Crossroads - Spring 2011

for some positive factor μ (i.e., we use μ = 1 so that Ind(w*; t0 / t0) = RNVal j(t0 / t0) = 1). The wi weights are only applicable to currency values in their normalized form. By contrast, the qi weights are in terms of the currencies themselves, where qUSD (for example) means the number of actual dollars in the basket, qEUR is the number of euros, etc. We will refer to this optimal currency basket as SAC (stable aggregate currency). Since SAC is a minimum variance basket of currencies, we consider it a form of stable money.

STABLE MONEY EXAMPLE Let us construct a stable basket of currencies using the four currencies {EUR, GBP, JPY, USD}. We form daily time series c(ABC/XYZ; t) for a “learning” period from t= 1 [January 1, 2006] to t=T=911 [June 30, 2008], with a time step of a single day. All data for exchange-coefficients c(ABC/XYZ; t) are taken from the website www.fxtop.com. In addition to these four currencies, we consider one composite currency— namely, the IMF’s SDR. Units of the SDR are designated by the symbol XDR comprised of the basket {q1*EUR, q2*GBP, q3*JPY, q4*USD}, where simple currency units’ amounts q1,...,q4 are defined by the IMF for the period from January 1, 2006 to December 31, 2010 as q1=0.41, q2=0.0903, q3=0.184, q4=0.632, q=q1+...+ q4= 1.3163. For convenience we define XDR’=XDR/1.3163 as scaled units of SDR, wherein XDR’ is comprised of the basket XRD’={v’ 1*EUR, v’ 2*GBP, v’ 3*JPY, v’ 4*USD}, v’ 1=0.311, v’ 2=0.069, v’ 3*=0.140, v’ 4*=0.480, v’=v’ 1+...+v’ 4= 1. Normalized daily time series of values, RNVal(X YZ;t/ t0), XYZ = EUR, GBP, JPY, USD, XDR are computed as in equation ( 1) above for the “learning” period [January 1, 2006 - December 30, 2008]. Subsequently, the optimal basket w*=(w1*,...,w4*) is found as described previously. Based on this optimizing procedure, we obtain the stable aggregate currency with optimal w* weights SAC={0.208*EUR, 0.166*GBP, 0.346*JPY, 0.280*USD}. Table 1 shows SAC has a much lower standard deviation in the learning period than the individual currencies. For instance, SAC is 18 times more stable than the U.S. dollar (USD). Also, it is more stable than the IMF’s SDR (XDR, see tables). This con-

Table 1: Basic Statistics of Different Individual and Basket Currencies. Normalized values RNVal(X YZ;t/t0), t0= 1. X YZ=EUR, GBP, JP Y, USD, XRD, SAC in the learning period [Jan. 1, 2006-Dec. 30, 2008].

Mean
Minimum
Maximum
EUR

0.8992

Range Standard Deviation STATISTICS

0.9707

0.1600 0.0592

0.0443

GBP

0.9831

1.0821

0.1684

1.1516

0.0372

JPY

0.8742

0.9392

0.1395

1.0137

0.0306

USD

0.9312

1.0166

0.1618

1.0929

0.0396

XDR

0.9335

0.9541

0.0695

1.0029

0.0061

SAC

0.9953

0.9977

0.0081

1.0033

0.0022

To highlight the stability of SAC, we provide Table 2 that divides volatility measures of differentcurrencies by the correspondent measures of SAC’s volatility. There we see that the dollar is17.9 times more volatile as SAC. It is clear that SAC is a better measuring stick for money thanthe component individual currencies as well as the SDR.

Table 2: Relative Volatility of Individual and Basket Currencies. Dividing by SAC’s volatility in the learning period [Jan. 1, 2006-Dec. 30, 2008].

JPY

13. 8 17. 3

XDR

2. 7

8. 6

SAC

We next examine the volatility of SAC during an out-of-sample “testing” period from t= 1[July 1, 2008] to t=T=822 [Sept. 30, 2010]. The results of these calculations are shown in Tables 3 and 4, which are analogous to Tables 1 and 2, respectively.

Range EUR

20.0

STATISTICS

19. 9

1

Standard Deviation

GBP

16. 8 20. 9

USD

17. 9

20. 1

Table 3: Basic Statistics of Different Individual and Basket Currencies. Normalized values RNVal(X YZ;t/t0), t0= 1, X YZ=EUR, GBP, JPY, USD, XDR, SAC, in the testing period [July 1, 2008-Sept. 30, 2010].

Mean
Minimum
Maximum
EUR

0.8472

Range Standard Deviation

S TATIS TICS

0.9252

0.1600 0.1602

0.0378

GBP

0.7585

0.8491

0.2492

1.0077

0.0535

JPY

0.9843

1.2013

0.3572

1.3415

0.0838

USD

0.9927

1.0657

0.1497

1.1424

0.0315

XDR

0.9672

1.0039

0.0999

1.0671

0.0138

SAC

0.9986

1.0078

0.0194

1.0180

0.0047

Table 4: Relative Volatility of Individual and Basket Currencies. Dividing by SAC’s volatility in the testing period[July 1, 2008-Sept. 30, 2010].

JPY

17.71 18.41

XDR

2.91

5. 15

SAC

Even in an out-of-sample testing period after its optimal weights are determined in the previous learning period, these results again show that SAC is very stable.