The Intertemporal Relation Between Expected Return and Risk on Currency

Abstract

The literature has so far focused on the risk-return trade-off in equity markets and ignored alternative risky assets. This paper examines the presence and significance of an intertemporal relation between expected return and risk in the foreign exchange market. The paper provides new evidence on the intertemporal capital asset pricing model by using high-frequency intraday data on currency and by presenting significant time variation in the risk aversion parameter. Five-minute returns on the spot exchange rates of the US dollar vis-à-vis six major currencies (the euro, Japanese yen, British pound sterling, Swiss franc, Australian dollar, and Canadian dollar) are used to test the existence and significance of a daily risk-return trade-off in the FX market based on the GARCH, realized, and range volatility estimators. The results indicate a positive but statistically weak relation between risk and return on currency.

Our empirical analysis relies on the maximum likelihood estimation of the GARCH-in-mean models as described in Appendix 1. We also use the seemingly unrelated (SUR) regressions and panel data estimation to investigate the significance of a time-series relation between expected return and risk on currency as described in Appendix 2.

Appendices

Appendix 1: Maximum Likelihood Estimation of GARCH-in-Mean Models

Modeling and estimating the volatility of financial time series has been high on the agenda of financial economists since the early 1980s. Engle (1982) put forward the Autoregressive Conditional Heteroskedastic (ARCH) class of models for conditional variances which proved to be extremely useful for analyzing financial return series. Since then an extensive literature has been developed for modeling the conditional distribution of stock prices, interest rates, exchange rates, and futures prices. Following the introduction of ARCH models by Engle (1982) and their generalization by Bollerslev (1986), there have been numerous refinements of this approach to estimating conditional volatility. Most of the refinements have been driven by empirical regularities in financial data.

Engle (1982) introduces ARCH(p) model:

$$ {R}_{t+1}\equiv \alpha +{\varepsilon}_{t+1} $$

(40.22)

$$ \begin{array}{l}\ {\varepsilon}_{t+1}={z}_{t+1}\cdot {\sigma}_{t+1\left|t\right.},{z}_{t+1}\sim N\left(0,1\right),\\ {}E\left({\varepsilon}_{t+1}\right)=0\end{array} $$

(40.23)

$$ E\left({\varepsilon}_{t+1}^2\left|{\Omega}_t\right.\right)={\sigma}_{t+1\left|t\right.}^2={\gamma}_0+{\gamma}_1{\varepsilon}_t^2+{\gamma}_2{\varepsilon}_{t-1}^2+\dots +{\gamma}_p{\varepsilon}_{t-p}^2 $$

(40.24)

$$ f\left({R}_{t+1};\mu, {\sigma}_{t+1\left|t\right.}\right)=\frac{1}{\sqrt{2\pi {\sigma}_{t+1\left|t\right.}^2}} \exp \left[-\frac{1}{2}{\left(\frac{R_{t+1}-\mu }{\sigma_{t+1|t}}\right)}^2\right] $$

(40.25)

where R _t+1 is the daily return for period t+1, μ = α is the constant conditional mean, ε _t+1 = z _t+1 ⋅ σ _t+1|t is the error term with E(ε _{t +1}) = 0, σ _t+1|t is the conditional standard deviation of daily returns, and z _t+1 ∼ N(0,1) is a random variable drawn from the standard normal density and can be viewed as information shocks or unexpected news in the market. σ _t+1|t ² is the conditional variance of daily returns based on the information set up to time t denoted by Ω _t . The conditional variance, σ _t+1|t ², follows an ARCH(p) process which is a function of the last period’s unexpected news (or information shocks). f(R _t+1;μ,σ _t+1|t ) is the conditional normal density function of R _t+1 with the conditional mean of μ and conditional variance of σ _t+1|t ².

Given the initial values of ε _t and, the parameters in Eqs 40.22 and 40.24 can be estimated by maximizing the log-likelihood function over the sample period. The conditional normal density in Eq. 40.25 yields the following log-likelihood function:

$$ \mathrm{Log}{L}_{ARCH}=-\frac{n}{2} \ln \left(2\pi \right)-\frac{n}{2} \ln {\sigma}_{t+1|t}-\frac{1}{2}{\displaystyle \sum_{t=1}^n{\left(\frac{R_{t+1}-\mu }{\sigma_{t+1|t}}\right)}^2} $$

(40.26)

Bollerslev (1986) extends the original work of Engle (1982) and defines the current conditional variance as a function of the last period’s unexpected news as well as the last period’s conditional volatility:

$$ {R}_{t+1}\equiv \alpha +{\varepsilon}_{t+1} $$

(40.27)

$$ {\varepsilon}_{t+1}={z}_{t+1}\cdot {\sigma}_{t+1|t},{z}_{t+1}\sim N\left(0,1\right)E\left({\varepsilon}_{t+1}\right)=0 $$

(40.28)

$$ E\left({\varepsilon}_{t+1}^2\left|{\Omega}_t\right.\right)={\sigma}_{t+1|t}^2={\gamma}_0+{\gamma}_1{\varepsilon}_t^2+{\gamma}_2{\sigma}_t^2 $$

(40.29)

where the conditional variance, σ _t+1|t ², in Eq. 40.29 follows a GARCH(1,1) process as defined by Bollerslev (1986) to be a function of the last period’s unexpected news (or information shocks), z _t , and the last period’s variance, σ ² _t . The parameters in Eqs. 40.27, 40.28, and 40.29 are estimated by maximizing the conditional log-likelihood function in Eq. 40.26.

Engle et al. (1987) introduce the ARCH-in-mean model in which the conditional mean of financial time series is defined as a function of the conditional variance. In our empirical investigation of the ICAPM for exchange rates, we use the following GARCH-in-mean process to model the intertemporal relation between expected return and risk on currency

$$ {R}_{t+1}\equiv \alpha +\beta \cdot {\sigma}_{t+1|t}^2+{\varepsilon}_{t+1} $$

(40.30)

$$ \begin{array}{l}\ {\varepsilon}_{t+1}={z}_{t+1}\cdot {\sigma}_{t+1|t};{z}_{t+1}\sim N\left(0,1\right);\\ {}E\left({\varepsilon}_{t+1}\right)=0\end{array} $$

(40.31)

$$ E\left({\varepsilon}_{t+1}^2\left|{\Omega}_t\right.\right)={\sigma}_{t+1|t}^2={\gamma}_0+{\gamma}_1{\varepsilon}_t^2+{\gamma}_2{\sigma}_t^2 $$

(40.32)

$$ f\left({R}_{t+1};{\mu}_{t+1|t},{\sigma}_{t+1|t}\right)=\frac{1}{\sqrt{2\pi {\sigma}_{t+1|t}^2}} \exp \left[-\frac{1}{2}{\left(\frac{R_{t+1}-{\mu}_{t+1|t}}{\sigma_{t+1|t}}\right)}^2\right] $$

(40.33)

where R _t+1 is the daily return on exchange rates for period t+1, μ _t+1|t ≡ α + β ⋅ σ ² _t+1|t is the conditional mean for period t+1 based on the information set up to time t, ε _t+1 = z _t+1 ⋅ σ _t+1|t is the error term with E(ε _t+1) = 0, σ _t+1|t is the conditional standard deviation of daily returns on currency, and z _t+1 ∼ N(0,1) is a random variable drawn from the standard normal density and can be viewed as information shocks in the FX market. σ ² _t+1|t is the conditional variance of daily returns based on the information set up to time t denoted by Ω _t . The conditional variance, σ ² _t+1|t , follows a GARCH(1,1) process as defined by Bollerslev (1986) to be a function of the last period’s unexpected news (or information shocks), z _t , and the last period’s variance, σ _t ². f(R _t+1;μ _t+1|t ,σ _t+1|t ) is the conditional normal density function of R _t+1 with the conditional mean of μ _t+1|t and conditional variance of σ ² _t+1|t .

Given the initial values of ε _t and, the parameters in Eqs. 40.30 and 40.32 can be estimated by maximizing the log-likelihood function over the sample period. The conditional normal density in Eq. 40.33 yields the following log-likelihood function

$$ \begin{array}{c}\mathrm{Log}{L}_{ARCH}=-\frac{n}{2} \ln \left(2\pi \right)-\frac{n}{2} \ln {\sigma}_{t+1|t}\\ {}-\frac{1}{2}{\displaystyle \sum_{t=1}^n{\left(\frac{R_{t+1}-{\mu}_{t+1|t}}{\sigma_{t+1|t}}\right)}^2}\end{array} $$

(40.34)

where the conditional mean μ _t+1|t ≡ α + β ⋅ σ ² _t+1|t has two parameters and the conditional variance σ _t+1|t ² = γ ₀ + γ ₁ε _t ² + γ ₂ σ ² _t has three parameters. Maximizing the log-likelihood in Eq. 40.34 yields the parameter estimates (α, β, γ ₀, γ ₁, γ ₂).

The interested reader may wish to consult Enders (2009), Chap. 3, and Tsay (2010), Chap. 3, for comprehensive analysis of ARCH/GARCH models and their maximum likelihood estimation. Chapter 3 in Enders (2009) provides a detailed coverage of the basic ARCH and GARCH models, as well as the GARCH-in-mean processes and multivariate GARCH in some detail. Chapter 3 in Tsay (2010) provides a detailed coverage of the ARCH, GARCH, GARCH-M, the exponential GARCH, and Threshold GARCH models.

Appendix 2: Estimation of a System of Regression Equations

Consider a system of n equations, of which the typical ith equation is

$$ {y}_i={X}_i{\beta}_i+{u}_i $$

(40.35)

where y _i is a N × 1 vector of time-series observations on the i th dependent variable, X _i is a N × k _i matrix of observations of k _i independent variables, β _i is a k _i × 1 vector of unknown coefficients to be estimated, and u _i is a N × 1 vector of random disturbance terms with mean zero. Parks (1967) proposes an estimation procedure that allows the error term to be both serially and cross-sectionally correlated. In particular, he assumes that the elements of the disturbance vector u follow an AR(1) process

$$ {u}_{it}={\rho}_i{u}_{it-1}+{\varepsilon}_{it};{\rho}_i<1 $$

(40.36)

where ɛ _it is serially independently but contemporaneously correlated:

$$ \mathrm{Cov}\left({\varepsilon}_{it}{\varepsilon}_{jt}\right)={\sigma}_{ij},\forall i,j,\mathrm{and}\ \mathrm{Cov}\left({\varepsilon}_{it}{\varepsilon}_{js}\right)=0,\mathrm{for}\ s\ne t $$

(40.37)

Equation 40.35 can then be written as

$$ {y}_i={X}_i{\beta}_i+{P}_i{\varepsilon}_i, $$

(40.38)

with

$$ {P}_i=\left[\begin{array}{ccccc}\hfill {\left(1-{\rho}_i^2\right)}^{-1/2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \dots \hfill & \hfill 0\hfill \\ {}\hfill {\rho}_i{\left(1-{\rho}_i^2\right)}^{-1/2}\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill \dots \hfill & \hfill 0\hfill \\ {}\hfill {\rho}_i^2{\left(1-{\rho}_i^2\right)}^{-1/2}\hfill & \hfill {\rho}_i\hfill & \hfill 0\hfill & \hfill \dots \hfill & \hfill 0\hfill \\ {}\hfill .\hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill .\hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill .\hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill {\rho}_i^{N-1}{\left(1-{\rho}_i^2\right)}^{-1/2}\hfill & \hfill {\rho}_i^{N-2}\hfill & \hfill {\rho}_i^{N-3}\hfill & \hfill \dots \hfill & \hfill 1\hfill \end{array}\right] $$

(40.39)

Under this setup, Parks presents a consistent and asymptotically efficient three-step estimation technique for the regression coefficients. The first step uses single equation regressions to estimate the parameters of autoregressive model. The second step uses single equation regressions on transformed equations to estimate the contemporaneous covariances. Finally, the Aitken estimator is formed using the estimated covariance,

$$ \widehat{\beta}={\left({X}^T{\Omega}^{-1}X\right)}^{-1}{X}^T{\Omega}^{-1}y $$

(40.40)

Where Ω ≡ E[uu ^T] denotes the general covariance matrix of the innovation. In my application, I use the aforementioned methodology with the slope coefficients restricted to be the same for all portfolios. In particular, we use the same three-step procedure and the same covariance assumptions as in Eqs. 40.36, 40.37, 40.38, 40.39, and 40.40 to estimate the covariances and to generate the t-statistics for the parameter estimates.

The interested reader may wish to consult Wooldridge (2010), Chaps. 10.4, 10.5, and 10.6 for recent developments on panel data estimation. Chapter 10 in Wooldridge (2010) presents Basic Linear Unobserved Effects Panel Data Models, Chap. 10.4 provides Random Effects Methods, Chap. 10.5 contains Fixed Effects Methods, and Chap. 10.6 First Differencing Methods. Bali (2008) and Bali and Engle (2010) follow SUR estimation to investigate the empirical validity of the conditional intertemporal capital asset pricing models (ICAPM).