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121 Hedging j Speculation Expected Return of Primary Investment Optimal Unhedged RISK Static Primary Hedge Investment FIGURE 11.5 Set of feasible portfolios available to an investor when he/she implements a currency hedging strategy. The efficient frontier lies on the upperleft edge of the set (gray area), or along the darkened edge in the figure. of the primary investment are denominated. In practice, this reduction in risk may be purchased at the price of slightly reduced returns due to transactions costs, thus the circle that intersects this risk line is lowered to slightly below the expected return of the primary investment in this example.14 This optimal static hedge has succeeded in reducing risk. However, note that there may still be some distance between the static point and the efficient frontier of the feasible set, which is defined by the use of dynamic allocation to foreign currencies through trading models. The distance between the statically hedged portfolios and the efficient frontier marks the improvement that can be attained through a dynamic currency overlay using trading models. The subject of currency hedging has been discussed in the literature for some time. In Froot (1993), full hedging is recommended for minimizing the risk due to shortterm FX volatility. On the other hand, it is also shown that a lower amount of hedging or even no hedging is better for minimizing the risk of longterm value 14 Whereas the transactions costs of a onetime, static FX transaction are minimal, the short positions in the foreign currencies may imply considerable costs of carry due to interest rate differential between the two currencies. These costs, which are sometimes also in favor of the investor, may lead to a distinct return difference between statically hedged and unhedged portfolios.
fluctuations of the investment. Depending on the time horizon, there is thus a certain range, a scope within which the hedging ratio can be chosen. Levich and Thomas (1993a) go one step further. They hedge a position dynamically by varying the hedge ratios over time. They show that this is profitable as compared to no hedging or static hedging. In their most successful strategy, a "currency overlay" with many currencies involved, they change the hedge ratios by following simple "technical trading signals." In the overlay strategy described here, the allowed ranges of static hedging and the exposure due to dynamic hedging are limited.15 Thus the main purpose of hedging, which is reducing the risk due to FX rate volatility, is maintained. On the other hand, we have a wellfounded additional profit expectation, based on the profitability of the trading models and trading model portfolios. 11.8.1 The Hedging Ratio and the "Neutral Point" Currency hedging means, for an investor who has bought a foreign asset such as equity of value s, holding a short position of size  st, in the foreign currency in order to minimize the volatility of the value of his/her total position due to FX rate fluctuations. The hedging ratio h is defined as h =  (11.31) The study by Froot (1993) shows that choosing the best h, the one that minimizes the total volatility, is not trivial and depends on the time horizon of the investor. For shortterm investors, the best h is 1 or slightly less; for longterm investors, who hold their position over many years, the best choice of h is about 0.35. In Froot (1993), the hedging ratio h is assumed to be constant over time. In our dynamic hedging approach (we follow here the method suggested in Muller et al., 1997b), we want to vary h over time, following some realtime trading models to reach an additional profit or to reduce the risk of the primary investment expressed in the home currency. This requirement will, when used during optimization, automatically set limits to the type of hedging strategy to use. In the lack of a clear criterion like reducing the risk, some rules could be introduced to achieve desirable features like, for instance, h would not depart too much from the best valuethat is, in most cases, between 0.35 and 1 according to Froot (1993). Each of the individual foreign currencies has its own hedging ratio hi, but there is also the option of taking only one global hedging ratio h for all currencies. The following discussion applies to both the individual / , and h. In lack of an objective risk criterion, most investors set some limits hm\n and / on the choice of the hedging ratio, often to satisfy some institutional constraints or to limit the risk if they have no other quantitative criterion to do this. A typical choice might be hm\n = 0 (no hedging) and / = 1 (full hedging). 15 Interested readers are referred to our internal paper where the methodology is described in detail, Muller el al. (1997b).
In the middle between the extreme h values, we define the neutral point, h, of the hedging strategy to be hm\d = (11.32) and the total possible range Ah of dynamic hedging is Ah = /!max/*min (11.33) We have seen that the O&A realtime trading models vary their gearings between two limits called "gearing1" and"gearing 1" as explained in Section 11.2.1. If the neutral point / ; is chosen for static hedging and if there is only one foreign currency and one trading model (for the FX rate between that foreign currency and the home currency), these trading limits directly correspond to the limits / ; and max However, the investor may decide to allow wider exposure limits for the dynamic positions than those of the static positions. The situation becomes more complicated in the presence of many currencies and many trading models. 11.8.2 Risk/Return of an Overlay with Static and Dynamic Positions To solve the allocation problem, a basis of portfolio theory has to be applied to our particular overlay problem. A portfolio can be written in terms of the sum of the primary investment (PI), a static foreign exchange position placed in order to hedge foreign exchange risk (SH) and a series of variable positions placed to dynamically hedge foreign exchange risk (DH) Tl = PI + SH + DH (11.34) In what follows, we refer to a portfolio diversified in / = 1 to n currencies and dynamically hedged by j:  1 to m trading models: n n m 1 = 1 1 = 1 j = \ Expectation of portfolio returns, or changes in value, can then be written as E[ ], or as a sum of the expectation values of its comprising parts: [ ] = [ /] + [ /?/ :]+ 1 7 ] (11.36) that is, n n m [ ] = a(EA/,] 4 ] / , [ /"*] 4 ] cojE[AR.J ] (11.37) /=1 ;=1 ;=1
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