B Portfolio mathematics

B.1 Marginal risk contribution

The portfolio beta interpretation of MRC:

\[\begin{align*} \frac{\partial}{\partial w_i} \Big [ \sigma(w) \Big ] & = \frac{\partial}{\partial w_i} \Big [ (w^\intercal \Sigma w)^{\frac{1}{2}} \Big ], & \\ & = \frac{1}{2} \cdot (w^\intercal \Sigma w)^{-\frac{1}{2}} \cdot \frac{\partial}{\partial w_i} \Big [ w^\intercal \Sigma w \Big ], & \text{(chain rule)} \\ & = \frac{\frac{1}{2}\cdot 2 \cdot (\Sigma w)_i}{\sqrt{w^\intercal \Sigma w}}, & \text{(as given in the main text)} \\ & = \frac{\mathbb{COV}[R_i, R_p]}{\sqrt{w^\intercal \Sigma w}}, & \text{($R_i$- returns on asset i)} \\ & = \mathbb{SD}[(R_p)] \cdot \frac{\mathbb{COV}[R_i, R_p]}{\mathbb{VAR}[(R_p)]}, & \\ & = \mathbb{SD}[(R_p)] \cdot \beta_{i, p}\;. & \end{align*}\]

Minimising the weighted sum of the of the MRC’s is equivalently the lowest beta portfolio with the single risk factor of the portfolio itself, which is the GMV portfolio. \[\begin{align*} \sum_{i = 1}^N w_i \cdot \frac{\partial}{\partial w_i} \Big [ \sigma(w) \Big ] & = \frac{\sum_{i = 1}^N w_i \cdot (\Sigma w)_i}{\sqrt{w^\intercal \Sigma w}}, & \\ \underset{w}{\text{argmin}} \Big \{\sum_{i = 1}^N w_i \cdot \frac{\partial}{\partial w_i} \Big [ \sigma(w) \Big ] \Big \} & = \underset{w}{\text{argmin}} \Big \{ \sqrt{w^\intercal \Sigma w} \Big\},& \text{(Taking the minimum on both sides)}\\ & = \underset{w}{\text{argmin}} \Big \{w^\intercal \Sigma w \} . & \text{($\sqrt{\cdot}$ monotonic)}\\ \end{align*}\]

B.2 Minimum weight constrained IHI

The maximum possible weight allocation for a single asset is \(\alpha\). This allocation should be given to \(\lfloor \frac{1}{\alpha} \rfloor\) assets where \(\lfloor \cdot \rfloor\) is the truncation function. An additional weight \(j\) should be set to satisfy the budget constraint, hence it should be \(w_j = 1 - \lfloor \frac{1}{\alpha} \rfloor \cdot \alpha\). All of the other weights should be set to 0. Then the lower bound is: \[\begin{align*} H_{\text{lb}}^{-1} & = \frac{1}{\lfloor \frac{1}{\alpha} \rfloor \cdot \alpha^2 + (1- \lfloor \frac{1}{\alpha}\rfloor \cdot \alpha)^2} \;\; . \end{align*}\]

B.3 Proof of the volatility order inequality

Consider a GMV framework where the IHI is bounded from below by a constant \(c\): \[\begin{align*} w^\diamond = \underset{w}{\text{argmin}} \Big \{ w^\intercal \Sigma w \Big \}, \end{align*}\] subject to the constraints: \[\begin{align*} \mathcal{C}(w) &= \begin{cases} H^{-1}(w) \geq c \\ w^\intercal \underline{1} = 1 \;\; \\ 0 \leq w_i \leq \alpha, \; \forall i \;\;\; . \end{cases} \end{align*}\]

If \(c \leq H^{-1}_{lb}\) then the constraint has no effect and the GMV portfolio is recovered. If \(c = N\) then the only feasible solution is the EW portfolio. If \(c > N\) there are no feasible solutions to the problem. The portfolio volatility is an increasing function of \(c\) therefore we can deduce that: \[\sigma(w_{gmv}) \leq \sigma(w^\diamond(c)) \leq \sigma (w_{ew}).\]

It remains to show that the ERC portfolio is a \(w^\diamond\) portfolio. Consider replacing the IHI constraint above with a log-constraint, \(\sum_i \ln(w_i) \geq c\). If \(c = - \infty\) then the GMV portfolio is recovered. If \(c = -n \ln(n)\) then the EW portfolio is recovered, and if \(c > -n\ln(n)\) then there are no feasible solutions. The portfolio volatility is also an increasing monotonic function of c therefore the inequality is replicated for a scaled choice of \(c\) and by extension the ERC portfolio.

B.4 Maximum Sharpe ratio risk-based portfolios

As mentioned in the text Scherer (2007) has shown that the marginal sharpe ratios are equalised for the MSR portfolio as below: \[\begin{align} \frac{\mu_i}{\text{MRC}_i} & = \frac{\mu_j}{\text{MRC}_j} \;\;\; \forall i, j \in \{1, ..., N \}, \tag{B.1} \end{align}\] where \(\mu_k\) represents the marginal excess return of asset k. Separately Jurczenko, Michel, and Teiletche (2013) showed that the EW, GMV and ERC portfolios could be found by the equalisation strategy: \[\begin{align} w_i^\gamma \sigma^{-\delta}_i \text{MRC}_i & = w_j^\gamma \sigma^{-\delta}_j \text{MRC}_j \;\;\; \forall i, j \in \{1, ..., N \}, \tag{B.2} \end{align}\] where the choice of \(\gamma\) and \(\delta\) defines the portfolio. Combining the portfolio condition from equality (B.2) with the optimality equality (B.1) yields the optimality condition for a given portfolio: \[\begin{align} w_i^\gamma \sigma_i^{(1- \delta)} \text{SR}_i & = w_j^\gamma \sigma_j^{(1- \delta)} \text{SR}_j \;\;\; \forall i, j \in \{1, ..., N \}, \end{align}\] where \(\text{SR}_k = \frac{\mu_k}{\sigma_k}\). Hence, a risk-based portfolio is optimal when constituents have equal weighted risk-adjusted SRs.

The EW portfolio can be analysed by setting \((\gamma, \delta) = (\infty, 0)\). For equlities (B.1) and (B.2) to be jointly true when \(w_i = \frac{1}{N}\) then: \[\text{MRC}_i = \text{MRC}_j \;\;\; \forall i, j \in \{1, ..., N \},\] and there needs to be identical excess returns between assets. This means there should also be identical volatilities and identical correlations between all assets (i.e.  \(\Sigma = \rho\sigma^2 \underline{1} \,\underline{1}^\intercal - \rho I\)). The GMV portfolio can be analysed by setting \((\gamma, \delta) = (0,0)\), which yields the optimality condition: \[\mu_i = \mu_j \;\;\; \forall i, j \in \{1, ..., N \}.\] Hence, identical excess returns are required only. The ERC portfolio can be analysed by setting \((\gamma, \delta) = (1, 0)\), which means the optimality condition becomes: \[w_i \sigma_i \text{SR}_i = w_j \sigma_j \text{SR}_j \;\;\; \forall i, j \in \{1, ..., N \}.\] Assuming constant correlation, \(\rho_{i, j} = \rho\), the ERC portfolio allocations are given by the weighted inverse volatilities in the portfolio \(w_i = \frac{\sigma^{-1}_i}{\sum_{k = 1}^N \sigma^{-1}_k}\) Maillard, Roncalli, and Teı̈letche (2010). The equality of SRs is then the requirement for the above shown condition to hold. Therefore, identical correlations and SRs ensure that the ERC portfolio coincides with the MSR portfolio.

References

Jurczenko, Emmanuel, Thierry Michel, and Jerome Teiletche. 2013. “Generalized Risk-Based Investing.” Available at SSRN 2205979.

Maillard, Sébastien, Thierry Roncalli, and Jérôme Teı̈letche. 2010. “The Properties of Equally Weighted Risk Contribution Portfolios.” The Journal of Portfolio Management 36 (4): 60–70.

Scherer, Bernd. 2007. “Can Robust Portfolio Optimisation Help to Build Better Portfolios?” Journal of Asset Management 7 (6): 374–87.