USD/JPY Falls 10%: Why the Yen Gains 11.11%, Not 10%

If USD/JPY falls from 150 to 135, the dollar loses 10% against the yen. Over exactly the same interval, the yen gains 11.11% against the dollar. Both statements are correct: the percentage calculation changes its starting denominator when the quote is inverted.

That difference is not an arbitrage opportunity, an extra return from a broker, or evidence that one chart is wrong. It matters when you compare currency strength, translate a cash balance, or read a return reported in the opposite quotation. The following prices are hypothetical, with no interest or fees unless expressly included.

Start with units, not the sign on the chart

USD/JPY at 150 means 150 yen per dollar. Its reciprocal, JPY/USD, is 1/150 dollars per yen. After USD/JPY reaches 135, each yen buys 1/135 dollars. The dollar buys fewer yen, while the yen buys more dollars.

The direct percentage is 135/150 − 1 = −10%. The reciprocal percentage is (1/135)/(1/150) − 1 = 150/135 − 1 = +11.11%. Dividing the 15-yen change by 150 answers the first question. The reverse calculation has a different economic unit and denominator; changing only the sign cannot answer it.

Use the return factor to reverse a percentage

Let Q be units of the quote currency per unit of the base currency, with positive initial and final rates Q0 and Q1. The direct simple return is r = Q1/Q0 − 1. For the inverse quote, s = Q0/Q1 − 1. Therefore s = −r/(1+r), and (1+r)(1+s) = 1. Enter returns as decimals: −10% is −0.10, not −10.

A 20% decline in a quotation corresponds to a 25% rise in its reciprocal. The further the change is from zero, the worse the shortcut s ≈ −r becomes. Small daily changes can make the shortcut look harmless; that does not make it exact or suitable for reconciling cash.

Do not confuse reversing the quotation with reversing time. A rate that falls 10% and then rises 10% ends at 99% of its initial value: 150 × 0.90 × 1.10 = 148.5. Returning from 135 to 150 requires an 11.11% increase. The same denominator issue explains both, but they are different comparisons.

Reconcile the percentage with an actual cash balance

Suppose a dollar-based holder exchanges USD 10,000 for JPY 1,500,000 at 150, holds the yen without interest, and later converts it back at 135. The ending balance is USD 11,111.11 before costs. Relative to the initial USD 10,000, the dollar return is 11.11%—the yen’s appreciation in dollar terms, not the −10% change shown by USD/JPY.

This example is a fully funded cash conversion, not a leveraged short-position calculation. A trading contract can settle profit in another currency, use different conversion rules, and divide return by equity or posted margin. Applying 11.11% to a margin deposit would silently replace the denominator and can be seriously misleading.

State all three items alongside a return: what was held, the currency in which the result is measured, and the initial amount used as the denominator. A quote percentage alone supplies none of the contract-specific financing or leverage information.

Then replace reference prices with executable sides

Now suppose the opening USD/JPY bid/ask is 150.00/150.04 and the closing quote is 135.00/135.04. Selling dollars initially uses the bid, 150.00. Buying dollars with the yen later uses the ask, 135.04. The ending balance becomes 1,500,000/135.04 = USD 11,107.82, a return of about 11.0782% before any separate charges.

The USD 3.29 difference from the frictionless example is already in the chosen execution prices. Do not deduct the same spread again as an additional cost. Separately charged commission, cash interest and conversion fees still belong in the ledger.

When a two-sided quote is inverted, the sides also exchange places: inverse bid = 1/original ask, and inverse ask = 1/original bid. For the transaction-side derivation, see constructing executable cross rates and reversing bid and ask. A central-bank reference rate or a chart midpoint can support a price comparison, but is not a promise that either conversion is available at that price.

Log returns reverse exactly, but answer another question

For a consistently measured rate, the log return is ln(Q1/Q0). Inverting the rate gives ln(Q0/Q1), exactly its negative. This can be useful when a research dataset needs one common currency orientation or when returns are added across periods.

It does not turn the ordinary cash return into an equal-and-opposite percentage. Convert a log return g back to a simple return using exp(g) − 1. Labelling a log change as a percentage gain on account equity mixes two measures. Nor does changing orientation make different currency pairs directly comparable if their observation times or reference currencies differ.

A practical check before comparing two numbers

First write the unit beside both rates—for example, JPY per USD—and align the opening and closing timestamps. Confirm that both observations use the same source and the same price convention. A bid series and a midpoint series are not interchangeable.

Next compute Q1/Q0 and its reciprocal before subtracting one. Their product should be one for an exact reciprocal pair. If it is not, check rounded inputs, reversed dates, mixed quote sides or different instruments before declaring a discrepancy. Preserve calculation precision and round only the reported result.

Finally, if the question is “what did my money earn?”, reconstruct the actual conversions and cash flows. Include the applicable bid/ask, commissions, interest and contract settlement rules, then divide the net result by the explicitly chosen starting capital. Inverting a quote fixes the direction of the price measure; it cannot reconstruct missing account economics.