October 2007

EDHEC RESEARCH: Non-linear risk

In the fourth of a series of articles on risk management techniques in asset management, we look at the benefits of hedging risks through non-linear beta management. By Noël Amenc and Lionel Martellini As recalled in the first article of this series (Funds Europe, May 2007), two different approaches to risk management can be followed. The first approach, which we described previously, consists of risk diversification, ie, reducing risk by optimal asset allocation techniques on the basis of imperfectly correlated assets. The second approach, which we will describe now, consists of risk hedging, ie, reducing risk by using some form of insurance contract or derivative instrument on a given underlying asset. Diversification and hedging are two different, and perhaps competing, forms of risk management. Asset pricing theory actually allows us to better understand the nature of the relationship between allocation and structuring from a conceptual standpoint. It can be argued that a structured product approach to risk management, based on hedging, can be regarded as the most general, dynamic, as opposed to static, form of asset allocation. It is indeed well known, since Merton’s (1973) replicating argument interpretation of the Black and Scholes (1973) formula, that a non-linear payoff based on an underlying asset can be replicated by dynamic trading in the underlying asset and the risk-free asset.  As a result, it appears that an investor willing and able to engage in dynamic asset allocation strategies will be in a position to generate the most general form of risk management possible, and this encompasses both static diversification and dynamic hedging. While the benefits of dynamic asset allocation strategies in a stochastically time-varying environment have been recognised since the late 1970s (see Hakanson (1969, 1971)  and Samuelson (1969), in a discrete-time setting, as well as Merton (1971), in a continuous-time setting, for the development of a multi-period approach to optimal asset allocation decisions, it is only recently that specific optimal asset allocation models that exhibit explicit time-dependency in the presence of stochastic opportunity sets have been introduced. With dynamic asset allocation, portfolio weights change through time, either as a response to changes in the investment opportunity set or in an attempt to generate non-linear payoff structures. This is opposed to a buy-and-hold strategy where weights evolve according to changes in prices, but also to a fixed weight strategy, where rebalancing is allowed to revert to the initial weights, hence severely restricting the kind of payoffs generated to simple linear functions of underlying asset classes. When it comes to implementation, a hedging strategy can be implemented either through direct trading in derivatives (eg, OTC options), a case we discuss first, or via the investment in structured products which result from the packaging of both the underlying asset and the insurance contract (or the suitably designed equivalent dynamic portfolio strategy), a situation we consider in next month’s article. Risk hedging with derivatives
The investment management process comprises a variety of tasks and involves both passive (indexing) and active (timing and picking) strategies. All these tasks may effectively be facilitated by using derivatives. In particular, futures instruments help investors or managers to neutralise biases in terms of factor exposure that may result from bond picking bets. Conversely, an investor may decide to protect himself from market risk in order to conserve only the return linked to the active bets taken by a manager. Another natural use of futures is for timing strategies between different styles within the equity universe or the bond universe. In what follows, we focus on how the use of options can further improve risk management thanks to their ability to generate non-linear, convex payoffs that offer downside risk protection. A favourite strategy with investors and asset managers alike is a protective put buying (PPB) strategy. This strategy consists of a long position in the underlying asset and a long position in a put option, which is rolled over as the option expires. It should be underlined that PPB is different from portfolio insurance, since the put is rolled over in each sub-period. Therefore, the payoff at the end of a total period with multiple sub-periods does not simply correspond to a guaranteed minimum payoff, as in the case of portfolio insurance. At the end of every sub-period, however, the long position in the put option offers a protection against downside risk, which leads to avoiding the left tail of the returns distribution. The PPB strategy has been widely studied in the context of equity portfolio management (see Merton et al (1982), Figlewski et al (1993), as well as in the context of fixed-income portfolio management (Goltz, Martellini and Ziemann (2006), from which we borrow the forthcoming illustration. Choice of strike price
We construct the PPB strategy in the following way, which is similar to Merton et al (1982, p.35). The portfolio held is made up of a number N of the underlying bond, plus the same number of put options. We then scale the initial investment to be equal to an amount I, say e100. Put options with time to expiration equal to   are bought at so that they expire at. Hence, an option pricing formula is only needed to establish the premium at the initial investment and when the strategy is rolled over. One outstanding question is the choice of the strike price of these options. Given the fact that exchange-traded options are issued with strike prices rather close to the current price of the underlying asset, an investor will choose from this proposed range of strike prices and end up with options that are not too far in or out of the money. The typical range of moneyness considered in the literature on options strategies is from 10% out of the money to 10% in the money. Hence, Merton, Scholes and Gladstein (1982) consider moneyness of -10%, 0% and 10%. Likewise, Figlewski, Chidambaran and Kaplan (1993) consider moneyness of -10%, -5%, 0%, 5% and 10%. As outlined in Merton, Scholes and Gladstein (1982), there is no single best alternative for the strike price. Instead, the choice depends on the preferences of the investor, such as his risk tolerance with a trade off versus the cost of the hedging strategy. In what follows, we decide to set the strike price to 10% out of the money. This corresponds to an investor who is willing to take on some downside risk in any sub-period and is concerned about decreasing profitability of the strategy when the strike price is increased (see Macmillan (2000, Chapter 17). Fabozzi (1996, Chapter 16) highlights that in the context of bond portfolio management, protective puts are usually implemented with out of the money puts on bonds or futures. The Bund futures strategy
The above table, extracted from Goltz, Martellini and Ziemann (2006), shows the results obtained in the context of a base case experiment with parameter values estimated stemming from a historical calibration of a two-factor interest rate model. The left part of the table indicates the percentiles of the return distribution over 1,000 scenarios. The right part of the table indicates some standard performance measures based on this distribution. The information ratio is calculated with respect to the Bund futures strategy. “Bond” denotes the Bund futures strategy. “PPB” denotes the protective put buying strategy.In order to assess the results for the PPB strategy with Bund futures and options on Bund futures compared to the position in the Bund futures only, we look at the portfolio returns after one year. The table shows both the percentiles of the returns distribution and typical performance statistics. It should be underlined that all these statistics are based on the distribution of the final portfolio value across the 1,000 scenarios that we generate. This is different from calculating such statistics from a time-series of asset returns, as is done in empirical studies.Favourable conclusion
Examining the performance statistics leads to the conclusion that the PPB strategy is largely favourable. In particular, the mean return also increases. This stems from the fact that the put option is exercised in scenarios with strongly negative returns. Consequently, the left tail of the returns distribution is cut off, which increases the mean return. This effect is equally apparent from the higher skewness of the PPB strategy. The figure below shows the return distributions for both the futures strategy (“Bond”) and the PPB strategy.Return distributions for both the futures strategy (“Bond”) and the PPB strategy.Inspection of the probability distribution functions of annual returns confirms the aforementioned results. Focusing on negative returns below 7%, it can be seen that the PPB strategy has less frequent losses of this magnitude.  In the final article of our series next month we shall examine risk hedging with structured products and draw some overall conclusions on risk management techniques in asset management.  • Noël Amenc is director of the Edhec Risk and Asset Management Research Centre, and Lionel Martellini is scientific director of the Edhec Risk and Asset Management Research Centre This article is based on research included in the EDHEC Publication, “The Impact of IFRS and Solvency II on Asset-Liability Management and Asset Management of Insurance Companies,” by Noël Amenc, Philippe Foulquier, Lionel Martellini and Samuel Sender, November 2006. This research was sponsored by AXA Investment Managers.

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