## Covered Interest Parity

In his 1923 work, “A Tract on Monetary Reform,” John Maynard Keynes introduced the concept of covered interest parity (CIP) that declares how spot and forward foreign exchange rates differ by the ratio of the nominal interest rates in the two currencies.

This seminal work, which assumes no arbitrage conditions, has provided the classic formulation for the calculation of forward foreign exchange pricing for many years:

*F* = The forward foreign exchange rate

*S *= The Spot foreign exchange rate

*i _{d}* = The domestic interest rate

*i _{f}* = The foreign interest rate

Forward Foreign Exchange Swap = F - S

Forward foreign exchange swap (FX forwards) prices are determined by subtracting S from F to find the difference in spot and forward FX rates.

While S refers to the spot rate in this formula, in practice it can also be a forward rate so long as F designates a longer dated forward rate, thus allowing the pricing of forward swaps from any forward date to any other longer dated forward date. In practice, most FX forwards are quoted vs. U.S. dollars. However, we must observe certain conventions: exchange rate such as EUR/USD and GBP/USD designate USD as the foreign currency, and therefore the foreign interest rate, while pairs such as USD/JPY or USD/CAD designate USD as the domestic.

## CIP uses today

CIP provides the basis of pricing, however, there is now much evidence that systemic frictions or specific demand and supply conditions can and often do cause forward FX pricing to deviate from the no arbitrage conditions that are assumed in CIP. The deviation from CIP implied pricing has become commonly referred to as the Cross Currency (X-ccy) Basis. It is commonly expressed in terms of basis points as an offset to the non-dollar currency in a pair. For example, 3-month EUR/USD x-ccy basis may be quoted as EUR -20bp meaning that the price of a 3-month FX forward swap in EUR/USD would be equivalent to pricing using CIP with an adjustment of the euro interest rate lower by 20 basis points.

Both CIP implied pricing and observed forward FX are therefore crucial to accurate pricing of cross currency basis.

This paper will show how CME Group Interest Rate futures can accurately provide forward interest rates in comparable benchmarks for both U.S. dollars and euros that can be used to formulate the base forward FX swap price implied by CIP. This paper will also illustrate how CME Group FX futures contracts offer highly accurate prices of forward FX that can be used to add transparency to observable x-ccy basis prices.

### €STR-SOFR inter-commodity spreads

CME Group launched Three-Month €STR futures (ESR) on October 31, 2022. They join the USD Three-Month SOFR (SR3) contracts listed since May 2018 and now trading over two million contracts per day. ESR contracts have many similarities with the SR3 sharing contract critical dates, final settlement compounding calculation, and contract size ($25 or €25 per contract per basis point). The only notable difference in the contracts is the underlying benchmarks.

Uniquely, CME Group also makes the contracts available to trade as inter-commodity spreads (ICS). These inter-commodity spreads provide functionality that allows customers to trade at a guaranteed price representing the difference in price of each leg, resulting in buying (selling) one leg vs. selling (buying) the other. ICS is available on a 1:1 basis closely matching current levels of EUR/USD= spot price.

Contract critical dates of ESR and SR3 are aligned as defined by the IMM calendar. Notwithstanding holiday differences within periods, the alignment of start and end dates allows comparison of expected interest rates in both currencies over the same period.

In addition to Interest Rate contracts, CME Group also lists for trading Foreign Exchange contracts in EUR/USD (6E), which also have expiry dates based on the IMM calendar. The difference between consecutive quarterly FX contracts represents the Forward Pips over that period. Consecutive contracts can be traded as calendar spreads, in increments of 0.2 FX pips, guaranteeing pricing on both legs.

#### Real-world example

Let us work through an example with the following observed prices in September Interest Rate contracts and both September and December FX contracts:

6EU4 (Sep 24 euro FX) 1.07955 *(S)*

6EZ4 (Dec 24 euro FX) 1.08445 *(F)*

ESRU4 (Sep 24 €STR) 96.5825 *(i _{d})*

SR3U4 (Sep 24 3M SOFR) 94.8450* (i _{f})*

From these prices we can determine that the forward swap price between the September IMM date and the December IMM date is the latter price less the former:

Sep 24 to Dec 24 Euro FX forward swap price = 1.08445 – 1.07955 = **0.0049**

This is commonly expressed as forward pips (multiply by 10,000), in this case 49 pips.

For the Interest Rate contracts we observe prices in SR3U4 at 94.8450 and ESRU4 at 96.5825. We can convert these prices to their equivalent yields using the simple formula of *R* = 100 – *P* where *R* is Rate and *P* is Price.

Sep 24 SOFR = 100 – 94.8450 = **5.1550%**

Sep 24 €STR = 100 – 96.5825 = **3.4175%**

Thus, we have the SOFR implied USD interest rate of 5.1550% and the €STR implied euro interest rate of 3.4175% both representing the IMM period from 18 Sep 2024 to 17 Dec 2024.

Using the combination of ESR, SR3 and 6E futures pricing along with the concept of Covered Interest Parity (CIP) we can calculate the forward value, as of 18 December 2024, of 1 euro and its equivalent in USD, which is 1.07955 according to the price of 6EU4. In each case the period is 91 days beginning on 18 September 2024.

Therefore applying 91 days of euro interest at a rate of 3.4175% to 1 euro we would end with:

Similarly in USD we apply 91 days of USD interest at a rate of 5.1550% to our starting $1.07955.

Thus, we calculate that **1.00864 EUR = 1.09362 USD**

Dividing both sides by the euro amount (1.00864) we can express the CIP implied forward rate as:

1 EUR = (1.09362 / 1.00864) = ** 1.08425 USD**

Thus, 1 EUR = 1.08425 USD as of 18 December 2024 and the forward swap rate is:

1.08425 – 1.07955 = 0.0047, or **47 pips**

We noted above that the observed forward swap from Sep 24 and Dec 24 FX futures is at 49 pips, which is 2 pips above the rate implied by the respective USD and euro interest rates alone. This difference is the **observed cross currency basis**. We can also, and more commonly would, express the difference in terms of basis points on the euro leg.

We can rearrange the basic equations of CIP to solve for the foreign interest rate if we know what the forward swap pips are. In our case 49 pips and the known USD interest rate of 5.1550% would imply that the required euro interest rate is 3.3442%.

To see how this rate is calculated, note that we have previously calculated forward value of $1.07955 as $1.09362. Also, our observed forward rate, consistent with 49 pips, is 1.08445. Thus, the forward value of 1 euro is calculated to be 1.09362/1.08445 = 1.00845. Converting that back to an annualized rate by subtracting 1 and then multiplying by 360/91 yields a result of 0.033442 or 3.3442%.

This calculation starts with solving for the EUR Rate:

Which can be re-arranged to:

And finally re-arranged to:

This rate is 7.3 basis points lower than the rate implied by the €STR futures price of 3.4175%. As a result, we would then expect to see the EUR/USD 3-month IMM x-ccy swap starting on 18 December 2024 quoted at EUR -7.3 bps.

*All examples in this report are hypothetical interpretations of situations and are used for explanation purposes only. The views in this report reflect solely those of the author and not necessarily those of CME Group or its affiliated institutions. This report and the information herein should not be considered investment advice or the results of actual market experience. *