.~innncinl Risk i'\fanagen1cnt fol· Australian W'hcat Gro\ve•·s

Greg Hertzler and Amanda Coad

Agticultural and Resource Economics 'I11c University ofWcstem Australia

Nedlands, WA 6907

How do you describe a revolution--"lt was the best of times. 1t was the worst of times?''

Perhaps tillS ts tme for Australian wheat growers Begmning m 1980, growers were guaranteed a

nummw11 price by the govemmcnt (Bardsley and Cashin). ln return, they \''ere required to sell all

t11etr wheat to the Australian \Vhcat Board (A \VB). ln 1989, the \\'heat Marketing Act removed

the guaranteed minnnum pnce and, followmg the Royal Conumssion into Gram Storage, Handling

and Transport (1\kC'oll), the domesttc wheat market was deregulated and growers could sell to

anyone withm Australia Fraser fmmd that the coefficient of vanatton m wheat prices rose from

8 2% pnor to deregulation to 17 4% after

Not all nsks af'c pnce lisks. 11m yield of whe.at is affected by climactic conditions of the

growmg season and vanes constderably from year to year, For example, Australian wheat

productton dropped from 16 million tonnes in I 993 to 9 million tonncs m 1994 due to widespread

drought (A\VB, 1994). Of course, some growers harvested nothing in 1994. Over the last 50

years. the coefficient of variation of Australian wheat yields has been about 29% (Campbell and

Fisher), 1l1is is prodm .. tion risk Variation in tllC climate also causes variations in the quality of

the whe.at (Anderson et a{.) A dry finish to the season may limit starch fom1ation more than

protein formation resulting in a higher protein content in the grain (Condon). A summer rain on a

dry crop will potentially downgrade nonnally millable wheat to feed wheat by activating enzymes

which reduce the elasticity of protein in the grain (Perry a11d Hilhnan). 11tis is quality or grade

risk.

Now. however. growers have new opportunities to hedge their risks in ways that were not

possible before. Although tlte A\VB has offered hedging.contracts for five years, ti\k l11cmagern.ent

Invited. paper presented at the 40th Annual Conference ofthe Australian Agricultural and Resource Economics Societyt University ofMetboumr~, Victoria, .February 1996,

2

is a new in11ovation for many growers and is bemg adopted slowly over time .(Rogers). Several

factors. mcludmg education, wealth and risk attitudes ''~ll affect the adoption of innovations

(Diilon and Sc..'lndizzo, Bond and \Vonder, M01ison, Ncwbeny). Most fam1ers are risk averse

(Bond and \Vender. Franctsco and Anderson) and should be willing to hedge. accepting lower

u1comcs on avcrngc IJl c..xchange for less nsk

Growers have several hedgmg strategies they can adopt l11oy can deliver at harvest for

the market price wtthout any hedging. l11ey can contract wheat to one of several A\VB pool~ for

ditlbrcnt grades of whc..1t. Growers arc then paid 80% of tho estnnated retum from a pool,

followed by a post-harvest payment and a final payment up to 18 mc.ntlls late1 Pools may even

out the highs and lows oftl1e marka 1s a hedge against price risk (Ryan). Most grain in Australia

1s pooled and the total pool payments hav~:. fallen below the original estimate only three times since

! 947 (A\VB, 1 Q95a) Growers can also warehouse wheat with tl1e A \VB or store wheat on-farrn in

thetr 0\\11 silos and sell sometime after harvest (ANZ f\1cCaughan). Other strategic~ are to adopt

forward contracts for a fixed pnce and a fixed grade of wheat, or forward contract£ for a fixed

pnce but multi pi~ grades of wheat {A \\• B, 1992). These contracts have no fees to growers. The

contract price that growers will receive is based upon the daily indicator price set by the A\\TBs

New York office. TI1is indicator price is detennined from the U.S. price for wheat futures and the

exchange rate (A \VB, 1995a). Fixed grade contracts are available all year round and multi grade

contracts are available from April until harvest (AWB, l 995b). Both fixed grade and multi grade

contracts are hedges against price risk, but multi grade contracts also hedge against grade risk w1th

bonuses or discOJ.mts for up to six oilier grades.

From 1 Q93 through 1995, growers could adopt theminimwn price contracts offered by the

AWB. Before these were discontinued due to lack of demand, growers could purchase minimum

price contracts which gave tl1em the right but not the obligation to sell their wheat at the minimum

price guaranteed by the contract. TI1e value at which minimlU11 price contracts could be purchased

was detem1ined by tl1e price of put options on futures and the minimum price growers wolllcl

receive for their wheat was set below the contract price for fixed price contracts (Ryan). Another

alternative growers can adopt is futures contracts on tlle U.S. wheat futures marktt, These will

require ~xpertise, effort and time to monitor tlle markets a11d. capital for making margin calls

3

(Petzel. Morison) Still anoUler altemativc is options on futures Growers can purchase call

options which gave tJ1em the right but not the oblig..'ltion to purchase futures contracts at the

exercise pnce guaranteed by the contract. BoU1 futures and options on ft;turcs arc subject to basis

nsk The basts ts the difference between U1c price of wheat in Australia and the futures price in the

U.S It has three components dtstance bctwLX.'fl markets, difference 111 twcs of wheat and

exchange rates between Austrnlian and U.S dollars If the basis changes tulcXpe<..icdly, there is

basts nsk. Basts nsk may be a deterrent to the adoption of fututcs and options on futures because

only exchange rate nsk can be hedged (1110mpson and Bond, Bond et al., Andersen). One way to

hedge exchange rate risk ts to use a curn .. 'ltcy futures contract and sell U,S dollars forward.

In Uus paper we evaluah~ the hedging strategies available t.o wheat growers, First we

constmct a model of optunal consumption. productiOn and hcdgin~ under risk which differs from

other models m allO\vltlg assets hkc wheat stocks, mint mum price contracts and options on futures

to be tncludcd l11en we denvc optimal doossons by a risk ... avcrse grower, analyse the individual

strategies that nught be adopted and dt.tenninc an optnnal pmtfolio amongst all strategies. Finall.y,

we denve a new optton pncmg mt."thod to show how mimmun1 ptice contracts should be valued.

Dynamic ~lodel of Decisions Under Risk

A model of consumptiOn, production and hedging decisions tmder risk will be fairly large.

In constructing and analysing the model there arc at least three methods to choose from: expected

utility theory, mean-variance analysis and stochastic dynamic programming. Expected utility

thea!)' can become intractable for large models. Mean-variance analysis is easy to apply to large

models, but restricts tl1e underlying probability distributions of the random variables. Both

methods are static, however, which hinders their applicability to dedsicns about wheat stocks,

mmimum price contracts and optiot\s oh futures. These are assets which change over time,

Stochastic dynamic programming overcomes the hmitations of expected utility theory and.meall•

variance analysis in modeling the altcmativcs available to wheat gtowers.

Hertzler developed a stochastic dynamic programming model for decisions in agricultur~~

In this model, a grower is assumed to behave as ifthe expected ·utility ofC(Jnslimption is m~xirnistXt

over time., ~ubjcct to .a budget constraintfor the change in wealth.

Financial .Risk M~nEigement

./( IJ'0 ) = max,E{!" "1'U(Q)dt}

subject to.

dW = (5(W. l'.S. C. F. G. J\1, N ~Ki Q}dt + tT(Y,S~ C. F, G, A·l. N ~K. Q}tl:..

Gtvcn inuj:lJ wt>..1lth, Wo, a grower who nmkc.<t opwmtl d<.,.>ctswns wtll have n ~xpccttxl utility, .J~

over n fanning c.urt.'Cr, T Expt.--ctcd utility ts the cxp<.X.'tt.Xi S\tUl of utihty, U, gn1rllxi from consuming

the quantity. Q. m ca<;>h year discounted at th9 rntc of time prcfcrl"~1cc, p TI1c budgt't constrau\t

gtvcs the change in wt.ulth, dW, ns the cxpcct.l'<i change. &li plus an error tcnn, ark. l11e expected

change ts a funct.t011 of current wealth. W. and both the cxpt'Ctcd chnngc and dtc error term are

f~mct1ons of ytcld. Y, storage, S, fh:txi grade contmcts, C. ftltutcs contracts. ~~~ muftt .. gradc

contracts, G. mmimum pncc contrnt-ts. At/. optJons on futures. N. currency futures, K~ and

consumption, Q 111c error tcnn 1s the product of a ve<,:tor of standard d~vtations, u. and a vt't."tOr

of normally dtstributt'd whtte notse over time, li:, Time subscnpts on all the variables have bt.•cn

onuttcd for brevity

11w major assumptton of this model ts that a grower's cart-er ts sufftcit'fltly long.

approxmmtely 30 y~1rs, for the central hnut. tht'Orcnt to hold and for mc.ans and vananccs to

become sufficient stattsttcs. regardless of the undcrlymg probability dtstributions. Higher moments

nrc not required and nonnal distributions can be assumed \\~thout. loss of _generality. \Vith thts

assumptton. maxunismg expected utility over a grower's caroor lS equivalent to maximising the

Hamilton .. Ja<:obi-Bellman equation in each producuon season 1l1e Hamifton•Jacobi-Bdlman

<XJUation Is ... partial differential t-quation in time and wealth which mtcgrates to CA:luar e.xpccted

utility.

11ie expression to maximise in brackets is discolUlteQ current utility of cottsmnption phls th~.

marginal utility of wealth, c11 J c-'lV, multiplied by the expect<;d change in W931th, 8, plus one-half

the change in the marginal utility of wealth r7"J/ r;JJ'2 , multiplied ·bytheicovariaJlCC matrix a!Jd~,

which is calculated by .squaring the error term for the change in wealtJt, Optimaljty conditions ate

fotmd by differcntiatingthe expression in br~ckcts at)d setth1$t11c .result t() zero.

5

(1)

·n1c tlrst optunahty condllmn •s for consuntptmn and th~ Sl>eond ts for• product.JOtl at\d lu,-dg1ng

dcctSIOflS To ~unphfv notntmn .. R. 1S dcfinl'<l ns t.ht1 coc01etl.,lt or :lbsolute nsk nvcrstQJl wludl

A more spc..-ctftc f'onn for tho changc.1 m \Vt.~1Hh 1ncludcs capttnt g.'m\s or lo~sus. on assets,

losses fn:uu dl>gra<l~Uon t'lf whc.ilt. St()()ks m stol'ngc, profit from product.lon and IK'<igmg doctsions

over tho s~sou. and error tcnns (S~-o the tt"clmlcil nppt.~1daccs awnlable from tho authors.)

f2) dW '!'!' '~1 (W nuH - nN ·~ pS} + dmAf + dnN + dpS- pl'i,~Srit- pr:rsSriZs

+[(P + g)r ... x(r) +(t"-- p)C+ (l-l)F +UJ- p- g)G +(k- k)k •,';S + pS -qQ]dt

+( dp + (1.~) r +( p + .t: )til' + (tip+ c(~}d>' - fiJiC -411::- (tip+ dg}G - dkK ·• dqQ

where 11 ~ U;S ~ O~pS;::; 0

1l1c first hue of tlus t.."iJ~tauon has C.1JJital sums and losses mtnus the V()lm~ of whc:tt dt.ogradcd m

storage 11u~ s~ond hno has profits from prtxim:tton ami hcdgtng. the costs of phys1c.'\Uy storing

whcat5 a non ncgauv1ty condttmn on the stocks U1 storage and cxpcs1dtturcs 011 coos.umphon, 11lc

thJrd hnc has the change m profits and c.xpcnd1tUrcs over the season~ and the fourth lmc has

complema1tnry slackness conditions whtch restrict, the stocks of whc.'lt in storage to be no!\

Quarihttcs m equation (2) ate dcs1ot«< by upper case letters and prices by lower casQ

letters In thK: .first line. a pottion of total wMith, W, is h¢1d it1. minirmmt price <:Qntr~cts with vahm

m nnd quantity A.f ot>tJons on futures With pJice .nand qllantt~y N ;.rid st~ks ofwh¢.1t WJth pric¢p

and quant.ity S 111e r(.•mail1dcr of total wc.1ltht W- mAl ... nN .. p$ is held ir) assets which rcwm ~1c

sarnc rate .as bond~, db l b Bcc~msc inflation and infl:iHoo tisk ar9 included ht the price. of

consuotptlorl goods, there is l1o· need to c.1l¢ulatc .a r~il 'f~tq; ofir1tcrctt ;~nd;<a .$rower Cc1tl ;1lways

chO<lsc bond$ as a risJ<4'roo. invc5~1tC.lt J\. grow¢r .cat} :d$o c:lt(Klsc capital gains or H>$~cs oo

6

mmmmm pncc contracts. dmo~\l, opuons ot\ filt\lrcs~ tlmV. ::u1d st.oc.ks m storagl\ dpS~ gtVerlthat

stocks arc expected to dt:'grado u • vttluo b~· pc}.;S, wnb crr·or Jur.,<I'Z.~.

In tho second I me of cqnahon (2);, wh~,t yJeld, J'. wtll sell for market p.nec~ pf tf it is Qf

Aust.mhnn Standard \Vlutc lAS\\') grade bqt may nttract a b011us or dtscount, g:, tf It ts of higher or

lower qualtty Yscld Will cost :\:( n to produce Some of the y1cld may be hedged wtth a forward

contract. C. m a fixoo pncc c• ·n1c contra~ ~~for AS\V alld futflflmg the ccmtrnc.1 ts equtvnlt>rlt to

bu;1ttg, th-.• quantity C at marker pnc<.~ p nod rcscllmg tt at the cotttmct prn~c c If the coHttact

tmcc cxc,'\:xfs the markt..-t rmcc there wttl bt~ a gmn on the ht:'(:isc Sumlarly. sonm of tho y1cld may

be hedged wHJl n ft}turc.«; comrnc.t~ F' 'I11c g,rowcr prtnnJSCS t.o sell tllch4.ng,,i qttnntit.>· of whc.,1t nt

pnct} l but rather thrm dchvcntlg. the wheat. the .grower ~'lnccls the onguml comrnct by buymg, at.l

oflst~t.mg contract at the futures pncc. l Sonw of the ~'lcld may he ht.'d£;~-d wtth a mult1"~8J1ldc

contract. G. at a fha.'Xi pnec. g a rs the quautny of ytdd whtdt no lon~cr st:Hs at the market prtec

plus grade boJtus. p .. g, bur sells at the flxtxi pnco. g. mstc.ad J;un,rcs contracts- are traded

ovets(>.as nnd lurv<.~ basts r1sk Basts nsk ts duo to dftfcralccs m locauon and types ofwhc.'it and to

exchange mu.~ nsk 111o grower n41Y h(.xi£,~! dns exchange rme nsk wtth a currctle}' futures

contract .• K ht>:dgcd at exchange rate k nfld offset at exchange r="te k FmnHy. the grower stor($

the qu;mt.uv of whcnt S at a cost of s per mut where S t.S non n~1u vc and J1 tS tho L~grnna,c

ntuluphcr on the non m.,-gaUVtty constmmt. and consumt"S g and pays the pncc; q

11tc thrrd hne of l'quat.ron (2) ts tho stochastic dcnvattve of profHs rn the second fmc (S~a

thu tcchmcal appendices ) ll1c nsks arc cont~unJ.Xi m tl1c st.ochastsc d•flcrcnuals tlpt tlg1 dl'~ dJ. ilk

and tlq for pncc. grade. yu::ld. fmurcs price, cxclmngc rntc, and mflation risks, Costs of

production, xO'). storage costs sS and complementary slt~ckne$s condition pS arc UltUrtcd at the

bcginnm~ of tl1c product1on sc.1son as,d. hence, aro a non stochnstic fiu\ctions of Y,. ar1d S, which

nrc the }'lcld and stock.;; at the bcgtm1ing. ofthc s«tson

A grower will fomt expectations about prices and· yield, but the otltcomC> wiU differ frcun

expectations. For example, the obscrvoo markel' price for wh~t aUh· beginning ofthe prQduct.lP'il

scasot.l IS p. the at:t.nal pri~ at: tho .end ofthc. season will be p 1· dp! andithe p.riceiliat. is cxpc~:u,'<i

by tho grower is p + lt(dp}. 1lu1s, ~toclla~tic differential oqt1atl0Jl$ for· pri~ CSlO ®.used to dcfh1~

actual and expected prices. It is typical h1 .firiatl~ to spoclfy ciiff'crendctl .cq~1atiQil~ \VftielJ1 if

7

pnccs 1s nofmrllly dtstnbntoo ::md prices thcmsclvt-s arc always positiVe.

(3) db= brr/1 ~·. ··~&;

dp ;-:; pli l,dt + pr::r,,d= p

dk = kli kflt ·t- kcr 4 dz•

dq = qt:5.-,dt +qCT,{I::1J

t(g = (T ~,d:. s

d}t = }'cr,dl1

d.l = p8 rdt + pa /1::. 1, +(p -f)dr +(p- f)li,dt +{p- /)a 4th•

f • 1 iin d r m = fll(.l ,/ 1 + . pa" :. "

rjJ

... ttn /ln . ~ 11 r~ m l 2 where 111(5 = .... + . p(5 + /.2 __ , ,e p (1' rn r/ /tJ I' i}')l p

dn = nc5,Jdt + ;,f p(a ,dz 11 +a ~.d: 1 )- m'i ~,dt-11ff,d:. 4 . rr: ttl ( ( • ., ) ) 11 ril-n 2{ .., 1) v·.rhcre m5, = -.·· + . p I + <~ P + <5 , - f + h ·;;::·r p a;, + a l a lf t;,

In tJ1ese dtffcrcnttnl lxtuauons. the expected changes are the terms multiplied by dt and the .ctTors

3rc tonns multlpht'd by d:. for pnccs or· dZ for the quanuty of yield. TI1c rctut'tl on nsk-frcc bonds,

r, tS known at the bcginnmg of the production season and the price, h. is not stochastic 111e

market pncc for wheat, p, the exchange rate. k, and the price of consumption goods, q~ cad1 h:w~

an expected change and an error term. TI1c grade bonus, g. and the yield, .Y, have no c.xpcctcd

change. only error tcnns. 1l1e grade bonus is tu1iquc in bcirtg nonnally distributed and can become

negattve as a discotmt for poor quahty wheaL

1l1c futures price, J, is a function of the 11mrkct price and the exchange rate and its

differential equation is dcnvcd from tJ1cse. (Soothe technical appendices.} The difl'erooc~ bctvv001

the market price and the futtu'cs price, p -f, is .the. basis. 1l1e futures prico is expected ·to chaog'!

v.~th the market price. but al~o to close the ~1p bctwca1 itself and the market price, cHminating dle

basi&. 111c error tenn [p • .flcr~,tf=~ is the basis risk. Basis risk is as~\.lmc.xi to be ~used solcy by

exchange tate risk whid1 C.1t\ be h¢dgcd .by futures contracts. Other components of ;b~sis risk arc

dista11cc and. type differences which C.'UlllQt be hf.'Qgcd.

Financiai··Ri$k. M$n~g~rnent

v '. ',- '' ' ~· ·,'' .:: :~::.:;:

;; •.. ;.,\.;. '·'·~"····"' .. : ... ,:", .. L:::.~;.,,,';,j .••. ::: .. ~ ·~·~··'·; i;:.,:.;:s·;,,;,i; ,,, .. :

8

The val uc of minimum price contracts depends ltpon the 1narkct price for wheat, p; and 'the

tin1e to matunt.y, r • t. where r 1S a fixed maturity date in the future.. In other words~ the value of

minimum pncc contracts is tl1e function, m(t,p). Its dttferenbal l'qUation is found by stochastic

differentiation. (Sec the tt>chnlcal appendices) BL>C.ause m 1s not11incar~ its cxpecti:.'d change, rm)".n,

dl'fK'nds upon the axpectcd change m the market price, p(~,_, ~( \Veil as the variat1ce ofthe market

pnce, p2a~ Its error tenn ts a hcteroskcdasbc transfonnation of the error term for the pnce of

wheat. (Sec the tl>chmcal appendices ) Snnilarly, the price of opttons on futures depends upon the

tune to matunt.y and the futures pncc It 1s the fum.tmn, n(IJ) Like fiiturcs contracts. optmns on

futures are traded m the U S and the pncc of options on futures has exchange rate nsk

Substttutmg the diffcrcnttal equations m (3} into the change m wealth m equ;;ttton (2) .gwes

the final fonn for the change m wealth For case in mterprctmg tln~ ophmaht)' com::iltions to follow.

all covanances, for example dpdY. are assumed to be zero except that between the yteld and the

grade bonus. In this case. dl'dg equals }' a}·a:~(l)1~, where (J),~ ib the correlation coeffic~ent bl't\veen

ytcld and grade

(4) dW ::-. [rw + n~<,)m - r)M + n(t5n • 8~. - r)N + {P{li P- t5 9 - r}-· s+ p)s + (1~1 + t~ ,)+!! + a,a KlOt~ )Y- X(}')+ (c- p{l + £5 p ))c + (l- p(l + c'i p )- (p- /)6k )F

+ (g- (p(l + 8 P )+ g))G + (f ~ k(l + /)t· )}K -q(l + liq }Q J'' + p[Y + S- C- F -- Ci + £ . ..!.;, Arf + a.

1 N]u dz qJ cr p p

-[(p- J)F +( 11- ~ pN )+f<K }-,dz,

+ (p+ g)l' arciZ1 + (J'- O}u fdz8 - pSa.'iciZs -qQa qdzq where:. p?: o~s;;: (),pS = o.

l11e expected change. in wealth consists of the temls in the first three lines \Vhich are. all muttipiioo

by dt. fn the optimality conditions of equation (1)) this e>,:pccted chaoge.in wealth is b: (S) li = t'W + 11~&m- r)M +1~8n -(5'4 - r)N+{p(o"- 88 .... r)-s+ Ji)S

+(p(l +op)+ g+aru,(l) 1x)~' -x(J'}+{c -p(t +o"))c+(J -p(t+8 P)-(p- f)ol)b"

+(g-{p(t +op)+ g))a+{f- k(l+a4))K --q{l+aq)Q where: Ji ~. 0; S ~ 0; p~ = 0.

The expected change in wealth cor1tains aU ·the expected changes .in pric~ f:fom .~uatiotl 0.). llJe

9

nt:lrkct pnce the grower cxpt~s to rcccavc ~t the end of' the sca~orl as p( J + b~), the e:~pcctcd

c~dmngc rate ts k( I + (t). at1d the cxp~"tc.xf price of consumptrcm is q(l + ,-~1} hl addition~ the

cxpt'Ctcd rcvt'flliC from product10n, (p( I ,., <)#) + g + rt~•o;_.w,~}J'. tllChldt.'.S the covariaucc between

ywld and gmde

11tc error tcmts m t'qUatmn ( 4) nrc fot pncc nsk, o.~clums,c rate nsk, ywld r1sk, grade nsk.

storage nsk and anfl:mon nsk 111c opttmtthty cond1tH:ms of tXJtmtton ( 1}. however·. r'(.,X]tnrc dtc

covanancc matnx uJ}a' Squanng tlw error terms and assunung thnt only )~cld and gradQ risks

am corrclauxi gwcs the covammccs (Sro tl1c tcchmcal appendices .)

(6) aDrr'::::: r/~ /{lal + Hl rrz -J· (' J12 + (,~ )J"2cr2 +{Y- G)2 rr, ' r p'll I 4 ~ 1' ' ,

+2(p + g}Y(Y- (t J<rrrr smt,r: + p2S2tr! +l/Q2tr~

I II )r s· .. r:.~ r• r:fn ~~ i11 '" w lcre ' = + t - ( -· r - \ .. 1 + .. n· + ' ,, r rtJ rj

and /1 1 = (p- .f)F +( 11- ~~ p )N + kK

To snnphfy notatton. 11,. ts tJ1c amount of )'lCid plus stocks on hand whtdl arc tuthlxigoo against

pnce nsk and llr ts the dollar~ worth of futures contracts plus opttons on futurtlS wluch arc

unhedgtxi agamst exchange rate nsk

Or>tim:d Decisions

Dectsions arc denvcd from the optimality conditions In cquatton (I), using the expected

change in we~lth in equation (5) and tltc covatianccs in equation (6). TI1e consumption decision

will not be analysed futthcr It is included .in tJ1c model to demonstrate that inflation risk affects

consumption but not investment decisions. A grower can always choose a risk..,frcc investment at

opportunity rate r. Yield, storage, fixed price contracts, wheat futures, multi~gradc contracts,

nunimum price contracts. options on futures and currency futures have the follO\.ving optimality

conditions.

Yield

(7) p(1 +tY1,)+ g+ara ~aJ1,.~ -x'- Rp2 H,a~,-I{(p+g)1 J'a;. +(p+g)i'a,u 8 (0rg]

-R[(r- G)c.r~ +(p+ g)(Y -G)cr,.a g<vr8 ]=o.

Fin~nciaf Risk Management

10

Storage

(8) p ~ O;S ~ O;p,S = 0.

Fixed Grade Contracts

(Q) c- p(t +c5 P)+ Rp1.11 ,cr~,:::: 0.

IVheat Futures

A4ulu-Grade Contracts

Alimmum Price Contracts

(12)

Options on Fuwres

Currency Fmures

The grower's optimality conditions all have expected marginal revenue above the marginal costs of

production and marginal risk premiums. 11te marginal risk premiums are - U R times tltc

derivative of the covariance in equation (6). Titerefore, tlte marginal risk premiums will be larger

or smaller, depending upon the grower's coefficient of risk aversion, R. A risk ... neutral grower has

a coefficient of risk aversion equal to zero and simply equates expected marginal revenue to

marginal costs. Such a grower may choose storage, fixe<i .price contracts and so forth, but to

speculate for profit and not to hedge risks. A risk-averse .grower has a coefficient of risk aversion

above zero and equates expected marginal revenue to marginal costs plus marginal risk premiums

Such a grower expects to make less profit for the sake of hedging risks and tnay adopt ·any of

several strategies.

11

De!zve!'for the .fdarkat Price at Harvest

11te sunplest strategy is no ht'<igmg at aiL A grower could deliver all of the season's

producllon at haJ'\ICSt for the market price. In this C..'l$0, the only optimality condition is equation

(7) for ytcld wluch IS slmplitit'<.i.

p(l +l) P) + g +c:r ,cr ~:mrs = x' + R[(P + g)l l'O'; +(p + g)Ya rcr ~OJ rx]

+R[rrr! +( p + g)l' cr 1.rr km ¥1~ )+ Rp2 Ycr~.

Tile expect:t.Xi margin~l tcvetnlc (l'vtR) on the Iefl-hand stdc of thts t'qllation is the expected pticc of

wheat plus nny grade bonus. mod1ficd by the cov:mance bt'twccn ytcld and grade. For example, if

the ~~eld rs uncxpect.edly htgh and. more often than not, the grade as unexpededly low., yteld and

grndc have a negatJve covmiance and the c>Xpt'ctt~ margmai revenue is lower. 11m marginal cost

(!\K') ts the derivative ofproduct1on costs wtth rcspt•ct to ytcld. x' To this IS added marginal risk

prcmtums. 111 'irst margmal nsk prcnmun IS for mnrgmal yteld nsk (MYR) 11m second is for

lllUrginal grade nsk (MGR) and the thtrd ts for margmaf pncc nsk (MPR). tf yield risk and grade

risk have a nc~'ltlvc covan:.mce, both MYR and MGR arc lower than othcl\Vtse

Usmg rtvrescntahve parameters for the whe..'lt mdustry from Table I, the optimal yield

with no hedgmg 1s demonstrated in Ftgurc 1 1l1e top line in the figure. at $162 I tonnet is expected

marginal revenue, MR. 1t. mtersct..is four curves 1l1e lowest curve is marginal cost, MC. Above

this, a second curve adds MYR. a third curve adds MGR and the highest curve adds 1Vtl>R. 1l1c

htghest curve is tl1e grower's supply curve under risk and equals MC' + MYR + MGY + MPR. A

grower who is risk neutral sets the margtnal nsk premiums to zero and the supply curve collapses

to equal the l\1C curve. A risk neutral grmver expects to produce 1,800 tomms of wheat where MR.

= MC. In Figure L, a grower who is nsk averse. with a

coefficient R of 0.02<t expects to produce only 1.500 tonnes

where MR = MC' + MYR + MGY + MPR.

Financial Risk ManagE)ment

Table 1: J>ara11.1cter Values.

p tso ~~ .OS J 140 t5A 0 g 5 (~\' .0) k 0.75 O"p 0.2 s 3 <TA 0.9 I' 0.1 O't o,2

c 140 <:Tg 30

l 140 a:~ 0.1

g 125 t.Otg .. Q.l

k 0.75 R 0~92

12

P1ice 200 ($/tonne)

100 I

0• r 2

Quantity ('000 tonnes}

J?igurc 1: l>clivcr for the 1\l:u'kct t•dcc at llarvcst.

l11e sum of all margmal nsk prcmmms m Fig~trc 1 is the veJtlcal dtstance between MR and the

MC of producmg l ,500 tonncs, or $85 I tonne. Divu:hng this vertical dtstancc by the honzontal

dtstance of I ,500. gives the stope of the dottc,d line nus slope cqttals the coefficient of risk

aversion multtplicd by tl1e vanances of ytcld, grade and price. Div1ding the slope by the variances

wves the grower's cocfftcJent of risk avers10n, R. l11creforc. if the grower's expected price for

wheat and expe<:tcd yield can be elicited and the MC curve measured. the grower's degrre of fisk

aversion is revealed An attcmative to elictting degrees of nsk aversion by asking growers to play

games and lottenes is by askmg growers about then· actual dectsions.

\Vhcat is harvested in the spring and summer bttt demanded all year round. Consequently,

wheat is stored and the price of wheat rises after harvest to offset the costs of storage. At the next

harvest~ the price of wheat falls again. \Varehousing of wheat with AWB and siloing on the fann

may be good strntegic:s for marketing before the ncx1: harvest. Deliverillg.to an AWa pool for an

inithd payment of 80% and receiving d1c final payment up to 18 rt1onths later is similar to

delivering 80% of the crop for the market price m1d storing 20% as a. hedge against the.price n$k of

growing a t1ew <;rop. Long-term storage as a ht.>dge can be analysed. by setting ~~~ other hedging

alternatives to zero and simplifying the optimality condition fot' storag¢ in equation (8).

p:O P +JJ= p(J,\. +r}+s+Rji2(l'+S)O'~, +Rp'-SC,~ = 0; p. ~O;S ~ O;pS =0.

13

'11H~ mnrgmnl rctum from stotngo is tho c:11>1tal g:tins frot\1 at' cxpt.'C1txf tisc in the wltMt pnco.

Agamst. tlus must bo wc1glwd the oxpct.::tcd vnluc of' stocks lost t:o dL'grndationt the oppmtunity

costs of mvcstmg m wheat. the storage costs nnd thu premiums fot· mnrginnl pncc tisk nnd

marg,mnl st.orng,c nsk From T;,blc I, n 5% nsl1 II\ the wheat. pncc by tho end of the season is not

enough to bnlnnco n 1 q·o loss to dcgradntton1 a I 0% opport.\nuty cost of uwcstnK'I.lt ond $3 I tonne

cost of storage. l'~t nlonc thl~ mnrgol<\t nsk prcmnJmS Jf stot·ngc was Ullconsu·mn(..'d, tl1o opt.imnl

ptcmmm for mnrgtnnl pncc nsk l3ecnuso stomgc t.s co,1st.mmcd f() be non m~'g.t·ltJvc, the t .. agmngo

mutophcr must be po~nttvc nnd, by conlplcnu.:ntnry slncknt:!&s, storngo must bl~ zero 111c opbmat

storage W<)n•t lK'Comc po~auvc unttl the cxpt."'Cicd nslrm thl.! whc~n pncc cxcctxls 35~'0 l!v<.~l for a

11Sk-ncult'nl grower, the cxpt,'Clcd nsc must cxc<.'<.Xl 13 11/o Only m UllliSUnl seasons Will stot'ngo be

:m optunnl htxfgc ng;tmst pncQ nsk nnd othl~r nltcmatwcs nrc nt'C<fcd

Price xx; ($ltonue)

100 J

o•

Ouahtity {'000 tonncs)

Ji'iJ~llr~ 2: Stm·:•gc using .l~()ols, \V:u·ehonsing n,ut ()n,..f~•nn Silo~.

N.v<rd Grade Cono·act

Perhaps the simplest htxiging stmtt'SY is an A\VB fixed grade contract. By :\doptfng, this

stratt-gy, a grower shifts b()th marginal revenue tmd th~ su(>ply cutvc. 111is catt bc. shown by

adding tho optimalif,y couditJon fbi' fix<.'d gtndc contracts in C<lUfltion (9} t.o d1c optimality ctmclitio11

for )'lt:!ld in ¢quntion (7),

l + g+rr,.rr 8(tJ ~"~: = ,,., + 11(p+ g) 1 Ya-~ +(/1-t•g)Y<T,.O' 8aJrg]+ 14J1a! +(p+g)t&rd,nf,Q:):*J

Fin~ncial Risk Management

14

11u~ cxpt.>ctcd mnrgmal revenue substitutes the contract pnce for tl1e expected wheat price and tl1c

supply curve no longer mdudes the premium for maa-ginal price risk. In Figure 3, rvfR shifts down

from $162 i tonne to $144 I tonne and the grower's supply curve shifts to l)cconte MC + MYR

MGR 11lC sh1ft in l\•1R and tl1e shift m the supply curve have oppostte effects on tlte production

dectston and the grower chooses to produce about the same quantity as before, 1,540 tonncs.

Gwen the optimal ~11cld, tl1c quanttty to hedge 1s calculated from the optimality condi.t1on

for fixed grade contracts

In Figure 3, the optunat hedge ts 570 tonnc.c; where the slopmg dotted line intersects MR. 1l1is 1s

tl1e quantity \Vhich sluft.s t.he supply curve so that 1t no longer includes l'vtPR. Shifting t11c supply

curve is not the same as hedging aU pnce risk 11te optunal hedge still leaves 970 tormes of

expected y10ld unhedged. l11e curve sho\\~ng MPR stuft.s because there is less price risk than

before, but it does not shift all the way to tl1e new supply curve. All price risk can be eliminated if

the contract pnce mcreascs and tl1e optmml hedge increases along the dotted line until it equals

expected )~eld. Other studies suggest tl1at the optimal hedge will be below expected )~eld when

tl1ere is productjon nsk. A grower is tl10ught to hedge conservatively in case tlmre is a production

shortfall and more wheat must be delivered than is produced. In this study. there are no restrictions

on the buying and selling of wheat. lf there is a production shortfall, a grower can purchase

enough to satisfy the contract. TI1erefore, productwn risk does not directly affect hedging,

although it could indirectly affect hedging if the wheat price and yield are correlated.

As in the case with no hedgmg, the grower's degree of risk aversion is revealed from actual

decisions. In this case, it is not ne<::essary to measure the marginal costs of production. From tl1e

expected p.rice, contract price, expected yield and quru'\tity hedged, the slope of the dotted line in

Figure 3 can be calculated. l11is slope equals the risk aversion coefficient times the variance of the

wheat prir-e. Dividing the slope by the variance gives the grower~s risk aversion coefficient.

''!'lillri

15

Price 2:0: ($/tonne)

100. ~t·' ,._.- -·----

0. (' 1 } 2

Quantity ('000 tonnes) .

•~igun~ 3: Fixe(f Grade Contra,~;. .:.:.J Hlheat Fmures Contract

Hedging w1th wheat futures contracts on U.S markets is similar to hedging with fixed

grade contracts except for basis risk Of the components of basis nsk, only exchange rate risk IS

considered. If the exchange rate can be hedged by selling US. dollars forward, then the optimality

condit1on for currency futures in equation (14) can be added to the condition for wheat futures in

equabon (I 0) and the result added to the condition for yield in equation (7). BOC<1Use the contract

prices for futures, k and j, equal the curreflt futures prices, k andf the optimahty condition for

yield 1s identical to that for a fixed grade contract except that the current futures price replaces the

contract price m expected marginal revenue. Figure 3 also appl.ies to an optimal portfolio of wheat

futures and currency futures.

If the exchange rate risk is not hedged, then equatton (14) is not added in to the optimality

conditions for yield and for wheat futures and a system of two simultaneous equations is solved for

the optimal yield and the optimal hedge.

f -(p- f)o A+ g+ O"rf]' llwrs = x' + ~(p + g)2 Y<Ti· +(p+ g)Yaro·smrx]

+R[Ya! +(p+ g)YarO' xru,.8 )+R(p- /}2 Fai

p(l +811 )- j +(p- f)o~. = Rp2(1'- F)a;- R(p-!)2 Fai.

Jn expected marginal revenue, MR, the current futures price minus the expected change in the basis

replaces the expected wheat price. As with fixed grade contracts, the supply no Ionger includes the

Financial Risk. Management

16

prcnmun for rnarguml IH'tCt1 nsk bur, 111 tlus cn$c. n prcnuurn for mnrgiital cxdt:ingo mlc nsk

(rvJf!R) JS nddt'Cl 11wrctbro.thc Slipply curve L'qtmls MC !· lVlYR 1 MOR + tvH!R E)(clUlllgc rato

nsk npplws to tho dlficrt·ncc bt1wt.X~l tho cut rl~nt whl:.\11 price and t.hc curn .. 'l\t f\tturcs pnco In th!l

U S the futurt$ pncc mum converge to the whl':.'ll pncc :tt mntunty :.111d tho lunas wtll bo chnmmtc.-d

In Austmhn .• the basts wtH nlso be cflm11mh . .-d tf the cx·.:h:mgc mtc stays constm1t but It wtH not be

chm1mth.xl tf tht~ ox chang<: nne dmngcs over the II f'o of the Htturcs contract Un~~XtK'Cft,'<f <:hangcs m

tht~ c~chtmgt~ Jalt) lund to t.'xchnugu mtv nsk

Solvmg the opunmhtv condttlons sumalt~mt'Ously gtvcs a yteld of 1.540 tonncs and a ht!<igt'

usmg wl\c~lt futml'S of 520 tonncs At thts level of hl'dgtll&, MER ts less than $1 I t.Oilllo In fitgurc

4. l\U!R ndds very httk• to the supplv curve and ts not shown l·k"XIgmg wtth whc:lt futures ts

almost tht:~ s•lmc ilS ht..'<lgm~ w1th fh,xf grad~ contracts llw optmtal htxfgc wnh f11tun.1s 1s 50

lonncs less due to the smnll etll>ct of cxch:mgc mtc nsk ·nns concluston cout.msts wtth other

studu:s wh1ch nssunu.:~ tJHll cxchnugc nH'' nsk npplu.~s to tho futures pncc 1tscJf mul not to t.ho

dttll.!I'L"llce bt'tW(,.'t.."'l \\'heat nnd futures pnccs l:!xch:lllg(,• rate nsk nppht'ti to tJte futures pnco would

SJVO n much tnrgor pn:.~nuum for m:ugtnnl exchange rotc rtsk

tf oxchnngc rnto nsk ts hcd&l'<f. thl! cont.mct pncc f'or fixl•d 8l:Hlo contracts should be the

sanlc as t.ho wheat futures pncc ·nus •s shown by subst.ltt1tmg the opttnmhty comhtHm fot·

currcucy futures IH lXJUatlon ()4) mto dm coudtuon tbr wlu.~t fliturcs m cquat.1on (to) and

subtrnctlllg tho comhtmn fbr fixt.~l gmdc contmcts m cquat.ion (9)

fr::::c

TIH,!retbrc, if both fixed grade contracts and futures contracts nrt' avatlablo. one or· the other ss

n .. xlundant Howlwcr. the model docs not com;idcr the difficulties in trading futures c¢ntmcts

Tradmg futures rt•quiro cxpmtisc. cfrort, time and avmhtbllity of capital for margin calls. Fot thusc

reasons, n grower may prefer iixl'<i gmdu contnu::ts

17

Pticc :;nl . ($/tonne)

100. ,. 1-••-•A••••-•n~

o· I·

auantity COOO 1onnes)

Figure 4: \\'h,~at Futures Contr~tct.

A1ultr-(1radt' Comract

Another sunplc hod~mg stratL'gV IS ~m A \VB mult .. • ... gmdc contract As Wtth fixed grndtl

contrncts, the grower's cxpcctt>d rnargmal nwt~mc nnd supply curve sh1fl In tlus case~ the supt>ly

curve tncludcs nc•dlci· the pn.muum for marp,mal pncc nsk no1 tho prcmnuu for margtnnl grad(!

nsk To sho\v th1s. the optttnahty condttlons for multt~grndc contracts m l.XJUahon ( 1 t) and y1cld 111

t:quat1on (7) arc cornbmed mto a system of equations

l! + rr r a' tm 1 !( l:!l x' + /~1 ( p + g) 2 J' cr f. + ( p + g)l' a, rr 1 01 r~]

p(r +li (!)+ g ·w .~ e:: Rp~(r ·~ O)(j;, + 14!Y- O)rr! +(p + g)l'rr,.a 11wr,] ~ 0

1110 expected margmal revenue, MR. 1s the c:ont.mct pncc of muJu .. gmdc contracts modtficd by any

covanancc between y1cld and grade 111c supply curve tncludcs only marguml costs and the

prcmnun for margmal ytcld nsk. or MC + MYR. lf multt .. gradc. contracts arc pncc..>d tho same as

fixc..xl grade COJltracts. 1J1c stun outward in the supply curve will cause a supply response aS U1c

grower d1ooscs more mtcnstvc production and highl~r yield. In Figure 5, multt ... gradc contracts arc

priced ic'Ycr, with a MR of $124 I tonne Solvmg d1c optimality conditions simultaneously gives

approxunatcly the same ytcld as before of 1,580 totmcs but an optimal hedge with mtilti-g,radc

contracts of only 450 tormcs. Although the supply curve no longer includes MPR and MGR; tlu.)

optimal hedge is less than yield and there is still price and grade nsk. ln. Figure 5~ the curves for

MPR and MOR shin to show that some, but not all prtcc and grade risk are elit1tinatcd. As the

FinanCial Risk Management

ts

I\ .. 1R of multtgrndc contrncts nscs toward tho marginal rew~mc. of fixed grade contratt:s~ the opumal

hl~igc wlll mcrt'.'lSc along the slopmg, dotttxi hnc unttJ tt <.:'(.)U:lls CXJR'Ctcd yield. cxcq>t for· any

dls<:tl.jlt'llCV caused bv corrclatt'<f grado and ytcld nsks ·n"s suggests that nuiH.t .. gradc contracts

arc.~ \lntquc ·nmy arc not redundant wtth ctU1c1 Hx(.xt grade or f\tturc.s contmcts and the A \VB has

some f1cxtbllttv m SL'ttmp. thctr pncc 11'o MR of mttlt.t g,mdc contracts can be lower than the

mnt"glnal rcvt'fntc offixl~i grade contracts and n nsk averse grower wtll shll usc them

Price ax> ($/torme)

O· ( i

Ouilnt.ity ('000 tonnes)

Figurt1 5: ~halti.-(:,rad~ Cnntr:tct..

lvlmmwm Prn't? Comract or an OptWil of l~"'tttur,~s

Mttlltll~Jin pncc contracts and optH)flS on futures arc altcmatJvcs to fix<..'<i grade and future..-;

contracts 1\ grO\vc.r may purchase those to chnunatc do\\11Stdo losses. but rt.'tam upstdc gams

Adding tho optunahty cond1tron for tnmmnun pncc contracts Ill (X}uahon ( 12) to tho condition for

flx<.i(l gfadc contracts m e<1uataon (8) shows that the cxpcct.cd wheat pnc:c tl\UlUS the contract pncc

for fixt'<i gtadc contrads equals the expected Cf\ptt.n1 gams above oppottllnity costs from mvcstiug

ut nunu11um price contract TI1c value of n'inmnun ptic.c CQntracts in umts of dollars per contract

IS corwcrtc;.xi lllto umts of dollars p(W tonne by diV1d!l1g by its derivative witll rcspt,>ct. to the price of

wheat.

(l.S)

If tho valllc of minmitm1 pnte contracts 1S pt.opcrl)l calculauxt as disctjsscd. bclowt. a grower is

mdiffen.•nt bcbVCC.l committing to fixtxi grad~ contracts, which clithirt~tc bQth the <;hance. of

19

do\vrlstdc loss<w~ from l{)wcr whc:n pnct~ and the chance of upstdc £,.1tnS from. htghcr poe~. or

purchasul.g nm1mnn1l price conttact.s. whiCh ehrmnatc dn1 chaoco. of downs1da h:>$scs but ,tct;,\U1thc

chnncc of Upstdc S.1UlS

Subsutuung t.ho opbmahty ~ondit.lon for currency futures 1n t>,qttatnm (l4) mto tho

<>putnnhty cond1ttons for wh~t t'\nun.~s nnd opuons on futures. m cquauons ( I 0) nnd { 13) tmd

c:orwcrstoJ.l mto mms of dollars P'~r tc:mna

lf d1o pncc .of optt(>ns on futures are properly calculated. n gt·o\vcr tS ttldtffcrcut be1\\'('Cfl usmg

£,.'ltns from a lowt~r fut:un."'S pncc ~1orcovcr. tf thu contrnct. pncc for fixed grade contracts c.qu!tls

the curtL11t futun.-s pncc. a grower tS mdtffcrcnt amot1£. fl~t~ grade contracts. wht~lt futures

Portfolw qf .41/lll!dgmg Altf!rilatrvos

lf alllu .. •c,igmg altcm:auv($ arc avadablc. only sttlrngc and multr-:gradc contracts a,rc utltqttc

nm:,-e otn of four of fixtxl grade contracts .. whc..·n futures .• nunm1um pncc cont.rncts arld opttons on

futures aro rcdundnl1t and at ts tmposstblc to pr~.SC11be winch a grower should dtoosc 1111s .ts

bc~nusc the e~ght. opumaht.y condtuons m cquauons (7) throt1H)1 { 14) ca11 be solvtxi for only five

mdcpcndcnt vannblcs Tl1csc arc y1cld. l, storage, s. mult.~gradc contracts. a. the amount htXigcd

n&•unst .p.ncc nsk, Y - II"' a11d the dollars worth ofwheat futltrcs and opttons on futon~s \\1ltch are

UJ:1hoo,gcd agamst exchange rate nsk, 111

07)

OSJ pt!O;S .. ~O;.{IS ~o

20

(lq)

(20)

(21)

For the tt\'lsons drscusscd. long·-icnn storogc as a .hedge agrunst price rtsk Will bo zero ul ntost

seasons Mutu .. gradc. contracts arc a ht'4gc agn111St pncc rtsk and arc mcludcd along w1d1 fixed

gt11dc conlracts. wheat fut:u1'cs. m1mmtun pncc c(lntmcts and opttons on futurt-s mthc t()tat amount

optunnl qu:mbty can bu calculated u;dcpcndcntly of other hoogms nltcmauvcs

F1gure 6 shows tho qptmnll pottfolro dloscr-1 from nll posstblc htX}gmg stratcgtcs TI1c

opt1Jllat pOrtfolio IS a COtnbU13l10fl of dtC Stt'atcgy for fiXOO grade contracts In f\gurc 3 and muJtt ..

grade contracts tn f'1gurc 5 111c optimal quanttty to hedge agamst pncc r1sk. l' • lip. is 600 tonnes

at the point \\here the uppcnno.~ ~· of the slopmg dotted ltnes mtcrsects the matgmal rcva)tle of fixed

grade contracts or $1441 tonne At tlus pomt, the supply turveno longer mch1d¢S MPRand shifts

to l>quat MC + l\1YR • MGR The optunat quanttty to hooge +t~unst grade risk, G. is only 300

tonnes at dtc pomt where tho lowcnnost of the .slopitlg dotted hne mtersccts the ~{R for ntulti•gt11de

contracts or $124 I tonne. 1l1e supply curve no lons~r tncludes MOR aod shifts further t() t\")ual

MC + ~1YR. Since mult1-grade contaacts hedge both price and grade risk but only half of the

hedging portfolio consists of multi-grade contracts. the other ·half of the .. portfolio n)ust bt1 cilhcr

fixed grade contracts, wheat futures, minimum pnce contracts or options on futures. Since the

optimal yield is the ~ame as with a multi,.,grade contqict, or 1580 tor1ncs) even a ·risk aver$c ~o.wer

leaves 980 tonnes unhe<:fged against either price orgn1de risk.

21

Price :ax> , ($/tonne)

t~tJ' (\.J•g. (TJ'O'tfl?:r ~ ·-· - r--!""' .. ~- ....!---

# l

• ~MR .~·~~+-----------~~~~ g • trr~;,ltlt» • MC

I

100 I, l ,, ..

O· ,\ fi t .1(1' } 2

Quantity COOO tonnes)

a;~igure 6: l'•lrtf«,Ji(l of ~\n llcdging t'lter".:lfi,,~~·

()ption l'ricing f"ormulas

In tho finauco htcmturc. the f:1mous lllack-Scholes opt.ton pnclllg fonnula follows from the

assumption tl1at mvcstors choose n 11sklcss portfoho ofhcdgmg alternatives (Merton. p 281) In

Ftgurc 6 above. the curve winch mcludes the prcmnun for margt11fil pncc nsk. MPR. shtfls to the

nght. but. only pan \vay toward clunmatmg an pncc nsk 11us ss bt-causc Ulo optunal .hedge IS a

fraction of cxpC(.;icd ytcld 1l1e optmml portfolio 1s not r1skk'.SS Ncvcr·th~lcss, the Bl:tck-Scholcs

and other opuon pncmg formulas cnn be dcnvcd from the optn11al dt>ciSions of a grower

Blac:k-Scholes Formu/(1 for Optrons on Stocks

1l1c origmal Black--Scholcs formula was dcvcfopcd to price options on financial stocks

(Black and Scholes} Storing finanC1al stocks is easy l11ey do not physically degrade, are costlcss

and rtskless to stor~ and can be posttive or negative A positive stock is an asset and a n~tivc

stock lS a liability. 1l1e optimality condi(ion f'or storage of wheat in equation (8) can be convcrtoo

to a condition for mvcstmg in financml stocks by sc«ingthe rate ofdcgradation~ stora~c costsJ risk

of degradation and the Lagrange multiplier to zeta. Because minirilim p.ricc. contracts arc

equivalent to put options~ tJ1c optirnality condition in cqu~ticm (1 t) can be c,.ombin¢d \Vifh the

OJitimality condition for t1nm1cial stocks, At the optin1um1. the ¢~p«ted ~pi~l gains pt1 st()Cks

above d1c opportltnity costs of hwcstmc!nt must cquat thr.. ;expected /capital gttins On (>.ut qptions

above oppmtunity costs, after conversion.~into units ofdoUa ts pet'Ullit.of stock.

·Fin~rlcial Risk Managemt:tnt

22

TI1c .expect~'(! c:apunl g..1ms on put ophorts~ nu)";.. evolve nccordittg to the diffcrt;flhnt LXJunbon

dcfhwd m t.'\.1UnUon (3). S.uhstttutirtg this diffcr<.'t1tHll cquauon. canccUtng. t.crms and rc.1rrartging

g.tVt\.~ ·a dtffcrcnt.tal (or the p1ice of opttons 1l1c price wtll chango over tunc until matunty at lH1tC.

r At miltt~rity. the owner has the nght but not the obhgttnon to sclt tl1c stoc . .k at tho c.xcrcisc.pnce,

p lf the cxcrctsc pflcc. t.lXC:l.'(xiS the prlcc of stocks~ opttons will be wotth the exercise pnce mh.tt.ts

thL~ stock pncc If t.he excrc1so pnce ts less tlum the stock pnce, options "~II be worthless. 111~1s.

the d1ffcn.'11ttnl t'qunuon for the chnngc m the Qptron pncc lS subJt>ct to boundary conditions at

mntunty

r ln f'lli t ")2m , , +"·., pr+)!: ·~y p'rr"',- mr=O d (:/} tp· f

wuh baundMy condtttons ~~~ r .p) = {P -p~ fl?. P 0. fJ < p

DtfferL'ntlnl cquatwns such os thJs mt~ratc to become probabihty d"1sit)1 fl•nctiotls. \ViU1out the

boundary conditiOn, th•s pntttcular L>quattQn ts a Koltnogorov equation whtdt intcgratt'S to tho

nonnal dt'llStty function \Vuh the bo~mdary condtttan. this cquatton integrates to t.'qual the

oxerctsc pncc, dtscolultt'd to the presf..•nt and nmltiplit.>d by n cumulative probability. minus the

current stock pnce, muluphed by a different cumulative probability.

qf, 9 11 •.

lt~t.p}= -p J (21t) '~e ,: 1 rlv+e~t(t~t)~ r(2ll}~e ''\.dv

lnp/~+{r+ ~2 cr1 Xr-t) where: d1 :::: -~-· · ,·-.' ...:.---

u('r- t} l

_ lnp:1 P+fr.,... \2u 2 Kr-t) d~--·~--~~----~~~

.. a( r- t)h

TI1elhnits of integration for the two cunndative probabilities .arc. d1 and t/1, It is c;tsy to check that

the .boundary conditions are satisfied at tllaturity. In the numcrat()ts of both limits of int~rati()n,

the naH1ral logarltllm of th(} ratio of the stock price to the exercise price, In pI.~ , is negative, for

au exerci$c price above the sto(!k price. As time~ t 11pproachcs the time of $uaturltY1 t, the

denominators go to zero and the. limits Qf integt;.:ltion go to minus hlfhtity. Tiie o~tiv~ otJh¢:

23

Jinuts of integration go to ph1s infinity. tl c cumulative probabilities go tb one and the price of put:

options equals tho exercise price lll' .us the price of stocks. l11e natural logarithnl of the ratio of

prices ts positive if the exercise 1 •rice is less than the price of stocks, tl1e limits of integration go to

plus mtimty, the cumulati vc probabilities go to zero and the price of put options equals zero. It is

tLxilous but straight forward to check that the differential equation is satisfied. The· solution is

diffcraltiatcd once witl1 respect to time, twtce w1th respc<,.,'t to tl1e ptice of stocks and the derivatives

substttutcd mto the differenttal <.XJuattOn to show tt equals zero.

l11e Black-Scholes fommla became famous because it docs not mclude expected capital

gams on stocks or nsk preferences EVCl)'One, regardless oftl1eir optimism or pessimism about the

pnce of stocks. and regardless of thetr degree of nsk aversion, agrees on the price of options

Unforttmately, the BlackpScholes fommla does not apply to wheat because wheat stocks are

difticult to store

Jdodified Black-Scholes Formula fm· Opt1ons 011 Futures

TI1e Black-Scholes fomllda can be modified for pricing options on futures (Melton, p.

347). TI1e dtfferentml t'qttattOn in cquatton (3) for the expected capital gains on options is

substituted into equation 16 to get a differential equation for the change in the option price. A

grower who hedges using futures contracts must buy futures contracts at the time of maturity to

oftset the hedge. In addition. a grower might purchase a call option on futures which gives the

right but not the obligation to buy futures at tlle exercise price. 1l1e boundary conditions for call

options are opposite to those for put option$. At maturity, the owner of an option has the right but

not the obligation to buy a futures contract at the exercise price. lf the futures price exceeds the

exercise price, the option is valuable, other\\'ise it is wmthless.

with bow1dary conditions; n( r,J) = f- f; f ~ ~· {

0 0

0~ J<J

This differential equation integrates to equal the futures pri.cc minus the exercise .price, both

discotmted to the present but multiplied by different cumulative probabilities.

Financial Risk M~n~gement

24

lnf J + '2rr2 (r-t) where: d1 = ~, ·

rr(r-t) ·

In f .f- 12 a

2(r- t) d, = ""'""'-'''""'"'""'···' _, .. , ....... --~·-·' • a{ r -t) 1

ln this case, tlle lttllits of intcgratton do not mcludc the intcn.~c;t rate, r, and their sign is positive in

the mtegrals As before, everyone, regardless of expc'Ctattons and 11sk preferences agrees on the

pnce of opttons on futures Unlike the Black-Scholcs fonnula, the modified Black-Scholcs fom1ula

can be applied to wheat futures tf exchange rata risk ts hedged

A-linimum Price Comract

l11e A \VB oflbrL-rl mimmum pncc contracts but has now discontinued them due to lack of

demand. Equation ( 15) above suggests that a grower may include fixed price contracts and

mm1mum price contracts mterchangcably 111 the optunal hedgmg portfolio, if minimum pnced

contracts arc correctly pnced. Substt.tutmg the expectoo capttal gams m equation (3) into equation

(15) gives the differential equatton for the value of minimum price contracts. l11e botmdary

conditions for a minimum price contract arc the same as for a put optton.

c'Jn t'ln ( _ ) t/ cjl m 2 2 -+-:- c-p +}2--,_-p aP-mr=O t1 (.p c,p

with boundary conditions: 11~ r,p) = {p- p~ f.~ P 0~ p< p

1l1e fonn of this differentml equation is similar to one for stocks which pay dividends {Merton, p.

297) and has no known analytical solution. In other words, the value of a minimum price contract

does not have a normal distribution, nor any other known distribution. It not possible to write the

fw-1ctional form for the probability density and then integrate it to find cumulative probabilities.

The value of minimum price contracts must be fow1d numerically, except for a special case. .lfthe

currE.'flt .price of wheat equals tl1e price for fixed grade contracts, the second tenn in the differential

equation is ~ero and tl1e value of minimum price contracts eqmtls the .price of put options on

futures.

25

p Inc. p+ ~2 a2 (r-t)

where: d, :::: ·-~···-·-~=-· ~·-·••e·> a{r -r) ·

lnc p- 12 u.2{r-r)

d2:::: ··-~·-·· .. ~---... ·""··--·· ,.../ .... - t\ ., ""\• 'I

F1gure 7 shows tl1e muncncal solution for the value of minimum price contracts when tho oxcrctse

price ts $140 I tonne 1110 solution was found by the finite clement method. At each grid point for

a particular wheat price and a part.tcular tunc to matunty. a differential equation was calculated. A

value for mmirnum pnce contracts was chosen at each gnd point to simultaneously sattsfy the

differential .c..'quat1ons at all nodes. At mntunty, tJ1c solutton for all wheat pnccs ts determined l>y

the boundary condttlons At n wheat pncc of $140 I tonne, which equals the price for fixed grade

contracts, tl1e solutton for all tunes to matunty tS calculated from the fonnula for the price of put

opt1ons on futures. The solution was tmplen1Ctlted m l'v1icrosofl Excel 5.0 using the mathematical

programnnng algonthm available as tl1e Solver opt10n. Solver uses the GRG2 algorithm for non

linear obJecuve functions and a Newton~Raphsonmethod for non linear constraints .. A copy of the

llllplemt."'1tation is available upon request.

Value Mintmum Price Contract

1.000

Years to Maturity

WheatPiice

Figure 7: Valpe of l\ljui~n.,.nl PJ-ic~ Coptr~~ts.

Although the tmderlying probability density is unknown, it still exists. 'I11e valqe ofminimwu price

contracts has the same interpretation as other optio11 prices. The value eq\lals the exercise price

weighted by a cumulative probability minus the wheat price weighted by a cumulative,probabiHty.

Financial Risk Management

·:,,'· ... ··,.,,

;,:,.···. ~·.~.:·~; ·~·. <l~;··:.•y·~·· '\ ~·, l~.;i

26

Everyone, r~nrdle.ss of their expcct,!ltions about whe.at. prices and their degree of risk aversion

agrees on the value of minimum pnce c,:mtracts.

1l1c nunwnc.'ll solut.ton shown in Ftgurc 7 ts new to the option pricing literature

Prcvtously. the value of minimum pnce contracts would have bl'Cil approximated, p.erhaps by the

fonnula for a pttt optton on stocks or by the fonnula for a put option on futures. Comparing

Ftgure 8 below w1th Figure 7 above shows that the pncc of a put option on stocks is a very poor

approximation which gets worse the longer tJ1c hmc nnt:tl maturity

Option Pnce

25·

1.000

Years t.'' Maturity

Figure 8: Approximation by Formula fo .. Options on Stocks!

On the other hand, Comparing Figure 9 with Figtlre 7 shows that the pnce of a put option on

futures is a11 excelk•nt approxunatton. Of course, at a wheat price of $140 I tonne, they are

1denttcal

1,000

Yearsto Maturity

' .·--- .........,..... ..

27

Jost as the pnce of fhcd grade contmcts should be ;;~)'.{ to t.'qWtl the futures pnco, tho valuo of

mm11num pnce coutmcts cnn be doscly approxtnlatcd by the pncc of put options on f\1turcs

Condusions

\Vht\1t growers now have SCVl!ml nltomnuves for ht'dgmg theu· nsks ·n\Csc mcludc long·

term storage of wheat. fix<..'<l grade contracts, wheat f\1t.urcs contracts, muJt.t ... gradu contract!'

mmtmum pncc cont.racts. optiOns on whc.at futures, and currency futures ronrracts. Storogc will

seldom be opt.amal A grower must cxpt.>ct a stgmficant pncc mcrc.1sc to JliStlf)t tlw dcgrndatton of

sto~..~ks, oppottumty costs of mvestmL'tlt and costs of phystcally storing l'1t.' wlw.at Fixed grade

contracts and futures contmcts arc almost l'qluvnk'llt hcdg,cs ngamst pno:.· nsl<., cxc<~pt for the

chfllcultJcs 111 C\<.~utmg fliturcs contmcts on U S markets lndtx'<i .. t.hc A\~'B could act ah a br~~lker

by oiTl't mg fi x<.~d ~rnde cant wets to growers wtth contract pntus set equal to the futures price

Both fixt>d grade contrncts and futures contracts chnunatc the dO\vnstdc losses and upstd{~ gnlllt: of

nsky wheat pncfJs Mmunum pncc contracts and opttons on futurt•s chmmata thfl downs•dc losses

but rcuun the upstdc gams For tills n.\-'lson, they :trc valuable asst>ts 11·:c A \VB tA>uld nlso act as

n broker by ofTcnng mmtmum pncc contracts to growers wtth values sd. cqt~at to the pncc of

opttons on futures Because they nrc eqUivalent \vays to hedge pncc nsk:· fixed price contracts,

futures contracts. mnumum pncc contracts nnd opt1ons on futures arc ~~:ichstmguishnblc from each ./

other m a grower's optnnnl pottfolto Mulll-grnde contracts hl.>dg; .... ,oth pncc nsk and grade .r·isk

and arc untquc 111 an optunal portfoho Because they also hr.~:tgc gmdc nsk. the pncc of m11~ti-

grade ccntmcts can be less than the pncc of fixed grndet_ontr;.lcts, but there arc no multi grode

t\tturcs contracts fm the A \VB to u:;p as a guide m sr:'1mgthc rn;if:c. Yield nsk can not be hedged

by any of the altemativcs ofTcrcd by the A\VB ;.ithough a possibility is the new options on yield

being proposed in the \JS

I-I edging shifls a growm · s c~p~1.txi marginal revenue and S\tpply cttrvr \Vith no hedging~

the matginal revenue equals the e.xptx:tcd wheat price plus :1ny bonus or discmmt r:llr grade plus the .

covanancc bct.wccn yield and gradr, ·nlr supply cuntc iricludcs the marginal costs of production

and premiums for marginal yicid risk, marginal grade rbk and margiual prico risk Hedging of

ptice risk substitutes either the contrac:t price of fixed gmde contracts or the future::; price tbr th~

Financial Risk Manageh~t:mt

28

expC~:tted wheat pnce in marg~nat revenue, and shifts thr supply curve so that it no longer includes

the prcnmun for marginal price nsk Hedging of grad~ risk substitutes the contract price of multi-

gradt~ contracts for the expected wheat ptice and any bonus for grade in marginal revenue, anrl

slllfts the l'upply curve so that 1t mcludes neither the premium for marginal price risk nor th~

premium for marginal grade risk 1l1e more risk averse a grower, the greater the supply shift

11ms, tf the A\VB st'ts lugh pnces for its contracts and growers are risk averse. there may be a

stgmficant supply response to the tlC\\' hcdgmg altematlvcs on offer.

Rcfcren\'es

AnNrSNl. T J ( l Q87) Currency and !merest Rate If edging. Prentice--Hall, New York lnstitute ofFinrmcc

Anderson. \V K . C'rosb1e. G B and Mason, M. G (I Q03) 4.'[fect o.f mnnagemetlf on wheat gram quality Grams Research and Development Corpor1t1on, \Vestem Australian Department of Agnculturc

ANZ McCaughan Futures Ltd ( 1995) Wheat Pnce Risk A1anagement

A \VB ( Jl)Q2) Annual Report, Melboume

A \VB (l QQ4) Annual Repm·t. J'vtclboume.

A\VB (1995a) /995 Growe~: .. Handbook, Melbourne.

A \VB ( 1995b). AJarketing Option: A1ultigrade. Fact Sheet No. 4, Melboume.

Bardsley, P and C~1~~wn. ?. (! 9~0} "Underwriting Assist:mce to the Australian \Vheat lndustl)'­an Applic;wivn of0ptiot1 Pricing TI1eory·. ~:;s/. J Agric. Econs. 34:212~222.

Black. F. and Scholes, M (19'3). "TI1e Pricing of Options and Corporate Liabilities." J. Polit, Econ., 81:637-654.

Bond, G.. and Wonder. 8. ( 1980) ... Risk attitudes amongst Australian growers". A ust . .1. Agric. Econs. 24·16-3 4.

Bond, G., TI10mpson, S. R. and Geldard, J. M. ( 1985). ".Basis risk and hedging strategies for Australian wheat exports'\ Aust. J. Agric. Econs. 29:199-209.

Campbell, K. 0. and Fisher, B.S. (1991) Agricu/tural.MarketingandPrices. 3rd ed. Longman Cheshire Pty. Ltd.

Condon, C. (1992). Quest for Quality- Operation Quality Wheat. Compiled by Sue Bestow, Department of Agriculture~ Moora, WA.

Dillon, J, L. and Scandizzo, P. L. (1979) nPeasant;agricultureandriskpreferences in norlh•east Brazil: a statistical sampling approach", in J.A. Roumassd, J;.:M, Boussard and l. Sit1gh

29

(Eds), Risk. l/ncertainty and Agricultural Development, SEARCA·ADC Publications, Lagtma. Pluhppines

Franstsco. E M nnd At1dcrson. J R. (I 972) "Choice and chance west of the Darling'\ A ItS/ . .1. Agnc. E('ons. 16:82-93.

Fmser, R ( 1 QQJ) ''1l1e welfare effects of deregulating producer prices". A mer. J. Agric. Econs. 74 21~26

Hertz.! cr. G (I QQ 1 ). "Dynanuc Dcctsions Under Rtsk Apphcatton of Ito Stochastic Control in Agnculture" A mer . .1. A gr. Eron., 73:1126-1 137

!\.fcC oil. J (I QSS) Report of the Royal Commissto17 mto Grain Storage. Handling and Transport Report AGPS. Canberra

i\•1enon. R C (I QqJ). Cominuous-Time Finance Blackwell, C'ambndge MA

I\·tonson, :M ( 1 Q92) tmpubhshcd Honours thes1s Agricllltural and Resource Econormcs. the Umverslty of\Vestem Australia

Newberry. D M.G. ( 1975) "Tenunal Obstacles to lnnovatton" J. l'Jevelop .• 1 I

Perry. ivt \V. and Hillman. B (Eds) (I 991) The Wheat Book -A Techmcal A1£mua1 for Wheat Producers 1l1e Dcptartment of Agnculturc. \VA

Petzel. T E (1984) .. Altcmativcs for l'vtanagmg Agricultural Pncc Risk Futures, Optmns Gnd Govcmment Programs", AEI Occasronal Paper, American Enterpnse Institute for Public Policy Research, \Vashington DC,

Rogers. E t\l ( 1983) Dtffhsion of Innovations 3td ed1tion, ll1c Fn.->e Press, New York

Ryan. T J. ( 1994) "Markctmg Australia.'s \Vheat Crop· 1l1e \Vay Ahead", RevieHl A1arketing & Agnc. Econs. 62:107-121

lltompson, S. R. and Bond. G. E, ( 1 Q87). ..Offshore commodity hedgtng tmder tloatmg e.xchange rates," A mer. J. Agric. Econs. 69'46-55.

Financial Risk Management