U.U.D.M. Project Report 2017:11

Examensarbete i matematik, 30 hpHandledare: Maciej Klimek Examinator: Erik EkströmMaj 2017

Department of MathematicsUppsala University

Liquidity Adjusted Value-at-Risk and ItsApplications

Peiyu Wang

LiquidityAdjustedValue-at-RiskandItsApplications

PeiyuWang

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1. IntroductionThe recent financial crises have been liquidity crises. Large hedge fundcompanies held positions thatwere too large to be liquidatedwithout causingsignificantpriceimpact.ThebreakdownofLTCMin1998andAmaranthAdvisorsin2006area fewexamples. Inthesub-primecrisisof2008,a lotofbanksandhedge funds also suffered from liquidity shortage and had to liquidate theirassets,whichdeterioratedthecrisisandcausedhugelosses.Thesearethesignsthatpeopleunderestimatetheliquidityrisk.Nowadays,hedgefundsstilluseValue-at-Risk(VaR)tomeasurethemarketrisk.However, this measure can cause problems because when the volume of thepositionis largeenoughtocausepriceeffectonthespread,thetradingpriceisnotatthemid-price.Thishasasaresultthattherealpricewilldependonboththevalueofthespreadandthepriceeffectofthetradingvolume.Andthus,themarket liquidity plays an important role. To sum up, we could see that thenormalVaRconceptlacksarigoroustreatmentofliquidityrisk.Regarding the liquidity risk, the fund company examines the fund position ineachholdingintermsofnumbersofshares.Thentheholdingiscomparedtotheaverage trading volumeof the latest 20days, to see howmuch of the positionthat canbe liquidated taking intoaccount thatnomore than10percentof theaveragetradingvolumeisusedinordernottoaffectthepricestoomuch.Butinthiswayweneitherquantifytheliquidityriskinamonetaryvaluenorexpressitasapercentage,whichmakesitlessconvenienttousewhencomparingbetweendifferentfundsandcontrollingtherisk.SothispaperisdevotedtoincorporationofliquidityriskintoVaRmodel,whichcouldbetterreflectpoorliquidityintheVaRframework.Inordertoincorporatetheliquidityriskwiththelimiteddata,thispaperusedaconceptofLIX,whichisintroduced by Oleh Danyliv, Bruce Bland, Daniel Nicholass(2014). It is a newmeasure of liquidity. With this measure, we could measure different stocksliquidityandpredicttheliquidityinthefuture.Whenweneededtoquantifytheliquidityrisk,weusedtheconceptofCostofLiquidity(COL),whichishalfofthespread. AsforVaR,weusedatfirstthevariance-covariancemethodincalculation.Theninordertoimprovetheaccuracy,weusedtheextremevaluetheory(EVT)withanewquantileestimator,whichincludedalltheinformationinthetail.Finally, we added the VaR and Cost of Liquidity in order to get the LiquidityadjustedVaR and compared the results from a large cap fund and a small capfund.The thesis isorganized in the followingway: the secondchapter introducesallthebasicconceptsweneedtoknowinordertounderstandthemodel.Fromthegeneralizedinversetoquantilefunction,andthedefinitionsofVaRandfattails.Itintroduces liquidity risk, LIXand costof liquidity.The third chapter shows thedetailsofLiquidityadjustedVaRmodelandanalyzeseachpartofthemodel.Thefourthchaptercontainstheempiricalresultandconclusion.

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2. Background

2.1 GeneralizedinversesFirstofall,quantile function in fact isanapplicationofgeneralized inverses infinancial mathematics. Therefore we will begin with a short introduction ofgeneralized inverses based on an article by Paul Embrecht and MariusHofert(2013).Theideaofageneralizedinversecomesfromthefactthat,althoughareal-valued,continuous, and strictly monotone function of a single variable have a uniqueinversefunctiononitsrange,sometimestherequirementistoostrong.Inordertoapplyitmoreeasilyinreallife,wehavetodroptheassumptionsofcontinuityandstrictmonotonicity,butstillweneed toget the inverse,which leads to thenotionofageneralizedinverse.

Let 𝑇:ℝ → ℝ be a non-decreasing function with 𝑇 −∞ = lim+→,-

𝑇(𝑥) and

𝑇 ∞ = lim+→-

𝑇(𝑥),thegeneralizedinverse 𝑇,1:ℝ → ℝ = [−∞,∞] isdefinedby

𝑇,1 𝑦 = inf 𝑥 ∈ ℝ|𝑦 ≤ 𝑇(𝑥) , 𝑦 ∈ ℝ.If 𝑇 is a distribution function and the target domain become 0,1 , 𝑇,1 iscalledquantilefunctionof 𝑇.Weusetheconventionthat inf∅ = ∞.

Figure:Anon-decreasingfunction 𝑇(left)anditsgeneralizedinverse 𝑇,1(right)Source:PaulEmbrechtandMariusHofert(2013,p.425)

We could easily observe the difference between the generalized inverse andnormal inverse from the figure. Firstly, we could drop the strictly increasingassumption, so 𝑇 could be flat. The flat part of 𝑇 corresponds to the jump inthegeneralizedinverse 𝑇,1.Secondly,wecoulddropthecontinuityassumption,so 𝑇 could have jumps. The jump part of 𝑇 corresponds to the flat in thegeneralizedinverse 𝑇,1.Thegeneralizedinversehasthefollowingproperties,whicharefrequentlyusedandprovedbyPaulEmbrechtandMariusHofert(2013).

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Proposition 1: Let 𝑇:ℝ → ℝ be a non-decreasing function with

𝑇 −∞ = lim+→,-

𝑇(𝑥) and 𝑇 ∞ = lim+→-

𝑇(𝑥),let 𝑥, 𝑦 ∈ ℝ. Then,

(i) 𝑇,1 is non-decreasing. If 𝑇,1(𝑦) ∈ (−∞,∞) , 𝑇,1 is continuousfromtheleftandhasrightlimits

(ii) If 𝑇 is continuous from the right, then 𝑇,1(𝑦) < ∞ implies

𝑇 𝑇,1 𝑦 ≥ 𝑦 . Furthermore, 𝑦 ∈ 𝑟𝑎𝑛𝑇 ∪ {inf 𝑟𝑎𝑛𝑇, sup 𝑟𝑎𝑛𝑇}

implies 𝑇 𝑇,1 𝑦 = 𝑦 . Moreover, if 𝑦 < inf 𝑟𝑎𝑛𝑇 then

𝑇 𝑇,1 𝑦 > 𝑦 and if 𝑦 > sup 𝑟𝑎𝑛𝑇 then 𝑇 𝑇,1 𝑦 < 𝑦 , where

randenotestheabbreviationofrange.(iii) 𝑇(𝑥) ≥ 𝑦 ⇒ 𝑥 ≥ 𝑇,1(𝑦). Furthermore, if 𝑇 is continuous from the

right, then 𝑇(𝑥) ≥ 𝑦 ⇔ 𝑥 ≥ 𝑇,1(𝑦) . Moreover, 𝑇(𝑥) < 𝑦 ⇒ 𝑥 ≤𝑇,1(𝑦).

(iv) If 𝑇1 and 𝑇N are continuous from the right and have samepropertiesas 𝑇,then 𝑇1 ∘ 𝑇N ,1 = 𝑇N,1 ∘ 𝑇1,1.

Proof(i) Let 𝑦1, 𝑦N ∈ ℝ and 𝑦1 < 𝑦N.Wehave

𝑥 ∈ ℝ|𝑦1 ≤ 𝑇(𝑥) ⊇ 𝑥 ∈ ℝ|𝑦N ≤ 𝑇(𝑥) ,so 𝑇,1(𝑦1) ≤ 𝑇,1(𝑦N), 𝑇,1 isnon-decreasing. Let 𝑇,1(𝑦) ∈ (−∞,∞) and 𝑦Q = 𝑦 . Suppose 𝑦R → 𝑦Q is a strictlyincreasing sequence. In order to prove 𝑇,1 is continuous from the left,

we need to prove limR→-

𝑇,1(𝑦R) = 𝑇,1(𝑦Q). Since 𝑇,1 is non-decreasing,

𝑥R ∶= 𝑇,1(𝑦R) ≤ 𝑥Q ∶= 𝑇,1(𝑦Q).Thus

limR→-

𝑥R = 𝑥 ≤ 𝑥Q forsome 𝑥 ∈ ℝ.

Therefore,weneedtoprove 𝑥 = 𝑥Q.Weassume 𝑥 < 𝑥Q:Fromthedefinitionof 𝑇,1,wehave∀𝜀 > 0𝑎𝑛𝑑𝑛 ∈ ℕQ = {0,1,2,3… },

𝑇 𝑥R − 𝜀 < 𝑦R ≤ 𝑇 𝑥R + 𝜀 .

let 𝜀 = +\,+N

, then for ∀𝑛 ∈ ℕ, 𝑦R ≤ 𝑇 𝑥R + 𝜀 ≤ 𝑇 𝑥Q − 𝜀 < 𝑦Q. So

𝑦Q = limR→-

𝑦R ≤ 𝑇 𝑥Q − 𝜀 < 𝑦Q, which is a contradiction. 𝑇,1 is

continuousfromtheleft.In order to prove 𝑇,1 has right limit, suppose 𝑦R → 𝑦Q is a strictlydecreasing sequence. 𝑇,1(𝑦R) is a non-increasing sequence and𝑇,1 𝑦Q > −∞.Fromthemonotoneconvergencetheorem, 𝑇,1 hasrightlimit.

(ii) We have 𝑇,1(𝑦) < ∞, then 𝐴 = 𝑥 ∈ ℝ|𝑦 ≤ 𝑇(𝑥) ≠ ∅. Suppose 𝑥R →

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𝑇,1(𝑦) isastrictlydecreasingsequenceand 𝑥R R∈ℕ ⊆ 𝐴,thus 𝑇 𝑥R ≥

𝑦. Since 𝑇 is continuous from the right, 𝑇 𝑇,1 𝑦 = limR→-

𝑇(𝑥R) ≥ 𝑦 .

Weprovedthefirstpart.At first we consider 𝑦 ∈ 𝑟𝑎𝑛𝑇 , then 𝐵 = 𝑥 ∈ ℝ|𝑦 = 𝑇(𝑥) ≠ ∅ . Wecould see 𝑇,1 𝑦 = inf 𝐴 = inf 𝐵 . Suppose 𝑥R → 𝑇,1(𝑦) is anon-increasing sequence and 𝑥R R∈ℕ ⊆ 𝐵, thus 𝑇 𝑥R = 𝑦. Since 𝑇 is

continuous from the right, 𝑇 𝑇,1 𝑦 = limR→-

𝑇 𝑥R =𝑦. Now let 𝑦 =

inf 𝑟𝑎𝑛𝑇 and inf 𝑟𝑎𝑛𝑇 ∉ 𝑟𝑎𝑛𝑇 (otherwisewe could just use the proofabove).Thus 𝐴 = 𝑥 ∈ ℝ|𝑦 ≤ 𝑇(𝑥) = ℝ,wehave 𝑇,1 𝑦 = inf 𝐴 = −∞.

So 𝑇 𝑇,1 𝑦 = 𝑇 −∞ = inf 𝑟𝑎𝑛𝑇 = 𝑦. At last let 𝑦 = sup 𝑟𝑎𝑛𝑇 and

sup 𝑟𝑎𝑛𝑇 ∉ 𝑟𝑎𝑛𝑇 (otherwise we could just use the proof of 𝑟𝑎𝑛𝑇).Thus 𝐴 = 𝑥 ∈ ℝ|𝑦 ≤ 𝑇(𝑥) = ∅ , we have 𝑇,1 𝑦 = inf 𝐴 = ∞. So

𝑇 𝑇,1 𝑦 = 𝑇 ∞ = sup 𝑟𝑎𝑛𝑇 = 𝑦. Weprovedthesecondpart.

If 𝑦 < inf 𝑟𝑎𝑛𝑇, we have 𝐴 = 𝑥 ∈ ℝ|𝑦 ≤ 𝑇(𝑥) = ℝ, 𝑇,1 𝑦 = −∞, so

𝑇 𝑇,1 𝑦 = 𝑇 −∞ = inf 𝑟𝑎𝑛𝑇 > 𝑦. If 𝑦 > sup 𝑟𝑎𝑛𝑇, we have 𝐴 =

𝑥 ∈ ℝ|𝑦 ≤ 𝑇(𝑥) = ∅, 𝑇,1 𝑦 = ∞, so 𝑇 𝑇,1 𝑦 = 𝑇 ∞ =

sup 𝑟𝑎𝑛𝑇 < 𝑦. Weprovedthethirdpart.(iii) From the definition of 𝑇,1 , we have 𝑇(𝑥) ≥ 𝑦 ⇒ 𝑥 ≥ 𝑇,1(𝑦). If 𝑇 is

continuous from the right, and ∞ > 𝑥 ≥ 𝑇,1(𝑦), from property (ii), we

have 𝑇 𝑇,1 𝑦 ≥ 𝑦. So 𝑇(𝑥) ≥ 𝑇 𝑇,1 𝑦 ≥ 𝑦. Thus 𝑇(𝑥) ≥ 𝑦 ⇔ 𝑥 ≥

𝑇,1(𝑦).If 𝑇(𝑥) < 𝑦 and let𝐴 = 𝑧 ∈ ℝ|𝑦 ≤ 𝑇(𝑧) , since 𝑇 is a non-decreasingfunction, 𝑇(𝑧) ≥ 𝑦 > 𝑇(𝑥) ⇒ 𝑧 > 𝑥.So 𝑇,1 𝑦 = inf 𝐴 ≥x.

(iv) Let 𝑇1 and 𝑇N are continuous from the right andhave samepropertiesas 𝑇, then 𝑇1 ∘ 𝑇N ,1 = inf 𝑥 ∈ ℝ|𝑦 ≤ 𝑇1(𝑇N 𝑥 ) . by property (iii), we

have 𝑦 ≤ 𝑇1 𝑇N 𝑥 ⇔ 𝑇N 𝑥 ≥ 𝑇1,1 𝑦 ⇔ 𝑥 ≥ 𝑇N,1 𝑇1,1 𝑦 . So

𝑇1 ∘ 𝑇N ,1 = inf 𝑥 ∈ ℝ|𝑥 ≥ 𝑇N,1(𝑇1,1(𝑦)) = 𝑇N,1 𝑇1,1 𝑦 = 𝑇N,1 ∘ 𝑇1,1.

∎

2.2 QuantilesLet 𝑋 bearandomvariable.If 𝐹f isthecumulativedistributionfunctionof 𝑋,thatis

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𝐹f 𝑥 = ℙ 𝑋 ≤ 𝑥 ,𝑥 ∈ ℝ.Wedefinethecorrespondingquantilefunctionasthegeneralizedinverseof 𝐹:

𝐹f,1 𝑞 = inf 𝑥|𝑞 ≤ 𝐹f 𝑥 ,𝑞 ∈ 0,1 .The number 𝐹f,1 𝑞 is called the q-quantile of 𝑋 . Note that 𝐹f is alwaysnon-decreasing,iscontinuousfromtherightandhasleftlimits.Theq-quantileof 𝑋 is then the smallest value 𝑥 such that the probability of 𝑋 notexceeding 𝑥 isnotsmallerthan 𝑞.We will prove the quantile function is equivariant under non-decreasing leftcontinuous transformations. In order to prove it, we need to prove 2 lemmasfirst.Lemma1:(QuantileValueCriterionLemma)𝐹f,1(𝑞) istheonly 𝑎 satisfying(i)and(ii),where (i) 𝐹f 𝑎 ≥ 𝑞;

(ii) 𝑥 < 𝑎 ⇒ 𝐹f 𝑥 < 𝑞.ProofSuppose 𝑥R → 𝐹f,1(𝑞) is a strictly decreasing sequence. According to thedefinition of 𝐹f,1(𝑞) and 𝑥R > 𝐹f,1(𝑞), we have 𝐹f(𝑥R) ≥ 𝑞. Since 𝐹f is rightcontinuous

limR→-

𝐹f 𝑥R = 𝐹f(𝐹f,1(𝑞)).

For ∀𝑛 ∈ ℕ, 𝐹f 𝑥R ≥ 𝑞 hence limR→-

𝐹f 𝑥R ≥ 𝑞 (i) holds. So the 𝐹f,1 𝑞 is the

smallest value satisfy 𝐹f 𝑥 ≥ 𝑞 , if 𝑥 < 𝐹f,1 𝑞 , then 𝐹f(𝑥) < 𝑞 . So 𝐹f,1 𝑞 satisfiesbothproperties. Assuming both 𝑎 and 𝑏 satisfy them and 𝑎 < 𝑏 , then 𝐹f(𝑎) ≥ 𝑞 by (i).However 𝑏 also satisfy both properties and 𝑎 < 𝑏 , then 𝐹f(𝑎) < 𝑞 by (ii),whichisacontradiction.

∎Let ℎ(𝑥) ∶ ℝ → ℝ beanon-decreasingfunction.Wedefine ℎ⋆(𝑦):

ℎ⋆ 𝑦 = sup 𝑥 ℎ 𝑥 ≤ 𝑦 .Lemma2:If ℎ(𝑥) ∶ ℝ → ℝ isanon-decreasingfunctionandleftcontinuous,then

ℎ ℎ⋆ 𝑦 ≤ 𝑦.

ProofSuppose 𝑥R → ℎ⋆ 𝑦 is a strictly increasing sequence. According to thedefinitionof ℎ⋆ 𝑦 and 𝑥R < ℎ⋆ 𝑦 ,wehave

∀𝑛 ∈ ℕ, ℎ 𝑥R ≤ 𝑦.

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Hence, limR→-

ℎ(𝑥R) ≤ 𝑦. ℎ(𝑥) isleftcontinuous ⇒ limR→-

ℎ 𝑥R = ℎ(ℎ⋆ 𝑦 ).

Finally,wehave ℎ ℎ⋆ 𝑦 ≤ 𝑦.

∎Theorem:(QuantileEquivariantTransformationTheorem)Suppose ℎ ∶ ℝ → ℝ isanon-decreasingfunctionandleftcontinuous,then

𝐹m f,1 𝑞 = ℎ 𝐹f,1 𝑞 .

ProofWe use Lemma 1 to prove this.We need to show ℎ(𝐹f,1(𝑞)) satisfies both (i)and(ii)inLemma1.

Firstly,wecouldsee 𝐹m f ℎ 𝐹f,1 𝑞 = ℙ ℎ 𝑋 ≤ ℎ 𝐹f,1 𝑞 . Since ℎ(𝑥) is

a non-decreasing function, 𝑋 ≤ 𝐹f,1(𝑞) ⇒ ℎ 𝑋 ≤ ℎ 𝐹f,1 𝑞 . Hence we have

ℙ ℎ 𝑋 ≤ ℎ 𝐹f,1 𝑞 ≥ ℙ 𝑋 ≤ 𝐹f,1 𝑞 andfromthedefinitionof 𝐹f,1 𝑞 we

have ℙ 𝑋 ≤ 𝐹f,1 𝑞 ≥ 𝑞. Therefore:

𝐹m f ℎ 𝐹f,1 𝑞 = ℙ ℎ 𝑋 ≤ ℎ 𝐹f,1 𝑞 ≥ ℙ 𝑋 ≤ 𝐹f,1 𝑞 ≥ 𝑞.

(i)holds.

For(ii),let 𝑦 < ℎ(𝐹f,1(𝑞)).Thenweneedtoshow 𝐹m f (𝑦) < 𝑞.Bythelemma2

wehave

ℎ ℎ⋆ 𝑦 ≤ 𝑦 ⇒ ℎ ℎ⋆ 𝑦 ≤ 𝑦 < ℎ 𝐹f,1 𝑞 .

Because ℎ is a non-decreasing function ⇒ ℎ⋆ 𝑦 < 𝐹f,1(𝑞) . Then we have

ℙ 𝑋 ≤ ℎ⋆ 𝑦 < 𝑞 and 𝐹m f 𝑦 = ℙ ℎ 𝑋 ≤ 𝑦 . According to the definition of

ℎ⋆ 𝑦 andlemma2,weknow ℎ⋆ 𝑦 isthebiggestvaluethatsatisfies ℎ(𝑋) ≤ 𝑦.Sowehave ℎ 𝑋 ≤ 𝑦 ⇒ 𝑋 ≤ ℎ⋆ 𝑦 .So:

𝐹m f 𝑦 = ℙ ℎ 𝑋 ≤ 𝑦 ≤ ℙ 𝑋 ≤ ℎ⋆ 𝑦 < 𝑞.

(ii)holds.∎

2.3 DefinitionofVaRFromChoudhry andAlexander(2013),we could have a basic understanding ofValue-at-Risk.VaRwasfirstintroducedtopublicinOctober1994whenJPMorganlaunched RiskMetrics free over the internet. With the development of riskmanagement,VaRbecameanacceptedmethodologyforquantifyingmarketrisk

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andbegantobeadoptedbybankregulators. In1997,majorbanksanddealerschose to include VaR information in the notes to their financial statements,because the U.S. Securities and Exchange Commission ruled that publiccorporations must disclose quantitative information about their derivativesactivity.Nowadays,mostofthebanksandhedgefundsuseVaRasameasureofmarketrisk,whichmakesVaRbecomeoneofthemostpopularriskmeasuresintheworld.Let 𝑇 > 0 denote the time horizon. Let 𝑃Q and 𝑃o denote the current priceand the future price at time 𝑇 of a stock. Let 𝑐 ∈ (0,1) denote the confidencelevel(which is typically 0.99 or 0.95). The quantity 𝛼 = 1 − 𝑐 is called thetolerance level. The profit/loss generated by the portfolio at time 𝑇 isrepresentedbytherandomvariable 𝑃o − 𝑃Q.Thevalue-at-riskwiththetolerancelevel 𝛼 isdefinedas

𝑉𝑎𝑅t = −𝐹uv,u\,1 𝛼 .

Therefore,VaRissimplythe 𝛼 quantileoftheloss/profitrandomvariablewithaminusinfront.Theminusmakesitpositiveasforsmall 𝛼 thequantileitselfisusuallynegative. Remark1:InviewofQuantileEquivariantTransformationTheorem

𝑉𝑎𝑅t = 𝑃Q − 𝐹uv,1 𝛼 = −𝑃Q𝐹wv

,1 𝛼 ,

where 𝑅o denotestherateofreturnfromthestock:

𝑅o =𝑃o − 𝑃Q𝑃Q

.

Proof

Let 𝑅o = ℎ 𝑃o = uv,u\u\

, ℎ be a non-decreasing and continuous function, from

theQuantileEquivariantTransformationTheorem,wehave:

𝐹m uv,1 𝛼 = ℎ(𝐹uv

,1(𝛼))

⟹ 𝐹wv,1 𝛼 =

𝐹uv,1 𝛼 − 𝑃Q

𝑃Q

⟹−𝑃Q𝐹wv,1 𝛼 = −𝑃Q

𝐹uv,1 𝛼 − 𝑃Q

𝑃Q= 𝑃Q − 𝐹uv

,1 𝛼 .

∎ThisiswhythealternativedefinitionofVaRdefinesitasthe 𝛼 quantileofthereturnrandomvariablewithaminusinfront.Remark2:Alternatively,wecanconsiderthelog-return

𝑟o = 𝑙𝑛𝑃o𝑃Q.

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TheninviewofQuantileEquivariantTransformationTheorem

𝑉𝑎𝑅t = 𝑃Q − 𝐹uv,1 𝛼 = 𝑃Q 1 − exp 𝐹|v

,1 𝛼 .

Proof

Let 𝑟o = ℎ 𝑃o = 𝑙𝑛 uvu\, ℎ isanon-decreasingandcontinuousfunction,fromthe

QuantileEquivariantTransformationTheorem,wehave:

𝐹m uv,1 𝛼 = ℎ(𝐹uv

,1(𝛼))

⟹ 𝐹|v,1 𝛼 = 𝑙𝑛

𝐹uv,1(𝛼)𝑃Q

⟹ 𝑃Q 1 − exp 𝐹|v,1 𝛼 = 𝑃Q 1 − exp 𝑙𝑛

𝐹uv,1 𝛼𝑃Q

= 𝑃Q 1 −𝐹uv,1 𝛼𝑃Q

= 𝑃Q − 𝐹uv,1 𝛼 .

∎Remark3:If 𝑃o 𝑃Q iscloseto1,then

𝑟o ≈ 𝑅o.Proof

𝑅o =𝑃o − 𝑃Q𝑃Q

=𝑃o𝑃Q− 1 ⇒

𝑃o𝑃Q= 1 + 𝑅o

AccordingtotheTaylorexpansion:

𝑟o = 𝑙𝑛𝑃o𝑃Q= ln 1 + 𝑅o = −1 R~1 𝑅o

R

𝑛 = 𝑅o − Ο 𝑅oN-

R�1

And 𝑃o𝑃Q→ 1 ⇒ 𝑅o =

𝑃o𝑃Q− 1 → 0

Then Ο 𝑅oN wouldbenegligible;wehave 𝑟o ≈ 𝑅o .∎

Buttherandomvariables 𝑟o and 𝑅o havedifferentdistributions.

2.4 FattailsEmpirical studies show that the distribution of asset returns is usually not

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normal, and insteadhas fatter tails thannormaldistribution.FromDaníelsson,Jorgensen,Samorodnitsky(2013),wecouldseeaformaldefinitionofafat-taileddistribution.Fat tailsmeans that the tailsvaryregularlyat infinityso that theyapproximately follow a Pareto distribution, which like power expansion atinfinity.Acumulativedistributionfunction 𝐹(𝑥) variesregularlyat −∞ withtailindex𝛼 > 0 if

limR→-

𝐹(−𝑡𝑥)𝐹(−𝑡) = 𝑥,t,∀𝑥 > 0,

andvariesregularlyat ∞ withtailindex 𝛼 > 0 if

limR→-

1 − 𝐹(𝑡𝑥)1 − 𝐹(𝑡) = 𝑥,t,∀𝑥 > 0.

Itgivesoutthataregularlyvaryingdistributionhasatailoftheform𝐹 −𝑥 = 𝑥,t𝐿 𝑥 ,𝑥 > 0.

Where 𝐿(𝑥) is a slowlyvarying functionwhichmeans ∀𝑥 > 0, 𝑡 → ∞ wehave𝐿(𝑡𝑥) 𝐿(𝑡) → 1. Andtheconstant 𝛼 > 0 isthetailindex.

2.5 LiquidityriskIn order to understand liquidity risk, we need to understand liquidity first.Liquiditydescribesthetrade-offbetweenhowquicklyanassetorsecuritycanbeboughtorsoldinthemarketandhowlargethedegreeofaffectingtheassetorsecurity’s price. In a liquid market, an asset or security can be sold quicklywithoutreducingthepricemuch.However,inarelativelyilliquidmarket,sellanassetorsecurityquicklywillneedtoreduceitspricetosomedegree.Therearetwotypesofliquidity:1. Marketliquidity,whichistheabilitythatamarketallowsassetstobebought

andsoldwithoutaffectingtheasset’spricetoasignificantdegree.Cashisthemostliquidasset.

2. Fundingliquidity,whichistheabilityaninstitutionensuresitspaymentwithimmediacy.

Therefore,liquidityriskistheriskthatanassetorsecuritycouldnotbesoldquicklyenoughwithoutcausinggreatchangeintheprice.Therearetwotypesofliquidityriskrespectively:1. Market(asset)liquidityrisk,whichistheriskthatapositioncannotbeclosed

(oranassetcannotbesold)quicklyenoughwithoutinfluencingthemarketpricetoasignificantdegree.

2. Fund(cashflow)liquidityrisk,whichistheriskthataninstitutionisunabletorepayitsliabilitiesormeetitsobligationswhentheycomedue,whichleadstodefault.

Hereinthispaper,wefocusedonmarketliquidityrisk.Traditionallyitismeasuredbythefollowing3aspects:1. ‘Width’,whichisalsocalledbid-askspread.Itisthedifferencebetweenthe

askpriceandthebidprice.Itcanalsobecalculatedinpercentageterm,

10

whichisproportionalspread.Itisthedifferencedividedbytheaverageoftheaskpriceandthebidprice.Itismeasuredinapricedimension.

2. ‘Depth’.Itisthemarketabilitytoabsorbtheexitofapositionwithoutchangingthepricedramatically,whichisthatthenumberofthesecuritiescanbeboughtwithouthavingpriceappreciation.Itismeasuredinaquantitydimension.

3. ‘Resiliency’.Itisthetimethemarketneedtogobackfromincorrectprice.Itismeasuredinatimedimension.

Thereisalsoanothermeasurebecomingpopularrecently,whichis‘volume’.Itistheamountofacertainsecuritytradedduringacertaintimeperiod.

2.6 ThedefinitionofLIXThis part is a summary of Oleh Danyliv, Bruce Bland, Daniel Nicholass(2014).Theygiveoutanewmeasureofliquidity,whichIwilluselaterinthemodel.Sohere I give a short summary about how they give out and define LIX(liquidityindex). Fromtheviewofa fundcompany,dealingwith liquidityriskcouldbeseenthesameasfindingananswertothefollowingquestion:whatamountofmoneycanonetrade/investwithoutmovingthemarket?Infact,itisaveryhardquestiontoanswer,especiallywhenwetrytoquantifyaspecificamountofit.Italsodependsonhowmuchtimethetraderhastoexecuteit,whichstrategythetraderusestoclose the position and how large the position is compared to average dailyvolume,etc.However,ifwelookattheprobleminanotherway,itwillgiveusamoreclearandprecisedefinitionofliquidity.Whatamountofmoneyisneededtocreateadailysingleunitpricefluctuationofthestock?

𝐿𝑖𝑞𝑢𝑖𝑑𝑖𝑡𝑦~𝐶𝑜𝑛𝑠𝑖𝑑𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝑃𝑟𝑖𝑐𝑒𝑅𝑎𝑛𝑔𝑒 ≡

𝑉𝑜𝑙𝑢𝑚𝑒×𝑃𝑟𝑖𝑐𝑒𝐻𝑖𝑔ℎ − 𝐿𝑜𝑤 .(1)

The liquiditymeasurewe got from the formula 1 ranges from thousands tobillions.Inordertohandlethenumbersmoreeasily,wetakethelogarithmoftheamount,whichreducestherangetomanageablenumbers.Sincethevaluewegotfromtheformula 1 doesnothaveunitsofmeasurement,itisreasonabletodoso.AnditiscalledLiquidityIndex(LIX):

𝐿𝐼𝑋� = log1Q(𝑉�𝑃���,�

𝑃���m,� − 𝑃���,�),(2)

where 𝑉� ! isthetradingvolumetoday, 𝑃���m,� isthehighestaskpricetoday,

𝑃���,� isthelowestbidpricetodayand 𝑃���,� istheaverageofaskpriceandbidprice.Alogarithmwiththebaseof10makesthe 𝐿𝐼𝑋� intoarangeroughlyfromveryilliquid5toveryliquid10.Ithasasimplemeaning:forSwedenstockstheamountofcapitalneededtocreate1krpricefluctuationscanbeestimatedas10��f� kr.Theliquiditymeasure (2) innon-loggedversion (1) hasalreadybeenusedinsome published literature. In the UK government’s Foresight report byLinton(2012),themeasure (1) wasusedtoinvestigatetheevolutionofmarket

11

liquidity with the following remark: “in the current environment a plausiblealternativetoclosereturnistouse intradayhighminus lowreturn,sincetherecanbeagreatdealofintradaymovementinthepricethatendsinnochangeattheendoftheday’”.LIXasaliquiditymeasurehasthefollowing2advantages:1. Thecurrencyvalueiseliminatedfromcalculation,sowecancomparestocks

ondifferentinternationalmarketsdirectly,2. The data that it requires is easy to have access to, comparing with other

measures.

2.7 CostofliquidityInordertocalculatethecostofliquidity(COL),weneedusebid-askspread.Itisthe difference between the highest ask price today and the lowest bid pricetoday:

𝑆𝑝𝑟𝑒𝑎𝑑� = 𝑃���m,� − 𝑃���,�.Itcanalsobecalculatedinpercentageterm,whichisproportionalspread:

𝑃𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛𝑎𝑙𝑠𝑝𝑟𝑒𝑎𝑑� =𝑃���m,� − 𝑃���,�

𝑃���,�,

where 𝑃���,� istheaverageofaskpriceandbidprice.Thefullspreadrepresentstheliquiditycostofaroundtrip,whichmeansthecostofbuyingandsellingthestocktoday.Here,weonlyneedtoconsidertheliquiditycostofsellingthestock,hencethecostofliquidity(COL)isonlyhalfofthespread.

𝐶𝑂𝐿� =12×𝑆𝑝𝑟𝑒𝑎𝑑� =

12 𝑃���m,� − 𝑃���,� ;

𝐶𝑂𝐿� =12×𝑃���,�×𝑃𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛𝑎𝑙𝑠𝑝𝑟𝑒𝑎𝑑�;

𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒𝐶𝑂𝐿� =12×𝑃𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛𝑎𝑙𝑠𝑝𝑟𝑒𝑎𝑑� =

12𝑃���m,� − 𝑃���,�

𝑃���,�.

12

3. LiquidityAdjustedVaRModel ThepurposeofaLiquidityadjustedVaRmodelistoincorporatetheliquidityriskintoVaRmodel.Bangia,Diebold,SchuermannandStroughair(1999)proposedamodel, which incorporated the exogenous liquidity risk into VaR model. Theymodeledthetransactionpriceasmid-priceplusahalfoftheproportionalbid-askspread:

𝑃¡��,�~1 = 𝑃¡��,� exp 𝑟�~1 −12𝑃�𝑆�~1,

where 𝑃¡�� is themiddle price of the ask price and the bid price, 𝑟�~1 is thedaily return between t and t+1 and 𝑆 is a time-varying proportional bid-askspread. Liquidity adjusted VaR is the sum of the normal VaR and Cost ofliquidity(COL):

𝐿𝑎𝑉𝑎𝑅 = 𝑃¡��,� 1 − exp 𝑧t𝜎|£�|¡¤¥¦¤w

+12𝑃¡��,� 𝜇¨ + 𝑧t𝜎

©ª�

,

where 𝜎| isthevarianceofthedailyreturn, 𝜇¨ isthemeanoftheproportionalbid-askspread, 𝜎« isthestandarddeviationoftheproportionalbid-askspread.𝑧t and 𝑧t are the 𝛼 -percentile of the daily return distribution and theproportionalspreaddistribution.LeSaout(2002)extendsthemodelofBangiaetal.forincludingtheendogenousrisk, by substituting the proportional bid-ask spreadwhich is used for Cost ofliquiditycalculationbyWeightedAverageSpread(WAS):

𝐿𝑎𝑉𝑎𝑅 = 𝑃¡��,� 1 − exp 𝑧t𝜎|£�|¡¤¥¦¤w

+12𝑃¡��,� 𝜇¨ ¦ + 𝑧t𝜎¨ ¦

©ª�

,

where 𝑆 𝑉 istheproportionalWeightedAverageSpread,whichisafunctionofthevolumeofthecertainstockwehaveintheportfolio.However, in order to getWeighted Average Spread (WAS), we need the orderbookdatafromthestockmarket,whichisquiteextensiveandexpensive.Basedontheideaabove,soinsteadofusingWeightedAverageSpread(WAS),weuse the liquidity measure LIX to forecast the spread. However, using LIX toforecast spread will produce unreasonable spread prediction under extremesituation.Inordertogiveoutmorerealisticprediction,weneedtoscaleit:

𝐿𝑎𝑉𝑎𝑅 = 𝑁𝑜𝑟𝑚𝑎𝑙𝑉𝑎𝑅 + 𝐶𝑂𝐿;𝐶𝑂𝐿 = 𝐴 ∗ 𝐶𝑂𝐿,

where 𝐴 > 0 isascalecoefficient,Costof liquidity(COL) ishalfof theSpread,the Spread is a function of LIX and volume of certain stock we have in theportfolio.

3.1 ModelofNormalVaRFirstlywecalculatedtheone-daystockreturn 𝑟� bytakingthelogarithmoftheratioof twoadjacentprices.Theprice 𝑃� isestimatedbytakingtheaverageofbidandaskprice:

13

𝑟� = ln𝑃�𝑃�,1

.

Thenweassumedthedailyreturnatt+1isnormallydistributedwiththemean𝐸 𝑟� andthevariance 𝜎�N

𝑟�~1~𝑁 𝐸 𝑟� , 𝜎�N .For the given confidence level we use both 99% and 95%. We calculate thestandard normal distribution percentile of 99% and 95%. The 1% left tail ofnormal distribution, Norminv(0.01)=-2.326. The 5% left tail of normaldistribution, Norminv(0.05)=-1.645. With the given 99% confidence level, theworstdailyreturnattimet+1willbe𝑟�~1� = 𝐸 𝑟� − 2.326𝜎� .Similarlywith95%confidencelevel𝑟�~1� = 𝐸 𝑟� − 1.645𝜎� .

Here, we consider the one-day horizon and thus we take the expected dailyreturntobe 𝐸 𝑟� = 0.Hence,theworstpricetomorrowwillbe

𝑝�~1� = 𝑝�𝑒|�²³´ .

Therefore,thenormalValueatriskforoneshareofthestockwillbe

99%𝑉𝑎𝑅 = 𝑝� − 𝑝�~1� = 𝑝� 1 − 𝑒|�²³´ = 𝑝� 1 − 𝑒,N.·N¸¹� ;

95%𝑉𝑎𝑅 = 𝑝� − 𝑝�~1� = 𝑝� 1 − 𝑒|�²³´ = 𝑝� 1 − 𝑒,1.¸º»¹� .

TheVaRinpercentagewillbe

99%𝑉𝑎𝑅 = 1 − 𝑒,N.·N¸¹�;

95%𝑉𝑎𝑅 = 1 − 𝑒,1.¸º»¹�.

We have the price directly on the market. The only parameter we need toestimatehere is 𝜎� .Fromtheempiricalanalysis, itwillgiveoutniceresultsbyusingexponentiallyweightedmovingaverage.

Exponentiallyweightedmovingaveragegives lessweight to the furtherdata inthepastandmoreweighttothemorerecentdata:

14

𝜎� = (1 − 𝜆1 − 𝜆o) 𝜆�,1 𝑟� − 𝐸 𝑟 N

o

��1

.

T is the history time period we considered. 𝜆 is the decay factor which isbetween (0,1).Fromthe industryexperience,wechoose thedecay factor tobe0.94.

FortheportfolioVaR,Wealsoneedtoestimatethecovariance 𝜎�½ .Herewestill

useexponentiallyweightedmovingaveragewaytodoit:

𝜎�½ = (1 − 𝜆1 − 𝜆o) 𝜆�,1 𝑟�� − 𝐸 𝑟�� 𝑟½� − 𝐸 𝑟½�

o

��1

.

Inordertogettheportfoliostandarddeviation 𝜎,wewillneedtheweightofeachstockintheportfolioandthecovariancematrix:

𝑊 = 𝑤1,𝑤N …𝑤R

𝑉 =𝜎11N ⋯ 𝜎1RN⋮ ⋱ ⋮𝜎R1N ⋯ 𝜎RRN

Asfortheweight,VaRofthecashandthemanagementfeeisconsideredtobezero.Soweonlyconsidertheweightofthestocks.

Thenwecouldget

𝜎N = 𝑤𝑣𝑤o.

TheportfoliopercentageVaR:

99%𝑉𝑎𝑅 = 1 − 𝑒,N.·N¸¹;

95%𝑉𝑎𝑅 = 1 − 𝑒,1.¸º»¹.

Howevertheassumptionthatthedailyreturnatt+1isnormallydistributedmaynot be true. Empirical studies show that the asset returns is not normallydistributedandinfactthedistributionhasfattails.Let’scheckthehistogramofdifferentstockreturnsincomparisonwithnormaldistribution.

15

Wecouldseefromthefigureabove,thedistributionofthestockreturnshasfattails. It is also called the tail coarseness problem. In order to deal with theproblem,byDaníelssonanddeVries(2000),wecoulduseextremevaluetheory(EVT),whichgivesoutanotherquantileestimator.

Foradistributionwith fat tails, the tail asymptotically followsapower law, i.e.theParetodistribution:

𝐹 𝑥 = 1 − 𝐴𝑥,t.

Wetakethesamplesizetobe 𝑛 andlet 𝑚 < 𝑛 sufficientlysmall.Herewetakethesamplesizetobe262,whichisoneyearhistoryinsteadof90days,becausethis method requires a relatively large sample size to accurately estimate thequantile. There is no good way to decide which m is the best, so we will trydifferentm.ThenwecouldusetheHillestimatortoestimatethetailindex 𝛼:

1𝛼(𝑚) =

1𝑚 𝑙𝑛

𝑋(�)𝑋(¡~1)

¡

��1

,

16

where 𝑋(�) indicates order statistics of stock daily return. So we need to

rearrangeallthehistorystockdailyreturnfromsmallesttolargest.ThenwewillgetthequasimaximumlikelihoodVaRestimator

99%𝑉𝑎𝑅 = 𝑋(¡~1)𝑚 𝑛0.01

1t ¡

;

95%𝑉𝑎𝑅 = 𝑋(¡~1)𝑚 𝑛0.05

1t ¡

.

AsfortheportfolioVaR, insteadofrearrangingallthehistorysinglestockdaily

return from smallest to largest and give the value to 𝑋(�), we calculate all the

history portfolio daily return. And let 𝑋(�) be the order statistics of portfolio

dailyreturn.ThentheformulawillgiveoutportfolioVaR.

The quasi maximum likelihood VaR estimator is estimated by using allobservations in the tail, which makes it less sensitive to the tail coarsenessproblem.Hence,itproducesmoreaccurateresults.

3.2 ModelofCostofliquidity(COL)Firstly,weknowthatCostofliquiditydependsonthespread

𝐶𝑂𝐿�~1 =12×𝑆𝑝𝑟𝑒𝑎𝑑�~1 =

12 𝑃���m,�~1 − 𝑃���,�~1 .

In order to estimate the cost of liquidity at 𝑡 + 1,we only need to predict thespreadat 𝑡 + 1.AccordingtothedefinitionofLIX,weseethat

𝐿𝐼𝑋� = log1Q(𝑉�𝑃���,�

𝑃���m,� − 𝑃���,�);

𝑆𝑝𝑟𝑒𝑎𝑑� = 𝑃���m,� − 𝑃���,� =𝑉�𝑃���,�10��f� .

Thenweseethecostofliquidityas

𝐶𝑂𝐿�~1 =12𝑉𝑃���,�~110��f�²³ ;

𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒𝐶𝑂𝐿�~1 =𝐶𝑂𝐿

𝑃���,�~1=12

𝑉10��f�²³ ,

where 𝑉 is the volume of stock we have in the portfolio, and 𝐿𝐼𝑋�~1 is theliquidity index at 𝑡 + 1. We could see it is a good measure to liquidity risk,becausethehigherthevolumewehaveintheportfolio,thehighertheliquiditycostwillbe.Andthehigher theLIXwehave(whichmeansthemore liquidthestockis),thelowertheliquiditycostwillbe.However,thevolumewehaveinthe

17

portfolio sometimes ismuch larger than the trading volume in themarket forsmall cap stocks under extreme condition, which will produce unreasonable𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒𝐶𝑂𝐿 over100%.Inordertodealwiththissituation,weintroduceascalecoefficienttoscaleit:

𝐶𝑂𝐿�~1 = 𝐴 ∗ 𝐶𝑂𝐿�~1;𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒𝐶𝑂𝐿�~1 = 𝐴 ∗ 𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒𝐶𝑂𝐿�~1,

where the constant 𝐴 > 0 is the scale coefficient. Thenwehave twoproblemshere: one is deciding A, the other is estimating the LIX at 𝑡 + 1. For the firstproblem,itisquitesubjectivetochooseascalesincewelackorderbookdatatocheck how the spread will look like under both crisis situation and colossaltransactionvolume.So,weusedtheexperiencefromindustryanddiscussedwiththeexperiencedriskcontrollerinthehedgefund.Wedecidedtoset 𝐴 = 1/10.Itis justanexperiencebasedchoice.Whenwecouldhaveaccess tomoredata inthefuture,thiscoefficientcanbecalibratedtoprovideabetterresult. Forthesecondproblem,weexperimentedtwoways:

1) Estimate 𝑳𝑰𝑿𝒕~𝟏 bytakingaverageNowadays inthe industry,weusetheratiobetweenthevolumewehave intheportfolio and past 20 days average daily market trading volume to measureliquidityrisk.Withthesimilaridea,wecouldusethepast20daysaverageLIXtoestimatetheLIXat𝑡 + 1:

𝐿𝐼𝑋�~1 =𝐿𝐼𝑋� + 𝐿𝐼𝑋�,1 + ⋯+ 𝐿𝐼𝑋�,1Ù

20 ;

𝐿𝐼𝑋� = log1Q𝑉�𝑃���,�

𝑃���m,� − 𝑃���,�,

where 𝑉! is the trading volume at 𝑡, 𝑃���m,� is the highest ask price at 𝑡 ,𝑃���,� isthelowestbidpriceat 𝑡 and 𝑃���,� istheaverageofaskpriceandbidprice.

2) Estimate 𝑳𝑰𝑿𝒕~𝟏 byassumingnormaldistributionInsteadofsimplytakingaverageofthepast20daysdata,wecouldassumethatthe 𝐿𝐼𝑋�~1 is normally distributed with the mean 𝐸 𝐿𝐼𝑋� and the variance𝜎��f,�N

𝐿𝐼𝑋�~1~𝑁 𝐸 𝐿𝐼𝑋� , 𝜎��f,�N .

Wealsoassumethatunderadversemarketsituation,theextremechangeinthestockpriceandliquidityindexhappenatthesametime.Therefore,similartothenormalVaRmodel,weuseconfidencelevelboth99%and95%.The1%lefttailofnormaldistribution,Norminv(0.01)=-2.326.The5%lefttailofnormaldistribution,Norminv(0.05)=-1.645.Withthegiven99%confidencelevel,theworstLIXattimet+1willbe

18

𝐿𝐼𝑋�~1� = 𝐸 𝐿𝐼𝑋� − 2.326𝜎��f,�.

Similarlywith95%confidencelevel

𝐿𝐼𝑋�~1� = 𝐸 𝐿𝐼𝑋� − 1.645𝜎��f,�,

wheretheequallyweighted 𝜎 estimator

𝜎��f,� =1𝑇 𝐿𝐼𝑋� − 𝐸 𝐿𝐼𝑋 N

o

��1

.

Asfortheportfoliocostofliquidity,itisjustthesumofeachstock’scostofliquidity.

19

4. Resultsandconclusion

4.1 NumericalresultsIn order to compare the liquidity effect on different funds, we perform thecalculationontwofunds,oneisalargecapfundandanotherisasmallcapfund.It isknown that thesmall cap fundhashigher liquidity risk than the largecapfund.WewanttoquantifytheliquidityriskandcomparetheresultsofLiquidityadjusted VaR between different funds. Running the MATLAB(R2016b) code inAppendix,wecouldgettheresults.Firstly, let us analyze the result from the large cap fund of calculating theVaRwithoutconsidering tail coarsenessproblemand the resultofbothmethodsofpredictingLIX.WecouldseefromtheTable1:thefirstresultofalargecapfund,‘NETIB’hasarelatively high Cost of liquidity comparedwith other stocks in the fund. If wecheck the ratiobetween thevolumewehave in theportfolio andpast20daysaveragedailymarkettradingvolume,‘NETIB’hasanratioof11.47,whichmeansthe share we have in the portfolio is 11.47 times of the average daily tradingvolume in the market. Indeed, it yields a relatively high liquidity risk. Thiscorroborateourwayofquantifying liquidity risk.Wecould see from the resultthatforthelargecapfundtheliquidityriskhasamuchsmallerimpactthanVaRduringadversemarketsituation. Theassumptionthatunderadversemarketsituation,theextremechangeinthestockpriceandliquidityindexhappensatthesametimemaynotbetrueinrealworld.Infact,whenthestockpricedropsdownextremely,theliquiditygoesupfirstlyandthen itgoesdown.This iswhythesecondmethodofcalculatingtheliquidity risk produces higher outcome comparing to the experience from theindustry.Therefore,wepreferthefirstmethodofestimatingthecostofliquidity.Whenwedealwith the small cap fund,weonlykeep themethodof taking theaverage.

20

Table1:thefirstresultofalargecapfundWhereLIX_avgm

eanstheLiquidityIndexgotbyusingmethod1)

99%LIX_norm

eanstheLiquidityIndexgotbyusingmethod2)w

ith99%

95%LIX_norm

eanstheLiquidityIndexgotbyusingmethod2)w

ith95%

COL_avgmeanstheCostofliquiditygotbyusingm

ethod1)99%

COL_normeanstheCostofliquiditygotbyusingm

ethod2)with99%

95%

COL_normeanstheCostofliquiditygotbyusingm

ethod2)with95%

21

Table2:thefirstresultofasmallcapfundWhereLIX_avgmeanstheLiquidityIndexgotbytakingtheaverage COL_avgmeanstheCostofliquiditygotbytakingtheaverageWecouldseefromthetables1and2thatbothfundshavesimilarVaR,butthesmall cap fundhasmuchhigherCostof liquidity than the large cap fund, from8.61%to0.16%. ‘OEMInternationalB’presentsaveryhighCostof liquidity. Ifwechecktheratiobetweenthevolumewehaveintheportfolioandpast20daysaverage daily market trading volume, ‘OEM International B’ has an ratio of701.16,whichmeans the sharewehave in theportfolio is701.16 timesof theaveragedailytradingvolumeinthemarket.Accordingtotheempiricalindustryexperience, selling 10%of the average daily trading volumewill not affect thestockpricesignificantly.Itmeansthatitwilltake7010tradingdays(26years)toexit the position, which definitely gives out a huge liquidity risk. Therefore,dealingwith largecapfund, the liquidityrisk isnotsosignificant.Dealingwithsmallcapfund,liquidityriskissomethingweneedtopayattention.Secondly, since we can see from the histogram of stock returns that thedistribution of the stock returns has fat tails. We used the quasi maximumlikelihoodVaRestimatortogetmoreaccurateVaR,andwedidsomeexperiments

22

ofdifferentmonbothfunds.WecanseefromtheTable3and4thatwhen 𝑚 𝑛 isnearthequantile,itgivesoutamoreaccurateestimationofVaR.For thesmallcap fund,sincewehaveanewly listed company, we only have a trading history of 173 days. Instead ofchoosing 𝑚 = 12,13,14 , we chose 𝑚 = 8,9,10 for small cap fund whenpredicting the 95%𝑉𝑎𝑅.We can also see that the VaR estimation usually is alittlebithigherwhenweconsiderthefattailofthestockreturndistribution.

23

Table3:thesecondresultofalargecapfund

24

Table4:thesecondresultofasmallcapfund

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Intheend,wedecidedtocombinethequasimaximumlikelihoodVaRestimatorandLIXestimatedbytakingtheaveragetogetthefinalliquidityadjustedVaRfortheportfolio.

Table5:thefinalresultofasmallcapfundWecanseefromthetable5and6thatthelargecap99%LiquidityAdjustedVaRis1.64%,which1.48%comesfromVaRand0.16%comesfromliquidityrisk,thesmallcap99%LiquidityAdjustedVaRis11.04%,which2.43%comesfromVaRand8.61%comes fromliquidityrisk.BycomparingthestockLIX intwofunds,wecanobservethat thestocks inthesmallcapfundaremore illiquidthanthestocks inthe largecapfund. Italsoexplainsthereasonwhythesmallcapfundsuffershigherliquidityriskthanthelargecapfund.

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Table6:thefinalresultofalargecapfund

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4.2 ConclusionInthisthesis,theLiquidityadjustedVaRmodelmanagedtoavoidtheweaknessofthenormalVaRmodelbyincorporatingtheliquidityrisk.Itmakestheresultmore realistic since it is considered that the real price of transaction deviatesfromthemidpriceofthespread.Duetothelimiteddataweaccessed,weusedthe LIX to predict the liquidity and spread of the stock. It yields results thatmatchtheindustryexperience.Bycomparingthecostofliquidity,wemanagedtodistinguish liquidportfolio and illiquidportfolio. So theLiquidity adjustedVaRreflectedthemarketriskmoreprecisely.Forafuturework,ifwehaveaccesstothelimitorderbookdata,wecanuseittopredicttheliquidityandspreadfromanotheraspect.Thenwecancomparetheresultswegetfrombothaspects.

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Appendix:TheMATLABcode% Peiyu Wang master thesis clear all close all clc %load the data from excel load StockReturnHistory load PriceHistory load weight load LIX_history load market_trading_volume load Volume_in_portfolio %%Model of normal VaR T=90; %The history period lamda=0.94; %The decay factor a=norminv(0.01);%The 1% or 5% left tail of normal distribution N=21;%how many stocks in the portfolio %Calculate the exponential weight w=(1-lamda)/(1-lamda^T)*lamda.^(0:T-1); %We get the last 90 days stock daily return SRhistory=StockReturnHistory(end:-1:end-(T-1),:); %Calculate the mean of the daily return of all the stocks E_sr=mean(SRhistory); M_sr=SRhistory-E_sr; %Calculate the Exponentially weighted moving average sigma sigma_sr=sqrt(w*M_sr.^2); %Then we can estimate the percentage VaR for each stock VaR=1-exp(a*sigma_sr); %Calculate the exponentially weighted Covarience matrix V V=zeros(N,N); for i=1:N for j=1:N V(i,j)=w*(M_sr(:,i).*M_sr(:,j)); end end

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sigma_portfolio=sqrt(weight'*V*weight); Var_portfolio=1-exp(a*sigma_portfolio) %for write the result into the excel %xlswrite('weight','VaR','C1:C21'); %%Model of Quasi maximum likelihood VaR m=14; %rearrange the Stock return history with one year Rear_Ln_dailyreturn=sort(StockReturnHistory); %rearrange the Stock return history with 90 days RA_SRhistory=sort(SRhistory); al=zeros(m,21); for i=1:m al(i,:)=Rear_Ln_dailyreturn(i,:)./Rear_Ln_dailyreturn(m+1,:); %al(i,:)=RA_SRhistory(i,:)./RA_SRhistory(m+1,:) end %calculate the one over alpha, which alpha is the tail index one_over_alpha=sum(al)/m; %calculate the quasi maximum likelihood VaR estimator QML_VaR=-Rear_Ln_dailyreturn(m+1,:).*(m/262/0.05).^one_over_alpha; %QML_VaR=RA_SRhistory(m+1,:).*(m/T/0.01).^one_over_alpha; %calculate the history portfolio daily return Por_Rhistory=StockReturnHistory*weight; %rearrange the Portfolio return history with one year Rear_Por_Rhistory=sort(Por_Rhistory); bl=zeros(m,1); for i=1:m bl(i)=Rear_Por_Rhistory(i)./Rear_Por_Rhistory(m+1); end %calculate the one over alpha for the portfolio, which alpha is the tail

index Por_one_over_alpha=sum(bl)/m; %calculate the quasi maximum likelihood VaR estimator for portfolio Por_QML_VaR=-Rear_Por_Rhistory(m+1).*(m/262/0.05).^Por_one_over_alpha

; %%Model of Cost of liquidity % 1.by taking the past 20 days average %set the scale coefficient A=1/10;

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%We get the last 20 days LIX Lhistory=LIX_history(end:-1:end-19,:); %get the past 20 days average LIX LIX_average=mean(Lhistory); %Calculate the cost of liquidity COL_average=A.*Volume_in_portfolio'./(2*10.^LIX_average); COL_average_port=COL_average*weight % 2.by assuming normal distribution %We get the last 90 days LIX LIhistory=LIX_history(end:-1:end-(T-1),:); %Calculate the mean of LIX of all the stocks E_LIX=mean(LIhistory); M_LIX=LIhistory-E_LIX; %Calculate the sigma of LIX of all the stocks w_LIX=ones(1,T)*1/T; sigma_LIX=sqrt(w_LIX*M_LIX.^2); %Then we can estimate the LIX for each stock LIX_nor=E_LIX+a*sigma_LIX; %Calculate the cost of liquidity COL_nor=A.*Volume_in_portfolio'./(2*10.^LIX_nor); COL_nor_port=COL_nor*weight %draw histogram of stock returns in comparison with normal distribution % for i=1:21 % figure('Name','stock returns in comparison with normal

distribution') %

histogram(SRhistory(:,i),[-0.08:0.005:0.08],'Normalization','pdf') % hold on % y = -0.08:0.005:0.08; % mu = mean(SRhistory(:,i)); % sigma=sqrt((ones(1,T)*1/T)*(SRhistory(:,i)-mu).^2); % f = exp(-(y-mu).^2./(2*sigma^2))./(sigma*sqrt(2*pi)); % title('histogram of stock returns in comparison with normal

distribution') % xlabel('stock returns') % ylabel('density') % plot(y,f,'LineWidth',1.5) % end

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