**Appendix A. Optimal Hedging**

The optimal hedge ratio is the one that minimises the variance of the hedged portfolio's return. To hedge quantity *Q*1 of an asset using quantities of representative combinations of hedge instruments denoted *Q*2, *Q*3 and *Q*4 gives the following hedge ratios (where *ρij* is the correlation between returns on assets *i* and *j*, and *σi*is the standard deviation of returns on asset *i*).

,

**One hedge instrument:** *Q*2 *Q*1 = −*ρ*12*σR*<sup>1</sup> *<sup>σ</sup>R*2 *σ*2*R*2 , **Two hedge instruments:** *Q*2 *Q*1 = −(*ρ*12−*ρ*13*ρ*23)*<sup>σ</sup>*1*σ*<sup>2</sup> (1−*<sup>ρ</sup>*223)*σ*22 and *Q*3 *Q*1 = −(*ρ*13−*ρ*12*ρ*23)*<sup>σ</sup>*1*σ*<sup>3</sup> (1−*<sup>ρ</sup>*223)*σ*23 **Three hedge instruments:**

$$\begin{split} \frac{Q\_{2}}{Q\_{1}} &= -\frac{(1-\rho\_{34}^{2})\rho\_{12} - (\rho\_{23}-\rho\_{24}\rho\_{34})\rho\_{13} + (\rho\_{23}\rho\_{34}-\rho\_{24})\rho\_{14}}{(1-\rho\_{23}^{2}-\rho\_{24}^{2}-\rho\_{34}^{2}+2\rho\_{23}\rho\_{24}\rho\_{34})\sigma\_{2}}, \\ \frac{Q\_{3}}{Q\_{1}} &= -\frac{(1-\rho\_{24}^{2})\rho\_{13} - (\rho\_{23}-\rho\_{24}\rho\_{34})\rho\_{12} + (\rho\_{23}\rho\_{24}-\rho\_{34})\rho\_{14}}{(1-\rho\_{23}^{2}-\rho\_{24}^{2}-\rho\_{34}^{2}+2\rho\_{23}\rho\_{24}\rho\_{34})\sigma\_{3}}, \\ \frac{Q\_{4}}{Q\_{1}} &= -\frac{(1-\rho\_{23}^{2})\rho\_{14} - (\rho\_{34}-\rho\_{24}\rho\_{23})\rho\_{13} + (\rho\_{23}\rho\_{34}-\rho\_{24})\rho\_{12}}{(1-\rho\_{23}^{2}-\rho\_{24}^{2}-\rho\_{34}^{2}+2\rho\_{23}\rho\_{24}\rho\_{34})\sigma\_{4}}. \end{split}$$

Correlations used in the the above calculations can be derived from dynamic estimates of variances as described in Gibson et al. (2017). For example, since the variance of (X + Y) = variance(X) + Variance(Y) + 2 Covariance(X,Y) the Covariance can be constructed from a rearrangemen<sup>t</sup> of estimates of variance(X + Y), variance(X) and Variance(Y).
