* Project: “Absolute Exchange Rates” *

The project “ Absolute exchange rate ” is engaged in the analysis of paired exchange rates, the allocation of absolute exchange rates from them and their analysis.

Within the framework of the project, a methodology has been obtained for converting from paired rates to absolute exchange rates. Absolute currency ABS is defined for this. Rates of all available currencies are expressed in relation to ABS.

The project explores the properties of absolute rates. Various applications of absolute exchange rates are explored.

To date, several articles have already been published on the application of the absolute exchange rate method. I give the last two.

The article “ A study of the coherence of world currencies through the correlation of absolute rates ” describes applications of technology absolute exchange rates. A formal method for calculating the relationship between different currencies is given.

In the article “ Markowitz’s Portfolio Method for the currency market

A detailed description of the technology is provided in the article “ From currency pairs to absolute currency rates <> <> >

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The pair exchange rate is the ratio of two absolute rates.

In order to get absolute rates, you first need to calibrate this equation.

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Accordingly, you can see that the logarithms of paired rates are linearly related to the logarithms of absolute exchange rates. And that means you can search between them a simple linear transformation. Absolute course logarithms can be multiplied by a direct transformation matrix and get logarithms of paired courses.

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And we should expect that there is an inverse linear transformation to go from pairwise to absolute courses.

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Accordingly, the entire method of obtaining absolute rates is written as follows.

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You just need to logarithm pair courses, multiply them by inverse matrix and apply the exponent .

It has been described above how to get absolute courses from steam rooms. This is done using a linear transformation of logarithmic pairs and return through the exponent.

A direct linear transformation from absolute courses to paired is made using a direct matrix. Get it very easy. It consists of zeros, ones and minus ones (“0” means no pair connection and currency, “+1” is the currency in the pair numerator, “-1” is the currency in the denominator of the pair). Below you can see it.

The biggest problem is getting inverse matrix (or rather, use pseudo-inverse conversion ). With its help, you can go from pair to absolute courses. But getting this matrix is nontrivial due to matrix degeneracy direct conversion.

The previous article describes one method for obtaining such a matrix. See “ details of the transition from paired (relative) exchange rates to absolute. Work on the bugs ”reveals the method of obtaining absolute rates. In the experiment, we used the technique of transition to linearly independent components. Thus, it was possible to get rid of the degeneracy in the pseudo-inverse transformation.

This paper proposes another method of avoiding degeneracy in the direct transformation matrix. Currently used paired courses provided by RBC . There are 88 currency pairs for export. For them, there is the following direct transformation matrix.

White is zeros, blue is -1 and red is one. But she is degenerate. Those. it will not make the inverse matrix.

In the previous experiment, we managed to notice that the diagonally similar matrix has the inverse transformation. Accordingly, an offer was made to go to cross rates and non-degenerate transformation matrix .

The entire calculation was carried out in the Wolfram Mathematica . The system allows you to get cross rates right inside the system .

There are 45 currencies in total. For clarity, we first consider the case with a small number of currencies and currency pairs. The calculation is available at link in the cloud Wolfram Cloud .

The following 10 currencies were selected: AUD, CAD, HKD, JPY, SEK, USD, CHF, EUR, CNY, CZK. Of these, 9 currency pairs were identified: AUD/CAD, CAD/HKD, HKD/JPY, JPY/SEK, SEK/USD, USD/CHF, CHF/EUR, EUR/CNY, CNY/CZK.

The direct transformation matrix has the following form.

The colorized version looks like this.

The size of the matrix - 10x9. Rank of the matrix - 9. The matrix is nondegenerate. You can freely look for the inverse of it.

As a result of applying pseudo-inverse transform method we get the inverse matrix.

In the colorized form so.

The size of the inverse matrix is 9x10.

Before moving on, check the resulting matrix. To do this, multiply the inverse matrix by a straight line. The result is as follows.

And in the colorized form.

Really got unit diagonal matrix . So the reverse is true.

Were taken daily cross rates for 30 days from 03/28/2019 to 04/27/2019.

Here is a matrix of paired exchange rates. These are pairs of exchange rates for the following pairs: AUD/CAD, CAD/HKD, HKD/JPY, JPY/SEK, SEK/USD, USD/CHF, CHF/EUR, EUR/CNY, CNY/CZK.

Here is the course chart for one of the currency pairs.

The average values for paired courses turned out as follows.

Now let's try to estimate the variability of the data. To do this, use standard deviation . But for standardization, we divide it by an average value.The results will display on the chart.

As you can see, the standard deviations for each currency pair are within 0, 2% - 0, 9% of the average values of the pair rates.

Now we calculate absolute courses for paired ones (the technique is described in the section Method of Obtaining Absolute Courses). After calculations, we get the following data series.

These are absolute exchange rates for the following currencies: AUD, CAD, HKD, JPY, SEK, USD, CHF, EUR, CNY, CZK.

We present a graph of the absolute exchange rate of one of the currencies.

Average values for absolute rates are as follows.

Evaluate the variability of absolute exchange rates.

Standard deviations for the absolute rates of each currency are within 0, 2% - 0, 7% of the average values of absolute rates. And this is consistent with the data for currency pairs.

Now consider the results on all available currencies. The source code is available at link .

The full list contains the following 45 currencies: AUD, CAD, HKD, JPY, SEK, USD, CHF, EUR, CNY, CZK, GBP, ILS, NOK, NZD, RUB, SGD, ZAR, AED, ARS, BRL, CLP, COP, DKK, EGP, HUF, IDR, INR, ISK, KRW, KWD, KZT, MXN, MYR, PEN, PHP, PKR, PLN, QAR, RON, SAR, THB, TRY, TWD, UAH, VND. The following 44 currency pairs are chosen for them: AUD/CAD, CAD/HKD, HKD/JPY, JPY/SEK, SEK/USD, USD/CHF, CHF/EUR, EUR/CNY, CNY/CZK, CZK/GBP, GBP/ ILS, ILS/NOK, NOK/NZD, NZD/RUB, RUB/SGD, SGD/ZAR, ZAR/AED, AED/ARS, ARS/BRL, BRL/CLP, CLP/COP, COP/DKK, DKK/EGP, EGP/HUF, HUF/IDR, IDR/INR, INR/ISK, ISK/KRW, KRW/KWD, KWD/KZT, KZT/MXN, MXN/MYR, MYR/PEN, PEN/PHP, PHP/PKR, PKR/ PLN, PLN/QAR, QAR/RON, RON/SAR, SAR/THB, THB/TRY, TRY/TWD, TWD/UAH, UAH/VND.

We obtained a direct transformation matrix of size 45 by 44. It has a rank of 44.

The inverse matrix is the result of applying reverse pseudo-transform. The matrix size is 44 by 45.

After multiplying the inverse matrix by the straight line, we get the identity matrix.

Loaded 44 pair currency cross-rate. An example of one is given in the following graph.

Here is the average exchange rate data for each currency pair.

AUD/CAD 0.951638

CAD/HKD 5.8662

HKD/JPY 14.2202

JPY/SEK 0.0834375

SEK/USD 0.107433

Usd/chf 1.00544

CHF/EUR 0.885125

EUR/CNY 7.54636

CNY/CZK 3.40522

CZK/GBP 0.0335481

GBP/ILS 4.69022

ILS/NOK 2.38106

NOK/NZD 0.173451

NZD/RUB 43.5338

RUB/SGD 0.0209621

SGD/ZAR 10.4641

ZAR/AED 0.259015

AED/ARS 11.7136

ARS/BRL 0.0907021

BRL/CLP 171.256

CLP/COP 4.72058

COP/DKK 0.00210715

DKK/EGP 2.60095

EGP/HUF 16.5291

HUF/IDR 49.5307

IDR/INR 0.00490364

INR/ISK 1.73912

ISK/KRW 9.44975

KRW/KWD 0.000266945

KWD/KZT 1248.05

KZT/MXN 0.050062

MXN/MYR 0.216251

MYR/PEN 0.803967

Pen/php 15.7631

PHP/PKR 2.71475

PKR/PLN 0.0269842

PLN/QAR 0.954411

QAR/RON 1.16298

RON/SAR 0.885697

SAR/THB 8.48908

THB/TRY 0.179564

TRY/TWD 5.39876

TWD/UAH 0.871089

UAH/VND 863.675

For each currency pair, we look at the variability as in the experiment above.

The standard deviation for all currency pairs is between 0.2% and 2.5% of the average.

After recalculation we get absolute rates. Here is a chart of the absolute exchange rate of one of the currencies.

The mean values of absolute rates are as follows.

AUD 12.4626

CAD 13.096

HKD 2.23247

Jpy 0.156996

SEK 1.88165

USD 17.5149

CHF 17.4213

EUR 19.6824

CNY 2.60821

CZK 0.765955

GBP 22.832

ILS 4.86814

NOK 2.04455

Nzd 11.7884

RUB 0.270822

SGD 12.9197

ZAR 1.23485

AED 4.76765

ARS 0.407218

BRL 4.49018

CLP 0.0262207

COP 0.0055548

DKK 2.63619

EGP 1.01359

HUF 0.0613224

IDR 0.00123809

INR 0.25249

ISK 0.145194

KRW 0.0153652

KWD 57.5605

KZT 0.0461203

MXN 0.921362

MYR 4.26106

PEN 5.30007

PHP 0.336241

PKR 0.123862

PLN 4.59025

QAR 4.8096

RON 4.1356

SAR 4.66938

THB 0.550046

TRY 3.06473

TWD 0.567676

UAH 0.651731

VND 0.000754602

The variability of absolute rates can be estimated from the chart.

The standard deviation of all absolute rates ranges from 0.2% to 2.5% of the mean. That is consistent with the data of currency pairs.

The experiment with obtaining absolute courses from paired cross courses was a success. A new method for calculating absolute rates has been obtained. The method works and is easy to apply. For further research it is easy enough to get absolute rates.

The accuracy of the method is limited only by the accuracy of the cross-rates issued.

Unfortunately, open sources of cross rates cannot be found online. And accordingly, this method will not work on the site. But during the initial comparison of absolute rates from the site and those obtained in the present experiment, differences were revealed only in the fourth decimal place. A detailed comparison will be carried out in the following papers.

The latest version of the website is: true "> link .

Enin A.V.

Orenburg.

05/02/2019

Source text: We get the absolute exchange rates from the paired cross-exchange rates