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Geman, Hélyette and Ohana, Steve (2005) Time-consistency in managing a
commodity portfolio: a dynamic risk. Working Paper. Birkbeck, University of
London, London, UK.
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Birkbeck Workin
g
Pa
p
ers in Economics & Finance
School of Economics, Mathematics and Statistics
BWPEF 0610
Time-consistency in managing a
commodity portfolio: a dynamic risk
measure approach
Hélyette Geman
Steve Ohana
measure approah
Helyette GEMANBirkbek College, University of London and ESSEC Business Shool Steve OHANA Birkbek College, University of London
Deember2005
Abstrat
We onsider the problem of themanager of astorableommodity (e.g. hydro,oal)portfolio faing demand risk while having aess to storage failities and illiquid spot and forward markets. In this setting,weemphasizethatadynamially onsistent wayofmanagingriskovertimemustbeintrodued. Inpartiular, wedemonstratethe temporal inonsisteny ofstati riskobjetivesbasedon nal wealth andadvoatetheuseofanewlassof reursiverisk measures suh asthose suggestedbyEpstein et al. (1989)and Wang (2000)for portfolio optimization and valuation. This typeof risk measures not only providetime-onsistentdeisionplannings butallow theportfoliomanagerto ontrol independently the ourreneofash-owsarosstimeandarossrandomstatesofnature. Weillustratethedisussioninan empirialsetionwherethetrade-obetweennalwealthriskandbankruptyriskatanintermediatedate isanalyzedandthesynergybetweenthephysialassetsomposingaommodityportfolioisassessed.
Weonsider thesituationofaretailer,whoisengagedinlong-termsaleontrats,ownsstoragefailities and an trade the ommodity in illiquid spot and forward markets. The retailer is faing a portfolio optimization problem, that translates into deiding at eah time step whih quantity to injet in or withdrawfrom herstoragefailitiesandtrade inthespotandforwardmarket,and aportfolio valuation problem, that onsistsin assessingthevalue oftheglobal portfolio andof eah assetomposing it. The optimizationandthevaluationtakeplaein theontextoftwotypesofrisk: thevolumeriskthatarises from therandomdemand oflong-term ustomersandis relatedto exogenousnon tradedvariables suh asweather,andtheprieriskthatislinkedtothevolatilityoftheommodityprie.
In this inomplete market setting, the value of the retailer's portfolio is not uniquely determined by arbitrageonsiderationsandanintegratedportfolioapproahisneededtohandleliquidityonstraints. Thestohastiprogrammingliterature,ontheonehand,hasessentiallytreatedsituationswhereportfolio managementisanalyzedthroughamean-varianeriterionapplied tonalor intermediatewealths,and fully dened at the rst deision date. In partiular, the risks arising at intermediate deision dates are not taken into aount, leading to possible onits betweendeisions taken over time. Examples of this approahare foundin Unger(2002), where aCVaRonstraint onthe nal wealth is addressed throughaMonte-Carlo approah, in Martinez-de-Albenizet al. (2005),where mean-varianetrade-os areonsideredandyieldexpliitsolutionsinaone-stepframework,andinKleindorferetal. (2004),where theaseofamulti-periodVaRonstraintonash owsis examined.
andtousedynamiriskobjetivesin inventoryandontratsportfolioproblems. Eihhornet al. (2005) usearestrition ofthe set ofoherent dynamirisk measures dened byArtzner et al. (2002)to solve aneletriityportfoliooptimizationproblembutdonotraisetheproblemoftimeonsistenyofoptimal strategies. Chen et al. (2004)dene their objetivefuntion as anadditiveintertemporal utility of the onsumptionproessoftheportfoliomanager. Instead,wehoosetheEpsteinetal. (1989)nonadditive intertemporalutilityobjetiveandapplyit diretlyto theashowproess. Theimpatof thishange issigniant: inoursetting,theinitial wealth isnotastatevariable,theonlystatevariablesbeingthe inventory level, and theumulative positions in the forward market for eah future delivery period; in addition,theretailer'sproblemappearsasaash-owstreammanagementoneratherthanaonsumption planningone;lastly,theexibilityofthenonadditiveintertemporalutilityallowstheportfoliomanager toseparatelyontrolthedistribution ofash owsarosstimeperiodsandarossstatesofnature,whih isnotallowedbyanadditiveutilityobjetiveontheonsumptionproess
1 .
Theontribution ofthis paperistwofold: i)on themethodologialside, wedene theonept of time-onsistenyof optimalstrategies,showthat thelassiallyused statiriskmeasures onnal wealth are nottime-onsistentandadvoatetheuseofreursiveutilitiesasatime-onsistentandexiblemeasurefor portfolioriskmanagementandvaluation;ii)ontheoperationalside,weprovideatratableframeworkto dynamiallymanagephysialassetsunderrandomdemandandevolutionofspotandforwardommodity pries, and show on a numerial example how the use of reursive utilities an help strike a trade-o between nal and intermediate wealth risk management and assess the synergy between the physial assetsomposingaommodityportfolio.
Theremainderofthepaperisorganizedasfollows. Insetion2,wedenethetime-onsistenyofoptimal strategiesandomparetwoobjetiveswithrespettotheissuesoftime-onsisteny,andrisk/substitution preferenes. Insetion 3,we presenttheretailer'sportfolio managementproblemand provide apriing formulaandbid/askpriesforphysialommodityassets. Setion4presentsanumerialillustrationof themainndings. Setion5ontainsonludingomments.
1
Theobjetiveof thissetion isto presenttwoexamplesdynami riskpreferenes andassesstheir time-onsisteny properties,whih weviewasanoriginalontributionofthepaper.
2.1 Stati risk measures
Intheaseofoneperiodsettings,anumberofstatiriskmeasureshavebeendenedtoexpresspreferenes ofriskaverseagents(seee.g.,Artzneretal. (2000)andFrittellietal. (2002)).Mathematially,a(stati) riskmeasureisafuntion,heredenoted,assoiatingtoaontingentlaimX arealnumber(X). (X) representsthepriethatitisaeptabletopayinordertopurhaseXand ( X)representstheapital thatmustbeprovisionedinordertomakeashortpositionin X aeptable.
2.2 Risk measure assoiated to a stream of ash ows 2.2.1 Possible riteria for ash owstreams assessment Denedonalteredprobabilityspae(;F;P;(F
t
)),thedisrete-timestohastiproessG=(G i
) i=1;:::;T
, representsasequeneofrandomashowsourringattimes(
i )
i=1;:::;T
. G isthesetofallF
i
-adapted ashowproessesfrom i=1toi=T. WehooseF
1
=f;;g (G 1
isdeterministi),andF
T
=F,so thatfullinformationisrevealedatdate
T . A dynami valuemeasure V =(V
i )
i=1;:::;T
onsistsof mappings V i
: G !R that assoiateto eah ash ow proess G 2 G and to eah ! 2 areal numberV
i
(G;!). The resulting stohasti proess (V
i )is F
i
-adapted. Finanially,it representsthevalueofthesequeneofashows(G k
) k =1;:::;T
or the apitalrequirementtoovertheliabilities( G
k )
k =1;:::;T
atdate i
.
Letusnowproposetwoategoriesofdynamivaluesmeasuresforstreamsofash ows:
1. The rst ategory onsists of extensions of stati riteria depending on the wealth aumulated betweendate
i
anddate T
:
W i;T
:= T X =i
G V
i
(G;!) = (W i;T
jF i
) (1)
Intheaboveequation,isaone-stepriskmeasureandthenotation(:jF i
onstrutedfrom theendofthetimeperiodbydening: V
T
(G;!) = G T V
i
(G;!) = W(G i
;(V i+1
jF i
)) 8iT 1 (2)
Intheaboveequation,isaone-stepertaintyequivalent 2
andthemappingW :R 2
!R isalled an aggregator. In this framework, the date
i
value is assessed reursively by aggregationof the urrentash owG
i
andertaintyequivalentofV i+1
seenfromdate i
. Animportantobservation isthattheproess(V
i )isF
i
-adapted.
2.3 Time onsisteny
Time-onsistenyisapropertywhihguaranteesthatpreferenesimpliedbyadynamivaluemeasuredo notonit overtime.
2.3.1 Examples of time-inonsisteny
ConsiderthetwoashowstreamsAandB,wherealltransitionprobabilitiesaresupposedtoequal0:5:
H
H H
H H
H
H H
H H 3
1(stateu)
0(stated)
7(stateuu)
1(stateud)
6(statedu)
1(statedd) A
H
H H
H H
H
H H
H H 3
2(stateu)
1(stated)
4(state uu)
1(state ud)
3(state du)
1
(state dd) B
LetusevaluatestreamAusingthedynamivaluemeasure(1)with(X)=u 1
(E[u(X)℄),u(x)=ln (x): V
2
(A;u)=exp(E( ln(W A 2;3
ju)))=exp(0:5(ln(8)+ln(2)))=4; V 2
(A;d)=exp(E(ln(W A 2;3
jd)))= p
6 2
V 1
(A)=exp(E(ln(W 1;3
)))=exp(0:25(ln(11)+ln(5)+ln(9)+ln(4)))=(5536) 4 NowevaluatestreamB:
V 2
(B;u)=exp(E(ln(W B 2;3
ju)))=exp(0:5(ln(6)+ln(3)))= p
18; V 2
(B;d)=exp(E( ln(W B 2;3 jd)))= p 8 V 1
(B)=exp(E( ln(W B 1;3
)))=exp(0:25(ln(9)+ln(6)+ln(7)+ln(5)))=(5435) 1 4 Wethushavesimultaneouslythefollowinginequalities:
V 2
(A;u)<V 2
(B;u); V 2
(A;d)<V 2
(B;d); V 1
(A)>V 1
(B)
Asaresult,thedynamivaluemeasureV denedin (1)qualiesB aspreferabletoAinallstatesofthe worldattime2andApreferabletoB attime1,heneitstimeinonsisteny.
Time onsisteny does not hold either if is a mean-variane instead of an expeted utility riterion in equation (1). Tosee this, onsider the two following ash owstreams A (left) and B (right),with transitionprobabilitiesbeingwrittenontopofeahar:
H H H H H 0
0(stateu)
0(state d)
1(stateuu)
0(stateud)
0 1 2 1 2 3 4 1 4 A 0
0(state u)
0(stated) 0.5 0 1 2 1 2 B
Letusevaluatestream Ausingthedynamivaluemeasure (1)with(X)=E(X) Var(X): V
2
(A;u) = E(W A 2;3
ju)) Var(W A 2;3 ju))= 3 4 ( 3 4 9 16 )= 9 16 V 2
(A;d) = E(W A 2;3
jd)) Var(W A 2;3
jd))=0 V
1
(A) = E(W A 1;3
V 2
(B;u) = E(W B 2;3
ju)) Var(W B 2;3 ju))= 1 2 V 2
(B;d) = E(W B 2;3
jd)) Var(W B 2;3
jd))=0 V
1
(B) = E(W B 1;3
)) Var(W B 1;3 ))= 1 2 1 2 ( 1 2 1 4 1 16 )= 3 16 = 12 64 Wethus havesimultaneouslythefollowinginequalities:
V 2
(A;u)>V 2
(B;u);V 2
(A;d)V 2
(B;d);V 1
(A)<V 1
(B)
2.3.2 Denition of time onsisteny and omparison of the two riteria Weassumethattheashowsdependondeisionsthataremadeateahdate
i
,usingtheinformation availableatthisdate. Deisionatdate
i
istheresultoftheoptimizationofadynamivaluemeasureof thetypedesribed above. This optimizationnot onlyyieldsthe rstdeisionatthat date, but awhole deision planning forall subsequent stages. The questionwepose in this setion is the following: are optimalplanningsonsistentovertime?
Letusdenetheproblemformally: onsider aash owsequene(G i
) 1iT
, ourringatdates( i
) i1
, depending ondeisions(q
i )
1iT
andonamulti-dimensionalrandomproess( i
) 1iT
: G i
:=f(q i ; i ). ( i
)is assumedtobeofthetype i+1
=g( i
; i+1
)forsomereasonablybehavedfuntion g, andawhite noisevetorproess(
i ).
We introduethe state variables x i
on whih depend deisionsat time i
and denote A(x i
) the set of admissiblestrategies(q
k )
ik T
attime i
. Wesupposethat,afterdeisionq i
ismadeattime i
,thestate x
i
leadsto x i+1 =h(x i ;q i ; i+1 ; i+1
),where h isadeterministi funtion and( i
)awhite noisevetor proess possibly orrelatedwith (
i
). Wedenote (F i
)the ltration generatedby theproesses ( i ; i ); (q i
)issupposed tobean(F i
)-adaptedproess.
Lastly,weonsider thefollowingoptimization problem,relatedtoadynamivaluemeasureV: J
i (x
i
):= Max (qk) k t 2A(xi) V i (G) (3)
Wedenote(q i k (x i )) k i
theresulting(F i
)-adaptedoptimalstrategydeidedatdate i
3
. Thequestionof onsistenyofoptimalstrategiesanbeformulatedin thefollowingway:
Isq i i+1 (x i ; i+1 ; i+1
)equalto(q (i+1) i+1
(x i+1
)),wherex i+1 =h(x i ;q i (x i ); i+1 ; i+1 )? 3
denedabove.
-First, letus onsider thenal wealth objetivedened in equation(1)with (X)=u 1
(E[u(X)℄),i.e, V
i
(G;!)=u 1
(E(u(G i
+G i+1
+:::+G T )jF i ))) 4 : J i (x i
): = Max (q k ) k i 2A(x i ) V i (G) = u 1 Max qi Max (q k ) k i+1 E i (E i+1 (u(G t +G i+1
+:::+G T ))) = u 1 Max qi E i ( Max (qk) k i+1 2A(xi+1) E i+1 (u(G i +G i+1
+:::+G T
)))
Thedate i+1
impliedproblem Max (qk) k i+1 E i+1 (u(G i +G i+1
+:::+G T
)))diersfromtheonederivedfrom thedynamivaluemeasure (V
i
),i.e., Max (qk)k i+1 V i+1 =E i+1 (u(G i+1 +G i+2
+:::+G T
)). As aresult,the optimalstrategydeidedattimeidiersfrom theoptimalstrategyexhibitedattime i+1.
Timeinonsisteny remainsifweuseamean-varianeobjetiveinsteadofan expetedutility. Inorder tofurtherinvestigatethisissue,letusonsiderasequeneofthreeashows(G
1 ;G
2 ;G
3
),dependingon the(F
i
)-adapted proess( i
) i=1;2;3
andF i
-measurabledeisions(q i
) i=1;2;3
, andletusdeompose the varianeofthesumoftheseashows. Asusual, wedenoteVar
i
(X):=Var(XjF i ). Var 1 (G 1 +G 2 +G 3
)=Var 1
(G 2
+G 3
)=E 1 [(G 2 +G 3 ) 2 ℄ [E 1 (G 2 +G 3 )℄ 2 =E 1 [E 2 ((G 2 +G 3 ) 2 )℄ [E 1 (E 2 (G 2 +G 3 ))℄ 2 =E 1 [E 2 ((G 2 +G 3 ) 2 )℄ E 1 ([E 2 (G 2 +G 3 )℄ 2 )+E
1 ([E 2 (G 2 +G 3 )℄ 2 ) [E 1 (E 2 (G 2 +G 3 ))℄ 2 =E 1 [Var 2 (G 2 +G 3
)℄+Var 1 (E 2 (G 2 +G 3
))=E 1
[Var 2
(G 3
)℄+Var 1 (G 2 +E 2 (G 3 )) Thelastequalityilluminateswhytotalvarianeistimeinonsistent: theF
1
-measurabletermVar 1 (G 2 + E 2 (G 3
))isontrolledbybothdeisionsq 1
andq 2
,inontrasttothetermG 1
,whihdependsonlyonthe deisionq
1
. Thisfat ompromisestheexisteneofanydynamiprogrammingequationlinking optimal strategiesat dates
1 and 2 : J 1 (x 1
): = Max (q k ) k =1;2;3 2A(x1) fE 1 (G 1 +G 2 +G 3
) Var 1 (G 1 +G 2 +G 3 )g = Max (q k ) k =1;2;3 fG 1 (q 1
) Var 1 (G 2 +E 2 (G 3
))+E 1 (E 2 (G 2 +G 3
) Var 2 (G 3 ))g 6= Max q 1 G 1 (q 1
) Var 1 (G 2 +E 2 (G 3
))+E 1
( Max (q k ) k =2;3 2A(x2) E 2 (G 2 +G 3
) Var 2 (G 3 )) 4
Fromnowon,wewilldenoteE(XjF i
)=E i
Asarstobservation,letusonsidertheaseofalinearaggregatorW(x;y)=x+y. Thedate i
objetive derivedfrom thevaluemeasureV
i
denedbyequation(2)isthen: J
i (x
i
): = Max (qk) k i 2A(xi) V i (G) = Max
(q k ) k i fG i (q i )+ i (V i+1 )g = Max qi G i (q i
)+ Max (q k ) k i+1 2A(x i+1 ) i (V i+1 )
ThequestionatthisstageistoknowwhetherpermutingtheoperatorsMaxandoperatorislegitimate inthelastequality,i.e.,ifthefollowingpropertyholds:
Max (qk) k i+1 i (V i+1 ) ? = i ( Max (qk) k i+1 V i+1 ) (4)
Ifthepermutationisvalid,thentheoptimalstrategieswillbetime-onsistentsinethedate i+1 implied problem Max (q k ) k i+1 V i+1
willoinidewiththeoptimizationproblematstagei+1;otherwise,theywillnot. LetustrytheaggregatorW(x;y)=
1
((x)+(y))andertaintyequivalent(X)=u 1
(E[u(X)℄), whereuandareinreasingfuntions and isapositivedisountingfator
5 : J i (x i
): = Max (q k ) k i 2A(x i ) V i
(G)= Max (q k ) k i 2A(x i ) 1 ((G i (q i )+( i (V i+1 ))) = 1 Max (q k ) k i 2A(xi) f(G i (q i ))+( i (V i+1 ))g = 1 Max qi (G i (q i
))+( Max (q k ) k i+1 i (V i+1 )) TheinversionbetweenoperatorsMaxandin thelastequalityispermittedas
Max (qk)k i+1 i (V i+1
) = Max (qk)k i+1 u 1 (E i (u(V i+1
)))=u 1 E i ( Max (qk)k i+12A(xi+1) u(V i+1 )) = u 1 E i (u( Max (qk)k i+12A(xi+1) V i+1 )) = i
( Max (qk)k i+12A(xi+1)
V i+1
) WeannowpresentageneralsuÆientonditionoftimeonsistenyforoptimalstrategies: Property 2.1: If thereexistnon dereasing funtionsab,, anddandpositive numbers
t
suhthat V
i
(G)=ahfb(G i (q i ))+ i [E i (d(V i+1
(G)) ℄gi (5)
thenthe dynami valuemeasure(V i
)leads totime-onsistentoptimal strategies.
For the reursivevalue proess dened by utility funtions and u, equation (5) holds with a = 1
, 5
b=,=Æu ,andd=u. Intheaseoflassialexpetationmaximization(risk-neutrality),equation (5)holdswitha=b==d=Id.
2.4 Risk and substitution
We have mentioned earlier that the problem of dynami optimization under unertainty involves two dimensions,one withrespettothedistributionofashowsarossstatesofnature,theotherover on-seutivetimeperiods. Therstdimensionhasaneetonthe nal wealthdistribution whiletheseond oneimpatsthe likelihood ofbankrupty withinthe timeperiod.
Dynamivaluemeasuresdened inequations(1) arenotappropriateto apturetheriskattahed to in-termediateashowssinetheyarebasedonnalwealth. Byontrast,reursivedynamivaluemeasures allowsone to disentangle randomnessand time omponents,via theertainty equivalent and the ag-gregatorW (respetivelyaountingfortheriskaversionandthesubstitutionpreferenesofthedeision maker). For instane, in the ase of reursive dynami valuemeasures based on utility funtions, the onavityofthefuntions uandleadstothesmoothingofash owsdistributionsin bothdimensions andinturn toajointontrol ofthenalwealthriskandbankruptyrisk.
Remark: Thehoieu=inreursivevaluemeasuresderivedfromutilityfuntionsuandleadstothe lassial objetive: V
i
(G) = u 1
(E i
( P
T k =i
k
i u(G
k
))), whih has beenwidely used in onsumption and portfolio hoie problems in nane (e.g., onsumption-based CAPM). Of ourse, this objetiveis time onsistentand aptures both riskaversionand substitution; itsdrawbakis that it doesnot oer as muh exibility as a more general reursive value measure sine risk aversion and substitution are representedbythesamefuntion u.
3.1 The model
Weadoptadisretetimesetting,withanitehorizon.Thedeisionperiodsaredenoted(p i
),i=1;:::;T (typiallymonthsorquarters). Thedates(
i
)aredeningtheperiods(p i
).
-date1 date2
...
dateT
1
2
T period1 period2
...
periodT
Weassumefromnowonthattheretailer'sportfolioisomposedofonesaleontratandonestorage reservoir. In addition, the ommodity is supposed to be traded, stored, and onsumed in the same loation (in order to avoid transmission osts and onstraints). The problem an be represented in a stylizeddiagram:
-6
6 ?
? retailer storage
market
lient
L max
is themaximallevelof storage,L min
is theminimal levelofstorage(at anydate), L init
is the initialstoragelevel,L
end
istheminimalstoragelevelattheendofthehorizon. L i
representsthestorage levelattheendofperiodp
i . Q
inj i
denotesmaximalinjetioninperiodp i
,Q draw i
maximalwithdrawal;we supposetherearenoinjetion/withdrawalostsnorholdingost. d
i
denotesthelient'srandomdemand inperiodp
i , K
s i
isthexed sellingprieoftheommodityforperiod i.
Onlyforwardontratsare onsidered;ash owsdueto forwardontratingaresettled atmaturity oftheontratandounterpartyriskignored. WedenotebyF(i;j)theforwardprieoftheommodity quoted during p
i
for delivery in period p j
6
(j i) and S i
the spot prie of the ommodity, where S
i
:=F(i;i). Remarks: 6
Here, F(i;j) an be onsidered as the average prie overall the quotation dates belonging to period pi of all forward ontratsfordeliveryinperiodp
2. Evenin the aseof illiquid markets, the retaileris assumedto bea prie-taker, meaningthat her tradingdeisionswillhavenoimpatonmarketpries
Storagedeisionvariablesorrespondingtoperiod p i
aresubjettothefollowingonstraints: 0q
inj i
Q inj i
; 0q draw i
Q draw i
i1 (6)
L 0
:=L init
; L i+1
=L i
+q inj i
q draw i
0iT (7)
L min
L i
L max
8i=1;:::;T; L T
L end
(8) n(i;j) denotes the net number of forward ontrats bought during period p
i
for delivery in period p j (ji),theasei=j beingaspot transation. N(i;j)representsthetotalforwardposition attheend ofperiodp
i
fordeliveryinperiodp j
andsatisestheonditions:
N(0;j):=0 8j1; N(i;j)=N(i 1;j)+n(i;j) 81ij (9) Wemodelthesequeneofeventsanddeisionsinthefollowingway: duringperiodp
i
,theretailerdisovers thelient'sdemandanddeidesondate
i
whihquantitiesn(i;j)tobuyonthespotandforwardmarket and q
inj i
or q draw i
to injet in or withdrawfrom storage, respeting the physial balane of ommodity owsduringperiodp
i i.e.,
N(i;i)+q draw i
q inj i
=d i
8 1iT (10)
Equation(10)expressesthatmarket andstoragearethetwowaysto servedemand atperiod p i
. We dene the disrete set of states of nature . Eah ! 2 represents a realization of the proess
i = (d
i ;F(i;j)
ji
), i = 1:::T. We denote by (F i
) the ltration generated by ( i
). Throughout the paper,weassumetheabseneofarbitrageopportunitiesintheommodityspotandforwardmarkets. On (;F;F
i
),wedenearisk-neutralprobabilitymeasureP, underwhih forwardpriesaremartingales 7
. WedenethesetAofadmissiblestrategiesas:
A:= n
(q i
) i1
=(q draw i
;q inj i
;n(i;j) ji
) i1
F i
measurableandverifyingonstraints(6) to(10) o 7
In this setion, it is assumed that there are neither onstraints nor osts assoiated to trading in the forwardmarket. Therisk-freeinterestrateris supposedonstant. Thegoalhereisto presenttwoases wherethepriingissuesand managementoftheportfolioarepartiularly simple:
-therstaseistheoneofaliquidmarketanddeterministidemand
-theseondaseinludesunertaindemandbut assumesrisk-neutralityoftheretailer,henetheuseof ariterionofexpetedprotmaximization
Inbothases,afulldeomposition oftheportfoliovalueandmanagementispossible. Thetotalash owduringperiod p
i
isdenoted asG i
andmaybewritten as: G
i
= d i
K s i
T X j=i e
r( j
i
)
F(i;j)n(i;j) (11) Remark: Cashowsduetoforwardtradingareinthispaperregisteredattransationdateanddisounted from deliverydate at the risk free interest rate r. We adopt this unusual rule beause we want ash ows at dates
i
to depend only ondate i
deisionsand not on previousones 8
, as would be the ase if ash ows from forward transation had been registered at delivery date. Sine interest rates are onsidereddeterministi,thisrepresentationhasnoonsequenesonthenalwealthbutmayhavesome onintermediatewealths
9 .
Assumingliquidspotmarkets,theouplingonstraint(10)anbetreatedas animpliitoneandwefae afullydeomposableproblem,withonstraintsonlyonindividualassets.
Derivingfrom(9)and(10)thevolumen(i;i)of spottransations,equation(11)beomes: G
i
= d i
K s i
n(i;i)S i
T X j=i+1
e
r(j i)
n(i;j)F(i;j)
= q draw i
S i
q inj i
S i
+d i
(K s i
S i
)+N(i 1;i)S i
T X j=i+1
e
r(j i)
n(i;j)F(i;j) Inthisform,G
i
appearslikethesumofthreeomponents: 1. q
draw i
S i
q inj i
S i
=periodp i
payofromthestoragefaility. Storagedeisionstakenovertime are inter-dependentdue totheapaityonstraintsexpressedinequation(6)
2. d i
(K s i
S i
)=periodp i
payofromthesaleontratdevoidedofanyoptionality,whihisinfata 8
inaordanewiththesettingdenedinsetion2.3.2 9
stripofswapsexhangingthesaleontratprieK i
forthespot prieS i
. Thevolumeinvolvedat periodp
i
iseitherxed(deterministidemand)or random(unknowndemand) 3. N(i 1;i)S
i P
T j=i+1
e
r(j i)
n(i;j)F(i;j)=periodp i
ashowfrom forwardontrats Under this form, the portfolio appears as a ombination of various options written on the ommodity spot priewhile theforwardmarket appearsas awayto hedge thespot prierisk. Theabovesplitting ofashowssuggestsadeomposition oftheportfolio'svalue. Infat,thelatterwill onlybepossiblein twopartiularases:
Portfoliodeompositioninaompletemarketsetting: here,weassumethatthedemandproess(d i
) isdeterministi(e.g., theontratsetsaxed volumeto bedeliveredin allfutureperiods). Then, thearbitrage prieof theportfolio is thesum ofmaximal expeted ash ows under the(unique) risk-neutralprobability measure; this value is the sum of thearbitrage priesof storage and sale ontrat. Inthis framework,the obviousstrategy forthe portfolio manageronsists in optimizing independently the storage faility against the spot market under the risk-neutral measure, and hedgingspotprieriskusingtheforwardmarket.
Portfoliodeompositionforarisk-neutralretailerinaliquidmarket:weassumeherethattheretailer faes both demand andprierisks but isrisk-neutral, i.e.,sheonly tries tomaximize herexpeted prot. Under the assumption that the physial measure is a risk-neutral measure, the optimal strategyfortherisk-neutralretaileronsists againin optimizingindependentlythestoragefaility againstthespot marketand doingnotradein theforwardmarket. Moreover,under deterministi demand, theoptimumof the risk-neutralretailer'sobjetive orrespondsto thearbitrage prieof theportfolio.
3.3 The retailer problem in an inomplete/illiquid market
Illiquidityismodeledbydeterministivolumeonstraintsonspotandforwardtrading,oftheform: n
b
(i;i+)n max b
(i;); n s
(i;i+)n max s
(i;) (12)
where n b
(i;j)and n s
(i;j) standfor thenumber ofbought and soldforwardontratsduring period p i fordeliveryinperiodp
j
(withn(i;j)=n b
(i;j) n s
i A(x i ):= n (q k ) k i =(q draw k ;q inj k
;n(k;j) jk
) k i
F k
measurableverifyingadmissibilityonstraints o
(13) and theanalogous set of illiquid market admissible strategiesA
liq (x
i
). The restritionsofthe previous deisionsetsto datet,deningtheadmissibilitysetsfordeisionsq
t
only,willbedenotedbyA t (x t )and A liq t (x t ).
Weannowformulatetheretailer'soptimizationproblemas: J
i (x
i
):= Max (q k ) k i 2A liq (xi) V i (G) (14)
wherethestatex i
isdened byx i
=(L i
;N(i;:); i
), Gby(11)andV i
(G) bythereursiveequation(2), with aggregator W and ertainty equivalent derived from onave inreasing funtions and u and positivedisountfators(
i ): W(x;y)=
1
((x)+ i
(y)); (X)=u 1
(E[u(X)℄) WedenotesuhadynamivaluemeasureasV
;u t
(G). TheoptimalvalueJ
i (x
i
)satises thedynamiprogrammingequation: J i (x i )= 1 ( Max qi2A liq i (xi) (G i (q i ))+ i Æu 1 (E i (u(J i+1 (x i+1
)))) ) (15) wherethestatex
i+1
isgivenbythetransitionequationx i+1 =(L i + q inj i q draw i
;N(i;:) + n(i;:);g( i
; i+1
)). The existene of equation (15) guarantees the time onsisteny of optimal strategies, as shown in the previoussetion.
3.4 A onavity property for J i Proposition 3.4.1:
ChoosingCARAtypeutilities(x)= e x
andu(x)= e x
suhthat0<,foralldatest,andall statesx
t
suhthatA liq t
(x t
)6=;,themaximizationproblem Max qt2A liq t (xt) (G t (q t ))+ t Æu 1 (E t (u(J t+1 (x t+1 )))) isonave withrespettodeisionsq
t
. Moreover,thedeisionsetA liq t
(x t
)isonvex. Theresultalsoholds for =Idanduof CARAtype.
i
In this setion, we show that, in omplete markets, J t
is thearbitrage prie of the portfolio under the twoonditons: (x) = x (no preferene for smoothversusirregular ash ows in time dimension) and i =e r( i+1 i )
(one period disountfator). These twoassumptionswill holdthroughout setion3.5. Property 3.5.1:
J i
(x i
)= Max (q k ) k i 2A liq (xi) V Id;u i
(G)isnevergreaterthantherisk-neutralobjetiveJ rn i
(x i
)= Max (q k ) k i 2A liq (xi) V Id;Id i (G) Proof: Theonavityofuimpliesthat forallrandomvariablesX:
u 1
(E[u(X)℄)E(X) (16) Itresults,byasimplereursion,that:
8G2G; 8i2T; V Id;u i
(G)=G i + i u 1 (E i (u(V Id;u t+1
)))G t + i E i (V Id;Id i+1
)=V Id;Id i
(G) andthepropertyholds.
Property 3.5.2:WhenonditionalvaluesV k +1
omputedatstagesk(k=i;::;T 1)arenonstohasti, thenV
Id;u i
isthe sum ofdisountedash owsfromstage itostage T Proof: In this ase, u
1 (E i (u(V Id;u k +1
))) = V Id;u k +1
for all k = i;:::;T 1, and, therefore, V Id;u i (G) = G i + i V Id;u i+1 = P T k =i e r( k i ) G k
,byasimplereursion.
Theonsequeneisthat, in aomplete market setting (i.e.,deterministi demandand noliquidity on-straints),J
i
isatleastequaltothearbitrage prieoftheportfolio.
Property 3.5.3: In a situation of market ompleteness, J i
(x i
) is equal to the arbitrage prie of the portfolioJ
ap i
(x i
)= Max (q k ) k i 2A(x i ) E Q i ( P T k =i e r(k i) G k
), whereQ isthe (unique)risk-neutralmeasure Proof: Thispropertyisderivedfromthefollowingobservations:
-J i
(x i
) Max (q k ) k i 2A(xi) V Id;Id i
(G),as exhibitedinproperty3.5.1 - Max
(q k ) k i 2A(xi) V Id;Id i
(G) = J ap i
(x i
), beause the optimal value of the risk-neutral retailer's portfolio is equalto itsarbitrageprie.
-J i
(x i
)J ap i
(x i
),as shownin property3.5.2.
strat-Proof: TheequalitybetweenJ i
(x i
)andJ i
(x i
)impliesanequalityinequation(16)foreahX =V i+1
, and,beausethefontionuisstrilyonave,theequalityispossibleonlyifunertaintyonallV
t
isnull. Consequently,weobtainthesatisfatorypropertythattheoptimizationprogrammealsoprovidesa hedg-ingstrategy.
To onlude this paragraph, we an note that the question of estimating the ask and bid pries of a physial asset or nanial ontratin inomplete markets remains to be solved. As often done in the literature,wedenetheask(bid)prieasthediereneofthevaluesofJ
i
,withandwithoutthebought (sold)asset. Underthisdenition,thebidandaskpriesofanassetdependnotonlyontheriskaversion ofthemanagerbut alsoonherinitial portfolio,alassialpropertyin asituation ofinompleteness.
3.6 A model for the evolution of the forward urve and demand Weassumealassialone-fatorevolutionmodel forthemarketforwardurveF(i;j):
F(i;j)=F(i 1;j)M i;j
exp(e k
i (
j
i )
X i
) 8ji8i2 (17) where(X
i )
i2
isadisrete-timestohastiproessomposedofindependentvariableswithlawN(0;( X i
) 2
), (k
i
)arepositiveparameters,and(M i;j
) ji
arepositiveonstantsensuringthatF(i;j) ij
aremartingale proesses. In this model, only one type of shok is allowed forthe forward urve, namely translations, withanamplitudevanishingwithtimeto delivery.
Regarding the demand proess (d i
) i2
, we assume that it is driven by a disrete-time stohasti pro-ess(Y
i
)(typiallythetemperature),omposedofindependentvariableswithlawN(0;( Y i
) 2
)positively orrelatedwiththeprieproesswithorrelationoeÆients(
i ): d
i
=max(f i
; d
i +Y
i
) (18)
where (f i
) are positive oors ensuring that the demand proess is positive, and ( d i
) are the average demandsateahperiod.
As a onlusion,to simulate the joint evolution of forward urve and demand at periods (p i
), we only needto jointlysimulate therandom variables(X
i
)and (Y i
4.1 The event tree
Weusehereastandardstohastiprogrammingtehniquetosolvetheproblem. Thesetofrealizationsof thedemandandtheforwardurveisrepresentedonaneventtreewithnodesn2N,thedeisionsq(t;!) areindexedon thenodesofthetree,and thetime-1objetiveismaximized numeriallywith respetto alldeisions(q
n )
n2N
usingalargesalenonlinearsolver.
Tobuildtheeventtree,weuseatwo-dimensionallattie(seeWebber(1997)),repliatingexatlytherst twomomentsoftheproess(X;Y)ateahtimestep.
Thefourvertexesoftheunitsquarerstprovidetheequiprobablejointrealizationsofavetor ~ Z =(
~ X;
~ Y) oftwounorrelatedzeromeanunit varianerandomvariables:
-6
Æ
Æ Æ
Æ
(1;1)
(1; 1) ( 1;1)
( 1; 1)
Figure 1: Senariosfortwo unorrelated randomvariables
The extension to two orrelated variables is straightforward: onsidering a vetor of two unorre-lated unit variane variables
~ Z = (
~ X;
~
Y), the vetor of random variables Z = (X;Y) = A ~ Z with A=
0 B B
x
0
y p
1 2
y
1 C C A
havezeromeanandovarianematrix= 0 B B
( x
) 2
x
y
x
y (
y )
2 1 C C A . Therefore,weproeedin thefollowingwaytobuildtheeventtreeontheprie/demandproess: - rst, using the matrix M =
0 B B
1 1 1 1 1 1 1 1
1 C C A
, whose olumns represent the four joint realiza-tionsofavetor(
~ X;
~
Y)oftwounorrelatedzeromean,unitvarianevariables,weformthe24matrix N =AM,whoseolumnsaretherealizationsofthevetor(X
1 ;Y
1
() Two-dimensional representation ofthe prie and demandproesses(X;Y) at eahtime step: the re-alizationsof theprie proess X anbe readonthe x-axis
Figure2: Event tree
thelastperiod
-nally,weapplyformulas(17)and(18)togettheforwardurveandthedemandateahnode,theterm M
i;j
beingdeterminedbythemartingaleonditionatnoden: F
n
(i 1;j)=E n
(F m
(i;j))= X m2S(n)
1 4 F
m
(i;j) (19)
whereS(n)isthesetofsonsof noden,whih gives: M
i;j =
1 P
m2S(n) 1 4
exp(e
ki(j i) X
m i
)
(20) It isimportant to pointout herethat theterm M depends onlyon i andj and notof node nbeause thevariables(X
i ;Y
i
)areindependentof (X i 1
;Y i 1
),henethesetsfX m i
; m2S(n)gare thesamefor everynodenofdate
i 1 . Weobtain4
T 1
Weassumethefollowingsetting:
-theretaileristradinganenergyprodut,whoseprieisexpressedin e/MWh
- there are veperiods of one quarter eah: during the rst quarter, the retailerfaes no demand and replenishesher storagefaility usingthe spot market in order to meet the unknown lient's demand in thefollowingyear
-thestoragehasaninitiallevelat20TWh,amaximalwithdrawalinjetion/withdrawalperperiod of10 TWh,amaximal(resp. minimal)storagelevelof50TWh(resp. 0),andaminimalendlevelof20TWh - theforwardpriedynamis are representedby the model desribed in equation (17)with parameters k
i
=2years 1
andvolatility X i
=0:28i2;theinitialforwardurveissupposedtobeatatthelevel 20e/MWh;inpartiular,theinitialspotprieequals20e/MWh
-themaximalallowedtradedvolumeinthemarketdereaseswithtime-to-delivery: itequals30TWhfor ontratsdeliveringin thepresent quarter("spot"transation), 10 TWhfor ontratsdeliveringin the nextquarter,5TWh forontratsdeliveringintwoquarters,and0TWhforontratsdeliveringin the followingperiods
-thesellingprieonthesale ontratis 21e/MWh(hene amarginof 5%withrespet totheaverage marketforwardprie);regardingthedemandharateristis,wesupposethatd
1
=0,and8i2: Y i
=10 TWh,
d i
=20TWh,f i
= di 3
,and i
=0:5. Therealizationsof(X;Y)ateahtimesteparerepresented ongure(2()): wenote that there are fourdierentrealizations forthedemand proess and twoonly fortheprieproess
- we adopt CARA utility funtions u(x) = e x
and (x) = e x
to represent risk aversionand substitution preferenes, with varyingrisk aversionand substitution parametersand ; interestrates aresetto0.
Figure(3(a))showsthemeanvarianetrade-ointhenalwealthobtainedwhenriskaversionvariesand thefuntionremainsequalto identity. Whentheriskisdened astheConditionalValueatRisk
10 on thenalwealth W
T 11
:
CVaR q
(W)=E( W T
j W T
>VaR q
(W)) (21)
theexpetedmeanisaninreasingfuntion ofrisk,asshowningure(3(a)). Forexample,adereaseof the0.5%(resp. 5%)CVaRonnalwealthfrom611(resp. 505)to371(resp. 291)Meimpliesaderease ofthe expetednal wealth from 67to 15Me. Figure (3(b))representsthetrade-o betweentherisks of thenal wealth andtemporal minimalwealth
12
. Figure (3(b))showsthat it ispossible toexhange bankruptyriskfornalwealthriskbydereasingtheratioofparametertoparameter. Forexample, to ut the0.5% (resp. 5%) CVaR on temporal minimal wealth from 1059 to 545(resp. 473) Me, one hastoaeptariseofthe0.5%(resp. 5%)CVaRonnal wealthfrom365(resp. 296)to516(resp. 458) Me. However,the exhangeof bankrupty riskfor nal wealth riskhas limits: Figure (3(b)) showsin partiularthat itis notpossibleto bringdownthe0.5%(resp. 5%) CVaRontemporalminimal wealth belowaertainthreshold,orrespondingto thepair(=0:1;=0:001)(resp. (=0:01;=0:0005)).
Figures (4(a)) showstheumulativefuntion ofthe nal wealth overthe 256tree senariosused in the optimization proedure under dierent values of risk aversion. In gure (4(a)), we observe that a risk aversion of 0:02 allows to signiantly redue the left tail up to 5% of the distribution obtained underarisk-neutralstrategy. Theostofahigherriskaversionis thatthemainpartofthenal wealth distribution (to the right of the 10%quantile) is signiantly moved upright. Figure (4(b)) shows the distributionoftheminimalwealthovertime: weseethatamoreonavefuntionsigniantlyredues the likelihood of a very negative minimal temporal wealth, whih is a onsequene of the smoothing of ash ows in the time dimension. However, as shown by gure (4(a)), if the ratio
beomes too high (e.g.(=0:01;=0:0005)),the nal wealth distribution exhibits alargeleft tail. Ifthe portfolio managerseekstostrikeabalanebetweennalwealthandbankruptyriskmanagement,hemayhoose (=0:1;=0:001) or (=0:01;=0:0001). Figure (5)representstheintermediatewealths obtained 10
VaRq(W)isthewell-knownValue-at-Riskassoiatedtoquantileq 11
thewealthW i
attheendofperiodp i
isdenedastheumulativesumofashowsfromperiodp 1
toperiodp i 12
Temporalminimalwealthisdenedasmin
i2f1;2;3;4;5g
orresponds to a dierent CVaR quantile and is onstruted withtakingthevaluesf0;0:001;0:005;0:01;0:02g
(b)CVaRofthetemporalminimalwealthintermsofCVaRof thenalwealth(inMe); eahurveorrespondstoadierent CVaRquantileandisonstrutedwith(;)takingthevalues
(0:1;0);(0:05;0:0001);(0:02;0:0001);(0:01;0:0001);(0:1;0:001);(0:01;0:005); (0:01;0:001);(0:001;0:0001)
=0(resp. =0)orrespondstoafuntion u(resp. )equaltoidentity
(b)Temporalminimalwealth(inMe)umulative fun-tionininompletemarkets;thease=0(resp. =0) orrespondstoafuntionu(resp. )equaltoidentity
Figure4: Finalandtemporalminimalwealthumulativefuntionsfordierentriskaversionandsubstitution parameters
() Wealthproleinthease (0.01,0.0001) (d)Wealthproleinthease (0.01,0.0005) Figure 5: Cumulative wealths(in Me)inthe dierentnodesof theevent tree fordierent pairs(;)
4.4 Portfolio value
Figure (6(a)) representstheportfolio valuedened in setion 3.5 fordierent riskaversionparameters. The portfolio value is a dereasing funtion of the risk aversion parameter. The spread between the risk-neutralandpositiveriskaversionvaluesanbeinterpretedasariskpremium,whosevalueinreases logiallywith theriskaversionparameter.
Thevalueofthesaleontrat,obtainedbysettingthestorageexibilitytozerointheoriginalportfolio 13
, behaves similarly. The storage value, obtained by setting the lient's demand to zero in the retailer's portfolio,doesnotdependontheriskaversionparameter: thisisduetothefatthat,undertheliquidity assumptionsmadeinsetion4.2,thestoragefailityhasauniquearbitragevalue(here55.26Me)whih an be seured by appropriate forward transations; in this ontext, the optimum J
1
of the storage management problem redues to the storage arbitrage value, as explained in setion 3.4. Thesynergy valuewhihisdenedasthespreadbetweentheportfoliovalue,ontheonehand,andthesumofthesale 13
parameters ferentdemandvolatilities Figure 6: Deompositionof J
1 (x
1
) = Max (q
k )
k 1 2A
liq (x
1 )
V Id;u 1
(G) (in Me) and synergy value for dierent risk aversionparameters and dierentdemandvolatilities
ontratandstorageseparatevalues 14
,ontheotherhand,isnullforarisk-neutralretailerandinreases with the risk aversion parameter, whih expresses the fat that the synergy between sale ontratand storagefailityisintermofriskmanagementratherthanin termofexpetedreturn.
Figure(6(b))representsthesynergyvalueintermoftheriskaversionparameterunderdierentdemand volatilities. Itisobservedthatthesynergyvalueinreaseswithdemand volatility,whihmeansthat the storagefaility'svalue-addedin theretailer'sportfolioinreaseswiththedemandunertainty. Figure(7) showsthatthestorage'svalueaddedbeomesnullinaontextofhigh forwardmarketliquidity,evenin thepresene of volume unertainty: the synergyeet arises onlyunder an illiquid forwardmarket. In addition,theportfoliovaluevariesfrom 89to37Me,dependingontheforwardmarketliquidity,whih pointsouttheimportane ofliquidityassumptionforportfoliovaluation.
14
intable (1)(with =0:01 and demandvolatility=10 TWh)
Q0 Q1 Q2 Q3 Q4 low liquiditysetting 30 10 5 0 0 mediumliquiditysetting 30 10 10 10 10
highliquiditysetting 30 30 30 30 30
Table1: Desriptionofthethreeliquiditysettings: Q0representsthemaximalvolumeof"spot"transations, Q1 the maximalvolume for delivery in the next quarter, Q2 the maximal volume fordelivery in the next followingquarter...
5 Conlusion
likelihoodofabankruptywithin thetimehorizon.
6 Referenes
Artzner P.,Delbaen F., EberJ.M., Heath D. (1999), Coherent Measures of Risk,Mathematial Finane 9: 203-228
Artzner P., Delbaen F., EberJ.M., Heath D. , KuH. (2002),Coherent Multiperiod Risk Measurement, WorkingPaper
Chen X., SimM., Simhi-LeviD., Sun P.(2004)Risk Aversion in Inventory Management, aeptedby OperationsResearh
DuÆe D., Epstein G. (1992), Stohasti Dierential Utility, Eonometria, Eonometri Soiety, vol. 60(2),pages353-94
EihhornA., RomishW.(2005),Polyhedral RiskMeasuresinStohasti Programming,SIAMJ.Optim. 16,pp. 69-95
Epstein G.,ZinS. (1989),Substitution, Risk Aversion, andthe Temporal Behavior of Consumption and AssetReturns: Atheoretialframework, Eonometria,Vol. 57,No. 4,pp. 937-969
FrittelliM., RosazzaGianinM.(2002),Puttingorderin riskmeasures,Journal ofBanking andFinane, Vol. 26pp. 1473-1486
Frittelli M.,RosazzaGianin M.(2004),Dynami onvex risk measures,NewRisk Measuresforthe21th Century,G.Szegoed., JohnWiley &Sons,pp. 227-248
LideLi,PaulR.Kleindorfer(2004),Multi-Period VaR-ConstrainedPortfolio Optimizationwith Applia-tionstothe EletriPowerSetor,EnergyJournal
Martinez-de-AlbenizV., D.Simhi-Levi(2003),Mean-VarianeTrade-os inSupplyContrats,Working paper,MIT
UngerG.(2002),Hedging StrategyandEletriity Contrat Engineering, PhDThesis,ETHZrih Wang,T.(2000),AClass of DynamiRisk Measures, underrevision
