I. Introduction

In the calculation of optimal fixed-income portfolios based on historical means and variances, which are often used by investors to determine the “best” allocation of their wealth in terms of fixed-income investment instruments and by some central banks in the management of their external reserve assets, the standard assumption is that historical volatilities of returns from investments in various currency-denominated government securities remain constant over time. The same assumption is also often made by the capital asset pricing model and the modern portfolio theory in the calculation of diversification benefits from investing in multiple assets and by the Black-Scholes model in the calculation of option-pricing formulas. The implications of incorrectly assuming constant variability of returns over time are loss of efficiency in terms of portfolio allocations and inappropriate inferences in terms of hedging the associated risk. Recent empirical work (see Section II) has shown that large changes in equity returns and exchange rates, with high sampling frequency, tend to be followed by large changes of either sign, that is, the variance of these returns and exchange rates is not constant and, therefore, this traditional assumption of many standard models of portfolio allocation does not hold. However, very few, if any, studies have dealt with returns from a multicurrency fixed-income portfolio.

A suitable model for dealing with non-constant volatilities of returns is the ARCH model. ^1/ This paper attempts to model total global fixed-income investment returns for a U.S. dollar-based investor as a univariate autoregressive conditional heteroskedastic (ARCH) process as an alternative to the more common portfolio model that assumes that the variance of returns from investing in various currency-denominated assets is stationary or time invariant. The ARCH model expresses the current conditional variance of a time series of total returns as a linear function of their past-squared residuals. ARCH models permit the prediction of changes in the conditional variance of asset returns on different currencies, but not the changes in the unconditional mean value of asset returns. Knowledge of the statistical properties of the process that generates the observed rates of return on various investments, and in particular of how the process depends upon developments in the financial markets, is important for characterizing asset-return volatility and for modelling optimal portfolios of assets denominated in various currencies.

Traditional models assume a constant one-period forecast variance and covariance matrix for the returns on different investments. This assumption can be generalized to introduce a new class of stochastic processes called ARCH processes. These processes imply serially uncorrelated errors which have zero mean and nonconstant variances that are conditional on past returns. To test whether the error terms follow an ARCH process, the autocorrelation of the squared ordinary-least-squares (OLS) residuals has been calculated. After the detection of ARCH effects based on the autocorrelation statistics, a factor-ARCH model ^2/ can be used to reduce the number of parameters to be estimated.

Our empirical results show that the variability of returns from investing in various currencies and currency composites is affected by lagged residuals, while the number of lags differs among currencies. The estimated constant terms in the ARCH specification, a₀, for all currencies are significant and non-negative, indicating a positive autonomous conditional volatility for currencies. Further, the estimated parameters of the lagged squared residuals in the ARCH process, a₁, exhibit strong statistical significance for all currency asset returns, indicating that the stochastic processes generating these currency returns are covariance-nonstationary. However, investment in SDRs provides greater insulation to investors than investing in any other currency with one lag of ARCH effects, as SDR returns demonstrate the lowest transmission of unexpected weekly shocks on their volatility. ^3/ In these circumstances and as already noted, an ARCH specification is a more efficient approach for modelling all considered asset returns than is the OLS approach. Finally, the Lagrange multiplier (LM) statistic for first-order ARCH specification is significant for all currencies, which also indicates that the conditional variances are not constant through time.

In addition, the results of several diagnostic checks on the distributional properties of total returns indicate that they deviate from the normal distribution. In particular, the kurtosis statistics of the probability distribution of returns on all currency investments are substantially larger than those from a standard normal distribution, indicating that these returns are leptokurtic (fat-tailed). This result is further supported by Bera-Jarque tests indicating not only a highly significant non-normality of the probability function for returns on these currencies but also a significant likelihood that the variance of these returns is not constant over time. The Bera-Jarque (BJ) test for normality ^4/ is highly significant for the deutsche mark, the pound sterling, the French franc, the Japanese yen, the ECU, the SDR, and gold with ARCH(1,1), as well as for the U.S. dollar with ARCH(6,2). Only the BJ test for the average composite index with ARCH(1,1) is not statistically significant.

The organization of this paper is as follows: Section II provides a brief review of the literature. Section III outlines the testing and estimation procedures for ARCH effects. Section IV presents an application of the standard univariate ARCH model on weekly returns from short-term investments in nine different currencies and analyzes the main results. Section V concludes by summarizing the evidence from modeling short-term fixed income returns as ARCH processes and discusses the practical implications for the management of reserves.

II. A Brief Literature Review

The uncertainty of speculative prices has been observed to be changing through time (Mandelbrot (1963) and Fama (1965)). The tendency of large (small) price changes in the high frequency financial data to be followed by other large (small) price changes is often called “volatility clustering.” One of the specifications that has emerged for characterizing such changing variances is the ARCH model (Engle (1982)) and its various extensions. In his seminal paper, Engle suggests that one possible parametrization for variances is to express them as a linear function of past-squared values of the errors of the model. With financial data, the ARCH model captures the tendency for volatility clustering, and numerous empirical applications of the ARCH methodology in characterizing asset return variances and covariances have already appeared in the literature. ARCH effects have generally been found to be highly significant in equity markets; for example, highly significant test statistics for ARCH models have been reported for various individual returns on stock market investments (Engle and Mustafa (1989)).

In a series of papers, the ARCH model has been analyzed, generalized, extended to the multivariate context, and used to test for time-varying risk premia in financial markets. These papers include Engle (1983) and Engle and Kraft (1983). The ARCH model was extended to the multivariate framework in Kraft and Engle (1982). Diebold and Nerlove (1989) use a multivariate approach exploiting factor structure, which facilitates tractable estimation via a substantial reduction in the number of parameters to be estimated. The factor structure captures commonality in volatility movements in financial time series. Engle, Ng, and Rothschild (1990) suggest using the factor-ARCH model as a structure for the conditional covariance matrix of excess asset returns. One- and two-factor ARCH models have been applied to pricing of treasury bills, with the results showing reasonable stability over time.

The importance of ARCH models in finance comes partly from the direct association of variance and risk, and the fundamental trade-off relationship between risk and return. The ARCH-M (in mean) model developed by Engle, Lilien, and Robins (1987) expresses the conditional mean as a function of the conditional variance of the ARCH process. It thus provided a tool for estimation of the (possibly) linear relationship between the return and variances of a given investment portfolio.

Another interesting observation in financial data is that the unconditional price and return distributions tend to have fatter tails than the normal distribution (i.e., are leptokurtic) (Mandelbrot (1963) and Fama (1965)). Ignoring the fat tails and the time-varying variances could lead to erroneous detection of abnormal returns (De Jong, Kemna, and Kloeck (1990)). Stock returns tend to exhibit non-normal unconditional sampling distributions, both in the form of skewness and excess kurtosis (Fama (1965)). For many financial time series, the leptokurtosis cannot be fully explained. The conditional normality assumption in ARCH generates some degree of unconditional excess kurtosis, but this is typically less than adequate to fully account for the fat-tailed properties of the data. One solution to the kurtosis problem is the adoption of conditional distributions with fatter tails than the normal distribution. Skewness and kurtosis are shown to be important in characterizing the conditional density function of returns on stocks (Engel and Gonzales-Rivera (1989)).

In addition to the leptokurtic distribution of stock return data, Black (1976) noted a negative correlation between current returns and future volatility. The linear generalized ARCH(p,q) model, where the variance only depends on the magnitude and not on the sign of the estimated residuals, is not able to capture this negative relationship since the conditional variance is only linked to past conditional variances and squared innovations, and hence the sign of returns plays no role in affecting the volatilities. This limitation of the standard ARCH formulation is one of the primary motivations for the exponential-GARCH(p,q) model by Nelson (1990). In these types of generalized ARCH models, the volatility depends not only on the magnitude of the past surprises but also on their corresponding signs.

Bollerslev, Engle, and Wooldridge (1988) first estimated a multivariate GARCH(p,q) process for returns on bills, bonds, and stocks where the expected return was assumed to be proportional to the conditional covariance of each return with that of a fully diversified or market portfolio. They found that conditional covariances are quite variable over time and are a significant determinant of the time-varying risk premia. Bollerslev (1990) applied also a multivariate GARCH model to short-run nominal exchange rates. Hall, Miles, and Taylor (1988), similarly applied a multivariate GARCH-M estimation of the capital asset pricing model. Pagan and Schwert (1990) compared several statistical models for monthly stock return volatility and showed the importance of nonlinearities in stock return behavior that are not captured by conventional ARCH or GARCH models. They also showed the nonstationarity of the volatility of stock returns. Vries (1991) compared stable and GARCH processes for modeling financial data and showed that the unconditional distribution of variables generated by a GARCH-like process, which models the clustering of volatility and exhibits the fat-tail property as well, can be stable. Further, Bollerslev (1987) had earlier introduced an extension of ARCH and GARCH models, called an integrated GARCH model, which tried to explain the persistence of variance over time. A detailed explanation of the major specifications and estimation procedures of the ARCH-family models is given in Appendix I.

III. Estimation and Testing for ARCH Effects

Autoregressive conditional heteroskedasticity (ARCH) models account for the empirical observation that large changes in asset returns tend to be followed by further large changes in these returns (though of unpredictable sign), and small changes in asset returns tend to be followed by small changes, thus leading to periods of persistently high or low volatility. ^5/ We utilize the ARCH model by assuming that the conditional mean, E (y_t|y_t-1), of each asset return, y_t, is linearly unpredictable, while its conditional variance, h_t, is predictable by an ARCH model. It is well known that the unconditional distribution of ARCH processes have thicker tails, even though their conditional distributions are normal. ^6/

The first order autoregressive, AR(1), model for y_t, combined with ARCH(1,1) errors, can be written as ^7/

where E (ε_t|y_t-1) = 0

and

where $\left|{\mathit{ɸ}}_1\right|$ is assumed to be less than one for the regression to be stable. To ensure that h_t is positive and that the process is stationary, we must have α₀ ≥ 0 and 1 ≥ α₁ ≥ 0. Intuitively, α₀ can be interpreted as a “normal” stationary element in the series on asset returns, and α₁ as the heteroskedastic coefficient, which shows the correlation between changes in returns and their corresponding volatilities in the previous period. For a U.S. dollar-based investor, for example, y_t denotes asset returns in U.S. dollars obtained by investing in various currency assets and is defined as the product of the three-month treasury bill rate for each asset multiplied by the percentage change of the corresponding U.S. dollar exchange rate, all expressed on a weekly basis. ^8/

The generalization of the ARCH(p,q) model for asset return y_t is as follows:

where ${ɸ}_p\left(L\right)=\underset{i=1}{\overset{p}{\mathrm{\Sigma }}}{ɸ}_i{L}^i$ is the polynomial autoregressive lag operator.

Note that

where ε_t follows a white noise process defined by E(ε_t) = 0 for all t, E(ε_t ε_s) = 0 for t ≠ s, and $E\left({\epsilon }_t^2\right)={\sigma }^2$ for all t. The conditional variance of ε_t, h_t, is allowed to vary over time and is a linear function of past squared residuals.

Testing for ARCH effects requires to test the null hypothesis that the conditional variance h_t is constant over time, or

H_o: α₁ = α₂ = … = α_q = 0 (no ARCH effects)

H₁: α₁ ≠ α₂ ≠ … ≠ α_q ≠ 0 (ARCH effects exist).

To test the null hypothesis H_o, we have utilized the following procedure: first, the residuals ${\hat{\epsilon }}_t$ are estimated from equation (3) by using ordinary least squares (OLS); second, the number of lags of ${\hat{\epsilon }}_t$ is determined by utilizing all available information, through maximization of its log-likelihood function; and third, estimates of α_o, …, α_q are obtained by applying OLS to

In order to test for an ARCH process of order q, we form the regression

where ${\hat{\epsilon }}_t^2$ is the square of the OLS residuals obtained from equation (3). ^9/ To test for higher order ARCH specifications and the joint significance of the coefficients attached to equation (6), the Lagrange multiplier (LM) statistic, is calculated as T·R² (where R² is the coefficient of determination and T is the sample size) which has an asymptotic x²(q) distribution. Significantly large values of the statistic will lead to the rejection of the null hypothesis H_o of homoskedasticity in favor of an ARCH process of order q.

Finally, if ARCH effects are detected, maximum likelihood estimation rather than OLS should be applied ^10/ and the additional assumption of (conditional) normality of returns should be employed in the third step of the above procedure. A detailed explanation of the various specifications and estimation procedures of the ARCH-type models is given in Appendix I.

IV. Empirical Results

In the univariate version of ARCH models, as described above, the conditional variance of a currency return is modelled as an autoregressive process of its lagged-squared residuals so that during periods of high variability of a return, its estimated variance will increase or remain high, and during periods of relative stability its estimated variance will decrease. In this section, the univariate ARCH model is estimated for a U.S. dollar-based investor using weekly data. The data set contains returns from investing in short-term securities in the U.S. dollar, the deutsche mark, the Japanese yen, the pound sterling, the French franc, Special Drawing Rights (SDR), European currency unit (ECU), an average composite index (ACI), and gold. The ACI represents the average share of currencies in official holdings of foreign exchange reserves of IMF member central banks for the period 1981-87. Our sample spans the period February 1982 to December 1991. The data set is described in Appendix II.

For given values of past-realized investment returns, α₀ and a_i, i=l, …, 5, were estimated for each total return on the above mentioned nine currency-denominated assets (see Table 1). All estimated coefficients, ${\hat{\alpha }}_{\mathrm{o}}$ and ${\hat{\alpha }}_{\mathrm{i}}$, are non-negative, thus ensuring that a positive variance is obtained. Furthermore, the summation of α_i’s (i≥1) is always less than one, indicating that the estimated variance is finite. The optimal lag length, q, is determined as the value that maximizes the log-likelihood function in a grid search over lags 1 to 6. To maximize the likelihood function, an iterative procedure based upon the method of BHHH ^11/ was used. As can be seen from Table 1, which presents the estimation results for the optimal lag lengths, the optimal lag length is one for the deutsche mark, the pound sterling, the French franc, the Japanese yen, ECU, SDR, and gold; two for the U.S. dollar; and five for the average composite index. The difference in the optimal lag length between the U.S. dollar and the rest of the employed currencies, except the average composite index, may be attributed to the fact that U.S. dollar returns comprise only interest rates on the respective assets, without taking into account any counteracting movements in the exchange rate as is the case for the other currencies. The ARCH coefficients for almost all currency returns are significantly greater than zero, as indicated by the asymptotic t-statistics given in parentheses.

Table 1.

ARCH (p,q) Estimation Results for Total Returns (Weekly, 1982:5-1991:50)

$\begin{array}{ccc}{\mathrm{y}}_{\mathrm{t}}={\mathit{ɸ}}_0+{\mathit{ɸ}}_1{\mathrm{y}}_{\mathrm{t}\mathrm{-}\mathrm{1}}+\dots +{\mathit{ɸ}}_p{\mathrm{y}}_{\mathrm{t}\mathrm{-}\mathrm{p}}+{\epsilon }_{\mathrm{t}}& \mathrm{and}& {\mathrm{h}}_{\mathrm{t}}={\alpha }_0+{\alpha }_1\end{array}{\epsilon }_{\mathrm{t}\mathrm{-}\mathrm{1}}^2+\dots +{\alpha }_{\mathrm{q}}{\epsilon }_{\mathrm{t}\mathrm{-}\mathrm{q}}^2$

* and ** denote statistically significant t-statistics at 0.05 and 0.1 levels, respectively.

^1/The Lagrange multiplier (LM) test is a test for the existence of ARCH effects in a series. The LM statistic is the product of the sample size, T, times the coefficient of determination, R². Its distribution is a x² with degrees of freedom equal to the number of lags.

Table 1.

ARCH (p,q) Estimation Results for Total Returns (Weekly, 1982:5-1991:50)

$\begin{array}{ccc}{\mathrm{y}}_{\mathrm{t}}={\mathit{ɸ}}_0+{\mathit{ɸ}}_1{\mathrm{y}}_{\mathrm{t}\mathrm{-}\mathrm{1}}+\dots +{\mathit{ɸ}}_p{\mathrm{y}}_{\mathrm{t}\mathrm{-}\mathrm{p}}+{\epsilon }_{\mathrm{t}}& \mathrm{and}& {\mathrm{h}}_{\mathrm{t}}={\alpha }_0+{\alpha }_1\end{array}{\epsilon }_{\mathrm{t}\mathrm{-}\mathrm{1}}^2+\dots +{\alpha }_{\mathrm{q}}{\epsilon }_{\mathrm{t}\mathrm{-}\mathrm{q}}^2$

	U.S. dollar	Deutsche mark	Japanese yen	Pound sterling	French franc	SDR	ECU	Average composite index	Gold
$\widehat{\mathit{ɸ}}$ 0	0.001	0.126	0.135	0.118	0.136	0.082	0.095	0.016	-0.059
	(2.889)*	(2.262)*	(2.406)*	(2.038)*	(2.458)*	(2.883)*	(1.714)*	(1.939)*	(-0.062)
$\widehat{\mathit{ɸ}}$ 1	1.191	0.331	0.347	0.311	0.332	0.330	0.302	0.362	0.248
	(27.026)*	(8.087)*	(7.394)*	(7.679)*	(8.119)*	(7.500)*	(7.134)*	(7.829)*	(5.454)*
$\widehat{\mathit{ɸ}}$ 2	-0.291							-0.105
	(-4.597)*							(-2.094)*
$\widehat{\mathit{ɸ}}$ 3	0.212							0.102
	(4.184)*							(1.999)*
$\widehat{\mathit{ɸ}}$ 4	0.009							-0.039
	(0.211)							(-0.816)
$\widehat{\mathit{ɸ}}$ 5	0.055
	(1.565)
$\widehat{\mathit{ɸ}}$ 6	-0.187
	(-8.734)*
$\widehat{\mathit{ɸ}}$ 0	0.000	1.355	1.25	1.374	1.270	0.369	1.163	0.018	3.329
	(11.199)*	(12.767)*	(15.332)*	(14.310)*	(14.431)*	(13.072)*	(10.963)*	(5.368)*	(22.932)*
$\widehat{\mathit{ɸ}}$ 1	0.538	0.144	0.17	0.181	0.197	0.114	0.203	0.106	0.194
	(9.665)*	(2.473)*	(4.742)*	(3.870)*	(3.107)*	(2.002)*	(3.065)*	(1.726)*	(4.369)*
$\widehat{\mathit{ɸ}}$ 2	0.162							0.074
	(2.944)*							(1.467)*
$\widehat{\mathit{ɸ}}$ 3								0.108
								(1.452)*
$\widehat{\mathit{ɸ}}$ 4								0.130
								(2.179)*
$\widehat{\mathit{ɸ}}$ 5								0.037
								(0.708)
Log likelihood	2234.24	-845.77	-829.02	-857.23	-841.03	-503.10	-755.69	158.17	-1081.92
Lagrange multiplier1/	112.72*	10.99*	6.09*	8.18*	19.15*	4.45*	10.21*	16.69*	0.60

* and ** denote statistically significant t-statistics at 0.05 and 0.1 levels, respectively.

^1/The Lagrange multiplier (LM) test is a test for the existence of ARCH effects in a series. The LM statistic is the product of the sample size, T, times the coefficient of determination, R². Its distribution is a x² with degrees of freedom equal to the number of lags.

View Table

Table 1.

ARCH (p,q) Estimation Results for Total Returns (Weekly, 1982:5-1991:50)

$\begin{array}{ccc}{\mathrm{y}}_{\mathrm{t}}={\mathit{ɸ}}_0+{\mathit{ɸ}}_1{\mathrm{y}}_{\mathrm{t}\mathrm{-}\mathrm{1}}+\dots +{\mathit{ɸ}}_p{\mathrm{y}}_{\mathrm{t}\mathrm{-}\mathrm{p}}+{\epsilon }_{\mathrm{t}}& \mathrm{and}& {\mathrm{h}}_{\mathrm{t}}={\alpha }_0+{\alpha }_1\end{array}{\epsilon }_{\mathrm{t}\mathrm{-}\mathrm{1}}^2+\dots +{\alpha }_{\mathrm{q}}{\epsilon }_{\mathrm{t}\mathrm{-}\mathrm{q}}^2$

	U.S. dollar	Deutsche mark	Japanese yen	Pound sterling	French franc	SDR	ECU	Average composite index	Gold
$\widehat{\mathit{ɸ}}$ 0	0.001	0.126	0.135	0.118	0.136	0.082	0.095	0.016	-0.059
	(2.889)*	(2.262)*	(2.406)*	(2.038)*	(2.458)*	(2.883)*	(1.714)*	(1.939)*	(-0.062)
$\widehat{\mathit{ɸ}}$ 1	1.191	0.331	0.347	0.311	0.332	0.330	0.302	0.362	0.248
	(27.026)*	(8.087)*	(7.394)*	(7.679)*	(8.119)*	(7.500)*	(7.134)*	(7.829)*	(5.454)*
$\widehat{\mathit{ɸ}}$ 2	-0.291							-0.105
	(-4.597)*							(-2.094)*
$\widehat{\mathit{ɸ}}$ 3	0.212							0.102
	(4.184)*							(1.999)*
$\widehat{\mathit{ɸ}}$ 4	0.009							-0.039
	(0.211)							(-0.816)
$\widehat{\mathit{ɸ}}$ 5	0.055
	(1.565)
$\widehat{\mathit{ɸ}}$ 6	-0.187
	(-8.734)*
$\widehat{\mathit{ɸ}}$ 0	0.000	1.355	1.25	1.374	1.270	0.369	1.163	0.018	3.329
	(11.199)*	(12.767)*	(15.332)*	(14.310)*	(14.431)*	(13.072)*	(10.963)*	(5.368)*	(22.932)*
$\widehat{\mathit{ɸ}}$ 1	0.538	0.144	0.17	0.181	0.197	0.114	0.203	0.106	0.194
	(9.665)*	(2.473)*	(4.742)*	(3.870)*	(3.107)*	(2.002)*	(3.065)*	(1.726)*	(4.369)*
$\widehat{\mathit{ɸ}}$ 2	0.162							0.074
	(2.944)*							(1.467)*
$\widehat{\mathit{ɸ}}$ 3								0.108
								(1.452)*
$\widehat{\mathit{ɸ}}$ 4								0.130
								(2.179)*
$\widehat{\mathit{ɸ}}$ 5								0.037
								(0.708)
Log likelihood	2234.24	-845.77	-829.02	-857.23	-841.03	-503.10	-755.69	158.17	-1081.92
Lagrange multiplier1/	112.72*	10.99*	6.09*	8.18*	19.15*	4.45*	10.21*	16.69*	0.60

* and ** denote statistically significant t-statistics at 0.05 and 0.1 levels, respectively.

^1/The Lagrange multiplier (LM) test is a test for the existence of ARCH effects in a series. The LM statistic is the product of the sample size, T, times the coefficient of determination, R². Its distribution is a x² with degrees of freedom equal to the number of lags.

The specification test for qth-order ARCH disturbances is based on the qth-order autocorrelation of the squared residuals. The LM test statistics for the presence of ARCH is asymptotically distributed as x²(q) under the null hypothesis of no qth-order ARCH effects. The critical values of the x²(1), x²(2), and x²(5) distributions at the 5 percent significance level are 3.84, 5.99, and 11.07, respectively. The presence of first-order ARCH effects is accepted at the 5 percent level for the deutsche mark, the Japanese yen, the pound sterling, the French franc, SDR, and ECU returns. The null hypothesis of no second-order ARCH effects is rejected for returns on the U.S. dollar, and that of no fifth-order ARCH effects is rejected for average composite investment returns. These results appear to provide statistical evidence that the variances of the respective currency returns are not independent over time, as they are related through the square of past residuals. As regards to gold returns, the evidence is inconclusive as to whether they follow an ARCH process since the t-statistic of the corresponding ARCH coefficient is statistically significant, while the LM statistic suggests the absence of ARCH effects.

The significance tests for the α_i; parameters and the LM statistics are corroborated by several additional diagnostic tests on the distributional properties of the total returns as given in Table 2. These statistics indicate that most currency returns, except for the ECU and the SDR, display negative skewness. This characterizes a distribution of returns where negative changes are less than positive ones. A small negative skewness statistic indicates that large positive changes are outnumbered by negative ones of smaller magnitude. Moreover, all of the currency returns considered here exhibit high leptokurtosis, which commonly appears in high-frequency exchange rate data. ^12/ As mentioned above, when the conditional distribution is normal with ARCH disturbances, its unconditional distribution is known to be leptokurtic, ^13/ implying that an ARCH model could partly eliminate the associated leptokurtosis. For distributions which are symmetrical and unimodal, kurtosis is a measure of flatness or peakedness of the distribution; leptokurtic curves tend to be sharply peaked curves relative to the normal distribution. The kurtosis statistics for all currency returns are significantly larger than three, which would not be consistent with a standard normal distribution of the total returns in these assets. ^14/

Table 2.

Preliminary Data Analysis on Total Returns ^1/

* Indicates statistically significant statistics at 5 percent level.

^1/The number of observation is 496.

^2/Skewness indicates how asymmetric the distribution is. The skewness statistic is zero in the population of a normal distribution.

^3/A high value of the kurtosis coefficient indicates the presence of a significant number of observations at the tails of this distribution. The kurtosis statistic is three in the population of a normal distribution. If the kurtosis coefficient is higher than 3, this indicates a higher number of observations at the tails than suggested under the normal distribution. For the calculation of the Kurtosis statistic, see Diebold (1988) PP. 8-11.

^4/The Bera-Jarque test for normality is a joint test for skewness and kurtosis.

^5/The augmented Dickey-Fuller statistic tests for unit roots in an autoregressive series and is robust to ARCH specification. If the augmented Dickey-Fuller statistic is significant, the existence of a characteristic root outside the unit circle cannot be rejected and therefore the respective series is considered non-stationary.

Table 2.

Preliminary Data Analysis on Total Returns ^1/

Statistic Skewness	U.S. dollar	Deutsche mark	Japanese yen	Pound sterling	French franc	SDR	ECU	Average composite index	Gold
Skewness 2/	0.998	0.281	0.899	0.443	0.074	-0.226	-0.209	0.070	0.651
Kurtosis 3/	10.506	3.133	3.190	3.218	3.264	3.081	3.282	3.245	3.2546
Bera-Jarque 4/	4817.583*	9.197*	220.964*	65.608*	14.496*	4.519*	10.524*	2.689	1158.28*
Augmented Dickey-Fuller 5/	-20.929*	-17.363*	-22.918*	-12.918*	-17.542*	-12.293*	-16.671*	-12.381*	-22.367*

* Indicates statistically significant statistics at 5 percent level.

^1/The number of observation is 496.

^2/Skewness indicates how asymmetric the distribution is. The skewness statistic is zero in the population of a normal distribution.

^3/A high value of the kurtosis coefficient indicates the presence of a significant number of observations at the tails of this distribution. The kurtosis statistic is three in the population of a normal distribution. If the kurtosis coefficient is higher than 3, this indicates a higher number of observations at the tails than suggested under the normal distribution. For the calculation of the Kurtosis statistic, see Diebold (1988) PP. 8-11.

^4/The Bera-Jarque test for normality is a joint test for skewness and kurtosis.

^5/The augmented Dickey-Fuller statistic tests for unit roots in an autoregressive series and is robust to ARCH specification. If the augmented Dickey-Fuller statistic is significant, the existence of a characteristic root outside the unit circle cannot be rejected and therefore the respective series is considered non-stationary.

View Table

Table 2.

Preliminary Data Analysis on Total Returns ^1/

Statistic Skewness	U.S. dollar	Deutsche mark	Japanese yen	Pound sterling	French franc	SDR	ECU	Average composite index	Gold
Skewness 2/	0.998	0.281	0.899	0.443	0.074	-0.226	-0.209	0.070	0.651
Kurtosis 3/	10.506	3.133	3.190	3.218	3.264	3.081	3.282	3.245	3.2546
Bera-Jarque 4/	4817.583*	9.197*	220.964*	65.608*	14.496*	4.519*	10.524*	2.689	1158.28*
Augmented Dickey-Fuller 5/	-20.929*	-17.363*	-22.918*	-12.918*	-17.542*	-12.293*	-16.671*	-12.381*	-22.367*

* Indicates statistically significant statistics at 5 percent level.

^1/The number of observation is 496.

^2/Skewness indicates how asymmetric the distribution is. The skewness statistic is zero in the population of a normal distribution.

^3/A high value of the kurtosis coefficient indicates the presence of a significant number of observations at the tails of this distribution. The kurtosis statistic is three in the population of a normal distribution. If the kurtosis coefficient is higher than 3, this indicates a higher number of observations at the tails than suggested under the normal distribution. For the calculation of the Kurtosis statistic, see Diebold (1988) PP. 8-11.

^4/The Bera-Jarque test for normality is a joint test for skewness and kurtosis.

^5/The augmented Dickey-Fuller statistic tests for unit roots in an autoregressive series and is robust to ARCH specification. If the augmented Dickey-Fuller statistic is significant, the existence of a characteristic root outside the unit circle cannot be rejected and therefore the respective series is considered non-stationary.

In addition, the Bera-Jarque (BJ) test statistic ^15/ for the deutsche mark, the pound sterling, the French franc, the Japanese yen, ECU, SDR, and gold with ARCH(1,1) is highly significant. Furthermore, the BJ test for the U.S. dollar with ARCH(6,2) and for the average composite index with ARCH(4,5) are also significant. The resulting high negative values for the augmented Dickey-Fuller tests implies that the characteristic roots of the Taylor expansion of the unconditional variance of ${\epsilon }_{\mathrm{t}}^2$ lie outside the unit circle for all currency returns. ^16/

The finding that the LM statistic for testing for ARCH effects is significant for all currency returns (except gold), suggests that investment in these currencies will result in increased estimated variances, and therefore increased risk, during periods of large unexpected shocks to returns, and diminished variances during periods of relative stability. This result indicates that it might be desirable to split the sample into periods of high and low variability of returns and examine whether ARCH effects persist for these currencies. The values for the parameter α₀ range from zero for U.S. dollar returns to 3.329 for gold and are all statistically significant. The parameter α_i (i=1, …, 5) represents the relative contribution of the respective previous period’s squared error of the return equation to the conditional variance in the current period. The parameter values of range from 0.106 for the average composite index to 0.538 for U.S. dollar returns and are all statistically significant at the 5 percent level. In the case of the U.S. dollar, this means that about 54 percent of a week’s disturbances is carried over into the following week. Among currency returns displaying one lag of ARCH effects, returns on the SDR demonstrate the lowest transmission of the weekly deviations of the SDR returns from their mean on the volatility of SDR returns, while the ECU returns had the highest. This suggests that shorter term investment periods are more likely to prove “successful” for the SDR, relative to the ECU and to the rest of the currencies, in terms of achieving a predicted risk/return tradeoff. The result for the ECU may be attributed to the fact that it contains currencies with larger volatility than those included in the SDR basket.

The constant term, ϕ₀, takes into account a possible non-stationary element (drift), like a time trend, in total returns. Under the null hypothesis, currency returns would allow an autoregressive structure with constant variance residuals. Under the alternative, an ARCH structure on the residuals is imposed. During the sample period, the constants in all currency returns, except that of gold, were positive and only gold returns were not significantly different from zero. These terms may be interpreted as mean weekly returns, thus the intercept for deutsche mark returns indicates that the annualized average weekly returns on deutsche mark investments were 10.28 percent. ^17/ Further, since ɸ₁ for all currency returns is less than one, it would seem that all returns are stable. Note that U.S. dollar returns exhibit a different pattern of generating process than other currency returns owing to the fact that returns on U.S. assets consist only of the respective interest rates. U.S. dollar returns at time t are found to be affected by a six-lag structure, although the coefficients of the fourth and fifth lags are not statistically significant. However, the coefficient of the sixth lag on the U.S. dollar-returns process is highly significant, indicating an apparent cyclicality of U.S. interest rates of similar periodicity. Since the sum of the six coefficients of the lag structure is 0.989, U.S. dollar returns appear to be also stable.

For most currency returns, the lagged squared errors provide a large contribution to the conditional variance, as indicated by the significant coefficients, which imply that currency returns are generated by processes with non-constant unconditional variance. Because the principal assumption of constant variances of the error terms in an OLS estimation is violated, the latter methodology should not be used as an estimation procedure for currency returns. Furthermore, the coefficient structure of the ARCH model for U.S. dollar returns is monotonically decreasing, indicating that a squared error from the previous week has a greater effect on current conditional variance than a squared error from two weeks ago.

Another important observation on the statistical properties of the stochastic process followed by SDR returns is that the conditional variance of the SDR returns displays an optimal lag length of one, as only the coefficient of the first period (squared) residual is statistically significant. Thus, an ARCH process of order one is indicated by the data. The estimated SDR returns have finite variance since the coefficient of the lagged residuals is less than one. The resulting ARCH effects for the SDR returns are further supported by their distributional properties of high leptokurtosis and the highly significant Bera-Jarque tests indicating a non-normal distribution for these SDR returns. These empirical findings indicate that the conditional variance of the SDR returns is not constant over time and that optimization of a portfolio which includes the SDR as an

V. Conclusions

This paper provides empirical evidence that would permit investors, including central banks, to make more accurate calculations of optimal strategies in the management of their reserves. The gains in accuracy depend directly on how the variance or risk measures on various currency investment returns change over time. To test whether the variability of returns is constant we employ the ARCH model, which was further utilized to provide estimates of the variability of the various currency investment returns.

On the basis of weekly data for investment returns on eight currencies and gold for the period February 1982 to December 1991, the empirical tests show that the variances of total returns on all currencies and gold change significantly over time. Evidence of positive and highly significant autocorrelation for weekly returns on those assets and their predictable level of changing variability is not consistent with the assumption of constant (time-invariant) risk that is often used in asset allocation models. The empirical results presented in this paper generally suggest that for a U.S. dollar-based investor, an ARCH model which takes into account the non-stationarity in return variability provides more accurate estimates of the variability of total returns on various currency-denominated investments than ordinary least-squared models.

In particular, the sign of the intercept in the ARCH specifications is non-negative for all currencies, indicating that the variability of asset returns would increase in the absence of any influences from previous period returns. A further interesting empirical result is that the volatility of returns from investing in the deutsche mark, the pound sterling, the French franc, the Japanese yen, ECU, SDR, and gold assets is affected by the square of one lag of their past residuals, while the variability of returns on U.S. dollar assets is affected by the square of two lags of their past residuals and that of a currency composite by the square of five lags of its past residuals. This result can be interpreted to mean that for individual currencies the effect of changes in variability on future variability lasts for only a short period and diminishes over time. The variability of returns on U.S. investments may be attributed to the fact that they comprise only interest rates on the respective assets. Interest rates, as opposed to exchange rates, tend to exhibit longer persistence, i.e., they are influenced by longer lags. For an average currency composite investment, however, the effect lasts for a significantly longer period.

The returns from investment in the SDR and ECU follow an ARCH process of order one and their probability distribution appears to be highly leptokurtic. These results are broadly similar to those for individual currencies. In particular, the coefficient of the past-period error introduced in the variance of SDR returns is smaller than in that of any other currency returns owing to a large extent to the built-in diversification of currencies in the SDR basket. In contrast, ECU returns exhibited the highest transmission of unexpected weekly shocks on their respective volatility during the sample period. The difference between the statistical results for the SDR and those for the ECU could be attributed to the fact that the ECU contains currencies of lesser importance and larger volatility in foreign exchange markets than those included in the SDR basket. Further study of the characteristics of the SDR and the ECU might shed some light on more fundamental comparisons between the performance of these two currency composites as a reserve asset. In addition, application of Monte-Carlo simulations with a traditional portfolio model and one using the ARCH specification may further quantify the deviations from optimal allocations and losses incurred in terms of risk-adjusted returns.