Hedging and Risk Aversion on Russian Stock Market: Strategies Based on MGARCH and MSV Models Valeriya Lakshina[0000−0003−1447−9377] National Research University Higher School of Economics, Nizhny Novgorod, Russia vlakshina@hse.ru Abstract. The paper studies the problem of dynamic hedge ratio calcu- lation for the portfolio consisted of two assets – futures and the underly- ing stock. We apply the utility based approach to account for the degree of risk aversion in the hedging strategy. Seventeen portfolios, consisted of Russian blue-chip stocks and futures, are estimated in the paper. In order to estimate the conditional covariances of hedged portfolio returns, such multivariate volatility models as GO-GARCH, copula-GARCH, asym- metric DCC and parsimonious stochastic volatility model are applied. The hedging efficiency is estimated on the out-of-sample period using the maximum attainable risk reduction, the financial result and the in- vestor’s utility. It’s shown that for 60% of portfolios ADCC surpasses the other models in hedging. Including the degree of risk aversion in the in- vestor’s utility function together with above-mentioned volatility models allows to reach hedging efficiency of 88%. Keywords: dynamic hedge ratio, stock futures, multivariate volatility models, risk aversion, hedging efficiency, copula, expected utility 1 Introduction Hedging is one of the most common tasks in finance. It requires knowing both hedging and hedged asset returns’ distribution. In other words, one should be aware of the multivariate distribution of returns, or, at least, the first two mo- ments of this distribution. Our paper is focused on modeling the second moment because the variance-covariance matrix is needed to solve the aforementioned financial task. There are two main approaches to modeling volatility: generalized autoregres- sive conditional heteroskedasticity or GARCH (a survey on multivariate GARCH models see in [2]) and models of stochastic volatility or SV (a review of multivari- ate SV models see in [1]). The latter take into account the volatility uncertainty directly by including the random term in the volatility equation. This assump- tion seems to be closer to the empirical evidence. But estimation of multivariate SV poses a challenge both due to dimensionality problem and the lack of closed- form likelihood function in general case.1 It’s worth mentioning that the first problem is also acute for GARCH models and implies that the number of pa- rameters grows quadratically (sometimes even faster as in VECH model, [8]) related to the number of assets in the portfolio. The second issue, arising from the use of SV for building the hedging strategy, is estimation. In contrast to GARCH, SV contains two sources of uncertainty and in most cases it is not possible to derive the likelihood function analyti- cally. However, there are a number of ways to estimate multivariate SV by the maximum likelihood method. For example the likelihood function can be ap- proximated by a Gaussian density (see, e.g., [19]), or simulated (see, e.g., [4, 13]). In this paper we attempt to propose a multivariate SV model in which both the aforementioned problems are remedied and apply it to stocks hedging. The model suggests that the demeaned returns follow Student’s t-distribution, whereas the volatility matrix is also random. This property follows from the fact that Student’s t-distribution can be represented as a mixture of normal distribu- tions. As a matter of fact, if the demeaned returns are distributed normally con- ditionally on volatility matrix and the volatility matrix itself has inverse Wishart distribution, then, according to [5], the volatility matrix can be marginalized out from the returns’ distribution, which results in Student’s t-distribution for the returns. Consequently, the demeaned returns distribution is known in contrast to the majority of other multivariate SV models. The paper contains the estimation of the described model via Markov chain Monte Carlo algorithm implemented in Stan software [18]. The parameters are obtained for seventeen stocks listed on Moscow Exchange futures market [14]. The sample covers the period from January 2006 to December 2016. All the positions are hedged with futures and the dynamic hedging coefficients are cal- culated using multivariate SV as well as several multivariate GARCH models. The resulted hedging strategies are compared via different criteria of hedging efficiency. The rest of the article is organized as follows. Section 2 describes the method- ology. Section 3 contains the estimation results and their discussion. Section 4 concludes. 2 Methodology 2.1 Hedge Ratio The aim of hedging is the reduction of portfolio value fluctuations. This can be achieved by opening the opposite position on the hedging instrument, usually a futures. The main task of building the hedging strategy is finding the optimal 1 The likelihood function for multivariate SV could be obtained under certain condi- tions, see [11]. amount of futures in the portfolio, i. e. calculation of optimal hedge ratio, which shows the relation of the hedged asset value to the hedging asset value. The return of the hedged position at time t is denoted by rt and equals to (1). rt = rS,t − hrt∗ · rF,t , (1) where rS,t – stock returns at time t; rF,t – futures returns at time t; hrt∗ – optimal hedge ratio at time t. We use the utility approach to implement the investor’s attitude to risk in building the hedging strategy. We obtain the optimal hedge ratio from maxi- mization of the investor’s expected utility E U (rt ), (2). V (rt ) E U (rt ) = E(rt ) − τ , (2) 2 where τ – positive parameter for risk aversion (large values indicate that investor dislikes risk), E(rt ) – expected portfolio returns, V(rt ) – portfolio variance. The optimal hedge coefficient hrt∗ is defined in (3). cov(rS,t , rF,t ) E(rF,t ) hrt∗ = − , (3) V(rF,t ) 2τ V(rF,t ) where cov(·) – covariance. Evidently, if τ → inf, then hrt∗ coincides with tradi- tional optimal hedge ratio, based on minimization of portfolio variance. 2.2 Multivariate Volatility Historically the first methods of hedge ratio calculation assumed that the ratio is constant. Further hedging strategies based on the dynamic hedge ratio appeared. They allow to implement heteroskedasticity of returns in the model. In this study, four volatility models have been taken to estimate the dynamic hedge ratio. There are two main approaches to the covariance matrix modeling: GARCH models and stochastic volatility models (MSV). Since the empirical evidence shows that volatility is volatile itself, it seems more appropriate to use SV, while modeling volatilty. SV has two sources of uncertainty – in mean and volatility equations. This fact results in challenging estimation procedure of SV, because it’s impossible to derive the likelihood function analytically in general case. Thus SV models are usually estimated within the Bayesian framework. Let xt , xt = (x1t , x2t , . . . , xnt )| be a portfolio consisted of n assets at time moment t. xt is represented as a sum of its mathematical expectation E (xt |Ft ), conditional on all available at t − 1 information, and innovations yt , (4). xt = E (xt |Ft−1 ) + yt , t = 1, . . . , T, xt − (n × 1)-vector, (4) Innovations yt conditional on volatility Σt are distributed normally with zero mean, (5). yt |Σt ∼ N (0, Σt ) (5) At the same time, Σt is a random process itself, generated by inversed Wishart distribution, (6). Σt ∼ IW (ν, Ht ) (6) Using properties of compound distributions, we obtain (7). yt ∼ t (ν, 0, Ht ) , (7) where tν (0, Ht ) – multivariate Student’s t-distribution with ν degrees of freedom and variance Ht , [5]. It’s worth mentioning that, since the distribution of yt is known in the model, the conjugate priors for the parameters could be derived. For univariate case con- jugate priors for parameters with fixed ν are derived in [21], where the conjugate prior for volatility is Fisher distribution with n − k (k – number of parameters) and ν degrees of freedom. According to [6], multivariate beta distribution is the generalization of Fisher distribution, which is a prior for Ht in (6). We also use several multivariate GARCH models, namely general orthogo- nalized GARCH (GO-GARCH), copula-GARCH and asymmetric dynamic con- ditional correlations (ADCC). The initial setup is analogous to MSV model, (see (4), (5)), except that the conditional distribution of innovations could be different. For the GO-GARCH model (8) holds. ΣtGO−GARCH = XVt X | , Vt = diag(vt ), (8a) vt = C + A(yt yt ) + Bvt−1 , (8b) where X is the matrix whose parametrization is based on the singular decomposi- tion of the unconditional variance of returns (for details, see [20]), Vt – a diagonal matrix, which nonzero elements are portfolio assets volatilities, given by any one- dimensional GARCH model. For example, in (8b) A, B are diagonal matrices of parameters, C is a n × 1 parameter vector, is an element-wise multiplication. As a result, each row of vt represents a standard univariate GARCH. ADCC volatility is modeled as in (9). ΣtADCC = Dt Rt Dt , (9a) Dt = diag(dt ), dt dt = vt (9b)     −1/2 −1/2 −1/2 −1/2 Rt = diag q11,t . . . qnn,t Qt diag q11,t . . . qnn,t , (9c) | | Qt = (1 − α − β)Q̄ + αyt−1 yt−1 + βQt−1 + γe yt−1 yet−1 , (9d) where Rt is conditional correlation matrix of returns, α, β, γ – parameters and γ is responsible for asymmetry effects in volatility, yet−1 are the zero-threshold innovations which are equal to yt when less than zero and are equal to zero otherwise. More details are in [3]. Copula-GARCH model is similar to (9), but differs by the fact that joint distribution of returns are modeled via Student’s t copula function, see [12, 16]. To sum up, our hedging strategies are based on four multivariate volatility models, three GARCH and one SV. We obtain the dynamic optimal hedge ratio from conditional covariance matrices, estimated by these models. Utility ap- proach allows to account for investor’s risk aversion, while building the hedging strategy. 3 Empirical Results 3.1 Data Description For our empirical study we take 17 companies, listed on Moscow Exchange [14], which stocks are also traded on the futures market. The companies included in the sample are presented in Table 1. Table 1. Companies under consideration Ticker Company name Ticker Company name chmf Severstal rosn Rosneft Oil Company fees Federal Grid Company rtkm Rostelecom gazp Gazprom sber Sberbank of Russia gmkn Norilsk Nickel sngs Surgutneftegas hydr RusHydro tatn Tatneft lkoh Lukoil trnf Transneft mgnt Magnit urka Uralkali nlmk Novolipetsk Steel vtbr Bank VTB nvtk Novatek It’s worth mentioning that there are about 100 participants on the stock section of MOEX futures market, but reasonable prices’ history is available only for stocks in Table 1. Each bivariate “stock-futures” price series has its own length and we do not take stocks with historical prices, which amount less than 200 observations. The longest series, belonging to lkoh, has 3586 observations and covers the period from the 9th of August 2002 till the 30th of December 2016. The rest of the series are within this period. The source of the data is Finam investment company website [9]. Short descriptive statistics of the data under consideration is presented in Table 2. 3.2 Estimation Results To calculate the dynamic hedge ratio, the following volatility models are eval- uated in this paper: ADCC, GO-GARCH, cop-GARCH and MSV. Conditional mean of returns is modeled using ARMA. For each asset the whole sample is divided into two parts: in-sample period includes the first 80% of the series, the Table 2. Descriptive statistics. N – number of observations, Mean – mean of daily logarithmic returns, St.dev. – standard deviation, Skew. – skewness coefficient, Kurt. – kurtosis coefficient Stocks Futures Ticker N Mean St.dev. Skew. Kurt. Mean St.dev. Skew. Kurt. chmf 1395 0.040 2.215 -0.366 6.237 0.040 2.402 -0.592 10.494 fees 1318 -0.050 2.941 -0.324 9.748 -0.048 3.184 0.189 8.993 gazp 2730 -0.013 2.458 -0.084 19.300 -0.013 2.591 0.213 24.388 gmkn 3041 0.058 2.747 -1.001 20.524 0.058 2.907 -1.112 26.078 hydr 1404 -0.032 2.172 0.199 6.239 -0.031 2.273 -0.092 6.818 lkoh 3586 0.055 2.317 -0.055 16.122 0.055 2.393 -0.335 25.889 mgnt 587 0.036 2.027 -0.089 5.140 0.040 2.202 -0.042 5.573 nlmk 231 0.244 2.030 0.229 3.906 0.242 2.879 0.936 12.178 nvtk 1973 0.084 2.925 -1.362 31.895 0.084 3.712 -0.530 14.097 rosn 2547 0.025 2.610 0.929 36.043 0.025 2.770 0.536 47.209 rtkm 3020 0.030 2.284 0.293 12.500 0.030 3.031 -0.676 26.798 sber 2753 0.068 2.995 0.129 17.146 0.068 3.122 0.210 17.946 sngs 3583 0.031 2.692 0.963 24.704 0.031 2.799 2.272 54.004 tatn 1402 0.065 2.170 -0.020 4.212 0.067 2.145 -0.407 7.613 trnf 2367 0.065 3.082 0.023 18.438 0.066 3.150 0.039 9.705 urka 1383 -0.028 2.136 -1.621 23.653 -0.024 2.537 -0.649 13.852 vtbr 2357 -0.026 2.919 0.576 45.755 -0.027 3.286 2.680 80.832 rest 20% are for out-of-sample period. The number of lags is chosen by mini- mizing Schwartz information criterion, according to which ARMA(1,0) for mean and GARCH(1,1) for all volatility models give the best fit. The parameters of MSV model are obtained by one of the Markov chain Monte Carlo methods – Hamiltonian Monte Carlo algorithm, also known as Hy- brid Monte Carlo [7, 15]. The convergence of Markov chains is checked by Geweke Z-test [10]. The test is based on the idea that the means, calculated on the first and the last parts of a Markov chain (usually 10% and 50% correspondingly), are equal. In that case the parameter samples are drawn from the stationary distribution of the Markov chains and Geweke’s statistics has an asymptotically standard normal distribution. The Markov chain converges under the null. For some important parameters (namely, covariance of stock and futures returns, futures variance and conditional return at a specific time point, see (3)) p-values are presented in Table 3. Evidently that Geweke Z-test reveals the convergence for the parameters under consideration. Mean hedge ratios for τ = 4 are presented in Table 4. The average hedge ratios range from 19% for lkoh to 94% for sngs. In order to compare hedging strategies, obtained from different volatility models, we calculate three measures of hedging efficiency – maximum risk re- duction, financial result (or profit) and investor’s utility. The first measure is Table 3. Geweke Convergence Z-testtext Ticker cov(rS,t , rF,t ) V(rF,t ) E(rF,t ) chmf 0.612 0.554 0.505 fees 0.832 0.829 0.937 gazp 0.844 0.929 0.557 gmkn 0.896 0.897 0.562 hydr 0.630 0.545 0.790 lkoh 0.914 0.944 0.759 mgnt 0.834 0.545 0.984 nlmk 0.709 0.548 0.822 nvtk 0.534 0.644 0.803 rosn 0.931 0.887 0.666 rtkm 0.664 0.598 0.611 sber 0.876 0.849 0.551 sngs 0.933 0.933 0.674 tatn 0.610 0.569 0.898 trnf 0.885 0.913 0.580 urka 0.591 0.690 0.969 vtbr 0.872 0.821 0.637 Table 4. Mean Hedge Ratios Ticker ADCC GO-GARCH cop-GARCH MSV chmf 0.812 0.778 0.799 0.768 fees 0.836 0.837 0.843 0.820 gazp 0.862 0.810 0.851 0.889 gmkn 0.893 0.817 0.897 0.905 hydr 0.867 0.874 0.861 0.844 lkoh 0.934 0.821 0.192 0.935 mgnt 0.914 0.825 0.911 0.805 nlmk 0.612 0.611 0.608 0.644 nvtk 0.673 0.732 0.678 0.567 rosn 0.883 0.815 0.863 0.864 rtkm 0.562 0.618 0.609 0.785 sber 0.891 0.824 0.899 0.883 sngs 0.910 0.816 0.910 0.939 tatn 0.935 0.903 0.246 0.896 trnf 0.885 0.820 0.879 0.797 urka 0.555 0.559 0.514 0.679 vtbr 0.871 0.753 0.864 0.835 defined as in (10). var(r) E =1− (10) var(rS ) Financial result is calculated as the sum of the logarithmic returns of the portfolio for the forecast period [17]. The formula for investor’s utility is described in (2). The comparison is conducted for the out-of-sample period and summarized in Table 5. We compute the measures of hedging performance for various levels of risk aversion τ . They vary from very small values, almost equal to zero, till ten and are presented in the first column of Table 5, denoted by “ra”. Maxi- mum risk reduction is abbreviated by “mrr”. The rest of the column labels are self-explanatory. Table 5 reveals the numbers of assets, for which the model in the corresponding column maximizes the corresponding criterion. According to Table 5. Summary of hedging efficiency ADCC GO-GARCH cop-GARCH MSV ra mrr profit util mrr profit util mrr profit util mrr profit util 0.00 4 0 1 4 2 2 7 6 8 2 9 6 1.11 10 0 13 7 2 3 0 3 1 0 12 0 2.22 10 2 10 7 1 6 0 5 1 0 9 0 3.33 10 2 10 7 1 6 0 6 1 0 8 0 4.44 10 2 10 6 5 6 1 5 1 0 5 0 5.56 10 2 10 6 5 6 1 5 1 0 5 0 6.67 10 2 10 6 5 6 1 5 1 0 5 0 7.78 10 2 10 6 6 6 1 5 1 0 4 0 8.89 10 2 10 6 6 6 1 5 1 0 4 0 10.00 10 2 10 6 6 6 1 5 1 0 4 0 Table 5, ADCC model clearly outperforms the other models by the maximum risk reduction and investor’s utility. MRR in average amounts to 74% for this model and ranges from 47% to 88%. GO-GARCH, copula-GARCH and MSV reach their maximum MRR at the levels of 84%, 85% and 83% respectively. The performance level of ADCC model seems to be stable and remains the same for τ larger than 2. The dynamics of GO-GARCH hedge efficiency criteria values also stabilizes for higher risk aversion levels. GO-GARCH performance is the same among different efficiency measures and is relatively lower than in ADCC case. Copula-GARCH demonstrates even lower performance by all criteria except the profit of the hedged position. Stochastic volatility clearly provides the highest financial result, if the investor prefers risk. At the same time, with the growth of τ performance level of MSV declines. It’s also worth mentioning that on small values of τ ADCC and MSV reach their maximum performance by utility and profit correspondingly. 4 Conclusion The article considers the development of a hedging strategy based on maximiz- ing investor’s expected utility, taking into account the level of risk aversion. The optimal hedge ratio is time-dependent and is calculated using four multivari- ate volatility models ADCC, GO-GARCH, copula-GARCH with the Student’s copula and multivariate stochastic volatility. The calculation is conducted for seventeen portfolios consisted of stocks and futures of seventeen Russian compa- nies. The efficiency of hedging strategies is assessed by maximum risk reduction, financial result of hedged position and investor’s utility with risk aversion param- eter varying from zero to ten. The most stable performance of hedging strategies according to the chosen criteria demonstrates ADCC model, which provides the highest maximum risk reduction and utility for about 60% of portfolios. MSV maximizes profit of the hedged position for small values of risk aversion in 70% cases. To summarize, ADCC and MSV models are recommended to use for con- structing hedging strategies on Russian stock market according to maximum risk reduction and utility for the former and profit for the latter. MSV gives better results for risk-lovers and ADCC outperforms the other models if risk aversion parameter is larger than 2. The possible directions of the future research include implementing time- varying degree of risk aversion, introducing the heterogeneity of investors by their attitude to risk and using other hedging instruments. References 1. Asai, M., McAleer, M., Yu, J.: Multivariate stochastic volatility: a review. Econo- metric Reviews 25(2-3), 145–175 (2006) 2. Bauwens, L., Laurent, S., Rombouts, J.V.: Multivariate garch models: a survey. Journal of applied econometrics 21(1), 79–109 (2006) 3. Cappiello, L., Engle, R.F., Sheppard, K.: Asymmetric dynamics in the correlations of global equity and bond returns. Journal of Financial econometrics 4(4), 537–572 (2006) 4. Danıélsson, J.: Multivariate stochastic volatility models: estimation and a compar- ison with vgarch models. Journal of Empirical Finance 5(2), 155–173 (1998) 5. Dawid, A.P.: Some matrix-variate distribution theory: notational considerations and a bayesian application. Biometrika 68(1), 265–274 (1981) 6. Dickey, J.M.: Matricvariate generalizations of the multivariate t distribution and the inverted multivariate t distribution. The Annals of Mathematical Statistics 38(2), 511–518 (1967) 7. Duane, S., Kennedy, A., Pendleton, B.J., Roweth, D.: Hybrid Monte Carlo. Physics Letters B 195(2), 216–222 (1987) 8. Engle, R.F., Kroner, K.F.: Multivariate simultaneous generalized arch. Economet- ric theory 11(01), 122–150 (1995) 9. Finam investment company. https://www.finam.ru/ (2017) 10. Geweke, J.: Evaluating the accuracy of sampling-based approaches to the calcula- tion of posterior moments, vol. 196. Federal Reserve Bank of Minneapolis, Research Department Minneapolis, MN, USA (1991) 11. Harvey, A., Ruiz, E., Shephard, N.: Multivariate stochastic variance models. The Review of Economic Studies 61(2), 247–264 (1994) 12. Jondeau, E., Rockinger, M.: The copula-garch model of conditional dependencies: An international stock market application. Journal of international money and finance 25(5), 827–853 (2006) 13. Kleppe, T.S., Yu, J., Skaug, H.J.: Simulated maximum likelihood estimation of continuous time stochastic volatility models. Advances in Econometrics 26, 137– 161 (2010) 14. MOEX Moscow exchange. http://moex.com (2017) 15. Neal, R.M.: MCMC using hamiltonian dynamics. In: Brooks, S., Gelman, A., Jones, G.L., Meng, X.L. (eds.) Handbook of Markov Chain Monte Carlo, pp. 113–162. Chapman & Hall/CRC. (2011) 16. Patton, A.J.: Modelling asymmetric exchange rate dependence. International eco- nomic review 47(2), 527–556 (2006) 17. Penikas, H.: Copula-based price risk hedging models. Applied Econometrics 22(2), 3–21 (2011) 18. Stan Development Team: The stan core library. http://mc-stan.org/ (2017), ver- sion 2.16.0. 19. Tsyplakov, A.: Revealing the arcane: an introduction to the art of stochastic volatil- ity models. Quantile 8, 69–122 (2010) 20. Van der Weide, R.: GO-GARCH: a multivariate generalized orthogonal GARCH model. Journal of Applied Econometrics 17(5), 549–564 (2002) 21. Zellner, A.: Bayesian and non-bayesian analysis of the regression model with mul- tivariate student-t error terms. Journal of the American Statistical Association 71(354), 400–405 (1976)