Download Measuring Historical Volatility

Measuring Historical Volatility
Louis H. Ederington
University of Oklahoma
Wei Guan
University of South Florida St. Petersburg
August 2004
Contact Info: Louis Ederington: Finance Division, Michael F. Price College of Business,
University of Oklahoma, 205A Adams Hall, Norman, OK 73019, USA. Phone: (405) 325-5591
Wei Guan: College of Business, University of South Florida St. Petersburg, 140 Seventh Avenue,
St. Petersburg, FL 33701, USA. Phone: (727)-553-4945
Comments may be sent to the authors at [email protected] or [email protected].
Measuring Historical Volatility
Abstract
The adjusted mean absolute deviation is proposed as a simple-to-calculate alternative to
the historical standard deviation as a measure of historical volatility and input to option pricing
models. We show that when returns are approximately log-normally distributed, this measure
forecasts future volatility consistently better than the historical standard deviation. In the markets
we examine, it also forecasts as well or better than the GARCH model.
Measuring Historical Volatility
In pricing options, anticipated volatility over the life of the option is the crucial unknown
parameter. While sophisticated volatility estimation procedures, such as GARCH, are popular
among finance researchers, these require econometrics software which is difficult for the average
undergraduate student or casual options trader to obtain or master so have not found their way
into most derivatives texts. Instead, almost all derivatives textbooks instruct students to use
historical volatility, specifically the historical standard deviation, over some recent period as the
volatility input.1 Of course implied volatility provides an alternative (and theoretically better)
estimate of future volatility but for option pricing purposes, this measure suffers from an obvious
chicken and egg problem in that to calculate implied volatility requires the option price and to
calculate the appropriate price requires a volatility estimate. Hence, the historical standard
deviation of log returns is the volatility estimator touted in most textbooks and most commonly
reported on options websites.
One well-known short-coming of the historical standard deviation as an estimator of
future volatility is that only the information in past returns is considered, ignoring other possible
information sets, like knowledge of future scheduled events that might move the markets such as
a quarterly earnings report or an upcoming meeting of the Fed’s Open Market Committee.2
Another well-known potential problem is that all past squared return deviations back to an
arbitrary date are weighted equally in calculating the standard deviation and all observations
before that date are ignored (Engle, 2004, and Poon and Granger, 2003). Evidence on volatility
clustering and persistence indicates that more recent observations should contain more
information regarding volatility in the immediate future than older observations. Accordingly,
more sophisticated models, such as GARCH and the exponentially weighted moving average
model utilized by Riskmetrics, employ weighting schemes in which the most recent squared
1
return deviations receive the most weight and the weights gradually decline as the observations
recede in time.3
While the GARCH weighting scheme in which the most recent squared return deviation
receives the greatest weight and the weights decline exponentially is theoretically more appealing
than that of the historical standard deviation in which all observations are weighted equally back
to an arbitrary cutoff point, evidence is mixed on whether GARCH actually forecasts better. In a
comprehensive review of 39 studies comparing the historical standard deviation (or variance) and
GARCH, Poon and Granger (2003) report that 17 find that GARCH forecasts better while 22
find that the historical standard deviation (or variance) forecasts better.
We see another potential drawback to the historical standard deviation: that since the
historical standard deviation and variance are functions of squared return deviations, they could
be dominated by outliers. For example, in the S&P 500 market over the period from 7/5/67 to
7/11/02, the 1% largest daily return deviations account for 24.5% of the total squared return
deviations. Suppose daily returns from one month (specifically 20 trading days) are used to
forecast volatility over the next month. 81.8% of the months will contain no observations from
this 1% tail while 18.2% will contain 1 or more. The 24.5% figure implies that, if the underlying
distribution does not change, the standard deviation for months that contain one observation in
the 1% tail will tend to be 60% higher than that for months with none. Consequently, if a month
with an observation in the 1% tail precedes a month with none, the forecast standard deviation
will tend to be much too high.
One way to avoid giving extreme observations undue weight is to use a longer period to
measure the historical standard deviation. For instance, one might use returns over the last year
instead of over the last month. The problem with this approach is that volatility tends to cluster.
Specifically high (low) volatility one week, or month tends to be followed by high (low)
2
volatility the succeeding week or month. Measuring historical volatility over a long period, such
as a year, smooths out the clusters and loses the information in recent volatility.
An alternative proposed by Andersen and Bollerslev (1998) and others which gives less
weight to extreme observations without lengthening the data period is to switch from daily to
intraday data. For instance, if the trading day is 6 hours in length, switching from one month of
daily return observations to one month of hourly return observations increases the number of
observations 6 times and cuts the weight on any one observation to one sixth the weight on daily
observations. For GARCH type models, Andersen and Bollerslev (1998), Andersen et al (1999),
and Andersen et al (2003) report considerably improved volatility estimates using such intraday
data. However, intraday data is not readily available to students and traders, is costly, and is
more difficult to handle. Authors of derivative texts apparently feel volatility estimates based on
intraday data are not usually a viable or cost effective alternative since they do not tout this
approach and their example calculations of historical volatility invariably use daily data.
In our view, a possible alternative based on daily data which is less sensitive to extreme
observations while keeping the sample period short enough to reap the benefits of volatility
clustering is to measure the historical standard deviation in terms of absolute, instead of squared
return observations. Specifically, we propose using the historical mean absolute return deviation,
instead of the historical standard deviation to measure historical volatility and forecast future
volatility. Relatively, extreme observations are simply less extreme when measured in absolute
rather than squared terms. Moreover, the mean absolute return deviation, is easily calculated
using rudimentary statistical software such as EXCEL.
The major drawback of the mean absolute deviation is that it is distribution specific. That
is, unlike the sample standard deviation, the relation between the mean absolute deviation and the
population standard deviation differs between normal, beta, gamma, and other distributions.
However, since the primary use of historical volatility is as an input into the Black-Scholes
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formula, which assumes log returns are normally distributed, basing the historical volatility
estimate on a log-normal distribution is a natural choice and imposes no additional assumption in
this application. Nonetheless, whether the historical standard deviation or the historical mean
absolute return deviation provides a better volatility forecast will likely depend on how closely
the actual distribution of returns approximates a normal distribution. This question is explored
below.
In this paper, we compare the ability of the historical mean absolute return deviation and
the historical standard deviation to forecast future volatility (measured as the standard deviation
of ex-post log returns) across a wide variety of markets, including the stock market, long and
short-term interest rates, foreign exchange rates and individual equities. Except for the
Eurodollar market at long horizons (and occasionally the Yen/Dollar exchange rate), the mean
absolute deviation forecasts future volatility better than the historical standard deviation in all
markets at almost all horizons. Not surprisingly, the major exception, Eurodollars, is the market
which is furthest from normality.
We also compare the mean absolute deviation volatility forecasts with those generated by
a GARCH(1,1) model. Their relative forecast ability seems to partially depend on how one
measures forecast accuracy. By one of our two measures, the mean absolute deviation clearly
dominates, by the other measure neither dominates. Since the GARCH model requires
specialized software, while the mean absolute deviation can be calculated using simple
spreadsheet software, such as EXCEL, and the mean absolute deviation forecasts as well or better
in most cases, there seems to be little payoff to the added expense and effort of GARCH type
models.
The remainder of the paper is organized as follows. In the next section, we explain the
adjusted mean absolute deviation measure of volatility and our measures of forecast accuracy.
4
Our data sets are described in section 2. Results are presented in 3 and section 4 concludes the
paper.
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1. Measuring and Forecasting Volatility
Let Rt = ln(Pt/Pt-1) represent daily returns on a financial asset.4 The historical
standard deviation over the last n days is measured as:
(1)
, the return deviation, and : is the expected return. Often : is replaced by the
where
sample mean and accordingly the r2 are divided by n-1, rather than n. The latter procedure
implicitly assumes that the expected return over the coming period equals the mean return in the
n-day period used to estimate STD(n). Given the low auto-correlation in returns there is no
justification for such an assumption and Figlewski (1997) shows that better forecasts are
normally obtained by setting :=0. In our calculations, we set : equal to the average daily return
over the entire data period. Since volatilities are normally quoted in annualized terms, the sum of
the squared daily return deviations is multiplied by 252, the approximate number of trading days
in a year.
As seen in equation 1, this volatility estimator assigns each squared past return deviation,
, after time t-n a weight of 1/n while observations before t-n receive a weight of zero. An
issue in applying this procedure is choosing the cutoff date n. While setting the length of the
period used to calculate historical volatility, n, equal to the length of the forecast period, s, is a
common convention, Figlewski (1997) finds that forecast errors are generally lower if the
historical variance is calculated over a longer period. Accordingly, we consider a variety of
sample period lengths and designate the length of the sample on which STD is based with (n).
For example, STD(20) indicates the annualized standard deviation of log returns over the last
twenty trading days.
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Our alternative measure of historical volatility is the adjusted mean absolute return
deviation over the same n days. The (unadjusted) mean absolute return deviation is:
(2)
However, in this unadjusted form, MAD(n) is a biased estimator of the population standard
deviation. While STD converges to F as sample size increases,5 MAD converges to F/k where k
depends on the distribution. If the distribution is normal,
(Stuart and Ord, 1998). If the
data follow another distribution, k takes another value. Consequently, we use the adjusted mean
absolute deviation defined as:
(3)
This yields an unbiased estimator of F if the rt are normally distributed. Like STD(n), AMAD(n)
is annualized by multiplying by the square root of 252.
In the early part of the 20th century there was debate among statisticians about the
relative merits of STD and MAD as measures of volatility but use of the sample standard
deviation dominated since it is a consistent estimator of F regardless of the data distribution and
Fisher showed that it is more efficient for normal distributions. Nonetheless, it has been argued
by Staudte and Sheather (1990), and Huber (1996) among others that MAD is more robust, i.e.,
that it yields better estimates if the data are contaminated.
In this paper, STD(n) and AMAD(n) are compared in terms of their ability to forecast
actual volatility over various future horizons, s, corresponding to likely option times to
expiration. We (like all other studies in this area) measure this realized volatility as the standard
deviation of daily log returns over the period s. Specifically,
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(4)
Note that STD(n) and AMAD(n) are both evaluated using the same measure of ex post volatility,
RLZ(s), the measure used in virtually all previous studies of forecasting ability.
We use two measures of the ability of STD(n) and AMAD(n) to forecast RLZ(s). The
first is the root mean squared forecast error:
(5)
where F(n)t designates STD(n)t or AMAD(n)t. The second is the mean absolute forecast error:
(6)
In this paper our main interest is which of these two simple volatility estimators, STD(n)
and AMAD(n), which are easily explained to and calculated by undergraduates, forecasts realized
volatility better. However, we are also interested in how AMAD(n) and STD(n) compare with
the more sophisticated ARCH-GARCH models proposed by academic researchers. By far the
most popular of the ARCH-GARCH models is Bollerslev’s original, the so-called GARCH(1,1)
model:
(7)
where vt represents the (unobserved) conditional variance at time t. Consequently, we also
measure how well this model forecasts actual volatility using RMSFE and MAFE.6
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2. Data and Procedures
We compare the volatility forecasting ability of these three models for four financial
assets with highly active options markets: the S&P 500 Index, the 10-year Treasury Bond rate,
the 3-month Eurodollar rate, and the Yen/Dollar exchange rate. We also collect data for five
equities chosen from those in the Dow-Jones Index: Boeing, GM, International Paper,
McDonald’s, and Merck. Daily return data for the five equities for the period 7/2/62 to 12/31/02
were obtained from CRSP tapes as were prices for the S&P 500 index. Daily interest rate and
exchange rate data were obtained from Federal Reserve Board files for the periods: 1/2/6212/31/03 for the 10 year bond rate, and 1/1/71-12/31/03 for the Eurodollar rate and the
Yen/Dollar exchange rate.
Using the three procedures, STD(n), AMAD(n), and GARCH, we forecast volatility over
horizons, s, of 10, 20, 40, and 80 market days or approximately two weeks, one month, two
months, and four months. Similarly, STD(n) and AMAD(n) are measured over historical periods,
n, of 10, 20, 40, and 80 market days. The GARCH procedure requires estimation of the
parameters " and $. This is done using 1260 daily return observations or approximately five
years of daily data. Consequently, our estimation periods for RLZ for the three models begin
approximately five years after the beginning dates for our data sets reported above, e.g., 7/5/67
for the S&P 500 index and the five equities. They end 120 trading days before the end of the data
sets, i.e., 7/11/02 for the S&P 500 index and the equities and 7/16/03 for the interest and
exchange rate series. To limit the computational burden, the GARCH model is re-estimated every
40 days.
Descriptive statistics for daily log returns are reported in Table 1 where the mean and
standard deviation are standardized for ease in interpretation. Skewness varies and is slight for
all except return on the S&P500 index. All except Merck exhibit excess kurtosis which is most
extreme for the S&P 500 index. Since the AMAD measure only provides a consistent estimate
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of the population standard deviation if the underlying distribution is normal, the KolmogorovSmirnov D statistic test for normality shown in the last column is of particular interest. In all
nine markets, the normality null is rejected at the .01 level. However, the K-S D is considerably
higher for Eurodollar returns than any of the other log return series indicating a more serious
deviation from normality in that market. This is somewhat surprising since both skewness and
kurtosis are more serious for the S&P 500 index emphasizing that skewness and kurtosis do not
completely account for deviations from normality.
3. Results
Root mean squared forecast error (RMSFE) (equation 5) and mean absolute forecast error
(MAFE) (equation 6), measures of how accurately the three models forecast actual future
volatility (RLZ), are reported in Tables 2, 3, 4, and 5 for forecast horizons, s, of 10, 20, 40, and 80
trading days respectively. RMSFE statistics are reported in panel A and MAFE in panel B.
STD(n) and AMAD(n) are calculated using sample period lengths n of 10, 20, 40, and 80 trading
days. For each n, the measure, STD(n) or AMAD(n), with the lowest RMSFE and MAFE is
shown in bold. Also, in each market, the cell of the model with the lowest RMSFE or MAFE is
shaded. When this is the GARCH model, the STD(n) or AMAD(n) model with the lowest
RMSFE or MAFE is more lightly shaded.
Consider first the results in Table 2 when the models are used to forecast volatility over a
relatively short horizon of 10 trading days or two weeks. As shown by the cells in bold, in all nine
markets for all four n (a total of 36 pairwise comparisons) , the historical adjusted mean absolute
deviation anticipates actual future volatility better than the historical standard deviation. This
results holds whether forecast accuracy is measured in terms of the RMSFE in panel A or the
MAFE in panel B. In 31of the 36 pairwise comparisons, AMAD’s volatility forecasting ability is
significantly greater than STD’s at the 10% level according to Diebold and Mariano’s (1995) S1
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statistic. In 16 of 36 it is significantly better at the 5% level. Clearly AMAD dominates STD at
this forecast horizon.
As shown by the darkly and lightly shaded cells, among the AMAD models, forecast
accuracy is best when the mean absolute deviation is measured over 40 (4 markets) or 80 (5
markets) trading days. This parallels Figlewski’s (1997) finding that the historical standard
deviation is best measured over a period longer than the forecast horizon.
As shown by the darkly shaded cells, the AMAD model beats the GARCH model in all
nine markets according to the MAFE criterion. According to the RMSFE criterion however, the
evidence is mixed with AMAD forecasting better in four markets and GARCH in five.
Considering that the adjusted mean absolute deviation is easily calculated with standard
spreadsheet software while the GARCH model requires time, study, and sophisticated software,
the AMAD model seems the most time and cost effective for all but professional traders and
finance researchers. Of course, since it is dominated by AMAD, STD fares worse against
GARCH. In a majority of markets, GARCH forecasts better than the historical standard deviation
by both measures.
Turning to Table 3 in which the models are used to forecast volatility over a 20 trading day
horizon, the same results generally hold. However at this horizon, STD(10) has a lower MAFE
than AMAD(10) for GM and STD(80) has a lower RMSFE than AMAD(80) for Eurodollars. The
former is of little consequence since our results indicate that in all markets one should never use a
sample period as short as 10 trading days to forecast future volatility regardless of the horizon.
The latter reversal is more serious since in the Eurodollar market, STD(80) turns out to have the
lowest RMSFE of all the models. Recalling from above and Table 1 that the deviation from log
normality is most serious in this market, it is not surprising that AMAD should perform least well
in this market. Nonetheless, AMAD(80)’s MAFE is considerably lower than STD(80)’s and
AMAD beats STD for all n in all other markets by both criteria. Moreover in 12 of the 36
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pairwise comparisons, AMAD’s forecasting ability is significantly better than STD’s at the .05
level.
As we move to horizons of 40 and 80 days in Tables 4 and 5 respectively, STD beats
AMAD by one or both criteria in the Eurodollar market for several n and STD(80) beats
AMAD(80) by both criteria in the Yen/dollar exchange rate market. However, in all of these
cases, the null that the forecasting ability of AMAD and STD is equal cannot be rejected at the .10
level. In all the other markets for all measures based on 20 on more trading days, AMAD beats
STD by both criteria. Moreover, in nine comparisons at the 40 day horizon and six at the 80 day
horizon, AMAD’s forecasting ability is significantly better at the .05 level.
As we move to progressively longer forecast horizons, another trend emerges in that
estimates of historical volatility based on longer sample periods tend to forecast volatility better
than estimates based on shorter periods. Specifically, as the forecast horizon is lengthened the
instances in which AMAD(40) has a lower RMSFE or MAFE than AMAD(80) declines. As
shown in Table 5, at a forecast horizon of 80 trading days (about 4 months), AMAD(80) beats
AMAD(40) in all markets.
At all forecast horizons, AMAD (or STD) forecasts future volatility better than the
GARCH(1,1) model in all markets except Merck according to the MAFE criterion while results
are mixed according to the RMSFE criterion.
4. Conclusions
We have sought to determine whether there is a simple alternative estimator, that is one
simple enough to calculate using standard spreadsheet software and easily explained to the average
undergraduate student, which provides a better volatility estimate to use in option pricing models
than the historical standard deviation. We have found that when log-returns are approximately
normally distributed there is - specifically, the adjusted mean absolute return deviation, which is
12
the absolute mean return deviation multiplied by the square root of B/2. In all except the shortterm interest rate market where the deviation from log-normality is most serious, the adjusted mean
absolute deviation forecasts actual volatility consistently better than the historical standard
deviation. We also find that this simple estimator compares very favorably with GARCH model
estimates. By one measure of forecast accuracy, the adjusted mean absolute return deviation beats
GARCH consistently, while by the other measure, they approximately break even.
13
REFERENCES
Andersen, Torben, and Tim Bollerslev, 1998, Answering the skeptics: Yes standard volatility
models do provide accurate forecasts, International Economic Review 39 (#4), 885-905.
Andersen, T.G., T. Bollerslev, F.X. Diebold, and P. Labys, 2003, Modeling and forecasting
realized volatility, Econometrica 7 (#2), 579-625.
Andersen, T.G., T. Bollerslev, and S. Lange, 1999, Forecasting financial market volatility: sample
frequency vis-a-vis forecast horizon, Journal of Empirical Finance 6, 457-477.
Diebold, Francis X., and Roberto S. Mariano, 1995, Comparing predictive accuracy, Journal of
Business and Economic Statistics 13, 253-263.
Ederington, Louis H. and Jae Ha Lee, 1993, How Markets Process Information: News Releases
and Volatility, Journal of Finance 48 (September), 1161-1191.
Ederington, Louis H. and Jae Ha Lee, 2001, Intraday Volatility in Interest Rate and Foreign
Exchange Markets: ARCH, Announcement and Seasonality Effects, Journal of Futures
Markets 21 (#6), 517-52.
Ederington, Louis H. and Guan, Wei, 2005, Forecasting Volatility, forthcoming Journal of Futures
Markets, and currently available on SSRN.com.
Engle, Robert, 2004, Risk and volatility: econometric models and financial practice, American
Economic Review 94(3), 405-420.
Figlewski, Stephen, 1997, Forecasting volatility, Financial Markets, Institutions, and Instruments
vol. 6 #1, Stern School of Business, (Blackwell Publishers: Boston).
Huber, Peter, 1996, Robust Statistical Procedures, 2nd ed., (Society for Industrial and Applied
Mathematics: Philadelphia).
Hull, John C., 2003, Options, Futures, and Other Derivatives, 5th edition, Englewood Cliffs:
Prentice Hall.
14
Poon, Ser-Huang and Clive Granger, 2003, Forecasting Volatility in Financial Markets: A Review,
Journal of Economic Literature 41 (#2), 478-539.
Staudte, Robert, and Simon Sheather, 1990, Robust Estimation and Testing, (John Wiley & Sons:
New York).
Stuart, Alan and Keith Ord, 1998, Kendall’s Advanced Theory of Statistics: Volume 1 Distribution
Theory, 6th edition, New York: Oxford University Press.
15
Table 1 - Descriptive Statistics
Statistics are reported for annualized daily log returns for the following periods: S&P 500 and
the five individual equities: 7/5/67-7/11/02, T-Bonds: 11/02/66-7/16/03, Eurodollars and the
Yen/Dollar exchange rate: 11/04/75-7/16/03.
Standard
Mean
Kolmogorov-
Deviation
Skewness
Kurtosis
Smirnov D
S&P 500
0.066402
0.1537
-1.6081
39.893
.0583
T-Bonds
-0.00602
0.1392
.0922
4.986
.0786
Eurodollars
-0.06424
0.2430
-.3859
8.311
.2395
Yen/Dollar
-0.03267
0.1043
-.5116
4.617
.0755
Boeing
0.098581
0.3369
.0212
5.242
.0556
GM
0.06834
0.2680
-.1853
7.351
.0470
Int. Paper
0.085882
0.2874
-.4092
13.128
.0459
McDonalds
0.12594
0.2898
-.3210
7.948
.0501
Merck
0.12818
0.2482
-.0516
3.375
.0444
16
Table 2 - Volatility Forecast Accuracy – 10 Trading Day Horizon
Root mean squared forecast errors (RMSFE) and mean absolute forecast errors (MAFE) are reported when the procedures listed in column 1 are
used to forecast the standard deviation of returns over the next 10 trading days. STD(n) denotes the standard deviation calculated over the last n
trading days. AMAD(n) denotes the adjusted mean absolute deviation over the last n days. Data periods are reported in Table 1. For each n, the
STD(n) or AMAD(n), with the lowest RMSFE and MAFE is shown in bold. In each market, the cell of the model with the lowest RMSFE or
MAFE is shaded. When this is the GARCH model, the STD(n) or AMAD(n) with the lowest RMSFE or MAFE is lightly shaded.
Markets
Forecasting
Model
10-year
90-day
T-Bond
Eurodollar
S&P 500
Yen/Dollar
Boeing
GM
Int’l Paper
McDonalds
Merck
Panel A - Root Mean Squared Forecast Errors (RMSFE)
STD(10)
0.07312
0.05717
0.12313
0.04790
0.15634
0.11596
0.12563
0.12338
0.10326
AMAD(10)
0.07023
0.05665
0.11976
0.04662
0.15101
0.11481
0.12003
0.12208
0.10197
STD(20)
0.07076
0.05325
0.11697
0.04327
0.14163
0.10928
0.11856
0.11314
0.09399
AMAD(20)
0.06616
0.05199
0.11317
0.04185
0.13465
0.10566
0.11244
0.11024
0.09162
STD(40)
0.06964
0.05216
0.11163
0.04215
0.13270
0.10321
0.11309
0.10750
0.08948
AMAD(40)
0.06477
0.05095
0.10834
0.04097
0.12638
0.09948
0.10798
0.10485
0.08729
STD(80)
0.07073
0.05296
0.11059
0.04148
0.12970
0.10113
0.10979
0.10957
0.08884
AMAD(80)
0.06661
0.05187
0.10885
0.04068
0.12504
0.09821
0.10473
0.10831
0.08712
GARCH
0.06461
0.05038
0.11192
0.04240
0.12716
0.09738
0.10741
0.10427
0.08586
Panel B - Mean Absolute Forecast Errors (MAFE)
STD(10)
0.04060
0.04142
0.08306
0.03469
0.10941
0.07839
0.08724
0.08448
0.07604
AMAD(10)
0.04021
0.04101
0.08253
0.03375
0.10750
0.07905
0.08528
0.08463
0.07576
STD(20)
0.03865
0.03823
0.08002
0.03143
0.09912
0.07336
0.08111
0.07608
0.06951
AMAD(20)
0.03719
0.03724
0.07747
0.02992
0.09570
0.07248
0.07861
0.07460
0.06817
STD(40)
0.03882
0.03789
0.07886
0.03045
0.09403
0.06979
0.07686
0.07211
0.06642
AMAD(40)
0.03679
0.03638
0.07342
0.02884
0.08943
0.06774
0.07440
0.06980
0.06412
STD(80)
0.04059
0.03896
0.07921
0.03027
0.09393
0.06856
0.07372
0.07241
0.06572
AMAD(80)
0.03797
0.03710
0.07378
0.02871
0.08877
0.06572
0.07050
0.07038
0.06366
GARCH
0.03721
0.03688
0.08217
0.03188
0.09482
0.06608
0.07428
0.07134
0.06452
Table 3 - Volatility Forecast Accuracy – 20 Trading Day Horizon
Root mean squared forecast errors (RMSFE) and mean absolute forecast errors (MAFE) are reported when the procedures listed in column 1 are
used to forecast the standard deviation of returns over the next 20 trading days. STD(n) denotes the standard deviation calculated over the last n
trading days. AMAD(n) denotes the adjusted mean absolute deviation over the last n days. Data periods are reported in Table 1. For each n, the
STD(n) or AMAD(n), with the lowest RMSFE and MAFE is shown in bold. In each market, the cell of the model with the lowest RMSFE or
MAFE is shaded. When this is the GARCH model, the STD(n) or AMAD(n) with the lowest RMSFE or MAFE is lightly shaded.
Markets
Forecasting
Model
10-year
90-day
T-Bond
Eurodollar
S&P 500
Yen/Dollar
Boeing
GM
Int’l Paper
McDonalds
Merck
Panel A - Root Mean Squared Forecast Errors (RMSFE)
STD(10)
0.07178
0.05346
0.11811
0.04358
0.14199
0.11011
0.11926
0.11448
0.09447
AMAD(10)
0.06830
0.05264
0.11521
0.04213
0.13579
0.10862
0.11292
0.11227
0.09296
STD(20)
0.06824
0.04855
0.10860
0.03910
0.12424
0.10092
0.10872
0.10127
0.08315
AMAD(20)
0.06291
0.04720
0.10520
0.03756
0.11644
0.09699
0.10217
0.09750
0.08097
STD(40)
0.06643
0.04694
0.10047
0.03714
0.11371
0.09306
0.10027
0.09631
0.07803
AMAD(40)
0.06109
0.04576
0.09876
0.03609
0.10730
0.08893
0.09456
0.09326
0.07611
STD(80)
0.06768
0.04722
0.09820
0.03572
0.10976
0.09008
0.09631
0.09805
0.07696
AMAD(80)
0.06302
0.04642
0.09864
0.03538
0.10580
0.08702
0.09059
0.09691
0.07547
GARCH
0.06141
0.04492
0.10012
0.03713
0.10638
0.08616
0.09506
0.09137
0.07286
Panel B - Mean Absolute Forecast Errors (MAFE)
STD(10)
0.03883
0.03888
0.08025
0.03161
0.09955
0.07424
0.08129
0.07676
0.07043
AMAD(10)
0.03835
0.03866
0.08198
0.03077
0.09787
0.07486
0.08012
0.07662
0.07027
STD(20)
0.03612
0.03522
0.07507
0.02832
0.08764
0.06676
0.07278
0.06555
0.06220
AMAD(20)
0.03456
0.03435
0.07455
0.02697
0.08422
0.06579
0.07027
0.06392
0.06105
STD(40)
0.03635
0.03425
0.07161
0.02655
0.08101
0.06135
0.06609
0.06210
0.05802
AMAD(40)
0.03434
0.03290
0.06830
0.02515
0.07701
0.05941
0.06332
0.06000
0.05641
STD(80)
0.03817
0.03495
0.07106
0.02617
0.07963
0.05903
0.06197
0.06241
0.05587
AMAD(80)
0.03545
0.03339
0.06798
0.02497
0.07583
0.05636
0.05852
0.06049
0.05442
GARCH
0.03468
0.03329
0.07588
0.02806
0.08102
0.05734
0.06346
0.06079
0.05442
Table 4 - Volatility Forecast Accuracy – 40 Trading Day Horizon
Root mean squared forecast errors (RMSFE) and mean absolute forecast errors (MAFE) are reported when the procedures listed in column 1 are
used to forecast the standard deviation of returns over the next 40 trading days. STD(n) denotes the standard deviation calculated over the last n
trading days. AMAD(n) denotes the adjusted mean absolute deviation over the last n days. Data periods are reported in Table 1. For each n, the
STD(n) or AMAD(n), with the lowest RMSFE and MAFE is shown in bold. In each market, the cell of the model with the lowest RMSFE or
MAFE is shaded. When this is the GARCH model, the STD(n) or AMAD(n) with the lowest RMSFE or MAFE is lightly shaded.
Markets
Forecasting
Model
10-year
90-day
T-Bond
Eurodollar
S&P 500
Yen/Dollar
Boeing
GM
Int’l Paper
McDonalds
Merck
Panel A - Root Mean Squared Forecast Errors (RMSFE)
STD(10)
0.07199
0.05278
0.11404
0.04291
0.13364
0.10540
0.11535
0.11053
0.09035
AMAD(10)
0.06808
0.05180
0.11119
0.04138
0.12666
0.10388
0.10864
0.10770
0.08904
STD(20)
0.06774
0.04739
0.10192
0.03758
0.11437
0.09430
0.10180
0.09792
0.07841
AMAD(20)
0.06210
0.04593
0.09948
0.03602
0.10606
0.09010
0.09462
0.09368
0.07629
STD(40)
0.06540
0.04480
0.09249
0.03440
0.10085
0.08574
0.09063
0.09223
0.07163
AMAD(40)
0.05993
0.04360
0.09266
0.03357
0.09525
0.08113
0.08407
0.08945
0.06996
STD(80)
0.06620
0.04422
0.08819
0.03206
0.09692
0.08228
0.08652
0.09377
0.07029
AMAD(80)
0.06095
0.04362
0.09085
0.03220
0.09360
0.07898
0.07990
0.09224
0.06870
GARCH
0.06009
0.04211
0.09092
0.03518
0.09216
0.07734
0.08623
0.08529
0.06436
Panel B - Mean Absolute Forecast Errors (MAFE)
STD(10)
0.04011
0.03855
0.07920
0.03112
0.09550
0.07098
0.07816
0.07492
0.06775
AMAD(10)
0.03954
0.03828
0.08204
0.03040
0.09322
0.07193
0.07691
0.07422
0.06792
STD(20)
0.03714
0.03464
0.07230
0.02691
0.08106
0.06228
0.06732
0.06422
0.05879
AMAD(20)
0.03589
0.03387
0.07335
0.02589
0.07741
0.06168
0.06498
0.06215
0.05804
STD(40)
0.03701
0.03285
0.06781
0.02469
0.07208
0.05664
0.05849
0.05926
0.05291
AMAD(40)
0.03509
0.03146
0.06770
0.02401
0.06919
0.05486
0.05560
0.05754
0.05176
STD(80)
0.03794
0.03326
0.06579
0.02368
0.07112
0.05392
0.05490
0.05997
0.05063
AMAD(80)
0.03517
0.03172
0.06488
0.02331
0.06831
0.05142
0.05092
0.05819
0.04905
GARCH
0.03531
0.03181
0.07232
0.02650
0.07231
0.05160
0.05736
0.05835
0.04788
Table 5 - Volatility Forecast Accuracy – 80 Trading Day Horizon
Root mean squared forecast errors (RMSFE) and mean absolute forecast errors (MAFE) are reported when the procedures listed in column 1 are
used to forecast the standard deviation of returns over the next 80 trading days. STD(n) denotes the standard deviation calculated over the last n
trading days. AMAD(n) denotes the adjusted mean absolute deviation over the last n days. Data periods are reported in Table 1. For each n, the
STD(n) or AMAD(n), with the lowest RMSFE and MAFE is shown in bold. In each market, the cell of the model with the lowest RMSFE or
MAFE is shaded. When this is the GARCH model, the STD(n) or AMAD(n) with the lowest RMSFE or MAFE is lightly shaded.
Markets
Forecasting
Model
10-year
90-day
T-Bond
Eurodollar
S&P 500
Yen/Dollar
Boeing
GM
Int’l Paper
McDonalds
Merck
Panel A - Root Mean Squared Forecast Errors (RMSFE)
STD(10)
0.07456
0.05419
0.11356
0.04231
0.13081
0.10505
0.11202
0.11459
0.09034
AMAD(10)
0.07063
0.05307
0.11158
0.04081
0.12424
0.10305
0.10467
0.11217
0.08913
STD(20)
0.07050
0.04827
0.10026
0.03638
0.11064
0.09348
0.09835
0.10184
0.07792
AMAD(20)
0.06468
0.04688
0.09923
0.03508
0.10325
0.08880
0.09000
0.09818
0.07574
STD(40)
0.06760
0.04482
0.08899
0.03232
0.09727
0.08470
0.08754
0.09601
0.07075
AMAD(40)
0.06141
0.04378
0.09073
0.03199
0.09207
0.07986
0.07931
0.09282
0.06873
STD(80)
0.06535
0.04307
0.08238
0.02940
0.08993
0.07881
0.08169
0.09272
0.06800
AMAD(80)
0.05926
0.04275
0.08678
0.03013
0.08680
0.07515
0.07367
0.09048
0.06607
GARCH
0.05994
0.04123
0.08351
0.03373
0.08315
0.07204
0.08343
0.08503
0.05890
Panel B - Mean Absolute Forecast Errors (MAFE)
STD(10)
0.04351
0.04004
0.08066
0.03119
0.09389
0.07250
0.07661
0.07836
0.06771
AMAD(10)
0.04288
0.03990
0.08443
0.03060
0.09199
0.07307
0.07488
0.07793
0.06752
STD(20)
0.04063
0.03569
0.07259
0.02683
0.07895
0.06348
0.06506
0.06724
0.05775
AMAD(20)
0.03934
0.03543
0.07576
0.02639
0.07604
0.06266
0.06253
0.06540
0.05725
STD(40)
0.03918
0.03346
0.06634
0.02413
0.07028
0.05695
0.05645
0.06175
0.05148
AMAD(40)
0.03727
0.03273
0.06946
0.02395
0.06828
0.05503
0.05307
0.06045
0.05045
STD(80)
0.03846
0.03314
0.06254
0.02215
0.06873
0.05270
0.05300
0.06047
0.04889
AMAD(80)
0.03567
0.03220
0.06516
0.02247
0.06586
0.05013
0.04884
0.05868
0.04767
GARCH
0.03638
0.03231
0.06763
0.02573
0.06653
0.05030
0.05684
0.06028
0.04366
ENDNOTES
1. Several textbooks do mention GARCH as a method for estimating volatility and reference
sources but do not attempt to explain or describe it. The only text that seeks to teach students
how to estimate volatility using GARCH that we have seen is Hull (2003).
2. See for instance Ederington and Lee (1993, and 2001). For an excellent review of the issues
in volatility forecasting see Poon and Granger (2003).
3. For a nice review of the highlights of this literature see Robert Engle’s Nobel prize acceptance
speech (Engle, 2004).
4. In the case of dividend paying stocks, the numerator is changed to Pt+Dt where Dt denotes any
dividends paid over the period from t-1 to t.
5. While the sample variance is an unbiased estimator of F2, the sample standard deviation is a
biased estimator of F in small samples. However, it is asymptotically unbiased and consistent.
6. The model in equation 7 yields a forecast variance for the next day, t+1. The forecast
variance over the next s days is obtained by successive forward substitution following the
procedure outlined in Ederington and Guan (2005).