estimating the var of a portfolio subject to price limits and non synchronous trading

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

mailto:[email protected]

http://dx.doi.org/10.1016/j.irfa.2005.03.002



mailto:[email protected]



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



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.




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




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




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.




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

Â TSt ;

PDt ¼ Int Ot −1ð1−0:07Þ

TSt

Â 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




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.




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.




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




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.




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.




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.




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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