91Ó°ÊÓ

A U.S. company is interested in using the futures contracts traded on the CME to hedge its Australian dollar exposure. Define \(r\) as the interest rate (all maturities) on the U.S. dollar and \(r_{f}\) as the interest rate (all maturities) on the Australian dollar. Assume that \(r\) and \(r_{f}\) are constant and that the company uses a contract expiring at time \(T\) to hedge an exposure at time \(t(T>t)\). a. Using the results in Chapter \(3,\) show that the optimal hedge ratio is $$e^{\left(r_{f}-r\right)(T-t)}$$ b. Show that, when \(t\) is one day, the optimal hedge ratio is almost exactly \(S_{0} / F_{0},\) where \(S_{0}\) is the current spot price of the currency and \(F_{0}\) is the current futures price of the currency for the contract maturing at time \(T\). c. Show that the company can take account of the daily settlement of futures contracts for a hedge that lasts longer than one day by adjusting the hedge ratio so that it always equals the spot price of the currency divided by the futures price of the currency.

Step by step solution

01

Establish the Base Formula

The hedge ratio relates to the hedging strategy to minimize risk. For currency hedging, the optimal hedge ratio is derived from the Covered Interest Rate Parity (CIRP), which states that...\[ F_0 = S_0 \cdot e^{(r - r_f)(T-t)} \]Thus, rearranging for the hedge ratio gives the base formula as:\[ \text{Hedge Ratio} = e^{(r_f - r)(T-t)} \]

02

Evaluate Short-term Hedge Limits

For part b, assuming that \(t = \text{1 day}\), the exponent \((T-t)\) becomes very small such that \[ e^{(r_f - r)(T-t)} \approx 1 + (r_f - r)(T-t) \approx 1 \]so the optimal hedge ratio can be approximated as\[ \frac{S_0}{F_0} \] which reflects the immediate adjustment of the spot to futures price without significant time influence.

03

Understand Daily Settlement Impact

For part c, recognizing the daily settlement of futures, the hedge needs recalibration daily. By setting the hedge ratio as \(S_{\text{current}}/F_{\text{current}}\), the hedge continuously adapts to reflect the current spot and future prices, ensuring the exposure matches adjusted market conditions, which negates the need for time dependency in ongoing contracts.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Futures Contracts

Futures contracts are agreements to buy or sell a specific quantity of a commodity or financial instrument at a predetermined price on a specified future date.
They are commonly used by businesses to manage risk or speculate on price movements.
For example, a U.S. company exposed to fluctuations in the Australian dollar may use futures contracts to stabilize financial outcomes. Here's how they work:

• The buyer agrees to purchase, and the seller agrees to deliver, a currency (or other asset) at the contract's expiration.
• The price is set at the time of the contract, providing certainty against future price changes.
• Futures are typically standardized and traded on exchanges, like the CME.

Using futures contracts in currency hedging enables companies to "lock in" current exchange rates for future transactions.
This assists in avoiding adverse effects from currency fluctuations.

Hedge Ratio

The hedge ratio is a critical concept in hedging, denoting the proportion of a position covered by a hedge.
It aims to minimize the risk of financial loss from adverse price movements.
In the context of currency hedging, the ideal hedge ratio can be determined using formulae based on the Covered Interest Rate Parity (CIRP). The common formula here is \[\text{Hedge Ratio} = e^{(r_f - r)(T-t)} \]where:

• \(r_f\) is the foreign interest rate (Australian dollar).
• \(r\) is the domestic interest rate (U.S. dollar).
• \(T-t\) is the time to contract maturity.

A hedge ratio allows companies to decide what portion of their exposure should be hedged to minimize risk.
A ratio approaching 1 implies a fully hedged position, matching the size of the asset or liability.

Covered Interest Rate Parity

The Covered Interest Rate Parity (CIRP) is a fundamental concept in international finance for pricing foreign exchange derivatives, including futures and forward contracts. This parity ensures no arbitrage opportunities exist by tying together:

• The spot exchange rate \(S_0\).
• The future exchange rate \(F_0\).
• The interest rates of two currencies.

The formula underpinning CIRP is:\[F_0 = S_0 \cdot e^{(r - r_f)(T-t)}\]This equation ensures the returns on hedged investments in different currencies are equalized.
CIRP provides a theoretical pathway to derive futures pricing based on current financial conditions, isolating the impact of changes in interest rates.

Spot Price

The spot price of a currency is the current exchange rate at which it can be immediately exchanged between two currencies.
It represents the actual market value of the currency at a given moment. Spot price is crucial for decisions involving currency transactions, as it sets the rate for:

• Immediate currency exchanges.
• Setting baseline comparisons for forward and futures contract pricing.

In currency hedging, the spot price helps determine how much currency exposure needs to be covered.
For hedges of longer durations, the spot price informs needed adjustments in hedge strategies based on current market activities.

Daily Settlement

Daily settlement is a key feature of futures contracts, ensuring that gains and losses are settled daily.
This mechanism promotes transparency and mitigates counterparty risk. How it works:

• Every trading day, futures prices are marked-to-market.
• Profits and losses are calculated based on contract variation from the previous day's prices.
• These are settled via margin accounts, where funds must be maintained above a required level.

For currency hedges longer than a day, daily settlement means adjusting the hedge ratio based on new spot and futures prices.
Continuous adjustment ensures that positions remain properly hedged, reflecting price realities daily.