Electronic copy available at: http://ssrn.com/abstract=1607722

1

Diversification, Volatilities and the Supercurrency:

The FX Markets Since Bretton Woods

Maxim Golts and Gregory C. Jones1

Diversification is arguably the oldest and the best way to manage risk. We review

the diversification measures appearing in the literature and suggest two new

statistics playing natural roles in the Grinold-Kahn Fundamental Law of Active

Management framework: the effective number of assets and the effective

dimension of the asset space. Further, we apply our diversification measures to

the FX markets and introduce a statistical supercurrency as an investigative tool

designed to help us estimate the effective number of currencies in the world. We

find that there are about 5 to 9 effective GDP-weighted currencies out of about

180 official currencies in the world serving 235 countries and subordinate states,

since many of these currencies are pegged, managed, or highly correlated. The

supercurrency concept has many additional applications including the estimation

of intrinsic volatility, covariance matrix estimation, and FX portfolio

management.

THIS VERSION: May 14, 2010

1 Golts: maxgolts@gmail.com, Jones: Gregory.Jones@gmo.com. We are grateful to Marc Groz, Yong Liu, Mark

Mueller, and Jason Ribando for useful discussions, and to J.P. Morgan for the FX rates data.

Electronic copy available at: http://ssrn.com/abstract=1607722

2

1 Introduction

Diversification is arguably the oldest and the most reasonable way to manage risk. The

degree of concentration in any market is a very important diagnostic; recent research suggests

that more concentrated markets and industries are more vulnerable to adverse shocks, see Billio,

Getmansky, Lo, and Pelizzon (2010) and Kritzman, Li, Page, and Rigobon (2010).

In this paper we review the diversification measures appearing in the literature and

suggest two new statistical measures: the effective number of assets and the effective dimension

of the asset space. These measures are based on the historical or implied correlations and

volatilities of asset returns, and possibly certain importance weights. Relating the effective

number of assets to the Grinold-Kahn framework, we argue that in markets with increasing

concentration it is increasingly hard for active managers to deliver a high information ratio

performance.

We focus on the FX market, which represents a particularly interesting case since the

diversification in this market is affected by fiat decisions of sovereign monetary authorities.

Historically, the world has gone through a few distinct monetary regimes.

Gold and other precious metals, in one form or another, performed the role of money for

millennia and until very recently. In particular, under the 1944 Bretton Woods agreements, most

currencies were pegged to the US dollar, which in turn was pegged to gold. In effect, the world

had one major currency, a very concentrated situation. In the early 1970s the remnants of the

gold standard were abandoned, and the exchange rates were allowed to float freely, increasing

diversification. However, in the late 1990s the Euro experiment has decreased the effective

number of currencies again, at least for a while. Overall, out of about 180 official currencies

serving 235 countries and subordinate states, many are pegged or managed, more or less rigidly,

3

to a major currency or currency basket. Other currency pairs are not officially pegged but are

nevertheless highly correlated. So, how many currencies are there in the world, really?

In order to answer this question while viewing all the currencies in a symmetric way, we

introduce a statistical supercurrency designed for diversity maximization. We use the

supercurrency concept to compute the effective number of currencies, estimate intrinsic

volatilities and optimize FX portfolios.

Rogoff (2001) argues that we should have 3 or 4 competing currencies, rather than a

single global currency. We show that this is what we do have effectively with liquid developed

currencies, and while the effective number of currencies has declined with the introduction of the

Euro, it has otherwise stayed remarkably stable between 3 and 5 since the Euro introduction.

Including less liquid emerging currencies, we find that the GDP-weighted effective number of

currencies in the world oscillates between 5 and 9.

2 Diversification Measures

In this section we will introduce and review a number of diversification measures

applicable to a wide variety of assets, markets, and style factors. Our measures are based on

(historical or implied) volatilities and correlations2 of returns, and importance weights, such as

GDP for currencies, capitalization for stocks, etc. The data is combined in the (weighted)

covariance matrix3 from which our diversification measures are extracted.

Condition number. This is the ratio of the largest and the smallest eigenvalues of the

matrix in question. Very high condition numbers, or poorly conditioned covariance matrices, are

2 The challenges of measuring and dynamically managing diversification with conditional correlations have recently

been discussed in Chua, Kritzman, and Page (2009). 3 See Appendix A for the formal definition of weighted covariance matrices.

4

indicative of extremely concentrated markets. Conversely, this measure would be equal to 1 in a

perfectly diversified market.

Effective number of assets. This is what Roy-Vetterli (2007) call the effective rank of

the matrix in question, formally introduced in Appendix A. This notion is based on the Shannon

entropy of the (weighted) covariance matrix. The effective rank could range from 1 in a perfectly

concentrated market to the nominal number of assets in a perfectly diversified market. Meucci

(2009) uses a related measure in a portfolio construction procedure.

Effective dimension. This notion, also formally introduced in Appendix A, arises from

geometry and is perhaps more relevant for active management. The effective dimension ranges

from 0 in a perfectly concentrated market to the nominal number of assets in a perfectly

diversified market. We suggest thinking about it as the effective size of the “free” asset space an

active manager can deal with while reasonably assuming that the constituent assets are

uncorrelated.

Herfindahl-Hirschman Index (HHI)4. This measure of industry concentration is widely

used in antitrust and competition law, where it is applied to the relative market shares of

companies. The Herfindahl-Hirschman Index of the distribution of eigenvalues of the

(weighted) covariance matrix would be a relevant measure of market diversification. This index

ranges from 1 in a perfectly concentrated market to the inverse of the nominal number of assets

in a perfectly diversified market.

Absorption ratio. In its simplest form, this is the ratio of the largest eigenvalue to the

sum of all eigenvalues of the matrix in question5. It ranges from 1 in a perfectly concentrated

market to the inverse of the nominal number of assets in a perfectly diversified market.

4 See for example Hirschman (1964).

5 Kritzman et al. (2010) use the first n eigenvalues to compute the absorption ratio.

5

Absorption ratio was introduced by Kritzman, Li, Page, and Rigobon (2010), and is used by

them to measure systemic risk.

Average absolute correlation. This measure is just the average absolute value of asset

return correlations, it ignores volatilities of returns. It ranges from 1 in a perfectly concentrated

market to 0 in a perfectly diversified market.

The statistical properties and relationships between these measures are formally

introduced and discussed in Appendix A.

Example: 22 correlation matrix

We will illustrate, compare, and contrast the diversification measures mentioned above in

the simplest situation of two equally volatile and important assets with the 22 correlation

matrix

1

1

.

In order to compare our measures, we adjust some of them to be on the same footing as

follows:

Adjusted condition number: N times the inverse condition number6

Adjusted Herfindahl-Hirschman Index (HHI): inverse HHI

Adjusted absorption ratio: inverse absorption ratio

Adjusted average absolute correlation: N times (1 – average absolute correlation).

6 Here 2N .

6

Figure 1: Diversification Measures in 2 Dimensions

Market concentration and the Grinold-Kahn’s framework.

Billio, Getmansky, Lo, and Pelizzon (2010), and Kritzman, Li, Page, and Rigobon (2010)

have used some of the diversification measures we described here to argue that the excessively

concentrated markets are more unstable and prone to crises. We would like to look at this idea

from the active portfolio manager‟s viewpoint.

Recall that in the Grinold and Kahn (1999) Fundamental Law of Active Management

framework the information ratio of the active strategy is given by the formula

BRICIR ,

where IC is the active manager‟s information coefficient and BR is the strategy breadth, defined

as the number of independent forecasts. If a strategy operates in a space with the effective

7

number7 of assets (streams, style factors, etc) equal to n , the active manager cannot make more

than n independent forecasts. Therefore, as the markets become more concentrated, the effective

breadth BR probably declines, and so does the information ratio IR, assuming that the manager‟s

information coefficient IC stays the same. In the extreme situation of a perfectly concentrated

one-dimensional asset space, the active manager could only be long or short the “market,” and

thus could only add value by timing the “market.” It is of course an irony that as managers all

try to diversify into the same “optimal” basket of assets, they correlate their flows and collapse

the effective dimensionality of the asset space.

3 Diversification in the FX markets

How many currencies are there in the world, really? If all the currencies in the world

were pegged to each other, we would effectively have just one global currency (Bretton Woods).

Conversely, if all the currencies were truly independent and equally important, their effective

number would coincide with the nominal number. The actual situation is somewhere in-between.

We will use two diversification measures here:

First, the GDP-weighted effective number (or rank) of currencies. The effective rank

could range from 1 (all currencies are pegged) to the nominal number of currencies. Adding to a

basket a currency pegged to another currency already in the basket does not change the effective

number of the currencies in the basket8.

Second, the effective dimension9 of the (weighted) covariance matrix ranging from 0 (any

7 In fact, Grinold and Kahn (1999) talk about the independent forecasts of exceptional returns, so perhaps our

effective dimension measure is more appropriate in this case. 8 There is however a subtlety here. From the point of view of GDP-weighted covariance, if e.g. China pegs to the

U.S. dollar, then the GDP of the USD+Pegs currency zone is increased and the allocation of GDP percentages

among the currencies is altered somewhat. Pegged currencies can only change effective rank by this indirect effect. 9 This metric is related to the rescaled volume of the parallelepiped in the high-dimensional Cholesky or “currency

8

two currencies are pegged) to the nominal number of currencies. Conceptually, it is the effective

size of the “free” asset space an active manager can deal with while reasonably assuming that the

constituent assets are uncorrelated. Any pegs must be taken into account, so the free space

dimension in that case is 0.

Let us illustrate with the following.

Example: Eurozone integration.

There were 12 individual currencies prior to the Euro‟s introduction in 1999.

Table 1: Pre-Eurozone currencies

Currency ISO Code Euro Conversion Rate

Austrian Schilling ATS 13.7603

Belgian Franc BEF 40.3399

Dutch Guilder NLG 2.20371

Finnish Markka FIM 5.94573

French Franc FRF 6.55957

German Mark DEM 1.95583

Irish Pound IEP 0.787564

Italian Lira ITL 1936.27

Luxembourg Franc* LUF 40.3399

Portuguese Escudo PTE 200.482

Spanish Peseta ESP 166.386

Greek Drachma** GRD 340.75

Notes: * The Luxembourg Franc was pegged 1:1 to the Belgian Franc.

** The Greek Drachma was replaced by the Euro on June 19, 2000, after the Euro’s introduction, but before the

introduction of the physical currency. Due to limited data we do not include it in these pre-Eurozone charts.

Ignoring LUF and GRD, we have 10 currencies. To warm up, we look at the history of

these now-defunct currencies using a USD numeraire. From this point of view, they all look

quite correlated, and the effective rank is much less than 10. We construct a realized covariance

matrix using a trailing 1-year half-life exponential weighted method10

. Except where noted, all

covariance matrices in this paper are computed from weekly returns. On each day, the

triangle volatility” space. 10

The exponential window is kept long enough that there are no “rear-window” effects.

9

covariance matrix is scaled, row and column, by the GDP‟s of the Eurozone countries (see

Appendix A1). GDP numbers are smoothed so there are no year-end jumps. We compute the

covariance eigenvalues and effective rank of this scaled covariance matrix, thus calculating the

effective number of constituents in a given collection of currencies as we move throughout

history. Moreover, we develop a methodology11

to compute how much a particular currency

contributes to the effective number.

The Eurozone countries, viewed as USD crosses, then have an effective rank history as

shown in Figure 2.

Figure 2: Pre-Euro Currencies and Eurozone Convergence, from the USD Numeraire Viewpoint

1975 1980 1985 1990 1995 2000 2005 20100

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Eff

ective N

um

ber

of

Curr

encie

s

DEMEUR

ATS

BEF

ESP

FIM

FRF

IEP

ITL

NLG

PTE

Notes: 10 pre-Euro currencies; 1-year half-life exponentially weighted covariance matrix, USD numeraire. The

period 1971-1975 is backfilled with the covariance matrix estimated in early 1975. Euro shares the Deutschemark

color, merely for convenience. See Appendix A for detailed methodology description.

Looking at Figure 2, we see that, prior to the Euro and since 1980, there were “at most” 2

11

We project the exchange rates onto the principal components to do a decomposition of the effective rank into

contributions from each FX. The contributions from the various FX sum up to the total. This methodology is

described in Appendix A2.

10

currencies from the point of view of a USD numeraire investor. One might be tempted to

conclude that the individual currencies were unnecessary, and that (should the Euro prove

insufficient) we might need only 2 currencies, rather than half a dozen. The “obvious” split

between northern Europe and southern Europe (especially Greece, which we have not included

in this long history) confirms this typical judgment of “two.” Figure 2, however, understates the

subtleties of the intra-Eurozone currencies during this time period; a USD investor sees them all

as too correlated, and the intra-European volatility as insignificant compared to the transatlantic

market. The USD numeraire effect dominates.

Now, we would like to analyze the situation from a perspective that is less USD-centric.

We could simply look at DEM crosses (hence 9 FX rates) but that would obscure any volatility

inherent in the DEM itself. Hence we introduce the supercurrency formalism, which will add a

hypothetical “outside” currency tailored to analyze the given set of currencies. The

supercurrency will show us how many effective currencies there really can be.

4 The Supercurrency

An exchange rate is, by definition, the rate between two currencies. To speak about a

covariance matrix of FX rates, we must fix a numeraire currency. Choosing a numeraire breaks

the symmetry inherent in FX markets and leads to a biased and sometimes cloudy picture of their

dynamics. We might desire not to pick an endogenous currency as numeraire (like DEM for

Eurozone convergence), because we lose the facility with making statements about that currency.

We might also not want to pick a particular exogenous currency (like USD for Eurozone

convergence) because of the dominating numeraire effect. When considering the whole world,

11

USD will clearly bias the picture, and yet no clear exogenous numeraire exists12

.

In order to deal with this issue, we introduce a concept of a supercurrency as a fictitious

(or constructed) currency, which is a basket of existing currencies, plus extra orthogonal

volatility. This extra volatility can be viewed as purely hypothetical, or as the real volatility

inherent in markets if we value paper currency using real assets as a numeraire (such as hard

commodities, equities, and/or real estate). We will argue the supercurrency is a statistical tool

useful in many applications. We will sometimes refer to the supercurrency by a “code” SUP.

A particular choice we make here is a supercurrency maximizing the effective dimension

viewed from the supercurrency perspective13

. Except where otherwise stated, in the examples in

this paper, we always maximize effective dimension of GDP-scaled covariance matrices.

Having chosen a statistical numeraire (supercurrency), we can treat all actual currencies

on an equal footing, removing the asymmetry inherent of the FX rates.

Now we return to the Eurozone convergence, and then look at other examples.

Example: Eurozone convergence from the intra-European SUP perspective.

We reanalyze the Eurozone convergence from the point of view of a supercurrency which

is defined to maximize effective Dimension for the 10-currency space of pre-Eurozone

currencies. This gives a pure analysis of the degrees of freedom, and avoids the overcorrelation

bias which comes from looking at them as USD crosses.

Using a 1-year half-life does not effectively show the effective rank of the currencies

12

One might consider baskets of currencies, or even IMF Special Drawing Rights. These are related to our concept

of barycenter in Appendices A4 and A7. However, unless one includes a real asset, equities, or some type of

orthogonal volatility, the resulting covariance matrix is singular. 13

A supercurrency could be chosen to maximize any diversification measure discussed in Section 2. However, the

effective dimension maximization problem has a closed form solution; see Appendices A4 and A7. It turns out that

maximizing the effective number gives a very similar answer to maximizing the effective dimension.

12

decreasing in the weeks leading up to EUR fixing. So we increase our resolution to a 3 month

half-life, and zoom in to 1986-2010.

Figure 3: Effective Number of Pre-Eurozone Currencies, from Eurozone Supercurrency Viewpoint

1986 1988 1990 1992 1994 1996 1998 2000 2002 20040

1

2

3

4

5

6

7

Eff

ective N

um

ber

of

Curr

encie

s

DEMEUR

ATS

BEF

ESP

FIM

FRF

IEP

ITL

NLG

PTE

Notes: 10 pre-Euro currencies; 3-month half-life exponentially weighted covariance matrix. Supercurrency defined

by maximizing effective dimension of the GDP-weighted covariance matrix. See Appendix A for detailed

methodology description.

Now we see that there was in fact more dimensionality in the Eurozone currencies prior

to the Euro‟s introduction; there were often 3 to 6. However, their independence was subject to

massive downward spikes when the ERM suffered a crisis. For example the ERM crisis in

October 1992 brought a devaluation of ESP, FIM, and ITL against DEM; this singular move

reduced the effective dimension of the covariance matrix massively from over 6 to just 3.

Moving into the late „90s, we see an emergence of independence of volatility in this

currency set. For a few months prior to the Euro‟s introduction, the intrinsic volatilities

13

(especially in Lira and Punt) did become reined in14

. We observe a massive collapse in the

dimensionality of the European currency space in 1999.

Example: Developed currencies since end of Bretton Woods

Taking the G8 currencies and adding the 10 pre-Eurozone currencies of interest, we

calculate the effective rank since the end of Bretton Woods era. We have nearly 40 years of

currency history, August 15, 1971, until the present (data runs to April 15, 2010).

Figure 4: Since Bretton Woods, GDP-weighted. Effective Number of Currencies, G8 and pre-Euro FX

1975 1980 1985 1990 1995 2000 2005 20100

1

2

3

4

5

6

7

8

9

10

Eff

ective N

um

ber

of

Curr

encie

s

USD

JPY

AUD

CAD

GBP

CHF

SEK

DEMEUR

ATS

BEF

ESP

FIM

FRF

IEP

ITL

NLG

PTE

Notes: 10 pre-Euro currencies and G8 liquid currencies; 1-year half-life exponentially weighted covariance matrix.

Supercurrency defined by maximizing effective dimension of GDP-weighted covariance. Currencies are color-coded

as bluish (presently funding currencies), grayish (Anglo and “commodity” currencies), copper (independent

continental currencies), and a green-to-red spectrum for the Eurozone. New Year’s Day is marked in 5-year

intervals. See Appendix A for detailed methodology description.

We see many currency events and trends in this depiction, including

14

Even though the Austrian Schilling never deviated too far from the Mark, it did undergo day-to-day (mean

reverting) volatility prior to 1/1/1999.

14

Devaluation of Escudo (March 1977) then Peseta (July 1977)

Devaluations of BEF, ESP, IEP, FIM, FRF, PTE, SEK against DEM, 1982-1983

Floating of the Australian Dollar15

, 1983

Breakdown of ERM 1, 1992, blow-ups in GBP (Black Wednesday) and SEK, subsequent

volatility in southern Europe (ITL, ESP)

USD/JPY risk event post-Russia default and LTCM crisis, October 1998

Significant and unrecovered loss in effective number upon EUR introduction in 1999

Secular increase in CAD volatility, 2002-2008

Lehman Brothers default and credit crisis risk episode, September-October 2008

Example: The global FX market, since 1999

We now expand our list of currencies to include the liquid G10 FX, currencies from the

G20 economies including most of the European Union, and some additional currencies ILS,

CLP, PEN, HKD, MYR, PHP, SGD, THB, and TWD from the J.P. Morgan ELMI+ index (see

Appendix D for the list of all currencies). Totaling 33 currencies, we achieve a fairly worldwide

view of currency volatility and covariance. We focus on the Euro epoch, 1999 to the present.

15

Note that AUD data comes on line in the late 1970s, and this is purely a data artifact.

15

Figure 5: Effective Number of Currencies in the world (G20 Economies and Emerging FX)

2000 2002 2004 2006 2008 20100

1

2

3

4

5

6

7

8

9

10

Eff

ective N

um

ber

of

Curr

encie

s

USDJPYAUDNZDCADGBPNOKSEKDKKCHFEURCZKHUFILSPLNRUBTRLZARARSBRLCLPMXNPENCNYHKDIDRINRKRWMYRPHPSGDTHBTWD

Notes: 33 world currencies; 6-month half-life exponentially weighted covariance matrix. Supercurrency defined by

maximizing effective dimension of GDP-weighted covariance. The colors are as follows: blues for funding

currencies USD, JPY, grays for Anglo and developed commodity currencies, coppers for developed European

independents (including NOK as a commodity currency), green for Euro, red-orange spectrum for EEMEA, blue-

purple spectrum for Latam; green-red spectrum for Asia. See Appendix A for detailed methodology description.

Again, we see several events and trends in this picture

World recovery from 1998 Russian default, and pre-1999 vol in BRL, IDR, RUB

Gradual increase in contribution from Emerging FX; growth of China; growth in

importance of Rupee (after 2004) and Baht (2007).

Turkish Lira devaluation, February 2001

South African Rand volatility emergence, late 2001

Argentina debt default and January 2002 Peso devaluation

Gradual increase in volatility independence of developed commodity currencies.

16

Sudden drop of effective number, carry trade blow up, after the “quant meltdown,”

August 15, 2007. Rougly one and a half “effective” currencies disappeared that day.

Lehman Brothers default / credit crisis risk episode, September-October 2008

The Greek default risk shows up at the turn of 2010 as a rocky trajectory in the global

number of currencies.

Figure 6: A Snapshot, April 15, 2010. Contributions to Total Number of Effective Currencies.

USD 8%

JPY 13%

AUD 4%

NZD < 1%

CAD 5%

GBP 8%

NOK 2%

SEK 2%

DKK < 1%

CHF 1%EUR 10% CZK, HUF,

ILS 1%

PLN 3%

RUB 5%

TRL 3%

ZAR 3%

ARG < 1%

BRA 8%

CLP 2%

MXN 5%

PEN < 1%CNY 3%

HKD < 1%IDR 2%

INR 3%

KRW 5%MYR,PHP,SGD

THB,TWD <1%USDJPYAUDNZDCADGBPNOKSEKDKKCHFEURCZKHUFILSPLNRUBTRLZARARSBRLCLPMXNPENCNYHKDIDRINRKRWMYRPHPSGDTHBTWD

Notes: 33 world currencies; 6-month half-life exponentially weighted covariance matrix. Supercurrency defined by

maximizing effective dimension of GDP-weighted covariance. See Appendix A for detailed methodology

description.

Figure 6 shows a snapshot of the contributions to effective number on April 15, 2010.

Interestingly, Japan has a larger contribution than USD or EUR to the total number of currencies.

This is partially due to other currencies being pegged or managed versus the latter two; Japan is a

far away vertex supporting the effective dimensionality and rank of modern currencies.

17

Developed currencies account for 52.8% of the overall dimensionality (22.7% including GBP is

developed Europe and 30% Pacific countries). EEMEA adds 16.8%, Latam 15.8%, and Asia

14.5%.

5 Supercurrency Applications

Intrinsic volatility

If all currencies appreciated by 1% against real assets in the course of a day, all exchange

rates would remain unchanged. Conversely, given daily FX moves, we cannot for certain ascribe

these to moves in the currencies themselves. The translation between currencies themselves and

the exchange rates is of fundamental importance to investors, especially those interested in real

value. There is an ambiguity in applying such a translation to FX covariance, and several efforts

have been made to resolve this ambiguity.

One concept of intrinsic volatility was introduced in Doust (2007). For a currency pair

),( ji let the exchange rate be jiij XXX / , where iX is the unobservable intrinsic value of the

currency i . The volatility of iX , or intrinsic volatility i could be estimated by running (panel)

regressions with all implied volatility tenors to determine i and intrinsic correlations ij for all

currency pairs ),( ji16

:

jiijjiij 2222 .

We propose using the supercurrency for the intrinsic volatility estimation. Indeed, the

16

In practice, Doust (2007) minimizes a slightly different objective function allowing the user to encourage zero

correlations ij between intrinsic currency values.

18

intrinsic volatility is simply the volatility of the currency i with respect to the supercurrency. An

an example we use trailing realized returns in a G8 universe to estimate intrinsic volatilities on

July 1, 2008, and report the results:

Table 2: A snapshot, July 1

st 2008, 1-year half-life exponentially weighted intrinsic and FX volatilities.

AUD CAD CHF EUR GBP JPY SEK USD

Intrinsic Volatility 9.89% 8.78% 6.33% 4.51% 5.83% 8.48% 6.49% 4.81%

AUD/ USD

CAD/ USD

CHF/ USD

EUR/ USD

GBP/ USD

JPY/ USD

SEK/ USD

FX Volatility 11.42% 9.36% 9.75% 8.08% 7.63% 10.18% 9.54%

Notes: G8 currencies; 1-year half-life exponentially weighted covariance matrix. Supercurrency defined by

maximizing effective dimension of GDP-weighted covariance. Reported volatilities do not use any GDP-scalings.

See Appendix A for detailed methodology description.

On this date, for example, EUR/USD is estimated to be more volatile than GBP/USD, but

EUR itself is less volatile than GBP. It is just that EUR‟s volatility is more anticorrelated with

USD. Similarly for CHF and SEK, which are significantly less volatile than CAD.

Chen and Doust (2009) update their method to look at entropy of the realized covariance

distribution, rather than involving a user-defined penalty function for intrinsic correlation. The

entropy of an N-dimensional Gaussian distribution integrates to give

constant) log(det2

1S

The scale and shape of the covariance matrix (or equivalently, its eigenvalues) give

additive contributions to the entropy with log(scale) entering into the “constant” term. If we

scale so that 1Σ tr , then S is the log effective dimension. Chen and Doust (2009) look at a

similar quantity, ))matrixn correlatiolog(det( . They define a supercurrency numeraire that

maximizes effective dimension of the correlation matrix.

19

Intrinsic currency covariance can be used to find correlations between the values of

intrinsic currencies. Without an outside numeraire, evaluating the correlation between USD and

EUR is not possible, and reasonable choices (JPY, GBP) will each give biased answers.

Supercurrency gives us a clear picture of the correlations.

Figure 7: Rolling Correlations Between Intrinsic Currencies.

2000 2002 2004 2006 2008 2010-100%

-50%

0

50%

100%

Corr(AUD,USD)

Corr(CAD,USD)

Corr(EUR,USD)

Corr(GBP,USD)

Corr(JPY,USD)

Notes: G8 Currencies AUD, CAD, CHF, EUR, GBP, JPY, SEK, USD. 1-year half-life exponentially weighted

covariance matrix. Supercurrency defined by maximizing effective dimension of GDP-weighted covariance. See

Appendix A for methodology description.

Qualitative features of the rolling correlations graph share much in common with Chen and

Doust (2009), namely the rise in Corr(JPY,USD) since the credit crisis, the long-term negative

Corr(EUR,USD) due to their being economic substitutes17

, and the gradual loss of

17

For years the Euro has promised to be a reliable store of value for reserves, continuing the traditional role of DEM

and allowing central banks to diversify from USD. Additionally, if EUR and USD are the only large and competing

currency block zones for satellite currencies (see Figure 8), it follows that if the USD-block appreciates, the EUR-

block depreciates. The viability of the Euro, however, has been a subject of debate and recently been called into

20

Corr(CAD,USD). Similar graphs show there has been a disruption in Corr(CHF,EUR), falling

from 85% pre-crisis to a low of 50%, before recovering to 65%.

Supercurrency-based minimal spanning trees

As pointed out by McDonald-Suleman-Williams-Howison-Johnson (2005), Minimal

Spanning Trees (MSTs) provide another way to look at the correlation structure18

. McDonald et

al. look at correlations between all pairs of crosses of FX. Obviously, there is some redundant

information in this setting, including reciprocals and FX triangles, but all possible correlations

are considered, and various economically meaningful structures emerge and exhibit survival

dynamics.

We can construct MSTs of intrinsic values19

rather than exchange rates. Consider N+1

currencies from the geometric perspective. We look at the vertices and the edges of the

covariance simplex (see Appendix A4), as a graph in the N-dimensional space. A spanning tree

is a tree connecting all vertices of this graph, and a Minimal Spanning Tree is the spanning tree

of minimal length. This graph can be visualized by projecting to a 2-dimensional space.

From the supercurrency viewpoint we can also construct MSTs for pairs of exchange

rates )/,/( SUPXSUPX ji by using the standard correlation (unit-sphere) distance function

))/,/corr(1(2 SUPXSUPXd jiij .

MSTs for exchange rates are used to answer the question “which currencies are in play.”

We can also obtain “in play” indicators using any of the diversification measures presented in

serious question. 18

See also Fenn et al. (2009). 19

This answers a different question than the MST of the exchange rates. For example, if NOK/EUR is highly

correlated with AUD/USD or CAD/USD due to an underlying commodity factor, this will not show up on the

intrinsic value MST.

21

Section 2. Consider the currency in question as a numeraire. If the rest of the market is

concentrated on some diversification measure from this viewpoint, then the currency is “in play.”

Block covariance matrices

In this section we show how to divide the currencies into a hierarchy of correlated

groups. Depending on a user‟s views, the covariance matrix could then be modeled separately

within each group and between groups. The block design could provide a more stable and

reasonable matrix in various applications.

Let‟s start with the simplex associated to the covariance matrix as in Appendix A4, and

attach the (equal, GDP or trade) weights to the vertices. Endow the supervertex with the average

weight of the existing currency verticies, times an arbitrary scale SUP , which we will dial up as

necessary. For every real currency vertex i (or “child”), connect it to the parent vertex j

maximizing the function 2/ ijj lw where ijl is the volatility (or edge length) between the currency

i and every other currency j, including the supercurrency. The heuristic idea is that i is a

“massive body” and other bodies j are pulling on it with a “gravity force” proportional to their

“weight” jw . If two bodies select each other as parent, the large one is defined parent, and must

choose the next largest 2/ ijj lw to be its parent (if necessary, the supercurrency itself). Projecting

this construction to a 2-dim space we obtain a directed graph.

If the supercurrency has multiple children, and we remove it, then the graph is broken

into disconnected components. So we scale up SUP (the “weight” of the supercurrency) until it

is strong enough to claim at least two children. The connected components of the resulting graph

are the currency groups. Here is the result for April 15, 2010.

22

Figure 8: Block Covariance Tree Structure, April 15, 2010.

Notes: 33 world currencies. 6-month half-life exponentially weighted covariance matrix. Supercurrency defined by

maximizing effective dimension of GDP-weighted covariance. See Appendix A for methodology description.

It was necessary to dial SUP up to 5.3 to claim both EUR and USD. If we continue to turn up

the scale constant, to SUP =5.5, the Singapore dollar detaches from the USD cluster, and prefers

the supercurrency as its direct parent. This is no surprise since SGD is managed against a basket

that includes both USD and Euro; so it should not fit neatly on our graph as being subordinate to

just one. As we dial up the scale constant even farther, to 5.7SUP , a slew of currencies break

off: CAD, RUB, TRL, INR, and MYR become subordinate to the supercurrency and thus

become stranded islands.

The network can change nontrivially day-to-day. On January 1, 2007, the situation was

different, as CAD, NZD, RUB, and ZAR attached to USD, which was a relatively high yield

currency, and JPY attached to EUR. Dynamics of currency networks are studied by McDonald

et al. (2005), Fenn et al. (2009).

Super Currency

EUR USD

CAD

RUB

TRL

INR

SGD

MYR

AUD

NZD

CHF

GBP

NOK

SEK

DKK

CZK

HUF PLN

ZAR

JPY

CNY

HKD IDR

ISL

TWD

KRW

PHP

THB

TWD

PEN

BRL CLP

MXN

ARS

23

In different environments, or with a different “force” function, the currency markets can

be described by higher-order hierarchies. Our block covariance matrices could also be used for

other asset classes, such as equity industry sectors.

Constructing portfolios from the supercurrency perspective

Golts and Jones (2009) propose a “Sharper Angle Optimization,” which is a robust

version of Markowitz mean-variance optimization. The Sharper Angle method reduces the

leverage of noise alphas by constraining the alpha-weight angle, thereby increasing the

reciprocal condition number and effective dimension.

Mean-variance optimization of an FX portfolio requires the choice of a numeraire

currency, say, USD or EUR. This choice is usually dictated by the accounting unit of the

investment portfolio. With classic MVO, tracking error is defined relative to the numeraire

currency, but an MVO-optimized strategy‟s performance is actually independent of the

numeraire. More specifically, if a USD-based investor has alphas and a covariance matrix with

USD as numeraire, and a EUR-based investor has equivalent alphas (each FX/EUR‟s alpha

shifted by the EUR/USD alpha from the first investor) and an equivalent covariance matrix with

EUR as the numeraire, then the two investors will have the same active weights and the same

percentage profit-and-loss on each and every day.

However, the situation changes, sometimes dramatically, in the presence of constraints,

Bayesian shrinkage, or other non-linearities. In particular, the magnitude of the alpha vector ||

is not an invariant of the change of numeraire, and neither is Sharper Angle objective function.

Intuitively, bringing the weight w in line with in the case where largely reflects a

foreign FX vs. USD bet, will magnify the risk allocated to this bet. The Bayesian reduction of

24

the USD covariance matrix toward its trace (the identity matrix) incorrectly models that the FX

are (less- or) un-correlated, and artificially removes the USD numeraire effect. In the worst case,

risk is allocated up to N times the appropriate amount. On the other hand, the EUR-based

investor will magnify FX vs EUR bets; and the results will differ.

As an alternative, we can optimize from the supercurrency perspective. Since the

supercurrency covariance matrix is by design of an increase (or maximum) effective rank, less

Bayesian reduction (or robust 10 parameter) is needed. The resulting trade is less one-

sided in terms of emphasizing any particular numeraire bet, and gives a more symmetric form of

Bayesian reduction for an FX system. By increasing the breadth of the FX coverage, we could

expect managers to perform at a higher information ratio, regardless of the Sharper Angle ideas.

Other applications

Several other applications of the supercurrency and intrinsic volatility have been

reviewed in Doust (2007) and Chen & Doust (2008,2009). The supercurrency or intrinsic value

framework can be used in a financial engineering setting to help price exotic FX options, or

options on illiquid FX crosses. Enlarging the covariance space helps us ask questions in risk

modeling owing to our portfolio‟s active numeraire FX exposure. Intrinsic volatility and options

also provide a framework for currency-by-currency portfolio Vega exposure (rather than FX-

based Vega exposure). Also, solving for the maximum likelihood intrinsic currency move

consistent with the observed FX move (Chen & Doust 2008, 2009), can give us clear definition

of what “happened” on a given day in the FX markets, and make possible a performance

attribution that consistently includes foreign and numeraire contributions.

25

Conclusion

Diversification is perhaps the best single risk management idea in existence. However,

nominal diversification among various assets and instruments may not produce any significant

benefits if the assets in question have very correlated returns, especially in a crisis. True

diversification is much harder to achieve.

In this paper we have reviewed and compared statistical measures of diversification based

on the correlations and volatilities of returns, and on importance weights. We have introduced

the effective number and effective dimension statistics and related them to the Grinold-Kahn

Fundamental Law of Active Management framework arguing that the active managers have a

much tougher time delivering high information ratios in very concentrated markets.

Further, we have focused on the foreign exchange markets, which have the following

unique feature: Exchange rates are ratios of value between pairs of currencies. Thus, in order to

consider FX returns, we must first fix a numeraire currency, say the U.S. dollar, the British

Pound, or the Euro. An active manager with clients based in different currencies must deal with

this issue. One idea is to manage the portfolio from, say, the US dollar perspective and swap out

the results into the client‟s currency. However, in the presence of non-linearities in the

manager‟s alpha or risk management systems, this could result in different outcomes for different

base currencies. Furthermore, the analysis underlying the FX active strategy management will

take different forms and exhibit biases depending on the choice of numeraire.

In order to address this asymmetry, we have introduced the concept of the statistical

supercurrency, a fictitious currency designed to maximize diversification in the existing currency

space. Viewing a portfolio from the supercurrency perspective should increase a strategy‟s

effective breadth and ultimately improve the active manager‟s information ratio. Managing an

26

FX portfolio from a supercurrency perspective would allow an active manager to serve clients

based in different currencies in a much more symmetric and straightforward manner. The

supercurrency can be used to calculate intrinsic volatility and correlations of currencies, and to

map out intrinsic currency tree networks and define currency blocks, without the complications

from considering all crosses. Coupling a supercurrency with macroeconomic importance

weights for currency zones gives us a quantitative breakdown for the effective number of

currencies in the world. This number has ranged between 5 and 9, far below the number of

nominal currencies, and experiences negative shocks as currencies correlate in a crisis.

Appendix A. The diversification measures

In this appendix we formally discuss ways to quantify diversification arising from

(historical or implied) covariance matrices.

A1. Effective rank, effective dimension, and the condition number

Suppose Σ is a (realized or implied) covariance matrix of asset returns, or in fact any

positive definite matrix of size NN , so it can be presented as

,),,,(diag= 22

2

2

1 QQ N (4)

where Q is an orthogonal matrix and 22

2

2

1 N are the eigenvalues of in the

decreasing order. Consider the probability distribution p of eigenvalues

tr

2

2

2

i

k

k

iip

.

Following Roy-Vetterli (2007) introduce the effective rank of Σ as

27

N

i

ii pp1

logexperank .

The effective dimension of Σ is given by the following formula

NN

N

/tr

)(detedim

/1

.

Effective dimension is an alternative to ( N times) the standard inverse condition number

2

max

2

minΣnicn

N .

The absorption ratio of the first n principal components, introduced by Kritzman et al. (2010),

is defined as

n

i

i

1

2

trar

.

The Herfindahl-Hirschman Index of the probabilty distribution p is defined as

N

i

N

i

iip

1 1

22

2

tr hhi

.

The idea behind HHI and some other measures is that whenever a set of N numbers adds

up to 1, the sum of a convex function of the numbers is minimized when all N of the numbers are

equal. So for a given set of numbers which are not all equal, the quotient to this theoretical

minimum, is a measure of their “dispersion” and can define how many numbers we have

“effectively.” For HHI, the convex function is just 2)( xxf , and the theoretical minimum is

N/1 .

The statistics described above could just as well be applied to correlation matrices, which

also yield naturally another diversification measure: the average absolute correlation.

Now, let ),...,( 22

1 Nuuw be a vector of importance weights. These could be market

28

capitalizations for stocks or GDP values for currencies. We could also consider the weighted

diversification measures by evaluating the effective rank and dimension of the matrix

),...,(diag),...,(diag)( 11 NN uuuuw ,

where ),...,(diag 1 Nuu is the diagonal matrix with ),...,( 1 Nuu on the diagonal.

A2. Individual assets contribution to the effective rank

Let us try to figure out how much individual assets contribute to the effective rank. First,

recombine the sum

i

ii pp log

by asset, rather than eigenvector. That is, for the asset k define

i

ii

k

i

k ppcs log)( 2 ,

where k

ic is the k -th asset correlation with the i-th eigenvector. Now define

kk sN)Σlog(erank

Σerank .

Then k

kNΣerank . We treat kN as the contribution of the asset k to the effective rank.

A3. Diversification measures and power means

Measuring diversification, we introduce a metric on the space of discrete distributions

and evaluate the distance between the distribution in question and the uniform distribution on N

points. Here we demonstrate how the metrics we described can be obtained from comparing two

29

power means. Recall that for real positive numbers Nxx ,,1 and real positive weights

N ,,1 , the weighted power p mean is defined as follows

pN

i

p

iip xD

xM

/1

1

1),(

,

and for 0p we have the limit

Di

n

i

ixxM

10 ),(

,

where ND 1 . It is well known that qp MM for qp .

Taking the vector of eigenvalues ),,,( 22

2

2

1 Ng of a matrix , and setting, we get

),(

),(erank

0

1

ggM

egMN ,

),(

),(edim

1

0

egM

egMN ,

),(

),(nicn

egM

egMN

,

2

1

2

),(

),(1 hhi

egM

egM

N,

and finally

),(

),( ar

1

1

egM

egM n ,

where )0,,0,1,,1( n

ne , and ),1,,1( N

Nee .

A4. The simplex associated to a covariance matrix

Suppose LL' is the Cholesky decomposition of . The matrix L is upper-triangular.

30

Treat the columns of L as vectors in the N-dimensional space. Notice that the dot-product of

these vectors give the elements ij of the matrix Σ

jiij ll ' .

The N vectors il together with the origin 00

l (representing the numeraire for ) are the

1N vertices of the simplex Δ associated to the matrix Σ. The length of the sides are volatilities

of the exchange rates, and the cosines of the angles between adjacent sides are the correlations of

those rates. The volume of the simplex is given by the formula

det vol .

Let us introduce the mean square edge length or MSEL of Δ as the average of the square

lenths of all 2/)1( NN edges.

2

11

2

1 1 1

12)(

)1(

2 msel

N

i

i

N

i

i

N

i Nkj

kjjkkkjjii lN

lNNN

.

From the FX perspective, MSEL is the average variance of all possible FX crosses.

The N+1 vertices of the simplex ( il and the origin) represent the N+1 currencies

themselves. As financial portfolios these points are special because they are associated with

governments or monetary authorities. One can consider the points

1 and 10 with ,1

10

0

N

i i

N

i

iiilN

in the convex interior of the simplex; these represent i -weighted baskets of currencies. The

(equal-weighted) barycenter

N

i

ilN 01

1 represents a portfolio divided equally in value amongst

the N+1 currencies. Points outside the simplex represent portfolios with short currency

positions.

31

A5. The matrices associated to other vertices

We can attach a positive definite matrix i to every vertex Ni 0 of the simplex. We

set 0 and it‟s not hard to see that under change of numeraire

'

ii

i AA ,

where

1100

0100

0100

0110

0101

A .

In other words, iA is the identity matrix except for the (-1)s in its i -th column. The eigenvalues

of are not preserved under this transformation, as the matrices iA are not orthogonal, but the

correct system of invariants here are permutation-invariant geometric quantities like mean

square edge length, mean square 2-face area, mean square 3-face volume, …, the N-simplex

square volume.

A6. The Supervetex and the Supercurrency

In this appendix we formally introduce the supercurrency as the supervertex of the

extended currency simplex maximizing the effective dimension. A supercurrency vertex could in

principle maximize a different diversification measure, such as effective rank or inverse

condition number. However, the effective dimension maximization problem has a closed form

solution. (In many cases, maximization of effective rank leads to a similar answer.)

32

There are two reasons for this maximization. First, the result is practical for

optimization; the supercurrency-numeraire covariance is well behaved in terms of eigenvalues.

Second, the procedure gives an upper bound on the effective dimension or number of currencies.

We would like to extend the simplex into the (N+1)-st dimension by introducing another

vertex, which we call the supervertex or supercurrency vertex. The extended simplex is denoted

by ~

. We choose the supervertex ),,...,,( 121 NN xxxx by maximizing the effective dimension of

the )1()1( NN matrix ~sup associated to the supervertex. The original symplex Δ lies in an

N-hyperplane embedded in the (N+1)-dimensional Cholesky space; and we can append a “0” as a

new coordinate for )0,( ii ll

Adding i=0 for the numeraire, such as USD, with )0,0(0

l , the

elements of this extended matrix are

.,0for )()'(~sup Njixlxl jiij

It is not hard to see (see A7 for the weighted case) that maximizing the effective

dimension we get the following

N

i

i

N lN

xxxx1

21

sup1

1),...,,(

,

that is, the vector ),...,,( 21 Nxxx is the barycenter of the simplex Δ. Furthermore,

msel)1(2

11

Nx N , (5)

and

msel

2

1~ msel

N

N, and

msel

2

1~tr sup N

,

) msel)((det)1(2

1)(det

~det 21sup

Nx N ,

33

)1/(

)1/(1

sup

)1/(1supsup

msel2

1

det1

1

)1()1/(

~tr

)~

(det)1(

~edim

NN

N

NN

NN

N .

Obviously, the supervertex and related numbers do not depend on the original vertex.

If we arrived at a supercurrency using a basket of real assets, the real asset would lie in an

(N+1)-dimensional space with angles to the FX simplex given by measured correlations. By

adding auxiliary FX exposure to the real asset, we can ensure that its projection to the FX

hyperplane lies precisely atop the barycenter. By increasing or decreasing leverage to this

adjusted real asset, using the barycenter as numeraire, we can match the volatility condition (5).

Thus we can construct the supercurrency as a tradable asset. This, of course, is the Gram-

Schmidt procedure of linear algebra.

A7. Solving the maximum effective dimension problem: Weighted Simplex

We now solve the edim maximization problem in the case where currencies are GDP-

weighted (or trade-weighted) with relative weights ),...,( 22

0 Nuuw with 10

2

N

i iu ; (here i=0

corresponds to USD). We want to solve for the supercurrency that maximises the edim of

))(()(~~ 2120supsup N

ijjiijjiijji

w

ij xxllxuuuu

(6)

The weights iu are not entirely trivial because a rescaling in the i-index vector algebra is not a

priori the same as a rescaling in the Cholesky space, indicated by the vector signs. We must find

the arg max of a properly invariant function (for example, trace, edim or erank) to connect these

symmetries together. The determinant is

212sup2sup )() (vol))det(diag()~

det())det(diag()~

det( N

iiji

w

ij xuu ,

34

and all factors but the last are constants. Taking the N-gradient of the trace of (6), we get the

weighted barycenter

N

i ii

w luBx0

2

sup

(7)

Then finding the edim-maximizing 1Nx coordinate, we get

N

i

w

ii

N BluN

x0

2221 )(1

, (8)

and

Nji jiji

wN

i ii lluuBlu0

2222

0

22 )()(

is a w-weighted MSEL.

Maximizing the effective rank of the supercurrency-based covariance matrix is not as

straightforward to solve; however, in the general situation with weights ),...,( 22

1 Nuuw the

location of the supercurrency will still project to the literal barycenter of the paper currency

vertices (7). The value of 1Nx is no longer given by (5), but is of the same order. So the edim

and erank optimizations only differ by a small amount of discrepancy in the orthogonal

volatility. For these reasons and for concreteness, we have just used the simple, closed form (7)-

(8) for 1Nx supercurrency in our supercurrency examples, even though we are discussing the

contributions to effective rank.

Appendix B. Shannon Entropy

The effective rank of the covariance matrix is related to the Shannon (or informational)

entropy of the market dynamics. Smaller entropy indicates more ex-ante knowledge about the

axis of returns. (Covariance says nothing about the sign of that axial return.)

35

Take a covariance matrix estimating the covariances of returns )',...,( 1 NRRR for the

next time period. Denote the eigenvalues 22

2

2

1 N and principal components

),,,( 21 Nqqq . We have

N

j

iii QQRRE1

22 ')diag()'( ,

with projections '= iii qq . Then )'( tr RRE is the estimate for the square-sum of principal

axis returns, and the expected square returns along the i -th component is

)'()'( 2 RREpRRE iii .

It is consistent to think of the total variance )'( RRE as being “distributed” among the principal

components in a probabilistic way, where each bit of total variance originates from principal

component i with probability ip .

The Shannon entropy, denoted by

)rank effectivelog(log1

N

i ii ppS

gives us the amount of additional information we need to know (on average) to explain the

ensuing securities returns R in terms of their decomposition on principal components. If all

covariance eigenvalues are equal, we have “no” information to describe R. Entropy is maximum

at NS log . On the other hand, if one principal component dominates, we are reasonably

confident that R will be much aligned with it. The sign of R (i.e. whether the market goes up or

down), and the scale of R (how much the market goes up or down on any particular day) are not

knowable, but the direction of the vector R is. When the return

N

i iiqRR1

~~ is realized, with

percentages i iii RRp 22 ~

/~~ the relative entropy (Kullback-Leibler divergence)

36

)~log(log1relative i

N

i ii pppS

measures the entropic divergence from the prior covariance estimate. This measure, when

multiplied by the size of the move 2|| R , is somewhat analogous to the turbulence measures of

Chow et al. (1999).

In the discussion above, the eigenvectors and eigenvalues stay fixed, and we analyzed

returns coming from the known distribution. In the real world, the eigenvalues and eigenvectors

are changing day to day, and the distribution of returns on the lagged eigenbasis is a more

dynamic problem.

While we choose the supercurrency to maximize the effective dimension, it could be

chosen to maximize effective rank, or in other words to maximize entropy. In optimization

problems, we desire to use a covariance matrix with an entropy larger than our straight estimate,

because the realized covariance out of sample will be different, and we will more likely do a

better job with a higher entropy estimator. By choosing a supercurrency in an enlarged space,

we can increase entropy while still staying consistent with the covariance matrix of the paper

currencies themselves. This may be preferable to a straightforward Bayesian reduction of a

given numeraire‟s covariance matrix, which (though better than nothing) will obviously bias

covariance estimates by under-assuming correlations. More practically, the supercurrency serves

as a higher-entropy starting point, where we can perform a milder Bayesian reduction.

Unlike closed collections of physical systems, which tend to chaos and always increase

total entropy when they interact, organized life (and human life in particular) has a tendency to

introduce order and decrease the entropy of the objects on which it acts, at the expense of heat

generated and distributed to the colder surrounding environment. It is not such a surprise that

investors interact with volatile systems in a fashion that not only removes arbitrage but aligns

37

risky trades, and we need look no further than the spike in correlations during a market crash,

and the sweat on our own brows. As new investment frontiers (emerging markets, volatility

products, higher frequency trading) are brought to the stage, the total entropy of the investment

universe is increased, and in time, human actions will “develop” these markets, removing

arbitrage opportunities and decreasing the entropy.

Appendix C. Covariance matrix alternatives and the 2007 shocks

Different methods for covariance matrix estimation can give a different picture of

effective rank. In the samples of Section 3, we used GDP-scaled covariance to show economic

significance of currency moves. In particular, for the G8 currencies post-Euro in Figure 4, we

have downplayed the importance of the Australian dollar. From the point of view of a currency

trader, Australian dollar is of extreme economic importance. So, we look again at the effective

rank of the 1-year half-life covariance matrix during the 2007-Present credit crisis. In Figure 9,

we compare GDP-weighted to unit-weighted (or simple econometric) covariance matrices, and

we look at the situation using both the exogenous supercurrency (hence an 8x8 dimensional

system) and the endogenous USD (7x7) as numeraires; see Figure 9.

Using the equal weighting, the supercurrency sees the precursors to the megacrisis of

2008. The moves in Australian dollar (in addition to JPY) gave sudden shocks to the covariance

matrix on August 15, 2007 (after the quant hedge fund crisis) and a repeat episode in November.

These events showed significant carry strategy drawdowns20

; Kritzman et al. (2010) have shown

that such measures of fragility are coincident as well as predictive signals for market

performance. These shocks do not show up on the radar of the GDP-weighted or any of the

20

It is tempting to superimpose the loss of effective rank with the profit-and-loss of the carry strategy itself.

38

Figure 9: Effective Rank of G8 Covariance Matrix, Weighting and Numeraire Comparison, 2007-2010

2006 2007 2008 2009 20100

1

2

3

4

5

6

7

8

Eff

ective N

um

ber

of

Curr

encie

s

Supercurrency, GDP-Weighted

Supercurrency, Unit-Weighted

USD-based, GDP-Weighted

USD-based, Unit-Weighted

Notes: G8 Currencies AUD, CAD, CHF, EUR, GBP, JPY, SEK, USD. Covariance matrices use 1-year half-life.

Supercurrency defined by maximizing effective dimension of GDP-weighted or unit-weighted covariance.

USD-numeraire schemes. The reason for this is that the principal eigenvector of the USD-

numeraire covariance is USD itself, i.e. the numeraire effect. This is an artificial effect and

obscures our view of the G8 space. If the numeraire effect dominates, a blow-up in the carry

trade, which may be the second principal component, will not register as a sudden drop in

effective dimension. As we see empirically on the graph, the unit-weighted supercurrency

alleviates this problem and shows in a numeraire-independent way the breakdowns in the

currency markets in 2007.

Appendix D. Currencies Table

We collect some information on the currencies we deal with in the table below.

39

Symbol Currency Name 2009 GDP, USD bn 2009 GDP, % of World GDP

USD U.S. Dollar 14,266 27.2%

JPY Japanese Yen 5,049 9.6%

AUD Australian Dollar 920 1.7%

NZD New Zealand Dollar 110 0.2%

CAD Canadian Dollar 1,319 2.5%

GBP Great Britain Pound 2,198 4.2%

NOK Norwegian Krone 369 0.7%

SEK Swedish Krona 398 0.8%

DKK Danish Krone 308 0.6%

CHF Swiss Franc 484 0.9%

EUR Euro 12,276 23.4%

CZK Czech Koruna 190 0.3%

HUF Hungarian Forint 124 0.2%

ILS Israeli Shekel 216 0.4%

PLN Polish Złoty 423 0.8%

RUB Russian Ruble 1,255 2.4%

TRL Turkish Lira 594 1.1%

ZAR South African Rand 277 0.5%

ARS Argentine Peso 301 0.6%

BRL Brazilian Real 1,482 2.8%

CLP Chilean Peso 150 0.3%

MXN Mexican Peso 866 1.7%

PEN Peru New Sol 127 0.2%

CNY Chinese Yuan (Renminbi) 4,758 9.1%

HKD Hong Kong Dollar 209 0.4%

IDR Indonesian Rupiah 515 1.0%

INR Indian Rupee 1,243 2.4%

KRW S. Korean Won 800 1.5%

MYR Malaysian Ringgit 207 0.4%

PHP Philippine Peso 159 0.3%

SGD Singapore Dollar 163 0.3%

THB Thai Baht 266 0.5%

TWD Taiwanese Dollar 357 0.7%

Notes: GDP data source: the IMF World Economic Outlook database. We define the World to

be the closed set of countries we are looking at.

40

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