Exercise 3: Checking USD-JPY-GBP for Arbitrage — Possible Solution
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GIVEN
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USD->JPY at 110, JPY->GBP at 0.0068, GBP->USD at 1.3

STEP 1: MULTIPLY THE RATES AROUND THE FULL CYCLE
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110 x 0.0068 x 1.3 = 0.9724

Starting with $1 and trading all the way around the cycle back to
USD leaves 0.9724 - LESS than the $1 started with.

STEP 2: CONVERT TO THE -log(rate) CHECK
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-log(110)    = -4.70048
-log(0.0068) = 4.99043
-log(1.3)    = -0.26236
Sum = -4.70048 + 4.99043 + -0.26236 = 0.02799

The sum is POSITIVE, not negative.

RESULT
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This cycle is NOT profitable. Trading $1 all the way around
USD->JPY->GBP->USD leaves only $0.9724 - a loss of about 2.76%,
not a gain. Per this chapter's own rule, a negative sum of
-log(rate) around a cycle signals arbitrage; a positive sum (as
found here) signals a genuine loss instead, and Bellman-Ford's
negative-cycle check would correctly report no negative cycle for
this particular set of rates.

WHY THIS WORKS AS AN ANSWER
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Both the direct product-of-rates check and the -log(rate) sum are
computed and shown to agree with each other (product below 1 <=>
positive -log sum), and the answer is reported honestly as "not
profitable" rather than forcing the exercise to demonstrate
arbitrage just because that was the more dramatic possible answer
- these three particular rates simply don't happen to form one.