ESTIMATING THE HEDGE RATIOS

This paper examines the problem of hedging portfolio returns. Many practitioners and academicians endeavor to solve the problem of how to calculate the optimal hedge ratio accurately. In this paper we compare estimates of the hedge ratio from a classical approach of a linear quantile regression, based on selected quantiles as medians, with that of a non-linear quantile regression. To estimate the hedge ratios, we have used a calibrated Student t distribution for the marginal densities and a Student t copula of the portfolio returns using a maximum likelihood estimation. We created two portfolios of the assets, one for equal weight and another for optimal weight in respect of minimal risk. Our findings show that an assumption of Student t marginal leads to a better estimation of the hedge ratio. JEL Classification Numbers: C21, C40, G11 DOI: http://dx.doi.org/10.12955/cbup.v4.874


Introduction
Contemporary financial markets are characterized by their instability due to many crisis periods. Academic literature is a rich source of trading strategies that are oriented to eliminate the risk of losses. Hedging is one of them. Hedging strategies have also been examined using models that incorporate tail dependence structure between the investment of interest and the hedging instrument (Lien, Shrestha, & Wu, 2016). Conventional method for the optimal hedge ratios estimate uses the regression method. This approach is mainly based on the expected relation between investment of interest and the hedging instrument returns, which neglect the relationships between these two returns at different quantiles. This paper uses the concept of a quantile hedge ratio. We use a quantile regression method in estimating these quantile hedge ratios using copula. The quantile hedge ratio allows investment of interest and the hedging instrument returns to have different relationships at different quantiles. Consequently, a higher investment return may require a different hedge ratio from a lower investment return (Brooks, Olan, & Persand, 2002;Alexander, 2008). When the relationship between investment and hedging instrument returns does vary across quantiles, the quantile hedge ratio can provide better hedging efficiency than the hedge ratio obtained by OLS.
The purpose of this paper is to determine the hedge ratios of two portfolios -Equally Weighted (EW) and Minimum Variance (MV) portfolio when the investor decides to reduce his risk. For this purpose 1. we calibrate both portfolios and the hedge instrument by an appropriate distribution using historical prices time series, 2. we compare the symmetric and asymmetric hedge ratio: • using simple linear regression based on OLS • using linear quantile regression • using non-linear quantile regressions based on Normal and Student t copula.
The paper is organized as follows. The next section presents the methodology used in estimating the optimal hedge ratios. Section 3 describes the analyzed data and section 4 discusses empirical results. The conclusion concludes the paper.

Methodology
Let St and Ft represent investment and hedged instrument prices at time t. Then, returns on investment and hedged positions can be defined as Δst = ln(St) − ln(St−1) and Δft = ln(Ft) − ln(Ft−1) respectively. The return on the hedged portfolio RHt is given by (Lien et al., 2016): where H is the so-called hedge ratio.
The MV hedge ratio HMV is obtained by minimizing the variance of RH with respect to H and is given by (Lien et al., 2016): In the conventional approach, MV hedge ratio is represented as the coefficient in a regression of investment returns on hedging returns: where the estimate of the MV hedge ratio, HMV, is given by the estimate of the slope coefficient β.
The OLS method has been frequently criticized because this technique considers only symmetric linear relationship between variables and cannot provide a distinction between dependence during up and down markets or between large and small stock price movements (Aymen & Mongi, 2016). Quantile regression technique (developed by Koenker & Bassett in 1978) is an extension of the traditional least squares estimation of the conditional mean to a compilation of models for different conditional quantile functions. Classical linear regression model that describes the dependence of Y on X assumes the variables are bivariate normal. If X and Y do not have a bivariate normal distribution, then it is needed to describe the conditional distribution F(Y|X). Quantile regression gives tools to describe the conditional distribution of the dependent variable using its quantiles (Alexander, 2008). Quantile regression expresses the conditional quantiles of Y given X based on an arbitrary joint distribution while it is assumed the errors of the quantile regression are i.i.d with specific error distribution function Fε.
Let q be a quantile of the error determined by and simultaneously q denote the conditional quantile of the dependent variable Y, which is found from the inverse of F(Y|X), by F −1 (q|X). Now, we take conditional quantile q of a simple linear regression model and we get a simple linear quantile regression model (Alexander, 2008): The aim of the simple quantile regression is to estimate the regression line parameters α and β based on a paired sample of X and Y, while quantile regression line passes through a quantile of the points. Quantile regression lines for different values of q are not parallel. Dependence of Y based on explanatory variable X could be constant when the values of (α, β) are not changed for different values of q, or monotonically increasing (decreasing) when (α, β) increases (decreases) with the value of q and symmetric (asymmetric) when the value of q is similar (dissimilar) for lower and upper quantiles (Aymen & Mongi, 2016). The coefficients (α, β) for a given q are estimated by minimizing the weighted sum of the absolute errors as , where (7) The solution    Copula quantile approach replaces the linear model in (6) by the q quantile curve of a copula. (Bouyé & Salomon, 2002). This enables us to model nonlinear relationships (Alexander, 2008).
Let the marginal distributions of X and Y be F(X) and G(Y). They are specified by their marginal parameters α= {αX, αY}, which we have already estimated using maximum likelihood method. Let F(X, αX) and G(Y, αY). We then specify some functional form for a bivariate copula, which depends on certain parameters θ (the normal bivariate copula has one parameter, the correlation coefficient ρ and the bivariate Student t copula has two parameters, the degrees of freedom ν and the correlation ρ).
Then the q quantile regression curve is   , ; , The explicit function Y = Yt of the normal copula quantile curves can be written as (Alexander, 2008): and Student t copula quantile curves has form

Data
This paper analyses the risk of investing in a portfolio created from these three stocks: Bayer (BAYN), Siemens (SIE) and Volkswagen Group (VOW). We have chosen the DAX index as the hedged instrument.
We have estimated the optimal hedge ratio for the risk of our portfolios. We have created two portfoliosequally weighted (EW) and mean-variance (MV) optimal portfolio. MV portfolio is composed from 24.153% of BAYN, 69.269% SIE and only 6.578% VOW.
Closing spot prices for both portfolios and the DAX index (Pt measured in time t) were rebased to 100 and the fluctuation (measured in terms of logarithmic return) was computed. The analyzed period was from September, 27, 2010 to June, 28, 2016, which comprised 1,500 trading days, or T=1,499 logarithmic returns (log returns are obtained by formula: rt = ln Pt /ln Pt-1, t = 1, …, T, excluding account dividends). A global financial portal provided the data (Fusion Media Ltd, n. d.; Finance Yahoo, n. d.). Our sample period is particularly relevant with several major changes occurring during that time. Volkswagen emission problem has caused a decrease in the value of the portfolio (see Figure 1). The time series of the closing prices and the effects of the Volkswagen emission scandal which erupted on 18 September 2015 are shown (Figure 1). Figure 1 shows the variation of the logarithmic returns in both portfolios and the DAX index. Figure 2 shows the distributions and quantile plots of the returns for EW and MV portfolios in comparison to the DAX respectively. Both portfolios and the DAX index returns do not have normal distribution (Figure 2). The distributions have fat tails, so we have used Student t distribution. Model parameters for Student t distribution were estimated using a maximum likelihood estimation (MLE) method and calibrated in Wolfram Mathematica software using our own procedures (see Table 1). Table 1 displays summary statistics for daily returns for both portfolios and DAX returns. Table 2 reports the results for Student t copulas parameters, as well as the corresponding AIC and BIC values.

Results and Discussion
Tables 3-5 compare the optimal hedge ratio computed by the classical approach (OLS), linear quantile regression approach (QR), normal copula quantile regression and Student t copula quantile regression based on 0.1, 0.5 and 0.9 quantile. In both cases, copulas marginal have Student t distribution with parameters as described in Table 1.   Our findings suggest that Student t copula quantile regression approach gives the smallest estimations of the hedge ratios for 3 out of 4 cases, while in the fourth case the estimation is very close to the lowest value. Small values of estimations mean we estimate the risks more precisely.

Conclusion
This paper shows how we can solve the hedge problem using conditional dependence between portfolio returns at a given quantile and the hedging instrument. As a reasonable example, we have analyzed the dependency for quantiles 0.1, 0.5 and 0.9. It follows from our analysis that we can use non-linear dependency (described by copula function) with returns that are not bivariate normal to obtain more precise results.