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Why Opposite Triangular Arbitrage Returns Are Not Equal in Magnitude

Article Quant Q&A · Author: Dercni
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Summary

The document examines a currency triangular arbitrage example and asks why traversing the exchange-rate loop in opposite directions produces a stated gain of 2.5% one way and a loss of 2.44% the other. The accepted explanation is that percentage gains and losses compound multiplicatively rather than canceling as equal additive amounts. A 2.5% increase followed by a 2.44% decrease returns the starting value, since the second percentage is measured against the increased amount.

The example illustrates a general arithmetic point about reciprocal trade paths and return measurement. It does not establish that an actual arbitrage opportunity exists: the quoted rates are assumed, and the discussion does not account for bid-ask spreads, fees, execution timing, or market impact. A second answer offers a spreadsheet, but provides no additional analytical explanation in the document.

Key ideas

  • Opposite currency conversion paths can show percentage returns with different absolute magnitudes.
  • A gain followed by a loss compounds on the changed balance, so equal percentages do not reverse each other.
  • The example’s gain and loss multiply to a round trip factor of one.
  • The calculation does not include transaction costs or demonstrate an executable arbitrage.

Tags

ForexArbitrageStatistics
Full text
# Triangular arbitrage formula error


# Triangular arbitrage formula error












I am struggling with a formula for calculating the above. I have been using the following example: https://www.youtube.com/watch?v=lKu2LAgEcpU

```
A = YEN82 / USD
B = USD1.6 / GBP
C = GBP128 / YEN
```

The implied JPY/GBP XRate is 131.2 (82 X 1.6).

As the difference between the implied JPY/USD XRate and the quoted XRate is a constant should my result not always be either positive or negative the same value? In my calculations I am getting a 2.5% margin when I calculate one way and a -2.44% margin when I calculate the other way. I would have expected the result to be either 2.5% or -2.5%

i.e. Starting with JPY100,000,000

```
Option A = 2.5% gain
====================
JPY to GBP = 781,250
USD to GBP = 1,250,000
JPY to USD = 102,500,000

Option B = 2.44% loss
=====================
JPY to USD = 1,219,512.20
USD to GBP = 762,195.12
JPY to GBP = 97,560,975.61
```

## Answer by dm63 (score 3, accepted)

https://quant.stackexchange.com/a/28117

It is correct because a 2.50pct gain followed by a 2.44pct loss gets you back to where you started. I.e. (1+0.025)(1-0.0244)=1.

## Answer by QuantK (score 0)

https://quant.stackexchange.com/a/28220

You can dowload the following Excel file that implements the triangular arbitrage. Just replace the numbers and currencies with yours and the solution to check your outcome.

Hope this helps :)

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.