By Armando Rodriguez

Indexes come in two flavors, Arithmetical and Geometrical. The first would the
average difference of the pairs quotes to a reference one or in math terms:

I_{a}
= S_{1}^{N}
(Q_{n} – Q_{ref})/N

Above I_{a} is the arithmetical index, Q_{n} the quote of the n^{th}
pair involving the specified currency and Q_{ref} is the reference value
of the n^{th} pair quote. The Geometrical is the Nth root of the
product of all the pair quotes to reference ratios or in math terms:

I_{g
}
= (P_{1}^{N}
(Q_{n}/ Q_{ref}))^{1/N}

Arithmetical indexes are straight forward, but may have negative values that
could be misleading. On the other hand, the geometrical indexes, though
involving a more complex calculation, can have no negative values, instead
values will be just less than one or greater than one. As with other ratios,
this index can be expressed in percent, resulting in the following expression:

I_{g
}
= 100*(P_{1}^{N}
(Q_{n}/ Q_{ref}))^{1/N}

Some indexes introduce weights, this is useful when not all the elements
contributing to the index are equally important. In an arithmetic indexes the
weight are introduced as follows:

Symbol |
Index Reference |

GBP/USD |
1.61993007745267 |

EUR/USD |
1.39650978005865 |

USD/CHF |
1.08450901988637 |

USD/JPY |
92.4404046920827 |

AUD/USD |
0.780169522240528 |

USD/CAD |
1.16181135869566 |

NZD/USD |
0.628516958174904 |

EUR/CHF |
1.51438709190672 |

EUR/JPY |
129.076219334245 |

EUR/GBP |
0.862326969476745 |

CHF/JPY |
85.2638280329801 |

GBP/CHF |
1.75724032051282 |

GBP/JPY |
149.744775809716 |

USD/HKD |
7.750876 |

EUR/CAD |
1.6233713908046 |

EUR/NZD |
2.2244984720862 |

EUR/AUD |
1.79057481142242 |

NZD/JPY |
58.1209682726204 |

AUD/CAD |
0.906692083333335 |

AUD/NZD |
1.24275102627258 |

I_{a}
= S_{1}^{N}
w_{n}(Q_{n}
– Q_{ref})/N

Weight in a geometrical index are introduced as:

I_{g
}
= (P_{1}^{N}
(Q_{n}/ Q_{ref})^{Wn})^{1/N}

In both case the weights must comply to:

S_{1}^{N}
w_{n}
= 1

L3Capital currency index is of the geometrical flavor. The references were
arbitrarily chosen to be the average quote on July the 1^{st} of 2009.
References may be arbitrary, since what is important for indicating changes in
the buying power (BP) of a currency, is the change in the value of the index,
not its absolute value.

As for the weights, it was decided to have none and here is why. Assume a pair
x/y that, say went up, this can mean one of the following: x increased its
buying power (BP); y decreased its BP or both moved, but still resulting in an
increase of the quotient. For an index of x, it would be best if y had remained
at a constant BP. So the pairs with the greater weights should correspond those
with a y less prone to changing its BP. This would lead to assign a greater
weight to those currencies with less inflation rates, but the currencies
involved in the index are all have a pretty low inflation rate, so there is no
point in assigning different weights.

The actual formula for the USD index is:

I_{USD} = 100*exp( (avg( log(Q_{USD/*}/_{ }Qref_{
USD/*})) - avg( log(Q_{*/USD}/ Qref_{ */USD}))/2 )

Generally, it is invalid to average the averages, but the above is a valid
expression for there being four pairs Q_{USD/*}/ and four pairs Q_{*/USD}
as well.

For the EUR:

I_{EUR}
= 100*exp(avg( log(Q_{EUR/*}/Qref_{ EUR/*}))

For the JPY

I_{USD}
= 100/exp(avg( log(Q_{*/JPY }/Qref_{ */JPY}))

The quotes used in the above formulas are not instantaneous but their averages
for the last ten minutes.