Currency attribution for bilateral exchange rates: a decomposition of the moments

A bilateral exchange rate is the price of a base currency in terms of a quote currency . In this situation, how do international investors decide the contributions of both the base currency and the quote currency to the mean, variance, skewness, and kurtosis of the movements in each bilateral exchange rate? For a group of currencies, each bilateral exchange rate can be decomposed into a difference between two multilateral exchange rates: a base-currency multilateral exchange rate minus a quote-currency multilateral exchange rate. In this paper, the decomposition of bilateral exchange rates is used to decompose the moments of the movements in bilateral exchange rates. The result is a quantitative methodology to perform currency attribution, where the mean, variance, skewness, and kurtosis of the movements in each bilateral exchange rate can be attributed to the base currency and the quote currency.


Introduction
Currency attribution is a quantitative methodology to decompose the mean, variance, skewness, and kurtosis of the movements in a bilateral exchange rate into the base currency and the quote currency of the bilateral exchange rate.A bilateral exchange rate is the price of a base currency in terms of a quote (or numéraire) currency.For a group of currencies, each bilateral exchange rate can be decomposed into a difference between two multilateral exchange rates: one associated with the base currency and one associated with the quote currency (Kunkler & MacDonald, 2015).The motivation of this paper is to provide a quantitative methodology to perform currency attribution, which allows international investors a finer level of granularity to uncover the contributions of both the base currency and the quote currency to the mean, variance, skewness, and kurtosis of movements in bilateral exchange rates.
The moments of a statistical distribution describe the shape of that distribution.With respect to bilateral exchange rate movements, the mean is the first moment and measures the central location of the bilateral exchange rate movements.After the first moment, the subsequent moments are usually adjusted by the first moment and are called central moments.The variance is the second central moment and measures the dispersion of the bilateral exchange rate movements relative to the mean.After the second central moment, the subsequent central moments are usually normalised by the second moment and are called normalised central moments.Skewness is the third normalised central moment of bilateral exchange rate movements and measures the symmetry of the bilateral exchange rate movements relative to the mean.A number of papers have reported significant asymmetry in the movements of bilateral exchange rates (see Broll, 2016;Brunnermeier et al., 2008, pp. 313-347;Jurek, 2014;Patton, 2006).Kurtosis is the fourth normalised central moment of bilateral exchange rate movements and measures the tailedness, or peakedness, of the bilateral exchange rate movements relative to the mean.There have been a number papers that reported significant leptokurtic behaviour in the movements of bilateral exchange rates (see Bollerslev, 1987;León et al., 2005;Wali & Manzur, 2013).
The decomposition of a system of N bilateral exchange rates into N + 1 currency factors was proposed by Mahieu and Schotman (1994).However, the system has infinitely many solutions, since there are fewer equations (N bilateral exchange rates) than unknown variables (N + 1 currency factors): thus the system of equations is underdetermined.Mahieu and Schotman (1994) assumed structure on the factor loadings and used ordinary least squares on a pooled times series sample to estimate the currency factors.In contrast, Kunkler and MacDonald (2015) extended the system of equations used in Mahieu and Schotman (1994) by introducing an equilibrium condition for the N + 1 currency factors, which added an extra equation to the system.The equilibrium condition changed the system from underdetermined to one where the number of equations (N bilateral exchange rates plus one equilibrium condition) E-mail address: m.kunkler@pgr.reading.ac.uk.In this situation, the system of equations has a single unique solution and the currency factors are tradeable equally-weighted multilateral exchange rates (see Kunkler & MacDonald, 2015).The multilateral exchange rates are independent of the choice of numéraire used for the system of N bilateral exchange rates and are equivalent to the invariant currency indexes of Hovanov et al. (2004).
This paper contributes to the literature by using the decomposition of bilateral exchange rates of Kunkler and MacDonald (2015) to further decompose the mean, variance, skewness, and kurtosis of the movements in bilateral exchange rates.This creates a quantitative methodology to perform currency attribution, which decomposes the mean, variance, skewness, and kurtosis of bilateral exchange rate movements into the base currency and the quote currency.It is shown that the mean can be decomposed into the difference between two terms associated with the two multilateral exchange rates: two mean terms.In addition, the variance can be decomposed into a weighted sum of three terms associated with the two multilateral exchange rates: two variance terms and one covariance term.Furthermore, the skewness can be decomposed into a weighted sum of four terms associated with the two multilateral exchange rates: two skewness terms and two co-skewness terms.Finally, the kurtosis can be decomposed into a weighted sum of five terms associated with the two multilateral exchange rates: two kurtosis terms and three co-kurtosis terms.
Currency attribution can be applied to the area of hedging exchange rate risk.When international investors buy foreign currencies, equities, or bonds, they indirectly hold positions in the foreign currency (Campbell et al., 2010).Thus, international investors are exposed to exchange rate risk and must decide whether or not to hedge the exchange rate risk.Risk-averse investors prefer positive skew to reduce exposure to extreme left-tail events and lower kurtosis to reduce exposure to extreme events generally (Kim et al., 2014).The standard approach to hedge bilateral exchange rate risk is by trading the associated bilateral exchange rate.However, the decomposition of bilateral exchange rates provides an alternative approach, where investors can hedge bilateral exchange rate risk by trading either or both of the two associated multilateral exchange rates: one for the base currency and one for the quote currency.In similar work, two optimal hedge ratios were estimated to adjust the currency exposures of international equity investors (Kunkler, 2021).

Sample moments
The sample moments are used to describe the shape of the unknown moments of a data sample.The kth sample moment of a data sample is given by: where k = 1, 2, 3, 4, …; m k is the kth sample moment; N is the sample size; and E is the expectation operator.For k > 1, the kth central sample moment of a data sample is given by: where k = 2, 3, 4,…; m k is the kth central sample moment; μ = m 1 is the first sample moment in Eq. (1); and N is the sample size.For k > 2, the kth normalised central sample moment of a data sample is given by: where k = 3, 4,…; √ is the square root of the second central sample moment in Eq. ( 2); μ = m 1 is the first sample moment in Eq. ( 1); and N is the sample size.
The central sample moments in Eq. ( 2) and the normalised central sample moments in Eq. ( 3) are biased, as they do not account for the degrees of freedom lost by estimating the moments used within the central sample moments or the normalised central sample moments.However, there are well known corrections that remove the bias.For example, the well-known Bessel's correction for the variance is: where m 2 * is an unbiased estimator of the second central sample moment.For simplicity, the biased versions will be used throughout this paper.

Decomposition of bilateral exchange rates
A bilateral exchange rate is the price of a base currency in terms of a quote (or numéraire) currency.Bilateral exchange rates are typically modelled in log terms to overcome the Siegel Paradox (see Taylor & Sarno, 1998).Consequently, all exchange rates throughout this paper will be written in log terms.For a group of N currencies, let p i/j (t) represent the ith/jth bilateral exchange rate at time t, which is the ith (base) currency in terms of the jth (quote) currency, where i, j = 1, …, N and t = 0, …, T. In addition, let Δp i/j (t) represent the log return of the ith/jth bilateral exchange rate at time t, where i, j = 1, …, N and t = 1,…, T.
The movements in a multilateral exchange rate are calculated by an equally-weighted sum of the movements in a group of bilateral exchange rates given by: where i = 1, …, N; t = 1, …, T; Δp i (t) is the log return of the ith multilateral exchange rate; and Δp i/j (t) is the log return of the ith/jth bilateral exchange rate (see Kunkler & MacDonald, 2015).The movements in a multilateral exchange rate in Eq. ( 5) are priced in terms of an equally-weighted basket of the N currencies.It should be noted that the multilateral exchange rates in Eq. ( 5) are independent of the choice of numéraire used for the group of N bilateral exchange rates (see Hovanov et al., 2004;Kunkler, 2022b;Kunkler & MacDonald, 2015).The movements in each bilateral exchange rate can be decomposed into a difference between the movements in a base-currency multilateral exchange rate and the movements in a quote-currency multilateral exchange rate by: where i, j = 1, …, N; t = 1, …, T; Δp i/j (t) is the log return of the ith/jth bilateral exchange rate; Δp i (t) is the log return of the ith (base-currency) multilateral exchange rate; and Δp j (t) is the log return of the jth (quotecurrency) multilateral exchange rate (see Kunkler & MacDonald, 2015).

Mean
The mean is the first sample moment of a data sample and measures the central location of the data sample.Using Eq. ( 1), the mean of the movements in a bilateral exchange rate is: where i, j = 1, …, N; μ i/j is the mean of the log returns of the ith/jth bilateral exchange rate; and Δp i/j (t) is the log return of the ith/jth bilateral exchange rate.Similarly, the mean of the movements in a multilateral exchange rate in is: where i = 1,…,N; μ i is the mean of the log returns of the ith multilateral exchange rate; and Δp i (t) is the log return of the ith multilateral exchange rate.
The mean of the movements in a bilateral exchange rate can be decomposed by substituting Eq. ( 6) for Δp i/j (t) into Eq.(7) to give: where i, j = 1, …, N; μ i/j is the mean of the log returns of the ith/ jth bilateral exchange rate; Δp i (t) is the log return of the ith (base-currency) multilateral exchange rate; and Δp j (t) is the log return of the jth (quotecurrency) multilateral exchange rate; μ i is the mean of the log returns of the ith (base-currency) multilateral exchange rate; and μ j is the mean of the log returns of the jth (quote-currency) multilateral exchange rate.

Variance
The variance is the second central sample moment of a data sample and measures the dispersion of the data sample relative to the mean.The variance of the movements in a bilateral exchange rate is given by: where i, j = 1, …, N; σ 2 i/j is the variance of the log returns of the ith/ jth bilateral exchange rate; Δp i/j (t) is the log return of the ith/ jth bilateral exchange rate; and μ i/j is the mean of the log returns of the ith/ jth bilateral exchange rate.The standard deviation is the square root of the variance: where σ i/j is the standard deviation of the log returns of the ith/ jth bilateral exchange rate; and σ 2 i/j is the variance of the log returns of the ith/jth bilateral exchange rate.
The variance of the movements in a multilateral exchange rate is given by: where i, j = 1, …, N; σ 2 i is the variance of the log returns of the ith multilateral exchange rate; Δp i (t) is the log return of the ith multilateral exchange rate; and μ i is the mean of the log returns of the ith multilateral exchange rate.The standard deviation is the square root of the variance: where σ i is the standard deviation of the log returns of the ith multilateral exchange rate; and σ 2 i is the variance of the log returns of the ith multilateral exchange rate.
The variance of the movements in a bilateral exchange rate can be decomposed by substituting Eq. ( 6) for Δp i/j (t) and Eq. ( 9) for μ i/j into Eq.( 10) to give: where i, j = 1, …, N; σ 2 i/j is the variance of the log returns of the ith/jth bilateral exchange rate; σ 2 i is the variance of the log returns of ith multilateral exchange rate; σ 2 j is the variance of the log returns of jth multilateral exchange rate; σ i,j = 1 is the covariance between the log returns of ith multilateral exchange rate and the log returns of the jth multilateral exchange rate; and ρ i,j = σ i,j /(σ i σ j ) is the correlation between the log returns of ith multilateral exchange rate and the log returns of the jth multilateral exchange rate.

Skewness
Skewness is the third central normalised sample moment of a data sample and measures the symmetry of the data sample relative to the mean and normalised by the standard derivation to the power of three.The skewness in the movements of a bilateral exchange rate is given by: where i, j = 1, …, N; s i/j is the skewness in the log returns of the ith/jth bilateral exchange rate; Δp i/j (t) is the log return of the ith/jth bilateral exchange rate; μ i/j is the mean of the log returns of the ith/jth bilateral exchange rate; and σ i/j is the standard deviation of the log returns of the ith/jth bilateral exchange rate.The skewness in the movements of the multilateral exchange rate is given by: where i, j = 1, …, N; s i is the skewness in the log returns of the ith multilateral exchange rate; Δp i (t) is the log return of the ith multilateral exchange rate; μ i is the mean of the log returns of the ith multilateral exchange rate; and σ i is the standard deviation of the log returns of the ith multilateral exchange rate.
The co-skewness in the movements of three multilateral exchange rates measures the symmetry of the multilateral exchange rates relative to their average.The co-skewness in the movements of three multilateral exchange rates is given by: where i,j = 1,…,N; cs i,j,k is the co-skewness in the log returns of the ith, jth, and kth multilateral exchange rates; Δp i (t), Δp j (t), and Δp k (t) are the log returns of the ith, jth, and kth multilateral exchange rate, respectively; μ i , μ j , and μ k are the means of the log returns of the ith, jth, and kth multilateral exchange rates, respectively; and σ i , σ j , and σ k are the standard deviations of the log returns of the ith, jth, and kth multilateral exchange rates, respectively.Note that when all the multilateral exchange rates in Eq. ( 17) are the same, the co-skewness is equivalent to the skewness, with cs i,i,i = s i .
The skewness of the movements in a bilateral exchange rate can be decomposed by substituting Eq. ( 6) for Δp i/j (t) and Eq. ( 9) for μ i/j into Eq.( 15) to give: where i, j = 1, …, N; s i/j is the skewness of the ith/jth bilateral exchange rate; σ i/j is the standard deviation of the ith/jth bilateral exchange rate; s i is the skewness of the ith multilateral exchange rate; σ i is the standard deviation of the ith multilateral exchange rate; s j is the skewness of the jth multilateral exchange rate; σ j is the standard deviation of the jth multilateral exchange rate; cs i,i,j is the co-skewness of the ith, ith, and jth multilateral exchange rates; cs i,j,j is the co-skewness of the ith, jth, and jth multilateral exchange rates; w i = σ 3 i /σ 3 i/j is the weight associated with s i ; w i,i,j = 3σ 2 i σ j /σ 3 i/j is the weight associated with cs i,i,j ; w i,j,j = 3σ i σ 2 j / σ 3 i/j is the weight associated with cs i,j,j ; and w j = σ 3 j /σ 3 i/j is the weight associated with s j .

Kurtosis
Kurtosis is the fourth central normalised sample moment of a data sample and measures the extremity of the data sample relative to the mean and normalised by the standard derivation to the power of three.The kurtosis in the movements of a bilateral exchange rate is given by: where i, j = 1, …, N; k i/j is the kurtosis in the log returns of the ith/ jth bilateral exchange rate; Δp i/j (t) is the log return of the ith/ jth bilateral exchange rate; μ i/j is the mean of the log returns of the ith/ jth bilateral exchange rate; and σ i/j is the standard deviation of the log returns of the ith/jth bilateral exchange rate.The kurtosis in the movements of the multilateral exchange rate is given by: where i, j = 1, …, N; k i is the kurtosis in the log returns of the ith multilateral exchange rate; Δp i (t) is the log return of the ith multilateral exchange rate; μ i is the mean of the log returns of the ith multilateral exchange rate; and σ i is the standard deviation of the log returns of the ith multilateral exchange rate.
The co-kurtosis in the movements of three multilateral exchange rates measures the extremity of the multilateral exchange rates relative to their average.The co-kurtosis in the movements of four multilateral exchange rates is given by: where i,j = 1,…,N; ck i,j,k,l is the co-kurtosis in the log returns of the ith, jth, kth, and lth multilateral exchange rates; Δp i (t), Δp j (t), Δp k (t), and Δp l (t) are the log returns of the ith, jth, kth, and lth multilateral exchange rate, respectively; μ i , μ j , μ k , and μ l are the means of the log returns of the ith, jth, kth, and lth multilateral exchange rates, respectively; and σ i , σ j , σ k , and σ l are the standard deviations of the log returns of the ith, jth, kth, and lth multilateral exchange rates, respectively.Note that when all the multilateral exchange rates in Eq. ( 21) are the same, the co-kurtosis is equivalent to the kurtosis, with ck i,i,i,i = k i .The kurtosis of the movements in a bilateral exchange rate can be decomposed by substituting Eq. ( 6) for Δp i/j (t) and Eq. ( 9) for μ i/j into Eq.( 19) to give: where i, j = 1, …, N; k i/j is the kurtosis of the ith/jth bilateral exchange rate; σ i/j is the standard deviation of the ith/jth bilateral exchange rate; k i is the kurtosis of the ith multilateral exchange rate; σ i is the standard deviation of the ith multilateral exchange rate; k j is the kurtosis of the jth multilateral exchange rate; σ j is the standard deviation of the jth multilateral exchange rate; ck i,i,i,j is the co-kurtosis of the ith, ith, ith, and jth multilateral exchange rates; ck i,i,j,j is the co-kurtosis of the ith, ith, jth, and jth multilateral exchange rates; ck i,j,j,j is the co-kurtosis of the ith, jth, jth, and jth multilateral exchange rates; w i = σ 4 i /σ 4 i/j is the weight associated with k i ; w i,i,i,j = 4σ 3 i σ j /σ 4 i/j is the weight associated with ck i,i,i,j ; w i,i,j,j = 6σ 2 i σ 2 j /σ 4 i/j is the weight associated with ck i,i,j,j ; w i,j,j,j = 4σ i σ 3 is the weight associated with ck i,j,j,j ; and w j = σ 4 j /σ 4 i/j is the weight associated with k j .

Summary
Currency attribution can be performed on the moments of the movements of a bilateral exchange rate, such as the mean, variance, skewness, and kurtosis.The mean can be decomposed into the difference between the mean of the movements of the base-currency multilateral exchange rate and the mean of the movements of the quote-currency multilateral exchange rate.
In addition, the variance can be decomposed into the variance of the movements of the base-currency multilateral exchange rate plus the variance of the movements of quote-currency multilateral exchange rate minus twice the covariance between the movements of the base-currency multilateral exchange rate and the variance of the movements of quotecurrency multilateral exchange rate.
Furthermore, the skewness can be decomposed into a weighted difference between the skewness of the movements of the base-currency multilateral exchange rate and the skewness of the movements of quotecurrency multilateral exchange rate minus a weighted difference between the co-skewness of the movements of the base-currency multilateral exchange rate and the co-skewness of the movements of quotecurrency multilateral exchange rate.
Finally, the kurtosis can be decomposed into a weighted sum between the kurtosis of the movements of the base-currency multilateral exchange rate and the kurtosis of the movements of quote-currency multilateral exchange rate, together with a weighted combination of three co-kurtosis terms between the movements of the base-currency multilateral exchange rate and the movements of quote-currency multilateral exchange rate.
The decomposition of the moments allow international investors a finer level of granularity to uncover the contributions of both the base currency and the quote currency to both the moments of the movements of a bilateral exchange rate.

Data sample
The data sample consists of monthly data from Bloomberg for a group of nine bilateral exchange rates against the US dollar, beginning on the January 1, 2000 and ending on the December 31, 2021.This results in a group of ten (N = 10) currencies, namely, the US dollar (USD), the Eurozone euro (EUR), the Japanese yen (JPY), the Australian dollar (AUD), the New Zealand dollar (NZD), the Swiss franc (CHF), the British pound (GBP), the Canadian dollar (CAD), the Norwegian krone (NOK), and the Swedish krona (SEK).

Multilateral exchange rates
Table 1 reports observed mean, standard deviation, skewness and kurtosis for the log returns of each multilateral exchange rate.The largest annualised mean is 2.19% for the Swiss franc (CHF), and the smallest annualised mean is − 1.14% for the British pound (GBP).The highest annualised standard deviation is 9.40% for the Japanese yen (JPY), and the lowest annualised standard deviation is 4.54% for the Eurozone euro (EUR).Both the Japanese yen and the Swiss franc have highly significant positive observed skewness.In contrast, movements in the Australian dollar (AUD), the New Zealand dollar (NZD), the British pound (GBP), the Canadian dollar (CAD), and the Norwegian krone (NOK) have significant negative observed skewness.All currencies have highly significant positive observed kurtosis.

US dollar bilateral exchange rates
US dollar bilateral exchange rates are chosen to provide consistent examples of currency attribution for one set of bilateral exchange rates.However, it is straightforward to use bilateral exchange rates that are priced in terms of another quote (or numéraire) currency.Table 2 reports mean, standard deviation, skewness and kurtosis for the log returns of each US dollar bilateral exchange rate.The largest annualised mean is 2.53% for the Swiss franc (CHF), and the smallest annualised mean is − 0.80% for the British pound (GBP).The highest annualised standard deviation is 12.98% for the New Zealand dollar (NZD), and the lowest annualised standard deviation is 8.64% for the British pound (GBP).
There are no bilateral exchange rates with positive observed skewness.In contrast, movements in the Australian dollar (AUD), the New Zealand dollar (NZD), the British pound (GBP), the Canadian dollar (CAD), and the Norwegian krone (NOK) all have significant negative observed skewness.All US dollar bilateral exchange rates have highly significant positive observed kurtosis, with the Canadian dollar (CAD) having the largest positive observed kurtosis of 8.07.Thus, there are contrasting results for the US dollar bilateral exchange rates in Table 2 compared to the multilateral exchange rates in Table 1.These results will be compared further in the sections below.

Mean
The mean is the first sample moment of a data sample and measures the central location of the data sample.Table 3 reports both the observed means for the log returns of the US dollar bilateral exchange rates and the observed means for the log returns of the multilateral exchange rates.Fig. 1 displays the observed means for both the US dollar bilateral exchange rates and the multilateral exchange rates.The observed means in the movements of the US dollar bilateral exchange rates are very intuitive when using the observed means in the movements of the multilateral exchange rates.The observed means in the movements of the US dollar bilateral exchange rates are simply the difference between the observed mean of the base-currency multilateral exchange rate minus the observed mean of the US dollar multilateral exchange rate.For example, the mean of the Australian dollar (AUD)/US dollar (USD) bilateral exchange rate is 0.48%.The decomposition of the mean in Eq. ( 9) can be used to understand this result by:   Notes: Table 3 reports the decomposition of observed mean for the log returns of the US dollar bilateral exchange rates, together with the observed mean for the log returns of the multilateral exchange rates and the observed mean (repeated) for the log returns of the US dollar multilateral exchange rate.
where μ AUD/USD is the mean of the Australian dollar (AUD)/US dollar (USD) bilateral exchange; μ AUD is the mean of the Australian dollar (AUD) multilateral exchange rate; and μ USD is the mean of the US dollar (USD) multilateral exchange rate.
For US investors, buying a base (foreign) currency involves selling the US dollar.Historically, the negative annualised mean of − 0.33% for the US dollar multilateral exchange rate when subtracted from the observed mean of the base-currency multilateral exchange rate shifts the observed mean of the US dollar bilateral exchange rates up by an annualised 0.33%.

Variance
The variance is the second central sample moment of a data sample and measures the dispersion of the data sample relative to the mean.Table 4 reports both the observed standard deviations for the log returns of the US dollar bilateral exchange rates, the observed standard deviations for the log returns of the multilateral exchange rates, and the observed correlation between the log returns of each multilateral exchange rate and the log returns of the US dollar multilateral exchange rates.Fig. 2 displays the observed standard deviations for both the US dollar bilateral exchange rates and the multilateral exchange rates.The observed standard deviations in the movements of the bilateral exchange rates are less intuitive than the observed mean in the previous section.For example, the observed standard deviation for the movements of the Japanese yen/US dollar (JPY/USD) bilateral exchange rate is less than the observed standard deviation for the movements of the Japanese yen (JPY) multilateral exchange rate.The decomposition of the variance in Eq. ( 14) can be used to understand this result by: where σ JPY/USD is the standard deviation of the Japanese yen/US dollar (JPY/USD) bilateral exchange; σ JPY is the standard deviation of the Japanese yen (JPY) multilateral exchange rate; σ USD is the standard deviation of the US dollar (USD) multilateral exchange rate; and ρ JPY,USD is the correlation between the Japanese yen (JPY) multilateral exchange rate and the US dollar (USD) multilateral exchange rate.The positive correlation of 0.456 when subtracted from the sum of the two variance terms shifts the observed standard deviation of the Japanese yen/US dollar (JPY/USD) bilateral exchange downwards.The opposite happens for the multilateral exchange rates that are negatively correlated with the US dollar (USD) multilateral exchange rate.For example, the observed correlation between the movements of the Australian dollar (AUD) multilateral exchange rate and the movements of the US dollar (USD) multilateral exchange rate is − 0.498.The decomposition of the variance in Eq. ( 14) can be used to decompose the standard deviation of the movements of the Australian dollar/US dollar (AUD/USD) bilateral exchange rate to give: where σ AUD/USD is the standard deviation of the AUD/USD bilateral exchange; σ AUD is the standard deviation of the Australian dollar (AUD) multilateral exchange rate; σ USD is the standard deviation of the US dollar (USD) multilateral exchange rate; and ρ AUD,USD is the correlation between the Australian dollar (AUD) multilateral exchange rate and the US dollar (USD) multilateral exchange rate.In summary, when two currencies are positively correlated the observed standard deviation of the associated bilateral exchange rate is shifted downwards.In contrast, when two currencies are negative correlated the observed standard deviation of the associated bilateral exchange rate is shifted upwards.

Correlation
The correlation between the log returns of base-currency multilateral exchange rate and the log returns of the quote-currency multilateral exchange rate plays an integral role in the magnitude of the variance of the associated bilateral exchange rate.Table 5 reports the correlation matrix for the log returns of the multilateral exchange rates.Fig. 3   Notes: Table 4 reports the observed standard deviations for the log returns of the US dollar bilateral exchange rates, the observed standard deviations for the log returns of the multilateral exchange rates, the observed standard deviation (repeated) for the log returns of the US dollar multilateral exchange rate, and the observed correlation between the log returns of each multilateral exchange rate and the log returns of the US dollar multilateral exchange rates.

displays the
off-diagonal correlation coefficients of the observed correlation matrix between the movements of each multilateral exchange rate and the movements of the other multilateral exchange rates.
The average of the off-diagonal observed correlations is − 0.105.In addition, the average observed correlations between the movements in each multilateral exchange rate and the movements in the other multilateral exchange rates are all negative.Thus, the interesting observed correlations are positive, which are driven by integrated economies from being geographically close, exporting commodities, and risk-off/risk-on currencies.For example, some geographically close currencies have positive observed correlation, such as 0.525 between the Australian dollar (AUD) and the New Zealand dollar (AUD), 0.395 between the Eurozone euro (EUR) and the Swiss franc (CHF), 0.375 between the Eurozone euro (EUR) and the Swedish krona (SEK), 0.356 between the Norwegian krone (NOK) and the Swedish krona (SEK), 0.161 between the US dollar (USD) and the Canadian dollar (CAD).In contrast, the significant positive observed correlation of 0.170 between the Canadian dollar (CAD) and the Australian dollar (AUD) is driven by Canada and  Notes: Table 5 reports the observed correlation for the movements of the multilateral exchange rates.Significance levels for correlations are denoted by * for 5%, and ** for 1%.Australia exporting commodities (see Chen & Rogoff, 2003).Finally, the significant positive observed correlation of 0.456 between the US dollar (USD) and the Japanese yen (JPY) is driven by both currencies being risk-off, or flight-to-safety, currencies.

Skewness
Skewness is the third central normalised sample moment of a data sample and measures the symmetry of the data sample relative to the mean and normalised by the standard derivation to the power of three.Table 6 reports the observed skewness for the log returns of the US dollar bilateral exchange rates, together with all the skewness and co-skewness terms that are used in the decomposition in Eq. ( 18).Fig. 4 displays the observed skewness terms for both the US dollar bilateral exchange rates and the multilateral exchange rates.The observed skewness of the movements of the bilateral exchange rates are much less intuitive.For example, the observed skewness of the movements of the Japanese yen/ US dollar (JPY/USD) bilateral exchange rate is − 0.071 compared to the observed skewness of 1.210 for the movements of the Japanese yen (JPY) multilateral exchange rate and the observed skewness of 0.251 for the movements of the US dollar (USD) multilateral exchange rate.The decomposition of the skewness in Eq. ( 18) can be used to understand this result by: where w JPY = 1.162; w JPY,JPY,USD = 2.756; w JPY,USD,USD = 2.179; and w USD = 0.574.The large co-skewness of the US dollar with the Japanese yen of c JPY,JPY,USD = 0.934, together with the associated large weight of w JPY,JPY,USD = 2.756 pulls the values of the skewness in the movements of the JPY/USD downwards.Thus, both the co-skewness and weight terms play important roles in the magnitude and sign of the skewness of the movements in bilateral exchange rates.

Co-skewness
The co-skewness between the log returns of base-currency multilateral exchange rate and the log returns of the quote-currency multilateral exchange rate plays an integral role in the magnitude of the skewness of the associated bilateral exchange rate.Table 7 reports the observed coskewness for the multilateral exchange rates.Fig. 5 displays the observed co-skewness between the movements of each multilateral exchange rate and the movements of the other multilateral exchange rates.The Japanese yen and the Swiss franc have positive observed coskewness against all other currencies, where the co-skewness terms are c JPY,j,j for the Japanese yen and c CHF,j,j for the Swiss franc.In contrast, only the Canadian dollar has negative observed co-skewness against all other currencies, where co-skewness terms are c CAD,j,j .

Kurtosis
Kurtosis is the fourth central normalised sample moment of a data sample and measures the extremity of the data sample relative to the mean and normalised by the standard derivation to the power of three.Table 8 reports both the observed kurtosis for the log returns of the US dollar bilateral exchange rates and the observed kurtosis for the log returns of the multilateral exchange rates.Fig. 6 displays the observed kurtosis in the movements for all ten multilateral exchange rates.The observed kurtosis in the movements of the bilateral exchange rates are less intuitive.For example, the observed kurtosis in the movements of the Canadian dollar/US dollar (CAD/USD) bilateral exchange rate 8.07 compared to the observed kurtosis of 3.78 for the movements of the Canadian dollar (CAD) multilateral exchange rate and the observed kurtosis of 4.38 for the movements of the US dollar (USD) multilateral exchange rate.The decomposition of the kurtosis in Eq. ( 22) can be used to understand this result by: where w CAD = 0.22; w CAD,CAD,CAD,USD = 1.10; w CAD,CAD,USD,USD = 2.03; w CAD,USD,USD,USD = 1.66; and w USD = 0.51.The large co-kurtosis of the Canadian dollar (CAD) with the US dollar (USD) of ck CAD,CAD,CAD,USD = 1.88, together with the associated large weight of w CAD,CAD,USD,USD = 2.03 pushes the values of the kurtosis in the movements of the CAD/USD bilateral exchange rate upwards.Thus, both the co-kurtosis and weight terms play important roles in the magnitude of the kurtosis of the movements in bilateral exchange rates.

Co-kurtosis
The co-kurtosis between the log returns of base-currency multilateral exchange rate and the log returns of the quote-currency multilateral exchange rate plays an integral role in the magnitude of the kurtosis of the associated bilateral exchange rate.For completeness, Table 9 reports the observed co-kurtosis (ck i,j,j,j ) matrix for the multilateral exchange rates and Table 10 reports the observed co-kurtosis (ck i,i,j,j ) matrix for the multilateral exchange rates.

Currency hedging
International investors and multinational corporations typically have exposure to exchange rate risk from foreign investments, such as currencies, equities, and bonds.For example, when US dollar investors buy foreign currencies they are simultaneously selling the US dollar, which can be seen from the decomposition in Eq. ( 6) by: where i = 1,…,N; t = 1,…,T; Δp i/USD (t) is the log return of the ith/USD bilateral exchange rate; Δp i (t) is the log return of the base-currency the (ith) multilateral exchange rate; and Δp USD (t) is the log return of the quote-currency (US dollar) multilateral exchange rate.
The standard approach to hedge bilateral exchange rate risk is by trading the associated bilateral exchange rate, which reduces the exposure to the bilateral exchange rate and consequently, the bilateral exchange rate risk.More specifically for US dollar investors, bilateral exchange rate risk is hedged by trading the associated US dollar bilateral Notes: Table 6 reports the decomposition of observed skewness for the log returns of the US dollar bilateral exchange rates, together with the observed skewness for the log returns of the multilateral exchange rates.
(h i/USD = 1) represents a full-hedge strategy, where investors are not exposed to bilateral exchange rate risk.Furthermore, a hedge ratio between zero and one (0 < h i/USD < 1) represents a fractional-hedge strategy, where investors are partially exposed to bilateral exchange rate risk.
The decomposition of bilateral exchange rates in Eq. ( 6) provides an alternative approach to hedge bilateral exchange rate risk by trading the two associated multilateral exchange rates: the base-currency (ith) multilateral exchange rate and the quote-currency (US dollar) multilateral exchange rate (see Kunkler, 2021).This can be seen by substituting the decomposition of US dollar bilateral exchange rates in Eq. ( 27) into Eq.( 28) and allowing for separate currency hedge ratios to give: where i = 1,…,N; t = 1,…,T; Δp i/USD (t) is the log return of the ith/USD bilateral exchange rate; h i is the base-currency hedge ratio for the basecurrency (ith) multilateral exchange rate; Δp i (t) is the log return of the base-currency the (ith) multilateral exchange rate; h USD is the quotecurrency hedge ratio for the quote-currency (US dollar) multilateral exchange rate; and Δp USD (t) is the log return of the quote-currency (US dollar) multilateral exchange rate.It should be noted that Eq. ( 29) is equivalent to Eq. ( 28) when the two currency hedge ratios are equal The alternative approach to hedging bilateral exchange rate risk in Eq. ( 29) provides extra flexibility by having two currency hedge ratios, rather than the standard approach that restricts investors to having a common exchange rate hedge ratio.Thus, investors can hedge either the base currency risk or the quote currency risk, or both.The decision on whether or not to hedge bilateral exchange rate risk could depend on many factors, some of which could be the mean, the standard deviation, skewness, and kurtosis of the underlying movements in the bilateral exchange rate.For example, risk-averse investors prefer positive skew to reduce exposure to extreme left-tail events and lower kurtosis to reduce exposure to extreme events generally (Kim et al., 2014).Thus, currency attribution can provide insight on whether to hedge the base currency Notes: Table 8 reports the decomposition of observed kurtosis for the log returns of the US dollar bilateral exchange rates, together with the observed kurtosis for the log returns of the multilateral exchange rates.Notes: Table 9 reports the observed co-kurtosis for the movements of the multilateral exchange rates, where the ith row and the jth column contains the observed cokurtosis of ck i,j,j,j .
risk or the quote currency risk, or both.
To simplify the analysis, it is assumed that US investors are happy with the exposure to the base (foreign) currency (h i = 0) but want to hedge the exposure to the quote currency (US dollar).In this situation, Eq. ( 29) can be rewritten by setting h i = 0 and by substituting the decomposition in Eq. ( 27) to give: where is the log return of the ith/USD bilateral exchange rate; h USD is the quote-currency (US dollar) hedge ratio; and Δp USD (t) is the log return of the quotecurrency (US dollar) multilateral exchange rate.Table 11 reports the mean, standard deviation, skewness, and kurtosis for both a zero-hedge strategy (h USD = 0) and full-hedge strategy (h USD = 1).A hedge ratio of zero (h USD = 0) represents a no-hedge strategy, where the US investor is fully exposed to bilateral exchange rate risk, where Eq. ( 30) is equal to Δp i/USD (t).In contrast, a hedge ratio of one (h USD = 1) represents a full-hedge strategy, where US investors are fully exposed to base currency (foreign) rate risk and no exposure to the quote currency (US dollar), where Eq. ( 30) is equal to Δp i (t).Fig. 7 displays the mean, standard deviation, skewness, and kurtosis for fractional-hedge strategies, where the range of quote-currency (US dollar) hedge ratios is between zero and one (0 ≤ h USD ≤ 1).
When US investors buy a foreign currency, they are also selling the US dollar.However, the annualised mean for the US dollar multilateral exchange rate is − 0.33% (see Tables 1 and 3).Consequently, the US dollar adds 0.33% to the annualised mean of all US dollar bilateral exchange rates (see Section 3.4).Thus, a larger US dollar hedge ratio corresponds to a smaller annualised mean for a fractional-hedge strategy.
Most of the annualised standard deviations are reduced as the US dollar hedge ratio increases.The exception is the Japanese yen (JPY), which has an annualised standard deviation of 9.40% for the fully hedged (h USD = 1) and 8.92% for the unhedged hedged (h USD = 0).The positive correlation of 0.456 between the comovements in the Japanese yen multilateral exchange rate (JPY) and the US dollar multilateral exchange rate (USD) shifts the observed standard deviation of the Japanese yen/US dollar (JPY/USD) bilateral exchange down (see Section 3.6).Thus, the positive correlation between the Japanese yen and the US dollar diversifies the bilateral exchange rate risk and reduces the need for hedging.
Risk-averse investors prefer positive skew to reduce exposure to extreme left-tail events (Kim et al., 2014).A full-hedge strategy for both the Japanese yen (JPY) and the Swiss franc (CHF) significantly increases the observed skewness from − 0.071 to 1.210 for the Japanese yen and from 0.148 to 1.482 for the Swiss franc.In contrast, a full-hedge strategy for the British pound (GBP) decreases the observed skewness from − 0.349 to − 1.084.
Risk-averse investors also prefer lower kurtosis to reduce exposure to extreme events generally (Kim et al., 2014).A full-hedge strategy for both the Japanese yen (JPY) and the Swiss franc (CHF) results in a higher observed kurtosis from 3.67 to 8.98 for the Japanese yen and from 4.74 to 10.22 for the Swiss franc.In contrast, a full hedge strategy for the Canadian dollar (CAD) results in a lower observed kurtosis from 8.07 to 3.78.
In summary, the decomposition of the bilateral exchange rates provides an alternative approach to hedging bilateral exchange rate risk.Investors change choose to hedge the base currency, the quote currency, or both.Currency attribution can act as a guide in the decision-making about whether or not to hedge bilateral exchange rate risk.More Notes: Table 10 reports the observed co-kurtosis for the movements of the multilateral exchange rates, where the ith row and the jth column contains the observed cokurtosis of ck i,i,j,j .

Table 11
No-hedge and full-hedge strategies for the quote currency (US dollar).Notes: Table 11 reports the observed mean μ i/USD , standard deviation σ i/USD , skewness s i/USD , and kurtosis k i/USD for the no-hedge strategy (h USD = 0) for the quote currency (US dollar) and the observed mean μ i , standard deviation σ i , skewness s i , and kurtosis k i for the full-hedge strategy (h USD = 1) for the quote currency (US dollar).

Limitations
A limitation of this paper is that it focused solely on a multicurrency numéraire that consisted of an equally-weighted basket of currencies, rather than considering other multicurrency numéraires.For example, a stable aggregate currency is an optimal multicurrency numéraire, which is estimated using a minimum-variance portfolio optimisation, subject to the index weights sum to one, and that all the index weights are positive (Hovanov et al., 2004).The International Monetary Fund's (IMF) Special Drawing Right (SDR) is another multicurrency numéraire that is an international reserve asset and consists of a weighted basket of international reserve currencies.Another limitation of this paper is that only one universe of currencies was considered in the Results section.The movements in a multilateral exchange rate are calculated based on the universe of currencies.Thus, any modification of the universe of currencies will impact the decompositions.In other work associated with the Frankel-Wei regression framework, it was recommended that large a welldiversified universe of currencies be chosen for the multicurrency numéraire (Kunkler, 2022a).

Conclusion
This paper presented a quantitative methodology to perform currency attribution, which decomposes the mean, variance, skewness, and kurtosis of the movements in a bilateral exchange rate into the base currency and the quote currency of the bilateral exchange rate.It is shown that the mean can be decomposed into the difference between two terms.In addition, the variance can be decomposed into a weighted sum of three terms.Furthermore, the skewness can be decomposed into a weighted sum of four terms.Finally, the kurtosis can be decomposed into a weighted sum of five terms.

Declaration of competing interest
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Fig. 1 .
Fig. 1.Mean of both the US dollar bilateral, and the multilateral, exchange rates.Notes: Fig. 1 displays the observed mean in the movements of both the US dollar bilateral exchange rates and the multilateral exchange rates.

Fig. 2 .
Fig. 2. Std.deviation of both the US dollar bilateral, and the multilateral, exchange rates.Notes: Fig. 2 displays the observed standard deviation in the movements of both the US dollar bilateral exchange rates and the multilateral exchange rates.

Fig. 3 .
Fig. 3. Correlation coefficients of the multilateral exchange rates.Notes: Fig. 3 displays the off-diagonal correlation coefficients of the observed correlation matrix between the movements of the multilateral exchange rates.

Fig. 6 .
Fig. 6.Kurtosis of both the US dollar bilateral, and the multilateral, exchange rates.Notes: Fig. 6 displays the observed kurtosis in the movements of both the US dollar bilateral exchange rates and the multilateral exchange rates.
hedging techniques are possible but are beyond the scope of this paper.The interested reader is referred to Campbell et al. (2010), Kim et al. (2014), and Boudoukh et al. (2019).

Table 1
Multilateral exchange rates.
Notes: Table 1 reports both the observed mean, observed standard deviation, observed skewness and observed kurtosis for the multilateral exchange rates.Significance levels for skewness and kurtosis are denoted by * for 5%, and ** for 1%.

Table 2
US dollar bilateral exchange rates.Notes:Table 2 reports both the observed mean, observed standard deviation, observed skewness and observed kurtosis for the US dollar bilateral exchange rates.Significance levels for skewness and kurtosis are denoted by * for 5%, and ** for 1%.

Table 3
Decomposition of the mean.

Table 4
Decomposition of the standard deviations.

Table 5
Correlation matrix for the movements of the multilateral exchange rates.

Table 6
Decomposition of the skewness.

Table 8
Decomposition of the kurtosis.

Table 9
Co-kurtosis (ck i,j,j,j) matrix for the movements of the multilateral exchange rates.

Table 10
Co-kurtosis (ck i,i,j,j) matrix for the movements of the multilateral exchange rates.