bounds?
• Exploit results from perturbation of positive definite matrices?

The main figure (Figure 1) shows the consequences of estimation error on the Sharpe ratio.

• Why not frame in terms of the Sharpe ratio?
• Ao, Li, and Zheng (RFS 2019) have already shown that as the number of assets grow at a slower pace compared to the number of observations, the ratio of the plugin Sharpe ratio to the theoretical maximum Sharpe ratio converges in probability to a limit strictly less than 1.
• Can we have non-stochastic bounds for the Sharpe ratio for perturbations of the return vector?

Is there a way to use the convex-analytic results to estimate the risk aversion parameter given observations on portfolio choices and asset returns?

• Highly speculative from my end
• Eq 13 feels like one of those moment inequality restrictions.
• Eq 14 feels like one of those moment equality restrictions.
• Akin to testing rationality via WARP restrictions.