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8/4/2019 Estimating the VaR of a Portfolio Subject to Price Limits and Non Synchronous Trading
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Estimating the VaR of a portfolio subject to price limitsand nonsynchronous trading
Pin-Huang Chou a,⁎, Wen-Shen Li a , Jun-Biao Lin a , Jane-Sue Wang b
a Department of Finance, National Central University, Jhongli 320, Taiwan
b Department of Economics, Ming-Chuan University, Taoyuan 333, Taiwan
Available online 20 March 2006
Abstract
Price limits and nonsynchronous trading are two main features in emerging markets. Price limits cause
stock returns to be restricted within a prespecified range whereas infrequent trading induces spurious
autocorrelation and biased estimate of the return variance. Both factors cause traditional measures of Value
at Risk (VaR) to be biased. In this paper, we propose VaR measures based on a two-limit type Tobit model
incorporating Scholes and Williams' [Scholes, M., & Williams, J. (1977). Estimating betas fromnonsynchronous data, Journal of Financial Economics 5, 309–328] estimator that adjusts for price limits
and nonsynchronous trading. Based on the simulation design of Brown and Warner [Brown, S., & Warner,
J. (1985). Measuring security price performance, Journal of Financial Economics 8, 205–258], we
compare the performance of our proposed methods with two traditional methods, one based on naive OLS
estimates and the other based on historical simulation. Using daily data of all stocks listed on the Taiwan
Stock Exchange and the OTC markets, the simulation results indicate that all methods perform reasonably
well. The only exception is that the naive OLS yields a slightly higher failure rate when the portfolio under
consideration is composed of only a few stocks. Thus, despite the potential problems induced by
nonsynchronous trading and price limits, their practical impacts seem limited.
© 2006 Elsevier Inc. All rights reserved.
JEL classification: G0; G1; G2
Keywords: Value at risk; Price limits; Nonsynchronous trading; Variance–covariance method; Historical simulation
1. Introduction
During the past decade, Value at Risk (VaR) has become one of the standard measures of risk
used by financial institutions and regulators. Conceptually, VaR measures the potential loss of a
International Review of Financial Analysis 15 (2006) 363–376
⁎ Corresponding author. Tel.: +886 3 4227151x66270; fax: +886 3 4252961.
E-mail address: [email protected] (P.-H. Chou).
1057-5219/$ - see front matter © 2006 Elsevier Inc. All rights reserved.
doi:10.1016/j.irfa.2005.03.002
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portfolio that will not be exceeded with a specified probability over a specified time horizon.
Thus, VaR is merely the quantile of the distribution of a portfolio's future returns, conditional on
any information available. Despite its conceptual simplicity, however, the measurement of VaR is
by no means an easy statistical exercise, but a very challenging statistical problem (see, e.g., thereview of VaR by Duffie and Pan, 1997).
The estimation of VaR may be even more difficult for emerging markets because trading in
those markets is typically subject to varying degrees of regulatory restrictions and market
imperfections. In this paper, we focus on the problems resulting from two most common and
important forms of regulatory constraints and market imperfection, namely price limits and
infrequent trading. Without considering microstructure factors such as infrequent trading and
regulations like price limits, the traditional estimates of distribution parameters may be biased,
thereby resulting in biased estimates of VaR measures.
Used ostensibly to prevent prices from fluctuating too much during a given trading session and
thereby preventing defaults and reducing the contract cost, price limits are a common regulationassociated with futures contracts and can also be found in many stock markets in Asia and Europe,
such as Austria, Belgium, China, France, Greek, Italy, Japan, South Korea, Malaysia, Mexico, the
Netherlands, Spain, Switzerland, Taiwan, and Thailand. Price limits cause two potential
problems. First, under daily price limits, changes in the value of a portfolio over a given day are
constrained to a prespecified range. But this does not mean that the VaR is effectively lowered
with the imposition of price limits. Chou, Lin, and Yu (2003) show that when a price limit is
triggered, the “unrealized” shock will be spilled over to the next trading days until it is fully
reflected in asset prices. Second, as the observed stock returns are limited to a certain range under
price limits, the usual estimates of risk and return are biased. Chou (1997), Lee and Kim (1997),
and Wei and Chiang (2002) show that the usual estimates of variance, covariance, and systematicrisk are biased downward, which results in a downward bias in the VaR estimate and a higher
failure rate.
Another feature we consider is infrequent trading, also known as nonsynchronous trading or
thin trading in the literature. Scholes and Williams (1977) show that estimates of variance and
systematic risk are inconsistent and biased downward in the presence of thin trading. Thus, as in
the case of price limits, the usual VaR measures without considering the impact of
nonsynchronous trading are also biased downward.
We propose portfolio VaR measures based on the variance–covariance method that adjusts for
the effects of price limits and/or nonsynchronous trading. Specifically, an estimation method
combining Scholes and Williams' model and the two-limit Tobit model is proposed to providemore accurate estimates of model parameters.
To measure the real effects of both factors, we follow the design of Brown and Warner (1985)
by conducting Monte Carlo simulations to compare the performance of our methods with some
traditional methods, including the variance–covariance method based on naive OLS estimate and
the historical simulation. The sample for simulations is real data constructed from the daily returns
of all stocks listed on the Taiwan Stock Exchange and the Taiwanese OTC markets from 1998 to
2003. The OTC data are included because the trading in the OTC markets is typically less
frequent than in the Exchange. During the sample period, Taiwan's stock markets impose a 7%
price limit regulation, and about 9% of the observations hit either up or down limits.
Our simulation results show that the naive OLS estimates of betas and variances–covariances
are biased downward. Nevertheless, all methods perform reasonably well, except in the case
where the portfolio is composed of only a few stocks, the VaR based on OLS yields a slightly
higher failure rate than the nominal value (i.e., 1% or 5%).
364 P.-H. Chou et al. / International Review of Financial Analysis 15 (2006) 363 – 376
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Our results have important implications for the calculation of VaR for stock markets whose
trading is subject to price limits and thin trading. Although many stock markets, especially the
emerging ones, also impose price limits and suffer from varying degrees of infrequent trading, the
frequency of limit hits is generally lower than in Taiwan (due to either lower market volatility or wider price limits). Thus, our results suggest that although theoretically traditional VaR measures
are biased in the presence of price limits and nonsynchronous trading, the practical relevance of
the two factors seems limited.
The rest of the paper is organized as follows. The next section presents the basic setting
describing the VaR of a portfolio. Sections 3 and 4 explain the econometrics with the analysis of
thin trading and price limits, respectively. Section 5 presents the simulation procedure a la Brown
and Warner (1985). The performance of our proposed method is compared with the traditional
delta-normal method and the bootstrap method. Section 6 presents the simulation results, while
the last section concludes the paper.
2. Methodology: the basic setting
Suppose the portfolio under consideration is composed of N assets. Denote R pt as the return on
the portfolio at time t , and r it as the return on the ith asset, then the portfolio return can be
represented as:
R pt ¼X N
i¼1
wir it ; ð1Þ
where wi is the weight of the portfolio on asset i, and normallyP N
i¼1 wi ¼ 1. Conceptually, wi'sare nonstochastic functions of past information and can be time varying. Without loss of
generality, the weights are treated as fixed in the paper.
Let Rt = (r 1t ,…, r Nt )′ denote an N -vector of returns on the assets whose mean and variance–
covariance matrix are μ and Σ, respectively.1 The portfolio return R pt of the N assets can be
written in matrix form as:
R pt ¼ w V Rt ; ð2Þ
where w = (w1,…, w N )′ are the vector of weights on the N assets. Thus, if the returns of the
individual assets, Rt , are independently and identically normally distributed (iid), then the
portfolio R pt has a normal distribution with mean and variance as w′μ and w′ Σw, respectively,
R pt f N ðw Vl; w VRwÞ: ð3Þ
Let V 0 denote the initial value of the portfolio, the relative (1−α)-VaR of the portfolio is
calculated as:
VaR p að Þ ¼ − Z a w VRwð Þ12V 0; ð4Þ
where Z α
is the z -statistic that corresponds to the 1−α percentile of a standard normal distribution.
The common choice of α is 1% or 5%. This measure of VaR is known as delta-normal or
variance–covariance method.
1 The mean and covariance matrix can also be time varying. The subscript t is ignored for notational simplicity.
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The precision of the portfolio VaR estimate for (4) relies crucially on the precision of the
estimate on the covariance matrix, Σ. Normally it is assumed that the equilibrium returns are
generated by a k -factor model in the spirit of Ross's (1976) arbitrage pricing theory (APT):
Rt ¼ b0 þ BF t þ et ; ð5Þwhere β 0 is an N -vector of intercepts, F t is a (k ×1) vector of factor realizations at time t , B is an
( N × k ) matrix of factor loadings or factor sensitivities, and εt is the N -vector of error terms. The
error terms are assumed to be cross-sectionally and serially uncorrelated, i.e.,
cov eis; e jt À Á
¼r2e;i if i ¼ j and s ¼ t
0 otherwise:
&ð6Þ
Thus, the error terms are distributed as follows:
et fiid
N ð0; DÞ;
where D =diag(σε,12 ,…, σ
ε, N 2 ) is a diagonal matrix, with each element being the error variance for
each individual asset. With this representation, the covariance matrix Σ becomes:
R ¼ BR F B Vþ D; ð7Þ
where Σ F is the (k × k ) covariance matrix for the k factors, i.e., Σ F = E ( F −μ f )( F −μ f )′.
The factors of course are unknown, and have to be identified. A simple treatment is to adopt a
single-index market model:
r it ¼ ai þ bir mt þ eit ; i ¼ 1; N ; N ; ð8Þ
where r mt is the return on a market index. Thus, the covariance matrix becomes
R ¼ bb Vr2m þ D; ð9Þ
where β = (β 1,…, β N )′ is the column vector of betas.
The single-index market model assumes that assets are correlated only through their
comovement with the market index. Without trading frictions and constraints like thin trading and
price limits, the covariance matrix estimates can be obtained by plugging the OLS estimates of β ,
σε,i2 , and σm
2 into Eq. (9). However, the usual estimates of expected returns and risks are biased in
the presence of thin trading and price limits. In particular, the estimate of beta is biased
downward, thereby causing a downward bias of the risks. To deal with the problems, somerefinements are needed. The adjustments of estimation for nonsynchronous trading and price
limits are introduced in the following.
3. Estimation in the presence of nonsynchronous trading
Scholes and Williams (1997) propose a rigorous treatment of the estimation problems
associated with nonsynchronous trading. They prove that nonsynchronous trading induces a
negative autocorrelation in stock returns, an overstatement of the return variance, and a downward
bias in the systematic risk (see also Lo and MacKinlay, 1990).
To deal with the problems, Scholes and Williams (1977) derive a consistent estimate for beta:
̂bi ¼b ̂
þ
i þ b ̂i þ b ̂−
i
1 þ 2q m̂; ð10Þ
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where b̂ i+, b̂
i and b̂ i− respectively are the OLS estimates of the slopes of regression of asset i's
returns on one-period lag, concurrent, and one-period ahead of the market index; ρ̂ m is the
first-order autocorrelation of the index return. Dimson (1979) and most following studies (for
example, the seminal paper of Fama and French, 1992) use a simplified estimate for beta.Specifically, the following extended market model is estimated:
r it ¼ ai þ bir mt þ b−i r m;t −1 þ eit ; ð11Þ
and the beta is estimated as sum of the ‘concurrent ’ and ‘lag’ betas:
̂bns
i ¼ b ̂i þ b ̂−
i : ð12Þ
The superscript ‘ns’ is used to denote the adjustment for nonsynchronous trading. Thus, the
covariance between asset i and asset j can be estimated as:
r ̂nsij ¼ bns
i bns j r ̂
2m: ð13Þ
According to Scholes and Williams (1977), the variance can be consistently estimated as the
following:
r ̂ns;2i ¼ s2
i þ 2g ̂i; ð14Þ
where si2 is the usual variance estimate for σ2, and γ̂ is the first-order autocorrelation of the
returns.
Nonsynchronous trading and price limits share a common feature in that in both cases, therecorded daily closing price might be a price traded before the market close. The difference
is that the trading interference due to nonsynchronous trading is “endogenous” in nature in
that the demand and supply determines the time when a last trade takes place. By contrast,
the trading interference caused by price limits is “exogenous” in that trading is interrupted
once the exchange-imposed limit is triggered. Another difference is that under price limits the
observed price is restricted to a price range, but it is not the case with nonsynchronous
trading.
4. Analysis under price limits
Suppose the equilibrium return is governed by a linear relationship like Eq. (5). Under
price limits, the daily closing price cannot either exceed the previous closing price plus a
certain percentage of the previous closing price (i.e., an up limit), or fall below the
previous closing price minus a certain percentage of the previous closing price (i.e., a
down limit). That is, the observed stock price at time t , Ot , must fall within the interval:
(Ot −1(1 + Ld), Ot −1(1 + Lu)), where Lu and Ld are, respectively, the daily up and down limits,
represented in terms of percentages. In other words, if the true stock price P t falls outside
the interval, one observes a limit price. The relationship can be characterized as follows:
Ot ¼Ot −1 1 þ Luð Þ if P t zOt −1 1 þ Luð Þ
P t if Ot −1 1 þ Ldð Þ < P t < Ot −1 1 þ Luð ÞOt −1 1 þ Ldð Þ if P t VOt −1 1 þ Ldð Þ:
8<: ð15Þ
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Dividing both sides by Ot −1 and then taking the natural logarithm, Eq. (15) can be
rewritten as the following:
z t ¼ l u if r t þ E t −
1z
l ur t þ E t −1 if l d < r d þ E t −1 < l ul d if r t þ E t −1Vl d;
8<: ð16Þ
where z t ≡ log(Ot / Ot −1) is the observed daily return at time t , l u≡ log(1+ Lu), and l d≡ log(1
+ Ld); E s≡ log( P s/ O s)=log( P s)− log(O s) is a leftover term that represents the unrealized
residual shock from trading day s.2
This model differs from the traditional two-limit Tobit model where the latent dependent
variable is censored as follows:
z t ¼l u if r t zl ur t if l d < r t < l ul d if r t Vl d:
8<: ð17Þ
By comparing the above two sorts of limit schemes, Chou (1999) indicates that the only
difference is that under price limits, there is an additional spillover term E t −1. He shows that the
two-limit Tobit model can still be applied to the case of price limits as long as one drops those
observations whose previous trading day is a limit day (i.e., the spillover term is non-zero). Based
on several simulation experiments, Chou (1999) shows that the two-limit Tobit model provides
very precise estimates of model parameters.
We apply the two-limit Tobit model to two model specifications. The first model is thesingle-index market model (8) which deals with the problem of price limits only. The second
model is the extended market model (11) with an additional lagged market return that
considers both the problems of price limits and thin trading. Parameter estimates from both
models are plugged into the formula for covariance matrix (9), which in turn gives the
estimates for VaR.
5. Data and Monte Carlo experiments
We use daily stock returns of all stocks listed on the Taiwan Stock Exchange (TWSE) and the
Taiwan over-the-counter (OTC) markets over the 1998–2003 sample period. The database is
compiled by the Taiwan Economic Journal (TEJ) Inc. Inclusion of stocks traded on the OTC
markets assures that our population contains stocks that are traded less frequently. The sample
contains 1081 firms, each of which has up to 1549 daily observations. In our sample, 646 firms
are from the Taiwan Stock Exchange and 435 firms are from the OTC markets. Of the 1081 firms,
about 70% of the firms belong to manufacturing industry. We compile a value-weighted index of
all stocks as the market index.3
Both the TWSE and OTC markets adopt a 7% price-limit regulation during the sample period.
However, because of the tick size, the return of stocks that touches the up limit (down limit) may
2 For demonstration simplicity, the analysis in this section is based on continuously compounded return for ease of
demonstration. Empirically, we actually use discrete-version returns.3 The value-weighted index is compiled as a capitalization-weighted average of the TWSE index and the OTC index.
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not exactly equal 7% (−7%). Let S t denote the potential up (down) limit price at time t , which is
the closing price at time t −1 times 1 + 7% (1−7%). The tick size (TSt ) is as follows,
TSt ¼
0:01 if S t < 5
0:05 if 5VS t < 5
0:1 if 15VS t < 50
0:5 if 50VS t < 150
1 if 150VS t < 1000
0:05 if S t z5
8>>>>>><>>>>>>:
ð18Þ
Considering the tick size above, at time t , the up-limit and down-limit prices are respectively:
PUt ¼ Int Ot −1ð1 þ 0:07Þ
TSt
 TSt ;
PDt ¼ Int Ot −1ð1−0:07Þ
TSt
 TSt ;
where Int(·) represents the integer function.
In our sample period (1998–2003), the government changes the price limits for several times
to accommodate some drastic market events. The first time is from September 27, 1999 to
October 7, 1999; because of 921 Chi-Chi Earthquake, a major earthquake attacking Taiwan on
September 21, the government changed the down-limit from 7% to 3.5%. After October 7, 1999,
the 7% down limit is restored. The second one is from March 3, 2000 to March 26, 2000, during
which the down limit is tightened to 3.5% due to presidential election and campaign. Due to theattack of 911, the government changes the price limit again from September 19, 2001 to
September 21, 2001; the down-limit changes from 7% to 3.5% and the up-limit remains the
same.
Some summary statistics are reported in Table 1. Table 1 reports the number of firms, average
number and percentage of up-limit hits, average number and percentage of down-limit hits,
average market value (in million NT dollars), average daily trading volume (in shares) and
average trading volume (in thousand NT dollars) for firms of different exchanges and/or different
industries. The OTC markets are about one fourth of the Taiwan Stock Exchange in terms of
average market value and trading volume.
Table 1 indicates that on average 9.52% of the observations hit limits, of which 6.16% areup-limit hits, and 3.37% are down-limit hits. The frequency and distribution of limit hits are
about the same for stocks in Taiwan Stock Exchange and for stocks in the OTC markets.
Among all industries, the food industry in the OTC markets and the construction industry in the
TWSE have the highest percentages of limit hits of 8.93% and 8.12%, respectively. The
percentage of limit hits in Taiwan's stock markets is comparatively higher than most of the
futures markets and stock markets worldwide probably because of the high volatility and
relatively tight price limits.
Tables 2 and 3 report some statistics concerning the thin trading problem. Table 2 reports
summary statistics on the durations of the last trade to market close for all firms over the year
2003. We limit the sample to the year 2003 only because identifying the last trade in each trading
day requires checking the whole intra-day transaction data, which is extremely large in size and
computationally time consuming. Table 2 shows that on average the last trade takes place at
7.39min before the market close. The non-trading duration is 3.21min for TWSE stocks, and
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13.30min for OTC stocks, suggesting that indeed the OTC markets suffer more from the thin-
trading problem. Among all industries, the food industry on the OTC markets has the longest non-
trading duration of 72.49min; this industry has the smallest average market value of 939 million
(in NT dollars) among all industries (see Table 1).
Table 3 reports the average number of non-trading days for the period 1988–2003 based on
daily data. The average number of non-trading days for all firms is 8.61days, which is about
0.83% of the total daily observations over the 5-year sample period. Again, the TWSE stocks
have a smaller number of 4.81 non-trading days than that of the OTC markets, which is
14.63days.We consider five estimates for the portfolio VaR; the first four methods are parametric, and the
last method is nonparametric.
1. Naive estimates: betas are estimated using naive OLS estimate of the market model (8).
2. Nonsynchronous-trading adjusted estimates: betas and variances are estimated based on
Scholes and Williams' (1977) estimator as outlined in the previous section.
3. Price-limit adjusted estimates: betas and variances are estimated based on a two-limited Tobit
model of the market model (8) as outlined in the previous section.
4. Nonsynchronous-trading and price-limit adjusted estimates: betas and variances are estimated
based on a two-limited Tobit model of the extended market model (11) as outlined in the
previous section.
5. Historical simulation: VaR is calculated as the empirical percentile of an artificial sample of
1000 random observations drawn from the portfolio returns.
Table 1
Descriptive statistics for Taiwan stock markets
Market Industry No. of firms Up % Down % Market value Volume Amount
All All 1081 68.32 6.16 39.58 3.37 11,677 3096 107,490TWSE All 646 77.71 6.02 45.59 3.44 17,431 4644 163,201
Food 25 66.96 5.33 38.04 2.67 5683 1838 41,013
Construction 41 116.68 8.12 73.93 5.09 6717 3387 43,849
Chemical 53 73.66 5.17 42.11 2.88 14,824 3990 106,138
Manufacturing 414 80.09 6.47 47.21 3.72 18,384 4842 203,656
Transportation 16 81.19 5.39 45.00 2.98 13,950 4308 78,304
Finance 42 49.71 3.72 25.50 1.90 43,480 10,398 206,008
Shops 12 65.75 4.28 38.08 2.48 10,682 1966 59,355
Other 43 58.28 4.08 33.56 2.31 6254 1622 47,641
OTC All 435 54.37 6.37 30.66 3.26 3133 797 24,757
Food 4 121.00 8.93 108.25 7.86 936 439 6708
Construction 26 80.58 6.75 57.31 4.72 1589 292 4024Chemical 32 42.84 4.80 19.88 2.23 1118 213 5603
Manufacturing 316 53.08 6.75 28.47 3.26 3253 873 29,207
Transportation 7 37.14 3.30 22.43 2.10 2675 139 2449
Finance 13 77.77 5.86 38.62 3.11 12,126 3112 54,753
Shops 5 48.80 3.93 31.60 2.39 2451 308 5562
Other 32 44.13 4.85 30.25 2.95 2041 368 14,760
This table reports the number of firms, average number and percentage of up-limit hits, average number and percentage of
down-limit hits, average market value (in million NT dollars), average daily trading volume (in shares) and average trading
volume (in thousand NT dollars) for firms of different exchanges and/or different industries. The sample covers daily data
of all firms listed on the Taiwan Stock Exchange (TWSE) and the Over-the-Counter (OTC) markets during the period from
January 3, 1998 to December 31, 2003.
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Table 3
Average number of no-trading days
Market Industry Mean S.D. 95th
All All 8.76 (0.83) 41.08 50.00 (3.89)
TWSE All 4.81 (0.36) 29.89 11.00 (0.78)
Food 19.32 (1.36) 40.86 115.00 (7.69)
Construction 4.05 (0.29) 10.77 23.00 (2.15)
Chemical 1.06 (0.09) 3.99 10.00 (0.96)
Manufacturing 4.11 (0.33) 30.42 5.00 (0.51)
Transportation 0.81 (0.05) 3.25 13.00 (0.84)
Finance 0.02 (0.00) 0.15 0.00 (0.00)
Shops 29.67 (1.98) 97.50 339.00 (22.62)
Other 7.70 (0.57) 26.08 21.00 (1.36)
OTC All 14.63 (1.52) 53.04 88.00 (9.64)
Food 37.00 (2.68) 50.42 112.00 (7.84)
Construction 33.00 (3.36) 67.79 134.00 (10.37)
Chemical 30.47 (2.51) 118.11 66.00 (8.46)
Manufacturing 7.49 (0.95) 30.33 60.00 (6.26)
Transportation 49.43 (3.64) 77.74 215.00 (14.14)
Finance 0.15 (0.04) 0.38 1.00 (0.44)
Shops 16.00 (1.08) 26.30 61.00 (4.01)
Other 49.72 (4.81) 90.13 277.00 (25.96)
This table reports average number of no-trading days for firms of different exchanges and/or different industries. The
sample covers daily data of all firms listed on the Taiwan Stock Exchange (TWSE) and the Over-the-Counter (OTC)
markets during the period from January 3, 1998 to December 31, 2003. 95th means the 95th percentile. The numbers in
parentheses are average percentages of no-trading days to exchange opening days.
Table 2
Duration of the last trade to exchange closed
Market Industry Mean S.D. Median 95th
All All 7.39 19.27 0.59 42.98TWSE All 3.21 10.46 0.15 18.85
Food 9.85 20.05 0.87 52.93
Construction 6.56 17.32 0.87 25.65
Chemical 2.06 4.39 0.12 9.83
Manufacturing 2.23 8.24 0.13 9.34
Transportation 0.97 2.03 0.05 7.78
Finance 1.19 3.96 0.02 5.47
Shops 12.84 24.91 0.59 78.77
Other 7.19 14.05 0.49 36.53
OTC All 13.30 26.13 2.61 72.14
Food 72.49 42.09 68.82 120.45
Construction 38.30 42.05 17.68 129.13Chemical 15.85 28.61 7.73 56.82
Manufacturing 8.23 18.98 2.00 42.61
Transportation 23.65 25.92 10.94 65.15
Finance 5.50 15.93 0.11 57.42
Shops 21.90 20.76 27.89 48.48
Other 34.11 39.18 24.67 115.16
This table reports the summary statistics of durations (in minutes) of the last trade to exchange closed for firms of different
exchanges and/or different industries. The sample covers intra-day transaction data of all firms listed on the Taiwan Stock
Exchange (TWSE) and the Over-the-Counter (OTC) markets during the period from January 2, 2003 to December 31,
2003. 95th means the 95th percentile.
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Since our purpose is to investigate how relevant the problems of price limits and thin trading
may be practically, we do not want to simulate the data from some assumed distributions, as
most Monte Carlo experiments do. To come up with portfolios that are more “realistic,” we
adopt the idea of Brown and Warner (1985) in which the simulation is drawn from real data toevaluate the performance of various event study methods.
Following the simulation design of Brown and Warner (1985), we generate 1000 samples of N
stocks from the population, and calculate the 1-day VaRs of the equal-weighted portfolio of the N
securities for various methods. We follow the convention by setting the estimation period as
250days (T =250), and consider two different portfolio sizes, N =10 and 30. The simulation
procedure is outlined as the following.
1. Randomly select an initial sample period [ s, s + 250], where s is an integer drown from the set
of numbers {1, 1299} with uniform probabilities.
2. During the sample period, draw N stocks without replacement from the population of 1081stocks. A stock is kept in the sample if it has a nonmissing price at the last day (i.e., at day
s + 250), and has more than 100 nonmissing observations during the sample period.
Otherwise, the stock is dropped. We considered two different values of N , N =10 and
N =30.
3. The 1-day 5% VaRs based on various methods for an N -stock portfolio are calculated.
4. The above steps are repeated for 1000 times, and the average VaR and the failure rate
for each method are calculated. The failure rate is calculated as the percentage of the
observations for which the corresponding true portfolio return exceeds the estimated
VaR.
As an implementation note, for each of the 1000 experiments we randomly draw N stocks from
the population of 1081 stocks over a randomly selected period, and compile an equal-weighted
portfolio of the N stocks. The N stocks may include stocks from both the TWSE and the OTC
markets. We believe such a portfolio is more realistic and practical than one that is simply drawn
from the OTC markets alone.
Statistically, the 99% confidence interval for the 5% VaR is (3.65%, 6.35%). A failure
rate that is smaller (greater) than the lower (upper) interval is considered to reject too little
(much).
6. Simulation results
The simulation results based on the design of Brown and Warner (1985) using real daily data
are reported in Tables 4–6.
6.1. Estimates of betas, variances, and covariances
Table 4 reports the average beta estimates based on naive OLS (denoted “OLS”), a two-limit
Tobit model adjusted for price limits (denoted “P.L.”), Scholes and Williams' (1977) method
adjusting for nonsynchronous trading (denoted “ N.S.”), and a two-limit model adjusted for both
price limits and nonsynchronous trading (denoted “P.L.+N.S.”) for firms of different exchanges
and/or different industries. The average betas are all equal-weighted.
Table 4 indicates that the naive OLS yields an average beta of 0.7564, the two-limit Tobit
model adjusting for price limits (the “P.L.” method) yields an average beta of 0.8225, and the
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Scholes and Williams' (the “ N.S.”) method yields an average of 0.8470. The last estimator that
adjusts for both effects yields an average estimate of 0.8359.4 Using the estimate based on the “P.
L.+N.S.” method that adjusts for both nonsynchronous trading and price limits as the benchmark,
the last three columns of Table 2 indicate that the downward bias in beta estimates based on the
naive OLS estimate is 9.51%, whereas the rest two methods, the “ N.S.” and “P.L.” methods, are
only slightly biased.As discussed in the previous section, biases in beta estimates will therefore cause biases in the
variance and covariance estimates. Table 5 reports the average variance, covariance, and the
absolute covariance for various estimation methods, all scaled by a number of 104. The average
variance, covariance and absolute covariance are calculated as the averages of the corresponding
statistics across all firms. We also report the average absolute covariance to assure that the average
bias is not cancelled out by biases of different signs.
The OLS estimates for variance and covariance are 9.63 and 1.73, respectively. In comparison
with the estimates based on the “P.L.+N.S.” method, the estimates are both severely downward
biased. The estimates of variance and covariance based on two-limit Tobit model that only adjusts
for price limits are only slightly downward biased. By contrast, the Scholes and Williams' (“ N.
Table 4
Beta estimates under different methods
Market Industry OLS (1) P.L. (2) N.S. (3) P.L.+N.S. (4) (1)− (4) (%) (2)− (4) (%) (3)− (4) (%)
All All 0.7564 0.8225 0.8470 0.8359−
0.0795(−9.51%)
−
0.0133(−1.59%) 0.0111(1.27%)
TWSE All 0.8271 0.9000 0.9239 0.9147 −0.0876 −0.0147 0.0092
Food 0.4999 0.5442 0.5692 0.6016 −0.1017 −0.0574 −0.0324
Construction 0.7193 0.7918 0.9155 0.9273 −0.2080 −0.1355 −0.0118
Chemical 0.7133 0.7569 0.8115 0.8039 −0.0906 −0.0469 0.0077
Manufacturing 0.9015 0.9906 0.9961 0.9820 −0.0805 0.0085 0.0141
Transportation 0.7358 0.7657 0.8557 0.8371 −0.1013 −0.0714 0.0186
Finance 0.8359 0.8769 0.8790 0.8672 −0.0313 0.0097 0.0118
Shops 0.6459 0.6915 0.7343 0.7456 −0.0997 −0.0541 −0.0114
Other 0.6195 0.6452 0.7033 0.6954 −0.0759 −0.0503 0.0078
OTC All 0.6514 0.7075 0.7327 0.7188 −0.0674 −0.0113 0.0139
Food 0.3129 0.3407 0.3823 0.3572 −0.0443 −0.0164 0.0251Construction 0.3227 0.3517 0.4106 0.4112 −0.0885 −0.0595 −0.0006
Chemical 0.3942 0.4308 0.4787 0.4763 −0.0821 −0.0455 0.0024
Manufacturing 0.7301 0.7957 0.8146 0.7974 −0.0673 −0.0017 0.0173
Transportation 0.3031 0.3041 0.3786 0.3597 −0.0566 −0.0556 0.0190
Finance 1.1138 1.2055 1.2044 1.2112 −0.0974 −0.0057 −0.0068
Shops 0.3786 0.3861 0.4271 0.4152 −0.0366 −0.0291 0.0118
Other 0.3721 0.3847 0.4168 0.4069 −0.0348 −0.0222 0.0099
This table reports the average beta estimates based on naive OLS (denoted “OLS”), a two-limit Tobit model adjusted for
price limits (denoted “P.L.”), Scholes and Williams' (1977) method adjusting for nonsynchronous trading (denoted “ N.
S.”), and a two-limit model adjusted for both price limits and nonsynchronous trading (denoted “P.L.+N.S.”) for firms of
different exchanges and/or different industries. The last three columns report the bias in beta estimates using the“
P.L.+N.S.” estimate as the benchmark. The numbers in parentheses are percentages in bias. The sample covers daily data of all
firms listed on the Taiwan Stock Exchange (TWSE) and the Over-the-Counter (OTC) markets during the period from
January 3, 1998 to December 31, 2003.
4 Since the average betas are calculated as equal-weighted, rather than capitalization weighted, the averages need not be
equal or close to one, which is the beta of the market.
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S.”) method yields a variance estimate of 12.34, which is severely biased upward in comparisonwith the estimate of 9.83 based on the “P.L.+N.S.” method. The covariance of the “ N.S.” method,
however, is very close to the estimate based on the “P.L.+N.S.” method.
Recall that an upward (a downward) bias in variance estimates will cause an upward
(downward) bias in VaR estimate and hence a lower (higher) failure rate. Likewise, an upward (a
downward) bias in covariance estimates will cause a downward (an upward) bias in the estimate
in the portfolio variance, thereby causing an upward (a downward) bias in VaR estimate and a
lower (higher) failure rate. Thus, the variance and covariance estimates reported in Table 5
suggest that the VaR estimate based on the naive OLS will be downward biased. The “ N.S.”
method might yield an overstated VaR estimate because the variance is upward biased. We report
the simulation results in the next subsection.
Table 6
VaR estimates and failure rates for various methods
Average
portfolio
return
(%)
OLS P.L. N.S. P.L. + N.S. H.S.
VaR (%) Failure VaR (%) Failure VaR (%) Failure VaR (%) Failure VaR (%) Failure
Panel A: equal-weighted portfolio of 10 firms
Mean −0.1240 −2.6345 0.0620 −2.7952 0.0520 −2.9255 0.0430 −2.8658 0.0450 −2.8483 0.0490
S.D. 1.7277 0.3595 0.4361 0.4068 0.4271 0.4455
Max 5.9220 −1.5587 −1.6385 −1.8033 −1.7442 −1.6910
Min −6.8870 −3.6840 −4.0857 −4.0728 −4.2327 −4.1660
Panel B: equal-weighted portfolio of 30 firms
Mean 0.0012 −2.4158 0.0530 −2.5964 0.0450 −2.6832 0.0410 −2.6684 0.0400 −2.6710 0.0400
S.D. 1.5359 0.3071 0.3671 0.3353 0.3566 0.3983
Max 5.5080 −1.4890 −1.5848 −1.6744 −1.6282 −1.4713
Min −6.6527 −3.2914 −3.7031 −3.5883 −3.6304 −3.8763
This table reports the average estimates of VaR and the failure rates based on naive OLS (denoted “OLS”), a two-limit
Tobit model adjusted for price limits (denoted “P.L.”), Scholes and Williams' (1977) method adjusting for nonsynchronous
trading (denoted “ N.S.”), and a two-limit model adjusted for both price limits and nonsynchronous trading (denoted “P.L.
+N.S.”), and historical simulation (denoted “H.S.”) for 1000 simulated portfolios based on the design of Brown and Waner
(1985). The sample contains daily observations for all firms listed on the Taiwan Stock Exchange (TWSE) and the Over-
the-Counter (OTC) markets during the period from January 3, 1998 to December 31, 2003. Panel A reports the result for an
equal-weighted portfolio of 10 firms, while Panel B reports the case of 30 firms.
Table 5
Variance and covariance properties under different methods
OLS P.L. N.S. P.L. + N.S.
Mean
(×10−4)
S.D.
(×10−4)
Mean
(×10−4)
S.D.
(×10−4)
Mean
(×10−4)
S.D.
(×10−4)
Mean
(×10−4)
S.D.
(×10−4)
σˆ i2 9.6390 3.4676 9.8194 3.9746 12.3425 5.6648 9.8283 3.9790
σˆ ij 1.7342 1.1106 2.0507 1.4029 2.1743 1.3983 2.1178 1.3718
|σˆ ij | 1.7439 1.0953 2.0576 1.3928 2.2021 1.3541 2.1380 1.3400
This table reports the average estimates of variance, and covariance, and absolute covariance based on naive OLS (denoted
“OLS”), a two-limit Tobit model adjusted for price limits (denoted “P.L.”), Scholes and Williams' (1977) method adjusting
for nonsynchronous trading (denoted “ N.S.”), and a two-limit model adjusted for both price limits and nonsynchronous
trading (denoted “P.L.+N.S.”) for all firms listed on the Taiwan Stock Exchange (TWSE) and the Over-the-Counter (OTC)
markets during the period from January 3, 1998 to December 31, 2003.
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6.2. Performances of VaR estimates based on various methods
Table 6 reports the VaR estimates and failure rates of various parametric methods discussed in
the previous subsection and the historical simulation method for two simulated equal-weighted portfolios of different sizes. Note that in the case of downside price movements, some stocks in
the portfolio may hit the limit, especially the down limit, thus causing the observed loss to be
understated. That is, the “true” return would have been lower without the imposition of price
limits. In this case, the “unrealized” negative shock is expected to be carried over to the next
trading day, and yet should be treated as the loss incurred in that certain trading day. Hence, the
failure rates calculated based on the observed return would be understated. In other words, a
failure rate that is slightly smaller than the nominal level (5% in our experiment) should be
preferred.5
Panel A of Table 6 reports the results for the equal-weighted portfolio of 10 stocks, and Panel B
reports the results for the 30-stock portfolio. The 10-stock portfolio has an average daily return of −0.1240% and standard deviation of 1.73, whereas the 30-stock portfolio has a positive return of
0.0012% with a smaller standard deviation of 1.54. The better performance of the 30-stock
portfolio is clearly due to its increased diversification by incorporating more stocks into the
portfolio.
Panel A of Table 6 indicates that the naive OLS method yields the smallest VaR estimate of
−2.63% (in absolute value) and the highest failure rate of 6.20% among all methods. The poor
performance of the naive OLS method is clearly attributed to the biased estimates of variances and
covariances. Although the failure rate of 6.20% still falls within the 99% confidence interval, we
consider it over-rejecting because an ideal failure rate should be slightly smaller than the nominal
level, as we discussed in the previous section. The remaining four methods, including thehistorical simulation method, all yield reasonable failures rates close to the 5% nominal rejection
rate.
Panel B of Table 6 reports the simulation for the 30-stock portfolios. Due to the increased
diversification effect, all methods yield smaller VaR estimates than in the 10-stock case. Among
the five methods, the OLS method still has the smallest VaR estimate and the highest failure rate,
but the failure rate of 5.30% is now much acceptable. In contrast, although the failure rates of the
remaining four methods are all within the confidence intervals, it seems that the two-limit Tobit
model adjusting for price limits alone (the “P.L.” method) has the closest failure rate to the
nominal value.
The failure rates of the historical simulation and our methods that adjust for nonsynchronoustrading and/or price limits all fall below the 5% nominal rate. One may attribute the lower
rejection rates to either the biased parameter estimates or improper choices of the critical value
(distributional percentile). But the “actual” failure rates would have been higher if one further
considers the spillover of residual shocks caused by price limits. Thus, practically a slightly lower
failure rate than the nominal rate can be viewed as a good property.
One might also conjecture the lower failure rates as the result of the fat-tailedness. However,
this is not the case because there would have been a higher failure rate if the returns were
generated by a fat-tail distribution like the t distribution. Thus, our simulation results suggest that,
at least in Taiwan's markets, there seems to be no need to further consider the GARCH-type effect
as long as the historical simulation or our methods are used.
5 Of course, it would be of interest to measure the unrealized loss that is carried over to the next trading day. This is,
however, difficult, and is beyond the scope of the paper.
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Overall, the simulation results suggest that although price limits and nonsynchronous trading
entail some potential estimation problems, its practical relevance seems limited.
7. Conclusion
We present portfolio VaR estimates based on the variance–covariance method that take into
account the effect of nonsynchronous trading and price limits—two features that are most
common in emerging markets. Based on the simulation design of Brown and Warner (1985), the
simulation results indicate that all methods, including the traditional naive OLS, the historical
simulation, and the proposed methods, perform reasonable well with real data. At least, in volatile
markets like Taiwan where the average limit hits are as high as 9%, all methods, including the
traditional methods like naive OLS and the historical simulation methods, perform reasonable
well. Maybe only the OLS method should be used with some caution when the portfolio at hand is
composed of only a few stocks.Our methods have been based on the simple market model specification. What is special in our
methods is that we explicitly take into account the parameter estimation problems induced by
price limits and nonsynchronous trading. As discussed in the paper, the analysis is easily extended
to a multi-factor setting. Also, we have only considered the 1-day VaR measure in the paper, the
extension to multi-day VaR measure, however, is easy. Finally, although our analysis is based on
portfolios of linear assets only, it is possible, and should not be too difficult, to extend the
proposed methods to deal with nonlinear assets.
Acknowledgements
We thank Chuan-Chang Chang, David Ding (the guest editor), and especially an anonymous
referee for helpful comments. We also thank Chih-Wan Yang for providing some of the programs
used in this study.
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