14. Currency Hedging for a Multi-national Firm
14.4 Comparison of Hedging Strategies Over a Single Year
The primary concern of the CFO is the probability distribution of the asset value w T at the end of the 12 month period. This represents a risk and return profile where preferences are subjectively formed based on the distributions of w T . For this reason, we consider a number of runs with the model exploring alternatives for such distribution for year 2004. Case 0 is without hedging. Case 1 mimics the current hedging practice. It is advanced in Case 2 relaxing previous hedging restrictions and using expected utility maximization. Speculative hedging with expected value maximization is presented in Case 3 and maximization of a weighted average of the worst case and the expected value is the
objective in Case 4. We discuss the results basd on E[w T ] and the pdf of w T . Expected hedging levels and asset values by time stage for Cases 0–2 are shown in Table 14.4 of the Appendix.
Case 0: No hedging. Consider the case without derivatives and $ positions fixed to zero. The distribution of the terminal value w T is shown in Fig. 14.5 (blue curves) using exchange rate models EqC (left) and TrE (right). Expected terminal value E[w T ] is 27.5 M€ for EqC and is 26.9 M€ for TrE.
Fig. 14.5 Probability distribution of terminal assets value w T (M€) with exchange rate models EqC (left figures) and TrE (right figures). In each figure Case 0 is depicted in blue and the red curve refers to Case 1 (first row), Case 2 (second row), Case 3 (third row), Case 4 (bottom row)
Case 1: Current hedging policy. This case mimics the current hedging practice: We prohibit both put and call options as well as forwards to buy dollars. Cash in dollars is kept at zero. Hence, we restrict cash d t , forwards b st put options p xst and call options c xst to zero. For t ≤ 3, 50% of
dollar revenues R t are hedged using forwards. For t > 3, we require that at most 50% of dollar
revenues are hedged. Furthermore, revenues at stages T and T − 1 can only be hedged at the preceding stages T − 1 and T − 2, respectively. Because revenues at stage 3, for instance, can be alternatively hedged at stages 0, 1 and 2, there is some freedom to choose. Therefore, we choose the hedging strategy employing the expected terminal value criterion (14.15).
Figure 14.5 (top, red curves) shows the distribution of the terminal value w T with EqC (left) and TrE (right). We observe that the distributions of Cases 0 and 1 quite similar, and expected terminal value w T increases from Case 0 by 0.1 M€ only. Such similarity may be explained by the fact that an unhedged dollar amount in Case 0 yields a loss if the dollar weakens and a gain in the opposite case compared with Case 1. When we take into account all scenarios and all dollar revenues over time, such gains and losses tend to cancel each other, and consequently at the end at stage T, the outcome in Case 0 without hedging is approximately the same as in Case 1 employing hedging with forwards.
Case 2: Optimal hedging with expected utility. Case 1 is advanced by allowing (but not requiring) all dollar revenues R t be hedged with forwards s st and puts p st , and forwards b st and calls c st to buy are available as well. Hence, in this case constraint (14.12) applies to selling dollars.
Similarly, buying dollars using forwards and calls is limited to R t . Expected utility criterion (14.14) is used with risk aversion coefficient γ = 0. 5/M€. Figure 14.5 (second row, red curves) show the distributions of w T with EqC (left) and TrE (right). Expected terminal value w T increases from Case 0 by 0.9 M€ for EqC and 0.7 M€ for TrN.
Case 3: Speculative hedging. Next, we consider an increasingly speculative hedging strategy.
We take Case 2 to begin with allowing all hedging instruments to be employed and use criterion (14.15) of maximizing expected terminal value w T . Figure 14.5 (third row, red curves) show the distributions of w T with EqC (left) and TrE (right). Expected terminal value w T increases from Case 0 by 1.5 M€ for EqC and 0.9 M€ for TrN.
As discussed in Sect. 14.3.4, expected value maximization can be an appropriate criterion if similar decision problems repeat over time. To illustrate, consider two and five independent 12 month hedging problems, each one being identical to the case above. If the optimal hedging strategy for a single year is adopted in each year, then the distribution of the average terminal value per annum has the same expected value as above. However, the variance decreases. Figure 14.6 shows the results of a single year (blue curve) as well as for its two and five independent repetitions (red and green curves, respectively).
Fig. 14.6 Probability distribution of terminal asset value w T (M€) in Case 3. Results are shown both for a single year (blue), as well as for two (red) and for five (green) independent repetitions
Case 4: Weighted objective. This case is obtained from Case 2 by switching to criterion
(14.16), where the objective is to maximize the weighted average of the expected value and the worst case of the terminal value w T . We use equal weights so that λ = 0. 5 in (14.16). Figure 14.5 (bottom row, red curves) show the distributions of w T with EqC (left) and TrE (right). Expected terminal value w T increases from Case 0 by 1.4 M€ for EqC and 0.7 M€ for TrN.