4. (12 pts) Hedged vs. unhedged outcomes (scenarios). To isolate the effect of hedging ( for simplicity), suppose the US bond pays exactly $\$ 100 \mathrm{~m} $ at $t = 1 $( we are abstracting awa

Question

4. (12 pts) Hedged vs. unhedged outcomes (scenarios). To isolate the effect of hedging ( for simplicity), suppose the US bond pays exactly $\$ 100 \mathrm{~m} $ at $t = 1 $( we are abstracting away from the effect of the bond's interest rate). Compare two spot outcomes at maturity. In $t = 1 $ there are two possible scenarios: (A) (sunny day) $S _{1} = $ 1.20 USD/EUR, and (B) (rainy day) $S_{1} = 1.00 $ USD/EUR. (a) ($\mathbf {4} $ pts) Unhedged (translation risk). Calculate the PF's proceeds (in EUR) under both scenarios. (b) ($\mathbf{2} $ pts) Hedged (with FX swap) Comparison. Compare the scenarios in $t = \mathbf{1} $ to the hedged amount in 3(c). (c) (3 pts) Profits/Losses on the forward leg. Compute the PF's euro gain using the agreed forward rate 1.13176 and the spot at maturity (1.20 or 1.00). (d) ($\mathbf{3} $ pts) Why hedge with an FX swap? Many NBFIs have domestic - currency obligations but hold global portfolios across currencies. In one sentence, explain why a European pension fund (PF) that buys USD assets would want to hedge currency risk, and how it uses FX swaps to do so.

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# Hedged vs. Unhedged Outcomes for a European Pension Fund

Suppose a European pension fund (PF) holds a US bond that pays \$100 million at $ t=1 $. The exchange rate at maturity ($ S_1 $) can be either 1.20 USD/EUR (scenario A: "sunny day") or 1.00 USD/EUR (scenario B: "rainy day").

---

## (a) Unhedged (Translation Risk): EUR Proceeds

**Unhedged**: The PF faces translation risk, converting USD proceeds to EUR at the spot rate at $ t=1 $.

### Calculation

$$
\text{EUR proceeds} = \frac{\$100\,\text{million}}{S_1}
$$

- **Scenario A** ($ S_1 = 1.20 $):
  $$
  \text{EUR proceeds} = \frac{\$100,000,000}{1.20} = \€83,333,333
  $$

- **Scenario B** ($ S_1 = 1.00 $):
  $$
  \text{EUR proceeds} = \frac{\$100,000,000}{1.00} = \€100,000,000
  $$

---

## (b) Hedged (FX Swap): Comparison

Assume the PF locks in a **forward rate** of 1.13176 USD/EUR (from part 3(c)).

- **Hedged amount at $ t=1 $**:
  $$
  \text{EUR proceeds (hedged)} = \frac{\$100,000,000}{1.13176} = \€88,336,000
  $$
  (rounded to the nearest euro)

- **Comparison**:
    - Scenario A (unhedged): €83.33 million \< **Hedged**: €88.34 million
    - Scenario B (unhedged): €100 million \> **Hedged**: €88.34 million

**Conclusion:**  
Hedging removes the risk: the PF receives €88.34 million regardless of the spot rate.

---

## (c) Profit/Loss on Forward Leg

Calculate the PF's euro gain/loss from the forward contract versus spot at maturity.

$$
\text{EUR gain/loss} = \text{EUR proceeds (hedged)} - \text{EUR proceeds (unhedged)}
$$

- **Scenario A** ($ S_1 = 1.20 $):
  $$
  \text{Gain} = 88,336,000 - 83,333,333 = \€5,002,667
  $$

- **Scenario B** ($ S_1 = 1.00 $):
  $$
  \text{Loss} = 88,336,000 - 100,000,000 = -\€11,664,000
  $$

---

## (d) Why Hedge with an FX Swap?

> A European pension fund with euro-denominated liabilities hedges USD asset currency risk with an FX swap to lock in a known euro value for future USD receipts, thereby eliminating uncertainty from exchange rate fluctuations.

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