Currency Indexes Explained
Indexes come in two flavors, Arithmetical and Geometrical. Arithmetical indexes average the difference of the pairs quotes to a reference or in math terms:
Ia = Σ1N(Qn – Qref)/N
Above Ia is the arithmetical index, Qn the quote of the nth pair involving the specified currency and Qref is the reference value of the nth pair quote. The Geometrical is the Nth root of the product of all the pair quotes to reference ratios or in math terms:
Ig = (Π1N (Qn/ Qref))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:
Ig = 100*(Π1N (Qn/ Qref))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:
Ia = Σ1NWn(Qn – Qref)/N
Weight in a geometrical index are introduced as:
Ig = (Π1N (Qn/ Qref)Wn)1/N
In both case the weights must comply to:
Σ1NWn=1
L3 Capital's currency indexes are of the geometrical flavor. The references were arbitrarily chosen to be the average quote on July the 1st of 2009 (see table below).
|
Symbol
|
Index
|
|
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
|
1.51438709190672
|
|
EUR/JPY
|
129.076219334245
|
|
EUR/GBP
|
0.862326969476745
|
|
CHF/JPY
|
85.2638280329801
|
|
GBP/CHF
|
1.75724032051282
|
|
GBP/JPY
|
149.744775809716
|
|
EUR/CAD
|
1.6233713908046
|
|
EUR/NZD
|
2.2244984720862
|
|
EUR/AUD
|
1.79057481142242
|
|
NZD/JPY
|
58.1209682726204
|
|
AUD/CAD
|
0.906692083333335
|
|
AUD/NZD
|
1.24275102627258
|
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 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 all have low inflation rates, so there is no point in assigning different weights.
The actual formula for the USD index is:
IUSD = 100*exp((avg(log(QUSD/*/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 QUSD/*/ and four pairs Q*/USD as well.
For the EUR:
IEUR = 100*exp(avg( log(QEUR/*/ Qref EUR/*))
For the JPY
IJPY = 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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