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Currency Indexes

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 = S₁^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 = (P₁^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*(P₁^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:

I_a = S₁^N w_n(Q_n – Q_ref)/N

Weight in a geometrical index are introduced as:

I_g = (P₁^N (Q_n/ Q_ref)^Wn)^1/N

In both case the weights must comply to:

S₁^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.

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