PROBLEM
The goal of hedging is to offset the risk inherent in a perhaps illiquid position by taking positions in liquid instruments. While in theory this is achievable, in practice it is possible to do so only in an approximate way. There are numerous considerations that need to be taken into account. More frequent rebalancing will reduce tracking error, but increase transaction costs. There is often a trade-off between liquidity and basis risk, for example, the futures contract that best matches a particular exposure may be relatively illiquid. Would it be better to use a different contract instead? If so, how do we adjust the hedge for the resulting basis risk?
APPROACH
In order to address the questions posed in the preceding paragraph in a systematic manner, we need a quantitative approach to the hedging problem. The success of value-at-risk methods over the past five years has demonstrated the feasibility of quantifying the risk of a financial portfolio by a single number using statistical methods. We thus propose the following paradigm. Formalize the hedging problem as one of minimizing the risk of the hedged portfolio as assessed by an appropriate VaR measure. In the first instance, an appropriate measure is simply the variance of the hedged portfolio. The residual risk of the hedged portfolio provides a metric by which the risk of alternative hedging strategies can be compared with their cost. As a simple example, we can compare the transaction cost of adding an additional hedge with the projected risk reduction.
The minimum-VaR approach is a particularly appealing alternative for hedging interest rate instruments. Duration-based hedging approaches are limited in that they only hedge against parallel shifts in the yield curve. Other common approaches hedge against shifts in all the par rates used to build the yield curve, but their prescriptions require taking positions in each of the curve-building instruments. The minimum-VaR approach allows the operator to specify the precise hedge instruments to be used and computes an optimum hedge based on the relative probabilities of various yield curve movements. The well-known observation that almost all yield-curve movements can be explained by two or three principle components suggests that it should be possible to construct efficient hedges with a relatively small number of hedge instruments. The minimum-VaR hedge can be efficiently computed for, but not limited to, parametric variance/covariance VaR methodologies using delta approximations; it amounts to nothing more than solving a weighted-least-squares problem (see box).
BENEFITS
One of the key benefits of the minimum-VaR approach is the use of a single meaningful objective function provides a consistent means of extending the methodology. For example, limits on the size of positions in certain hedging vehicles can be accommodated simply by minimizing the risk measure subject to those constraints. A practical solution to the otherwise daunting problem of gamma hedging of interest-rate instruments is to minimize the quadratic or delta-gamma approximation of portfolio VaR. Using a statistical approach to address the problem of basis risk is nothing new; Professor John Hull uses correlations to compute hedge ratios in chapter 2 of his popular Options, Futures and other Derivatives. In the past, however, deployment of large-scale multivariate methods for this purpose would have entailed a substantial investment in development and maintenance. Today, the widespread availability of standardized VaR systems in the middle office makes such deployment possible with a relatively small incremental investment by the front office.
IMPLICATIONS
Nobel laureate Merton Miller commented on the Metallgesellschaft debacle in a talk given at the University of Munich June 12, 1995, "Finance professors also insist MGRM should have done a so-called 'strip' or maturity-matched hedge rather than a stack and roll. That would, indeed, have eliminated any basis risk, assuming, of course, that appropriate, low-cost futures and swaps of up to 10-years tenor had been available (which they were not). But MGRM was trying to manage basis risk, not eliminate it. After all, if the requisite, low-cost long-term hedging vehicles were readily at hand, the customers wouldn't have needed MGRM. They could have hedged their delivery requirements directly!" The minimum-VaR approach does provide a practical alternative between a rock and a hard place: hedging with every single hedging vehicle creates some obvious difficulties in both funding and execution, while exposure to basis risk by rolling a single hedging vehicle may lead to painful consequences. In a sense, this methodology is another piece of ammunition for risk practitioners to make the most out of a sub-optimal situation.
This week's Learning Curve was written by Alvin Kurucand Bernard Leeof Infinity, a SunGard company.
