Risk measurement of index portfolios on financial markets – a copula approach

Nội dung text: Risk measurement of index portfolios on financial markets – a copula approach

1. RISK MEASUREMENT OF INDEX PORTFOLIOS ON FINANCIAL MARKETS – A COPULA APPROACH Nguyen Thu Thuy - Dinh Thi Hai Phong*1 ABSTRACT: In this paper, we propose a method for risk measurement of a portfolio containing indexes on financial markets, including stock and foreign exchange markets. Specifically, copula method is applied to estimate the Value at Risk (VaR) and the Conditional Value at Risk (CVaR) of some optimal portfolios consisting Vietnam stock index VNINDEX, some international stock markets such as the USA, Chinese, Korean, Japanese, French, Australian markets and exchange rates. The results show that, in the first scenario, in Vietnam, when one invests on both these markets, risk on stock market is higher than on foreign exchange. Therefore, in order to reduce the risk, one may probably invest on stock market with a lower proportion than on foreign exchange market. In the second scenario, when one invests on international and Vietnamese stock markets, one may diversify portfolio to some international stock markets such as Japanese, Korean or Australian ones beside Vietnamese market. Investing only on Vietnamese stock market is risky. In the third scenario, when one invests on international and Vietnamese stock markets and foreign exchange market, risk on stock market is still higher than on foreign exchange. In addition to investing on foreign exchange market, one may diversify portfolio to some international stock markets such as Japanese or Chinese ones. One should never invest only on single Vietnamese stock market. Keywords: Risk measurement; portfolio; stock market; foreign exchange market. 1. INTRODUCTION “Index-based investment” is quite popular in foreign markets, but relatively new in the Vietnamese stock market. In recent years, some companies, gold exchange in Vietnam has implemented products trading VN-Index to investors. “Index-based investment” means the purchase by the investors of an investment fund whose portfolios consist of a part or all of the securities constituting an index on the market. An investment fund that provides such a fund is called the index fund. On the basis theory, (Hoang Dinh Tuan, 2010) asserts that the portfolio of the portfolio is indeed a portfolio, so the measurement of portfolio risks is practicable. Index investment has several advantages such as: First, it is easy to use. The biggest advantage of index investing is that it does not require any presence of investors in the transaction. All the investors have to do is choosing the index which they want to invest and then the fund management will do the rest. The return that the investor receives will be the performance of the index during that investment period. The most popular index funds in the world are S&P500, DowJones and Nasdaq. Second, it is low-cost. Since the management of index funds is not too complex, the cost to investors for fund managers will be lower than that of conventional mutual funds. Dynamic funds are more profitable but compensate for the high costs associated with risk. Third, it is higher efficiency. Index funds have historically performed better than other dynamic funds. They are proud that they bring higher profits while investment costs as well as participation fees are low. 1 Academy of Finance, 58 Le Van Hien, Duc Thang, Bac Tu Liem, Hanoi, Vietnam.
2. INTERNATIONAL CONFERENCE STARTUP AND INNOVATION NATION 329 Index funds are highly profitable thanks to the diversification of their portfolio at a low cost to investors. High profitability makes index funds an attractive tool for investors. However, index investment has some drawbacks: First, there is no ability to beat the market. If index funds pursue “moments” of the market as other investors, they will show that they are not capable of doing better than the market. Active investment funds have the ability to adjust their portfolio to take advantage of the times when the market is “parallel”. In the context of economic growth, index funds will outperform the active funds. Second, the performance of the fund depends on the state of the economy. Index funds may also perform poorly in a stagnant economy. For example, if investors invest in the S&P500 in 2000, investors would actually lose money over the past decade. If the US economy had continued to grow slowly or had not grown, index funds would continue to perform poorly. As we can see, index funds have their advantages and disadvantages as well. However, they are really a good investment tool for passive investors looking for quick and easy diversification at low cost. This investment method already popular in most advanced markets around the world, it will inevitably be applied in other emerging markets in the future, for example in Vietnam. In fact, there are also a number of investment funds in Vietnam’s stock market such as FTSE Vietnam Index ETF, Market Vectors Vietnam ETF (VNM), MSCI Frontier Markets Index, iShares MSCI Vietnam Investable Market Index Fund VFMVN30. Vietnam stock market is in the process of perfecting and developing, the annual growth rate of the market is quite high and larger than the world average. An index investment strategy is suitable for long-term investors with low risk tolerance, especially for new entrants who are lack of investment knowledge and experience or who have not enough time to closely monitor the market developments. Therefore, their goal is to achieve profitability as well as market risk. This investment approach is to reduce the risk while “putting eggs into many baskets” and easy to apply as well as cutting research costs when just tracking the macro information to predict the market trend. It save time for investors in basic and technical analysis on individual stocks in the short term. Therefore, the formation of index funds will give domestic retail investors a relatively safe financial product with relatively high profitability. This paper focus on presenting a tool for measuring indicator risk, so that investors are somewhat aware of the degree of risk when selecting indicators, and so that decision makers have confident information in selecting indicators to create index portfolios. In recent years, the collapse of many financial institutions has spill-overed between markets. After those collapses, the role of risk management in financial investment is of particular interest. In particular, the importance of risk measurement is considered to be crucial. To measure the maximum amount of losses that could occur when holding a financial asset or a financial portfolio during an investment period, the “value of risk” measure is usually used (Value at Risk - VaR) at various levels of confidence (see(Hoang Dinh Tuan, 2010)). VaR model is widely used in market risk and credit risk management. However, according to a study by (Artzner et al., 1999), VaR is not a coherent risk measure, so diversification rules in investment are disrupted. In order to overcome this weakness, a new approach to risk measurement of the portfolio thanks to Conditional Value at Risk (CVaR) has been recently used. Due to some superior properties, the CVaR presents a more complete risk measurement than VaR. In this paper, the study proceeds to estimate both measures of risk. Typically, due to certain technical conditions, the VaR, CVaR of the asset returns or the portfolio returns are calculated, so that the VaR, CVaR of the asset or portfolio are deduced. Let rt denote the asset or portfolio returns. VaR at the probability level p, denoted by VaR(p), is defined as follows: Pr<=VaR () p p (1) ( t )
3. 330 HỘI THẢO KHOA HỌC QUỐC TẾ KHỞI NGHIỆP ĐỔI MỚI SÁNG TẠO QUỐC GIA CVaR at the probability level p, denoted by CVaR (p), is defined as follows: CVaR()p= Er | r > VaR () p (2) ( tt ) In practice, there are several methods for estimating the VaR, CVaR. Some typical studies in Vietnam such as: Study of (Hoang Dinh Tuan, 2010) on traditional methods based on a hypothesis that the asset returns is normally distributed with unchanged mean and variance; study of (Tran Trong Nguyen, 2012) on the extreme value theory method; study of (Tran Trong Nguyen, 2013) on quantile regression model with heteroskedasticity; study of (Hoang Duc Manh, 2015) on the copula method. Copula method is a useful method in estimating depence structure and measuring financial risk. The co-author used copula method in a recent study, (Nguyen Thu Thuy, 2015). In this paper, the authors use the copula method to estimate the VaR, CVaR of the portfolio due to the advantages of this method (see Section 2). This study chooses the Student copula that describes the dependence structure of the assets in the portfolio, and then use this copula to estimate the VaR, CVaR of the portfolio. With such research problems, the paper focuses on two main ideas: Firstly, the marginal distribution is built for each market index return. After that, the marginal distribution is fitted with Student copula functions so that this copula is chosen to describe the dependence structure between markets. Secondly, perform the risk measurement for an optimum portfolio between markets at a given expected return by following such steps: (i) choosing the probability distribution for the market index returns; finding the optimal portfolio thanks to the Mean – Conditional Value at Risk (M-CVaR) model as in (Hoang Duc Manh, 2015); (ii) Estimating the VaR and CVaR of the optimal portfolio using the Student copula function chosen in the previous idea. 2. RESEARCH METHODOLOGY 2.1. Copula method During the risk measurement of a portfolio, the asset returns are often assumed to be independent and identically distributed. However, in practice, these assumptions are often not satisfied. This fact makes it difficult to determine the joint distribution of assets and their dependence structure. The copula is a strong method to solve this problem. In other words, one can use the copula functions to determine the joint distribution of assets in the portfolio as long as their marginal distribution functions are known. Especially, copula method is quite useful in the study of extreme dependence and nonlinear dependence of assets. Copula is a joint distribution built from the marginal distribution functions of one-dimension-random variables. Within the scope of this paper, the study mainly uses 2-dimension-copulas. In fact, higher- dimension-copulas are also constructed similarly to the two-dimension-copulas (see (Cherubini et al., 2004) and (Nelsen, 1998)). Two-dimension-copula function A two-dimension-copula function (refers to a copula) is a function C whose domain is [0;1]× [0;1] , and value domain is [0;1] and C satisfies the following properties: 1) Cx( )= 0, ∀∈ x [0;1]2 if at least one component of x equals 0. 2) C(1; x )= Cx ( ;1) = x , ∀∈ x [0;1] . 2 3) ∀∈(aa1 ; 2 ), ( bb 12 ; ) [0;1] where a1≤≤ ba 12, b 2, it follows: Ca(;2 b 2 )−−+≥ Ca (; 12 b ) Ca (;) 21 b Ca (;) 11 b 0 (3) The problem of modeling dependence structures is that this feature does not always show out of the joint distribution function under consideration. It would be of some help to separate the
4. INTERNATIONAL CONFERENCE STARTUP AND INNOVATION NATION 331 statistical properties of each variable from their dependence structure. Copula functions provide us with a viable way to achieve this goal. According to Sklar’s theorem (McNeil et al., 2005, p.200), for Fx11(), Fx22() are, respectively, the marginal distribution functions of the random variables XX12, , there exists a copula function C such that: 2 Fx(12 ; x )= CF ( 11 ( x ); F 2 ( x 2 )) , for ∀∈(;xx12 ) R (4) If FF12, are continuous, C is uniquely exists. Conversely, if C is a copula and FF12, are, respectively, the marginal distribution functions of the random variables XX12, ,the function F defined in (4) becomes the joint distribution function, which is built from marignal distribution functions FF12, . Clearly, the Sklar theorem is particularly useful in studying the dependence structure and the risk measurement of portfolio whose assets are non-independent and non- identically distributed. 2.2. Research Methods Based on the Sklar theorem and the formula of the VaR and the CVaR of a portfolio, this empirical research performs copula method on the data of the Vietnam stock market and the foreign exchange market in following steps: (i) Data analysis: Descriptive statistics, preliminary investigation the properties of market index returns, test the normality distribution of returns. (ii) Determine a copula function to describe the dependence structure of each porfolio of index returns. Firstly, this study build the marginal distribution functions for returns. According to (Patton, 2012), there are two methods to build marginal distribution functions. Those are parameter method and non-parametric method. In this paper, the research uses non-parametric method, i.e., the empirical probability distribution functions are used as marginal distribution functions. After that, copula Student is chosen to fit the data, in the way that we estimate the parameter of Student copula. (iii) Estimate the VaR and CVaR values of some optimal portfolio including: - The VNIndex and an exchange rate. - The VNIndex and some international stock market indices, without exchange rate. - The VNIndex and some international stock market indices, and exchange rate. The Student copula which was chosen in step (ii) for each portfolio of marginal distrubution functions is used here to estimate the VaR and CVaR values of the responding optimal portfolios. 3. DATA AND RESEARCH RESULTS 3.1. Research data The study uses daily closing data of the Vietnam stock market index (VNINDEX), and daily adjusted closing data of some international stock market indices (S&P500, SSE, KOSPI, JPX, CAC, ASX of corresponding USA, Chinese, Korean, Japanese, French and Australian stock markets) and the exchange rates of Vietnam Dong by the US dollar (VND/USD). The VNIndex was collected from vndirect.com.vn. International stock market indices were collected from These international stock markets are chosen thanks to some evidence of contagion of these markets to Vietnamese stock market, which was shown in (Nguyen Thu Thuy, 2015) and (Tran Trong Nguyen et al., 2016). The foreign exchange rates of Vietnam Dong (VND) and United State Dollar (USD) was collected from https:// vn.investing.com. The data series were collected from May 2nd, 2013 to April 27th, 2018, including 1245 observations, in 5 years. As deviation of the holidays and trading hours between international markets, the data is adjusted suitablely. The returns of indexes are defined as follows:
5. 332 HỘI THẢO KHOA HỌC QUỐC TẾ KHỞI NGHIỆP ĐỔI MỚI SÁNG TẠO QUỐC GIA rt = ln(It/It-1), (5) where It is the value of the index at the closing time of the day t. In practice, for simplicity, we denote the yield of the XYZ index as RXYZ. Descriptive statistics of the index returns of the two markets are computed in Table 1. Table 1. Summary of descriptive statistics of index returns. RVNINDEX RASX RCAC RJPY RKOSPI RSP500 RSSE REX Mean 0.000641 0.000120 0.000278 0.000279 0.000194 0.000419 0.000281 6.73E-05 Median 0.001343 0.000360 0.000426 0.000188 5.38E-05 0.000370 0.000615 0.000000 Maximum 0.037784 0.032849 0.040604 0.047444 0.029124 0.038291 0.056036 0.013710 Minimum -0.060512 -0.041765 -0.083844 -0.060821 -0.032270 -0.041843 -0.088732 -0.013547 Std. Dev. 0.009836 0.008113 0.011128 0.010867 0.007241 0.007828 0.014706 0.001270 Skewness -0.759107 -0.318438 -0.563045 -0.435735 -0.282337 -0.572022 -1.284819 0.787636 Kurtosis 7.034920 4.881264 7.754415 7.213951 4.938263 6.400727 10.67885 34.43362 Jarque-Bera 963.3511 204.4703 1237.393 959.7896 211.2582 667.2908 3398.596 51343.71 Probability 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 Sum 0.797655 0.148890 0.346447 0.346937 0.241726 0.521799 0.349029 0.083736 Sum Sq. Dev. 0.120249 0.081824 0.153922 0.146789 0.065182 0.076168 0.268827 0.002004 Observations 1244 1244 1244 1244 1244 1244 1244 1244 In general, the average value of returns are relatively close to 0. All of Vietnam stock market, international stock markets and the foreign exchange market have positive investment efficiency since the average of returns are all positive. The most of skewness coefficients of returns are negative, except for the skewness coefficient of exchange return. The Q-Q normality distribution graphs of the returns show that these returns are all not normallity distributed (Figure 1). As can be seen, the tail of the distributions are separated from the straight line which shows the normallity distribution. It means that the returns have fat tail distributions, not normality distribution. The kurtosis coefficients of the returns are all high, from 4.9 to 34.4, which also show that the returns are not normality distributed. The abnormally distribution of returns is also supported by the Jaque-Bera statistic in Table 1. The Jaque-Bera test, with very small probability, rejects the hypothesis H0: “the returns are normality distributed”. Therefore, the normality distribution is not appropriate to study these returns. RVNINDEX RASX RCAC .04 .03 .04 .02 .02 .02 .01 .00 .00 .00 -.01 Quantiles of Normal Quantiles of Normal -.02 Quantiles of Normal -.02 -.02 -.04 -.03 -.04 -.08 -.04 .00 .04 -.06 -.04 -.02 .00 .02 .04 -.10 -.05 .00 .05 Quantiles of RVNINDEX Quantiles of RASX Quantiles of RCAC RJPY RKOSPI RSP500 .04 .03 .03 .02 .02 .02 .01 .01 .00 .00 .00 -.01 -.01 Quantiles of Normal -.02 Quantiles of Normal Quantiles of Normal -.02 -.02 -.04 -.03 -.03 -.08 -.04 .00 .04 .08 -.04 -.02 .00 .02 .04 -.06 -.04 -.02 .00 .02 .04 Quantiles of RJPY Quantiles of RKOSPI Quantiles of RSP500 RSSE REX .06 .006 .04 .004 .02 .002 .00 .000 Figure 1. Q-Q graph of yield chains. -.02 -.002 Quantiles of Normal Quantiles of Normal -.04 -.004 -.06 -.006 -.10 -.05 .00 .05 .10 -.02 -.01 .00 .01 .02 Quantiles of RSSE Quantiles of REX
6. INTERNATIONAL CONFERENCE STARTUP AND INNOVATION NATION 333 3.2. The estimation procedures of VaR and CVaR In this section, the study uses the Student copula chosen aboved to measure the risk of portfolios, including the VNindex and an exchange rate. The estimation procedures of VaR and CVaR as follows: Step 1: Using Generalized Pareto Distribution (GPD) to estimate the lower and upper tail of the probability distribution for each return; The middle distribution of each return is approximated by its empirical distribution thanks to the Kernel method. In fact, to estimate the upper tail distribution, observations of at least 90% quantile of the returns and to estimate the lower tail distribution, observations at most 10% quantille of the returns. In addition, the previous research also uses the Maximum Likelihood Method to estimate the parameters of the GPD, and must ensure the convergence of numerical solutions. In this paper, the study applies the sampling rule as above to estimate the parameters of GPD distribution for the lower and upper tails for each return. Estimated results by Matlab software. The results are in Table 2. Table 2. Results of GPD parameter estimation for the lower and upper tail coefficients. Returns Lower tail Upper tail Shape (ξ ) Scale (σ ) Shape (ξ ) Scale (σ ) Rvnindex 0.186758 0.00653135 -0.0202895 0.00572845 Rasx -0.0560729 0.00624768 - 0.082239 0.00506877 Rcac 0.122396 0.00768653 -0.162154 0.00870068 Rjpy 0.125597 0.00767447 0.0215807 0.007497 Rkospi -0.132716 0.00601704 0.045943 0.0039494 Rsp500 -0.0233668 0.00691633 -0.0204268 0.00449804 Rsse 0.301419 0.0113643 -0.0932815 0.0106485 Rex 0.303303 0.000767517 0.314174 0.000973257 In addition, we can observe the probability plot of the distribution patterns of the yield chains. Based on the probability distributions of the return series, we observe how the probability distribution of the series of marginal differs from the normal distribution and the distribution of GDP is being used, especially the distribution of the left and the right tails. It partially help us decide to use the GDP distribution or standard distribution to describe the distribution of the returns. To be more precise, we perform the following distribution test: As a result in probability theory, if the random variable X has a probability distribution function F(x), then the random variable U = F(X) is distributed uniformly in [0, 1]. This result has been applied in randomized simulation to generate the values ​of a random variable following a given distribution (e.g., standard distribution, Student distribution, exponential distribution, ). Some predict of suitability of GPD-Kernel-GDP probability distribution to returns can be explored from Figure 2.
7. 334 HỘI THẢO KHOA HỌC QUỐC TẾ KHỞI NGHIỆP ĐỔI MỚI SÁNG TẠO QUỐC GIA Figure 2. Some Probability distribution of returns. Figure 2 shows that it would be significantly biased if we used usual normal distributions to describe the probability distribution of these returns, particularly the left and right tail of distribution. At the same time, if we use GPD to estimate the left and right tail of distributing these returns, it is quite consistent with the actual data. Because the plot of actual return is nearer the Pareto Tail Distribution than the Normal Distribution. To confirm the suitability of GPD-Kernel-GDP (if exists) in the probability distribution estimates of the returns, we perform a distributed test for standardized yield chains, which is the series transformed by the probability distribution function (GPD-Kernel-GPD) of the original return series. It is also possible to prove that the correct GPD-Kernel-GPD form is indeed a probability distribution of the original returns. In concrete, to confirm the suitability of GPD-Kernel-GDP distribution in estimating the probability distribution of these returns, the study performed a uniform distributed test for series, such as: URVNindex, URusd-vnd, These series transformed by the probability distribution function (GPD-Kernel-GPD) of the original return series. The Anderson-Darling tests results are presented in Table 3. Table 3. Probability values ​​in the Anderson-Darling test for transformed series. Urvnindex Urasx Urcac Urjpy Urkospi Ursp500 Ursse Urex 0.8177 0.6934 0.8013 0.5414 0.6331 0.6172 0.5234 0.0012 This result shows that the GPD-Kernel-GPD distribution is appropriate used to approximate the distribution of almost returns series, except for the returns of exchange rate (since the probability value of the Anderson-Darling test in the case of exchange rate is really equal to 0). Therefore, we will use the empirical distributions thanks to Kernel function of normal distribution for exchange rate, while GPD- Kernel-GPD distribution is used for the others. Figure 3. Density function graph by Kernel normal for exchange rate return. Step 2: After selecting the probability distributions for the returns, we construct joint distributions to estimate risk measures and select the optimal portfolio.
8. INTERNATIONAL CONFERENCE STARTUP AND INNOVATION NATION 335 Previously, the study performed empirical analysis to select the appropriate distribution for each return. Based on the Copula methodology, research can build the joint distributions of returns of the portfolio. In this approach, in order to estimate the VaR, CVaR and solve optimal problem, the study use the randomized simulation to generate the returns ​of the joint distribution. After that, the authors calculate the return of the portfolio and the risk measures. First, the author takes steps to calculate the VaR and CVaR measures for a particular category. For the series which were transformed into those whose value domain is [0; 1], the study estimates the parameters of Student Copula. Thanks to the Inference Functions for Margins (IFM) method, the study estimates the two parameters of Student Copula. Step 3: Find the optimal portfolios at given expected returns thanks to the M-CVaR model. To do so, the study proceeds to estimate the effecient boundary according to M-CVaR model by calculating 20 marginal portfolios without bear sale. Therefore, the optimal portfolios are found. Finally, once we get the weight of each assets in the portfolio, we estimate the VaR and CVaR of each portfolio. In empirical results, we built 3 scenarios: first, one invests on Vietnamese stock market and foreign exchange market, index portfolio consists of 2 indices; second, one invests on some international stock markets relating to Vietnamese stock market and Vietnamese stock market, but no foreign exchange market index, index portfolio consists of 7 indices; third, one invests on some international stock markets, Vietnamese stock market, and also foreign exchange market index, index portfolio consists of 8 indices. Following above steps, we estimate risk of some portfolios of indices on financial markets as in below subsections. 3.3. Risk measurement of portfolio including of Vnindex and exchange rate In the first situation, we consider a portfolio of two indices of Vnindex and exchange rate. The Student copula parameters are: the number of degrees of freedom, which is 53.2561, and the Correlation coefficient matrix, which is: 1− 0.04246  −0.04246 1 We continue to estimate the efficient boundary of the portfolio: Figure 4. Efficient boundary for portfolio of Vnin- dex and exchange rate thanks to M-CVaR model. Thanks to the results of the efficient bound of the portfolio, we calculate some optimal portfolios corresponding to some given returns, thanks to M-CVaR model. Here is the results: Table 4. Some optimal portfolios of Vnindex and exchange rate returns by M-CVaR model. Porfolio 1 Porfolio 2 Porfolio 3 Preturn 0.03% 0.05% 0.07% Proportion RVnindex 0.3705 0.6852 0.9999 Rex 0.6295 0.3148 0.0001 Prisk 0.0267 0.05 0.0733
9. 336 HỘI THẢO KHOA HỌC QUỐC TẾ KHỞI NGHIỆP ĐỔI MỚI SÁNG TẠO QUỐC GIA The results in Table 4 show that, in Vietnam, if one invests on both the stock market and the foreign exchange market, one should spend on a smaller proportion on the stock market than on the foreign exchange market to reduce risk. At a relative low return, such as 0.03%, one may invest 37.05% of the capital on Vietnamese stock market and 62.95% of the capital on foreign exchange market. When one increases the proportion of capital on stock market, and reduce the proportion of capital on foreign exchange market, the risk increases. Particularly, if one just invest on stock market only, the situation is the most risky. The results are given in Table 5, which corresponds exactly to the portfolios in Table 4. Table 5. Risk measurement of portfolio Vnindex and exchange rate returns by Copula Student. Confidence level Porfolio 1 Porfolio 2 Porfolio 3 VaR CVaR VaR CVaR VaR CVaR 90% 0.4% 0.68% 0.72% 1.24% 1.05% 1.82% 95% 0.57% 0.88% 1.06% 1.62% 1.55% 2.37% 99% 1.06% 1.43% 1.93% 2.64% 2.82% 3.87% 3.4. Risk measurement of portfolio including of Vnindex and international stock market indices In the second situation, we consider a portfolio of Vnindex and six international stock markets, but not exchange rate. The Student copula parameters are: the number of degrees of freedom, which is 6.8828, and the Correlation coefficient matrix, which is: 1 0.06 0.146 0.035 0.128 0.186 0.100  0.06 1 0.204 0.233 0.219 0.225 0.115 0.146 0.204 1− 0.005 0.289 0.532 0.096  0.035 0.233− 0.005 1 0.249 0.009 0.118 0.128 0.219 0.289 0.249 1 0.338 0.232  0.186 0.225 0.532 0.009 0.338 1 0.144  0.100 0.115 0.096 0.118 0.232 0.144 1 We continue to estimate the efficient boundary of the portfolio: Figure 5. Efficient boundary for the portfolio of Vnindex and international stock market indices thanks to M-CVaR model. Thanks to the results of the efficient boundary of the portfolio, we calculate some optimal portfolios corresponding to some given returns, thanks to M-CVaR model. Here is the results:
10. INTERNATIONAL CONFERENCE STARTUP AND INNOVATION NATION 337 Table 6. Some optimal portfolios of Vnindex and international stock market indices by M-CVaR model. Preturn Porfolio 4 Porfolio 5 Porfolio 6 0.02% 0.04% 0.06% Proportion RVnindex 0.0897 0.2565 0.8050 Rasx 0.2308 0 0 Rcac 0.0655 0.1116 0 Rjpy 0.1043 0.2123 0.0770 Rkospi 0.3924 0 0 Rsse 0.0931 0.4196 0.1180 Rsp500 0.0242 0 0 Prisk 0.0256 0.036 0.0682 The results in Table 6 show that, in a situation, if one invests on international stock markets and the domestic stock market, one should not spend only on the domestic market, but also international stock markets such as Korean, Australian and Japanese ones. Maybe, because these international stock markets have dependence structure with Vietnam stock market at some levels. Investors should notice that if they increased the proportion of capital on Chinese and Vietnamese stock market at the same time and reduced the proportion of capital on the other stock markets, the risk of the index portfolio would increase dramatically. Again, one should not invest on Vietnam stock market only. The results are given in Table 7, which corresponds exactly to the portfolios in Table 6. Table 7. Risk measurement of portfolio Vnindex and international stock market indices by Copula Student. Confidence level Porfolio 4 Porfolio 5 Porfolio 6 VaR CVaR VaR CVaR VaR CVaR 90% 0.6% 0.95% 0.62% 1.07% 0.9% 1.51% 95% 0.84% 1.2% 0.92% 1.38% 1.27% 1.96% 99% 1.4% 1.81% 1.64% 2.17% 2.28% 3.38% 3.5. Risk measurement of portfolio including of Vnindex, international stock market indices and exchange rate In the last scenario, we consider a portfolio of Vnindex, six international stock markets, and also exchange rate. The Student copula parameters are: the number of degrees of freedom, which is 8.3507, and the Correlation coefficient matrix, which is: 1 0.064 0.143 0.036 0.128 0.187 0.010− 0.033  0.064 1 0.208 0.236 0.225 0.231 0.114− 0.08 0.143 0.208 1−− 0.0006 0.290 0.529 0.093 0.024  0.036 0.236−− 0.0006 1 0.252 0.011 0.120 0.025 0.128 0.225 0.290 0.252 1 0.335 0.235− 0.033  0.187 0.231 0.529 0.011 0.335 1 0.141− 0.0004 0.010 0.114 0.093 0.120 0.235 0.141 1− 0.012  −−−0.033 0.08 0.024 − 0.025 −− 0.033 0.0004 − 0.012 1 We continue to estimate the efficient boundary of the portfolio:
11. 338 HỘI THẢO KHOA HỌC QUỐC TẾ KHỞI NGHIỆP ĐỔI MỚI SÁNG TẠO QUỐC GIA Figure 6. Efficient bound for the portfolio of Vnindex, international stock market indices and exchange rate thanks to M-CVaR model. Thanks to the results of the efficient bound of the portfolio, we calculate some optimal portfolios corresponding to some given returns, thanks to M-CVaR model. Here is the results: Table 8. Some optimal portfolios of Vnindex, international stock market indices and exchange rate by M-CVaR model. Preturn Porfolio 7 Porfolio 8 Porfolio 9 0.025% 0.04% 0.06% Proportion RVnindex 0.1798 0.3493 0.7737 Rasx 0.0121 0.0374 0 Rcac 0 0 0 Rjpy 0.1179 0.1885 0.2021 Rkospi 0 0 0 Rsse 0.1766 0.2943 0.0242 Rsp500 0 0 0 Rex 0.5136 0.1305 0 Prisk 0.0163 0.029 0.0567 The results in Table 8 show that, in another scenario, if one invests on not only international stock markets, Vietnamese stock market but also foreign exchange market, foreign exchange market is once again a safe haven. We can see, if one spent over a half (51.36%) of the capital on foreign exchange market, some 17.98% of the capital on Vietnamese stock market and some 17.66% on Chinese stock market and some 11.79% on Japanese stock market, it is safer than that one spent some 34.93% on Vietnamese stock market, some 13.05% on foreign exchange market, some 29.43% on Chinese stock market and some 18.85% on Japanese stock market. And it is so risky when one just invests on Vietnamese stock market and Japanese stock market, and without on foreign exchange market. The results are given in Table 9, which corresponds exactly to the portfolios in Table 8. Table 9. Risk measurement of portfolio Vnindex, international stock market indices and exchange rate by Copula Student. Confidence level Porfolio 7 Porfolio 8 Porfolio 9 VaR CVaR VaR CVaR VaR CVaR 90% 0.31% 0.53% 0.56% 0.93% 0.87% 1.48% 95% 0.44% 0.68% 0.78% 1.2% 1.24% 1.91% 99% 0.81% 1.12% 1.41% 1.97% 2.24% 3.2%
12. INTERNATIONAL CONFERENCE STARTUP AND INNOVATION NATION 339 4. CONCLUSION The study used the empirical distributions as the marginal distributions for the returns of indexes of the stock market and the foreign exchange market. The study also fitted the data with Student copula function. The study also used the GPD distribution and Kernel method to select the appropriate probability distribution GPD- Kernel-GPD pattern for all abnormality distributed returns. The results are those: exchange rate return is suitable with kernel model, while all the other returns are appropriate with GPD-Kernel-GPD pattern. The paper applied the M-CVaR model, Copula method and the random simulation to construct efficiency boundaries. For a given expected return of each investor, one can build an “optimal portfolio”, and estimate its VaR, CVaR values. From 3 scenarios which we assume, the results homogeneously show that: First, if investors want high yields, they must accept high risk. Second, the foreign exchange market is a safe haven for investors in financial investing. One should select foreign exchange market index as one component of the index portfolio. If there is not a foreign exchange market index in the index portfolio, one should diversify the portfolio on some international stock markets. Which international stock markets should be chosen? This is an interesting question that the authors are on the way searching. Some international stock markets which were studied in the paper is not really enough. Finally, some future research can be extended from this paper are as follow: First, we can study more copula functions to fit the data. Second, some more international stock markets should be taken into consideration. Third, we can expand the research to commodity markets such as world oil market, world or domestic gold markets, etc REFERENCES Andrew J. Patton (2012). A review of copula models for economics time series. Journal of Multivariate Analysis, Vol. 110, September 2012, pp. 4-18. Artzner P., Delbaen F., Eber J.M., Heath D. (1999). Coherent measures of risk, Mathematical Finance, Vol. 9, No. 3, pp. 203–228. Cherubini U., Luciano E., Vecchiato W. (2004). Copula methods in Finance, John Wiley & Sons, October 22, 2004, - 310 p. Hoàng Đình Tuấn (2010). Mô hình phân tích và định giá tài sản tài chính, Nhà xuất bản khoa học và kỹ thuật, - 316 tr. Hoàng Đức Mạnh (2015). Lựa chọn danh mục đầu tư tối ưu khi lợi suất tài sản không phân phối chuẩn: áp dụng cho thị trường chứng khoán Việt Nam, đề tài NCKH cấp cơ sở. McNeil A.J., Frey R., Embrechts P. (2005). Quantitative Risk Management: Concepts, Techniques and Tools, Princeton and Oxford. URL: Nguyen Thu Thuy (2015). Testing for contagion from some Asian emerging stock markets to Vietnamese stock market during the Global Financial Crisis using Copula. The first International Conference proceedings for Young Researchers in Economics and Bussiness, Volume 3, pp. 71-90. Roger B. Nelsen. An Introduction to Copulas, Springer Verlag, New York, 1998. Trần Trọng Nguyên (2012). Đo lường rủi ro tỷ giá bằng tiếp cận lý thuyết cực trị, Tạp chí Kinh tế & Phát triển, số 182, trang 51-59. Trần Trọng Nguyên (2013). Phân tích rủi ro trong đầu tư vàng – sử dụng hồi quy phân vị, Tạp chí khoa học và công nghệ, số 19, trang 51-54. Tran Trong Nguyen and Nguyen Thu Thuy (2016). Testing for contagion to Vietnam stock market during Global Financial Crisis using Copula. Enlightenment of Pure and Applied Mathematics, Volume 2, Issue 2, pages 187-210.