FX Forward Pricing and Covered Interest Parity

Worked Solution

How to Think About It: Covered interest parity (CIP) is the no-arbitrage link between FX spot, FX forwards, and interest rates in two currencies. The idea is simple: borrowing in one currency, converting to the other, investing, and locking in the forward to convert back should yield exactly zero profit. If it does not, you have a risk-free arbitrage (ignoring credit risk and spreads). In practice, CIP holds very tightly in liquid G10 pairs -- deviations are usually within bid/ask spreads. When they are not (as happened during the 2008 crisis), it signals serious funding stress.

Key Insight: The forward must be set so that the cost of synthetic borrowing in JPY via the USD market equals direct JPY borrowing. The forward/spot ratio reflects the interest rate differential.

The Method:

*Part (a): Deriving the fair forward*

Consider these two strategies for converting JPY at time $T$:

• Strategy 1 (direct): Invest 1 JPY at rate $r_{\text{JPY}}$ for time $T$. You receive $e^{r_{\text{JPY}} T}$ JPY at maturity.

• Strategy 2 (synthetic): Convert 1 JPY to $S_0$ USD today, invest at $r_{\$}$ for time $T$ to get $S_0 \, e^{r_{\$} T}$ USD, then convert back to JPY at the forward rate $F_{0,T}$ (USD per JPY).

In Strategy 2, the JPY you receive at maturity is: $\frac{S_0 \, e^{r_{\$} T}}{F_{0,T}}$

No-arbitrage requires these to be equal: $e^{r_{\text{JPY}} T} = \frac{S_0 \, e^{r_{\$} T}}{F_{0,T}}$

Solving: $\boxed{F_{0,T} = S_0 \, e^{(r_{\$} - r_{\text{JPY}})T}}$

When $r_{\$} > r_{\text{JPY}}$, the forward is above spot (USD depreciates forward -- the higher-yielding currency trades at a forward discount in terms of itself).

*Part (b): Implied JPY rate*

Rearranging the CIP formula: $F_{0,T} = S_0 \, e^{(r_{\$} - r_{\text{JPY}})T}$ $\frac{F_{0,T}}{S_0} = e^{(r_{\$} - r_{\text{JPY}})T}$ $\ln\!\left(\frac{F_{0,T}}{S_0}\right) = (r_{\$} - r_{\text{JPY}})T$ $\boxed{r_{\text{JPY}}^{\text{implied}} = r_{\$} - \frac{1}{T}\ln\!\left(\frac{F_{0,T}}{S_0}\right)}$

This implied rate is what the forward market "thinks" the JPY rate is. If it differs materially from quoted JPY deposit rates, either the deposit market or the forward market is mispriced (or there are credit/liquidity premia).

*Part (c): Checking arbitrage with bid/ask spreads*

Suppose you suspect the forward is too high relative to CIP. The arbitrage would be: borrow USD, convert to JPY at spot, invest in JPY deposits, and sell JPY forward. To check if this is real:

1. Borrow USD: You pay the USD ask (offer) rate $r_{\$}^{\text{ask}}$.
2. Buy JPY spot: You sell USD and buy JPY at the spot ask price $S_0^{\text{ask}}$ (the price you pay in USD per JPY when buying JPY).
3. Invest in JPY deposits: You receive the JPY bid rate $r_{\text{JPY}}^{\text{bid}}$.
4. Sell JPY forward: You lock in the forward bid $F_{0,T}^{\text{bid}}$ (the price you receive when selling JPY for USD).

The arbitrage profit per JPY notional is: $\pi = F_{0,T}^{\text{bid}} \cdot e^{r_{\text{JPY}}^{\text{bid}} T} - S_0^{\text{ask}} \cdot e^{r_{\$}^{\text{ask}} T}$

If $\pi > 0$ after accounting for all four bid/ask spreads, the arbitrage is real. In practice, you also need to consider settlement timing, counterparty credit lines, and balance sheet costs.

For the reverse trade (forward too low), flip all sides: lend USD (bid rate), sell JPY spot (bid), borrow JPY (ask rate), buy JPY forward (ask).

Practical Considerations: - Post-2008, CIP violations of 20-40 bps have persisted in some currency pairs due to bank balance sheet constraints and regulatory costs. - The implied rate from part (b) is widely used as a cross-currency basis indicator. The "basis" is $r_{\text{JPY}}^{\text{implied}} - r_{\text{JPY}}^{\text{quoted}}$. - Always use the worst-case side of each spread when checking arbitrage -- if the trade still works under worst-case pricing, it is real.

Answer: (a) $F_{0,T} = S_0 \, e^{(r_{\$} - r_{\text{JPY}})T}$. (b) $r_{\text{JPY}}^{\text{implied}} = r_{\$} - \frac{1}{T}\ln(F_{0,T}/S_0)$. (c) Use worst-case bid/ask on every leg: borrow at ask, lend at bid, buy FX at ask, sell FX at bid. If the round-trip still yields positive PnL, the arbitrage is real.