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Cornish-Fisher Expansion on Estimation and Forecasting Models of Stock Return Volatility

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Cornish-Fisher Expansion on Estimation and Forecasting Models of Stock Return Volatility Cornish-Fisher Expansion on Estimation and Forecasting Models of Stock Return Volatility Dr. Leila Y. Calderon Financial Management Department, De La Salle University-Manila INTRODUCTION The ability to forecast financial market volatility is important for portfolio selection and asset management. However, predicting volatility is a challenge and there are different volatility models available to choose from. Understanding the stock return volatility will result into better investment strategies. In most cases, the widespread popularity of mean-variance analysis is due to the fact that it is very simple and powerful. Most of the theoretical and empirical work on portfolio selection and the pricing of financial assets use the mean-variance analysis. Many investors limit their decisions on the mean-variance analysis as a simple exercise to risk-return tradeoff. The variance is usually unconditional as computed by the standard deviation of the sample period. However, it has been proven by studies that stock return volatility is time varying as was shown by Autoregressive Conditional Heteroscedasticity (ARCH) (Engle, 1982). DLSU Business & Economics Review Volume 13 No.1 2001-2002 22 CORNISH-FISHER EXPANSION The daily volume and prices of the ten selected index issues and that of the daily Philippine Stock Exchange Composite Index (Phisix) from 01 April 1994 - 31 March 1999 were taken from the research department of the Philippine Stock Exchange (PSE). During this period, the One-Price One Market exchange has been achieved through the successful link-up of the two trading floors, PSE-Ayala and PSE-Tektite (Fact Book, 1996). At present, the Phisix has 30 constituent stocks; however, this study is limited only to ten stocks included in the Phisix. Selection of the ten issues was done randomly; however, all the stocks selected were included in all the recompositions of the Phisix for the years 1994, 1996, and 1998. This means that these stocks have satisfactorily complied with the requirements of the PSE to be included in the Phisix (Appendix A). In the estimation and forecasting models of stock return volatility, focus was only on the closing prices. The closing prices were used because investors and/or fund managers look at these prices to determine their positions. At times, they drive the market to close at a certain level by entering the market a few minutes before the close of the trading period. It is noted that investors may not have the time to monitor the market at all instances, thus forcing them to make investment decisions on closing prices. The daily returns (R,) for the Phisix and the selected issues were computed as follows: R, = In (P,/ P ,_, ) (1.1) Where P, today's closing price P ,_, previous closing price In is the natural logarithm Stock return volatility was determined through the computation of the standard deviation. 2 <J = _ {I(r. -r) (1.2) V · n -1 where N is the number of observations r1 is each observation r is the average observation r.v summation sign which means that we add up over all observations DLSU Business & Economics Review Volume 13 No.1 2001-2002 Leila Y. C&lderon 23 As was mentioned earlier, stock return volatility as measured by the unconditional standard deviation is time-varying. Thus, Alexander (1998) suggests the "Generalized Autoregressive Conditional Heteroscedasticity (GARCH) to estimate and forecast volatility. "Generalized," because it is a general class of ARCH model; "Autoregressive," the variances generated by ARCH models involve regression on their own past; and "Conditional Heteroscedastic," means changing variance or volatility clustering. In this GARCH model, there are two equations: (1) the conditional mean and (2) the conditional variance equations. The parameters in both equations are estimated simultaneously, using maximum likelihood estimation. The standard input for a GARCH model for financial market volatility is a series of daily returns, r,, the dependent variable. It is important to note the difference between an estimate and a forecast. In general, historical data is used to estimate volatility or correlation, and the estimate is then used to construct the forecasts. In this case, the GARCH volatility estimates are different from the forecasts. GARCH forecasts of volatility of any maturity can be computed in a simple iterative manner. Put: (1.3) Assuming returns have no autocorrelation, the GARCH forecasts of variance of h-day returns is h a",,,= I cf,,, (1.4) i=1 and the GARCH h-day volatility forecast is 100cr,, • v'250%, assuming 250 trading days per year. An important element of the ARCH model is that it more readily explains ''fat tailed" and leptokurtic distributions of asset price changes. The major applications of ARCH models, have, to date, been modelling correlations between assets and forecasting volatility (Das, 1998). The GARCH model is an infinite order ARCH model. One member of the family of ARCH processes, GARCH (1, 1) has been especially popular because of its parsimony. In this study, the GARCH (1, 1) is used to estimate volatility of the index issues' stock returns. DLSU Business & Economics Review Volunu 13 No.1 2001-2002 24 CORNISH-FISHER EXPANSION Alexander (1996) considered the unconditional standard deviation over a long data period as a benchmark against which to test different volatility forecasting models. In this study, the unconditional standard deviation will be the one used to compare against the different volatility forecasting models. The test of forecasting power lies in out-of-sample (usually post-sample) predictive tests. A certain amount of historic data is withheld from the period used to construct the forecast model. The model is evaluated through the use of two forecast error statistics, namely: root mean squared error and mean absolute error. The smaller the error, the better the forecasting ability of that model according to that criterion. (Take directly from Eviews 3 User's Guide). As mentioned earlier the unconditional standard deviation will be used to compare against the forecasts of the other volatility models. In most cases, the widespread popularity of mean­ variance analysis is due to the fact that it is very simple and powerful. Most of the theoretical and empirical work on portfolio selection and the pricing of financial assets use the mean-variance analysis. However, an inadequacy of relying on the variance alone is that it fails To measure the symmetrical treatment of gains or loses, if these deviations are pleasant or unpleasant surprises. These surprises need not be evenly distributed, however. There may be a large probability of a slightly above average return and a small chance of a catastrophic loss, or there might be a large probability of a small loss and a small chance of a bonanza. To capture this asymmetry, skewness may be a rational explanation. (Reid, 1991 ). The asymmetry of the distribution is called the skewness, or the third central moment, given by n M' = l:Pr(s) [r(s)-E(r)]' (1.5) S=1 where s =return Pr = probability E(r) = expected value of return Cubing the deviations from expected value preserves their signs, which allows us to distinguish good from bad surprises. DLSU Business & Economics Review Volume 13 No.1 2001-2002 Leila Y. Calderon 25 Because, this procedure gives greater weight to larger deviations, it causes the long tail of the distribution to dominate the measure of skewness. Positive numbers are associated with positive · skewness and hence is desirable (Bodie, Kane, Marcus , 1999). As for kurtosis, the fourth central moment, along with the variance (second moment) represent the likelihood of extreme values. Larger values for these moments indicate greater uncertainty (Bodie, Kane, Marcus , 1999). n M4 = IPr(s) [r(s)·E(r)]4 (1.6) S=1 where s =return Pr = probability E(r) = expected value of return Kurtosis is the degree of peakedness of a distribution, usually taken relative to a normal distribution. A distribution having a high peak, and having a value greater than 3 are called leptokurtic; whereas, values of Kless than 3 are platykurtic (flat-topped); and a kurtosis value of 3 is known as mesokurtlc. Most statistical problems rely in some way on approximations to densities or distribution functions derived from asymptotic theory. The central limit guarantees that for most underlying . distributions, the distribution of sample mean Y can be approximated by a normal distribution. This approximation may be improved by transforming the Y's before averaging or by incorporating an adjustment for skewness and kurtosis. In adjusting for skewness and kurtosis, the Cornish-Fisher expansion has been utilized. 2 3 3 z + (1/6) (z 1) p + (1/24) {z ·3 z ) p• (1.7) w.= 0 0 0 0 where z = probability of normal distribution at 5% 0 significance level p 3 = skewness p4 = kurtosis (source for equations: N. Reid, 1991) DLSU Business & Economics Review Volume 13 No.1 2001·2002 26 CORNISH-FISHER EXPANSION In the model, three forecasts will be made: one for the whole sample period from 04 April1994 to 31 March 1999; a 90-day in­ sample period from 18 November 1998 to 31 March 1999; and a 90-day out-of-sample per iod from 05 April 1999 to 10 August 1999. The unconditional standard deviation will be added to or subtracted from the mean to determine the forecasted stock price using the formula: x ± 1.96*s/vn (1.8) Where X - the mean of the closing prices of the stock s - unconditional standard deviation of the returns n - sample size, in this case 1,250 observations for the period 4 April1994- 31 March 1999. The forecasted stock price is based on the actual closing price of the day since under the Martingale Property, "the unconditional expectation of your winnings at any time in the future is just the amount you already hold: (Wilmott, 1998) The same procedure will be done using the mean­ unconditional standard deviation, the mean-conditional standard deviation.
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