Motivation
Empirical Study
Summary
Liquidity and the Value-at-Risk
Lidan, Li
1 University
2
of Konstanz
Webex Presentation, 10 Dec 2010
Lidan, Li,
Liquidity and the Value at Risk
Motivation
Empirical Study
Summary
Outline
1
Motivation
Fundamental Problem
Denition of Liquidity
Eects of Illiquidity
Liquidity Proxies
2
Empirical Study
Our Model
Data
Estimations
Causality between liquidity and volatility
Lidan, Li,
Liquidity and the Value at Risk
Motivation
Empirical Study
Summary
Basic Problem
Denition of Liquidity
Eects of Illiquidity
Liquidity Proxies
Motivation
Why care about liquidity?
aects prices even when fundamental value remains constant
(Amihud, Mendelson & Wood (1990) )
high liquidity improves market eciency (Amihud, Mendelson
& Lauterbach (1997))
aggregate measures predictable, exhibit commonality (Chordia,
Roll & Subrahmanyam (2001), Hasbrouck & Seppi (2001),
Amihud (2002), Jones (2002), Huberman & Halka (2001),
Korajczyk & Sadka (2007))
it is priced (Pastor & Stambaugh (2003), Acharya & Pedersen
(2004, 2005), Sadka (2005))
experiences shocks (Huang (2003))
Lidan, Li,
Liquidity and the Value at Risk
Motivation
Empirical Study
Summary
Basic Problem
Denition of Liquidity
Eects of Illiquidity
Liquidity Proxies
Problem
Liquidity is not observable
Transmission mechanism into price process is unclear.
Lidan, Li,
Liquidity and the Value at Risk
Motivation
Empirical Study
Summary
Basic Problem
Denition of Liquidity
Eects of Illiquidity
Liquidity Proxies
Market Liquidity
Wikipedia: market liquidity is an asset's ability to be sold
without causing a signicant movement in the price and with
minimum loss of value
Common denition: ability to buy or sell signicant quantities
of a security quickly, anonymously, and with minimal or no
price impact
Price of immediacy
Kyle (1985) 3 dimensions- tightness, depth, resiliency
Lidan, Li,
Liquidity and the Value at Risk
Motivation
Empirical Study
Summary
Basic Problem
Denition of Liquidity
Eects of Illiquidity
Liquidity Proxies
Reasons for market illiquidity
asynchronous arrivals of buyers and sellers, role for
market-makers
bid-ask spread - inventory holding (Garman (1976), Stoll
(1978), Amihud & Mendelson (1980), Ho & Stoll (1978)).
But costs is found to be small! (Stoll (1978), George et. al.
(1991), Madhavan & Smidt (1991))
Chacko et al. (2008) assumes zero inventory costs, shows that
the asynchronous arrivals of buyers and sellers give market
makers transitory pricing power
Lidan, Li,
Liquidity and the Value at Risk
Motivation
Empirical Study
Summary
Basic Problem
Denition of Liquidity
Eects of Illiquidity
Liquidity Proxies
Reasons for market illiquidity (II)
adverse selection due to information asymmetry: Bagehot
(1971)
Kyle (1985), Glosten (1989), Easley & O'Hara (1987), Glosten
& Harris (1988) show that the eect of asymmetric
information is most likely captured by price impact of trades
PIN type measures:
No of speculators in the market: Grossman & Miller (1988),
Brunnermeier & Pedersen (2008)
Lidan, Li,
Liquidity and the Value at Risk
Motivation
Empirical Study
Summary
Basic Problem
Denition of Liquidity
Eects of Illiquidity
Liquidity Proxies
Relationship between liquidity and volatility
rst order eects of liquidity: expected stock returns
∝
illiquidity proxies (Amihud (1986), Brennan & Subrahmanyan
(1996), Datat et. al. (1998), Easley et. al. ,Pastor &
Stambaugh (2003),etc )
second order eects of liquidity
Trading activity
∝liquidity, ∝-volatility
(Tinic (1972), Benston
& Hagerman (1974) )
Constantinides (1986) equilibrium model: nds that higher
volatility corresponds to higher trading costs, and hence a
wider no-trade zone.
Modern automated auction markets: state of limit order book
describes liquidity supply of asset (Biais, et al (1995), Handa &
Schwartz (1996), Ahn et. al. (2001), Beltran et. al. (2005) )
Lidan, Li,
Liquidity and the Value at Risk
Motivation
Empirical Study
Summary
Basic Problem
Denition of Liquidity
Eects of Illiquidity
Liquidity Proxies
Liquidity adjusted VaR
VaR :Quantile forecast: model for forecasting volatility +
method of computing quantiles
Bangia et. al. (1999):
LVaRt ,q = σt ,return Zq,return + σt ,spread Zq,spread
Strong assumptions. Berkowitz (2000) : impact of transaction
sizes on prices
Lidan, Li,
Liquidity and the Value at Risk
Motivation
Empirical Study
Summary
Basic Problem
Denition of Liquidity
Eects of Illiquidity
Liquidity Proxies
Liquidity Proxies
Bid-ask spread type measures: inside bid ask spread, quote
slope
QS = log BS
Depth
Volume related measures: volume, number of transactions
Price impact measures: Kyle (1985), Admanti & Peiderer
(1988):
∆pt = λ qt + ψ[Dt − Dt −1 ] + yt ,
Hasbrouck (1991), Foster & Viswanathan (1993) :
qt
∆pt
n
n
= αp + ∑ βj ∆pt −j + ∑ γj qt −j + τt
j =1
j =1
= αp + ψ[Dt − Dt −1 ] + λ τt + vt
Lidan, Li,
Liquidity and the Value at Risk
Motivation
Empirical Study
Summary
Our Model
Data
Estimations
Causality between liquidity and volatility
Joint distribution of returns and liquidity
Angelidis & Benos (2006): decompose components of spread.
LVaR has intraday U shape pattern
LVaR should incorporate instantaneous eects of liquidity on
returns:
f (rt , lt |Ft −1 ) = f (rt |lt , Ft −1 )f (lt |Ft −1 )
p = −∞ fq (rt , lt |Ft −1 )dldr
Estimating f (rt |liqt , Ft −1 )−Incorporating liquidity in conditional
R LVaR R
1.
density of returns
Linear:
Garch (1,1) :
ht = ω + αεt2−1 + β ht −1 + γ liqt
quantile garch:
Qut (τ|Ft −1 ) = β0∗ + ∑pi=1 βi∗ Qut−i (τ|Ft −i −1 ) + ∑qj=1 γj∗ |ut −j |
nonlinear :
exponential garch: ln
2. Estimating
ht = ω + αεt2−1 + β ln ht −1 + γ ln liqt
f (liq |F
Lidan,
Li,
Liquidity
and the of
Value
at Risk states
)-Dynamic
modeling
liquidity
Motivation
Empirical Study
Summary
Our Model
Data
Estimations
Causality between liquidity and volatility
Data
NYSE TAQ data ltered by rules adapted from
Barndor-Nielsen et. al. (2008)
27 stocks
2005-2008: 1007 days.
Table: descriptive statistics
mean
sdev
skewness
kurtosis
JB stat
aa ret
-0.002219
0.023831
-1.132675
12.742556
4197
c ret
-0.002926
0.030443
-2.705678
38.880062
55244
Lidan, Li,
ibm ret
0.000744
0.013055
-0.443328
7.383169
839
aa qs
0.015206
0.006741
1.337592
5.824882
635
c qs
0.009254
0.007832
5.148196
39.97825
61821
Liquidity and the Value at Risk
ibm qs
0.050768
0.049192
4.895691
39.052015
58557
Motivation
Empirical Study
Summary
Our Model
Data
Estimations
Causality between liquidity and volatility
Line graph
Lidan, Li,
Liquidity and the Value at Risk
Our Model
Data
Estimations
Causality between liquidity and volatility
Motivation
Empirical Study
Summary
standard Garch tests
25 out of 27 reject for homoskedasticity, 16 out of 27 reject test for
linear garch
Table: linear GARCH test
pvalues
aa
aig
axp
ba
bac
c
1
0.11596215
4.6121941e-011
4.3572475e-005
0.21836067
5.1321147e-006
4.3712979e-005
5
4.6864754e-005
2.9456266e-015
1.8173076e-008
0.0063437401
1.4207275e-010
9.6559111e-007
10
3.3693466e-007
1.1832488e-034
1.0403632e-006
0.020689865
4.8448195e-012
1.405496e-006
Table: nonlinear GARCH test
aa
aig
axp
ba
bac
c
SB test
-1.179
0.8214
2.519
1.446
2.06
0.3814
pvals
0.238
0.411
0.011
0.148
0.039
0.7028
NSB test
1.96
-2.469
-2.820
-1.338
-4.877
6.081
pval
0.049
0.0135
0.0048
0.1806
1.07e-006
1.19e-009
Lidan, Li,
PSB test
0.299
-7.562
-3.699
-0.653
-2.60
-0.8981
pval
0.764
3.963e-014
0.0002
0.513
0.009
0.369
Liquidity and the Value at Risk
General
4.11
60.954
17.395
2.51
27.28
43.00
pval
0.249
3.675e-013
0.0005
0.473
5.13e-006
2.454e-009
Motivation
Empirical Study
Summary
Our Model
Data
Estimations
Causality between liquidity and volatility
Garch(1,1) estimations
aa
aig
axp
c
dd
gm
hpq
hon
ibm
jpm
ω1
variance regressor-daily ave quote slope
0.000063
(0.334954)
0.001592
( 1.959817)
0.003653
( 3.068729)
0.004808
(4.295362)
-0.001142
(-1.040142)
0.008403
(0.997375)
0.0019726
(1.038327)
0.004711
( 1.540569)
0.003999
(3.991991)
0.003861
(3.889822)
α1
0.077991
(3.562869)
0.182233
(4.026930)
0.111610
(4.091173)
0.183844
(5.665177)
0.082773
(2.406288)
0.079120
(1.764832)
0.039546
( 2.397216)
0.076807
( 2.663340)
0.101512
(4.030168)
0.129481
(4.010669)
Lidan, Li,
β1
0.920312
(36.081008)
0.832144
(24.215470)
0.882698
(29.541562)
0.816131
(27.672662)
0.881455
(13.315929)
0.921642
(21.122267)
0.898113
(24.128927)
0.864799
(16.760280)
0.842225
(19.832386)
0.867526
(26.715588)
γ1
0.000478
(0.538177)
0.000078
(0.251349)
-0.000226
(-0.408095)
-0.001528
(-1.227071)
0.001141
( 1.065728)
-0.000614
(-0.405460)
0.002905
(1.409540)
0.001372
(1.311829)
0.000415
(1.506518)
-0.000351
(-0.475982)
Liquidity and the Value at Risk
Motivation
Empirical Study
Summary
Our Model
Data
Estimations
Causality between liquidity and volatility
Shifting focus to extreme quantiles
Longin (2000) and Bali (2000) : volatility measures based on
asset return distributions cannot produce accurate measures of
market risk during volatile periods.
Allow for regressors to impact conditional tail quantiles
dierently from central quantiles
Koenker & Basset(1978) regression quantiles: least
asymmetric absolute deviation problem
Lidan, Li,
Liquidity and the Value at Risk
Our Model
Data
Estimations
Causality between liquidity and volatility
Motivation
Empirical Study
Summary
Quantile Garch(1,1)
Table: aa- quantile regression results
τ
0.01
0.05
0.1
0.2
0.3
ω
β
α
γ
-0.032949683
-0.3133369
-2.2400412
-0.01384044
-0.013332437
-0.68619769
-0.99403384
-0.0073056359
-0.016070251
-0.18213752
-0.41068334
-0.0065702043
-0.011544173
-0.10697263
-0.26433129
-0.0044813663
-0.0059909927
-0.077650486
-0.19348258
-0.0025501655
0.4
qgarch (90 %
-0.0046672071
-0.011825408
-0.026522968
-0.0017563458
Table: c- quantile regression results
τ
0.01
0.05
0.1
0.2
0.3
ω
β
α
γ
-0.032607317
-1.2018977
-0.85429112
-0.016805871
-0.0067585568
-1.1193023
-1.0093918
-0.011114935
-0.0036311977
-0.74001279
-0.80873688
-0.0065269578
-0.0027424452
-0.49995051
-0.4553683
-0.0045542156
-0.0019580268
-0.29255192
-0.21614284
-0.0010507862
Lidan, Li,
Liquidity and the Value at Risk
0.4
qgarch (90 %
-0.0009154247
-0.17398816
-0.16251356
-0.0016848939
Motivation
Empirical Study
Summary
Our Model
Data
Estimations
Causality between liquidity and volatility
Exponential Garch (1,1)
aig
aa
axp
ba
c
ibm
hpq
c
1.081412
(0.724338)
1.210381
(2.386101)
-0.053551
(-0.156410)
2.645763
(2.417373)
0.580475
(0.592333)
0.905157
(1.697931)
2.223891
(1.983677)
ω
α
β
γ
-0.774965
(-3.336173)
-0.176737
(-2.787408)
0.053638
(1.192938)
-0.141037
(-2.178208)
-0.075331
(-4.754929)
-0.116812
(-3.560465)
-0.119265
(-1.486515)
0.070650
(1.274415)
0.032776
(2.366915)
0.079636
(2.512507)
0.017978
(2.304863)
0.013632
(5.060816)
0.046812
(2.250848)
0.026403
(2.340134)
0.741077
(6.312925)
0.916418
(28.404282)
0.847086
(13.629401)
0.959246
(41.845684)
0.985397
(409.584020)
0.893339
(16.699990)
0.914802
(33.813713)
-0.374399
(-2.245243)
-0.076850
(-2.316936)
0.424063
(2.311409)
0.045636
(0.896446)
-0.045602
(-8.114259)
0.018581
(0.185414)
0.080283
(1.927078)
Lidan, Li,
Liquidity and the Value at Risk
Motivation
Empirical Study
Summary
Our Model
Data
Estimations
Causality between liquidity and volatility
Granger Causalities between liquidity and volatility
out of 27 stocks, 22 reject for nongranger causality bet liq and
realized vol
liq to realized vol
realized vol to liq
lags
5
10
15
20
5
10
15
20
aa
25.7680
18.4836
14.5109
12.7633
33.8057
25.8839
21.0978
18.0936
aig
56.2347
42.6982
34.5208
30.3883
52.9909
38.7367
31.8736
27.1626
axp
10.2775
6.4562
4.83922
3.9788
11.0912
10.6735
10.9238
10.3448
ba
14.4012
10.8708
9.78064
8.1009
14.2308
10.6756
10.2098
8.48768
c
9.8891
6.7312
5.6338
6.8044
14.2877
11.5201
10.2247
10.1749
cat
0.8948
0.5821
0.2083
0.62697
0.0044
-0.4372
-0.1587
-0.3014
dd
9.9097
8.4811
7.5559
7.4770
16.5684
14.5606
12.6370
11.5005
dis
11.3784
7.92746
7.3078
6.1666
12.7966
11.3234
10.9793
9.4725
ge
2.2663
1.6180
1.4740
4.2036
4.8376
3.1967
2.2948
1.2522
gm
0.5730
1.2118
1.3306
2.4816
1.8345
1.0424
0.2204
-0.0610
Lidan, Li,
Liquidity and the Value at Risk
Our Model
Data
Estimations
Causality between liquidity and volatility
Motivation
Empirical Study
Summary
Granger causalities in tails
Hong et. al. (2009): tests for Granger causality in risk.
H10 : P (Y1t
H1A : P (Y1t
Risk indicator:
< −V1t |I1,t −1 ) = P (Y1t < −V1t |I1,t −1 , I2,t −1 )
< −V1t |I1,t −1 ) 6= P (Y1t < −V1t |I1,t −1 , I2,t −1 )
Z1t ≡ 1(Y1t < −V1t ).
Investigate
cross-spectrum between
Z1t , Z2t ,f (ω) ≡ 21π ∑∞j =−∞ ρ(j )e −ij ω , ω ∈ [−π, π], i =
H10 : f10 (ω) ≡
0
1
2π
∑
j =−∞
Lidan, Li,
√
−1
ρ(j )e −ij ω , ω ∈ [−π, π]
Liquidity and the Value at Risk
Motivation
Empirical Study
Summary
Our Model
Data
Estimations
Causality between liquidity and volatility
Granger Causality in tails (2)
Using kernels for downweighting lags:
fˆ(ω) ≡ 21π ∑jT=−11−T k (j /M )ρ̂(j )e −ij ω
Comparing:
L2 (fˆ, fˆ10 ) ≡ 2π
Rπ
fˆ(ω) − fˆ10 (ω)|2 d ω = ∑Tj =−11 k 2 (j /M )ρ̂ 2 (j ).
−π |
Test statistic:
Q1 (M ) ≡ [T ∑Tj =−11 k 2 (j /M )ρ̂ 2 (j ) − C1T (M )]/D1T (M )1/2
Lidan, Li,
Liquidity and the Value at Risk
Motivation
Empirical Study
Summary
Our Model
Data
Estimations
Causality between liquidity and volatility
Granger causality in tails
Qtrun
Qreg
lags
15
20
5
10
15
20
aa 0.01
-2.7385391
-3.1622076
2.2121
8.2582
10.347
10.234
aa 0.05
-2.7385424
-3.162197
2.9476
3.3150
4.1031
3.6476
aa 0.10
-2.7385339
-3.1621925
1.5139
1.7369
1.1432
1.3408
aa 0.15
-2.7385242
-3.1621837
4.2837
2.8853
1.9985
1.3405
aa 0.20
-2.7385572
-3.1622173
2.1726
2.7303
1.4026
1.0893
aa 0.25
-2.738533
-3.1621903
2.9063
2.5707
2.0531
1.7652
c 0.01
-2.7385973
-3.1622637
0.9503
4.9678
3.3964
2.3592
c 0.05
-2.7385925
-3.1622525
1.1715
-0.2205
1.5387
0.8250
c 0.10
-2.7386058
-3.1622708
-0.8000
-1.296
-1.2263
-0.5628
c 0.15
-2.7386072
-3.1622727
2.3162
1.388
1.3304
0.5815
c 0.20
-2.7386036
-3.1622693
1.1267
0.9242
0.4232
0.3453
c 0.25
-2.738607
-3.1622721
0.7047
0.0710
-0.3740
-0.6736
ibm 0.01
-2.7386114
-3.1622761
-1.500
-2.146
-2.645
-3.060
ibm 0.05
-2.7385247
-3.1621989
6.158
4.489
3.546
3.466
ibm 0.10
-2.7385326
-3.1622048
3.267
2.611
2.406
3.554
ibm 0.15
-2.7385238
-3.1621834
3.064
3.327
2.156
2.931
ibm 0.20
-2.7385331
-3.1621874
3.879
3.821
3.359
2.92
ibm 0.25
-2.7385434
-3.1622023
1.143
3.959
3.406
2.763
Lidan, Li,
Liquidity and the Value at Risk
Motivation
Empirical Study
Summary
Summary
Causality of liquidity in the means and tail dier
Direct incorporation of liquidity measure into tails is more
robust (no distributional assumption)
Outlook
testing for instantaneous causality
dynamic model for liquidity states
dierent liquidity proxies
Lidan, Li,
Liquidity and the Value at Risk