How does this work?

• Where are you getting your exchang rate info from? Are these bank rates or currency trade rates? – Aias Nov 25 '16 at 18:13
• Imagine taking 1 dollar and converting it back and forth. Do you think you should be making money both ways? – JB King Nov 25 '16 at 18:21
• This is just arithmetic. – Nick Nov 25 '16 at 19:04
• comparing the % numbers will never agree because one is a % of a USD and the other is a % of a CAD. – Kate Gregory Nov 25 '16 at 19:38
• How did you get 27.0% from 0.73988? – Federico Poloni Nov 26 '16 at 7:57

USDCAD: 1.35206 as a %, equals 1.35206 / 1 = 135%

CADUSD: .73988 as a %, equals 1 / .73988 = 135% [another way to say this is that 1 USD equals 135% of 1 CAD, or that 1 CAD equals 73.9% of 1 USD]

It is common to think of the % being "1.00 - .739 = 27.1%". However, that is only a close approximation, when the rates are close to 'parity'. The further out from \$1:\$1 you are, the less correct that is.

Consider an example where the USDCAD rate is 2 CAD for every 1 USD.

USDCAD would be 2.00, which as a % equals 200%. This means 1 USD is worth 200% of 1 CAD [ie: it is worth twice as much].

CADUSD would be 0.50, which as a % equals 50%. This means 1 CAD is worth 50% of 1 USD [ie: it is worth half as much].

Well, let's first do a bit of math:

``````1.35206 * .73988 = 1.0003621528
``````

Now round that to two decimal places and you get 1.00. In other words, if you convert both ways, you get back to where you started (with reasonable rounding).

Another example, with easier math.

Add 25% one way and subtract 20% to get back to where you started. The reason is that you start from different bases. Or subtract 20% and then add 25%.

Take \$100. Add 25% or \$25 and you get \$125. 80% of that is \$100 because 20% of \$125 is \$25.

Or take \$100. Subtract 20% or \$20 and you get \$80. 125% of that is \$100 because 25% of \$80 is \$20.

You always have to add a higher percentage than you subtract going the other way. Because when you add, you make the base bigger. So if you did the same percentage, you'd subtract a bigger number. E.g.

``````\$100 * 25% = \$25
\$100 + \$25 = \$125
\$125 * 25% = \$31.25
\$125 - \$31.25 = \$93.75
``````

Or when you subtract then add, you add a smaller number than you subtract

``````\$100 * 20% = \$20
\$100 - \$20 = \$80
\$80 * 20% = \$16
\$80 + \$16 = \$96
``````

The only time the currency percentages should match is when they are both 0%. Otherwise, the exchange rates should be multiplicative inverses.

If as a US dollar holder, it costs me \$3 to buy 4 Canadian dollars, (.75), does it stand to reason that my friend to the north would need to spend \$4 CAD to buy the \$3 US (1.33)?

Converting to percents or premium only obfuscates the issue. it's not magic, it's simple arithmetic.

• The astounding thing is many folks are asking questions like this, in the context: "Because I have a trading idea quants haven't thought of yet! PS what time does the market open" – Fattie Nov 26 '16 at 12:03
• ha. thanks. I had no idea why this 7th grade math issue was being asked this way. Makes sense now. – JTP - Apologise to Monica Nov 26 '16 at 13:04

## protected by Chris W. ReaOct 30 '18 at 16:46

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