Improving The Risk Return Performance Of Portfolios
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Published: Wed, 28 Feb 2018
With the development of the Chinese capital market, more and more investors start to look for a more rational way to invest. To increase the investment return and decrease the risk, investors must learn to allocate their funds in order to diversify risk. However, due to the limited assets that can be invested in, the convenience and effectiveness of portfolio diversification must be studied. This paper mainly explores the function of futures in the ordinary stock portfolio through the study of risk-return performance. By comparing the efficient frontiers of different portfolios, the risk-return performance of the futures portfolio and mixed stock-futures portfolio is better than the stock only portfolio. Futures play an important role in upgrading the integrated portfolio of stock and futures. The results of this study provide investors with a feasible way to diversify their funds in multi-type investment portfolios, which is of great theoretical and practical significance.
I. An introduction to Chinese capital market
Ever since December 19, 1990, when Shanghai stock exchange opened, people become more and more interested in investing in the security market to make money. After twenty years, investing in stocks is a quite popular and important way for ordinary Chinese people to manage their money. However, stock market itself can not meet investors’ needs of diversifying risk and increase capital return, and investment diversification becomes a natural solution and guiding concept.
Although twenty years have passed since Shanghai stock exchange came in existence, development of Chinese capital market is quite slow, with limited kinds of investment products. Lack of varieties of trading tools and incomplete structure of capital market products make it difficult to diversify in Chinese capital market. In developed capital markets such as Hong Kong, over 80% of financial derivative instruments in international financial market have been introduced. In stock market, the trading of index futures, options and warrants is quite active with a trending of exceeding the trading of spot market. Hong Kong bond market is even more diversified. Based on three basic kinds which are bond, note and certificate of deposits of fund-raising tools, many more complicated derivatives such as floating rate bonds, variable rate bonds, convertible bonds, credit card receivables, and the current debt instruments traded on the Hong Kong Stock Exchange listing has been increased to 129.(2009)
On the contrary, despite of stocks, there are few more than five years investment instruments in mainland China capital markets. The trading of 1-5 year instruments is also confined so that the available trading instruments are quite limited. As an emerging market the risk of stock market is higher than normal, both systematic risk and market risk. The systematic flaws in Chinese stock market such as no trades of state owned and corporation owned stocks and lack of index futures  or other kinds of hedging instruments make the whole stock exchange system more uncertain. The strong influence of state policy changing is also a reason for high uncertainty. As for the market risk, stock market is in sharp adjustment since the end of 2007. On the one hand, the overall risk has lowered a little; it is still too high compared with the mature capital markets. On the other hand, the low self-control ability of the participants involved in stock market makes the unsystematic risk higher than average. Investing only in stock market can not successfully diversify risk. Considering the incompleteness of Chinese warrant market, futures have been chosen to diversify risk.
Chinese future market also started in 1990. After six years of cleaning up and reconstruction (1995-2000), future market is in good development. In 2002, stock market turned down, which made part of the stock market capital switch to future market and made it a hot deal. This situation is quite similar to what happened in 2007-2008. Chinese future market developed from first pilot reform to rectification and now has entered a new stage of stable development. The legal operation and market discipline have been significantly improved. These features make futures possible as a component of portfolio.
At present, research of the role of futures in the portfolio is focused on index futures and its hedging properties, while the research of commodity futures is focused on its function of price discovering. Adding futures into ordinary stock portfolio has not been well discussed so that this article will research on the performance of portfolio with commodity futures to see whether futures can effectively diversify risk and raise the return.
How to optimize investment portfolio becomes the first and most important question that investors need to consider. Thus, modern portfolio theory becomes quite widely applied in practice. Portfolio means investors allocate certain amount of money to different kinds of assets in order to gain as much as possible return or to get the lowest possible risk.
II. Past literature review in portfolio selection theories
In 1959, Markowitz published his paper named Portfolio Selection: Eficient Diversification of Investments, which conducted a pioneering study of optimizing portfolio in the security market. Ever since then, modern finance and investment decision making comes into a quantitative stage. Portfolio theory is a set of theories and methods to help investors choose certain types and allocate their money from varieties of instruments to form efficient portfolio. In Markowitz theory, mean-variance model can be applied to any class of financial assets, as long as its expected return and the correlation of each asset can be accurately estimated (Markowitz, 1959). In his model, mean represents the expected return of an asset and its risk is represented by the variance. In order to use the Markowitz mean-variance method, we need to find the expected rate of return and risk. However, considering the ineffectiveness of Chinese stock market, the simple mean-variance is not applicable. Thus, more appropriate method of evaluating return and risk needs to be found.
Among these different evaluating methods, people tend to agree using expected return as a representative of future earnings. The return of a financial asset is consisted of two parts: intertemporal cash flows and capital premium (asset price changes during the holding period). The return that this article is going to use is the daily logarithmic rate of return, so the intertemporal cash flows can be ignored. The yield can be expressed as:
Because logarithmic rate of return can be simply added which facilitate the data processing by software and its value can be any real numbers, this article will use logarithmic rate of return as the evaluation of asset yields.
The simplest way to get the expected rate of return is calculating its average. Its flaws are also quite obvious: the result is far from accurate. In order to find more accurate estimation, we need to fit time series data to appropriate model and find the unconditional expectation of asset return. In 1980s and 1990s, lots of literatures have discussed the predictability of stock market and suitable model of predicting asset returns. M.Hashem Pesaran and Allan Timmermann (1995) found that the predictable components of stock returns are highly correlated with business cycle and the magnitude of shocks influences the model more than expected. But because what they studied is a long term relationship in the stock market, the results can only be a consultation. As for the daily stock return, many researches suggest that it shows significant dependence on former returns. Vedat Akgiray found in his paper about the conditional heteroscedasticity in stock returns that the probability distribution of return lag of s days is dependent on return today for several values of s (1989). He used daily returns on the CRSP (Center for Research in Security Prices) value-weighted and equal-weighted index from January 1963 to December 1986 to find that GARCH (1,1) shows the best fit and forecast ability among the econometric models. Noticing that the return he used is also logarithmic rate, the features of logarithmic rate in this article can be expected to be just like that in his study.
Similar results can be obtained from other literatures. There is a positive relation between the expected risk premium and the predictable level of volatility and a negative relation between unpredictable component of stock market risk and excess holding period return (K. R. French et al, 1987). Although they can not determine a certain model to describe the exact relation (difficult to choose between ARIMA and GARCH-M), the relation between return and risk is quite significant.
Studies about Chinese stock market also show evidence of fitting stock return data in ARMA or GARCH models. The daily returns of Shanghai and Shenzhen index indicates significant ARCH effect and the data fit in GARCH-M model well (Hua Tian and Jiahe Cao, 2003). It is reasonable to choose ARMA or GARCH model to simulate the actual stock movement.
But as for the measurement of risk there are comparably various methods. Markowitz explained the mean-variance theory in his 1959 portfolio selection paper which introduced the statistical concept of expectation and variance into the study of investment portfolio. Under a certain probability distribution of returns, he used the average deviation from the average return of all the random returns. Thus, risk can be quantified with the expectation of return as return expected and standard deviation as the measurement of risk.
Although variance has some easy to use features such as simple calculating and easy understanding, it is only an approximate measurement of risk. Using variance needs the distribution to be systematic and does not take the investors different feeling about capital gain and loss into consideration. Given the same amount of gain and loss, the pain of loss is usually larger than the happiness of the capital earnings. Variance ignores this asymmetry while LPM (lower partial moments) would be a better measurement. Harlow proposed this new indicator as a more accurate way to describe risk (1991).
LPM is an abbreviation of lower partial moment, which P (partial) stands for its measuring only one side of the returns compared with the fundamental rate and L (lower) stands for less than fundamental rate (downside risk). LPM is a risk measurement which meets the requirements of Von Neumann – Morgenstern utility function and can cover almost all people’s risk preference. It shows a new way to describe risk apart from the traditional utility measurement which is the function of variance or the standard deviation. The expression of LPM is:
where n is called the order of LPM indicators, representing the risk aversion of investors, and z is called fundamental rate of return which is the minimum return that investors would accept. Different values of n would change LPM into different measurements of risk and therefore meet different investors’ risk preference, from risk preference to risk neutral, then risk aversion. One advantage of LPM is that it can show only the pain or loss possibility when the return is lower than the expected. The other is it can show what investors’ different risk preference can affect the feelings to the same asset by simply changing the order n.
LPM is less popular in evaluating volatility than variance as the calculation of LPM is more complicated. Another reason is that LPM must be calculated separately for each variable while variance can be added or processed under certain assumptions. This means people need to program it in order to use LPM with computer data processing programs. On the contrary, all the data processing programs have a default function of calculating variance.
The way to evaluating the performance of asset portfolios is its efficient frontier. Every combination of risky assets can be plotted in a risk-return space, and those combinations with the highest return under the same risk or with the lowest risk under same return are called efficient portfolios. Usually, the upper part of the curve which describes risk-return features of efficient portfolios is called efficient frontier. Ordinary efficient frontier of investment portfolio is calculated by Markowitz’s mean-variance method. This article will use LPM to substitute variance to calculate efficient frontier which makes it more like investors’ thoughts of risk.
Merriken suggested that variance and LPM are suitable for the study of short-term investment (1994), which is quite popular in Chinese capital market. Based on the review of the related literatures, this article will use econometric models to get expectation daily return of stock and futures and both variance and LPM to calculate efficient frontiers to see whether adding futures into stocks would improve the performance of portfolios.
III. Theoretical study and empirical data results
i. Theories of econometric models and multi-type asset portfolio
The econometric models used to estimating the expected return and risk are ARMA and GARCH models depending on the features of different stock and futures time series. ARMA is an abbreviation of autoregressive and moving average model, which is typically used in estimating autocorrelated time series. As what is mentioned in the literature, auto-correlation in daily logarithmic return is shown by theoretical study, and the empirical study of the realistic data also suggests this result. Typical ARMA model is consisted of two parts: AR (auto-regressive) part and MA (moving average) part. It is normally notified as ARMA (p, q) where p is the order of autoregressive part and q is the order of moving average part.
AR part is written as:
where are the parameters and is the error term (usually white noise). The value of p suggests how many lags of are regressed on and therefore is a measurement of autocorrelation. For the need of stays stationary, usually we need the absolute value of is less than unit.
MA part is written as:
where are the parameters, is the expectation of , and is still the error term (usually white noise). The value of q suggests how many error terms are included in the smoothing process of average and MA process is always a stationary time series.
Thus, ARMA model is written as:
which is a combination of autoregressive part and moving average part. The value of parameters is generally determined by the least square method which minimized the residual error term. The value of p and q is chosen to better fit the model without too much lags or smoothing terms. The method used in this article is through the value of ACF (autocorrelation function, which is used to determine the order of moving average) and PACF (partial autocorrelation function, which is used to determine the order of autoregressive part).
In spite of autocorrelation, there are other special features of financial time series data such as fat tails, extreme values and volatility clustering. Simple ARMA models assume that the error term is independently and identically distributed which does not meet the fact. Thus, Engle (1982) posed ARCH (Autoregressive Conditional Heteroscedasticity) model to analyze this volatility feature of financial data. Four years later, T.Bollerslev improved this model and made it GARCH which is a generalized ARCH model. GARCH model is developed specially for financial data and is widely used in the study of volatility. In addition to the normal econometric model, people use GARCH to better analyze and forecast volatility.
GARCH model can be written as:
where the first equation is a simple ARMA model, but this time is not an independently and identically distributed normal error term. is an independently and identically distributed error term and is called conditional variance which is estimated by the third equation (also an ARMA model). and are independent of each other and the distribution of is not restricted as normal but can be changed to satisfy actual situation. This makes GARCH a more accurate model in estimating the expected rate of return and risk.
Hiroshi Konno and Katsunari Kobayashi (1997) made an attempt to add bonds into ordinary stock portfolio to find a new way of allocating investment. Their purpose is to extend the mean-variance model normally used in optimizing stock portfolios to integrated bond-stock portfolios. At that time, big scale mean-variance models were restricted in stock portfolios although the computer technology and mathematical methods in financial engineering developed fast.
Although bonds seem always to be considered separately when people intend to invest in financial market, Hiroshi and Katsunari still want to add bonds into portfolios. The reason is that before 1980s, the return of bond was far less risky than that of stock due to the stable interest rate. However, after 1980s, interest rate became much more volatile and investors bore heavily loss and huge risks. Actually, the volatility of bonds at that time was even higher than that of stocks. Considering this, combining bonds and stocks into the same portfolio is of great realistic meanings. The method they used is mean-variance and mean-absolute deviation models where variance and absolute deviation are as the different measurement of risk. The results are also quite satisfied as adding bonds into stock portfolios can increase the expected return under the same risk level.
Never the less, Raimond Maurer and Frank Reiner in 2001 also used this idea of multi-type asset portfolio to discuss the possible outcomes of adding real estate securities into international asset portfolios under a shortfall risk frame. They noticed the fact that financial time series data had its own features and the tradition way of evaluating risk using variance can not reflect what investors think in the reality. Therefore, LPM was introduced as the way of measuring risk to reflect the asymmetry in the rate of return of asset.
They compared the situation in Germany and in US by calculating the efficient frontiers of common portfolios, then calculating the efficient frontiers of adding real estate securities into portfolios. Because they studied between different countries, Raimond Maurer and Frank Reiner also calculated the effects of hedging. The results are also quite satisfied as the efficient frontiers move to the left, especially for those high risk-averse investors in Germany. Also, hedging could improve the performance of portfolios, especially for the US investors. With hedging they can build investment portfolios with higher rate of return under a relatively low risk level.
But as mentioned above in the introduction part, there are few commercial bonds besides the government bonds; the only possible type of asset besides stocks that can be added into investment portfolios is futures. This article will also calculate the efficient frontiers of stocks, futures and combined portfolios separately, using both variance and LPM as the measurement of risk.
As to the number of assets that should be held in one portfolio, investors have different opinions. Most mutual funds in the US market hold more than 100 stocks. Although these over-sized investment portfolios may well diversified risks, the expected return can be just acceptable as higher operational fee are needed to maintain such a huge portfolio and these stocks usually contains some low return ones. Xianyi Lu (2006) discussed this question that how many stocks are suitable for Chinese investors to hold in a single portfolio. He constructed portfolios with different number of stocks to compare their risk-return performance. The measurement of risk he used is variance. He came to the conclusion that 20 stocks would be enough to diversify most of the risk.
The close-up price of stock is quite easily obtained while to find suitable closing price of futures is somewhat tricky. Futures are contracts which specify certain quantity and quality of fundamental assets between two parties to trade at a specified date in the future with a price agreed today. Thus there can be various contracts with the same kind of fundamental asset in different delivery date. Considering the trading characteristics of Chinese future market, Chengjie Ge and Yong Liang from a Chinese fund called Guotai Junan tried to construct a continuous future contract to get the daily closing price in 2008. When a contract first comes into market, the transactions are quite few. One contract is traded most actively just three or four months before delivery date, as the coming of specified date the trading volume begins to fall quickly. Those investors, especially the speculators would only trade those contracts that so-called â€œdominant contractâ€?. Thus, each future contract is in good liquidity only for a short time period. A continuous future contract is selecting the most actively traded contract of same fundamental asset at the same time to form a new, artificial contract to get the continuous price time series of one asset.
ii. Data collection and analysis
This article uses daily closing price of stock and futures from the time period 04/01/2007 to 31/12/2008. The data is obtained from RESSET database 
Futures chosen are copper, aluminum, rubber and fuel oil from Shanghai Future Exchange, corn and soybean meal from Dalian Future Exchange and cotton and wheat gluten from Zhengzhou Future Exchange. In order to get daily return we need to construct continuous future contracts by selecting the most active contracts. As to the 8 futures used in this article, the most active contracts of wheat gluten, soybean meal, cotton, fuel oil and corn are those contracts with delivery date four months before the current month (not accounting current month); the most transacted contracts of rubber, aluminum and copper are those with delivery date two months before the current month (still not accounting current month).
For example, current time is 19970201, so the contract which should be selected for cotton is the 199705 contract whose delivery month is May 1997. When it comes to 19970301, the contract selected for cotton should be 199703, and so on, so forth. After constructing eight continuous future contracts, we can get the time series of close-up price. The calculation of logarithmic rate of return, variance and LPM is just like the stock data.
Table 1 shows the descriptive statistics of futures like the mean, the standard deviation, and some others. As the bond market is not mature in China, the risk free rate that used in this article is the three-month central bank bill rate which is also from the RESSET database, same database as the closing price of stocks and futures. From the statistics in the table we can find that the logarithmic daily return of futures shows asymmetry and fat tails, far from the assumption of mean-variance model that the distribution of returns should be normal distribution, or at least a symmetric bell-shaped distribution. Thus, using variance or standard deviation or any other kind of symmetric statistics would be less accurate. Fitting data into econometric models should provide a better estimation of expected rate of return and risk.
Table 2.1-2.4 and Table 3.1-3.3 show the estimation of coefficients using ARMA and GARCH models. The models of stock returns are mostly ARMA models, but of futures are half GARCH models and half ARMA models. Table 2 is the results of future data and table 3 is the results of stock data. From the table we can see that there are four futures which are better fit in GARCH models and for the other four, ARMA is enough as the residual series after ARMA does not show significant heteroscedasticity in error terms. As for stocks, none of the 19 stock time series show significant heteroscedasticity which means ARMA could describe the features of stock price series. One interesting finding is that only 11 stock price time series show the correlation effect while the other 8 stock price series seem to be random walk.
Table 2.1 and Table 2.2 are the GARCH results of future returns. Cotton, soybean meal, aluminum and copper show significant auto correlated heteroscedasticity. The basic model that used to estimate the return is ARMA model, and the first two lags show the most correlation with current logarithmic rate of return. The null hypothesis for all the coefficient in the model is the coefficient equals zero. The constant terms in the models are not significant despite that of soybean meal whose p-value is 0.0202, which means we can reject the null hypothesis under a 5% confidence level. The reason for not able to reject the null hypothesis of constant terms equaling zero may be the absolute value of daily logarithmic rate of return is too small, usually under 0.01. In such a low level the normal test to calculating p-value may become not suitable. So the value of constant terms is still used in the ultimate model to calculate the estimation of expected return although we can not reject the possibility that it actually equals zero.
Table 2.3 and Table 2.4 show the ARMA results of future returns. Wheat gluten, corn, fuel oil and rubber daily logarithmic rate of return are estimated by ARMA model. The null hypothesis is also that any coefficient equals zero with p-value stands for the probability of making mistakes when rejecting the null hypothesis. The problem is the same with that of GARCH models as the p-values are too large to reject. But still we accept this result and make forecast using the present model. In spite of the not-so-satisfying results in the constant term, the coefficients of AR term and MA term are quite significantly different from zero which can be tell from the p-values. This is also true in futures GARCH model and stocks ARMA models. The significance of correlations in logarithmic rate of return series matches the features of financial time series and is what we would like to expect when estimating these coefficients.
There are 19 stock return series to be modeled, but only 11 of them shows autocorrelation with their lags. None of these shows significant heteroscedasticity in the error terms so the model chosen is ARMA model. The constant terms of each stock return model is smaller than that of future return model, and the p-value is bigger than 0.05 as expected. The current return of four stocks out of this eleven shows significant correlation with the six and seven lags, showing the existence of cycle effects in the stock market. For these four stocks, what happened in the week before affects the price of this week more compared with other time. Other seven stocks show the ordinary one or two lags correlation. The coefficients of AR and MA part are also of great significance and the null hypothesis can be rejected.
For those 8 stocks which do not show the existence of autocorrelation, the processing method is to calculate the basic descriptive statistics such as mean and variance. This method may ignore the asymmetry and fat tails of the data, but as there is no good econometric model to estimate random walk series, this simple way has its own advantage and also of quite high accuracy in estimating the expected rate of return and risk.
This article use the forecast value of each model as the expected rate of return, and the variance of the sample as the expected risk for the mean-variance model of investment portfolios. For those 4 GARCH future models, the expected risk is the forecast value of the error part model. As for those eight stocks whose logarithmic daily return series are random walk, simply use the mean as the expected rate of return and the variance as the expected rate of risk. LPM1 is using the three-month central bank bill rate as fundamental rate of return because of its risk-free characteristic. The mean-LPM model also uses the results of expected rate of return from the forecast of GARCH and ARMA models as the only change in this new model is the risk measurement from variance to LPM.
Someone may argue that different econometric models could cause different estimation of expected rate of return, thus the results of efficient frontiers become not so convincing. The purpose of this article is to compare the efficient frontiers of different asset portfolios, trying to find the possible improvement of adding futures into the ordinary stock portfolios. The econometric estimation is used to construct Markowitz’s mean-variance model. What can be seen from Table 2 and Table 3 is that most of the assets can be fitted into ARMA model. As a matter of fact, because the absolute value of daily logarithmic rate of return is too small, the difference of constant terms between GARCH and ARMA model for the same asset is very small that can be ignored.
The calculation of efficient frontiers is using MATLAB financial tool box, and the original data is what has been done above. After calculating the correlation coefficient matrix of 19 stocks and 8 futures, there is not much correlation of each asset. In fact, most of the correlations coefficients are between 0.1 to 0.3, with some of them even to be negative correlated. It suggests that the risk diversify of investment portfolios should successful using these 27 assets according to the statement of Markowitz.
8 futures portfolio
Stock and future portfolio
(The green line is the efficient frontiers of 19 stocks portfolio, the purple line (in the middle) is of 8 futures portfolio and the blue line is of the mixed stock and future portfolio.)
Compared these three efficient frontiers, we can find that adding futures into the ordinary stock portfolio can greatly improve the performance of portfolios, which is even greater under lower risk level. Single future portfolio also performs well compared with single stock portfolio as it can offer higher rate of return under the same risk level. From Figure 1 we can find with the same expected return of 0.4Ã—10-3, the mixed stock and future portfolio can reduce the risk from 0.012 of single stock portfolio to less than 0.006. This more than fifty percent of risk reduction shows great practical meaning of multi-type asset investment portfolios.
Figure 2: the efficient frontiers of stock, future and mixed portfolios using mean-LPM model
Figure 2 shows the same results as the Figure 1. The mixed stock and future investment portfolio can improve the risk-return performance of portfolios. Similarly, future portfolio performs much better than stock portfolio, and it can greatly raise the expected return under higher risk level. The mixed portfolio’s improvement is mainly under low risk level, as the risk becomes bigger, the performing difference between future portfolio and mixed portfolio are not so significant, for the efficient frontiers overlap each other.
The efficient frontiers are straight lines in Figure 2 while they are curves in Figure 1. The different risk measurement may result in this. Because LPM only calculates the downside risk, the risks of the portfolios which provide same return are not the same. Every single LPM must be calculated separately. So the shape of the new efficient frontiers may look different from the traditional hyperbola-shaped curves in mean-variance models.
Both the mean-variance model and the mean-LPM model show that only investing in stock market can not get as much return as investing only in future market under the same risk level because the efficient frontier of stock portfolio is to the right of that of future portfolio and the distance between the two efficient frontiers is quite large. It reveals a fact that investing only in stock market can not guarantee ideal revenue. Although twenty years has passed since the establishment of
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