International Finance

The Forward Market for Foreign Exchange

Main issues

• Understanding forward exchange contracts

• Hedging with forward contracts

• Covered interest rate parity (CIP)

• Valuation of forward contracts

• The cross-currency basis: when CIP “fails”

Forward Contracts

Definition of the forward exchange rate

The forward exchange rate \(F_{t,T}\) is the price agreed upon today (\(t\)) at which one currency can be exchanged for another at a future date (\(T\)).

• Rate is set today — no uncertainty about the price
• No cash flows at initiation; exchange happens only at \(T\)

A forward contract is like agreeing today on the price at which you’ll trade currencies at a future date. Unlike a spot transaction (settled in T+2), the forward locks in a rate now for future delivery. The key insight: the rate is known today, so there is no exchange rate uncertainty for the hedged position.

Forward quotes

CIP-implied from spot rates and 3M interbank interest rates. Source: FRED.

This table shows spot rates and CIP-implied forward rates for three major pairs. Note: - EUR forward is above spot (EUR at a premium) because US rates exceed EUR rates. - GBP forward is roughly equal to spot (rates are similar). - JPY forward is below spot (JPY at a premium, i.e. fewer JPY per USD) because JPY rates are far below US rates. We will derive exactly why this is the case when we cover CIP.

Hedging with forward contracts

• FC inflows at a future date \(\rightarrow\) sell FC forward

• FC outflows at a future date \(\rightarrow\) buy FC forward

Hedging translates:

• A future FC cashflow whose HC value depends on the random \({\tilde S}_T\)

• Into a known HC amount equal to \(F_{t,T}\)

The forward contract eliminates exchange rate uncertainty.

Example: hedging a foreign currency outflow

Suppose:

• You live in the US
• You are planning a vacation to the UK in 12 months
• Your budget is GBP 1,000

Questions:

1. What is your exchange rate exposure?
2. How can you hedge using a forward contract?
3. What is the cost of hedging?

1. You need GBP in the future. If GBP appreciates (USD/GBP rises), you’ll need more USD. You are short GBP forward.
2. Buy GBP 1,000 forward at the 12M forward rate. This locks in the USD cost.
3. The cost of hedging is NOT the difference between the spot and forward rate. The true cost is the difference in bid-ask spreads: the forward spread minus the spot spread. The forward premium/discount reflects interest rate differentials, not a “cost.”

Covered Interest Rate Parity

The CIP formula

In frictionless markets:

\[F_{t,T} = S_t \frac{1 + r_{t,T}}{1 + r^*_{t,T}}\]

where

• \(S_t\) is the spot rate (HC per FC)
• \(r_{t,T}\) is the domestic (HC) interest rate
• \(r^*_{t,T}\) is the foreign (FC) interest rate

Where does this formula come from? We prove it via a replication argument.

CIP: numerical example

Given:

\(S_0\) (USD/GBP)	1.26
\(r^{USD}_{0,1}\)	4.3%
\(r^{GBP}_{0,1}\)	4.5%
\(T\)	1 year

CIP gives us:

\[F_{0,1} = 1.26 \times \frac{1.043}{1.045} = 1.2576\]

GBP is at a forward discount: \(F < S\).

Why? Because \(r^{GBP} > r^{USD}\).

CIP proof: the replication argument

We compare two strategies that both deliver GBP 1 at time \(T\) with zero cash flow today:

Strategy A: Buy GBP 1 forward.

Strategy B:

1. Borrow \(\frac{S_t}{1 + r^*_{t,T}}\) in USD at rate \(r_{t,T}\) for period \(T\)
2. Convert to GBP \(\frac{1}{1 + r^*_{t,T}}\) at spot rate \(S_t\)
3. Invest this in GBP at rate \(r^*_{t,T}\) \(\rightarrow\) delivers GBP 1 at \(T\)

CIP proof: the payoff table

Strategy	Cashflow at \(t\)	Cashflow at \(T\)
A: Forward	0	Pay \(F_{t,T}\); Receive GBP 1
B: Replication	Borrow USD \(\frac{S_t}{1+r^*}\)	Pay USD \(S_t \frac{1+r}{1+r^*}\)
	Buy GBP \(\frac{1}{1+r^*}\) spot
	Invest GBP at \(r^*\)	Receive GBP 1
	Net: 0

Both strategies: zero cost at \(t\), deliver GBP 1 at \(T\). No arbitrage \(\Rightarrow\) same cost at \(T\):

\[F_{t,T} = S_t \frac{1 + r_{t,T}}{1 + r^*_{t,T}}\]

Implications of CIP

A forward contract can be replicated using three markets:

1. The spot FX market
2. The domestic money market
3. The foreign money market

Practical consequence: If you cannot trade forwards (e.g., restricted currency), you can create a synthetic forward using spot and money markets.

Conversely: If you observe spot, forward, and two interest rates, CIP pins down the relationship among all four. Any deviation is an arbitrage opportunity.

Forward premium and discount

We can write CIP as:

\[\frac{F_{t,T}}{S_t} = \frac{1 + r_{t,T}}{1 + r^*_{t,T}}\]

• \(F > S\) \(\Rightarrow\) FC is at a forward premium \(\Rightarrow\) \(r > r^*\) (HC rate is higher)

• \(F < S\) \(\Rightarrow\) FC is at a forward discount \(\Rightarrow\) \(r < r^*\)

Strong currencies offer low interest rates; weak currencies compensate with high rates.

Forward premium over time

Source: 3M interbank rates from OECD via FRED, 1994–2026.

This shows the annualized 3M CIP-implied forward premium for EUR/USD and GBP/USD. When the line is positive, the EUR (or GBP) is at a forward premium, meaning US rates exceed European rates. The dramatic swing around 2008 reflects the collapse in interest rates. The post-2022 rise reflects the aggressive Fed tightening cycle relative to the ECB/BoE.

CIP and corporate borrowing

The CIP can be rewritten as:

\[\underset{\text{cost of borrowing HC 1 abroad}}{S_t^{-1}(1 + r^*_{t,T}) \, F_{t,T}} \;=\; \underset{\text{cost of borrowing HC 1 at home}}{(1 + r_{t,T})}\]

In frictionless markets, you cannot save money by borrowing in one currency rather than another (once you hedge the FX risk).

The low interest rate in JPY does not make JPY borrowing “cheap” — the forward premium offsets it exactly.

This is a crucial insight for corporate treasury. Many firms think they are “saving” by borrowing in a low-rate currency. But the CIP says the all-in cost (after hedging) is the same. The only way to truly save is if (a) you don’t hedge (taking FX risk), or (b) the cross-currency basis creates a deviation from CIP.

CIP: the mnemonic

In the ratio \(\frac{1 + r_{t,T}}{1 + r^*_{t,T}}\), the numerator interest rate belongs to the numerator currency of \(S_t\).

Example: If \(S_t\) is quoted as EUR/JPY:

\[F_{t,T} = S_t \frac{1 + r^{EUR}_{t,T}}{1 + r^{JPY}_{t,T}}\]

EUR is in the numerator of the exchange rate \(\Rightarrow\) EUR rate goes in the numerator.

Valuation of Forward Contracts

Valuing an existing forward contract

At \(t+1\), value a forward initiated at \(t\) (maturity \(T\), rate \(F_{t,T}\)) to buy FC 1.

Cashflows at \(T\): Receive FC 1, Pay HC \(F_{t,T}\). Present value at \(t+1\):

\[V_{t+1} = \frac{S_{t+1}}{1 + r^*_{t+1,T}} - \frac{F_{t,T}}{1 + r_{t+1,T}}\]

Using CIP: \(\frac{S_{t+1}}{1+r^*_{t+1,T}} = \frac{F_{t+1,T}}{1+r_{t+1,T}}\), so:

\[V_{t+1} = \frac{F_{t+1,T} - F_{t,T}}{1 + r_{t+1,T}}\]

Value at maturity

\[V(T) = S_T - F_{t,T}\]

At maturity, the value is simply the difference between the spot rate and the contracted forward rate. The long position (buyer of FC) profits if the spot rate exceeds the forward rate; the short position (seller) profits in the opposite case. This is a zero-sum game.

Value at initiation

At the initiation date \(t\):

\[V(t) = \frac{S_t}{1 + r^*_{t,T}} - \frac{F_{t,T}}{1 + r_{t,T}}\]

But CIP tells us \(F_{t,T} = S_t \frac{1 + r_{t,T}}{1 + r^*_{t,T}}\), which means \(\frac{S_t}{1 + r^*_{t,T}} = \frac{F_{t,T}}{1 + r_{t,T}}\).

Therefore: \(V(t) = 0\).

The forward rate is chosen so the contract is fair at initiation — neither party pays the other.

Numerical example

3M forward for GBP at \(F_{0,T}\) = USD/GBP 1.26. At initiation: \(V_0 = 0\).

After 1 month, spot is \(S_{t+1}\) = 1.28:

\[V_{t+1} = \frac{1.28}{1 + r^*_{t+1,T}} - \frac{1.26}{1 + r_{t+1,T}} \approx 0.02 > 0 \quad \text{(positive value to buyer)}\]

At maturity, spot is \(S_T\) = 1.30:

\[V_T = 1.30 - 1.26 = 0.04 \text{ per GBP}\]

The Forward Rate as Certainty Equivalent

What is the market’s certainty equivalent for \({\tilde S}_T\)?

Suppose you are offered FC 1 at future date \(T\). The present value can be computed two ways:

\[PV = \underset{\text{risky CF at risk-adjusted rate}}{\frac{E_t[{\tilde S}_T]}{1 + {\tilde r}_{t,T}}} \;=\; \underset{\text{safe CF at riskless rate}}{\frac{F_{t,T}}{1 + r_{t,T}}}\]

Therefore: \(F_{t,T} = CEQ_t({\tilde S}_T)\)

The forward rate is the certainty equivalent of the future spot rate.

What does this mean?

• CIP links the current spot rate, the current forward rate, and interest rates.

• CEQ links the future spot rate and the current forward rate.

Important: \(F_{t,T} = CEQ_t({\tilde S}_T)\) does not imply \(F_{t,T} = E_t[{\tilde S}_T]\).

Whether the forward rate equals the expected future spot rate is the question of the Unbiased Expectations Hypothesis (UEH). This is tested separately and generally fails — which we will cover in a later lecture.

Empirical Evidence on CIP

CIP deviations: the cross-currency basis

In theory: CIP holds exactly, and forward rates are pinned by interest rate differentials.

In practice: Small but persistent deviations exist, especially since 2008.

The cross-currency basis measures the deviation:

\[\text{basis} = \text{implied USD rate from FX swap} - \text{actual USD rate}\]

A negative basis means it is more expensive to obtain USD synthetically (via FX swaps) than to borrow USD directly.

CIP deviations: evidence from FX swap markets

Source: Du, Tepper and Verdelhan (2018), “Deviations from Covered Interest Rate Parity,” Figure 2.

This figure from Du, Tepper and Verdelhan (2018) shows the ACTUAL cross-currency basis computed from FX swap data. Key observation: before 2008, the basis was essentially zero for all currency pairs. After 2008, the basis became persistently NEGATIVE (meaning synthetic USD is more expensive than direct USD borrowing). This is NOT a transient crisis effect — it persists for years after the GFC, reflecting the structural impact of bank regulation (Basel III leverage ratios, supplementary leverage ratios) on banks’ willingness to intermediate in FX swap markets. The paper data ends in 2016; the pattern has continued.

Why the basis exists

• Balance-sheet constraints: Banks need capital to intermediate FX swaps. Basel III leverage ratios limit supply.

• Funding stress: Dollar shortage forces foreign banks to pay a premium for synthetic USD (GFC, COVID).

• Regulatory costs: Quarter-end window-dressing creates predictable spikes.

• Segmentation: Not all market participants can arbitrage freely (insurance companies, central banks).

For the firm: The basis is a risk signal. When it widens, hedging costs rise.

Summary

Summary

• Forward contracts allow hedging of future FC cashflows at a known rate.

• CIP: \(F_{t,T} = S_t \frac{1 + r}{1 + r^*}\) — proved via replication (no arbitrage).

• Forward valuation: \(V(t) = 0\) at initiation; \(V(T) = S_T - F_{t,T}\) at maturity.

• CIP implies you cannot save money borrowing in another currency (after hedging).

• Forward rate = certainty equivalent of the future spot rate (but \(\neq\) expected spot).

• Cross-currency basis: Real-world frictions create persistent CIP deviations. The basis matters for hedging cost and funding decisions.