Covered Interest Parity and FX Forward Points

This relationship is used to check why the forward exchange rate of an FX forward sits at its level, and to see how far the cost of raising dollars through an FX swap deviates from covered interest parity. It is also the groundwork for thinking about the cost of hedging the exchange rate on foreign currency assets (the hedging cost).

Building the forward exchange rate from deposits

The forward exchange rate is set by the cost of exchanging currencies at today's rate (the spot exchange rate) and keeping the money on deposit until the delivery date.

A rate used for a trade whose delivery is not on the second business day after the trade is agreed is called a "forward" rate, as distinguished from the "spot" rate. This rate is set by the interest that each currency earns over the period of the trade¹. This site calls them the spot exchange rate and the forward exchange rate.

Consider a hypothetical example in which the spot exchange rate is 150.00 yen per dollar, the 180-day yen interest rate is 0.50% a year, and the US dollar interest rate is 4.50% a year. A promise to receive USD 1 million in 180 days can be built today from the following three trades.

The amounts in the table show receipts as positive and payments as negative². Today's payments and receipts net to zero in both yen and dollars, and in 180 days you pay 147,060,991 yen and receive USD 1 million. This has the same result as a promise to buy each dollar at 147.061 yen. The forward exchange rate for delivery in 180 days is therefore 147.061 yen. This is the cost of carry model, which prices a forward from the price of the underlying asset and the cost of holding it, applied to currencies. In other words, the forward exchange rate is the rate at which investing in yen gives the same result as converting into dollars and investing in dollars³. The cost of carry model is covered in Forwards and Futures.

With 146,699,267 yen on hand, the same 180-day investment can be laid out as two routes, as Figure 1 shows. The 147,060,991 yen in the figure is calculated with the forward exchange rate before rounding, 147.060991 yen, so it differs by 9 yen from 147,061,000 yen, which is 147.061 yen times USD 1 million, because of the fractions.

Figure description

The figure lays out the same 180-day investment as two routes: depositing in yen, and converting to dollars and depositing them. The starting figures are 146,699,267 yen on hand, a spot rate of 150.00 yen, a yen rate of 0.50%, and a dollar rate of 4.50%; converting gives USD 977,995, and the forward exchange rate is 147.061 yen. On the dollar route, at the same time as depositing the dollars, you agree to sell the USD 1 million due at maturity at the forward exchange rate. In 180 days, both routes give the same amount of yen, 147,060,991 yen. The forward exchange rate at which the two routes give the same result is the value from the covered interest parity formula.

A trade that exchanges currencies on a set future date at a set rate is called an FX forward.

The covered interest parity formula and day count conventions

The relationship that expresses the forward exchange rate in terms of the spot exchange rate and the interest rates of the two currencies is called covered interest parity. Writing the combination of trades in the previous section as a formula gives the following forward exchange rate.

$$F = S \times \frac{1 + r_{\text{JPY}} \times d / 365}{1 + r_{\text{USD}} \times d / 360}$$

$S$ is the spot exchange rate and $F$ is the forward exchange rate (both in yen per dollar). $r_{\text{JPY}}$ and $r_{\text{USD}}$ are the annual interest rates of the yen and the dollar, and $d$ is the number of days in the period. The numerator is the factor by which the amount repaid grows when you borrow yen for $d$ days, and the denominator is the factor by which the amount received grows when you deposit dollars for $d$ days. A formula of the same form is used as covered interest parity in an analysis of EUR/USD FX swaps⁴.

The two currencies use different numbers of days in a year to match the day count conventions of their money markets. Under the US dollar money market convention, interest is calculated on the actual number of days, assuming a 360-day year⁵. For the yen, the floating-rate calculation of a Japanese yen OIS (overnight index swap) converts the period to which the uncollateralized overnight call rate (TONA) applies into years by dividing the actual number of days by 365⁶.

Because the day count conventions differ, the forward exchange rate is not equal to the spot exchange rate even when the yen and dollar interest rates are the same number. If both are 3.00% a year, dollar interest divided by 360 days is larger, and the forward exchange rate shows a stronger yen than the spot exchange rate. The forward exchange rate equals the spot exchange rate when the dollar interest rate is 360 ÷ 365 times the yen interest rate.

Deep diveAn approximation using the interest rate differential

Divide both sides of the covered interest parity formula by the spot exchange rate, subtract 1, and treat the products of the rates and days as small. The approximation shows that the difference between the forward and spot exchange rates, as a fraction of the spot exchange rate, is nearly equal to the difference between the two currencies' interest for the period.

$$\frac{F - S}{S} \approx r_{\text{JPY}} \times \frac{d}{365} - r_{\text{USD}} \times \frac{d}{360}$$

Taking the natural logarithm of both sides gives the same approximation, so the difference between the spot and forward exchange rates of an FX swap, annualized, can be read as the interest rate differential between the two currencies⁴. In the example with a spot exchange rate of 150.00 yen, 0.50% for yen, 4.50% for dollars, and 180 days, the approximation gives a difference of 150.00 × (0.002466 − 0.022500), or −3.005 yen. That is 0.066 yen larger in absolute value than the −2.939 yen from the exact formula. For the same interest rates, the approximation error grows with the period: under the conditions of this example it is 0.017 yen for 90 days and 0.259 yen for 360 days. For this reason, the worked example and the simulator use the exact formula, not the approximation.

Worked example: the 180-day forward exchange rate

With a spot exchange rate of 150.00 yen, a yen interest rate of 0.50%, a dollar interest rate of 4.50%, and a period of 180 days, the forward exchange rate is 147.061 yen. The intermediate values are as follows.

The difference between the forward and spot exchange rates is called the forward points. Forward points reflect the interest rate differential between the two currencies, and they are also called the spot-forward spread or swap points⁷. In this example, USD/JPY forward points are stated in sen (one hundredth of a yen). There are also examples that state the spot-forward spread in sen, such as 30 sen⁸. The forward points in this example are −2.939 yen, or −293.90 sen⁹.

The annualized rate is the forward points divided by the spot exchange rate, multiplied by the ratio of the days in a year to the days in the period. With a 365-day year, the annualized rate in this example is −3.97%. It is close to the difference (−4.0625%) between the yen interest rate of 0.50% and the dollar interest rate restated on a 365-day basis, 4.5625%. The two do not match because the annualized rate is a value after dividing by the dollar deposit factor of 1.0225.

The sign of forward points

When the dollar interest rate is higher than the yen interest rate, the forward exchange rate shows a stronger yen and a weaker dollar than the spot exchange rate. When the USD/JPY forward exchange rate shows a weaker dollar and a stronger yen than the spot exchange rate, it is called a dollar discount, and the opposite is called a dollar premium¹. The forward points in the worked example are negative because the dollar interest rate of 4.50% is higher than the yen interest rate of 0.50%, which makes a dollar discount.

The forward exchange rate does not show how market participants see the future spot exchange rate. In practice, it is set not by expectations that a currency will rise or fall, but by adding the spot-forward spread, derived from the interest rate differential between the two currencies, to the spot exchange rate or subtracting it. Apart from exceptional periods, dollar interest rates have been higher than yen interest rates, so the USD/JPY forward exchange rate has almost always been at a stronger yen and a weaker dollar than the spot exchange rate⁷. The cost of hedging the exchange rate on foreign currency assets (the hedging cost) comes up in institutional investors' portfolios.

How the interest rates and the period change the forward exchange rate

The difference between the forward and spot exchange rates grows as the interest rate differential between the two currencies widens and as the period lengthens. Extending only the period in the worked example to 360 days gives a forward exchange rate of 144.249 yen and forward points of −575.15 sen, nearly double those for 180 days. If the yen interest rate is set above the dollar interest rate restated on the same number of days in a year (the dollar rate × 365 ÷ 360), the forward points turn positive (a dollar premium).

Arbitrage when the forward exchange rate deviates from the formula

When the market forward exchange rate moves away from the value of the covered interest parity formula, you can combine deposits and FX forwards into trades that earn a profit regardless of exchange rate movements (arbitrage). Covered interest parity is said to hold when the spot-forward spread, annualized, equals the interest rate differential. When the two do not match, spot and forward trades increase and narrow the gap, so covered interest parity normally holds⁸.

Suppose that, under the conditions of the worked example, only one bank's 180-day forward exchange rate deviates from the formula's value of 147.061 yen. In either direction, the following difference remains per USD 1 million.

At 146.50 yen, the USD 1 million received under the forward repays the borrowed dollars (USD 1 million of principal and interest). At 147.50 yen, the USD 1 million received when the dollar deposit matures is delivered under the forward. In both cases, today's payments and receipts net to zero.

This profit, however, is available only when you can borrow and deposit at the rates in the table and the counterparties keep their promises. Deviations that arbitrage cannot explain can remain for several reasons: lower liquidity in the dollar money market prevents participants from raising enough dollars for arbitrage, there is counterparty credit risk that interest rates do not reflect, and there are transaction costs and measurement errors¹¹.

Deviations from covered interest parity: the cross-currency basis

An FX swap is a trade that makes a spot delivery and a forward delivery in opposite directions at the same time. When you raise dollars through an FX swap using yen or another currency as collateral, the amount added on top of the dollar market interest rate is called the dollar funding premium¹². A deviation from covered interest parity is expressed in two ways: as this positive amount (the dollar funding premium) and as a signed value (the cross-currency basis).

Suppose that, under the conditions of the worked example, the market forward exchange rate is 146.881 yen (forward points of −311.90 sen) instead of 147.061 yen. Say you borrow yen at 0.50%, buy dollars at the spot exchange rate, and enter into an FX swap that sells the dollars at 146.881 yen in 180 days. The cost is the same as borrowing dollars at 4.75% a year: 150.00 × 1.002466 ÷ 146.881 − 1, multiplied by 360 ÷ 180, is 4.7506%, rounded to two decimal places. The difference of 0.25% (25bp) from the dollar market interest rate of 4.50% is the dollar funding premium in this example.

The cross-currency basis expresses the same deviation as a signed value added to the interest rate of the yen or other currency. In this example, replace the yen interest rate so that the forward exchange rate of 146.881 yen matches the value of the covered interest parity formula. The result is 0.50% plus about −0.25% (−25bp), found by solving 150.00 × (1 + yen interest rate × 180 ÷ 365) ÷ 1.0225 = 146.881. When borrowing dollars is expensive, the cross-currency basis seen from the side of the yen or other currency is negative, and the stronger the demand for dollar funds, the more negative it becomes¹³¹². The cross-currency basis is also used in quoting the interest rates of cross-currency swaps. Which currency's interest rate it is added to, and how, is covered in Cross-Currency Swaps.

Under conditions such as no segmentation between the FX swap market and the dollar money market and no imbalance in participants' credit risk, arbitrage theoretically brings the dollar funding premium to zero. The premium widens sharply when concern about credit risk rises or when markets become segmented¹². The periods when the premium widened and the background are as follows¹⁴.

In the widening from 2014, unlike in the earlier crises, credit risk indicators did not rise¹⁴. From around mid-2014, dollar funding rates also tended to rise at quarter-ends in markets such as the FX swap market, driven by banks in the United States, Europe, and elsewhere responding to leverage ratio regulations¹⁵. When international financial markets became unstable in the spring of 2020, the cost of raising dollars rose sharply. Against the dollar, the cross-currency basis has been negative for the Japanese yen and European currencies, and positive for the Australian dollar and the New Zealand dollar¹³.

Notes

1. Bank of Japan, "「外国為替市況」の解説" (Explanation of the "Foreign Exchange Market Rates", in Japanese), "1. 3ヶ月フォワード・スプレッド" (1. Three-month forward spread) under "（参考）「時系列統計データ検索サイト」に過去計数のみ掲載の項目" (Reference: items published only as past data on the Time-Series Data Search Site). ↩ ↩²

2. Amounts are rounded to the nearest yen or dollar. Calculations use the values before rounding (a dollar deposit of USD 977,995.11 and a yen borrowing of 146,699,266.50 yen). ↩

3. Institute for International Monetary Affairs, "直先スプレッドと先物相場の算出方法" (How the spot-forward spread and the forward rate are calculated, in Japanese). The source's example uses a spot rate of 112 yen per dollar, a 3-month yen interest rate of 0.01%, and a dollar interest rate of 1.1%, and finds a forward rate of 111.70 yen with a period of 3 ÷ 12 years. ↩

4. Bank of Japan, "為替スワップを利用した米ドル資金の調達コストの動向について" (Bank of Japan Review, March 2012, in Japanese), note 2 (page 7). The formula for EUR/USD FX swaps uses a 360-day year for both interest rates. An English version is available: "Recent Developments in U.S. Dollar Funding Costs through FX Swaps" (Bank of Japan Review 2012-E-3, April 2012). ↩ ↩²

5. Federal Reserve Bank of New York, "Additional Information about Reference Rates Administered by the New York Fed," explanation of the SOFR Averages and Index. ↩

6. Study Group on Risk-Free Reference Rates, "日本円ＯＩＳ（Overnight Index Swap）─取引の概要と活用事例─" (Yen OIS: overview of transactions and examples of use, April 2018, in Japanese), how the floating rate is calculated under "基本的な取引方式" (Basic trading terms), page 5. It sets the period to which the uncollateralized overnight call rate applies as "実日数（カレンダー上の日数）/365 日" (actual days (calendar days)/365 days; our translation). ↩

7. Institute for International Monetary Affairs, "直先スプレッドと先物相場の算出方法" (How the spot-forward spread and the forward rate are calculated, in Japanese), the opening paragraph. ↩ ↩²

8. Institute for International Monetary Affairs, "金利平価" (Interest rate parity, in Japanese). With an example of a spot rate of 112 yen per dollar and a 3-month forward rate of 111.70 yen, it gives the spot-forward spread as 30 sen (▲0.30 yen), and says that the annualized value of 1.07% is almost equal to the dollar-yen interest rate differential of 1.09%. ↩ ↩²

9. The forward points and the annualized rate are calculated from the forward exchange rate before rounding (147.060991 yen). ↩

10. Bank of Japan, "「外国為替市況」の解説" (Explanation of the "Foreign Exchange Market Rates", in Japanese), "1. 3ヶ月フォワード・スプレッド" (1. Three-month forward spread). The formula is "年率＝スプレッド×（360÷日数）÷スポット相場（1年＝360日で換算）" (annualized rate = spread × (360 ÷ days) ÷ spot rate (converted with 1 year = 360 days); our translation). The source describes this item as one whose past data only are published on the Time-Series Data Search Site. ↩

11. Bank of Japan, "為替スワップを利用した米ドル資金の調達コストの動向について" (Bank of Japan Review, March 2012, in Japanese), note 4 (page 7). ↩

12. Bank of Japan, "グローバルな為替スワップ市場の動向について" (Bank of Japan Review, July 2016, in Japanese), the summary (page 1), "最近のドル資金調達プレミアム拡大" (Recent widening of the dollar funding premium, pages 1–2), and Chart 1. The summary describes the amount added on top of dollar LIBOR as the dollar funding premium, and the body of that section and the title of Chart 1 also call the same amount the cross-currency basis. An English version is available: "Recent Trends in Cross-currency Basis" (Bank of Japan Review 2016-E-7, September 2016), which calls the amount added to USD LIBOR the cross-currency basis. ↩ ↩² ↩³

13. Bank of Japan Financial Markets Department, "店頭デリバティブ取引データからみた通貨スワップ市場：感染症拡大の影響とその後の回復を中心に" (The cross-currency swap market seen through OTC derivatives data, focusing on the impact of the spread of COVID-19 and the subsequent recovery) (Bank of Japan Review, May 2021, in Japanese), "はじめに" (Introduction, page 1) and "通貨スワップ取引の概要" (Overview of cross-currency swap trades, page 2). The source states that the cross-currency basis corresponds to the dollar funding premium, and that a negative cross-currency basis (written α in the source) indicates relatively strong demand for dollar funds. ↩ ↩²

14. Bank of Japan, "グローバルな為替スワップ市場の動向について" (Bank of Japan Review, July 2016, in Japanese), the summary (page 1) and "最近のドル資金調達プレミアム拡大" (Recent widening of the dollar funding premium, page 2). ↩ ↩²

15. Bank of Japan, "グローバルな為替スワップ市場の動向について" (Bank of Japan Review, July 2016, in Japanese), "ドル資金市場でのマーケットメイク活動" (Market-making in the dollar money market, page 4). ↩

16. Bank of Japan, "為替スワップを利用した米ドル資金の調達コストの動向について" (Bank of Japan Review, March 2012, in Japanese), "はじめに" (Introduction, page 1), "ドル転コストの構成要素" (Components of the dollar conversion cost, page 2), and notes 1 and 2 (page 7). The source expresses the dollar conversion cost for a euro area financial institution as "ユーロ調達金利＋為替スワップのコスト（β）" (euro funding rate + FX swap cost (β); our translation). ↩