Trade and cross hedging exchange rate risk

Abstract

This paper examines the behavior of a competitive exporting firm that exports to two foreign countries under multiple sources of exchange rate uncertainty. The firm has to cross hedge its exchange rate risk exposure because there is only a forward market between the domestic currency and one foreign country’s currency. When the firm optimally exports to both foreign countries, we show that the firm’s production decision is independent of the firm’s risk attitude and of the underlying exchange rate uncertainty. We show further that the firm’s optimal forward position is depending on whether the two random exchange rates are correlated in the sense of expectation dependence. Our results refine the literature on cross-hedging by introducing the expectation dependence structure. The existing of risk-sharing institutions, such as forward markets, significantly modify the impact of uncertainty on international trade in the economy.

Appendix

We formulate program (2) as a two-stage optimization problem. In the first stage, the firm chooses the optimal amount of exports to country 1, x ₁(x ₂), and the optimal forward position, h (x ₂), for a given amount of exports to country 2, x ₂. In the second stage, the firm chooses the optimal amount of exports to country 2, $x_{2}^{*}$, taking x ₁(x ₂) and h (x ₂) as given. The complete solution to program (2) is thus $x_{2}^{*}$, $x_{1}^{*}=x_{1}(x_{2}^{*})$, and $h^{*}=h(x_{2}^{*})$.

The solution to the first-stage optimization problem must satisfy the following Kuhn-Tucker conditions:

$$ \mathrm{E}\left\{u^{\prime}[\tilde{\pi}(x_{2})]\{\tilde{e}_{1}p_{1}-c^{\prime}[x_{1}(x_{2})+x_{2}]\}\right\}\leq 0, $$

(A.1)

and

$$ \mathrm{E}\{u^{\prime}[\tilde{\pi}(x_{2})][\mathrm{E}(\tilde{e}_{1})-\tilde{e}_{1}]\}=0, $$

(A.2)

where $\tilde {\pi }(x_{2})=\tilde {e}_{1}p_{1}x_{1}(x_{2})+\tilde {e}_{2}p_{2}x_{2}-c[x_{1}(x_{2})+x_{2}] +[\mathrm {E}(\tilde {e}_{1})-\tilde {e}_{1}]h(x_{2})$. If x ₁(x ₂) > 0, condition (A.1) holds with equality. Multiplying p ₁ to Eq. A.2 and adding the resulting equation to condition (A.1) yields

$$ \mathrm{E}(\tilde{e}_{1})p_{1}-c^{\prime}[x_{1}(x_{2})+x_{2}]\leq 0, $$

(A.3)

since u′(π) > 0. For x ₂ sufficiently small such that $c^{\prime }(x_{2})<\mathrm {E}(\tilde {e}_{1})p_{1}$, it follows that x ₁(x ₂) > 0 and inequality (A.3) holds with equality. Thus, when x ₂ = 0, we have

$$ \mathrm{E}(\tilde{e}_{1})p_{1}-c^{\prime}[x_{1}(0)]=0. $$

(A.4)

In this case, h(0)=p ₁ x ₁(0) solves (A.2) since $\pi (0)=\mathrm {E}(\tilde {e}_{1})p_{1}x_{1}(0) -c[x_{1}(0)]$, which is non-stochastic.

Let EU be the objective function of program (2) with x ₁=x ₁(x ₂) and h = h (x ₂). Totally differentiating EU with respect to x ₂, using the envelope theorem, and evaluating the resulting derivative at x ₂ = 0 yields

$$ \frac{\mathrm{d}EU}{\mathrm{d}x_{2}}\bigg|_{x_{2}=0}= u^{\prime}[\pi(0)]\{\mathrm{E}(e_{2})p_{2}-c^{\prime}[x_{1}(0)]\}. $$

(A.5)

Substituting (A.4) into the right-hand side of Eq. (A.5) yields

$$ \frac{\mathrm{d}EU}{\mathrm{d}x_{2}}\bigg|_{x_{2}=0}= u^{\prime}[\pi(0)][\mathrm{E}(\tilde{e}_{2})p_{2}-\mathrm{E}(\tilde{e}_{1})p_{1}]. $$

(A.6)

If $\mathrm {E}(\tilde {e}_{1})p_{1}\geq \mathrm {E}(\tilde {e}_{2})p_{2}$, Eq. A.6 implies that $x_{2}^{*}=0$. We then know from Eq. A.4 that $x_{1}^{*}$ solves $c^{\prime }(x_{1}^{*})=\mathrm {E}(\tilde {e}_{1})p_{1}$ and $h^{*}=p_{1}x_{1}^{*}$. This proves part (i) of Proposition 1.

If $\mathrm {E}(\tilde {e}_{1})p_{1}<\mathrm {E}(\tilde {e}_{2})p_{2}$, Eq. A.6 implies that $x_{2}^{*}>0$. In this case, inequality (4) holds with equality. Let us reformulate program (2) as a two-stage optimization problem. In the first stage, the firm chooses the optimal amount of exports to country 2, x ₂(x ₁), and the optimal forward position, h (x ₁), for a given amount of exports to country 1, x ₁. In the second stage, the firm chooses the optimal amount of exports to country 1, $x_{1}^{*}$, taking x ₂(x ₁) and h (x ₁) as given. The complete solution to program (2) is thus $x_{1}^{*}$, $x_{2}^{*}=x_{2}(x_{1}^{*})$, and $h^{*}=h(x_{1}^{*})$.

The solution to the first-stage optimization problem must satisfy the following first-order conditions:

$$ \mathrm{E}\bigg\{u^{\prime}[\tilde{\pi}(x_{1})]\{\tilde{e}_{2}p_{2}-c^{\prime}[x_{1}+x_{2}(x_{1})]\}\bigg\}=0, $$

(A.7)

and

$$ \mathrm{E}\{u^{\prime}[\tilde{\pi}(x_{1})][\mathrm{E}(\tilde{e}_{1})-\tilde{e}_{1}]\}=0, $$

(A.8)

where $\tilde {\pi }(x_{1})=\tilde {e}_{1}p_{1}x_{1}+\tilde {e}_{2}p_{2}x_{2}(x_{1})-c[x_{1}+x_{2}(x_{1})] +[\mathrm {E}(\tilde {e}_{1})-\tilde {e}_{1}]h(x_{1})$. Let EU be the objective function of program (2) with x ₂=x ₂(x ₁) and h = h (x ₁). Totally differentiating EU with respect to x ₁, using the envelope theorem, and evaluating the resulting derivative at x ₁ = 0 yields

$$ \frac{\mathrm{d}EU}{\mathrm{d}x_{1}}\bigg|_{x_{1}=0} =\mathrm{E}\{u^{\prime}(\tilde{\pi}^{0})[\tilde{e}_{1}p_{1}-c^{\prime}({x_{2}^{0}})]\}, $$

(A.9)

where ${x_{2}^{0}}$ and h ⁰ are defined in Eqs. 6 and 7. Substituting Eq. 7 into the right-hand side of Eq. A.9 yields

$$ \frac{\mathrm{d}EU}{\mathrm{d}x_{1}}\bigg|_{x_{1}=0}= \mathrm{E}[u^{\prime}(\tilde{\pi}^{0})][\mathrm{E}(\tilde{e}_{1})p_{1}-c^{\prime}({x_{2}^{0}})]. $$

(A.10)

If $c^{\prime }({x_{2}^{0}})\geq \mathrm {E}(\tilde {e}_{1})p_{1}$, Eq. A.10 implies that $x_{1}^{*}=0$. Thus, in this case we have $x_{2}^{*}={x_{2}^{0}}$ and h ^∗=h ⁰. This proves part (iii) of Proposition 1.

Finally, if $c^{\prime }({x_{2}^{0}})<\mathrm {E}(\tilde {e}_{1})p_{1}$, Eq. A.10 implies that $x_{1}^{*}>0$. In this case, condition (3) holds with equality:

$$ \mathrm{E}\{u^{\prime}(\tilde{\pi}^{*})[\tilde{e}_{1}p_{1}-c^{\prime}(x^{*})]\}=0. $$

(A.11)

Multiplying p ₁ to Eq. 5 and adding the resulting equation to Eq. A.11 yields $c^{\prime }(x^{*})=\mathrm {E}(\tilde {e}_{1})p_{1}$, since u′(π) > 0. The optimal amounts of exports, $x_{1}^{*}$ and $x_{2}^{*}$, and the optimal forward position, h ^∗, then solve conditions (3) and (4) with equality and Eq. 5 simultaneously. This proves part (ii) of Proposition 1.