A value-at-risk analysis of carry trades using skew-GARCH models
-
Yu-Jen Wang
,
Huimin Chung
Abstract
We carry out a value-at-risk (VaR) analysis of an extremely popular strategy in the currency markets, namely, “carry trades,” whereby a position purchased in high interest rate currencies is funded by selling low interest rate currencies. Since the natural outcome of the truncated normal distribution of interest-rate spreads combined with the normal distribution of exchange rate returns is a skew-normal distribution, we consider a skew-normal innovation with zero mean for our analysis of carry trade returns using generalized autoregressive conditional heteroskedasticity (GARCH) models. The stress testing results reveal that skew-normal or densities are suitable for the measurement of VaR for carry trade returns involving, for example, taking up a long position in Australian Dollars or Argentine Peso which are funded by selling Japanese Yen.
- 1
One of the most important events, as reported by Melvin and Taylor (2009), was the carry trade unwind occurring on 16 August 2007; the 1-day change in the Japanese Yen (JPY) price of the Australian Dollar on that date was –7.7%.
- 2
A comprehensive introduction to the subject is provided by several of the early review articles presented by Duffie and Pan (1997), along with the books written by Dowd (1999) and Jorion (2000).
- 3
See Azzalini and Dalla (1996), Azzalini and Capitaino (1999, 2003) and Jones and Faddy (2003).
- 4
See, for example, Fernandez and Steel (1998).
- 5
The parameter estimates for the ML estimates were introduced by Lin (2009, 2010), along with certain numerical techniques, such as the EM-type algorithm.
- 6
The generalized error distribution (GED) was first proposed by Nelson (1991), while the normal inverse Gaussian (NIG) distribution was subsequently proposed by Barndorff-Nielsen (1997); the tails of the distribution decrease more slowly than those of the normal distribution.
- 7
The actual losses that exceed VaR are usually referred to as “exceptions” or “exposed.”
- 8
Because the tails of the normal and skew-normal densities are more slender than the t and skew-t densities, respectively, the parameter estimates are obtained using the normal and skew-normal models which are highly affected by a single outlier, whereas the influence for the t and skew-t models is limited within a short range.
- 9
We observe in Figure 2 that the left tail of the t distribution is fatter than that of the skew-t distribution when the skewness parameter is positive.
Appendix
EM algorithm for the skew-normal ARMA(1,1)-GARCH(1,1) model
There are some similarities between the EM algorithms for the skew-normal and skew-t distribution introduced by Lin (2009, 2010). If carry trade returns follow the skew-normal distribution, then zt~SN(μt, σt, λ). By Equation (1), zt has the following hierarchical formulation:
where μt= φ0+ φ1zt–1+ψ1εt–1 and 
Hence, the conditional complete-data log-likelihood, ignoring additive constant terms, is expressed as follows:
where Θ=( φ0, φ1, ψ1, ω, α, β, λ).
Furthermore, the posterior distribution of τ1 is:
The Q function is:
where Z=z1, . . . , zn.
The EM algorithm is as follows:
E-step:
Given 
we compute:
for t=2, . . . , n.
M-step:
Maximize the numerical 
with respect to Θ and obtain updated estimates of the parameters, denoted by 
Repeat the E-step and M-step until the parameters converge. The iteration is stopped if 
EM algorithm for the skew-t ARMA(1,1)-GARCH(1,1) model
The carry trade returns are said to follow a similar skew-t distribution, which uses the notation zt~ST(μ, σt, λ, v). Equation (2) leads to the following hierarchical formulation:
Letting Ξ=(μ,ω, α, β,λ,v) and Z=z1, . . . , zn, the conditional complete-data log-likelihood, ignoring additive constant terms, is expressed as:
giving Z=Z; the Q function is then expressed as:
where
With
and DG(x)=Г′(x)/Г(x) is the di-gamma function.
E-step:
Compute the expectations 
and 
M-step:
Maximize the numerical 
with respect to Ξ and obtain updated estimates of the parameters, denoted by 
Repeat E-step and M-step until the parameters converge.
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©2013 by Walter de Gruyter Berlin Boston
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- Maximum likelihood estimation of continuous time stochastic volatility models with partially observed GARCH
- A value-at-risk analysis of carry trades using skew-GARCH models
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Articles in the same Issue
- Masthead
- Masthead
- Bayesian adaptively updated Hamiltonian Monte Carlo with an application to high-dimensional BEKK GARCH models
- Off-the-record target zones: theory with an application to Hong Kong’s currency board
- Nonlinear and nonparametric modeling approaches for probabilistic forecasting of the US gross national product
- Maximum likelihood estimation of continuous time stochastic volatility models with partially observed GARCH
- A value-at-risk analysis of carry trades using skew-GARCH models
- Income taxes and endogenous fluctuations: a generalization
