South Korean Won to U.S. Dollar 3-Month Forward-Implied Interest Rate Differential

The forward-implied interest rate differential is the annualized log premium of the 3-month won-dollar forward over the spot rate, the interest rate differential embedded in currency forward prices. When covered interest parity holds a forward trades at the spot rate adjusted for the two currencies' interest rates, so this premium measures the won-minus-dollar rate gap over the contract's horizon, and it is also the carry of a won position against the dollar, namely what the position accrues if the spot rate stays unchanged (Koijen et al. 2018). The series is the numerator of the carry-to-risk ratio of Galati, Heath, and McGuire (2007), which Aquilina et al. (2024) define as the differential implied by currency forwards, published as a series of its own.

Here $F_d(3\mathrm{M})$ and $S_d$ are the 3-month forward and spot rates in won per dollar on day $d$, and $\bar c_w$ is the mean over the valid days of the Monday–Friday week $w$, dated on its Friday and expressed in annualized percent, so each value is a weekly average rather than an end-of-day reading.

The sign follows a position long the won and short the dollar. A positive value means the won's forward-implied rate exceeds the dollar's, so a long-won position receives carry, and a negative value means that position pays it. This is the forward discount on which Lustig, Roussanov, and Verdelhan (2011) and Menkhoff et al. (2012) sort currencies into carry portfolios, and the sign is never flipped with the regime.

The differential is read from forward prices, not from money-market rates. Covered-parity deviations separate forward-implied from money-market differentials (Du, Tepper, and Verdelhan 2018), and the won's offshore non-deliverable forwards trade apart from the onshore market, where their implied yields can diverge from onshore rates (Ma, Ho, and McCauley 2004), so the series is an offshore forward-implied differential and not the gap between the two countries' money-market or policy rates. Fama (1984) further showed that the forward premium carries a time-varying risk premium, so the differential is a price and not a gauge of the depreciation to come. The inputs are public trade records of offshore non-deliverable forwards subject to U.S. reporting requirements, a subset of the offshore market, and weekly averaging induces positive autocorrelation in week-to-week changes (Working 1960), while non-overlapping weeks avoid the correlation of overlapping windows (Hansen and Hodrick 1980).

The differential is estimated daily from forward trades and averaged weekly, using transaction prices from public trade records rather than dealer mid quotes.

(1) Trade selection. Offshore non-deliverable forward trades with 7–366 days to settlement are kept after the reported trade history is applied, with cancellations removed, corrections substituted and reinstatements restored, and swaps, compressions, post-priced, non-standard and package trades are excluded. Trades executed offshore on Korean holidays are kept. The contracts belong to the offshore non-deliverable market described by Ma, Ho, and McCauley (2004), and only forwards enter because the definition in Aquilina et al. (2024) is the differential implied by currency forwards, which differs from a money-market differential by the covered-parity basis (Du, Tepper, and Verdelhan 2018).

(2) Daily curve. Log forward prices are regressed on a quadratic in time to settlement $\tau$, measured in years on an ACT/365 basis, with a fixed effect for each 15-minute execution window $j$.

Residuals beyond four times the scaled median absolute deviation are trimmed iteratively. The window effects absorb the spot level within each window, so the intercept plays the role of the spot rate and no external spot series is used.

(3) Daily carry. The 3-month carry is the annualized log forward premium at $\tau=0.25$, the 3-month horizon of Galati, Heath, and McGuire (2007) and the carry definition of Koijen et al. (2018).

It is computed only on days whose trade count, coverage near 3 months and number of populated windows meet fixed minimums. The premium is recorded as a price and is not converted into a depreciation gauge, since forward premia contain a time-varying risk premium (Fama 1984).

(4) Weekly mean and revisions. The published value is the simple mean over a Monday–Friday week.

A week with fewer valid days is left empty and never filled. Aggregating daily data to a lower frequency follows Menkhoff et al. (2012), who build global currency volatility from daily data by month, and weekly sampling costs little for questions of persistence because test power depends on the span of the sample rather than the sampling frequency (Shiller and Perron 1985). Non-overlapping weeks avoid overlap correlation (Hansen and Hodrick 1980), and the within-week mean induces positive autocorrelation in weekly changes (Working 1960). Trades are processed in a fixed order so that repeated computation reproduces the same values (Demmel and Nguyen 2013). Corrections, cancellations and late reports can arrive after a week closes, and the affected weeks are then recomputed, so past values can be revised and a first-published value can differ from the final one (Orphanides and van Norden 2002).

The ratio places the won within the carry-trade literature, where a position borrows in a low-interest currency and lends in a high-interest one. Fama (1984) showed that the forward premium is a biased guide to subsequent spot changes, the forward-premium anomaly, so a forward-implied differential embeds a time-varying risk premium and is not a gauge of the depreciation to come, and Lustig, Roussanov, and Verdelhan (2011) trace the payoffs of currency portfolios sorted by forward discount to a common carry factor tied to global risk. Scaling the differential by option-implied volatility, as Galati, Heath, and McGuire (2007) do, restates carry in units of the risk the option market prices, the sense in which Bhansali (2007) describes a carry position as economically close to a short position in currency volatility.

The scaling matters because carry and volatility move against each other in stress. Brunnermeier, Nagel, and Pedersen (2008) document that carry positions build up gradually and unwind abruptly as risk and funding illiquidity rise, which leaves high-interest currencies exposed to crash risk, and Menkhoff et al. (2012) identify global foreign-exchange volatility as the risk to which high-interest currencies are exposed. A jump in implied volatility therefore moves the ratio toward zero even when the differential is unchanged, the configuration Aquilina et al. (2024) examine around a sharp unwinding of yen-funded carry positions in which the won sits on the long side of the basket, and Daniel, Hodrick, and Lu (2017) document how deep and prolonged carry drawdowns can be.

For the won the ratio condenses two Korea-specific features into one number. Its numerator is an offshore non-deliverable forward-implied differential, and because the offshore market trades apart from the onshore one its implied yields can diverge from onshore rates (Ma, Ho, and McCauley 2004), while the covered-parity basis further separates it from the onshore money-market spread (Du, Tepper, and Verdelhan 2018). Its sign records whether, on forward prices, the won is a target or a funding currency against the dollar, so a negative reading means that a position long the won pays carry. Koijen et al. (2018) stress that carry is what accrues only if prices stay unchanged, and Burnside et al. (2011) relate carry payoffs to peso problems, rare adverse states that a short sample may not contain, so the ratio characterizes current pricing and is not a forecast.

The ratio is an exact identity, the forward-implied differential divided by the at-the-money implied volatility, so every weekly change is fully accounted for by changes in those two companion series, which show which of them moved. The three series are one construction and are not independent evidence about one another.

For currency desks and investors with won exposure, the ratio states on one scale how much forward-implied rate differential a won position carries relative to the exchange-rate volatility that options price, whose global component Menkhoff et al. (2012) identify as the risk priced in carry. A reading near zero means the differential is small relative to priced volatility, and a change of sign marks a switch in which side of a won-dollar forward receives the carry (Koijen et al. 2018).

Carry and option protection are two sides of one trade-off. Burnside et al. (2011) and Jurek (2014) hedge carry positions with currency options and find that removing crash exposure leaves part of the average carry payoff in place, and Bhansali (2007) shows that the carry position itself behaves like a short volatility position, whose exposure is priced by the implied volatility in the denominator. A won-based holder of dollar assets hedges by selling dollars forward, a position long the won, so a negative differential is a hedging cost and a positive one a hedging pickup, and the ratio sets that 3-month forward-hedge carry against the price of hedging with options at the same horizon (Garman and Kohlhagen 1983; Reiswich and Wystup 2010).

The series is a descriptive record and is not a forecast, nor a trading rule. Regressions of asset payoffs on persistent financial ratios suffer the small-sample bias documented by Stambaugh (1999), rules chosen after inspecting a sample invite the data-snooping biases analyzed by Lo and MacKinlay (1990) and White (2000), and carry strategies have suffered deep and prolonged drawdowns (Daniel, Hodrick, and Lu 2017) that a short sample may not contain.

The records cover only offshore trades subject to U.S. reporting requirements, a subset of the offshore market, and recent weeks can be revised as late reports arrive, so their values are provisional in the sense of Orphanides and van Norden (2002). Desks that read the ratio beside the forward-implied differential and the implied volatility should treat the three as one identity and attribute a change in the ratio by checking which leg moved.

KRFXIRD3M is the weekly mean of the daily 3-month forward-implied interest rate differential of the won against the dollar, the numerator of the carry-to-risk ratio that Aquilina et al. (2024) define through currency forwards, and as a difference of two rates it is a spread-type object whose level is tested on the pinned sample of 104 complete weeks from 2024-10-04 to 2026-09-25 recorded for sign-off. The deterministic term is pinned to a constant, since a difference between two interest rates cannot trend without bound and a fitted trend would be a specification error. The object is the differential read from forward prices, which covered-parity deviations separate from a money-market spread (Du, Tepper, and Verdelhan 2018), so nothing in this battery carries over to the gap between the two countries' money-market rates.

Over this span of about two years the three procedures agree against the stationary prior of a spread. The augmented Dickey and Fuller (1979) test, lag-augmented as in Said and Dickey (1984) with the lag chosen by an information criterion (Ng and Perron 2001), does not reject a unit root with a statistic of 0.240807 at p = 0.97446, the Phillips and Perron (1988) test concurs at −0.195018 with p = 0.939113, and the KPSS test rejects level stationarity at 1.08298 with p < 0.01 (Kwiatkowski et al. 1992), an honest agreement that carries a clean I(1) verdict. The GLS-detrended escalation of Elliott, Rothenberg, and Stock (1996) is reserved for ambiguous readings and was not needed. Unit-root power depends on the span of the sample rather than on the number of observations (Shiller and Perron 1985). The verdict therefore describes this two-year window only, over which the differential cannot be told apart from a series with a stochastic trend, and it says nothing about how rate differentials behave over a full policy cycle.

On the level the first-order autoregressive coefficient is 1.00318, at or above unity, so the half-life is undefined, a descriptive persistence summary and not an integration claim, read with the small-sample bias of Andrews (1993) in mind. The Ljung and Box (1978) portmanteau on the level, refining the Box and Pierce (1970) form, rejects the no-autocorrelation null at lags 13 and 26 on the weekly ladder, with Q = 833.968 and Q = 1002.53 and p = 0.000 in both cases, which on a level with a unit-root reading restates that persistence. Because the triplet gave a clean integrated reading, the Bai and Perron (1998) search, computed by the Bai and Perron (2003) algorithm with fifteen-percent trimming, moves to the 103 weekly differences from 2024-10-11 to 2026-09-25 and finds two breaks in the mean of the differenced series, dated 2025-02-14 and 2025-05-30 (Perron 1989). These mark shifts in the average week-to-week change of the differential between sub-periods, not shifts in its level, and the segment they bound is short, so the dates describe this sample rather than identify events. The differences are changes in weekly averages, which carry an induced positive first-order autocorrelation even when the daily path has none (Working 1960).

The three verdicts of this group are tied by an exact identity, since the carry-to-risk ratio equals this differential divided by the at-the-money implied volatility in every week, so they are not independent evidence, and the clean integrated reading of the ratio is in large part this series' persistence passed through the quotient. Late trade reports can revise past weeks after first publication. Replaying the archive in file-date order over 104 weeks, the 95th percentile of the absolute gap between each week's first-published and final differential is 0.0054 percentage points, below the median weekly standard error of 0.0109 that serves as the gate's tolerance, and no week was empty at first publication (Orphanides and van Norden 2002).

The series is weekly and no seasonal quantity is posited for a rate differential of this kind, so the seasonal-unit-root machinery of Hylleberg et al. (1990) and the Canova and Hansen (1995) seasonal-stationarity test are not applicable and were deliberately not run (Beaulieu and Miron 1992; Ghysels and Osborn 2001).