Optimal hedge ratio under a subjective re-weighting of the original measure

ABSTRACT

In this article we study a risk-minimizing hedge ratio with futures contracts, where the risk of the hedged portfolio is measured through a spectral risk measure (SRM), thus incorporating the degree of agent’s risk aversion. We empirically estimate the optimal hedge ratio (OHR) using a long time series of UK and US equity indices, the EURUSD and EURGBP exchange rates and four liquid commodities (Brent crude oil, corn, gold and copper), to represent different asset classes. Comparing the results with common OHRs (such as the minimum variance and the minimum expected shortfall), we find that the agent’s risk aversion has a material impact, and should not be ignored in risk management.

KEYWORDS:

• Risk management
• spectral risk measures
• expected shortfall
• risk aversion

JEL CLASSIFICATION:

• G30
• G32

Acknowledgements

We would like to thank the editor (Mark P. Taylor) and two anonymous referees for their helpful suggestions. We also thank Emanuele Bajo and the participants at the 2012 MAF conference in Venice. All errors are our responsibility.

ORCID

Massimiliano Barbi http://orcid.org/0000-0003-1250-5148

Silvia Romagnoli http://orcid.org/0000-0002-4945-0673

Notes

¹ Quadratic preferences imply negative marginal utility after a certain level of wealth and increasing absolute risk aversion, and therefore they are not representative of investor’s behaviour. Also, empirical financial literature strongly rejects the assumption of normally distributed returns, and there is evidence that moments higher than 2 are priced by the market (Kraus and Litzenberger 1976; Harvey and Siddique 2000; Dittmar 2002).

² Also, a number of downside risk measures have been used to compute the hedge ratio. To name the more relevant in the literature, we cite the mean extended-Gini coefficient and the generalized semi-variance (or lower partial moment) (see Chen, Lee, and Shrestha 2003).

³ The conditions under which maximum expected utility hedge ratio is equal to MVHR are the following: (a) futures prices are unbiased (i.e. futures prices follow a martingale), and (b) spot returns can be written as a linear function of futures returns, and futures returns are independent of residual randomness of spot returns. In particular, condition (b) holds in case spot and futures returns are jointly normally distributed.

⁴ According to the dual theory of choice, preference relations are represented comparing the expected value of random variables computed with respect to a transformed (changed) probability measure.

⁵ We ignore the different sizes of the spot and the futures position. In this way, h can be interpreted as the relative size of the futures position with respect to a unitary spot position. We do not lose in generality, as the problem can be straightforwardly modified to incorporate the unequal size of both positions.

⁶ The differentiability (or at least a forced differentiability) of a SRM depends on the differentiability of the quantiles, and requires some technical assumptions on the joint probability distribution. We refer the reader to Tasche (2001) for the case of VaR and expected shortfall. For a general SRM, Tsanakas (2009) proves that the Gateaux differentiability is guaranteed for convex risk measures, a set including also SRMs.

⁷ We use Brent oil futures traded at the ICE (Futures division), corn futures traded at the CBOT, gold futures traded at the NYMEX (COMEX division) and copper futures traded at the LME.

⁸ Spot and futures gold time series exhibit a notably lower correlation relative to other assets. This is likely due to the fact that futures are traded and exchanged in the United States, whilst spot prices are dominated by trading in London. Consequently, spot prices have less time to respond to daily news in the United States due to the 5 hours difference (Roache and Rossi 2009).

⁹ The bounds of this arbitrary interval are never binding in our optimization procedure. The lower bound cannot be exceeded, as for positive correlations between spot and futures returns (a necessary condition to hedging) a long (short) spot position requires a short (long) futures position. The upper bound is sufficient in practice even for extremely volatile asset classes.