Forecasting index volatility: sampling interval and non-trading effects

Transcription

1 Applied Financial Economics, 1998, 8, Forecasting index volatility: sampling interval and non-trading effects DAVID M. WALSH and GLENN YU-GEN TSOU Department of Accounting and Finance, ºniversity of ¼estern Australia, Nedlands ¼A 6907, Australia A detailed comparison is made of volatility forecasting techniques on Australian value-weighted indices. The techniques compared are the naïve approach (historical volatility), an improved extreme-value method (IEV), the ARCH/GARCH class of models and an exponentially weighted moving average (EWMA) of volatility. The study suggests that the EWMA technique appears to be the best volatility forecasting technique, closely followed by the appropriate GARCH specification. Both the IEV and historical volatility approaches are poor by comparison. The diversification benefit that arises from indices with larger numbers of stocks appears to make forecasting the volatility of larger indices more accurate. However, as the sampling interval is reduced, the non-trading effects evident in the larger indices start to counteract this benefit. I. INTRODUCTION Much research has been conducted to assess the ability to forecast volatility of asset returns. The importance of being able to forecast volatility accurately (or at least, more accurately) is apparent in many different areas. For example, option pricing has traditionally suffered without accurate volatility forecasts, since most option pricing models have a forecast of volatility as one of the most (perhaps the most) critical input. Volatility forecasting is also important in controlling for estimation error in portfolios constructed to minimize ex ante risk. The more accurate the forecast, the better the ability to take advantage of the correlation structure between assets. Finally, building and understanding asset pricing models must also take into account the nature of volatility and its ability to be forecast, since risk preferences will be based on market assessment of volatility. In an Australian context, study of the volatility of major indices is clearly important for all of the above four reasons. However, it is also important since the emergence of index funds and index tracking requires substantial knowledge of volatility, its forecastability and the sample interval over which we consider it. We compare and contrast four different volatility forecasting methods; the naı ve approach (historical volatility), an improved extreme-value method, the ARCH/GARCH class of models and an exponentially weighted moving average of volatility. The first of these is a simple historical average, and weights all previous data equally. The second includes the extra information that the extreme values of the sample period can provide, effectively weighting observations with greater extreme values more heavily. The third allows the conditional volatility to evolve, weighting previous volatilities in an autoregressive way. The final is similar (at least in appearance) to the third method, in that it uses an optimized mix of historical volatilities the weighting scheme is hence the optimum over some prior data set. Much of this has been attempted before; see Section II for a discussion of this. However, there are two principal contributions from this article. Firstly, we consider the volatility forecasting methods in the context of a major issue in the use of financial data; the importance of the sampling interval and the resulting difficulty associated with non-trading periods. Secondly, since indices include a diversification effect relative to individual stocks, we study the trade-off between this diversification effect and the sampling interval/non-trading effect. These two issues have not been addressed with regard to volatility forecasting in any previous literature. By choosing index data in the way that we do, we ensure that these issues are drawn out. We use Australian index Routledge 477

2 478 D. M. ¼alsh and G. ½.-G. ¹sou data across three value-weighted accumulation indices (the 20 Leaders, the 50 Leaders and the All-Ordinaries (roughly 300 stocks)), and across different sampling intervals (hourly, daily and weekly). We then assess the ability of each volatility forecasting technique to predict in each of the nine possible combinations of index and sampling interval. (Note that since these indices are market-value-weighted, we can make no conclusions about other types of indices (for example, price-weighted or equally weighted) because by construction the volatilities will be different.) In Australia, more than half of overall market capitalization resides in the top 20 stocks the 20 Leaders. The volume of trading is similarly concentrated. Hence, for a given sampling interval, as we move from 20 Leaders to 50 Leaders to All-Ordinaries, we expect a reduction in volatility (and, we expect, forecast error) due to the diversification across many stocks. However, at the same time, the nontrading problem will increase dramatically, since lessfrequently-traded stocks are being included. This will potentially increase forecast error. Studying the trade-off between the diversification effect and the non-trading effect is made much more interesting by varying the sampling interval for a given index. In this case, there is no diversification effect, so any change in the forecast errors will be due purely to non-trading. This will then allow us to draw conclusions regarding both effects. Section II discusses the volatility measures we use and historical evidence on volatility forecasting. Section III gives the data and methodology that we apply, the results we obtain, and then discusses these results. Section IV gives concluding remarks. II. VOLATILITY MEASURES USED Below we describe in brief the volatility measures used here. When we implement each volatility measure, we use it to forecast as follows. The first forecast is calculated using an initial out-of-sample period (in our case, 12 months). The forecast is then compared to the actual outcome (observed volatility, as squared price changes), the forecast error is recorded and then the estimation rolls forward one interval. This repeats until the data set is exhausted. Naïve historical model This method uses past sample volatility to forecast future volatility and is appealing in its simplicity. The naïve forecast of volatility is given by:» " 1 (u!u*) (n!1) (1) where n#1" number of observations in estimation period, t " forecast interval immediately following estimation period, u " ln(s /S ), S " index price at end of ith interval (i"0, 1,2, n), u* " mean of the u. Note that this technique is an equally weighted model and does not place more emphasis on more current volatility figures. Clearly, this will be inefficient if recent observations are more important that long-distant observations. Alford and Boatsman (1995) studied historical volatility in predicting long-term stock return volatility. They found, like others, that when using historical volatility, lower frequency sampling should be used. That is, the naïve historical method seems to break down with finer time partitioning, when dependency on lagged values of volatility appears to become greater. Improved extreme-value (IE») method Parkinson (1980) suggested that the average difference between maximum and minimum values over a given period would get larger (smaller) as the volatility increased (decreased). Intuitively, the high and low prices encompass important information about the volatility that closing prices do not. A measure which incorporates this extra information will presumably be a better predictor than historical volatility. Kunitomo (1992) extended Parkinson (1980) to incorporate closing prices into the original Parkinson equation by allowing the geometric Brownian motion to have a nonzero drift term. By including a drift term in the stochastic process of stock return variances, Kunitomo was able to markedly improve the efficiency of Parkinson s estimation method. The forecast calculated by the IEV method of Kunitomo (1992) is: where» " 2 R (2) n π¹ R "Max(½ )!Min(½ ) Note that the 50 Leaders is a subset of the All-Ordinaries, and the 20 Leaders is a subset of the 50 Leaders. To calculate the maximum and minimum values over the hourly, daily and weekly intervals, we use samples every 5 minutes. (3) ½ "S!(i/¹ )S (4) n#1 " number of observations in estimation period, t " forecast interval immediately following the estimation period, ¹ "length of each interval.

3 Forecasting index volatility 479 Again, this weights all observations equally, so we would expect it to perform less well than a method that allows weighting of more important (that is, more recent) values. Exponentially weighted moving average (E¼MA) The EWMA method was first used by Akgiray (1989) to forecast the volatility of stocks on the NYSE. It involves forecasting volatility as a weighted average of previously observed volatilities. The most recent observation receives the largest weight, while earlier observations are weighted (geometrically) less according to their age. Commonly, and it is also the case here, only a single lag is used. The application of this technique is slightly different to the others. Here, we use an estimation period of 6 months, a validation period of 6 months and a forecast window of 6 months. The EWMA forecast uses the following equation: where» "θ» #(1!θ)» * θ " smoothing parameter,» " historical volatility calculated over the estimation period (which is 6 months), * t " forecast interval immediately following validation period. Note that Equation 5 contains only a single lagged forecast value. The EWMA technique operates in three steps. The first step involves calculating the historical volatility estimate» over the initial 6 month estimation * period, and then rolling it forward one interval for 6 months. This yields an historical estimate for each interval in the 6 months following the initial estimation period. This second 6 month period is known as the validation period. The second step is to calculate» values for each interval in the validation period using Equation 5, with each value dependent on the (unknown) parameter θ. These values are then compared with the actual values to generate a one-step ahead forecast error. Each error is dependent on θ. To complete the second step, the value of θ that minimizes the mean squared error (MSE) over the validation period of 6 months is chosen to be the correct value of of θ. The third step is to use this optimum value of θ in a 6 month forecast window following the validation period. (5) Now that we have generated an estimate of θ, Equation 5 can be used to forecast out-of-sample, and one-step-ahead forecast errors can be generated. At the end of the 6 month forecast window, the process halts, and we repeat from step 1 with the data set moved forward 6 months. Generalized autoregressive conditional heteroscedasticity (GARCH) Merton (1980) criticized the failure of some researchers to account for the effects of changes in the level of risk when estimating expected returns. One of his conclusions was that estimators which use realized returns should take account of heteroscedasticity and further research should develop accurate variance estimation models which take account of the errors in variance estimates. Engle (1982) took up this idea and developed the autoregressive conditional heteroscedasticity (ARCH) model which specifically allows for changing conditional volatility. The ARCH model rests on the presumption that forecasts of volatility at some future point in time can be improved by using recent information. In particular, ARCH models rely on volatility clustering. This means that if large changes in financial markets tend to be followed by large changes, in either direction, then volatility must be predictable. The GARCH model, developed by Bollerslev (1986), generalized the ARCH model of Engle (1982) by allowing the current conditional volatility to be a function of the first p past conditioned volatilities as well as the q past squared error terms (or innovations) covered by ARCH(p) models. Hence, this model relies on the tendency for extreme returns to be followed by other extreme returns. The GARCH (p, q) process is given by: where ε φ &N(0,» )» "α # α ε # β» (6) ε " error term which provides for uncertainty in the stochastic process and results from regressing the return on a constant, φ " information set at time t!1, α " constant term, α and β " coefficients estimated by the GARCH process. The first lagged value needed in Equation 5 is taken to be the historical estimate over that period. The search technique used was the Newton Rhapson algorithm, applied to the MSE formula. Typically, for this study, the value of θ ranged from 0.66 to 0.92 which is in line with previous studies which used EWMA and reported their θ value. For example, Akgiray s (1989) estimated θ values ranged from 0.76 to Tse (1991) used a θ value of On the other hand, Brailsford and Faff (1996) had a minimum of 0 (measured after October 1987) to a maximum of 0.90.

4 480 D. M. ¼alsh and G. ½.-G. ¹sou Table 1. iterature review of the performance of volatility measures in different countries Best Country Author(s) forecast method Other measures of volatility compared against USA* Akgiray (1989) GARCH (1, 1) Historical and EWMA USA Heynen (1995) Stochastic volatility Random walk, GARCH (1, 1) and EGARCH (1, 1) UK Dimson and Marsh EWMA Historical, ARCH and GARCH (1990) UK Poon and Taylor GARCH-M ARCH and EWMA (1992) UK Corhay and Rad GARCH (1, 1) Various orders and models of ARCH and GARCH UK Heynen (1995) Stochastic volatility Random walk, GARCH (1, 1) and EGARCH (1, 1) UK Vasilellis and Meade GARCH Historical and EWMA (1996) Australia Brailsford and Faff GARCH (3, 1) Various orders of ARCH and GARCH (1993) Australia Heynen (1995) Stochastic volatility Random walk, GARCH (1, 1) and EGARCH (1, 1) Australia Brailsford and Faff (1996) GJR-GARCH (1, 1) Historical, Random walk, 5 and 12 year moving average, Exponential smoothing, EWMA, Regression, GARCH (1, 1), GARCH (3, 1) and GJR-GARCH (3, 1) Canada Calvet and Rahman IGARCH Various orders of ARCH and GARCH (1995) Finland Booth et al. (1992) GARCH (1, 1) Various orders of ARCH and GARCH France Corhay and Rad GARCH (1, 1) Various orders and models of ARCH and GARCH Germany Corhay and Rad GARCH (1, 1) Various orders and models of ARCH and GARCH Holland Corhay and Rad GARCH (1, 1) Various orders and models of ARCH and GARCH Hong Kong Heynen (1995) Stochastic volatility Random walk, GARCH (1, 1) and EGARCH (1, 1) Italy Corhay and Rad IGARCH (1, 1) Various orders and models of ARCH and GARCH Japan Tse (1991) EWMA Historical, ARCH and GARCH Japan Heynen (1995) Stochastic volatility Random walk, GARCH (1, 1) and EGARCH (1, 1) Singapore Tse and Tung EWMA Historical, ARCH and GARCH (1992) *For further examples of US studies, see Bollerslev et al. (1992). Generally, the GARCH (1, 1) is the most appropriate at forecasting market returns compared to various orders and models of ARCH and GARCH Precedent literature There has been considerable precedent literature on this topic, conducted in various countries, to measure the volatility of the securities of that country. The more relevant studies which have utilized the same volatility measures as this paper are presented in Table 1. The consensus of previous research is that methods of volatility estimation that have weighted recent observations more heavily that older observations (that is, EWMA and GARCH) are more successful. This is clear from a number of studies including Tse (1991), Tse and Tung (1992) and Corhay and Rad. However, the benefits of one of these two methods over the other has not been resolved. Comparative studies by Akgiray (1989), Vasilellis and Meade (1996), Brailsford and Faff (1993, 1996) and Kroner (1996) suggest that GARCH and EWMA perform comparably. However, Tse (1991) showed that GARCH forecasts were slow in reacting to changes in volatility compared to the EWMA method. Therefore, the GARCH models may not be superior in periods when volatility changes vary abruptly. Little comparison has been conducted between the extreme value and GARCH or EWMA methods. One of the few examples is Lin and Rozeff, who find that when a high low spread is added to the GARCH equation, GARCH effects disappear. They concluded that the extreme-value method could partly explain conditional

5 Forecasting index volatility 481 volatility. Hence, the clear difference between the GARCH and EWMA methods and the extreme value method may not be as clear as first thought. Most studies that examine the extreme value method in comparison with the historical method have found that it is considerably better than the historical method. Evidence of this includes Garman and Klass (1980), Parkinson (1980), Bookstaber (1991), Kunitomo (1992), and Turner and Weigel (1992). However, Beckers (1983) found that the information content and estimation error of the high-low estimates were not always superior to the close-to-close variance, and Cho and Frees (1988) find that price discreteness caused the extreme-value method to be significantly downward biased relative to the close-to-close method. These results were supported by Marsh and Rosenfeld (1986). Also, Garman and Klass (1980) demonstrated that non-continuous trading (that is, non-trading periods) cause the extreme-value estimators to be biased downwards, and Wiggins (1991) showed that the extreme-value method is sensitive to outliers in daily high and low prices, especially for those securities with relatively low trading volumes. In summary, the expected forecasting ability is not clearly ranked among the four methods. The historical method is normally the worst, followed by the extreme value method, but the exact ranking of the extreme value method, GARCH and EWMA has not been determined: it seems to vary between data sets and across countries. A plausible explanation of the inability of one method to dominate is the sampling interval used and the non-trading effects that are observed as a result. No studies have examined these effects, so it is our aim here to throw light on how these methods are affected by this. To eliminate spurious effects specific to individual stocks, we use diversified portfolios (market-value-weighted indices) and concentrate on forecasting these volatities across different sampling intervals. III. DATA, METHODOLOGY AND RESULTS Data Time- and date-stamped samples from the three price indices were taken every five minutes from 1 January 1993 to 31 December From this, hourly, daily and weekly values were found as the value at the close of each hour, day or week. Problems with the construction of the accumulation indices prior to these dates prevent satisfactory analysis, although three years of data appears to be more than enough for our purposes. Also, problems with missing data were relatively small over this period (only several days), and daily missing values were easily inserted. The data were obtained directly from the SIRCA database. Unfortunately, the SIRCA database does not contain the corresponding accumulation values of these indices at an intraday level (only the price indices, which include everything except dividend payments). Hence dividend yield was obtained from the Australian Stock Exchange (ASX) Monthly and Weekly Index Analysis reports, and the accumulation indices reconstructed from the price index and the dividend yield. Comparison of the forecast techniques takes place over the last two years of the three-year sample in each technique, the first year is used as an initial out-of-sample period for calculation of parameters. One-step-ahead forecast errors In this study we examine the ability of the four volatility forecasting techniques listed above to predict volatility. For this, we use Australian index data across three valueweighted accumulation indices (the 20 Leaders, the 50 Leaders and the All-Ordinaries), and across different sampling intervals (hourly, daily and weekly). We then assess the ability of each volatility forecasting technique to predict in each of the nine possible combinations of index and sampling interval. As noted in Section I, we construct the tests in this way to compare the abilities of the various techniques and to test and contrast the trade-off between diversification and nontrading effects. Each volatility forecast technique has hence generated a set of forecasts for each interval, throughout the data set. We then calculate the forecast error for each technique for each interval, by comparing the forecast to actual volatility. Actual volatility is measured simply by taking the square of price changes. The forecast errors are then compared using four loss functions; mean squared error (MSE), root mean squared error (RMSE), mean absolute error (MAE) and the mean absolute percentage error (MAPE). Their calculations are as follows: MSE" 1 n (»!» ) (7) Ball and Bowers (1987) found errors in the construction of the accumulation indices by the ASX; daily dividends were not reinvested in the index portfolios until the end of that year, meaning that accumulation and prices indices were identical. Rather than reconstruct the accumulation indices, more recent (correct) index values are used here. Obtained from the ASX Monthly and Weekly Index Analysis reports. Securities Industry Research Centre of Asia-Pacific.

6 482 D. M. ¼alsh and G. ½.-G. ¹sou RMSE" n 1 (»!» ) (8) MAE" N 1 (»!» ) (9) MAPE" 100 n ((»!» )/» ) (10) The method with the lowest forecast error for all four techniques will be the best (least error) predictor of return volatility across that particular time interval. The relative values of each forecast method are also compared to the worst method, by normalizing the error of the worst method to a value of 1.00 and using it as a benchmark to assess the rest. Hence, the lowest relative value will signal the best forecast method. Results Hourly data analysis results. The results of the intraday (hourly) data analysis are presented in Table 2. Unfortunately, intraday data do not support the calculation of the MAPE error statistic, since it is possible to obtain zero actual volatility in an interval. Note also that the relevant GARCH specification was the GARCH(1, 1). From Table 2, the best predictors of volatility are the EWMA and GARCH(1, 1) techniques, depending on the loss function used. The difference between them is, in any case, slight. Interestingly, IEV is worse at forecasting than even the naïve model as it has larger forecast errors for two of the two three indices, namely the 20 Leaders and All Ordinaries Accumulation Index. This result is unexpected given that the original extreme-value method was found to be better than the historical method by Parkinson (1980), Garman and Klass (1980) and later by Kunitomo (1992). As mentioned above, as we move from 20 Leaders to 50 Leaders to All-Ordinaries, we have two effects. Diversification will, we expect, tend to make the forecast errors smaller, but we expect the non-trading effect to increase forecast error. We see here strong evidence of this trade-off. The 50 Leaders has, in general, greater forecast error than either 20 Leaders or the All-Ordinaries, no matter how we measure this error. We can reasonably ascribe the increase in error from 20 Leaders to 50 Leaders as being due to the nontrading effect, since the stocks unique to the 50 Leaders are quite thinly traded. The decrease in error from 50 Leaders to All-Ordinaries would seem to be due to the increased diversification effect evident in the All-Ordinaries. In fact, the diversification effect has outweighed the non-trading effect completely, because forecast errors in the All-Ordinaries are smaller than those for the 20 Leaders. Daily data analysis results. The daily results are summarized in Table 3. Both GARCH (1, 1) and GARCH (1, 2) have been included because the likelihood ratio test failed to reject GARCH (1, 1) in favour of GARCH (1, 2), but the second lag of conditional variance was still significant in the conditional variance equation. Hence, including both simply ensures that we are more confident of our results. Surprisingly, the larger number of coefficients in the GARCH (1, 2) does not improve on the prediction of Table 2. Forecast error statistics from hourly volatility MSE RMSE MAE Actual Relative Actual Relative Actual Relative 20 LEADERS Historical IEV EWMA GARCH (1, 1) LEADERS Historical IEV EWMA GARCH (1, 1) ALL-ORDS Historical IEV EWMA GARCH (1, 1) Bold entries indicate the best technique.

7 Forecasting index volatility 483 Table 3. Forecast error statistics from daily volatility MSE RMSE MAE MAPE Actual Relative Actual Relative Actual Relative Actual Relative 20 LEADERS Historical IEV EWMA GARCH (1, 1) GARCH (1, 2) LEADERS Historical IEV EWMA GARCH (1, 1) GARCH (1, 2) ALL-ORDS Historical IEV EWMA GARCH (1, 1) GARCH (1, 2) Bold entries indicate best technique the GARCH (1, 1); in fact, the prediction is worse than the GARCH (1, 1) model. Again, the results are relatively similar to those discussed in the hourly results section above. The EWMA and GARCH techniques are superior, but it is difficult to distinguish between them because the results depend on the loss functions used. This is interesting because previous studies have either found EWMA to be much better (for example, Dimson and Marsh, 1990; Tse, 1991; and Tse and Tung, 1992) or GARCH was better (for example, Vasilellis and Meade, 1996 in UK and Akgiray, 1989). This discrepancy is curious. A possible explanation could be, as suggested by Brailsford and Faff (1996), that the error statistics are not symmetric. Therefore, to properly assess this problem, it would be necessary to use an asymmetric measure of forecast error. Again, the poor performance of the IEV technique is a mystery, since previous authors have found it to be useful. We find it to be less successful even than the historical estimation technique. The comparison across the different indices is different at the daily level to the hourly level. The errors decrease uniformly as we move from 20 Leaders to 50 Leaders to All-Ordinaries. Since at a daily level the non-trading problem would be much less than at the hourly level, this is strong evidence that the decrease is due to diversification, and that the non-trading problem has less effect. Also note that the errors are less at the daily level than at the hourly level, and that this reduction is most for 20 Leaders and least for the All-Ordinaries. ¼eekly data analysis results. Table 4 shows the results of the weekly data analysis. No GARCH (p, q) models converged, so only three of the four volatility measures are presented here. Not surprisingly, EWMA is again the best predictor of weekly volatility. However, an interesting point to note is that its superiority is marginal (0.748 to to the worse model) as shown in the relative column under each error statistic. For hourly data, the EWMA relative was to a maximum of and to for daily data. In other words, the predictive superiority of the EWMA model declines as the sample interval widens. Furthermore, the historical model is now a worse predictor than IEV, unlike intraday and daily data. These results suggest that the benefits of the superior methods decreases as sampling interval increases, because we are less likely to have the strong clustering of volatility that we observe at high frequencies. It also suggests that extreme values had greater information content (or contained less noise) at longer sampling intervals. One last point is that the error measures again fall (in general) from the 20 Leaders to the All-Ordinaries. This is similar to the result from the daily sample, and shows that the non-trading problem becomes much less as the sample interval increases. Even though many stock included in the All-Ordinaries do not trade in any one day, the diversification effect strongly outweighs the non-trading problem. When considering the non-trading problem in the assessment of volatility forecasts, it would seem that the problem

8 484 D. M. ¼alsh and G. ½.-G. ¹sou Table 4. Forecast error statistics from weekly volatility MSE RMSE MAE MAPE Actual Relative Actual Relative Actual Relative Actual Relative 20 LEADERS Historical IEV EWMA LEADERS Historical IEV EWMA ALL-ORDS Historical IEV EWMA Bold entries indicate best technique is greatest for very short sampling intervals, and diversification tends to reduce the impact. The problem could be particularly bad when forecasting volatility using high frequency sampling of an individual stock. IV. SUMMARY AND CONCLUSIONS The results we obtain are interesting. For any sampling interval, we find that EWMA and GARCH models dominate for all of the loss functions used. If EWMA did not dominate, it was a close second to the optimal GARCH model fitted to the data. Historical volatility and the IEV method were dominated by EWMA and GARCH in every case. However, for hourly and daily data, the IEV was often out-performed by historical volatility, although this effect was not evident in the weekly analysis. This supports the work of Wiggins (1991) who notes that the IEV method is very sensitive to non-trading effects and, in particular, to outliers. These effects will reduce as we increase the sampling interval, and so we see its improvement as a forecast (although it was never a better forecasting tool than GARCH or EWMA). In general, however, our results support those of previous researchers. The conclusions regarding the non-trading effect are also interesting. All forecast error measures decrease as sampling interval increases, for all indices. Further, for a given index, sampling more frequently in every case increases the forecast error. This suggests strongly that the non-trading effect evident in more thinly traded stocks has the effect of reducing our ability to forecast volatility. However, for daily and weekly sampling, the forecast error (by every measure) decreased from 20 Leaders to 50 Leaders and from 50 Leaders to the All Ordinaries. It appears that diversification effects reduce the volatility forecast error of the index. Although non-trading effects must increase, they are not large enough to counteract the diversification benefit that arises from increasing the number of stocks in the index at daily and weekly sampling. This result is different for the hourly sample; the largest forecast error is attributed to 50 Leaders, followed by 20 Leaders, followed by the All Ordinaries. Again, the index containing the most stocks has the least forecast error, but the order of the other two has reversed. This result suggests that the non-trading problem has increased to such a level that it is now starting to affect the diversification benefit that we saw in the daily and weekly samples. In summary: The EWMA technique appears to be the best volatility forecasting technique, closely followed by the appropriate GARCH specification. Both the IEV and historical volatility approaches are poor by comparison. The diversification benefit that arises from indices with larger numbers of stocks appears to make forecasting the volatility of larger indices more accurate. However, as we reduce the sampling interval, the non-trading effects evident in the larger indices start to counteract this benefit. In the absence of any diversification effect, for example, when attempting to forecast volatility of a single stock, the non-trading effect will not be counteracted at all and the forecast errors will increase as the frequency of trading decreases. This must have severe implications for the accuracy of forecasting volatility of infrequently traded stocks.

9 Forecasting index volatility 485 REFERENCES Akgiray, V. (1989) Conditional heteroscedasticity in time series of stock returns: evidence and forecasts, Journal of Business, 62, Alford, A. and Boatsman, J. (1995) Predicting long-term stock return volatility: Implications for accounting and valuation of equity derivatives, Accounting Review, 70, ASX Market Information Department, ASX Monthly and ¼eekly Index Analysis, various issues. Ball, R. and Bowers, J. (1987) A corrected Statex-Actuaries daily accumulation index, Australian Journal of Management, 12, 1 8. Beckers, S. (1983) Variances of security price returns based on high, low, and closing prices, Journal of Business, 56, Black, F. and Scholes, M. (1973) The pricing of options and corporate liabilities, Journal of Political Economy, 81, Bollerslev, T. (1986) Generalised autoregressive conditional heteroscedasticity, Journal of Econometrics, 31, Bollerslev, T., Chou, R. Y. and Kroner, K. F. (1992) ARCH modelling in finance: a review of the theory and empirical evidence, Journal of Economics, 52, Bookstaber, R. (1991) Option Pricing and Investment Strategies, 3rd edition, Probus Publishing Company. Booth, G., Hatem, J., Virtanen, I. and Yli-Olli, P. (1992) Stochastic modelling of security returns: evidence from the Helsinki Stock Exchange, European Journal of Operational Research, 56, Brailsford, T. and Faff, R. (1993) Modelling Australian stock market volatility, Australian Journal of Management, 18, Brailsford, T. and Faff, R. (1996) An evaluation of volatility forecasting techniques, Journal of Banking and Finance, 20, Calvet, L. and Rahman, A. (1995) Persistence of stock return volatility in Canada, Canadian Journal of Administrative Sciences, 12, Cho, D. and Frees, E. (1988) Estimating the volatility of discrete stock prices, Journal of Finance, 43, Corhay, A. and Rad, A. Statistical properties of daily returns: evidence from European stock markets, Journal of Business Finance and Accounting, 21, Dimson, E. and Marsh, P. (1990) Volatility forecasting without data-snooping, Journal of Banking and Finance, 14, Engle, R. (1982) Autoregressive conditional heteroscedasticity with estimates of variance of United Kingdom inflation, Econometrica, 50, Garman, M. and Klass, M. (1980) On the estimation of security price volatilities from historical data, Journal of Business, 53, Heynen, R. C. (1995) Essays on Derivatives Pricing ¹heory, Thesis Publishers, Amsterdam. Kroner, K. F. (1996) Creating and using volatility forecasts, Derivatives Quarterly, Winter, Kunitomo, N. (1992) Improving the Parkinson method of estimating security price volatilities, Journal of Business, 65, Lin, J. and Rozeff, M. Variance, return and high-low prices spreads, Journal of Financial Research, 17, Marsh, T. and Rosenfeld, E. (1986) Non-trading, market making, and estimates of stock price volatility, Journal of Financial Economics, 15, Merton, R. (1980) On estimating the expected return on a market: an exploratory investigation, Journal of Financial Economics, 8, Parkinson, M. (1980) The extreme value method for estimating the variance of the rate of return, Journal of Business, 53, Poon, S. and Taylor, S. (1992) Stock returns and volatility: an empirical study of the UK stock market, Journal of Banking and Finance, 16, Tse, Y. (1991) Stock returns volatility in the Tokyo Stock Exchange, Japan and the ¼orld Economy, 3, Tse, Y. and Tung, S. (1992) Forecasting volatility in the Singapore stock market. Asia Pacific Journal of Management, 9, Turner, A. and Weigel, E. (1992) Daily stock market volatility: , Management Science, 38, Vasilellis, G. and Meade, N. (1996) Forecasting volatility for portfolio selection, Journal of Business Finance and Accounting, 23, Wiggins, J. (1991) Empirical tests of the bias and efficiency of the extreme-value variance estimator for common stocks, Journal of Business, 64,