Forward Exchange Rate Formula: A Forward Is Not a Forecast

The forward exchange rate formula is the one equation anyone buying yield in a foreign currency needs. At the close on 2 October 2026, three-month money in Brazil paid about 13.5% a year; in the United States it paid about 4%. Lately that gap has been moving on-chain: a former central-bank director has announced a real-pegged stablecoin that would pass Brazilian government-bond yield to its holders (CoinDesk, January 2026), and tokenized Brazilian receivables and on-chain credit pools offer reais to investors who report in dollars. For anyone whose books are kept in dollars, every one of these products raises the same question: what is 13% in reais worth in dollars?

The answer turns on a single decision—whether you fix the rate at which the reais come back. Leave it open, and the dollar result can land almost anywhere. Fix it today, and the result is about 6% a year: the dollar rate, not the real rate. This article explains why. The forward exchange rate is not a forecast of the currency. It is arithmetic on two interest rates, and it absorbs the gap between them exactly. We check that arithmetic on Brazil’s exchange data for 2 October 2026.

A million dollars, ninety days

An investor brings $1,000,000 to Brazil and converts it at the spot rate—the rate for immediate exchange, settled in a day or two—of 5.2265 reais per dollar. That gives R$5,226,500. The money goes on deposit for 90 calendar days, which is 60 Brazilian business days, at the interbank DI rate of 13.533% a year. On day 90 the account holds R$5,386,855: plus 3.07% in reais.

Now the reais have to become dollars again, and the result depends on an exchange rate nobody knows yet:

From +8.8% to −10.2% in one quarter: that spread is currency risk, and it dwarfs the 3% of interest. A fall of about 3% in the real, to 5.3869, is enough to wipe out the whole quarter’s interest. Moves far larger than that are on record. Between April 2014 and September 2015 the real lost 47% of its dollar value; between January and May 2020 it lost almost 32% (FRED H.10 daily rates).

A hedge removes the unknown. On day zero the investor agrees to sell the reais on day 90 at 5.3103. The outcome is then known in advance—$1,014,425, plus 1.44% over 90 days, about 6% a year—whatever happens to the currency.

Look again at the third row. 5.3103 is the forward rate: the exchange rate for a future date, agreed today. If the real weakens exactly to the forward, hedged and unhedged end level. The hedge wins if the real ends weaker than the forward and loses if it ends stronger.

Try it: where will the real be in 90 days?

The calculator below runs the same $1,000,000. The slider is the unknown rate on day 90; the two other sliders are the interest rates the forward is built from.

Drag the first slider and the purple curve moves while the green line stays put: that is what a hedge buys. Now change the DI rate instead. The crossing point—the forward—moves, even though nothing about anyone’s view of the real has changed. That single gesture is the argument of this article.

Why the forward is arithmetic, not a forecast

There are two ways to turn dollars today into dollars in 90 days.

• Path A. Keep the dollars and earn the dollar rate.
• Path B. Convert to reais, earn the real rate, and on day zero sell the future reais back at a forward rate agreed now.

Both paths are riskless: every number on them is known at the start. So they must end at the same place. If path B paid more, a bank could borrow dollars, run path B, repay the loan and keep the difference with no risk, and banks would do that until the forward moved back into line. That no-arbitrage equality is called covered interest parity. “Covered” means the currency leg is locked by the forward; its cousin, uncovered parity, is a claim about expectations and does not hold well in the data.

Nothing in that formula describes where anyone thinks the real is going. The forward discount on the real is the interest gap and nothing else. A market that expected the real to strengthen would still quote it weaker for future delivery, for as long as Brazilian rates sat above dollar rates.

The formula in Brazil’s market conventions

In practice the two rates come with their own conventions. The real rate is DI, the interbank rate; its term structure, the “pré” curve, is built by Brazil’s exchange, B3, from DI futures and compounds over 252 business days a year. The dollar rate that matters is the dollar interest rate inside Brazil, the cupom cambial, implied by B3’s onshore dollar-interest-rate futures (DDI and FRA de cupom) and quoted as a simple rate on a 360-day year (B3 contract specification).

Here is the calculation on B3’s curves for 2 October 2026, 90 calendar days (60 business days):

The match to within 0.0001 is not a lucky test of theory, and it should not be sold as one. It is how the exchange builds its number: B3’s reference dollar curve comes from exactly these two rate curves, and the spot in the first row is the one consistent with them. The central bank’s official fixing that day, PTAX, was 5.2238. These are prices at the close on 2 October 2026. On 5 October, after the first round of Brazil’s presidential election, the real strengthened about 4.6% against the dollar in a single day (PTAX 5.2238 → 4.9859), and the curves moved with it; the arithmetic did not. What the exercise does prove is the point of this article: the forward you are quoted is the output of a formula, and the formula has no input for anyone’s view of the real.

The difference between forward and spot is quoted as forward points:

Here, 5.3103 − 5.2265 = 0.0838 reais. Annualized, the same gap is the number usually called the cost of the hedge:

(5.3103 / 5.2265)^(365/90) − 1 = 6.66% a year. Why that is not a fee, who pays it, and why the market price sits well below the gap between the two central-bank rates is the subject of the next article in this series.

What a hedged real earns

Follow the money through path B. The hedged investor ends with 1.4425% over 90 days—exactly the onshore dollar rate, about 6% a year once compounded. The 13.5% did not vanish; it went into the forward rate, which pays out the real’s higher interest as a worse exchange rate on day 90. You cannot hold 13% in reais and remove the currency risk at the same time. They are the same trade seen from two sides.

# B3 curves, 2 October 2026, 90 calendar days = 60 business days
S = 5.2265          # spot consistent with B3 curves, BRL per USD
r_di = 0.13533      # DI "pre" curve, annual, 252 business-day compounding
r_cup = 0.0577      # onshore dollar rate (cupom cambial), annual, simple, ACT/360
du, dc = 60, 90

g_brl = (1 + r_di) ** (du / 252)     # 1.030681
g_usd = 1 + r_cup * dc / 360         # 1.014425
F = S * g_brl / g_usd                # 5.3103 (B3 publishes 5.3102)

brl_end = 1_000_000 * S * g_brl      # R$5,386,855
print(f"forward {F:.4f}, points {F - S:.4f}")
print(f"hedge cost {((F / S) ** (365 / dc) - 1):.2%} a year")
print(f"hedged: ${brl_end / F:,.0f}  ({brl_end / F / 1e6 - 1:+.2%} over {dc} days)")
for s_t in (4.95, S, 5.50, 6.00):
    usd_end = brl_end / s_t
    print(f"unhedged at {s_t:.4f}: ${usd_end:,.0f}  ({usd_end / 1e6 - 1:+.2%})")

Where readings of the forward go wrong

Treating the forward as a forecast

“The market expects the real to weaken by 1.6% in three months” is the most common misreading of a forward quote. The quote says only that Brazilian rates are higher than dollar rates. As a predictor of the future exchange rate, the forward is poor: it does no better than assuming today’s rate will hold (Meese and Rogoff, 1983), and the forward discount is a biased guide to the actual move (Fama, 1984). Brazil shows the pattern: over 2006–2026 the dollar rose against the real by about 4.5% a year, while three-month forwards priced in a median of about 7% a year. The gap between the two is the carry—and what carry earned and cost in the tails is the fourth article of this series.

Taking spot and forward from different moments

PTAX, the central bank’s reference rate, is the average of four dealer polls taken in windows starting at 10:00, 11:00, 12:00 and 13:00 Brasília time (Banco Central do Brasil). B3’s curves are struck at the end of the trading session, hours later. Divide the end-of-day forward by the midday PTAX and you get a hedge cost of 6.88% instead of 6.66% for 90 days, and 6.82% instead of about 6.2% for 32 days. The shorter the term, the larger the distortion, because a small mismatch in the spot is annualized over fewer days. Take spot and forward from the same moment, or skip the spot entirely and compare two forwards.

Expecting a hedged local yield

If an offer shows a Brazilian-rate yield next to dollar-denominated safety, one of the two is not there in full. Either the currency risk is still with the holder, or the extra return comes from somewhere other than the interest rate: credit risk, a liquidity premium, the onshore-offshore basis, or a subsidy. The question to ask is which one.

Mixing day-count conventions

DI compounds over 252 business days; the onshore dollar rate is simple on a 360-day year; most investors annualize on 365 calendar days. The same trade gives 6.66% annualized on calendar days and 6.91% on business days. Comparing hedge costs is only meaningful within one convention.

What this means for a token that pays Brazilian yield

There are three ways to hand a real-denominated yield to a holder who counts in dollars, and the forward prices all three.

1. Unhedged. The holder receives DI minus fees, plus the full currency move. A real-pegged stablecoin held by a dollar investor is exactly this position. Historically the average has favored it; the tails—a 47% fall over 17 months to 2015 and 32% in four months of 2020—erase years of interest.
2. Hedged. The holder receives the onshore dollar rate plus whatever credit premium the asset pays over DI, minus hedging friction. A farm loan at 20% in reais, hedged at this example’s 6.66%, becomes about 12.5% in dollars: 1.20 / 1.0666 − 1. The 7.5 points between the loan rate and 12.5% are not lost; they are the interest gap, and they belong to the hedge counterparty.
3. USD-native. The structure lends dollars, and the borrower carries the currency risk. The holder’s yield then has to be built from credit up, layer by layer, as in our tokenized agri-debt adjudication.

Two refinements matter for structuring. First, the term. The forward curve on 2 October ran to 5.4007 at 180 days and 5.5801 at 360 days, so a one-year hedge locked in about 6.9% a year at that day’s prices; a vault that funds 9-month loans with rolled 3-month hedges takes on the risk that the next hedge costs more. Second, where the dollar rate lives. The onshore dollar rate of 5.77% sat well above the Federal Reserve’s 3.75–4.00% range, so a hedged real held inside Brazil earned more than Treasury bills in the United States. That premium is the cross-currency basis. It is reachable only through the onshore market; offshore investors hedge the real with non-deliverable forwards that can price differently. Since 2008 such deviations from parity have been large and persistent even in major currencies (Du, Tepper and Verdelhan, 2018). Both refinements decide whether a structure’s currency layer costs what its model assumes; the four funding levers show where that layer sits in the rest of an RWA deal.

The takeaway

A forward exchange rate is the spot rate adjusted by two interest rates. On 2 October 2026 that arithmetic turned 5.2265 reais per dollar into a 90-day forward of 5.3103, a hedge cost of 6.66% a year, and a hedged return equal to the onshore dollar rate: about 6%. The real’s 13.5% went into the forward, as covered interest parity says it must. For a dollar holder there is no version of a Brazilian yield that keeps the rate and drops the currency risk. There is an unhedged position with a wide distribution of outcomes, a hedged one that earns dollar rates plus credit, and a USD-native one priced from credit up. Knowing which of the three a product actually is comes before any judgment of whether its yield is fair.