What Interest Rate Parity Actually Says
Parity is a statement about not being able to get something for nothing across currencies. Holding a euro deposit for a year at 2.50% versus holding a dollar deposit at 4.50% differ by exactly two percentage points, so the currency market must build that two-point gap into the price at which you can lock a future conversion. If it did not, an investor could borrow the cheap currency, convert, invest the dear one, and lock the reconversion forward for guaranteed profit with zero net investment.
Formally, the covered condition equates the domestic return to the hedged foreign return: 1 + rq = S(1 + rb)/F. Rearranged, the forward equals the spot scaled by the ratio of gross returns, so the forward premium or discount is mechanically the interest differential adjusted for compounding. The rate gap is the cause and the forward points are the effect; the exchange rate level itself plays almost no role in the difference.
The condition earned its keep as a building block of international finance because it holds tightly wherever money can move freely. Before testing any parity claim on real quotes, it helps to confirm your spot quote itself is derived correctly; the cross exchange rate calculator does that for pairs quoted through a common leg such as USD. A mis-derived spot contaminates every downstream forward point.
The Parity Formula and Its Three Conventions
The money-market form F = S(1 + rq x t)/(1 + rb x t) matches how interbank deposits actually accrue: simple interest on an actual-over-360 or actual-over-365 basis for tenors up to a year. The annually compounded form replaces the linear terms with powers, and the continuous form collapses to S x e^((rq - rb)t), the version most academic papers use. All three agree in direction and nearly in size; at the default 6-month setup they price 1.09067, 1.09049, and 1.09085 respectively, a range of fewer than four points.
The choice matters more as term or rate levels grow. At the one-year defaults the simple and compounded forms coincide exactly at 1.10107 because t equals one, while continuous prices 1.10182; stretch the term and the wedge widens. For tenors beyond a year, dealers quote swap points off annually compounded curves, so keeping the simple convention active will systematically misprice long forwards by an amount that grows with the square of the rate gap times term.
A dedicated forward pricer with day-count switches handles contract-level detail like ACT/360 accrual; the currency forward calculator is built for that job. Use this page when the question is the parity relationship itself: how far a quote sits from no-arbitrage, and what that gap is worth in basis points and cash. The two tools answer complementary questions about the same formula.
Forward Points, Premiums, and Discounts
Dealers quote forwards as spot plus swap points, and parity tells you exactly how many points are justified. At the defaults, the 2.00% dollar-over-euro differential justifies 210.73 points on a 12-month forward: a spot of 1.0800 becomes a forward of 1.10107. Points scale roughly linearly with term at short horizons, 17.96 at one month, 53.66 at three, 106.67 at six, and 210.73 at twelve, with slight convexity from the denominator as terms lengthen.
Points also scale one-for-one with the rate gap. Moving the dollar rate from 4.50% down to 3.50% cuts the justified premium from 210.73 to 105.37 points, while lifting it to 5.50% raises the requirement to 316.10 points. This linearity is why swap points are quoted in the market as a rate product: their price tracks central bank policy differentials almost tick for tick, and re-pricing after a Fed or ECB move is immediate.
Annualizing the premium makes quotes comparable across tenors: 106.67 points at six months is a 1.98% annualized premium, essentially the same information as 210.73 points at twelve months. The forward premium calculator specializes in that annualization and the premium-versus-discount framing. Note that the exact annualized premium, 1.95%, sits just under the raw 2.00% differential, and that 4.9 basis point convexity gap is the compounding cross term, not an arbitrage signal.
Covered Versus Uncovered Parity
The covered version involves a forward contract, so every cash flow is locked and the condition is a pure no-arbitrage statement that holds almost perfectly in liquid markets. The uncovered version drops the forward and asserts that the expected future spot equals the parity forward, converting an arbitrage identity into a theory about expectations. That is the version that fails: exchange risk cannot be arbitraged away, so expected returns can and do deviate.
The empirical failure is famous enough to have its own name, the forward premium puzzle. High-interest currencies depreciate less than the differential predicts, and sometimes appreciate outright, which is exactly the condition under which the carry trade calculator shows persistent profits. At the defaults, parity says the euro should fall about 2% against the dollar over a year; in many historical years it did nothing of the sort, and the difference was the carry trader's compensation for bearing crash risk.
Practical takeaway: treat the covered forward as a price and the uncovered claim as a forecast with a poor track record. Corporations that hedge at the forward are not making a directional bet, they are paying the interest differential to eliminate risk. Investors who skip the hedge are implicitly selling insurance, collecting the differential in quiet periods and paying it back during currency crises when capital suddenly flees.
Testing a Dealer Quote for Arbitrage
The arbitrage test is where this tool earns its keep. Take the defaults, a 1.10107 parity forward, and suppose a dealer quotes 1.1100 for twelve months. The quote is rich by 89.27 points, or 81.07 basis points annualized, and the covered strategy is mechanical: borrow 1,080,000 dollars, buy 1,000,000 euros spot, invest the euros at 2.50% so they grow to 1,025,000, and sell that amount forward at 1.1100 for 1,137,750 dollars. The dollar loan repays 1,128,600, locking in 9,150 dollars before costs, about 915 per 100,000 of notional.
A cheap quote flips every leg. At a quoted 1.0900 the quote is 110.73 points cheap, and borrowing euros, selling them spot, investing dollars at 4.50%, and buying euros forward at 1.0900 locks in 11,350 dollars on the same million. The tool computes both directions automatically and states which four transactions to run, so the only judgment left is whether the deviation survives bid-offer spreads and whether your credit lines can carry both legs to maturity.
For real quotes the deviation is usually small and persistent rather than large and fleeting. A quote of 1.0983 against the 1.10107 parity rate is 27.73 points cheap, a -25.19 basis point annualized basis, worth 2,842.50 dollars per million if fully captured. Persistent gaps of that size are the signature of cross-currency basis rather than an execution error, which is the subject of the next section. To stress the dollar figures at different principal amounts, the interest rate calculator handles the underlying deposit math.
Why Covered Parity Sometimes Breaks
Textbooks present parity as an iron law, and before 2008 the deviations in G10 forwards were measured in fractions of a basis point. The financial crisis changed that: when dollar funding seized, banks that needed dollars were willing to pay measurably above parity for the ability to swap euros into them, and the euro-dollar cross-currency basis blew out to more than 80 basis points. Dealers still quote forwards off the swap curve, so the old textbook identity now prices through a basis term that reflects funding stress.
The basis persists because exploiting it requires balance sheet, and balance sheet is not free. A bank capturing a 25 basis point basis on a million-euro trade books the trade, uses short-term funding, and consumes regulatory leverage capacity; at post-crisis capital costs, small bases sit below the hurdle. Central banks noticed and built swap lines precisely to compress the basis when it widens, making the basis itself a widely watched stress indicator.
Stale or convention-mismatched data causes most apparent violations you will find yourself. Rate quotes from different times of day, a 360 versus 365 day-count mismatch, or comparing a mid quote against a deal-side forward can each fake a 5 to 20 point deviation. Checking the interest-rate side of the trade with the forward rate calculator helps confirm whether your curve inputs, not the FX market, are the outlier.
Where the Calculation Shows Up in Practice
Corporate treasury teams live inside this formula every quarter. A US importer owing euros in six months compares the forward quote against parity to check fairness, then decides how much exposure to lock versus leave open; a 100-point move in the points on a 5 million euro payable is 50,000 dollars of budget impact. The same desk uses the rate-differential logic to choose between hedging with forwards or simply holding euro deposits, since the compound interest calculator shows the opportunity cost side of that choice.
Investors meet parity in share classes and fund wrappers. A dual-listed fund hedging a euro share class back to dollars pays the dollar-euro differential as a drag on returns, and that drag is visible in the fund's documentation as swap points. Options and futures courses derive every early result from parity, and the continuous-convention forward price is the same S x e^((rq-rb)t) expression the continuous compound calculator evaluates for pure rate problems.
Policy analysis rounds out the uses. Real-rate versions of parity connect inflation and nominal rates across countries, and the fisher effect calculator decomposes each currency's nominal rate into its real and inflation components before the cross-rate comparison. When central banks diverge, parity arithmetic is how analysts translate a policy path into an expected path for forward points, quarter by quarter.
Worked Examples and Sensitivity Checks
The default scenario prices a full textbook problem end to end: 1.0800 spot, 2.50% euros, 4.50% dollars, twelve months. The parity forward lands at 1.10107, a premium of 210.73 points, an annualized premium of 1.95%, and a 4.9 basis point convexity gap versus the naive 2.00% differential. Extend the same pair to ten years and the gap stops being trivia: the exact annualized premium is 1.60% against the 2.00% linear claim, a 40 basis point overstatement for anyone quoting forwards off the naive rule at long tenors.
Sensitivity to the dollar rate is close to linear in points: at rq of 3.50%, 4.00%, 5.00%, and 5.50%, the justified twelve-month premium runs 105.37, 158.05, 263.41, and 316.10 points. Each 50 basis point move in the differential shifts the forward by about 53 points, which is why forward traders watch central bank guidance as closely as spot traders watch news. The term dimension is nearly linear too at short horizons, roughly 17.6 points per month at these rates, bending slightly as compounding accumulates.
The arbitrage examples quantify execution reality. The rich 1.1100 quote banks 9,150 dollars per million but vanishes the moment anyone can trade it; the cheap 1.0983 quote books 2,842.50 and is exactly the size of gap that persists as basis when funding costs block the trade. Run your own numbers through the fields above with live mid rates, then re-run with deal-side rates; the collapse in apparent profit between the two runs is the honest measure of how much free money the market actually offers, which is usually none.