Market Segmentation and Pricing of Closed-End Country Funds: An Empirical Analysis

Abstract

This paper finds that for closed-end country funds, the international CAPM can be rejected for the underlying securities (NAVs) but not for the share prices. This finding indicates that country fund share prices are determined globally where as the NAVs reflect both global and local prices of risk. Cross-sectional variations in the discounts or premiums for country funds are explained by the differences in the risk exposures of the share prices and the NAVs. Finally, this paper shows that the share price and NAV returns exhibit predictable variation and country fund premiums vary over time due to time-varying risk premiums. The paper employs generalized method of moments (GMM) to estimate stochastic discount factors and examines if the price of risk of closed-end country fund shares and NAVs is identical. GMM is an econometric method that was a generalization of the method of moments developed by Hansen (Econometrica 50, 1029–1054, 1982). Essentially GMM finds the values of the parameters so that the sample moment conditions are satisfied as closely as possible.

The views expressed in this paper are strictly that of the author and not of the OCC or the US department of Treasury.

Appendix 1: Generalized Method of Moments (GMM)

GMM is an econometric method that was a generalization of the method of moments developed by Hansen (1982). The moment conditions are derived from the model. Suppose Y_t is a multivariate independently and identically distributed (i.i.d) random variable. The econometric model specifies the relationship between Z_t and the true parameters of the model (θ₀). To use GMM there must exist a function g(Z_t, θ₀) so that

$$ \mathrm{m}\left({\uptheta}_0\right)\equiv \mathrm{E}\left[\mathrm{g}\left({\mathrm{Z}}_{\mathrm{t}},{\uptheta}_0\right)\right]=0 $$

(25.19)

In GMM, the theoretical expectations are replaced by sample analogs:

$$ \mathrm{f}\left(\uptheta, {\mathrm{Z}}_{\mathrm{t}}\right)=1/\mathrm{T}\sum \mathrm{g}\left({\mathrm{Z}}_{\mathrm{t}},\uptheta \right). $$

(25.20)

The law of large numbers ensures that the RHS of above equation is the same as

$$ \mathrm{E}\left[\mathrm{f}\left({\mathrm{Z}}_{\mathrm{t}},{\uptheta}_0\right)\right]. $$

(25.21)

The sample GMM estimator of the parameters may be written as (see Hansen 1982)

$$ \Theta = \arg\ \min\ {\left[1/\mathrm{T}\sum \mathrm{g}\left({\mathrm{Z}}_{\mathrm{t}},\uptheta \right)\right]}^{\prime}\left.{\mathrm{W}}_{\mathrm{T}}1/\mathrm{T}\sum \mathrm{g}\left({\mathrm{Z}}_{\mathrm{t}},\uptheta \right)\right] $$

(25.22)

So essentially GMM finds the values of the parameters so that the sample moment conditions are satisfied as closely as possible. In our case for the regression model,

$$ {\mathrm{y}}_{\mathrm{t}}={\mathrm{X}}_{\mathrm{t}}\prime \upbeta +{\upvarepsilon}_{\mathrm{t}} $$

(25.23)

The moment conditions include

$$ \mathrm{E}\left[\left({\mathrm{y}}_{\mathrm{t}}-{\mathrm{X}}_{\mathrm{t}}\prime \upbeta \right){\mathrm{x}}_{\mathrm{t}}\right]=\mathrm{E}\left[{\upvarepsilon}_{\mathrm{t}}{\mathrm{x}}_{\mathrm{t}}\right]=0\ \mathrm{for}\ \mathrm{all}\ \mathrm{t} $$

(25.24)

So the sample moment condition is

$$ 1/\mathrm{T}\sum \left({\mathrm{y}}_{\mathrm{t}}-{\mathrm{X}}_{\mathrm{t}}\prime \upbeta \right){\mathrm{x}}_{\mathrm{t}} $$

and we want to select β so that this is as close to zero as possible. If we select β as (X′X)⁻¹(X′y), which is the OLS estimator, the moment condition is exactly satisfied. Thus, the GMM estimator reduces to the OLS estimator and this is what we estimate. For our case the instruments used are the same as the independent variables. If, however, there are more moment conditions than the parameters, the GMM estimator above weighs them. These are discussed in detail in Greene (2008, Chap. 15). The GMM estimator has the asymptotic variance

$$ {\left(\mathrm{X}\prime \mathrm{Z}\kern0.24em {\left(\mathrm{Z}\prime \Omega Z\right)}^{-1}\mathrm{Z}\prime \mathrm{X}\right)}^{-1} $$

(25.25)

The White robust covariance matrix may be used for Ω as discussed in appendix C when heteroskedasticity is present. Using this approach, we estimate GMM with White heteroskedasticity consistent t-stats.