Insurance Futures and Options 5
1993 General Insurance Convention
InsuranceFutures
The Browns relaxed,knowing they had hedged theirlosses with Insurancefutures.
Members of the Working Party:
John Ryan (Chairman) Helen Murphy PeterClark Michael Lomax JulianLowe StephenRix
373 1.Executive Summary
On 11 December 1992 trading One objectiveofthe Working Party commencedin Insurancefutures at the hasbeen to overcome the learning curve ChicagoBoard Of Trade.The graphshows thatinvestors face, be theyspeculator or thetrading experience ofthe March National hedger. futurescontract inthe first 6 monthsafter We considerthe uses of Insurance launch.Two pointsstand out. Firstly,the and futuresand optionscontracts as futuresprice had doubledbefore the end of proxiesfor reinsurance - futures behave 1992 whilethe marketfound itsfeet. likeproportional reinsurance whereas Secondly,the trading volumes were initially optionsbehave like non-proportional quitelow but have reinsurance. pickedup over Wethe Pricehave And also Volume summer. looked at The launch pricing, of the CBOT althoughmuch contractscould workremains to not have been be done in this better timed. area (but willit Globalcapacity in be published?!). the reinsurance The marketwas low dynamicsof an after Insurance unprecedented futuresmarket lossesin the late1980’s and early90’s. havebeen a particularinterest for us, HurricaneAndrew had justcreated the sincethe survival of the CBOT contracts largestinsured natural catastrophe loss in appears tolie in the balance at the time USA history,at over$15bn. toof going press. There are valuable As a proxyfor conventional reinsurance, lessonsto be learntfrom other futures Insurancefutures and optionsshould have markets. hadplenty of natural buyers. We hope that the outcome of So why werethe initial trading volumes CBOT’s excitinginitiative is positive. solow? We offerthe following reasons : We are encouragedby the recent increasein trading volume and the fact • regulatoryobstacles thatLIFFE iscurrently considering a UK • steepicamingcurves /European contract, although few details l lackof interested sellers are yet available. • imperfectrisk transfer In any event,it seems that the convergenceof finance and insurance is Thereare barriers tothe market on both unstoppable.We welcome this trend and sides:the finance specialists and insurers believeitwill offer exciting challenges as wereone culture separated by two languages. wellas some threatsfor the actuarial profession.
374 Contents
1. ExecutiveSummary 2. The CBOT ContractsAnd TheirExperience To Date 3. Hedging And Market Players 4. PricingInsurance Futures And Options 5. Development Of New FuturesContracts 6. Wider Repurcussions Of Trading In Insurance Futures And Options 7. RegulatoryAspects 8. Data Sources 9. Glossary 10. Bibliography
Appendices
I Example Of The StatisticalInformation Available II.1 Summary Of The Trading PeriodsAnd ReportingPeriods For ContractsCurrently Traded II.2 Lines OfBusiness Included In The CBOT Contracts II.3 Companies ContributingTo The ISO Data II.4 CBOT LimitationsOf PriceMovements And Holdings III FuturesAnd Options Hedging Examples IV.1 PricingMathematics - SimulatingThe Expected SettlementValue IV.2 PricingMathematics - Developing Formulae For PricingOptions
375 2.The CBOT ContractsAnd TheirExperience To Date
2.0Overview
Section2 summarisesthe Insurance futures and optionscontracts traded on the ChicagoBoard of Trade(CBOT) exchange.Details of thetrading to dateand further technical information regarding the contractsare provided in Appendices I andII.
2.1The CBOT InsuranceFutures And Options Contracts
RegionsCovered
Insurancefutures and optionscan be purchasedcovering three different regions of theUnited States: Eastern, Mid-Western and Western.There is also a National contractcovering the whole of theUnited States.
The followinggraph shows the states covered by each contract:
376 PeriodOf Loss
For each region,futures and optionscontracts are availablein respectof losses incurredduring one of a number of calendarquarters. Each calendarquarter is identifiedby the “contract month”. The contractmonths being traded at the time of writingare as follows: Contract Loss Month Period Sep 93 Apr -Jun 93 Dee 93 Jul- Sep 93 Mar 94 Oct - Dec 93
TradingAnd LossReporting Period
Tradingcommences at an ad hoc date,well in advance of the loss period, as decided by CBOT. Tradingceases six months and fivedays after the end of theloss period. Lossesreported during the loss period and duringthe three months following the end of the lossperiod are includedin the calculationof the settlementvalue. This informationissummarised for the contracts currently trading in Appendix II. 1. The followinggraph summarises the trading and lossreporting periods:
377 ContractValue
The futurescontract value depends on an indexreflecting the ratio of catastrophe lossesto estimatedpremium:
IndexValue of Future = $25,000x ReportedLoss Ratio
= $25,000x (IncurredCatastrophe Losses) (EstimatedProperty Premium ) wherethe estimated property premium isbased on themost recentstatutory annual statementsfiled by theISO reportingcompanies.
Pricesare quoted in tenths of a point(i.e.: 0.1% of$25,000 or $25).Thus a priceof 5.2points is equivalent to 0.052 x 25,000= $1,300,and theminimum amountby whichthe price of a contractcan move (theUnit of Trading) is $25.
The finalSettlement Value will be determinedat the close of the last trading day. An interimvalue will also be computedas atthe end of the loss quarter.
CompilationOf LossData
ISO DATA Inc.a subsidiaryof The InsuranceServices Office Inc. (ISO) compiles lossdata for certain causes of lossfor certain lines of business. Essentially the losses arisingfrom Wind,Hail, Earthquake, Riot, and Floodare recorded. Details of the losscauses for each line of business are shown inAppendix II.2.
We understandthat losses included in the SettlementValue calculationare not restrictedto eventswith aggregatelosses in excessof a given threshold.It is assumedthat the catastrophe element of thelosses recorded will be equal to a pre- statedpercentage of all losses arising from the specified causes.
The informationisprovided by a numberof reporting companies. ISO DATA select thereporting insurers based on insurersize, diversity of property insurance business, and thereceipt of complete and usableloss data on a quarterlyor monthlybasis in a formspecified by the Insurance Services Office.
ISO DATA selectat least 10 insurancecompany groups to be thereporting insurers. A listof the reportingcompanies to be used for any contractmonth will be announcedby theexchange prior to thelisting ofthat contract month.
378 WeightingOf Premium and Losses
The distributionof companies who report to ISO DATA is not entirely representativeofthe US asa whole.In orderto achievemore representativenational and regionalresult ISO DATA willweight both premium and lossesby stateand line.Weights are announced prior to thestart of tradingin any contract month, and willnot be changedduring the life of the contract.
RestrictionsOn PriceMovements
To maintainan orderlymarket the CBOT imposerestrictions on the maximum value of a futurescontract and thedaily price movement. Furthermore,the CBOT has a reportinglimit of 100 contractsinone month,and limitsthe maximum netlong or shortposition that any person may own, control,or carryto 10,000contracts.
Furtherdetails of themaximum dailyprice movement and Futurecontract value are givenin Appendix II.4
Security
At theend of eachday’s trading the Board of TradeClearing Corporation (BOTCC) substitutesitself asthe opposite party to everytransaction. That is it interposes itself asthe buyer to everyclearing member seller,and theseller to everyclearing member buyer.Consequently, a buyer does not need to locate the original seller.
The BOTCC operatesa margin system whereby the clearing members are required to maintaina deposit with the Futures Commission Merchants in respect of theiropen positions.The marginsystem has two components;the initialmargin, which isin effecta deposit, and a variationmargin.
At theend of eachday eachclearing member’s account is marked to market. In this processthe BOTCC collectsmoney fromthose that lost money and redistributesitto thosethat gained. The amountcollected isknown asthe variation margin. Failure to pay therequired variation margin in cash by 10 am thefollowing day resultsin the investorsposition being closed out. The BOTCC guaranteesthe performanceof everycontract.
A resultof themargining system and theBOTCC’s guaranteeis that the contracts shouldbe highlysecure. The CBOT emphasisethat since the start of this system, in 1925,no customerhas lost money due todefault on a futuresposition.
379 2.2Reasons For LaunchingInsurance Futures And Options Contracts
The launchwas a resultof marketingby CBOT, not demand from insurance companies.So why didCBOT decideto setup a marketin insurance futures in the firstplace? The reasonis quitesimple: financial reward. Futures exchanges throughoutthe world are in a continualrace to discovernew productsthat will generatelarge trading volumes and, therefore, large profits for the exchange.
The stakesare very high. Growth inthe world’s derivative markets has been truly phenomenalin the last 20 years.According to the Bank forInternational Settlements, therewere $4.5trillion-worth of outstanding derivatives contracts (of ail types) tradedon exchanges(such as theCBOT andLondon’s LIFFE) as at the end of 1992.
[What istotal world-wide insurance premium volume and premium volume by way ofcomparison ?]
On thedownside, the failure rate of new futurescontracts is quite sobering (see Carlton(1934)). Products that fail to achieveadequate trading volumes are mercilesslyshelved. The majorityof futurescontracts are withdrawn within 10 years of theirintroduction. Not surprisingly,the longevity of new contractsis quite skewed,since those products that do succeedlast for many years.One thirdof all new contractsdo notmake itbeyond 2 yearshowever.
380 2.3 The TradingHistory Of CBOT CatastropheFutures And Options
The followinggraphs show thefutures prices and volumes for the March National and JuneEastern contracts:
Itcan be seenthat although the number of tradesper day hasbeen increasing,there isstill very little trading.
381 2.4 The BenefitsOf A Market In InsuranceFutures And Options
We needto distinguishbetween the benefits to speculatorsand hedgers.For the naturalhedgers - theinsurers and reinsurers- a marketin Insurance futures offers two potentialbenefits, namely risk transfer and price discovery.
The risktransfer benefit isvery familiar, since conventional reinsurance achieves this already.However, conventional reinsurance may be unacceptablyexpensive or itmay notbe availableat all. An open,liquid, competitive market in reinsurance based on standardised,guaranteed contracts could potentially attract new capitalfrom outside theinsurance industry and provideguaranteed capacity at a fairprice.
Some have suggestedthat the attractionof capitalfrom outsidethe insurance industryvia an organisedmarket could dampen theunderwriting cycle. The agency relationshipsin traditional reinsurance could be broken.Traditional long term relationshipsbetween reinsurerand cedant,founded on mutual trustand retrospectivepricing mechanisms, could give way to an era in which normalising pricingmechanisms cannot be reliedupon becausethe barriers to themarket have beentorn down.
Forthe speculator, an Insurance futuresand options market offers the opportunity of reapingspectacular rewards from insurance without making a long term commitment to thebusiness and withoutincurring the up-front fixed expenses of overcoming regulatoryobstacles and developingthe necessary operational expertise. There are timeswhen the risk-rewardrelationship in the traditionalreinsurance market is enoughto tempt even the most risk averse speculator.
2.5 LIFFE’sReaction To The CBOT initiative
The workingparty are not aware of any plansto issuea correspondingcontract to theNorth AmericanInsurance futures and optionscontracts. However thereis a workingparty, including some ofthe major brokers and some ABI members,which islooking at the possibility ofputting together a European index. This would allowa similarcontract to be offeredin Europe to coverEuropean catastrophes. Most ofthe majorLondon futuresbrokers would haveno problemsin actually organising trades inNorth American futures and optionscontracts. Consequently from a reinsurance pointof view there is no particularneed to havea specialfacility inthe UK forUS exposures.
382 2.6 A ComparisonOf InsuranceFutures With Other Futures Contracts
We can learna lotabout Insurance futures by comparingthem to existingfutures productsand markets.Experience of otherproducts and marketscan point towards answerson questionssuch as :
What makesa good Insurancefutures contract ?
How shouldwe valueInsurance futures and options ?
Commodity FuturesAnd FinancialFutures
Futuresare traded on a verylarge number of commoditiesand financialassets on exchangesthroughout the world,the largest of which arethe ChicagoBoard Of Trade,the Chicago Mercantile Exchange and LIFFE.
The commoditiesinclude types of grainand oilseeds,livestock and meat, food and fibre,metals and petroleumand lumber.Financial “assets” include currencies, bonds and varioussecurities indices (although in the case of an indexthere is no physical asset).
An Insurancefuture should strictly be labelled a financial future since it is based on an underlyingfinancial “asset”, namely the value of an indexof insurancelosses. However,the characteristics of an Insurance futures market are perhaps more akin to a marketin commodityfutures than a marketin financialfutures. A market in commodityfutures responds to thesupply and demand characteristicsof a particular industrysegment that meets itsrisk transfer needs by buyingand sellingin the market.
Hedgers And Speculators
Ailfutures markets are made up of hedgersand speculators.Hedgers are meeting theirrisk transfer needs and speculatorsare in it for the profit. The characteristicsof a futuresmarket are determined by theexistence or otherwiseof activehedgers on thebuy and/ or sellside.
383 A marketin Insurancefutures and optionsis likelyto be dominated by natural hedgers- inthis case insurers and reinsurers- meeting their specific risk transfer needs.Compare thisto otherfinancial futures, where the buyers and sellersare very diverseand numerousand speculators play an activepart in the market.
One couldsay thatan insurancefuture is a futurein the commodity “insurance capital”.
AssetsHeld For InvestmentVersus Assets Held For Consumption
Many commodityfutures are held for the purpose of consumption.Copper futures arean example.Financial futures on theother hand arebased on underlyingfinancial assetsthat are generally held for the purpose of investment by a significantnumber of investors.Bond futuresare an example.Some commodityfutures such as futuresin goldand silverare also held by a significantnumber of investorsfor the purpose of investment,rather than consumption.
The distinctionbetween futures on assetsheld for different purposes - investmentor consumption- isan importantone, much more importantthan the “commodity”/ “financial”label. Insurance futures are assets held for consumption - the consumption of“insurance capital”. This distinction isdiscussed further insection 4.1.
384 3. Hedging And Market Players
3.0Overview
Section2 describedwhat Insurancefutures are, this section describes, politely, what you cando withthem and who mightwant to do it,or not.
3.1Who Might Do What And Why ?
The mainpotential doers are insurers, reinsurers, and speculators.As with any such contract,professional “market makers” are needed to providethe bufferbetween buyersand sellers.
Thereare two main thingsan insurermight want to do withthem. First they might want to buy Insurancefutures to “hedge”away, that is reduce, the insurance risk they are exposedto for some of theirportfolio; this is akin to Quota Share reinsurance.Secondly, they might want to protectthemselves against catastrophic lossesbeyond a certainamount by buyingInsurance options; this is akin to Excess of Loss Reinsurance.
In a similarvein, reinsurers might want to sellInsurance futures to allowthem to writeprofitable high level reinsurance covers whilst protecting their capital position. They may alsowant to buy Insuranceoptions as a proxyfor reinsuring their own portfolio.
Speculatorsmight sell or buy Insurancefutures or optionsto make money. On a slightlyless blunt level, they can usethem to diversifytheir portfolio and obtainan exposureto insurance risk. They can do thiswithout going through the troublesome businessof settingup an insurancecompany, or, throwing caution to the wind, buyingshares in an insurancecompany.
385 3.2How Do They Do It?
Doing ItWith Futures- The Theory
Considersomeone who hastaken out an Insurancefutures contract to buy at a price X. The holderwill make a profitif the price goes up aboveX and a lossif the price goes below X. Now theprice of an Insurancefuture increases as the amount of Catastrophelosses for a quarterincreases. This means thatthe holder will make a profitif Catastrophe losses exceed a certainamount and a lossif they are less than a certainamount. This is basically good news forinsurers ! If they can pitchthe number of Insurancefutures and theprice at which they are taken out, so that the gainson theInsurance futures balance out thelosses due to greaterthan expected catastrophes,they can immunise themselvesagainst higher than expected Catastrophelosses - they have “hedged”away thatpart of theirexposure to Catastrophelosses.
Thereare two downsidesto the hedging process. The firstis that they immunise the hedgeragainst lower than expected Catastrophe losses - ifCatastrophe losses are lowerthan expected they make a loss on theirInsurance futures position, which cancelsout the “gain” they would otherwise have made. The secondis that the “If’ in “Ifthey can pitch the number of Insurancefutures and theprice....” isreally more of an “If" More on thisin section 3.3.
Doing ItWith Futures- The Practice
A simpleexample illustrates how a hedgemight be achieved.Take theindustry as a whole as havinga premiumbase of $3,000mper quarter (for lines of businessthat areexposed to Catastrophes)and expectingultimate Catastrophe losses of $400m per quarterfor the third quarter. Consider an insurerwhose portfoliorepresents preciselyone percentof the market as a whole,i.e. premiums of $30m and expected ultimateCatastrophe losses of $4m. Assume thatby theend of thereporting period forthe Futures Contract, 75% ofthe ultimate losses have been reported.
386 At or beforethe start of the quarter, we expectthat the Catastrophe loss ratio for the July-Septemberquarter will be 4/30,and thatonly 75% of thiswill have been reportedby the end of December.The priceone putson theInsurance future is therefore:
$25,000 x 4 / 30 x 0.75 = $2,500
$25,000is the baseprice of the contract,equivalent to a l00% loss-ratio.The numberof contractsthe company needsto takeout issuch that the futures gains / lossestie to the company’s own premiumbase, that is:
$30,000,000/ $25,000 / 0.75 = 1,600 contracts.
Ifthe Catastrophe losses are more thanone assumed,the gain from the Insurance futuresoffsets those losses, and viceversa. For example,if the actualindustry ultimatelosses are $600m, we havethe following situation:
UltimateIndustry losses: $600m (andso Company lossesof $6m) ReportedIndustry losses by Dec.:$450m FuturesSettlement Price: $3,750= 25,000x 450 /3,000 Gainper Futures Contract: $1,250 TotalCompany Gain: $2m = $1,250x 1,600 Company losses- Futures Gain: $4m = $6m - $2m
Thus theactual losses were $2m higherthan expected, but this amount has been off- setby thefutures gain. To helpthe reader see how thislogic works for other loss sizes,Appendix III, Futures Example 1,shows theresultant gains and lossesfor a rangeof Industrylosses. The resultsare summarised graphically as follows:
387 FuturesExample 1 EffectofFutures on overall losses
As theabove graph shows, whatever the actual size of industry losses, assuming that thecompany has losses of exactlyone percent of theIndustry, the overall losses ( = Cat losses+ FuturesGain / (Loss)) arenow fixedat the$4m levelthe company expected.Ithas immunised itself against higher (or lower) than expected losses.
Companiesthat expect to have loss-ratioshigher or lowerthan the Industry as a wholecan also achieve a hedgeby adjustingthe number of contractsthey purchase. FuturesExamples 2 and 3 in AppendixIII illustrate how this can be done. Companiesmay ifthey wish hedge more or lessthan 100% of theircatastrophe risk by buyingmore or lesscontracts. Ifa companyhas a differentreporting speed than theIndustry as a whole,they will have to use theIndustry reporting speeds when calculatingthenumber of contracts, and theirown reportingspeeds when estimating theirown losses,paying particular reference to the estimatedreporting speeds of largeCatastrophe losses, such as those arising from a hurricane.
The problemwith hedging in this fashion arises because an individualcompany is not preciselymatched to theIndustry as a whole:the company may sufferlarge losses when as a whole lowerthan average losses are experienced and viceversa. These difficultiesarediscussed insection 3.3
388 Doing ItWith Options
Doing ItWith Options- The Theory
Considersomeone who hasbought an Insuranceoption at an Exerciseprice of X. The buyerwill make a profitifthe price goes up aboveX, and willmake no money atall if the price goes below X. The priceis in this case that of theInsurance future, and so willincrease as the amount of Catastrophelosses for a quarterincreases. This means thatthe holderwill make a profitif Catastrophe losses exceed a certain amount,but make no money atall if Catastrophe losses are below a certainamount. Again,the trick is to pitchthe number of Optionsand theprice at whichthey are exercisable,so that the profits on theInsurance options balance out the losses due to Catastrophesbeyond a certainsize. If they can do this,insurers have effectively obtaineda proxyfor Excess of Loss reinsurance,covering losses above a certain size.
Again,there are practicaldifficulties, as it is not possibleto exactlymatch the company’sportfolio with that of the Industry as a whole.The difficultiesthisthrows up arediscussed insection 3.3.
Doing It With Options - The Practice
A simpleexample illustrates how buyingan Insuranceoption protects a company againstCatastrophe losses in excess of a certainamount. Take thesame industryand company dataconsidered in the Insurance futures example above. Say thecompany wishesto protectitself against Catastrophe losses in excess of 20% of itspremium. The priceat which one wouldwhich to have an InsuranceOption exercisingis:
$2,500x 0.20/ (4/ 30)= $3,750($2,500 is our currentestimate of the futures price)
The numberof contracts the company would wishto purchaseis the same as forthe futuresexample, namely 1,600
389 Ifthe Catastrophe losses for the industry (and hence the company in this example) aregreater than 20% ofthe premium, the company will make a gain on the Insurance optionwhich offsets losses in excess of 20% ($6m inthis case). For example,if the actualindustry ultimate losses are $1,200m, we havethe following situation:
UltimateIndustry losses: $1,200m(and so Company lossesof $12m) ReportedIndustry losses by Dec.:$900m FuturesSettlement Price: $7,500= 25,000 x 900 / 3,000 Gainper Options contract: $3,750 TotalCompany Gain: $6m = $3,750x 1,600 Company Losses-OptionsGain: $6m = $12m - $6m
Thusthe actual losses were $6m higherthan $6m butthe amount in excess of $6m hasbeen off-set by theoptions gain. To helpthe reader see how thislogic works for otherloss sizes, Appendix III, Options Example 1,shows theresultant gains for a rangeof Industry losses. The resultsare summarised graphically as follows: OptionsExample 1 Effect of Option Overall losses
As theabove graph shows, whatever the actual size of IndustryCatastrophe losses, thecompany hasnow limitedits’ Catastrophe losses to a maximum of 20% of its’ premium($6m). It has of course had topay forthis privilege - the cost of buying the Insuranceoption.
390 Companiesthat expect to havedifferent loss-ratios than the Industry as a wholecan adjustthe number of contractsof theInsurance option in a similarfashion to that illustratedinFutures Examples 2 and 3 inAppendix III. Similarly, ifa companyhas a differentreporting speed than the Industryas a whole, they will have to use the Industryreporting speeds when calculatingthe numberof contracts,and theirown reportingspeeds when estimatingtheir own losses,as before. In practicea company willnot havelosses that are a precisemultiple of thoseof theIndustry as a whole which willaffect the efficiencyof thismethod of protectingagainst losses, as discussedin section 3.3.
The protectionpurchased by suchan Insuranceoption goes up to the 200% Loss- ratiolevel, at which losses are capped - inpractice then, this is effectively an “open layer”.Even HurricaneAndrew didnot lead to the 200% levelbeing breached for the EasternContract for the simulatedfutures prices calculated by CBOT for that quarter.
Itis possible to obtaina proxyfor a “layer”of reinsuranceby sellingan Insurance optionat a pricecorresponding to a higherLoss-Ratio. The followinggraph shows a similarsituation tothe Options Example 1, but with the addition of a Short position at an Exerciseprice of $9,375.This corresponds to a layerof protectionfor the companyin the range 20%-50%.
OptionsExample 2 Effect of buying and selling options on overall losses
391 In thiscase, the protectionin excess of $6m works as before but now for Industry losses in excess of $1,500m (company losses in excess of $l5m), the company will sufferadditional losses in excessof $6m. These lossesare in excessof the layerof protectionafforded by the combinationof buyingand sellingInsurance options. This reducedlevel of protectionhas a reduced costas the company now receivesmoney forselling the option.
The calculationsbacking the numbers inthe OptionsExample 2 graph are alsogiven inAppendix III,Options Example 2.
Doing It Together !
A company can adopt a combinationof Insurancefutures and optionspositions to combine hedging and capping,in a similarfashion to an insurerwho takesout both Quota Share and Excess of Loss reinsurancetogether.
3.3 A Comparison With TraditionalReinsurance
Tailor-Made Products
Conventionalreinsurance has the advantagethat it directlypasses the riskto the reinsurer.It is a tailormade productand, subjectto any coveragedisputes and the securityof the reinsurer,will indemnify the reinsuredin respectof the lossesthat arise.It is therefore much easierfor the reinsuredto determinehis requirements and to be certainthat he willbe reimbursed in the event of a covered loss.
PricingTransparency
The pricepaid for, say, Insurance options as a proxy for excess-of-lossreinsurance willbe more clearlyvisible than for reinsuranceand comparisonsmore easilymade between the costof the optionsand the expectedcosts of the underlyingrisks. The currentcost of the optionsappears, as one might expect,to liebroadly between the costof conventionalreinsurance and the expected pure risk costs of the underlying events.
392 Lack Of Corrclation
Insurancefutures have the obviousdisadvantage that they only pay basedon an industryindex. To theextent that the reinsureds losses are not 100% correlatedwith thatindex the coverage may notpay when itis required. Of course,there may be a windfallgain for losses that the reinsured have not participated into the extent of the marketas a whole.Clearly this is an issuethat an individualcompany needsto considerfrom its own pointof view.
The lackof correlationrefers not only to eventsbut also to beingmuch more heavily exposedto a specificloss. This seriously hampers the use of Insurancefutures to hedgeaway insurancerisk. Consider a company thatfeels it has taken on too much exposurein a givenregion, and wishes to reduce this by buyingInsurance futures in a similarfashion to Quota-Sharereinsurance. Itcould be thecase that it may make a losson itsfutures position, while still having higher than expected losses itself. PurchasingInsurance futures would inthis case increase the maximum amount the companywas exposedto throughCatastrophe losses, rather than the intended effect ofdecreasing its exposure to Catastrophe losses. Conversely however, it could make a profiton a futuresposition atthe same time as itmade smallerthan expected losses on its’own book ofinsurance business.
To estimatethe extent to whichan insurer’scatastrophe losses are correlated with thoseof theindustry as a wholecan be a complexprocess. In theextreme it may involveassessing an insurer’sexposure at, say, post-code level relative to some total insuredexposure and comparingthese figures to storm-tracksof hurricanes.
Itis possible that markets may ariseproviding coverage for the difference between theindex and a particularcompany’s account. This would havethe advantage of being able to have the flexibilityof a futures or options and without the disadvantagesof the lack of correlation.IfInsurance futures become widely used, it isnot difficultto imagine how thesebooks of businesscould be put together. Essentiallya major player who provideda lotof thiscover would have a correlated index.The parallelisexact with diversifying an investment portfolio.
FlexibilityOf Futures Contracts
Assuming the market developsfrom itspresent state, then a futurescontract becomesmuch easierto organiseduring the year. There is also a greaterdegree of flexibilityof cover. There is no needto worry so much aboutcontinuity or indeed aboutthe volumes of business that one islikely to writein the current year. Against thisof course the reinsured needs to evaluate the exposure he needs.It may be more difficulttolayer unless the option market develops.
393 Security
The securityissues are differentin that it is unlikelythat a major catastrophewill createany securityproblems in respectof a futurescontract. It is unlikely, at leastfor the major exchanges, that these contractswill ever become a major part of their business.Given the importance of financialderivatives and theiruse by the banking system, itis almost inconceivablethat the financialauthorities could allow a major securitiesfutures exchange defaultand the securityof the contractsis a functionof the exchange as a whole. Consequently securityis less likelyto be a problem. Clearly some of the major reinsurersare very soundly based, but obviously in the event of a major catastrophe,their security is going to be tested to its limits.
Underwriting Cycle
The cycle of the futureswill be different.The price on the futuresmarket willbe purely market driven and indeed will provide some secondary check on company pricing.There willnot be any element of pay-back as clearlyplayers can eitherenter or leave the market according to itsperceived attraction without any difficulty.This is not the case with a reinsurancecompany. A futurescontract will not provide access to the other servicesthat a reinsurermight provide such as claims handling advice and so on.
Evaluation Of Exposure
A futures contract puts a greater onus on the ceding company to determine their actualexposure to any particularloss in determining the amount of cover required, whereas the reinsurerwill need to evaluate that more when undertaking pricing.
Transaction Costs
Transactioncosts are alsolikely to be lessin terms of futuresdriven market than with a conventionalmarket. In part thiswill be due to the factthat futurescontracts are not tailormade to the reinsured,whereas reinsurance contracts are. It is also likely thatintroductory or brokerage costs would be less.
3.4 The Players In An Insurance Futures Market
A market is formed by buyers and sellers coming together to trade. A “natural” market is one with a lot of automatic buyers and sellers.For example, a market in grain is a naturalmarket, because producers and users are on differentsides of the market. Farmers want to fixthe priceat which they selltheir produce and the users want to fix the price at which they buy their raw materials.
394 Ifthere are only “natural” market participants on one sideof a market- buyersor sellers- then tradingvolumes may suffer,and an illiquidmarket is a bad market. Basiceconomics suggests that the forcesof supplyand demand willgenerate a marketif someone is willing to trade.For example,if hedgers desperately want to hedgethen they will be willingto pay a pricethat is attractive enough to attract speculators.This argument assumes well-informed investors.
We suggestthat a marketin Insurance fiutures isnot a “natural”market. It is likely tobe a one-sidedmarket. There are natural hedgers - theinsurers and reinsurers- but one onlyexpects them to be on one side of the market at any point in time, dependingon the currentposition in the underwritingcycle. When traditional reinsuranceprices are highand capacitylow, the insurersand reinsurerswill be buyersof insurancefutures, each meetingtheir risk transfer needs. In theory, reinsurerscould be alsobe sellers,but why shouldthey write insurance via an organisedfutures or optionsmarket when thereis the conventionalroute ? Speculatorsare likely to be thesellers, although the high possibility of a smallprofit alongsidethe small probability of a largeloss does not fit the normal perception of a speculator’sutility curve. This risk profile could however be diversifiedaway.
395 4. Pricing Insurance Futures And Options
4.0 Overview
Boththe modelling of insurance losses and thepricing of futuresand futuresoptions arehuge and complicatedsubjects, so in a generalpaper such as this,an outlineof theapproaches that can be adoptedis all we canrealistically hope to achieve.To put our thoughtson valuationof Insurancefutures and optionsinto context, it is necessaryto givea briefresume of thetechniques for valuing conventional futures and options.The reader’sattention is drawn to the Glossaryin section9 for a descriptionoftechnical finance terms.
4.1Valuation Of TraditionalFutures Contracts
Price Versus Value Of A Futures Contract
Itis important to appreciate the distinction between a futuresprice and the value of a futurescontract.
The futuresprice at any given time during the trading period of a particularcontract isthe delivery price of a contractentered into at that time. When the contractis initiated,thedelivery price is agreed between buyer and seller,bearing in mind that thereis no initialcash outlay; therefore in theory, the initial value of the contract shouldbe zero.This provides the basis for determining the futures price at all times.
Forward AgreementsAre EasierTo Value....
Forwardagreements are easierto valuethan futures contracts because they only involvea singlepayment at maturity.In contrast,futures contracts are “marked to market”,that is, they are settled daily during the life of the contract.
Forward and futuresprices are generallyvery close to one another when the maturitiesofthe two contractsare the same. Itis therefore fairly safe to make lifea littleeasier and to consider forward prices as a proxyfor futures prices.
396 In fact,when therisk-free interest rate is constant and the same for all maturities, forwardprices and futuresprices can be shown to be exactlythe same, using a dynamicarbitrage argument (see Hull (1993)). This argument can be extendedto coversituations where the interest rate is a known functionof time.
When interestrates vary unpredictably, forward and futuresprices are no longerthe same and the relationshipbetween them iscomplex (see Cox, Ingersolland Ross (1981)) and dependson thecorrelation of the underlying asset with interest rates.
We findthat when theunderlying asset is strongly positively correlated with interest rates,futures prices will tend to be higherthan forward prices. This is because when theprice of theunderlying asset increases, an investorwith a longfutures position makes an immediategain that is invested at a higherthan average rate of interest,and viceversa. An investorholding a forwardis not affected inthis way by interestrate movements.
AssetsHeld For InvestmentVersus Assets Held For Consumption
Futuresand forwardprices on assetsheld significantly forinvestment are fairly straightforwardto calculate, whereas this is not the case for contracts on assetsthat are heldfor consumption.This is because if a marketis strongly influenced by hedgers,then levelsof inventoryand storage costs and supply and demand characteristicswilltend to influence pricing to a potentiallygreat degree.
ValuingForwards On AssetsHeld For Investment
The generalapproach to valuingforwards on assetsheld for investmentby a significantnumber of investors isto use a simplearbitrage argument. A portfolio in the underlyingsecurity plus cash is constructed that replicates the pay-offof the forwardat maturity.In theabsence of arbitrageopportunities, the value of the forwardmust be thesame as the value of the replicating portfolio.
Thisis a common approachused in modern financialeconomics to value many types ofderivative securities, including option contracts.
397 Forwards On Non-income Paving Securities
Appendix IV.2 considers a forward on a simple financialasset that provides the holder with no income, such as a discountbond. A forward price at time t,F(t), is derivedas:
where S(t) is the price at time t of the asset underlying the forward contract and r(t) isthe risk-freerate of interestper annum at time t for an investment maturing at time T.
Forwards On Other Types Of Assets Held For Investment
The valuationapproach is the same as for non-income producing assetsheld for investment.Relationships between the forward price, delivery price, risk free interest rate,time to maturity and the value of the contract can be derived to produce similar resultsto thatquoted above. These are outlinedin Appendix IV.2.
Valuing Forwards On Assets Held For Consumption
The pricingof forward on assetsheld for consumption is complicated by the factthat storageof the asset (commodity) may be costlyand also spot markets may be non- existentor too thin for arbitrage.
For commodities that are not significantlyheld for investment purposes, arbitrage valuationarguments need to be treatedwith extreme care. Individualsand firms require the commodity for its consumption value - it is utilised as part of their businessrather than held as an investment.Holding the commodity may be preferred to holding a futures contract because futures contracts cannot be consumed. Benefitsof ownership might includethe abilityto meet unexpected demand for the commodity or keeping a productionprocess going.
These benefitsare sometimes referredto as the convenience yield provided by the underlyingasset. Convenience yield is thereforelike a liquiditypremium. When inventoriesof a commodity are plentifulwe expect low convenience yieldsand vice versa.
The usual approach to valuingforward contractson assetsheld for consumption is based on “convenience yields”and “storage costs"
398 Extending The Arguments To Indices
We have shown thatfutures can be valuedby lookingat the forward price for one security/ commodity. The argumentscan be extendedto an index;the security is just substitutedby a portfolioof stocks comprising the index, and thedividend stream, whereappropriate, replaced by thedividends on thestocks in the portfolio.
How Do FuturesPrices Compare To ExpectedFuture Spot Prices?
The futuresprice at a pointin time is the delivery price of a contractentered into at thattime. This may be differentfrom theexpected future Spot Price. If the buyers arehedgers, and thesellers are speculators, one can arguethat the futures price will be belowthe expected future Spot Price. This is because the hedgers are prepared to acceptslightly reduced returns, as theyare happy to havethe benefit of the hedge, whereasthe speculators require compensation for the extra risk they are taking on. The positionwhere the futuresprice is below the expectedfutures spot price is known as Normal Backwardation.The conversesituation is known as Contango. Thesetwo casesare illustrated below:
Futures Price Drift
399 Now the more riskyan investmentis, the more an investormight expect to be rewardedby higherreturns. Ina similarfashion to theargument developed above, by equatinga security with a Long Forwardposition plus cash, one can show that:
E(S(T))is our expectationof theSpot price at timeT, r(t)is our risk-freerate of return,as before, k(t) is now a measureof the systematic risk of the investment over theperiod (T-t), i.e. the risk that cannot be diversifiedaway. One canthen go on to equateforwards with futures to develop the result for futures.
Clearly,as k 4.2How Can We ExtendThese Arguments To InsuranceFutures? We have alreadyseen that Insurance futures are somewhat differentin nature to otherfinancial futures. We havealso made theimportant distinction between futures contractson assetsheld for consumption and assets held for investment. Ifwe thinkof theasset underlying an Insurancefuture contract as being “insurance capital”,we can seethat this is akin to an assetheld for the purpose of consumption (i.e.it is employed, or utilised,inthe business of insurance).This puts Insurance futuresin the “tricky”valuation category. Just as levelsof inventorywill affect commodityfutures prices, so levelsof industrycapital will affect insurance futures prices,unless a significantnumber of speculators can be attractedto the market. 400 As a startto valuing Insurance futures, we canlook at how theprice may varywith theexpected losses during a quarter.There are three separate periods to considerfor arrivingat an InsuranceFutures price - before,during and afterthe Loss quarter. Beforethe Loss quarter, the buyer will simply have an expectationof what the losses willbe atthe end of thequarter to work on.During and afterthe Loss quarter,one willhave further information about losses that have happened during the quarter, and thena viewon how much of theselosses have been reported three months after the quarterhas finished. Beforethe Loss quarter then, one couldexpect the price to be a functionof the form: Duringand afterthe quarter, on would expectthe price to be of thesimilar form: F(t)= E(S(T) perceivedI Incurred losses to date )*e(r-k)*(T-t) The r and k makingallowance for the time value of money and the non-diversifiable riskinessof the investment.Following the argumentsoutlined previously, as the buyersare expected to be hedgersand thesellers are likelyto be speculators,we would expectthis approach to lead to the futuresprice exhibiting “Normal Backwardation”,that is the futures price would tend to be less than the expected SettlementPrice at the end of the period. 401 The expectedvalue of the Settlement Price at the end ofthe period can be arrivedat by makingassumptions either about the Catastrophe Loss-ratio as a whole,or by buildingup a model forthe frequency and severityof theCatastrophe losses one expectsduring a givenquarter for a givenfutures contract (Eastern, National etc.) In AppendixIV. 1 we haveoutlined an approachto derivingthese expected prices at theend of thetrading period, i.e. the expected Catastrophe losses at the end of the period,by making broad assumptionsabout the frequencyof differenttypes of Catastropheand theseverity ofdifferent types of Catastrophe.For illustrationthisis done forthe National Contract, with no attemptto modifythe results for seasonal effectsby makingthe model specific to a particularLoss quarter. This is in no way meantto be a Working Party“view” on the likelylevel of Futuresprices, and is intendedjust to illustrate thetype of technique that could be applied. As mentionedat the start of this section and insection 4.1, care needs to be takenin applyingan arbitragetype of valuationto investmentsheld for otherthan pure investmentpurposes. In thecase of Insurancefutures, one would want to tryand make some allowancereflecting the added valueor otherwiseof being exposed directlyto insurancerisk compared to obtainingthat exposure through Insurance futures. 4.3Valuation Of TraditionalOption Contracts BriefOverview The valuationof traditionaloption contracts ismore complexthan valuing Futures. In orderto examinethe applicability of some of thecurrent models, we willgive a quickdescription ofhow some ofthe current models work and theassumptions they make about the underlyingstochastic process. This willenable us to give an indicationasto why some ofthese models cannot be appliedto value an optionon an Insurancefuture and make suggestionsas to which existing models seem to havethe mostdesirable features for being adapted to copewith Options on InsuranceFutures. We haveindicated in thelist of referenceswhere theinterested reader may obtain detailsofthe models mentioned here if they wish to investigatethem further. 402 Black-ScholesModel And Variations Thisis the classic option-pricing model (see Black and Scholes(1973)) around which many othermodels are based. In itsbasic form itis a model forpricing European optionson non-dividendpaying securities. The key startingassumption is that the stockprice follows a Wienerprocess. This is defined in the Glossary,section 9. Broadlythis means that the changes in stock price follow a Brownianmotion type of process.The processassumes that returns over a periodare normally distributed and it can be shown thatthis implies that given a startingstock price now, the distributionofthe stock price at some futuretime is Log-normally distributed. Blackand Scholesshowed that,making the assumption, amongst others, about the Stochasticprocess above, the option price could be arrivedat in a closedform by solvinga differential equation S followingthe notation adopted previously, beingthe volatilityof the stock) givencertain boundary conditions. This depends only on therisk-free interest rate, so theexpected return over the period does not affect the results, and it is the volatility of the stock,amongst other factors, which affectsthe priceof the option.Other assumptionshave to be made, such as continuoustrading, no transactioncosts, interestrates and constant volatility forall maturities and so on. Thismodel can be extendedto commoditiesand dividendpaying stocks. It can also, withsome persuasion,be adaptedto provideinformation about American options. One can alsoassume that an indexfollows the geometricBrownian motion, and applythe methodto variousindices. This is not quiteconsistent, as if individual stocksfollow this Brownian motion process, a weighted average of those stocks would not.Never-the-less, itis just a model forthe processand works wellin practice. Blackalso showed that, assuming the futures price was ofthe form one couldliken futures to a securitypaying dividends and, with certainassumptions about onecould produce a pricingmodel for options on futures. Variationsinclude a model wherethe interest rate follows a stochasticprocess (see Merton(1973)) and one where thevolatility follows a stochasticprocess (see Hull and White(1990)). Can, Say,The BlackModel Be AppliedTo InsuranceOptions (On Futures)? NO. 403 Why Not ? The underlyingprocess in allthe modelsabove is the Brownianmotion typeof process.In English,this means thatchanges in the priceof the derivativeunder considerationvary up and down in some random way. Thisdoes not seem to be consistentwith the behaviour we mightexpect of Insurancefutures, that is to say Catastrophelosses. At thestart of a periodwe mighthave an expectationof the final SettlementPrice at the end of theperiod. We mightexpect this to graduallydecline overthe Loss quarter iflosses do notoccur, as shown below: Insurancefutures and Wiener processes As lossesdo occur,the futures price might be expectedto jump. This will not be a Markov process,which is part of theassumptions of theWiener process. Once the pricehas jumped to a givenlevel, the price in the future will depend on thepast. The stochasticvariation will not be thesymmetric variation underlying the movement in stockprices -there will be a much greatertendency to increase,aslosses occur, than to decrease,as lossesdo not occur.The linch-pinof theassumptions that lead to much of thecurrent option-pricing theory does not, it seems to us,apply in the case ofInsurance options (on futures). 404 Thatis not to saythat current models do not add some value.In realitystocks and indicesdo notprecisely follow the Wiener process. If one hasa feelfor the bias that isintroduced by applyinga modelto a processfor which theassumptions are not quitecorrect, use can still be made ofthat model. Say,for example, one thoughtthat in realitythe “tail”of the distributionof the instrumentunder consideration was fatterthan the Log-normaltail implied in the Black-Scholesmodel and its’variations. This would mean thatthe model would tend to under-pricethe option - thisis still useful information as itprovides information aboutthe minimum priceof theoption. It may alsobe thatthe Insurance futures markettakes on a lifeof itsown, and the “noise”from generalfutures trading becomes such thatthe insurancenature of the spikesin priceare no longera significantfeature and themodel can be more readilyapplied. Are ThereOther Models One Could Try And Apply ? Yes. What Are They ? Thereare models that assume jumps inthe stochastic process, rather than just the driftup or down. One such model is the Pure Jump Model (seeCox, Ross & Rubinstein(1979)). The gistof thismodel isthat in time dt, say, the current stock price,S, can jump from: S to Su,with probability dt S to Se with probability1 - dt Thishas a lotmore appealthan the Wiener process, as an explanationofmovements inInsurance futures prices. We expectthe futures price to driftdown as lossesdo not happen,but to “jump”up as a Catastropheloss occurs. In thisbasic case the jumps areof fixedsizes; extensions of the model can allow for jumps of varying sizes.In thecontext of stockprices, this model has the disadvantagethat it only allowsjumps up - inan Insurancefutures context, this becomes a desirablefeature. Thereis also a Jump DiffusionModel (seeMerton (1976)),which superimposes the typeof Jump structureabove onto an underlyingdiffusion process. 405 Where Do We Go From Here ? The pricingof options, especially the more exotictype of optionwe are considering, isa highlyspecialised area. All we havetried to do hereis to indicatewhy we think thatmost standardoption-pricing models cannot be validlyapplied in an Insurance Futurescontext and to indicateother types of modelthat could perhaps be a more fruitfulsource in trying to price Insurance options. The detailsgiven in Appendix IV.1 describe an approachto simulatingthe price of Insurancefutures at the end of theperiod, by simulatingnumbers and amountsof losses.This type of approachcan leadone to arriveat the E(S(T)) indicated inour suggestionof valuingInsurance futures above, and couldbe the startingpoint for tryingto valueInsurance options by some sortof Jump process.The indicated abovein the Pure Jump model,is akin to thePoisson probability of a Catastrophe occurring;the “u” in the Pure Jump modelis akin to the amount of loss this gives rise to;a modelwhich forms a proxyfor Insurance Options would haveto havethe “u” itselfa stochastic variable. Alternatively one could make a simplerassumption about thedistribution ofthe overall Loss-ratio, and try to build a model around this. 406 5. Development Of New Futures Contracts 5.0Overview Effortsare currently underway to developInsurance futures and optionscontracts basedon UK and Europeaninsurance losses, for example windstorm losses. The main efforthas been in trying to constructa reliable index of European windstorm losses.Can theEuropeans learn anything from theCBOT experienceto date? Could CBOT have marketedthe contractsbetter ? This sectionlooks at the likely ingredientsof a successfulInsurance futures contract and pointsout a few unpalatableones along the way. 5.1 What Makes A Good FuturesContract ? Quitesimply, a good futurescontract is one thathas a large,liquid market. The experienceof theCBOT contractsto datehas not so farbeen encouragingin this respect,although CBOT haveprofessed themselves happy with the progress to date and pointedout thatother futures markets have takenseveral years of market familiarizationbefore the trading volume took offNever-the-less, Insurance futures havebeen tradedvery little and thereis, as yet,no significantmarket in the futures option.Is thisbecause the Insurancefutures as currentlydesigned are inherently unsuitablefor this type of market or becausewe areseeing initial reluctance to trade a new type of derivative? FeaturesOf The UnderlyingAsset Thereis a lotof empirical evidence on what makes a successfulfutures contract (see Carlton(1984)). Carlton cites five important features that a commoditytraded on a futuresexchange should possess to be successful: (a) uncertainty (b) price correlationsacross slightly different products (c) a largepotential number of interested participants andindustrial structure (d) largevalue of transactions (e) price freelydetermined and absence of regulation Insurancescores on allof thesecounts, with the possibleexception of the price correlationrequirement. Since insurance risk can be very specific,a hedging instrumentsuch as a futureor an optionbased upon a standardised,industry exposuremay onlybe weaklycorrelated with an insurersrisk transfer needs. 407 The CBOT catastrophefutures are based on broadgeographical zones - Western, Midwestern,Eastern and National- and averageindustry experience in each zone. Thesezones are not well matched to most insurersinwards risks (although natural- hazard-proneTexas is included in both the Eastern and Midwesternexperience so a suitabletrading strategy can isolate Texan experience) Futureson an insuranceindex also present a numberof particularproblems that we canadd to Carlton’s list : (i) no spot market (ii)complex nature of the underlying asset (iii) concernsabout “insider” information (iv) high marginrequirements (v) existenceof a naturalsubstitute - conventional reinsurance (vi) naturalhedgers only on one sideof the market Theseproblems are dealt with in turn below. No SpotMarket Incommon withcertain other commodity futures, there is no establishedspot market in which insurancecan be traded.For example,there is no spotmarket in the underlyingasset for frozen orange juice concentrate futures when theoranges are stillon the trees! An underlyinginsurance “price“ index must first be identifiedor constructed before a market in insurancederivatives can be established.CBOT’s solutionwas to commissionthe ISO to createa lossratio index based on catastrophelosses and premiums.A UK orEuropean initiative would need to go down a similarroute. Investors’lack of familiarity with a new indexis a realbarrier. Itsteepens the learning curvefor a new productand alsomakes arbitrage opportunities difficult to identify. ComplexityOf The UnderlyingAsset The complexityof theunderlying “asset” in thecase of an insurancederivative is itselfa barrier to marketliquidity. A large contributory factor to futuresmarket liquidityisthe presence of “locals” - local exchange specialists that make theirliving out of tradingnarrow arbitrage anomalies. Locals tend to be attractedto simple contractsthat are easily understood, with well established spot markets. 408 “Insider”Problems CBOT isnot using an indexdetermined by an outsidebody butthe published returns ofabout two dozeninsurance companies. This inevitably raises the question of moral hazard.This is especially the case for the final three months of tradingof a contract afterthe lossperiod. Outside investors may considerthemselves to be at a disadvantageintrading these contracts, however this perceived risk should be largely eliminateddue to the inclusion of a considerablenumber of insurers, no one ofwhich constitutesa substantial part of the index. High Margin Requirements The volatilenature of catastrophe insurance means that margin requirements must be veryhigh. This might discourage potential speculators. ConventionalReinsurance As A Substitute A hurdlefor insurance futures as a hedginginstrument is the existenceof a well developedand understoodsubstitute market for hedging risk - the conventional reinsurancemarket. Why use a standardisedrisk transfer tool when a “bespoke” solutioncan be negotiatedwith another party ? NaturalHedgers Will Dominate One SideOf The Market We havealready commented in section 3 on thelikely characteristics of a market in Insurancefutures. One problemis that there do notseem to be naturalhedgers on both sidesof themarket, buying and selling.While we mightexpect insurers and reinsurersto be on differentsides of themarket, the existence of the conventional reinsurancemarket casts some doubt on this. Summary Perhapsthe most important attribute of a marketin Insurance futures would be its liquidity.The one advantagethat a futuresmarket can offerover the conventional reinsurancemarket is guaranteed capacity and a fairmarket price,by accessing capitaloutside the insurance industry. Unfortunately,we seem to have a Catch22 (seeHeller (1962)) that Yossarian would be proudof. To attractinvestors the market must be liquid,to be liquidthe market must attractinvestors! The challengefor CBOT and,perhaps at a futuredate, LIFFE,is how toovercome this negative feedback. 409 6. Wider Repercussions Of Trading In InsuranceFutures And Options 6.0Overview Thissection looks at some of the potentialimplications of an Insurancefutures market. 6.1Smoothing Of The UnderwritingCycle If Insurancefutures were to be widelytraded, it is likelyto smooth out the underwritingcycle. This would be to thebenefit of allconcerned. The reasonfor smoothingout the underwriting cycle is that it would be verymuch easierto enteror leavethe futures market than the reinsurance market. 6.2Implications For The ReinsuranceMarket TransactionCosts Forreasons mentioned above, it is likely that the transactional cost would be lesson thefutures, which would tendto favourthe use of futuresespecially ifa market developsbetween the differences between a particularcompany and an index.This wouldnot be good news forreinsurance brokers, unless they took some intermediate rolein this process. CostOf Cover If the use of Insurancefutures contracts became wide-spread,it may make conventionalreinsurance more expensiveand notless. This is because the contracts wouldtake a percentageof the bread and butterbusiness out of themarket, leaving reinsurerswith a lesswell diversifiedportfolio. It could alsomean thatthe reinsurancemarket may reducefor more esotericrisks. A furtherconsequence of a reductioninthe call on traditionalreinsurance may be thatreinsurers actually become sellersofInsurance futures, to obtaina diversified portfolio. Converselyone couldargue that in times of low capacity,ifInsurance futures lead to an influxof new capitalinto the insurance industry, this can only serve to reducethe priceof reinsurance. 410 SellingThe Market Share The issuesof insurancecompanies manipulating the market might need to be considered. 6.3Insurance Futures At Lloyd’s Ithas been suggested, though not widely, that one way of solvingthe Lloyd’s open yearproblem would be to use futures.Essentially one couldderive a contract wherebyshares of pollutionand asbestosrisks were simplytraded. This is an idea thathas some interestingconcepts within this area. However at this stage this workingparty has not made anyattempt to analyseit further. 411 7. Regulatory Aspects 7.0 Overview Thissection briefly describes some of theareas where clarification of the regulations regardingInsurance futures and optionsis required and some ofthe work beingdone inthis area. 7.1 InsuranceFutures And OptionsAre Not InsuranceContracts Insurancefutures can achievesimilar risk transfer objectives as proportional reinsuranceand optionson Insurancefutures can behave like non-proportional reinsurance(see section 3 for more details)However Insurancefutures and options arenot contracts ofinsurance. This has implications for: (i)accounting treatment (ii)statutory solvency (iii)taxation treatment 7.2Institute And FacultyWorking Party The Instituteand Facultyof Actuaries have a workingparty, chaired by BillAbbott, thatis discussing with the DTI valuationregulations (from an insurancecompanies pointof view)for derivative instruments including futures and options. However thisis largelyinvestment driven and the issuesinvolved are not the same for Insurancefutures, although initial regulations are likely to cover them in the same way as any othercontract. It is likelythat information will be releasedby the Instituteand Faculty working party to a wideraudience in the not too distantfuture. NorthAmerican regulators have not in general approved the use of futures contracts, At thetime of writing Illinois isthe only state to have authorised the use of Insurance futuresand options,whilst New York and Californiaare in the processof passing legislationto allow their use. American regulators have more draconianpowers than UK authorisationinterms of actuallybanning the use of suchinstruments, so the authorizationofthe use of Insurance futures as hedging instruments by severalstates isa positivemove. Now thatseveral states have given them the greenlight, it is likelythat other states will follow. 412 Itshould be recognisedthat even if the various regulatory authorities do not allow creditto be takenfor statutory purposes for these instruments, they could be usedto offsetthe accountingmechanisms. We are not aware of any furtheraccounting standardstaken on thisarea. Ideally profit / loss on reinsurancefutures and options wouldbe addedto the underwriting result. 7.3 Other Considerations As faras theStock Exchange is concerned, it would be open to a company to point outthat although it may be likelyto show a largegross underwriting loss, this has no materialnet effect on itsoverall financial position because of thefutures contracts it holdswhich will provide corresponding investment profits. On theassumption that theStock Market behavessensibly, one would expectthe share prices to recognise thisbenefit. 413 8. Data Sources Due tothe low volumesof contractstraded none of the major financial journals are listingthe Insurance futures contracts at present. The CBOT maintainextremely good informationservices. The CBOT Educationand MarketingDepartments have compiledan explanatoryfolder and providea query answeringservice, which are described below: London Explanatoryfolders can be requestedfrom and questionscan be directedto: SuzanneMatus 52-54Gracechurch Street London EC3V OEH Tel:07l-929-0021 Chicago Explanatoryfolder can be obtainedfrom the Literature Services Department: Tel:0101-312-435-3558 Questionscan be directedto: Dena Karras The Chicago Board of Trade 141 West Jackson Suite2280 Chicago Illinois60604 Tel:0101-312-435-3674 414 Historicalstatistics forthe contracts can be obtainedon diskettein various formats for a fee from: Robert Pfannkuche The ChicagoBoard of Trade 141 West Jackson InformationSystems Dept. Suite 940-A Chicago Illinois60604 Tel:0l0l-800-288-CBOT (2268) The typeof statistical information available isillustrated inAppendix I. Informationregarding the contracts can alsobe obtainedfrom the usual Information agencies,such as Reuters. 415 9. Glossary American Option Contract These are optionsthat may be exercisedat any time up to the expirationdate (maturity). Most of theoptions traded on exchangesare American, but they are more difficultto valuethan European Options. Arbitrage An arbitragestrategy is a tradingstrategy in whichthe arbitrageurmakes a profit whiletaking no riskat all.Forces of supply and demand mean that arbitrage opportunitieswillnot exist for long in an open,liquid market. Call Option A CallOption gives the holder the right to buy theunderlying security by a certain datefor a certainprice. Contango The futuresprice at a pointin time is the delivery price of a contractentered into at thattime. This may be differentfrom the expected future Spot Price.If the buyers arespeculators, and thesellers are hedgers, one can arguethat the futures price will be abovethe expected future Spot Price. This is because the hedgers are prepared to acceptslightly reduced returns, as theyare happy to havethe benefit of the hedge, whereasthe speculators require compensation for the extra risk they are taking on. The positionwhere the futuresprice is above the expectedfutures spot price is known as Contango. DeliveryDate (Futures/ Forward Contracts) The dateon whichthe holder of theshort position delivers the underlying asset to theholder of the long position, or settles the cash difference between Spot Price and DeliveryPrice. 416 DeliveryPrice (Futures/ Forward Contracts) Thisis the price at which the holder of a shortposition ina contractagrees to sellthe underlyingasset to the holder of the long position atmaturity of the contract. Itis fixed at the commencement of thecontract and isdetermined on thebasis that thereis no costto entering into the transaction at the outset. European Option Contract Theseare options that may onlybe exercisedatmaturity of the contract. Forward Agreement Thisis a bilateralagreement to buy or sellan assetat a certainfuture Delivery Date fora certainDelivery Price. The agreementis usually between two financialinstitutions or between a financial institutionand one ofits corporate clients. Itis not normally traded on an exchange. No cashchanges hands initially. Therefore at the time the contract isentered into, the DeliveryPrice negotiated should in theory be such that the value of the forward contractto both parties is zero. A Forward Agreementis settled entirely at maturity,when theholder of the short positiondelivers the amount of theunderlying asset specified inthe contract to the holderof thelong position (if possible) in returnfor a cash amount equal to the DeliveryPrice. Forward Price Thisis the Delivery Price for a contractnegotiated on thesame day. In practiceit is negotiatedby buyerand seller. In theory this is the Delivery Price that would make a contracthave zero value, As time passes,the Forward Price is liableto change,while the DeliveryPrice remainsfixed during the lifeof each contract(although it will be differentfor differentcontracts). 417 FuturesContract Thisis the same asa ForwardAgreement, except that : -futures contracts are normally traded on an exchange; - tomake tradingpossible, the exchange specifies certain standardised features of the contract; - sincethe two partiesto thecontract do not necessarilyknow one another,the exchangeprovides a mechanismwhich gives the two partiesa guaranteethat the contractwill be honoured; - an exactDelivery Date is not always specified; - thereis marking to market, or daily settlement. FuturesOption Contract Theseare Options on FuturesContracts. FuturesPrice SeeForward Price, reading “Futures” for “Forward”. The FuturesPrice is the price at whichtrades occur on a futuresexchange. It is determinedby theforces of supply and demand. Hedger An investorwho buysor sellsa contract for the primary purpose of risk transfer. InitialMargin (FuturesContract) Thisis the amount that an investormust depositin his Margin Account at thetime the contractisfirst entered into. The InitialMargin will depend on thevolatility ofthe underlying asset. 418 MaintenanceMargin (FuturesContract) To ensurethat the balancein the Margin Account neverbecomes negative,a MaintenanceMargin (a.k.a. Variation Margin) is set that is somewhat lower than the InitialMargin. If the balance in the MarginAccount falls below the Maintenance Margin,the investor receives a “margin call” and isrequired to top up theMargin Accountto the Initial Margin level immediately. The MaintenanceMargin is usually 70-40% of theInitial Margin for most typesof futurescontracts. Margin Account(Futures Contract) A brokerwill require an investorto depositfunds in a marginaccount to guarantee theinvestor’s performance. Normal Backwardation Thisis the oppositeto Contango.If the buyersare hedgers,and the sellersare speculators,one can arguethat the futures price will be below theexpected future Spot Price.This is because,following the same argumentas for Contango,the hedgersare prepared to accept slightly reduced returns, as they are happy to have the benefitof the hedge, whereas the speculators require compensation for the extra risk theyare taking on. The positionwhere the futures price is below the expected futures spotprice is known as Normal Backwardation. Option Contract Thereare two basictypes : Call Option and Put Option (see definitions). An optiongives the holder the right to do something.The distinguishingfeature of an optioncontract isthat the holder does not have to exercisethis right. An investormust pay somethingto enterinto an optioncontract, the “option price”, unlikeFutures and Forward Contracts. Open Interest Thisis the total number of outstanding contracts ina securityata certaintime 419 Put Option A CallOption gives the holder the right to sellthe underlying security by a certain datefor a certainprice. SettlementPrice (Futures) Thisprice is set by theexchange at the end of eachday. It is the official price to be usedfor determining daily gains or losses and margin requirements. Speculator An investorwho isbuying or sellinga contract for investment purposes, rather than asa hedgeagainst unwanted risk. Snot Price The isthe price of an assetfor immediate delivery. StrikePrice (Options - a.k.a.Exercise Price) The priceat which the long call/put option holder can exercisehis right to buy/ sell theunderlying security. Terminal Value Thisis the value of a longposition ina contractat maturity. Wiener Process Thisis the process stock prices are assumed to followin most option-pricing models. Itis a stochasticMarkov process;in English this means it is a combinationof fixed and randomvariables, forwhich only the present value of thevariable is needed to determinethe futurevalues (it doesn’t matter what levelsthe variablehas been before,just where it is at now). The behaviourof a variablefollowing a Wiener processhas two key properties dependingon how smallchanges in the variable are related to smallchanges in time. Firstlythat a smallchange in thevariable is proportional to (the small change in time) (1/2)x N(0,1),where N(0,1)is a standardisedNormal Random Variable. Secondlythat changes in the variableover two differenttime intervalsare independent. 420 In more common parlance,the model for stock pricebehaviour is known as BrownianMotion, that familiar term from O’ levelphysics days! In a similarfashion tosmoke particles,stock prices are assumed to wobble up and down, with an equally likelychance of goingup or down, butcan have an overalldrift imposed on them one way or theother. It can be shown thatstocks whose changein price follows a Wienerprocess have a futurevalue whose distribution is Log-normal. 421 10. Bibliography Black,F. and Scholes,M. (1974).‘The pricing of Optionsand CorporateLiabilities’, Journalof Political Economy, Vol.8 1 (May-June)(pp.637-659) Carlton,D.W. (1984),‘Futures Markets: Their Purpose, Their History, Their Growth,Their Successes And Failures’,The JournalOf FuturesMarkets Vol.4 No.3 (pp.237-271) Chang, E.C (1985),‘Returns to Speculatorsand the theory of Normal Backwardation’,Journal of Finance, Vol.40 (Mar.) (pp. 193-208) The ChicagoBoard Of Trade(1992), ‘Catastrophe Insurance Futures And Options: A ReferenceGuide’ Cox,J.C., Ingersoll J.E. and Ross S.A(1981), ‘The relation between forward and futuresprices”, Journal Of FinancialEconomics, Vol.9 (Dec.) (pp. 312-346) Cox,J.C., Ross, S-A., Rubinstein, M. (1979)‘Option pricing: a simplified approach’, Journalof Financial Economics, Vol.7 (Sep.) (pp.229-263) Cox,S.H. and Schwebach,R.G. (1992),‘Insurance Futures And Hedging Insurance PriceRisk’, Symposium On InsuranceFutures, The JournalOf Risk And Insurance, Vol.LIX,No.4 (Dec.) D’Arcy,S.P. and France,V.G., (1990), ‘Will Insurance Futures Fly ?‘, Journal Of Commerce (Oct.) D’Arcy,S.P. and France,V.G., (1992), ‘Catastrophe Futures : A BetterHedge For Insurers’,Symposium On InsuranceFutures, The JournalOf Risk And Insurance, Vol.LIX,No.4 (Dec.) D’Arcy S.P and France,V.G. (1993),‘Catastrophe Insurance Futures’, Faculty WorkingPaper 93-0132, Bureau of Economic and BusinessResearch, College of Commerce andBusiness Administration, University ofIllinois at Urbana-Champaign Duffie,D. (1989),‘Futures Markets’, Prentice-Hall Inc. Dusak,K (1973),‘Futures Trading and InvestorReturns: an Investigationof CommodityRisk Premiums’, Journal of Political Economy, Vol.69 (Dec.) (pp. 250- 260) 422 Hogg andKlugman (1984),‘Loss Distributions’, J.Wiley & Sons Inc. Houthakker,H.S (1957),‘Can Speculators Forecast Prices?’ Review of Economics and Statistics,Vol 39 (pp. 143-151) Hull,J.C. (1993), ‘Options, Futures And OtherDerivative Securities’, Prentice-Hall Inc.,Second Edition Hull,J. and White,A. (1990),‘Pricing Interest Rate Derivative Securities’, Review of FinancialStudies, 3,4 (pp.573-592) Kidder,Peabody & Co.Inc. (1992), ‘Introducing Catastrophic Insurance Futures‘ LeviC. and PartratC. (1989),‘Statistical Analysis of Natural Events in the United States’,presented at ASTIN Colloquium,New York (Nov. 1989) Merton,R.C (1973),‘Theory of Rational Option Pricing’, Bell Journal of Economics and ManagementScience, 4 (pp.141-183) Merton, R.C (1976), ‘Option Pricing when Underlying Stock Returns are Discontinuous’,Journal of Financial Economics, Vol.3 (Mar.) (pp. 125-144) Niehaus,G. and Mann, S.V.(1992), ‘The TradingOf UnderwritingRisk : An AnalysisOf InsuranceFutures Contracts And Reinsurance’,Symposium On InsuranceFutures, The JournalOf RiskAnd Insurance,Vol.LIX, No.4 (Dec.) Renshaw,A.E (1991),‘The Interpretation and Implementation of Generalized Linear Models (withapplications)‘, CityUniversity London Robertson,J.P. (1991), ‘insurance Futures‘, The Actuarial:Review (March) Roll,R. (1984),‘Orange Juice And Weather’,American Economic Review, Vol.74 Sherman,R.E. (1991),‘Actuaries And InsuranceFutures’, The ActuarialReview (Feb.) Smith,C.W. and Stulz,R.M. (1985),‘The Determinants Of Firms’Hedging Policies’, JournalOf FinancialAnd QuantitativeAnalysis, Vol.20 No.4 (Dec.) Telser,L.G (1958)‘Futures Trading and theStorage of Cottonand Wheat’,Journal ofPolitical Economy, Vol.66 (Jun.) (pp. 233-255) 423 Appendix I Example Of The StatisticalInformation Available 423 Appendix II.1 Summary Of Trading PeriodsAnd Reporting Periods For ContactsCurrently Traded Contract Loss Reporting Trading LastDay of Month Period Period Commences Trading_ Sep 93 Apr - Jun 93 Apr 93 - Sep 93 4 Jan 93 5 Jan 94 Dec 93 Jul - sep 93 Jul 93 - Dec 93 4 Jan 93 5 Apr 94 Mar 94 Oct - Dec 93 Oct 93 - Mar 94 4 Jan 93 5 Jul 94 425 Appendix II.2 Linesof Business Included In The ContractBy Cause ofLoss Causeof Loss Linesof Business Wind HailQuake tRiot Flood Homeowners x x Commercial x x X MultiPeril Earthquake X Fire x x X1 Allied x x X1 Auto,Physical x x x x x Damage (Private & Commercial) Farmowners x x x CommercialInland X x x xx x Marine 1commercial element only 426 Appendix II.3 CompaniesContributing To ISO DATA AmericanFinancial Group LibertyMutual AMICA MutualInsurance Company LincolnNational Group CIGNA Group RoyalInsurance Group CNA InsuranceCompanies SafecoInsurance Group CU InsuranceCompanies St PaulGroup ContinentalInsurance Companies TransamericaCorporation Group EmployersMutual Companies UnitedStates F&G Group Fireman’sFund Companies USAA Group GeneralAccident Group WestfieldCompanies HanoverInsurance Companies ZurichInsurance Group - US ITT HartfordGroup Kemper CorporationGroup / Kemper NationalInsurance Companies 427 Appendix II.4 CBOT LimitationsOf’ Price Movement And Holdings The maximum valueof an Insurancefutures contract is $50,000(because the catastrophelosses would usuallybe expectedto compriseonly part of thetotal loss ratio,the limitationof $50,000 represents considerably more thana doublingof expectedcatastrophe losses, as witnessed by HurricaneAndrew notcausing this level to be breached). Shoulda contracthave a dailysettlement price greater than 90% of itsmaximum finalsettlement price, a new contractfor delivery inthe same calender month may be listed,with a maximum finalsettlement amount to be determinedby the CBOT Board. The dailyprice should not move by more thanten points (i.e.: 10/o of $25,000or $2,500),though a variablelimits of fifteenpoints may be applied(i.e.: 15% of $25,000or $3,750) The maximum netlong or shortposition that any person may own, control,or carry is10,000 contracts. There will be no additionallimits imposed in the spot months. Thereis a reportablelimit of 100contracts in any month. 428 Appendix III FuturesExample 1 ForecastsinJuly: $m (1) Company IncurredLoss 4 (2) Company Premium 30 (3) Company ReportedLoss 75% by Dec. (4) IndustryIncurred Loss 400 (5) IndustryPremium 3,000 (6) IndustryReported Loss 75% by Dec. (7) FuturesPrice in July $2,500 ( = 25,000x (4)x (6)/ (5)) (8) To hedge100% ofpremium 1,600contracts need to be bought ( = (2)/ 25,000x 100% / (6)) $m $m $m $m $m $m $m IndustryReported Loss 200 250 300 350 400 450 500 IndustryUltimate Loss267 333 400 467 533 600 667 (9) FuturesPrice ($) 1,6672,083 2,500 2,917 3,333 3,750 4,167 Company ExpectedLoss 4.00 4.00 4.00 4.00 4.00 4.00 4.00 (10)Company ActualLoss 2.67 3..33 4.00 4.67 5.33 6.00 6.67 (11)Futures Gain / (Loss) (1.33)(0.67)0.00 0.67 1.33 2.00 2.67 (12)Overalllosses ((10)-(11)) 4.00 4.00 4.00 4.00 4.00 4.00 4.00 FuturesExample 2 Forecastsin July: $m (1) Company IncurredLoss 2 (2)to (7)as previously (8) To hedge50% ofpremium: 800 contractsneed to be bought ( = (2)/ 25,000x 50% / (6)) $m $m $m $m $m $m $m IndustryReported Loss 200 250 300 350 400 450 500 IndustryUltimate Loss 267 333 400 467 533 600 667 (9) FuturesPrice ($) 1,6672,083 2,500 2,917 3,333 3,750 4,167 Company ExpectedLoss 2.00 2.00 2.00 2.00 2.00 2.00 2.00 (10)Company ActualLoss 1.33 1.67 2.00 2.33 2.67 3.00 3.33 (11)Futures Gain / (Loss) (0.67)(0.33)0.00 0.33 0.67 1.00 1.33 (12)Overall losses ((10)-(11)) 2.00 2.00 2.00 2.00 2.00 2.00 2.00 429 Appendix III FuturesExample 3 Forecastsin July: $m (1) Company IncurredLoss 6 (2) to (7) as previously (8) To hedge150% ofpremium: 2,400contracts need to be bought (= (2)/ 25,000 x 150% /(6) ) $m $m $m $m $m $m $m IndustryReported Loss 200 250 300 350 400 450 500 IndustryUltimate Loss 267 333 400 467 533 600 667 (9) FuturesPrice ($) 1,6672,083 2,500 2,917 3,333 3,750 4,167 Company ExpectedLoss 6.00 6.00 6.00 6.00 6.00 6.00 6.00 (10)Company ActualLoss 4.00 5.00 6.00 7.00 8.00 9.00 10.00 (11)Futures Gain / (Loss) (2.00)(1.00)0.00 1.00 2.00 3.00 4.00 (12)Overall losses ((l0)-(1 1)) 6.00 6.00 6.00 6.00 6.00 6.00 6.00 OptionsExamples 1 And 2 Forecastsin July: $m (1) Company IncurredLoss 4 (2) to (7) as previously (8) To hedgeLoss-ratio inrange of 20-50% 1,600( = (2) /25,000 x 100% / (6))contracts need to be boughtat a priceof $3,750( = (7)x 0.2/ ( (1)/ (2) ) ) and soldat $9,375 ( = (7)x 0.5/ ( (1)/ (2) ) $m $m $m $m $m $m IndustryReported Loss 113 450 788 1,125 1,463 1,800 IndustryUltimate Loss 150 600 1,050 1,500 1,950 2,400 (9) FuturesPrice ($) 938 3,750 6,563 9,37512,188 15,000 Company ExpectedLoss 4.00 4.00 4.00 4.00 4.00 4.00 (10)Company ActualLoss 1.50 6.00 10.50 15.00 19.50 24.00 (11)Long OptionGain 0.00 0.00 4.50 9.00 13.50 18.00 (12)Overall Effect (( 1O)-( 1l))* 1.50 6.00 6.00 6.00 6.00 6.00 (13)Short Option Loss 0.00 0.00 0.00 0.00 (4.50)(9.00) (14)OverallEffect ((12)-(13))+1.50 6.00 6.00 6.00 10.50 15.00 * Figuresfor Options Example 1 + Figuresfor Options Example 2 430 Appendix IV.1 PricingMathematics - SimulatingThe ExpectedSettlement Value The followingindicates a simple approach that can be takenin building a model for theexpected final Settlement Value of an Insurancefuture, based on assumptions aboutthe frequency and severityoflosses. This expected value can then be usedas a basepoint in arriving atthe value of an Insurancefuture, as indicatedinsection 4.2. The Model The followingcrude model isfor the National contract and takesno accountof seasonalityofthe losses. In practiceone would want to adjustthe model for different quartersand extendit to the three different regions. The majorone-off events that causes Catastrophe losses are Hurricanes. These are thereforemodelled separately. There is a considerablebody of work on the modelling of Catastrophelosses, particularly forHurricane losses (see Levi & Partrat(1989), Hogg & Klugman(1984) and Renshaw(1991)). Otherlosses are mainly due totornadoes/windstorms; inaddition there are losses due to Earthquakes,Hail and Riots.The studyand modellingof any one of these Catastropheswould be a paperin itself, so we havehere adopted a fairlybroad-brush approachto their modelling. The Parameters The model we haveconstructed translates into an expectationof slightlyover one Hurricaneper year. The othersources of Catastrophe are expected to occurbetween fourand twelvetimes per quarter,with an overallexpectation of thirty-twolosses per year. The Hurricaneshave an expectedloss of about$1 billion.The “other”Catastrophes havelosses ranging between roughly $200m to $1,800m per quarter,overall about $3.2billion per year. 431 We have useda Poissonmodel forthe frequency of theseevents. The severityof Hurricanesis modelledby the Log-Normaldistribution, and the severityof the “other”Catastrophes ismodelled by theNormal distribution.The parameters used are as follows: HurricaneLosses Quarterlyfrequency Poisson: mean 0.2803 SeverityLog-Normal: mean 5.4912,Standard Deviation 1.74297 “Other”Losses Quarterlyfrequency Poisson: mean 8 SeverityNormal: mean 100,Standard Deviation 10 Othermodels were consideredfor frequency and severity.For example,Pareto or Weibulldistributions could be usedto model theHurricane severity. The fitof the model was consideredat two levels.First, that the frequency and severitywere a good model forthe known Hurricanedata, which was drawnfrom severalsources, includingthe referencesmentioned above. Secondly, that the overalllosses, and hencesimulated Insurance Futures prices, were a satisfactorymodel for CBOT’s simulatedFutures settlement prices over the last fourteen quarters. We willnot dwell on the modellingprocess, as thispaper is not meant to be a learnedtreatise on modellingCatastrophe losses. The modelis simply meant to be an illustrationof the approachthat could be adopted. The Results Simulationsbased on theabove model produce a mean and a standarddeviation of the Insurancefutures price at the end of the period.These are compared to the CBOT simulationof Insurancefutures prices for the fourteen quarters prior to the launchof the contract below: Model CBOT Data CBOT Data (withHugo (withoutHugo andAndrew) and Andrew) Mean $2,200 $4,000 $2,100 SD $2,900 $5,900 $1,100 432 The Model hashurricanes as 26% of totallosses, compared to 27% in theCBOT simulateddata. The simulationallows one toexamine the distribution of the expected finalSettlement Value at the end ofthe period, as illustrated below: Simulationinsurance Futures SettlementPrice:Expected Value$2,210 The model describedabove can be usedto examinethe effect on a hedgingstrategy of a company’slosses not being perfectly correlated with the Industry as a whole. One can simulateCompany lossesat the same time as Industrylosses, either by havingcompany lossesas a variablepercentage of Industrylosses (say in the range 0.5% - 1.5%,rather than precisely l%), or one couldsimulate completely separate lossesfor the Company, but have them correlatedto varyingdegrees with the Industryas a whole. 433 UsingThe Model To Make ComparisonsWith ReinsuranceCosts Section4.3 described how thepricing of options,particularly exotic options of the kindrepresented by Insuranceoptions (on futures)is bedevilled with problems and uncertainty.In order to make a roughcomparison with the priceof conventional reinsurance,we have examinedthe simulatedexpected pay-off of an Insurance Option,using the simple model above, in a risk-neutralworld. The resultsof these broad-brushcomparisons are describedin section 3.3. The method involves simulatinga series of Catastrophelosses, and examiningthe expected pay-offs on Insuranceoptions with varying Exercise levels. This is really just akin to comparing typicalcurrent reinsurance costs with the underlying expected values of the losses. Justas reinsurancecosts will reflect more thanthe pure expected losses underlying them,so Insuranceoptions would reflectthe riskiness ofthis type of investment,as perceivedby differenttypes of investor. Examples of the distribution ofthese sample Optionpay-offs are reproduced below: Simulation ofPay-off OptionXS 20%: Expected Value $360 434 Simulationofoption Pay offXS 40%: Expected Value $220 Simulation ofPay-off OptionXS 60%: Expected Value $150 435 Appendix IV.2 PricingMathematics - DevelopingFormulae For PricingForwards Forwards On Non-income Paving Securities Definethe following variables: T timewhen theforward contract matures t currenttime to timewhen contractwas firstinitiated S(t)price at time t of the asset underlying the forward contract K deliveryprice in the forward contract F(t)forward price at time t f(t)value of a longforward contract at time t r(t)risk free rate of interest per annum attime t, with continuous compounding, foran investmentmaturing at time T Consider2 portfoliosas follows : PortfolioI * Long forwardcontract on theunderlying security plus cash equal to Ke-r(1)*(T-t). * Valueis * Pay-offat time T is PortfolioII * One security. * Value is S(t). * Pay-offat time S(T). 436 From a simpleno-arbitrage argument, portfolios I and IIhave the same value,that is: When thecontract was firstinitiated, K should have been negotiated so thatf(t0) = 0, thatis By definition,thisis also the forward price at time t0, so F(t0)= K. More generally, theforward price at time t, F(t) should be givenby : At maturity,F(T) = S(T),that is, the spot price of the underlying security attime T. Of coursein practice F(t) is determined by marketforces in a futuresmarket and it may notequal the theoretical value S(t)*er(t)*(T-t). Ifthis is the case then arbitrage opportunitieswill exist and a speculatorcan takea risklessprofit by adoptinga suitabletrading strategy. ForwardsOn OtherTypes Of AssetsHeld For Investment The valueof the contractcan be derivedusing arbitrage arguments in many situations,including : (a) underlyingsecurities that provide a known cash income (b) underlyingsecurities that provide a known dividendyield (c) forwardcontracts on currencies (d) forwardson gold and silver 437 Usingthe same notationas previously, itcan be shown thata generalformula linking F,K, r and finthese situations isas follows (see Hull (1993)): where: I(t)is the present value at t of the known cashincome produced by the underlyingasset from t toT; and q isthe annual dividend rate on theunderlying asset, paid continuously Returningto the fourexamples just mentioned, we can interpretthe equation as follows: (a)underlying securities that provide a known cashincome Put q=0 in the equation. (b)underlying securities that provide a known dividendyield PutI(t)=0 in the equation. (c)forward contracts on currencies Putq=0 and I(t)= 0 S(t)is the current price in Sterlingof one unitof theforeign currency at maturity, namelyS’(t)*e-p(t)*(T-t), where S’(t) is the current price in Sterling of one unit of the foreigncurrency and p(t) is the risk-free rate prevailing inthe foreign currency. (d)forwards on gold and silver Ifwe want to allowfor storage costs, we can use appropriatenegative values for eitherq or I(t)at thesame timeas zeroingthe other, depending on thenature of thesecosts. 438