Cornish-Fisher Expansion on Estimation and Forecasting ......simultaneously, using maximum likelihood estimation. The standard input for a GARCH model for financial market 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.

<J = _ {I(r. -r)2

V · n -1

where N is the number of observations r

1 is each observation

r is the average observation

(1.2)

r.v summation sign which means that we add up over all observations


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,,, i=1

(1.4)

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


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 meanvariance 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)]'

S=1

where s =return = probability

(1.5)

Pr E(r) = expected value of return

Cubing the deviations from expected value preserves their signs, which allows us to distinguish good from bad surprises.


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

S=1

where s Pr E(r)

=return = probability = expected value of return

(1.6)

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.

w.= z0+ (1/6) (z2

0 1) p3+ (1/24) {z30

·3 z0

) p• (1.7)

where z0

= probability of normal distribution at 5% significance level

p3 = skewness p4 = kurtosis

(source for equations: N. Reid, 1991)

DLSU Business & Economics Review Volume 13 No.1 2001·2002


In the model, three forecasts will be made: one for the whole sample period from 04 April1994 to 31 March 1999; a 90-day insample 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 meanunconditional standard deviation, the mean-conditional standard deviation. The period will cover the out-sample period of 01 April 1999 to 01 June 1999.

DISCUSSION OF RESULTS

The (1, 1) in GARCH (1, 1) indicates that a>" is based on the most recent observation of !!2, and the most recent estimate of the variance rate. The more general GARCH (p,q) model calculates a•, from the recent p observations on !!2 and the most recent q estimates of the variance rate (Hull, 2000:90).

The GARCH (1, 1) specification was used for all the stocks. (TABLE 1). The GARCH (1, 1) specification was used since it has been shown to be a parsimonious representation of conditional variance that adequately fits many economic time series (e.g., Bollerslev, i 987). A succinct measure of the persistence of



variance as measured by GARCH is the I( a, + pf)< 1 (where i >0): as this sum approaches unity, the greater is the persistence of shocks to volatility (Lamourex and Lastrapes, 1990). The GARCH (1, 1) results for all the stocks show that there is a large GARCH lag coefficient which indicates shocks to conditional variance take a long time to die out so volatility is persistent. This means that the market takes time to absorb the impact of information arrival before adjusting to the true price of the stock. As was mentioned earlier, in the GARCH models, volatility clustering is evident. This means that "large changes tend to be followed by large changes of either sign, and small changes tend to be followed by small changes .... " (Kon and Kim, 1994; Fama, 1965; and Mandelbrot, 1963).

On the other hand, large GARCH error coefficients mean that volatility is quick to react to market movements and volatility tend to be spikier. The size of the GARCH lag coefficient and GARCH error coefficient determine the shape of the resulting volatility time series in this case, the returns of the stocks.

Table 1. GARCH (1 1) Results '

ARCH(1) GARCH(1) ABS 0.241703 0.521692 AC 0.146486 0.767395 ICTSI 0.590300 0.104320 MBTC 0.576499 0.507600 MER 0.108574 0.863562 PCOR 0.111118 0.812984 PNB 0.253918 0.681568 PX 0.024829 0.973087 SMC 0.065370 0.929910 TEL 0.160349 0.

Given the set Q0

of all information about past and present returns (R., R,, ... ),forecasts of variance of future returns (either var (R,I Q.,), or var (R, + ... +R. I Q.,) for some N) may be obtained. Forecasts of future variance are useful for several reasons. First of all, the predictive capabilities of ARCH and GARCH models constitute further evidence as to their overall usefulness as



practical models of stock returns and also about their relative merits as such. Second, since risk is inherently related to volatility, expected future volatility in a major factor in the pricing of securities {Akigiray, 1989).

Based on the {G) ARCH models, a volatility forecast is constructed for each index stock and PHISIX. Appendix B presents the forecast using the in-sample for 1 ,250 days. 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 {Eviews User Guide). The results show that the errors are small.

The G {ARCH) model is more advantageous than the standard deviation since it models time-varying volatility as compared to the unconditional standard deviation. However, the above models considered only the first two moments, the mean and the standard deviation. A limitation of the standard deviation or variance is that it does not matter if these deviations are pleasant or unpleasant surprises, since the variance squares the mean.

At present, investors do a portfolio selection through the mean-variance analysis. Such analysis is considered simple. However, investors can iook beyond the mean-variance analysis by including skewness and kurtosis.

Skewness is computed by cubing the deviations from expected value which preserve their signs, and allows one to distinguish good from bad surprises. 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 are desirable {Bodie, Kane and Marcus, 1999). This means that investors can look at the trading band of a stock for a certain period and decide whether to include the stock in the portfolio. If there are sudden deviations, the investor must look whether the skewness is positive or negative. If the skewness is positive, then this means that after a sudden surge in prices, the trading band remains at positive, with little deviations. However, if the skewness is negative, this means that a sudden surge in stock price is reversed, wiping out all possible gains. Stock prices may continue to fall until it settles down. However, in the Philippine stock exchange, there is a mechanism to halt sudden surge or drop. At present, the price freeze rule says that when the stock


Lelia Y. Galderon 29

price has gone up 50% or has gone down 40% from its previous close, trading of this stock will be halted at the ceiling or floor price. The trading halt will only be lifted if the corporation has come out with the proper disclosure on why there was a sudden surge or drop. However, such price freeze rule may not be able to capture the volatility of the stock return for the day. Investors who come in late in the market are penalized because they cannot immediately profit if the price freeze rule is implemented. However, the investors who see that prices are going up fast will have gains if they have sold the stocks before the price freeze.

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 and Marcus,1999). This is evident on the spikes that are exhibited in the returns. Investors must be able to look beyond the standard deviation. Favorable or unfavorable information arrival may cause stock prices to move up or down. If an investor buys at a low price, a surge in the price because of good news means that he/she gains profit. On the other hand, if the investor buys the stock at a high price, and the bad news comes in, this may result to losses. If the return is leptokurtic, i.e., fat tails and tall peaks, the stock prices are trading a certain band, a support and resistance level. However, there will be a time that the stock price will break the resistance level and move to the next support level or even higher or lower than expected.

However, an investor can possibly be moving ahead of other informed investors, if aside from the possible expected information, he/she has determined the four moments: mean, standard deviation, skewness, and kurtosis. In an efficient market, there is mean-reversion, which means that eventhough prices move up or down, these will eventually settle at the mean price. Investors must use the standard deviation-recognized as time varying-in evaluating their portfolio selection. Likewise, if the investor looks at the stock returns' skewness, he/she will note that most of the returns stay in a certain band. Nevertheless, unexpected news may cause stock prices to surge or plummet. If the investor has positioned himself/herself early on, then he/ she will have possible gains. Table 2 shows the results of forecasted stock prices based on the mean-constant standard deviation, meantime-varying standard deviation, and the meantime varying standard deviation, skewness, and kurtosis (four


.30 CORNISH-RSHER EXPANSION

moments) on a 90-day out-of-sample. The four moments forecasts have a larger range which takes into account the surprises and sudden jumps in stock prices. This method is deemed to be better than the time-varying standard deviation and mean analysis since it takes into account both the skewness and kurtosis.

Table 2 Forecasted Stock Prices ACTUAL Projected Projected Projected Projected Projected Projected

CLOSING Prices {P) Prices (P) Prices (P) Prlces(P) Prices{P) Prices(P) PRICE Based on Based on Based on Based on Based on Based on

(P) Mean-std Mean-std Mean-std Mean-std Mean-std Mean-std {constant) {constant) (time (lime (four (four

(·) (+) varying) varying) moments} moments} (·) (>) (·) (+)

ABS 39 38.9 41.7 38.9 39.0 36.1 41.8

AC 10.50 10.52 10.51 10.47 11.18 11.16 11.16

ICTSI 4.15 4.15 4.15 4.17 4.16 3.91 4.38

MBTC 320.00 319.36 320.35 319.55 320.16 296.22 343.49

MER 98.5 98.28 98.50 98.71 98.51 78.01 118.72

PCOR 3.50 3.49 3.50 3.49 3.50 3.17 3.82

PNB 138 137.49 138.11 137.53 138.10 124.96 150.65

PX 0.46 0.46 0.46 0.46 0.46 0.43 0.49

SMC 66.50 66.34 66.55 66.34 66.55 59.71 73.18

TEL 1100.00 1097.69 1101.32 1097.47 1101.54 1072.72 1102.53

Note. Forecasted stock pnces are based on Aug 10, the 90" day out~of·sample.

The following table summarizes the average price range of the selected stocks based on the four moments models for a ninety-day period. Investors can look at this price range to consider in their investment decisions since aside from the mean and timevarying volatility, skewness and kurtosis have been accounted for.

Table 3 Four Moments Model

Stock Low (P) High (P) ABS 28.94 33.54 AC 10.47 17.032 ICTSI 3.84 4.30 MBTC 340.45 394.77 MER 77.90 118.55 PCOR 3.87 4.66 PNB 96.05 115.81 PX 0.38 n44 RMC 57.13 7n.n::>

TEL 1,114.87 1,145.85



To determine whether the stock return volatility is overvalued or undervalued, the study benchmarked the index return volatility. If the issue's return volatility is greater than the index return volatility, then it is overvalued. At this point, the investor must not invest in this issue if he/she is avoiding risk. However, such volatility results will also provide the investor better profits at a shorter time. On the other hand, if the issue's return volatility is less than the index return volatility, then it is undervalued. Investors may look at this issue and invest on it but it may take time before they can recognize possible gains. Also, in doing a portfolio selection, investors must also look at stock return volatility forecast on a short-term basis, at most 91 days, parallel to the treasurybill rate. Such short-term forecasts are more accurate than longterm volatility forecasts since information keep on arriving and investors can immediately react on them.

An investor will do well in his portfolio selection if he/she looks at the four moments. Since reversals do not occur very quickly, it is hard to attribute the changes in volatility to a trading-related phenomenon. Ex1reme return days occur on average following substantial losses, and jumps in stock prices are intertemporally clustered.

"If these changes persist, many investment strategies must be re-evaluated. Rebalancing strategies of various kinds, especially short-term rebalancing periods, will be more likely to be whipsawed than in the past. Strategies which require fast trading on short notice may fail, on extreme-move days, and strategies which are sensitive to price jumps such as certain option, convertibles, or dynamic strategies may not perform as expected. Such strategies are, in effect, riskier than previously believed." (Turner and Weigel1982, p.1703).

How does a trader/ market player use this information for trading? In case of a foreign fund manager, the standard deviation maybe a tool in asset allocation. Foreign fund managers allocate funds by regions. And the alloted amount in the region, for example, Asia, is divided among the different markets in the region. The fund manager can allocate a greater portion to the country that has the lowest standard deviation since this is a less volatile market. Or in cases that the volatility is almost the same for all countries since they are in a regional crisis, then fund managers can move from equities to the fixed income market. The fund manager can look at the skewness and kurtosis of the stock

DLSU Business & Economics Review Volume 13 No.] 2001-2002


returns and combine positively skewed with negatively skewed to minimize risk.

For the Filipino fund manager, likewise, the same trading strategy can be done. If the market is too volatile as compared to previous years, then the allocation on the equities market can be reduced and invested more in the fixed income.

For stock brokerages, if concentration is focused on the index issues, then they can invest in those, which have the lowest standard deviation. They can also look at the price range forecasted using the four moments, buy at the lowest price, and wait for the stock to move up.

In portfolio selection, the investor can combine stocks that are negatively correlated, moving in opposite direction. The investor can look at the four moments, particularly skewness, to determine whether the deviation is favorable or not. The kurtosis of the stock return shows the magnitude of this deviation. The investor can take advantage of these four moments by choosing stocks belonging to different industries to minimize risk. In our sample of ten selected stocks, the investor can combine food and real estate or banking. This maybe a good combination since food is consumer-generated and real estate or banking will be affected by interest rates. If for example, banking stocks are down, it could be possible that food stocks are up since this depends on consumer consumption. The investor can also look at the skewness and kurtosis of each stock, and to minimize risk, combine positively skewed with negatively skewed stock returns. If the investor has bought the stock early before an unanticipated good news that would result into an increase in stock price, then the investor will have possible profits. However, if the investor bought the stock at a high price, unfavorable news will cause the stock price to plummet and result to possible losses. To minimize on possible losses, an investor must be able to diversify his/her portfolio. However, an investor may only minimize his/her risk through diversification. The investor may not be able to totally eliminate risk since the investor has to deal with non-diversifiable risk.


BIBLIOGRAPHY

BOOKS


Alexander, C. {Ed.) The Handbook of Risk Management and Analysis. N.Y.: John Wiley and Sons, Ltd., 1996.

Bodie, Z., Kane, A., & Marcus, A. Investments. 4111 Edition.

Campbell, J., Lo, A., & MacKinlay, {1997). The Econometrics of Financial Markets. Princeton University Press.

Damodaran, Aswalth.lnvestment Valuation. N.Y.:John Wiley & Sons, Inc., 1996.

Das, S. {Ed). Risk Management and Financial Derivatives: A Guide to the Mathematics. N.Y.: McGraw-Hill, 1998.

Eviews 3. 1 User's Guide Manual. 2"" edition.

PSE FACT BOOKS. {1997; 1996; 1995; 1994).Philippine Stock Exchange Research Department.

N. Reid. {1991 ). Statistical Theory and Modelling. In Honor of Sir David Cox, FRS, Chapter 12: Approximations and asymptotics. Chapman & Hall.

Smith, G. {1991). Statistical Reasoning. Allyn and Bacon. Wilmott, P. {1998). Derivative. John Wiley & Sons.

JOURNALS

Bollerslev, T. "Generalized Autoregressive Conditional Heteroscedasticity." Journal of Econometrics 31 { 1986) 307-327.

Engle, R. "Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation. • Econometrica, Vol. 50, No.4 (1982) 987-1007

Turner, A. & Weigel, E. "Daily Stock Market Volatility: 1928-1989." ManagementScience, Vol.38, No.11 {1992):1568-1608.



Appendix A PSE Composite Index (PHISIX) FOR THE YEARS 1994, 1996, 1998

1998 PSE Composite Index 1. Metropolitan Bank and Trust Company 2. Philippine Long Distance Telephone Company 3. San Miguel Corporation 4. Jollibee Foods Corporation 5. Manila Electric Company 6. International Container Terminal Services, Inc. 7. Ayala Corporation B. Belle Corporation 9. DMCI Holdings, Inc. 10. Southeast Asia Cement Holdings, Inc. 11. lonics Circuits Inc. 12.C& P Homes, Inc. 13.Ayala Land, Inc. 14. SM Prime Holdings, Inc. 15. Lepanto Consolidated Mining Company 16. Patron Corporation 17. Phil. Commercial International Bank, Inc. 1 B. Filinvest Development Corporation 19. Ben pres Holdings Corporation 20. Filinvest Land, Inc. 21.JG Summit Holdings, Inc. 22. Philippine National Bank 23.ABS-CBN Broadcasting Corporation 24. Metro Pacific Corporation 25. La Tondetia Distillers, Inc. 26. Fii-Estate Land, Inc. 27. Digital Telecommunications Phils, Inc. 2B. Pilipino Telephone Corporation 29.Aboitiz Equity Ventures, Inc. 30. Megaworld Properties and Holdings, Inc.

1996PHISIX 1. Metropolitan Bank and Trust Company 2. Philippine Long Distance Telephone Company 3. San Miguel Corporation 4. Jollibee Foods Corporation 5. Manila Electric Company 6. International Container Terminal Services, Inc. 7. Ayala Corporation B. Belle Corporation 9. DMCI Holdings, Inc.


10. Southeast Asia Cement Holdings, Inc. 11.1onics Circuits Inc. 12.C& P Homes. Inc. 13.Ayala Land, Inc. 14.SM Prime Holdings. Inc. 15. Philex Mining Company 16. Petron Corporation 17. Phil. Commercial International Bank, Inc. 18. Filinvest Development Corporation 19. Ben pres Holdings Corporation 20. Filinvest Land, Inc. 21. JG Summit Holdings, Inc. 22. Philippine National Bank 23.ABS-CBN Broadcasting Corporation 24. Metro Pacific Corporation 25. Pilipino Telephone Corporation 26. Fii-Estate Land, Inc. 27. Universal Robina Corporation 28. Pilipino Telephone Corporation 29. Aboitiz Equity Ventures, Inc. 30. Mega world Properties and Holdings, Inc.

1994PHISIX Commercial and Industrial Sector

1.ABS-CBN Broadcasting Corporation 2.Ayala Corporation 3.A. Soriano Corporation 4. Bacnotan Cement Corporation 5.Benpres Holding Corporation 6.EEI Corporation ?.Globe Telecom GMCR Inc. 8.Guoco Holdings (Phils) Inc. 9 .International Container Terminal Services. Inc.

10. Jollibee Foods Corporation 11. Manila Electric Corporation 12. Metro Pacific Corporation 13.Metropolitan Bank and Trust Co. 14. Petron Corporation 15. Philippine National Bank 16. San Miguel Corporation 17. Union Bank of The Phil. 18.Universal Robina Corp.

DLSU Business & Economics Review Volume 13 No.1


1001-1002

36 CORNISH-ASHER EXPANSION

Property Sector 1 . Ayala Land Inc. 2. Filinvest Land Inc. 3. Kuok Phil Properties Inc 4. SM Prime Holdings Inc.

Mining Sector 1. Lepanto Cons. Mining Co. 2. Manila Mining Corporation 3. Philex Mining Corporation

Oil Sector 1. Basic Pet. & Minerals, Inc. 2. Oriental Pet. & Mineral Corp 3. The Philodrill Corporation

Source: PSE Research Department (PSE Fact Books, 1997, 1996, 1995)

The randomly selected issues included in the study are as follows :* 1 . San Miguel Corporation (SMC) 2. Ayala Corporation (AC) 3. Philippine Long Distance & Telephone Company (TEL) 4. Philex Mining Corporation (PX) 5. Patron Corporation (PCOR) 6. Manila Electric Company (MER) 7. International Container Terminal Services, Inc. (ICTSI) 8. Metropolitan Bank and Trust Company (MBTC) 9. Philippine National Bank (PNB)

10. ABS-CBN Broadcasting Corporation (ABS)

*COMMONLY FOUND IN THE INDICES FOR 1994, 1996AND 1998.



AppendixB Garch {1, 1) Forecast

.~T•Tr'''"''•"'l'"'W"l)"r1r,'i'!·~~~'~J ' ~ ' I ' ~ ~ II ' ' ' 1 : : '' '

<>15 ~~'Oiili)IT""~"'<iili\~·,n-r '"6t\O''rJnT~ 1

•.os .0.10

I ..ees UEETOF - :t2~1

AC LNRE1D1f --- -- ~ - t2ftD

... .-------------

...

IF<lleC8st ,aas ~ Ll'f£fllF I .llctuai:}Bs_!H'EJ01 I Forecast sample: 1 1250 i Adj\I!Sted sampte: 2 1250 I Included obseNaHons: 1249

R>ot Mean &juarad Enor Mean Ab$~ute Brnr M&Bn Jibs. f'etcent Bror Thelllneq..oality Onfficient

Bas Roportlon 'Jariance Proportion :O:Wariance Pruportion

FUa:so;t:AC_I.NRE101F Ad:IS:AC_L.NRE1D1 Faa:a;tEf~Pe; 11:BI Aqutadsatpa212!fl lrdu:!Ed~: 1~

Ro:tMBI Sq.sa:l Enu MB!AbadLJaEIItl' M!enAII:<.Piln'll:riEm::r lhlillneq.illttyCotftci.n

BissPrcpRo:61 'laAn::ePqutim Cwaiin:eP~


O.Q19764 0-013450 67.97438 ,.000000 0.000034 , ....... O.OCXlCICICI

""""' , . ., "'""" :I ......

2001-2002


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010

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Fa8.:alt 1.1£RlNRE1F Ac:Uil: t.ERi..NRET fUec:a;tsatpa: 11250 Actl.lillld SIITJM: 21250

I

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Bi• PlqXllkn Vat.aPqxdm --

-K>llNRETF Actl:ft:IC11..NRET Rnlca!tanPe; 1129:1 ~sa-rPa:2123:l h:luBldl!awli!IB: 1:!49

..,..., .., ... ,..,., umm . .,..,. . ....., . .,.,.,

Pat~Sq.Jn:tEru 0011$4 Ma1AllldtmEm:r 04?re29 Mal AI:& P6certElltf ffi61:S lheillrlq.llllilyCo:HiidErt 1.ooml

Bis P!t:p:ftm rum:I:a Vailn::ePrqntia'l Q!S7947

'----="""'""'=·=c_p-==~c.___ OfWOX) ;

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- PCOAlHAElF ,.,... """""""' Faecasl:ssrpe: t 1200 ft4llill:dSS'I"Pe:11312D lrdu:B:Id:aN!Iims; 1138 ......

'"""" 74.516]1) l.<DDXI ...... "'""" ......

--·-·-~~

ForecastPNBLNRETF ~ Aclual: PN BLN RET Forecasts ample: I 1250 Adjus...,d sample:2 1250 klcluded observations: 1249

: RootMeanSquared Err<ft.03357(l

LMeanAbsolutl'l Error 0.021609 Mean Ab. s. PercentErrorS4.7S783 Theillnequahty Coe11ic ie111100DOOG

Bias Proporkn 0.002644 Var:ance Proportion 0997356

___£_o~ariance Proporlic:OlOO~ODD _j

2001-2002

40 CORNISH-ASHER EXPANSION

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2001-2002

Documents

Cornish-Fisher Expansion on Estimation and Forecasting ......simultaneously, using maximum likelihood estimation. The standard input for a GARCH model for financial market volatility