Com monSourc esofVariationinNaturalG as
FuturesP ric es
O w enB eeld ers¤
Dec emb er 19 9 8
R evised : M arch 19 9 9
A bstract
W eidentifythecommonsources ofvariationintheterm structure
ofnaturalgas futures prices and the term structure ofconvenience
yields using principalcomponentanalysis. In the term structure of
futures prices werequireatleast5 components toexplain 90 % ofthe
variation:the¯rstcomponentisalevelcomponentthata®ectsfutures
prices equallyacross maturityandtheremainingcomponents areseasonal. For the term structure of convenience yields, a level, slope
and curvature componentexplain 93%, 4% and 1 % ofthe variation,
respectively.
1
In
trod uc tion
NaturalG aspric esare notoriously volatile. During a c old spellinFeb ruary 1996, the spot pric e spiked to $ 12 af
ter averaging$ 2 .90 inJ anuary and
the im plied volatility ofthe trad ed options c ontrac ts w as over 150 % (Sim ons (19 97)). F itzgerald and P okalski (199 5) attrib ute this volatility to
aninterplay ofthe in°exib ility ofstorage and transportationf
ac ilitiesand
the extreme w eather c ond itionsthat arise unexpec ted ly. Follow ing the rec ent d eregulationofthe naturalgasm arkets, d istrib utionc om paniesare no
¤
M ailing A ddress: D epartmentofEconomics, Emory U niversity, A tlanta G a 30 3222240 .e-mail:obeelde@ emory.
edu.I thankJohnY unandB obSubrickfortheircomments.
T heremainingerrors are allmine.
1
longer guaranteed a \f
air return"b y the Fed eralP ow er Com m ission;instead
their returnsare likely to °uc tuate w ith the naturalgaspric e.O ne m ethod
ofred uc ing the risk and uncertainty oftheir returnsished ging. T he tw o
m ainob jec tivesofhed gingare a red uc tioninearningsvolatility and a low er
prob ab ility of¯nanciald istress.Ad d itionalb ene¯tsac c rue inthe f
orm ofenhanced c red itw orthiness, higher m arket c apitaliz ationand a low er af
ter-tax
c ost ofc apital.T here are also b ene¯tsto sharehold ersinthe f
orm ofhigher
and lessrisky returnsto sharehold ers' equity.
T o hed ge the risk inherent innaturalgaspric esw e ¯rst need to d eterm ine the sourc esofvariationinnaturalgaspric es.For exam ple,one sourc e
ofvariationisthe seasonald em and f
or gas; the pric e respond ssharply to
unexpec ted W inter c old spellsand Sum m er heat w aves.Although the id enti¯c ationofthe sourc esofvariationisanem piric alprob lem , w e need to b e
guid ed b ytheory. T he naturalstartingpoint isthe c om m od ity optionspric ing m od elintrod uc ed b y B lack (19 76). Like the B lack and Scholes(1973)
stock optionpric ing m od el, it hasonly one sourc e ofstochastic variation.
How ever, one sourc e ofvariationisinad equate f
or pric ing c om m od ity options.T hisb egsthe question: ifone sourc e ofvariationisnot enough, how
m any d o w e need to ad equately m od elthe variationofc om m od ity pric es?
T he theoretic alguid e to answ eringthisquestionisa m od elofc om m od ity
pric ing d eveloped b y Am in, Ng and P irrong (19 95); it c anac c om m od ate
m ore thanone sourc e ofvariation.T he d i®erence b etw eenthe B lack (1976)
m od eland the mod elproposed b yAm in,Ngand P irrong(199 5) isanalogous
to the d i®erence b etw eenthe c apitalasset pric ing m od el(CAP M ) and the
arb itrage pric ingtheory (AP T ).W hereasthe CAP M only hasone sourc e of
systematic riskor variation,the AP T c anac c om od ate m ore thanone sourc e,
b ut issilent onthe numb er ofsourc esofsystem atic risk.
T he statistic altoolthat w e use to id entif
ythe num b er ofsourc esofvariationisprincipalc omponentsanalysis(P CA).P CA isa m ethod f
or id entif
ying
the numb er ofsourc esofvariationw ithina group ofvariab lesand the relative
c ontrib utionofeac h sourc e to the totalvariation.Litterm anand Sc heinkm an
(19 91) introd uc ed the m ethod ofP CA inthe c ontext ofhed ging¯xed incom e
portf
olios.W hereasstoc ksare know nto have a lot ofunsystem atic riskthat
c anb e measured b y the volatility oftheir returns, ¯xed incom e sec urities
have a lot m ore systematic riskthat isc om m onac rossm aturities.Litterm an
and Sc heinkm an¯nd that at least tw o sourc esofvariationare need ed to ad equately explainthe variationinthe term struc ture ofinterest rates.M ore
rec ently, Cortaz ar and Sc hw artz (199 4 ) applied P CA to the term struc ture
2
ofc opper pric es.T hey id entif
y three sourc esofvariationthat explain9 8%
ofthe variation.
T he ob jec tive ofthispaper isto replic ate the Cortarz ar and Schw artz
(19 94 ) analysisf
or naturalgaspric esand extend the analysisto c onvenience
yield s.B ased onthe theory ofasset pric ingund erlyingf
uturespric ing,P CA
should id entif
y one less sourc e ofvariation. Insec tion2 w e d isc uss the
theory off
uturespric ingand the role ofa sec ond state variab le.Insec tion3
w e d isc ussthe d ata,provid e f
urther b ackground to the m ethod ofP CA and
d isc ussthe resultsofour analysis.W e c onclud e w ith sec tion4 .
2
T he T heory ofFuturesP ric ing
T he naturalstartingpoint f
or the pric ingofc om m od ity f
uturesc ontrac tsis
the c ost-of
-c arrymod elthat relatesthe f
uturespric e to the c ost ofpurchasing
1
the c om m od ityonthe spot m arket and c arryingor storingit untilm aturity.
T he m od elc onsistsoftw o c om ponents: the ¯rst c om ponent isa stochastic
d i®erentialequationthat c harac terizesthe d ynam ic softhe spot pric e,
d S(t)
= ¹ (t)d t+ ¾ 1 (t)d W 1(t);
S(t)
(1)
w here S(t) isthe spot pric e,¹ (t) isthe expec ted instantaneouspric e c hange,
¾ 1 (t) isthe instantaneousstand ard d eviation,W 1 (t) isa W iener proc essand
isthe sourc e ofvariance ofthe spot pric e. T he sec ond c om ponent isthe
theory ofstorage relation,
·Z T
¸
F (t;T ) = S(t)exp
(r(u) ¡c
(t;u))d u
(2 )
t
w here F (t;T ) isthe f
uturespric e at tim e tf
or d elivery at T, r(t) isthe
risklessrate ofinterest and c
(t;u) isthe d eterm inistic c onvenience yield , in
exc essofthe c ost ofstorage, that the c om m od ity hold er earns at tim e u
b ased oninf
ormationat time t. T he c onvenience yield c anb e thought of
asthe b ene¯t ofhaving the c om m od ity onhand to sm ooth prod uc tionin
c ase a shortage arises(K ald or (1939 )).T he storage relationisanarb itrage
c ond itionthat equatesthe f
uturespric e to the opportunity c ost ofb uying
1
A min,N gand P irrong(1 995)haveaverydetailed exposition ofthedi®erentfutures
pricingmodels in thecontextofpricingoptions on futures.
3
the c omm od ity onthe spot m arket, i.
e. the opportunity c ost includ esthe
c ost ofb uyingthe c omm od ity onthe spot m arket,the interest f
oregone and
the c ost ofstorage lessthe c onvenience yield ofhold ingthe c om m od ity f
rom
tto T .
B ysub stituting(1) into (2 ),w e ob tainthe risk-ad justed d ynam ic sf
or the
f
uturespric e,
d F (t;T )
= ¡c
(t;T)d t+ ¾ 1(t)d W 1¤(t);
F (t;T )
(3)
w here W 1¤(t) = W 1 (t) ¡¸ 1 (t) and ¸ 1 (t) ´ ¹(t) ¡r(t) isthe pric e ofrisk
assoc iated w ith the spot pric e. T he d ynam ic softhe f
orw ard pric e suggest
another interpretationofthe c onvenience yield : it is\d ivid end "stream that
ac c ruesto the hold er ofthe c om m od ity,b ut not to the hold er ofthe f
utures
c ontrac t. T he d raw b ac k ofthism od elisthat f
uturespric esare perf
ec tly
c orrelated ac ross maturities b ec ause there is only one sourc e ofvariation
c om monto allf
uturespric es,the W iener proc ess,W 1 (t).
O ne m ethod ofred uc ing the c orrelationb etw eenthe f
utures pric es of
d i®erent maturitiesisto allow the c onvenience yield to b e stochastic : the
introd uc tionofa sec ond sourc e ofvariationthat isim perf
ec tly c orrelated
w ith the ¯rst,b reaksthe perf
ec t c o-m ovem ent b etw eenthe f
uturespric esof
d i®erent m aturities.Assume that the d ynam ic softhe c onvenience yield are
d eterm ined b y the stoc hastic d i®erentialequation,
dc
(t;T ) = ¯ (t;T)d t+ ±(t;T )d W 2 (t);
(4 )
w here ¯ (t;T ) isthe expec ted instantaneouschange inthe c onvenience yield ,
±(t;T ) isthe instantaneousstand ard d eviation, W 2 (t) isa W iener proc ess
that isc orrelated w ith W 1(t) and w e d enote the c orrelationb y½.T he W iener
proc ess, W 2 (t); isthe sec ond sourc e ofvariationinthe m od eland isim perf
ec tly c orrelated w ith W 1(t), thusit c anred uc e the c orrelationoff
utures
pric esac rossthe term struc ture.
Inord er to ob tainthe risk-ad justed d ynam ic soff
uturespric es, w e turn
to m ore rec ent d evelopm entsinthe theory ofasset pric ing. Harrisonand
K reps(1979) and Harrisonand P liska (1981) have show nthat inthe ab sence
ofarb itrage opportunitiesthe risk-ad justed asset pric e isa m artingale and
the assoc iated prob ab ilitymeasure isref
erred to asthe equivalent m artingale
m easure.U nd er the equivalent m artingale m easure, F (t;T) isa m artingale
4
and itsd ynamic s are c harac terized b y the f
ollow ing stochastic d i®erential
2
equation,
·Z T
¸
d F (t;T )
¤
= ¾ d W 1 (t) ¡
±(t;u)d u d W 2¤(t):
(5)
F (t;T )
t
U nd er the equivalent m artingale m easure the d rif
t ofthe c onvenience yield
isrestric ted b y the c orrelationw ith the spot pric e and the requirem ent that
the c onvenience yield equalzero at the m aturity ofthe c ontrac t.3 T he riskad justed d ynam ic softhe c onvenience yield are givenasf
ollow s,
·Z T
¸
dc
(t;T ) = ±(t;T )
±(t;u)d u¡¾ ½ d t+ ±(t;T)d W 2¤(t);
(6)
t
w here W 2¤(t) ´ W 2 (t) ¡¸ 2 (t) and ¸ 2 (t) isthe pric e ofrisk assoc iated w ith
the stoc hastic c onvenience yield .T he im m ed iate payo® to the introd uc tion
ofa stoc hastic c onvenience yield isthat f
uturespric esare no longer perf
ec tly
c orrelated , i.
e. the ad d itionalsourc e ofvariationrelaxesthe d egree ofc om ovement b etw eenf
uturespric esalongthe term struc ture.
E quations (1), (2 ) and (4 ), and the theory ofasset pric ing provid e a
c om plete mod elf
or the pric ing off
uturesand optionsonf
uturesc ontrac ts.
B ef
ore implementingthism od elina hed gingprogram ,w e need to d eterm ine
iftw o sourc esofvariationare su± c ient to m od elthe variationthat isob served
inthe term struc ture off
uturespric es.Ifthe m od elisinad equate insom e
d imension,a signi¯c ant sourc e ofrisk w illnot b e pric ed and not b e hed ged .
For exam ple, inthe ¯xed incom e literature, Canab arro (199 5) show sthat
one-f
ac tor interest rate mod elshave poor hed ging charac teristic sespec ially
ina tw o-f
ac tor ec onomy.
W e c and etermine the numb er ofsourc esofvariationusingprincipalc om ponentsanalysis(P CA).P CA isa purely statistic alm ethod f
or d eterm ining
the numb er ofstoc hastic term sthat generate the variationinf
uturespric es.
W ithinthe asset pric ing f
ram ew ork that w e have used ab ove, there are tw o
2
A min, N g and P irrong (1 995)develop a modeloffutures pricing using the H eath,
Jarrowand M orton (1 992)frameworkwhich leads tothe same stochasticdi®erentialfor
futures pricingwith multiplestochasticstatevariables.
3
T he requirementthatthe futures price be amartingale imposes aconstrainton the
driftofthe dynamics ofthe convenience yield underthe equivalentmartingale measure,
similarin spirittothe H eath,Jarrowand M orton (1 992)restrictions on the driftofthe
term structureofforward rates ofinterest.
5
ob servationallyequivalent w aysofd eterm iningthe numb er ofsourc esofvariation.F irst,ifw e regard the spot pric e asc ontainingthe ¯rst sourc e ofvariation,P CA ofthe c onvenience yield sc and eterm ine the numb er ofad d itional
sourc esofvariation.Sec ond , P CA ofthe f
uturespric es, exc lud ing the spot
pric e, should id entif
y the sam e numb er ofsourc esofvariationasthe ¯rst
m ethod (Cortazar and Sc hw artz (19 94 )).Inthe next sec tionw e d isc ussthe
m ethod ofP CA and itsapplic ationto naturalgasf
uturespric es.
3 Appl
ic ation
3.
1
Data
Daily spot and f
uturespric e d ata f
or the Henry Hub c ontrac t are availab le
4
f
rom Datastream International.
Futurespric esare availab le f
rom the inceptionofthe m arket onApril3,19 90 .Spot pric e seriesare only availab le f
rom
Novemb er 5,1993,c oincid ingw ith the d ate that the pric e w as¯rst pub lished
inthe W allStreet J ournal. W e c onstruc t a seriesofc onstant m aturity f
uturespric esw ith 1 through 12 m onthsto m aturity b y choosing the Frid ay
c losingpric e ofthe c ontrac t w ith expirationd ate c losest to w ithintw o w eeks
ofthe required m aturity.5 T hism ethod issim ilar to Schw artz (1997).
T he naturalgasf
uturesc ontrac t hasthe f
ollow ingc ontrac t spec i¯c ations.
E ac h trad ing unit is10 ,0 0 0 m illionB ritish therm alunits(M M B tu). T he
c ontrac t istrad ed f
rom 10 :0 0 A.
M .- 3:10 P .
M .(E .
S.
T .), f
or the openoutc ry session.Af
ter-hourstrad ing isc ond uc ted via the NY M E X ACCE SS°R
elec tronic trad ing system f
rom 4 P .M .to 7P .M .
, M ond ay through T hursd ay. T he c ontrac t trad esf
or 36 c onsec utive m onthsc om m encing w ith the
next c alend ar m onth (f
or exam ple,onO c tob er 3,1998,trad ingoc c ursinall
m onthsf
rom Novemb er 199 8 through O c tob er 2 0 0 1) although trad ingislight
inthe ¯rst 2 4 m onths.T rad ing term inatesthree b usinessd aysprior to the
¯rst c alend ar d ay ofthe d elivery m onth and d elivery ism ad e at the Sab ine
P ipe Line Co.'sHenry Hub inLouisiana.Ifc ontrac tshave not b eensettled
prior to expiry,d elivery takesplac e no earlier thanthe ¯rst c alend ar d ay of
4
T heD atastream codesforthespotpriceandfuturespricesareN N G and N N G M M Y Y ,
respectively,where M M denotes the month and Y Y the yearofmaturityofthecontract.
Forexample,thefutures contractmaturingin D ecember1 999 is N N G 1 299.
5
H erbert(1 995)has documentedthematurityand volumee®ects ofcontracts closeto
maturity sowe donotchoose a\ one-month"contractthathas less than three weeks to
maturity.
6
the d elivery m onth and isc om pleted no later thanthe last c alend ar d ay of
the d eliverym onth.Alld eliveriesare to b e m ad e at anhourly and d aily rate
of°ow asunif
orm aspossib le over the c ourse ofthe d elivery m onth.
T he c onvenience yield is c alculated b y inverting the theory ofstorage
relation,i.
e.
ZT
ZT
c
(t;T ) ´
c
(t;u)d u= ln(F (t;T)=S(t)) ¡
r(u)d u;
(7)
t
t
w here c
(t;T ) isthe c onvenience yield earned f
rom tto T. Asanapproxim ationto the c ontinuoustime d i®erential, w e analyse the d isc rete, w eekly
c hangesinthe f
uturespric esand c onvenience yield s, i.e.f
rom equation(5)
w e have,
d F (t;T ) F (t;T ) ¡F (t¡1;T)
¼
;
F (t;T )
F (t¡1;T)
and f
rom (6) w e have,
dc
(t;T ) ¼¢ c
(t;T ) ¡c
(t¡1;T);
t ´c
w here ¢ c
otesthe d isc rete change inthe c onvenience yield that isused
t d en
to approxim ate the d i®erential,d c
(t;T).
Inthe next sec tionw e provid e a d isc ussionofthe P CA m ethod .
3.
2
P rincipalCom p onen
tsAn
al
ysis
P CA isa statistic altec hnique f
or d eterm iningthe numb er ofstochastic c om ponentsthat ad equately sum m arize the system atic variationina group of
variab les.P CA isisb ased onthe assum ptionthat there isa ¯nite and unknow nnumb er ofstoc hastic c om ponentsthat generate the variationw ithin
a group ofvariab les.T he ¯nancialec onom ic analogto thisassum ptionisthe
d istinctionb etw eenthe numb er ofsourc esofsystem atic variationofa stock
returnsthat are emb od ied inthe CAP M and the AP T .W hereasthe CAP M
only hasone sourc e ofsystem atic risk,the AP T c anac c om od ate m ore than
one sourc e ofsystem atic risk,b ut issilent onthe numb er ofsourc esofrisk.
Inthissec tionw e d isc ussthe P CA ofc onvenience yield s, b ut the m ethod
also appliesto the term struc ture off
uturespric es.
7
W e c onjec ture that the c onvenience yield sare a linear c omb inationofk
unknow nstoc hastic state variab lesor f
ac tors,i.e.
¢c
Z t+ "t;
t¡¹ = L ¢
(8)
w here ¢ c
gesinthe c onvenience yield sinperiod
t isthe m £1 vec tor ofchan
t,¹ isthe m £1 vec tor ofexpec ted valuesofthe changesinthe c onvenience
yield s,L isa m £kmatrixoff
ac tor load ings,Z t isthe k£1 vec tor off
ac tors
at period tw ith expec tationz ero and "t isa m £1 vec tor ofrand om shocks.
B y c onstruc tion, the c ovariance m atrix ofZ t isthe id entity m atrix, and Z t
isuncorrelated w ith "t.
B ased onthe assumed f
ac tor struc ture in(8), the c ovariance m atrix of
the c onvenience yield s,§ ,c anb e d ec om posed asf
ollow s:
§ = LL 0+ ª ;
w here c
ov("t) ´ª .G ivenanestim ate of§ ; P CA provid esanestim ate ofL.
How ever, P CA isa d ec om positionofthe c ovariationofthe m c onvenience
yield s,thusthe estimate ofL isa®ec ted b y the relative siz e ofthe ind ivid ual
variances.Instead ofd ec om posingthe c ovariance m atrixw e pref
er to use the
c orrelationm atrix, - , b ec ause allthe variab lesare sc aled to have the sam e
variance.W e proc eed asf
ollow s: ¯rst, w e ob tainthe eigend ec om positionof
the c orrelationmatrix,
- = U 0DU;
w here U isanorthonormalm atrix c ontaining the eigenvec torsand D isa
d iagonalmatrixw ith the eigenvalues,d i,ind esc end ingord er d ow nthe d iagonal.T he sum ofthe eigenvaluesisa m easure ofthe totalvariationw ithin
the group ofvariab les.T he proportionofvariationexplained b y each principalc omponent c anb e measured b y the ratio ofeach eigenvalue to the sum .
O ne rule-of
-thumb f
or d eterm ining k, the num b er ofstochastic c om ponents
that ad equately sum mariz esthe variation,isthe ratio ofthe sum ofthe ¯rst
k eigenvaluesand the sum ofallthe eigenvalues.Like the c oe± c ient ofd eterminationor R 2 inregressionanalysis, thisratio isnot a statistic altest,
b ut a m easure ofthe variationthat isexplained b y the ¯rst k c om ponents.
For exam ple,Litterm anand Sc heinkm an(199 1) and Cortaz ar and Schw artz
(19 94 ) ¯nd that that f
or k= 3; m ore than95% ofthe variationisexplained .
Although the principalc om ponentsare purely a statistic alc onstruc tion,w e
8
c ananalyz e eac h c olum nof U to d eterm ine the w eighting that isattached
to eac h ofthe originalm variab les.T he analysisofthe w eightshelpsto ad d
some ec onomic c ontent to each c om ponent.For exam ple, inLitterm anand
Sc heinkm an's(199 1) analysisofthe term struc ture theyid entif
ythe ¯rst f
ac tor asa levelf
ac tor that hasthe sam e e®ec t ac rossm aturies,i.e.a positive
shoc k to the ¯rst f
ac tor increasesallinterest ratesb y the sam e increm ent.
T he sec ond and third f
ac torsare related to the slope and c urvature ofthe
term struc ture,respec tively.
T he principalc om ponentsare linear c omb inationsofthe originalvariab les:
Z = X U;
w here X isthe n £m d ata matrixsuc h that c
orr(X ) = - .T he ¯rst principal
c om ponent isthe linear c omb inationofthe c onvenience yield sthat explains
the greatest proportionoftheir variation. T he sec ond c om ponent isc onstruc ted to b e orthogonalto the ¯rst c om ponent and explainsthe greatest
proportionofthe rem ainingvariation.T hisproc ed ure isrepeated m tim esto
prod uc e m orthogonalprincipalc om ponents.Inessence,w e have c onstruc ted
a set ofm orthogonalprincipalc om ponentsf
rom m c orrelated c onvenience
yield s.
Insum m ary,although w e require m stoc hastic c om ponentsto explainall
the variationinthe m c onvenience yield s, P CA provid esa m ethod ofred uc ing the d im ensionality ofthe prob lem w henthere are k < m stochastic
c om ponentsor sourc esofvariationthat explaina large proportionofthe variationofa group ofvariab les.R eturningto the ¯nancialec onom ic analogy at
the b eginningofthissub sec tion,the kstochastic c om ponentsare analogous
to the k sourc esofsystematic risk inthe AP T and the rem aining m ¡k
c om ponentsre°ec t unsystem atic risk or noise.
4
4.
1
R esul
ts
Forw ard R ates
T ab le 1 c ontainsthe c orrelationm atrixofchangesinthe f
uturespric esf
or the
period J anuary 31,19 92 to Novemb er 2 0 ,199 8.T he d ec rease inthe c orrelationc oe± c ientsasthe time intervalb etw eenm aturitiesincreases, ind ic ates
9
that w e need more thanone sourc e ofsystem atic variationto m od elthe term
struc ture off
uturespric es.T ab le 3 c ontainsthe perc entage ofvariationexplained b y the principalc om ponentsofthe c orrelationm atrixinT ab le 1.It
isim m ed iately c lear that w e need 5 principalc om ponentsto explainat least
90 % ofthe variationinthe f
orw ard rates.
T urning to F igure 1, w e see that the ¯rst c om ponent isa \level" e®ec t,
i.e.it e®ec tsthe term struc ture off
uturespric esunif
orm ly ac rossm aturity
and ac c ountsf
or 52 % ofthe variationinthe f
utures pric es. T he sec ond ,
third and f
ourth c omponentsac c ount f
or 15% , 13% and 8% ofthe variation,respec tively, and c apture the seasonality inf
uturespric es.T he sec ond
c om ponent c apturesthe d om inant annualc yc le ofthe naturalgasm arket,
nam ely,the d emand f
or gasinthe W inter Heatingperiod .T he third c om ponent also re°ec tsa seasonalc om ponent and the f
ourth c om ponent c aptures
b oth the W inter heatingand Sum m er c ooling.Insum m ary,the P CA ofthe
term struc ture off
uturespric es id enti¯es the d om inant levele®ec t ac ross
m aturitiesand the strongseasonalc yc le ind em and .
4.
2
Con
ven
ien
c e Y iel
ds
T ab le 2 c ontainsthe c orrelationm atrixofchangesinthe c onvenience yield s
f
or the period Novemb er 5, 1993 untilNovem b er 2 0 , 199 8. W e om it f
our
w eeklyob servationsf
rom Feb ruary2 to Feb ruary2 3w hena W inter c old spell
c aused the spot pric e to spike to $ 12 . E xc ept f
or the c onvenience yield of
the one-m onth c ontrac t,the c orrelationsare allgreater than0 .
84 .It isw ellknow nthat the one-m onth c ontrac t hasm ore id iosyncratic or unsystem atic
variation(Herb ert (199 5)) so the large positive c orrelationac rossm aturities
suggests that there is one d om inant stochastic c om ponent inc onvenience
yield s.T ab le 3c ontainsthe perc entage ofvariationexplained b ythe principal
c om ponentsofthe c orrelationm atrixinT ab le 2 .At m ost three f
ac torsare
need ed to explain98% ofthe variationinthe c onvenience yield s.T he ¯rst
c om ponent ac c ountsf
or 93% ofthe variationw hile the sec ond and third only
ac c ount f
or 4 % and 1% ofthe variation,respec tively.Aninterestingf
eature
ofthe ¯rst three f
ac tor load ingsisthat they are id entic alto those ob tained
b y P hao (1998) inthe principalc om ponent analysisofU .S.T reasury yield s.
T he ¯rst c om ponent isa \level"e®ec t that a®ec tsthe f
uturespric esunif
orm ly ac rossthe term struc ture.T he sec ond c om ponent hasb eenc alled a
\slope" e®ec t: a shoc k to the sec ond c om ponent c ausesc onvenience yield s
10
at the short end to d ec rease and those at the long end to increase, thusinc reasing the slope ofthe term struc ture. T he third f
ac tor isref
erred to as
a \c urvature" or \tw ist" e®ec t: a shoc k to the third f
ac tor c ausesthe term
struc ture ofc onvenience yield sto increase at the short and long end , w hile
d ec reasingat the interm ed iate m aturities.
None ofthe ¯rst three f
ac tors have any seasonalc om ponent to them .
Although the f
ourth f
ac tor d oeshave a seasonalc om ponent, it explainsless
than1% ofthe variationso it playsa verysm allrole.T husthe resultssuggest
that ifw e are goingto m od elthe seasonality inthe term struc ture,w e w ould
have to includ e it inthe spot rate (P ilipovic (1998)) and not the c onvenience
yield s.
5 Con
cl
usion
Ad equate m od elling ofthe system atic risk inthe term struc ture off
utures
pric es, exc lud ing the spot pric e, requiresat least 5 sourc esofvariationand
they are allseasonal.Ad equate m od ellingofthe system atic risk inthe term
struc ture ofc onvenience yield srequiresone d om inant sourc e ofvariationand
at m ost 2 other sourc es.T he ab sence ofa d om inant seasonalc om ponent in
the c onvenience yield ssuggeststhat a sim ple m od elofthe spot pric e and
the term struc ture ofc onvenience yield sc anb e c onstruc ted b y includ ing a
seasonalc om ponent inthe d rif
t ofthe spot pric es(P ilipovic (199 8)) and the
d om inant levele®ec t ofthe c onvenience yield s.T he em piric alperf
orm ance
ofthismod elislef
tf
or f
uture researc h (B eeld ers(199 9)).
R ef
erences
[1] Amin,K aushik,Vic tor Ngand S.CraigP irrong(19 95) \ValuingE nergy
Derivatives,"inM anagingE nergy P ric e R isk,R isk P ub lic ations.
[2 ] B eeld ers,O w en(199 9) \T he P ric ingofNaturalG asFuturesContrac ts,"
w orkingpaper,E m ory U niversity.
[3] B lac k,F isc her and M yronScholes(1973),\T he P ric ingofO ptionsand
Corporate Liab ilities,"J ournalofP olitcalE conomy,81,M ay-J une,637659.
11
[4 ] B lac k, F isc her (19 76) \T he P ric ing ofCom m od ity Contrac ts," J ournal
ofF inancialE conomic s,3,1/2 ,p.167-179 .
[5] B liss, R ob ert R .(19 97) \M ovem ents inthe T erm Struc ture ofInterest rates," Fed eralR eserve B ank ofAtlanta E conom ic R eview , Fourth
Quarter,p.16- 33.
[6] Canab arro,E d uard o (1995),\W here d o O ne-Fac tor Interest R ate M od elsFail?" J ournalofF ixed Incom e,Septemb er,31-52 .
[7] Cortaz ar,G onzalo and E d uard o S.Sc hw artz (199 4 ) \T he E valuationof
Comm od ity Contingent Claim s,"J ournalofDerivatives,1,2 7-39.
[8] Harrison, M ic haelJ .and David M .K reps(1979 ) \M artingalesand Arb itrage inM ultiperiod Sec urity M arkets,"J ournalofE conom ic T heory,
2 0 ,381-4 0 8.
[9] Harrison, M ic hael J . and Stanley P liska (1981) \M artingales and
Stoc hastic Integralsinthe T heory ofContinuousT rad ing," Stoc hastic
P rocessesand T heir Applications,11,2 15-2 60 .
[10 ] Heath, David , R ob ert A.J arrow and And rew M orton(19 92 ), \B ond
P ric ingand the T erm Struc ture ofInterest R ates: A New M ethod ology,"
E conometrica,60 ,p.77-10 5.
[11] Herb ert, (19 95), \T rad ing Volum e, M aturity and NaturalG asFutures
P ric e Volatility,"E nergy E conom ic s,17,4 ,2 9 3-2 99.
[12 ] K ald or,Nic holas(1939),\Spec ulationand E c onom ic Stab ility,"R eview
ofE conomic Stud ies,O c tob er,VII,1-2 7.
[13] Litterman,R ob ert and J ose Scheinkm an(1991),\Com m onFac torsAf
f
ec tingB ond R eturns,"J ournalofF ixed Incom e,1,J une,p.49 -53.
[14 ] Ng, Vic tor K .and StephenCraig P irrong (19 94 ), \Fund am entalsand
Volatility: Storage,Spread sand the Dynam ic sofM etalsP ric es,"J ournalofB usiness, 67,2 ,2 0 3-2 30 .
[15] P hao, W esley (199 8), Ad vanced F ixed Incom e Analytic s, pub lished b y
Frank J .Fab oz z iAssoc iates.
12
[16] P ilipovic , Dragana (1998), E nergy R isk: valuingand M anagingE nergy
Derivatives,New Y ork,M c G raw -Hill.
[17] Sc hw artz, E d uard o S.(1997), \T he Stochastic B ehavior ofCom m od ity
P ric es: Implic ationsf
or Valuationand Hed ging", J ournalofF inance,
LII,3,9 2 3-973.
[18] Simons,How ard L.(199 7) \It'sa G as,"Futures,J une,4 4 -4 8.
13
T ab le 1: T he CorrelationM atrixofthe perc entage c hangesofthe FuturesP ric es
M aturity 1
2
3
4
5
6
7
8
9
10
1
1.00 0 0 .895 0 .752 0 .
60 6 0 .
513 0 .
4 37 0 .
4 09 0.
384 0 .34 8 0 .34 2
2
1.00 0 0 .914 0 .
715 0 .
54 3 0 .
4 33 0 .
4 02 0.
4 0 3 0 .374 0 .32 8
3
1.00 0 0 .
879 0 .
683 0 .
4 89 0 .
397 0 .
4 0 1 0 .40 1 0 .354
4
1.
000 0.
887 0 .
64 2 0 .
422 0.
362 0 .372 0 .362
5
1.
000 0.
851 0 .
567 0 .
39 5 0 .318 0 .32 4
6
1.
000 0.
82 9 0 .
562 0 .339 0 .239
7
1.
000 0.
82 9 0 .485 0 .230
8
1.
0 0 0 0 .80 1 0 .44 6
9
1.00 0 0 .767
10
1.00 0
11
12
14
11
0 .41
0 .35
0 .31
0 .33
0 .33
0 .26
0 .14
0 .17
0 .38
0 .75
1.00
T ab le 2 : T he CorrelationM atrixofthe Changesofthe Convenience Y
M aturity 1
2
3
4
5
6
7
8
1
1.00 0 0 .94 7 0 .879 0 .
833 0 .
811 0 .
797 0 .
787 0 .
776
2
1.00 0 0 .968 0 .
92 5 0 .
892 0 .
874 0 .
866 0 .
862
3
1.00 0 0 .
9 80 0 .
9 52 0 .
92 8 0 .
9 16 0 .
914
4
1.
000 0.
9 88 0 .
9 65 0 .
94 6 0 .
939
5
1.
000 0.
9 89 0 .
9 69 0 .
956
6
1.
000 0.
9 90 0 .
976
7
1.
000 0.
99 2
8
1.
000
9
10
11
12
15
ield s
9
0 .759
0 .84 8
0 .90 7
0 .935
0 .94 8
0 .962
0 .975
0 .99 1
1.00 0
10
0 .757
0 .84 0
0 .89 9
0 .932
0 .94 6
0 .954
0 .961
0 .975
0 .99 2
1.00 0
11
0 .77
0 .84
0 .89
0 .92
0 .94
0 .95
0 .95
0 .96
0 .97
0 .99
1.00
T ab le 3.P roportionofVariationE xplained b y eac h P rincipal
Component
Forw ard R ates
Component Ind ivid ual Cum ulative
1
52
52
2
15
67
3
13
80
4
8
88
5
6
94
6
2
96
> 6
4
10 0
16
Convenience Y ield s
Ind ivid ual Cumulative
93
93
4
97
1
98
<1
<1
<1
<1
10 0
F igure 1:
17
F igure 2 :
18