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2016immc优秀论文3


Team?#20160921?Page?1?of?20?

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Summary?Sheet?? ?
Screaming? at? the? top? of? our? lungs,? we? eagerly? witness? world?class? athletes? set? world? record? after? world ? record? from? the?spectators’? stand.?However,? behind? the?adrenaline?induced? races,? gripping?their?armrests? tightly? and? watching? the? races? tensely,? are? the? organizing ? committee? and? their? insurance? company? ??? the? organizing? committee?bears?the?risk?of?paying?a? significant? amount?of? money? as? a?bonus?to?the?winner?if?he?or?she?succeeds ? in?setting?a?new?world?record,?while?the?insurance?company?may?suffer?a?loss?from?insuring?such?a?race.?? ? In? order? to ? help? the? insurance? company ? determine? the? price? of ? the? premium? they?offer?that?ensures?their? financial? stability,? and? the? organizing?committee?to? make?a? the? correct?decision?on? when?considering?purchasing? insurance,?we?looked?into?the ?complexity?of?the?probability?of?record?breaking,?premium?calculation ?and?decision ? making? under? risk? in? this? paper,? and? came? up? with? three? different? mathematical? models? targeting? different? scenarios?when?making?the?above?decisions.?? ? The?most? important? factor? affecting ?both?parties’ ?decisions?is?the?risk?of?such?an?investment,?which?can?be? seen?as? the?probability?of?an ?athlete?breaking?the?world?record.?We?developed?a?model?using ?regression?to?analyze? past? data? collected? on? athletes? and? their? past? records,? then? computed? the? standard? deviation? and? normal ? distribution?to?predict?the?next?record?breaking? as?a? reference?for?both?the?insurance?company ?and?the?organizing? committee. ? The? model? was? first? built? around? one? potential? world ? record?breaker,? and? was? advanced? to? accommodate? multiple? potential ?record?breakers.? Since?subsequent?models? are?largely?based?on?the?result?of?this? model,?to?access?our?model,?we?tested?it?to?ensure?its?feasibility.?? ? To? help ?the? insurance?company?determine?the?price?of?the?premium,?we?developed?another?model?that?takes? expected? claim? amount,? safety?loading? and? profit?gaining? loading,? administrative? costs,?commission?for?agents,? settlement? of? compensation?and?other? factors?into?consideration.? After? deriving? the?equation?to?do?so,?we?applied ? the?eigenvector?centrality?concept?to?rank?the?importance?of?each?factor.?? ? Our?third? and?final? model? helps?the?organizing?committee ?decide?whether?to? purchase? insurance?or?not ?by? considering? the? profit? they? gain? after? each? competition? and? the? likelihood ? of?paying?the ?bonus ?across? a?certain ? period.?The?model?was?later?generalised?so?that?it?can?be?applied?for?multiple?sport?events?using?a?single?equation.? ? Afterwards? we? analyzed? the? strengths? and? weaknesses? of? our? models,? covering? areas? including? but? not? limited ? to?feasibility,? efficiency?and?ease? of? usage,? through?which?our? team?hopes?to? provide? a?clearer ?overview? and? further? understanding? of? the? models ? and? each ? of? their? the ? merits? and? limitation.? With? real?life? simulating ? models? that? consider?a? variety? of? scenarios?and?different ?factors,?it? is?our ?sincere?wish?to?provide?models?that?are? helpful ? and? useable? for? the? insurance?company? and?the?organizing? committee?to? make? better?decisions? for? their? own?benefit.??

Team?#20160921?Page?2?of?20?

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Introduction?? ?
In? athletics,?athletes?are?often?given?a ?significant?amount?of?bonus?for?breaking?the?world?record?of?an ?event? by?the?organizing? committee?of? the? competition,?so?as? to ?attract? top?athletes?to?join?said?competition.?As?the?risk? of? bearing? a?heavy?financial?burden?if?such? an? event?takes?place,?the?organizing?committee?has?to?decide?whether ? to? purchase? insurance? or? not? (in? other? words,? self?insure).? To? aid? the? decision?making? processes? for? both? the? organizing? committee ? and? the? insurance? company,? we? have? developed? several? mathematical? models? that? are? presented?in?this?paper,?and?test?cases?to?assess?the?viability?and?sensibility?of?our?models.?

? Problem?Restatement?
1. 2. ? This?year’s?problem?requires?us?to?do?5?things:? Calculate?the?average?cost?of?the?bonus?for?the?Zevenheuvelenloop.? From?the?insurance?company’s?perspective,?determine?the?insurance?premium?for?the? Zevenheuvelenloop?by?considering?different?factors?such?as?the?average?cost?of?the?bonus,?operating? costs,?time?value?of?money,?etc.? From?the?organizing?committee’s?perspective,?a)?identify?the?criteria?used?to?determine?the?necessity?of? purchasing?the?insurance?for?the?Zevenheuvelenloop,?and?b)?determine?whether?or?not?they?should.? From?the?organizing?committee’s?perspective,?determine?the?way?the?organizing?committee?should? weight?each?factor?in?deciding?whether?they?should?purchase?the?insurance?for?each?of?40?events.? From?the?organizing?committee’s?perspective,?develop?a?mathematical?model?for?the?decision?making? process.?

3. 4. 5. ?

Overall?Assumptions?and?Justifications??
? Assumption? Accidents?do?not?happen?before?or?during?the?competition,?i.e.? the?runners?do?not?get?injured?or?quit?the?competition?under? any?circumstances?once?they?are?enrolled?in?the?competition.? The?performance?of?the?runners?are?consistent?and?do?not? deviate?too?much?from?his?or?her?average?performance?level.? All?runners?engaged?in?the?race?aim?to?break?the?world?record.? Each?competition’s?result?is?independent?of?others.? Peak?period?of?runners?is?similar?for?all?athletes?participating? in?the?same?event.? This?assumption?is?made?with?references?to? past?medical?research1 ?on?the?peak?periods?of? runners,?muscle?mass?declinations?and?other? factors.?? This?is?for?the?ease?of?calculation?and?to? simplify?the?problem,?which?also?resembles? real?life?situations?where?the?organizing? committee?will?do?so?to?avoid?a?large?financial? burden.?? Justifications? This?is?for?the?ease?of?calculation?and?to? simplify?the?problem.?? ?

Only?the?new?record?holder?can?receive?the?bonus.?

1

??Schulz,?R.,?&?Curnow,?C.?(1988).?Peak?Performance?and?Age?Among?Superathletes:?Track?and?Field,? Swimming,?Baseball,?Tennis,?and?Golf.?? Journal?of?Gerontology,? ?? 43? (5).?? ?

Team?#20160921?Page?3?of?20?

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The?first?few?records?set?in?the?earliest?races?are?excluded? from?consideration?during?calculation.??

Since?the?competitions?might?not?have?gained? enough?popularity?to?attract?top?runners,?the? records?are?not?valid?for?projecting?the? probability?of?breaking?records?in?the?future.?? This?is?to?simplify?our?calculation?and?ensure? the?model?can?be?used?in?the?following?years?to? come.?? The?model?is?designed?to?be?used?for?not?only? this?year,?but?anytime?in?the?future??seeing?the? year?of?usage?of?the?model?as?year?0?will?allow? calculations?to?be?simplified.??

All?the?money?terms?and?prices?in?the?paper?are?calculated?in? real?terms?without?the?effect?of?inflation.?
th? The?year?of?usage?of?this?model?is?seen?as?year?0?(or?the?0? year).?

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Definition?of?Important?Terms?? ?
Terms?in?the?paper? Dynamic?information? Constant? Top?athletes?(Potential? winners)? Peak?age? Definition?(explanation)? Variables?that?change?throughout?the?years?? Variables?that?stay?constant?throughout?the?years?? Athletes?who?have?the?potential?to?break?the?world?record?? Period?in?an?athlete's?life?during?which?his?or?her?performance?is?at?the?highest? level??

Average?cost?of?the?bonus? The?ratio?of?the?amount?of?bonus?and?the?expected?number?of?times?the?event?is? replicated?before?the?current?record?is?broken?? Insurer? Insured? Insurance?company,?the?party?responsible?for?collecting?the?premium?and? determining?the?amount?of?the?premium?? The?organizing?committee?in?the?competition?in?this?paper,?the?party?responsible? for?paying?the?premium?and?making?the?decision?of?whether?to?purchase?the? premium?or?not? When?the?world?record?is?broken?by?an?athlete?and?the?organizing?committee?has?to? pay?him?or?her?a?bonus? The?amount?the?insurer?can?claim?back?once?accident?occurs? The?period?in?which?the?insurer?is?protected?by?the?insurance?plan?

Event? Claim? Coverage?period?

? Models?
? 1.?Finding?the?expected?average?cost?(Question?1)? ? 1.1?Additional?assumptions?? ?

Team?#20160921?Page?4?of?20?

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Assumptions? Top?athletes?have?the?chance?of?breaking?the?world? record?for?more?than?once?in?successive?years.? The?performance?of?athletes?obey?the?normal? distribution.? Chance?of?athlete?breaking?world?record?is? exponentially?proportional?to?the?percentage?difference? of?their?personal?best?result?and?the?world?record.? The?average?cost?will?not?remain?constant.?

Justifications?? This?is?to?simplify?the?problem?and?for?the?ease?of? our?calculation??

As?society?changes?and?technology?develops,?chances? of?nurturing?top?athletes?also?change?and?so?does? chance?of?breaking?the?world?record.?? This?is?to?simplify?our?calculation?and?ensure?the? model?can?be?used?in?the?following?years?to?come.??

The?peak?performance?period?of?different?athletes?are? the?same.? ? 1.2?Addition?Definition?of?Terms?? Term? Definition?

Average?cost?of?year?? y?? The?cost?times?the?chance?of?recording?breaking?in?the?said?year? ? 1.3?Method?analysis? ? According? to? the? definition? of? the? average? cost? (the? amount? of? bonus? divided? by?the?expected?number?of? times? the? event? is? replicated? before? the? current? record? is? broken),? since? the? former? is? fixed,? the? question? then? becomes? finding? the ? latter,? which? solely? depends? on? the? frequency/probability? of? breaking? the? world? record.? Therefore,?we?develop?models?to?estimate?the?probability?of?the?world?record?being?broken.?? ? Since? the? models? of? the? following? questions? also?rely?on? this ?probability,? we?have? developed? models?that? calculate?not?only?the?average?cost?of?the?15K?run?as?mentioned?(i.e.?Zevenheuvelenloop),?but ?also?that?of?all?other? athletics?events.?Two?different? models,? the? historical?data? model?and?the?real?time?data?model, ?are?developed,?and? are?optimised?for?different?situations.?? ? 1.4?Historical?data?model? ? The?historical? data?model? directly?calculates?the?probability?of?a?top?athlete?breaking?the?world?record?using? regression? and? analysis?of?historical? data,? and?is? best?used?for?competitions?that?have? few? potential ?world?record? breakers?(for?example,?one?in?five?years).?? ? Data?about? past?world?records?are?collected,?along?with?the?year?of?achieving ?the?record?and?the?information? of? the? athlete? that? set? the? record.? To?come? up?with?a?formula?that?can?estimate?when?the?next?top?athlete?will?be ?at? peak? age,? By? taking? linear,? exponential,? logarithmic? and? quadratic? regression? on? the ? data? points? of? h(x) = the?year?the?xth?world?record?was?broken? and?using?the?one?with?the?least?error,?we?will?be?able?to?come?up? with?a?formula?that?can?estimate?when?the?next?top?athlete?will?be?at?the?peak?age.?? ? Suppose? the? identified? regression?function?is? f (x) .? Then? the? estimated? time? is? μ = f (y + 1) ,?where?? y? is?the? number? of? data? points? currently? on? the? graph.? (Note:? If? more? than? one? value? needs ? to? be? estimated,?

Team?#20160921?Page?5?of?20?

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f (y + 2), ?f (y + 3)? et? cetera ? can? also? be? calculated,? but? one? should? note? that?the? error?increases?as?the?number ?of? estimations?increase.)? ? We?will?calculate?the?deviation?of?the?regression?(denoted?by? σ )?of?the?derived?function:? ?

σ=
?



( ∑ |f (i) ? h(i)|2) ÷ n ?
i=1

n

After? the? calculations? of? the? desired? values,? we? will? progress? to? calculate? the? probability? of? the? said? top? athlete? breaking? the? record? in? each? of? the? coming? years.? According? our? research2,? it? is? assumed? that? the? performance?of? the? athletes?obey? the?normal?distribution?with?mean?at?the?peak?age?(that?is,?the?top?athlete?has?the? highest? probability? of?breaking? the?record? at?their?peak?age),? the? probability?of?the?next?top?athlete?breaking?the? world?record?in?a?certain?year?is:? ?

q(x0 : date?of ?calculation) = ?

x0+0.5

x0?0.5



(x?μ) 1 e? 2σ2 ?σ√2π

2

dx ?

The? reason? why? we? can? take? the? σ ? as? the? standard? deviation? in? the? above? equation? is? that? the? physical? meaning? of? σ is? the? error?between?the?estimated?peak? age?of?past ?top? athletes?and?their?actual?peak?age??the?more? accurate?our? regression?is,?the?more?accurate?our? estimation? of?the ?peak? age?of?athletes ?is?(and?thus?a?smaller? σ ),? and? the?higher ?possibility?the? peak? age? of?the?future?athlete?can?be?estimated?correctly,?and?vice?versa.?Thus,?the? error?in?our?previous?calculations?can?be?used?as?the?standard?deviation?in?the?above?equation.?? ? Then,?the?average? cost?of?the?competition?for?a?year,? x ,? will?be? C ? q(x) , ??where?? C?denotes?the?prize?money? of?the?competition?should?the?world?record?be?broken.?? ? 1.5?Real?time?data?model? ? The? real? time? data? model? analyzes? the? probability ? of? an? athlete? with? a? certain? personal?best? record?(PB)? breaking? the? world? record,? and?counts?the?number?of?athletes ?that ?stand?a?chance?at?breaking?the?world?record?and? their?respective? probabilities,? thus?finding?the?overall?probability?of?having?at ?least?one?athlete?breaking?the?world? record.?This?model?is?optimised?for?events?where?there?are?frequently?multiple?top?athletes?competing?every?year.?? ? We? first? narrow? down? the? pool? of? athletes? who? apply? to? the? event?to? so?called?“top? athletes”,? those? who? stand?a? higher?chance?of?breaking?the?world? record,? for? the? ease? of?calculation.?This?is? done?by?comparing? their? current? performances? to? the? past? performance? of? all? world?record?holders,? as?we?believe?them? to?be ?reasonable? benchmarks? for? estimating? an? athlete’s? future? likelihood? for? breaking? a? record.? The? closer? an? athlete ? is? to? the? benchmark,?the?higher? his/her?chance.? Athletes? whose?performance?is? too? far? off ?stand?a?minimal?chance ?and?are? filtered? off,?as? the? possibility? of? them?improving?to? reach?world?record? standard? in ?such? a?short?period?of?time?is? close? to ?zero.? (Due?to?the?nature?of?this?model,?this?procedure?has?to?be?carried ?out?only?after?the?list?of?applicants? has ?been? confirmed,?and? so? we? can?safely?assume?the?period? of? time? from?the?application?and?the?actual?event?is ? too? short? to? cause? significant? errors.)?All? that? remains?is? to?quantify?the?probabilities?of?the?remaining? potential? winners?breaking?the?record,?then?the?expected?value?and?thus?average?cost?can?be?calculated.? ?

2

??Schulz,?R.,?&?Curnow,?C.?(1988).?Peak?Performance?and?Age?Among?Superathletes:?Track?and?Field,? Swimming,?Baseball,?Tennis,?and?Golf.?? Journal?of?Gerontology,? ? 43? ? (5).?? ?

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Team?#20160921?Page?6?of?20? 香港拔萃女書院 Hong Kong Diocesan Girls' School

Data?about? past?world?records?are?collected,?along?with?information?of?the ?athletes?breaking?the?record.?The? performance?curve?of?the?past? athletes?is?analyzed.?First,? we?define?the?peak?performance?period?as?the?age?range? of? world?record?breakers? when?they?broke?the?record,?as?athletes?are?in?their?peak?physical?status?when?they?break? the?world?records.? For? example,?if ?the? youngest?athlete?to?ever?break?the?world?record?is?22 ?years?old?when?he?or? she?broke?it?and?the?oldest?is?30,?then?the?peak?performance?period?is?22?to?30?years?old.?? ? We? then?compare?the?performance?of? all?world?record? holders? in?the?peak?performance ?period?to?the?world? record? they?hold,? and ?find?the?average?percentage?over?the?period.?Finally?we?average?the?number?over?all?athletes? to? obtain?the?benchmark?percentage,?denoted?by? Δp .?For?example, ?suppose?there?were ? n past?record?holders,?with? record? breaking?time? a1, ?a2, ..., ?an respectively.?Also?suppose?the?identified?peak?performance?period?is?from? y1 to? y2 years?old,? a?total?of? y years,?and?the?personal? best?of? athlete? i ?in?age? x is? bix .?Then,?the?benchmark?percentage? is?calculated?as:? ?

Δp = [ ∑ ( ∑
i=1 j=y1

n

y2

|bij?ai| ai )

÷ y] ÷ n ?

?
Afterwards,? current? athletes? who? have? achieved? good? results? and? can? be? reasonably? deemed? as?potential? winners? will? be?analyzed? according?to?the?benchmark.?If?their?personal?best?result?is?better ?than? W × (1 ± Δp) ,?the? sign? of? Δp depending? on? the? nature? of? the? event? (e.g.? in? track? events? it?is? 1 + Δp ,? whereas? in?long?jump,?high? jump? etc.? it? is? 1 ? Δp ),where ? ?? W? is? the? current ? world? record,? then? they? should? be? deemed? as? potential? record? breakers,? or ?top? athletes.?The? filter?is?created?as? there ?is? an? average?difference?between?a? world?record?breaker’s ? usual? performance? and? best? performance,? so? there? is? a? need? to? consider? a? range? so? that? athletes? whose? performances?fall? into?the?range ?defined,? (i.e.?those?who? have?a? reasonable?chance?of?breaking?the?world?record) ? would?also?be?considered?when?we?calculate?the?probability?of?the?occurrence?of?the?next?record?breaking.??? ? We? will? then? use? the? method? presented? in? the? historical?data? model?to? calculate? the? performance? of? a?top? athlete?during? a?certain?year,? q(x) . ?? The?steps? of ?calculations ?are? the? same? as?that?in?the?previous?model,?just?that? the?samples? will?include?not?only?past? world?record?breakers,?but?also?past?top?athletes?whose?personal?best ?result? is? within? Δp × 100% of? the? closest? future? world? record? at? their? time.? (A? wider?range? of?data?allows? for? a?more? accurate? estimation.)? For? example,? suppose? a? world? record? is? broken? at? 2003? and? 2008?? then? athletes? whose? personal?best? performance? between? year? 2004?and?2007? is?within? Δp × 100% ? of?the? 2008?record? should? also?be? included?in?the?data?set?for?calculating? q(x) ?.? ? Next,? we? assume? the? probability?of?an?athlete?breaking?the?world?record ?is?exponentially?proportional ?to?the? percentage? difference? between? their ? personal? best? result? and ? the? world? record,? as? this? follows? the? probability? density? function? and? resemble? real? life? situations? closely.? The? rationale? behind? why? we? need? the? following? calculation? is?because? there?is ?a?difficulty?for?an?athlete ?to?improve?his/her?performance?to?reach?the?world?record:? a ?30?second?difference?between?his/her?PB?and?the?world?record?and?a?3?second?difference?do?not?imply?a ?10?time? difference? in? their? chances? of? breaking ? the? world? record?? the? athlete? with? the? 3?second? difference? has? an? exponentially? higher? chance? of? breaking? the? record? than? the? one? with? the? 30?second? difference.? Hence,? the? probability? of ?a?top?athlete i ?with?a? personal?best?result? di breaking?the?world?record?in?the? xth year ?following ?year? w can?be?estimated?as:? ?? ?|di?W |

pi (w, x) = e

W

? q(w + x) ?

?
Lastly,? we? will? consider? the? worst? case ? scenario,? where? an? estimated? m top ? athletes? will? all? join? the? competition,?with?winning?probability? p1, ?p2, ?..., ?pm respectively. ?The?probability?of?at?least?one?of?them?breaking? the?record?in?the? xth year?following?year? w is:? ?

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Team?#20160921?Page?7?of?20? 香港拔萃女書院 Hong Kong Diocesan Girls' School

p(w, x) = 1 ? ∏(1 ? pi) = 1 ? (1 ? p1(w, x))(1 ? p2(w, x))...(1 ? pm(w, x)) ?
i=1

m

?
Naturally,? one? may? question? that? even? in? the? foreseeable? future,? the? performance? of? top? athletes? will? deteriorate? while? new? top? athletes? would? appear.? Thus,? for ? each? top? athlete? that? grows? out? of? the? peak? performance? period,? we? can? assume? another? athlete? with? performance? pm+1(w, x) = (∑ pi(w, x)) ÷ m ? ? with?age?
i=1 m?

the?young ?end?of?the?peak?performance ?period?will?appear?for?a?more ?accurate?model.?Since?the?award?money?will? be? given? if? any ? one? of? them? breaks? the? record? (the ? more? potential? winners? there? are,? their? individual? chances? remain? unchanged? but? the? higher? the? chance? of? the? record? itself? being? broken)? ,? the? average? cost ? of? the? competition?will?be? C ? p(w, x) ,?where?? C? ?denotes?the?award?amount.?? ? 1.6?Comparison?of?Models? ? Both ?models?serve?for?different? functions?and?are?optimised?in? different?situations.? In?situations?where?the ? data? set?is? smaller?(for?instance,? there ?is? only? one ?top? athlete,?or? the? database?is? not?as? large),? the?historical ?data? model?will? be? optimal?as? it? utilizes?a? small?amount?of? information? to?give?a? relatively?accurate?projection?of? the? future.? This? is? especially?useful?when?insurance?companies? have? nothing?but?data?about?that?particular?event? on? their?hands.?? ? On?the?other?hand,?when?information?and? data?is? sufficient,? the? real? time? data?model? will? definitely? give ?a? more? reliable? answer? compared? to? the? historical? data? model,? as? it? utilizes? data? not? only? from? history,? but? the? performance? curve? of? real?time? athletes.? Although? the? calculations? of? the ? real? time ? data? model? is? more? complicated,? it? can?also? tackle? situations ?such? as?multiple? potential?top? athletes,?which?is ?in?most?cases,?closer?to? the?real?world?situation.?? ? As? such,? the? models? to? the? following? problems? will? be ? based? on? the? real? time? data? model,? although? the? variables?in?the?following?models?can?also?be?easily?replaced?to?be?based?on?the?historical?data?model.?? ? 1.7?Testing?of?Models?? ? The?real?time?data?model?is?used?for?the?test?case.? World?Record? Breakers? Robert?de?Castella? Valdenor?dos?Santos? Philemon?Hanneck? Josephat?Machuka? Worku?Bikila? Felix?Limo? Leonard? Komon? Personal?Best?of?World?Record?Breakers?in?Age?x?(seconds)? World?Record? 21? ?? ?? ?? ?? ?? 2489? ?? 22? ?? ?? ?? 2543? ?? ?? 2473? 23? ?? ?? 2555? 2586? ?? ?? 2486? 24? ?? 2561? ?? 2543? ?? ?? 2506? 25? ?? ?? ?? ?? ?? 2612? 2535? 26? 2567? ?? ?? ?? ?? ?? 2605? 27? ?? ?? ?? ?? ?? ?? ?? 28? ?? ?? ?? ?? ?? ?? ?? 29? 2726? ?? ?? ?? 2540? ?? ?? 2567? 2561? 2555? 2543? 2540? 2489? 2473??

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Team?#20160921?Page?8?of?20? 香港拔萃女書院 Hong Kong Diocesan Girls' School

?? ? The?benchmark?percentage?is?calculated?as:? ?

Δp = 0.011532104? ? *?
? *Due? to? insufficient?data,?we?were ?not? able?to?find?out? the? personal?best?of?world?record? breakers?for? every?age? from? 21? to ?29.?Therefore,? ? y?in? Δp
n y2 |bij?ai| ai )

= [∑( ∑
i=1 j=y1

÷ y] ÷ n ?? denotes?the?number?of?years?we?have ?data?for?

the?personal ?best?of?world?record?breakers.?For?example,?in?the?case?of?Robert?de?Castella,? y = 2 ? ,?as?we?only?have? his?personal?best?data?for?ages?26?and?29.? ? Past?top?athletes?whose?PB?is?within? Δp × 100% of?the?closest?future?world?record?at?their?time:? ? ? 1? 2? 3? 4? 5? 6? 7? 8? 9? 10? 11? 12? 13? 14? 15? 16? 17? Year? 1983? 1984? 1985? 1986? 1989? 1993? 1994? 1994? 1995? 1995? 1996? 1997? 2001? 2005? 2007? 2007? 2007? Top?Athlete? Robert?de?Castella? Mike?McLeod? Mike?McLeod? John?Treacy? Keith?Brantly? Valdenor?dos?Santos? Haile?Gebrselassie? Philemon?Hanneck? Josephat?Machuka? Josephat?Machuka? Jon?Brown? Worku?Bikila? Felix?Limo? Sileshi?Sihine? Samuel?Kamau?Wanjiru? Samuel?Kamau?Wanjiru? Zersenay?Tadese? Time?(seconds)? 2567? 2575? 2582? 2579? 2570? 2561? 2580? 2555? 2557? 2543? 2562? 2540? 2489? 2498? 2489? 2490? 2494? World?Record? ?? ? ? ? ? ?? ? ?? ? ?? ? ?? ?? ? ? ? ?

?

Team?#20160921?Page?9?of?20? 香港拔萃女書院 Hong Kong Diocesan Girls' School

18? 19? 20? 21? 22? 23? 24? 25? 26? 27?

2007? 2007? 2007? 2009? 2009? 2009? 2010? 2010? 2011? 2011?

Deriba?Merga? Patrick?Makau?Musyoki? Evans?Kiprop?Cheruiyot? Deriba?Merga? Patrick?Makau?Musyoki? Wilson?Kipsang?Kiprotich? Zersenay?Tadese? Leonard?Patrick?Komon? Leonard?Patrick?Komon? Samuel?Tsegay?

2494? 2494? 2489? 2489? 2490? 2495? 2493? 2473? 2486? 2491?

? ? ? ? ? ? ? ?? ? ?

? By?taking?exponential?regression?on?the?data?points?of? h(x) = the?year?the?xth?athlete?broke?the?world?record?or?came?close?to?breaking?the?world?record , ??? the?formula?that?can?estimate?when?the?next?top?athlete?will?be?at?the?peak?age?is:?? ?

y? = ?1.136141636?x? + ?1984.390313 ?
? ? Then ?q(2016)
2016?+0.5 2016?0.5

=



?μ) 1 ?(x2σ 2 e ?σ√2π

2

dx = 0.1569 ?

?
If?the?personal?best?result?of?current?athletes?(2010?2015)?is?better?than? W × (1 + Δp) = 2473 × (1 + 0.011532104) = 2501.5 ? seconds,?then?they?are?deemed?as?potential?record?breakers. ?? ? ? ????1? 2? 3? 4? Year? 2010? 2010? 2011? 2011? Potential?Record?Breakers? Zersenay?Tadese? Leonard?Patrick?Komon? Leonard?Patrick?Komon? Samuel?Tsegay? ? Time? 2493? 2473? 2486? 2491?
??

World?Record? ? ?? ? ?

p1(2010, 6) = e p2(2010, 6) = e

?|2493?2473| 2473 ?|2473?2473| 2473

? q(2010 + 6) = 0.155636213 ? ? q(2010 + 6) = 0.1569 ?
??

?

Team?#20160921?Page?10?of?20? 香港拔萃女書院 Hong Kong Diocesan Girls' School

p3(2010, 6) = e ? p4(2010, 6) = e

?|2486?2473| 2473 ?|2491?2473| 2473

? q(2010 + 6) = 0.156077376 ? ? q(2010 + 6) = 0.155762132 ?
??

??

The?probability?of?at?least?one?potential?record?breakers?breaking?the?record?in?the? 6th year?following?year? 2010 is:?

p(2010, 6) = 1 ? ∏ (1 ? pi) = 0.492803618 ?
i=1

4

The?average?cost?of?the?15K?race?is:? ?

? C ? p(2010, 6) = 25000 × 0.492803618 = 12320.09046 ? ?
However,? the? actual? probability?that?at? least?one? of? the? potential ?record?breakers?break?the?record ?in?the?6th ?year? following?2010? should? be? lower?than?0.49,? as? the? test?case?demonstrated?above?is ?the?worst?case?scenario,?where ? all? potential? record? breakers? join? the? same? competition.? Moreover,? there? are? two? major? 15K ?competitions? held? each?year????the?Zevenheuvelenloop?and?the?Gasparilla?Distance?Classic,?and?by?using?the?fact?that?most?potential? record?breakers?only?train?for?one?15K?competition?per?year,?the?probability?can?be?further?reduced?by?half.? ? 2.?Criteria?for?determining?the?cost?of?insurance?(Question?2)? ? 2.1?Additional?assumptions?? ? ? The?insurance?company?is? sensible,? able?to? identify?high?risk?and?low?risk?investment,?aims?to?gain?a?profit? and?avoids?losing?money.? ? The?insurance?market?is?imperfectly?competitive,?and?all?the?insurance?policies?are?homogenous?.? ? All?the?clients?or?insured?are?able?and?willing?to?pay?the?whole?sum?of?premium?at?one?time.?? ? All?monetary?values?in?the?following?calculations?are?using?real?values?without?the?effect?of?inflation.? ? The?insured?will?report?the?incident?once ?occurs?and?the?insurer ?will?settle?down?the?payment?as?soon?as?the? report?is?made.? ? The?commission,?marketing?expenses?and?other?administrative?costs?are?constant?.? ? 2.2?Method?analysis? ? To? analyze? the? criteria? the? insurance? company? should? consider?and? the?method?they? should? use?to? weigh? each? factor? that? determines? the? amount? added? to ? the? average? cost,? we? first? separate? the? whole? premium? into? different? components,?then? analyze?the?factors?affecting?the?value? of?each?component?and?weigh?their?importance? using?the?eigenvector?centrality?concept.? ? 2.3?The?model? ? To? calculate? the? amount? of? premium? the? organizing?committee?has? to?pay, ?the? insurance?company? has?to? consider?a?number?of?factors?in?order?to?offer?an?insurance?that?is?attractive?to?potential?clients?while?establishing?a? safety?loading?(an?amount?the ?insurance?company? charges? to?lower?the?probability?of?loss?if?the?actual?number?of? claims? is?greater?than?its?expected?value)?that?has?the?twofold ?purpose?of?preventing?monetary?loss?and?subsequent? bankruptcy? in? the? case? of? an? underestimation? of? claim? amount,? and? guarantees? a? profit? in? the ? case? of? an? overestimation.? Therefore,? it? is? vital? for? the? insurance? company? to? consider? both? the? statistical? probability? of? claiming,? which? directly ? affects? the? expected? value? of ? the? claim? and? consequently? the? value? of? the? premium? (according? to? the? equivalence? principle),? as? well?as? the? relevant? risk? factors? (i.e.?the?accuracy?of? the?probability? calculated)?that?affect?the?size?of?the?safety?loading?(i.e.?the?reserve).??

?

Team?#20160921?Page?11?of?20? 香港拔萃女書院 Hong Kong Diocesan Girls' School

? A?few?additional?factors?identified?that?will?affect?the?premium?price?include?but?are?not?limited?to:? ? Expected ? interest? rate? in? the? economy? which? affects ? the? proportion? of ? the? premium? to? be? used? for ? investment?by?the?insurance?company?? ? Expenditure?of?the?insurance?company,? also?known?as?the?insurer’s?expenses,?which?include?administrative? costs?such?as?commission?for?agents,?marketing?expenses?for?promoting?the?offer,?etc?? ? Market? forces,? which? affect? the? actual? amount? of? the? premium? in? real? life.? For? example,? as? the? competitiveness? of? the ? market? increases,? the ? amount? of? the?actual?premium? may?decrease? to?attract? more? clients,?and?vice?versa,?ceteris?paribus.?? ? Figure?2.3.1?shows?the?typical?process?for?determining?the?price?of?the?premium:?

? Figure?2.3.1?process?of?determining?the?amount?of?premium? ? There? are? a? number? of ? ways? to? calculate? the? actuarial? premium.? The? one? presented? in? this? paper? is? the? expense?loaded? calculation? method,?which?takes?into?consideration?the?administrative ?cost,?the?insurer’s?expenses? and? profit,? as? well? as? the? safety? loading? portions.?Thus,?this?method?is? a?better? simulation? of? real? life?situations? than? others? (e.g.? equivalence?premium?and? net?premium?calculation,?expected?which?only?consider ?the? expected ? claim?value?and?the?risk?of?each?investment.?? ? Before?looking?at?the?expense?loaded?method,?we?first?look?at?the?calculation?of?? ? the?expected?aggregate?claim? amount,? which?is?the?expected?number? of?claims?times?the?claim?amount?per? claim?(i.e.?the?total?amount?claimed)?? ? the? net? premium,? which,? as? mentioned,? considers? the? risk? of? offering? such? an? insurance? and? the? safety?loading?portion?on?top?of?equivalence?premium??finally?? ? the? expenses?loaded? premium,? which? is? the? net? premium? plus? the? added? administrative? costs? and? extra? components?that?consider?the?insurer’s?expenses?and?profits.?? ? In? pricing?the?premium,? the? insurance? covers ?a?period? of?time?known?as?the?coverage?period,?during?which? the?insured?can? make? a?maximum?of? ? n?? number? of?claims. ?We?can?express?the?terms?of?the?plan?as? (y, n) ,?where?? y? is?the?duration ?of?the?coverage ?period?in? terms?of? years.?The ?claim?amount?is?also?fixed ?according?to?the?terms?in? the?plan,?and?will?not?change?over?the?coverage?period?regardless?of?the?inflation?rate.?? ? Assume? that? all? claims? are? immediately? reported? and? settled ? in? all ? circumstances,? the? expected? claim? amount?by?the?insurer?throughout?the?coverage?period?is:? ?

E[X ] = E[N ] ? X ? ?

?

香港拔萃女書院 Team?#20160921?Page?12?of?20? Hong Kong Diocesan Girls' School

where?? E[N]?? is?the?expected?number? of?claims?per?insured?and?? X?is?the?previously? fixed?claim? amount?per?claim.? Note? that? ? E[N]? ? is? ? n? ? times? the? probability?of?athletes?breaking?record? over? the? coverage? period,? i.e.? ? p(x)?? or?? q(x)? ,? depending? on? the? model? used? (refer? to? sections? 1.3? and? 1.4).? The? expected? claim? amount? is? also? part? of? the? consideration?for?safety?loading?of?the?insurance?company.?? ? Apart? from?the?expected?number?of?claims?and? the?claim?amount,?a?couple?other?factors?will?also?affect?the ? value?of?? E[X]? ? (the?expected? claim?amount).? For ?example,?in?practice,?the?insurance?company?would?take?a?part?of? the? premium? received? for ?investment?to? gain? an? income? that? can?at? least? cover?the?promised? claim?amount.? This? can?be?represented?by?the?following:?

R = Π + ?kΠ ? (1 + i) ?
? where?R?is ?the? total? revenue?made?as?a? result?of?the?insurance? plan,?which?includes?the?collected ?premium? Π ? and? the?profit? gained ?through? investing?part?of?the?premium?received? ? kΠ? ,? ? k?is?a?fixed?percentage, ?? i?? is?the?interest?rate? as?determined?by?market?forces,?and?? y? ?is?the?coverage?period?in?terms?of?years.?? ? According?to? the? usual?practices? of? the? insurance? industry,? insurers ?usually?aim?to? generate?a? profit?that?is? larger? than? the? claim? amount? to? ensure? sufficient? reserve? in ? the? company? and? avoid? monetary? loss? should? the? actual?claim?be?larger?than?the?expected?claim.?In?short,?? ?

y

kΠ ?

[X]

? (1 + i ) ≥ X ? .??

y

In? this? case,? kΠ[X] should? be?considered?as?the?safety?loading?of?the?premium,?as?a?reward?for?the?risk?borne? by? the? insurer,? as? well? as? the? expected? profit? of? the? insurer.? To? include? the? risk? factor? and? calculate? a? safety?loading?proportional?to?the?severity?of?the?risk,?the?net?premium?is?calculated?as?thus:? ?

Π = E [X ] + λ?V ar [X ] ? ? ,? ?
Where? λ ? is?the?given?intensity,?and?? ?

?

? V ar[X ]? = E[X ] ? (E[X ]) ? ?????????????????????????????????????????????????????????????????(3 )? ?????????????????????????????????????=? E[N ]V ar[X 1] + E[N 2](E[X 1])2 ? (E[X ])2 ? 2 ????????????????????? = E[N ]V ar[X 1] + V ar[N ](E[X 1]) ?

2

Aside?from?considering?the?risk?factor,?calculating?a?safety?loading?proportional?to?severity?of?the?risk,?the? addition?part, λ ?Var[X]? ,?variance?of?the?amount?the?insurer?has?to?pay?back?to?the?insured,?also?takes?the?time?value? of?the?money?into?consideration?while?calculating.?? ? Lastly,? administrative? costs,? commission? costs,? marketing? costs? et? cetera? have? to? be? included? in? the? calculation? of ? the? premium.? Let ΘA, ?ΘG be? the? fixed? amount? of ? commission? and ? administrative? costs? and?other? expenses? of? the? insurer? as? a? result?of? offering? this?insurance?service? respectively,?and? ΘSE[N ] be?the?amount?of? expenses? for? each? settlement? made,? where? E[N ] ?is? the? expected? frequency? of? claims?made.? Based? on?the?flow? chart? and? the? factors? to?be?considered,?for?a?fixed?claim?amount?? X,?the?equation?for?calculating?the?expense?loaded? [X] actuarial?premium? Π ?is?given?by:? ? ?? With? accordance? to? a? theorem? introduced? in? Olivieri,? A.,? &? Pitacco,? E.? (2011).? ? Introduction? to? insurance? mathematics:?Technical?and?financial?features?of?risk?transfers? .?Berlin:?Springer?Verlag.?? ??
3

?

Team?#20160921?Page?13?of?20? 香港拔萃女書院 Hong Kong Diocesan Girls' School

Π[X ] = E [X ] + λV ar[X ]? + ΘA + ΘG + ΘsE [N ]? ? ?
2.4??Weighting?? ? Based? on? section? 2.3,? we? can? see? that? different ? components? involved? in? the? calculation? of? the? price? of? premium?depend? on?a?group?of?other?variables.?To?determine?the?importance?of?each?variable,?thus ?the?importance? of? each?component,?we?came?up ?with?a?network?showing?the?relationship?between?variables.?Note?that?importance? in? this?paper? is?determined?by? how?much? each? component?can?vary,?thus?increasing?or?decreasing?the?price?of?the? premium.??

? Figure?2.4.1?Relationship?between?each?component?? (nodes?are?not?drawn?according?to?the?importance?yet)? ? In? the? above? diagram,? we?can? see?that?a? number? of? components,?such?as?the?amount?of?compensation? the? insurer? promised? to? pay? (? X? ),? the? expected? number? of? claims? over? the? years? (? E[N]) ? ,? and? the? interest? rate? for? investment?(? i? )?affect?the?different?components?above?according?to?the?calculation?of?each.?? ? To? find? out? the ? importance? of? each? factor,? we? are? going? to? use? the ? eigenvector? centrality? for? the? final? ranking.? The? reason? we? chose? the? eigenvector? centrality ? instead? of? other? centralities? such ? as? the? closeness? centrality? and ?degree ?centrality?is?because?it?can ?measure?the?influence?of?each?node?in?the?whole?network?as?a?big? picture.?According?to?the?diagram,?we?can?develop?the?following?adjacency?table:?

?

Team?#20160921?Page?14?of?20? 香港拔萃女書院 Hong Kong Diocesan Girls' School

? Figure?2.4.2?Adjacency?Table? ? Where? 1? indicates? a? directed? relationship? from? one? component? to ? another?? whereas? 0? indicates? no? directed? relationship.?The?process?for?calculating?the?eigenvector?can?be?simplified?as?below:?

?

X → M → M (x) = λx ?
? Where?? M? is?the?adjacency ?matrix,?? M(x)?is?the?re?distribution?of?nodal?value?after?re?shuffling,? λ ? is?the?eigenvector? number?or?the?eigenvector?and?? x? ?is?the?nodal?value.?? ? First, ?let? us? set ?the? nodal?value?of?all?nodes?to? 1,? that? is?? x? ? =1,?then?we?calculate?the?nodal ?value?of?? q?nodes? based?on?the?connection?of?each?node?they?are?connected?to,?that?is? ?
?1 ?1

M (x) = (value?of ?connected?node?1) + (value?of ?connected?node?2) + ... + (value?of ?connected?node?q) ? ?
After? the? first? round? of?reshuffle?of? nodal?value,? M (x) = / λ(x) ? ,?? M(x)? ? becomes ?the? new?nodal? value? of? the? node,? however? it?does? not? reflect?the ?ranking? of ?importance? of?each? node,?thus?it? is?not?the?eigenvector.?Therefore,?we? then? conduct?another?round?of?reshuffle?of?nodal?value.?The?nodal?value?after?each?shuffle?tends?to?the?eigenvector? as?a?limit.?? ? We?used?Excel?for?the?calculation?and?the?result?is?shown?in?the?figure?below:? ??

?1

?

Team?#20160921?Page?15?of?20? 香港拔萃女書院 Hong Kong Diocesan Girls' School

? Figure?2.4.3?Result?table?of?eigenvector? ? From?the?above?result? table,? we? can?see ?that?the?most?important?factor?that?affects?the?value?of?the?premium?is ?the? amount?the?insurer?has?to ?pay?back?to?the?insured?(ie.?? X? ),?the?second?most ?important?factor?is?the?expected?number? of?claims?the?insured?can?get?during?the?coverage?period?calculated?by?the?insurer?(ie.?? E[N]? ).?? ? 3.??Risk?analysis?of?self?insuring?and?general?decision?scheme?(Questions?3?5)? ? 3.1?Additional?Assumptions? ? Insurance?company?will?be?responsible?for?and?willing?to?pay?the?ensured?award?amount? ? The?annual?profit?of?the?organizing?committee?will?be?able?to?cover?the?annual?insurance?cost? ? The?organizing?committee?would?prioritise?a?stable?financial?status?over?profit? ? 3.2?Method?analysis? ? Based? on? the? results? of? the? models? of? question?1,?we?have?developed?a? model?that?allows? the? organizing? committee?to?flexibly?calculate?whether?they?should?purchase?insurance.?? ? 3.3?Model?for?calculating?risk?factor? ? This? model? allows? the? organising? committee? to? determine? how? risky? it? is? to? self?insure.? Evidently,? the ? organizing? company? would? want ? to? save? the?added?cost? (that? is,?to? not?buy?insurance?as? much?as?possible),?but? would? first? need? to? ensure? that ? their? financial? situation? is? healthy? (i.e.? that? they? will? not? suffer? from? a? sudden? monetary? loss? as? somebody?breaks?the?record).?Thus,?whether?or? not?insurance?should?be?purchased? depends? on? the?organizing?committee’s?financial? situation,? the?amount?of ?bonus ?and?the?probability?of?record?breaking?during? the? coverage? period.? The? problem,? then,? is? to? find? out? the ? probability? of? record?breaking? during? the? coverage? period,?and?the?number?of?record?breaks?the?committee’s?financial?situation?allows.?? ? The? probability? of? record?breaking? of? the? ? xth? year? following? year? ? w? ,? p(w, x) ,? is? already? computed? in ? question?1.?For?example,?the?probability?of?the?record?being?broken?for?1?time?in?the? y years?following?year? w ?is:? ? ?P (w, y, 1) so?the?expected?amount?of?bonus?is:? ?

= ∑ (p(w, i) ∏?(1 ? p(w, j )))
i=1 j= /i

y

?

?

? C ? ∑ (p(w, i) ∏?(1 ? p(w, j )))
i=1 j= /i

y

?

?

? the?probability?of?the?record?being?broken?for?2?times?in?the? y years?following?year? w ?is:?? ?

?

Team?#20160921?Page?16?of?20? 香港拔萃女書院 Hong Kong Diocesan Girls' School

P (w, y, 2) = ∑ ∑ (p(w, i1)p(w, i2)
i1=1 i2>i1

y

y

∏ (1 ? p(w, j )))
j= / i1,?j= / i2

?

?

? and?the?expected?amount?of?bonus?is:?
y

?? 2C ? ∑ ∑ (p(w, i1)p(w, i2)
i1=1 i2>i1

y

∏ (1 ? p(w, j ))) ?
j= / i1,?j= / i2

?

? ?In?general,?the?probability?of?breaking?the?world?record?for? z times?in?the? y years?following?year? w is:? ?

P (w, y, z) = ∑ ∑ ∑ ... ∑ (p(w, i1)p(w, i2)...p(w, iz)
i1=1 i2>i1 i3>i2 iz>iz?1

y

y

y

y



?

(1 ? p(w, j )))

?

j= / i1,?j= / i2,...,j= / iz

?

Then,? P (w, y, z) ? is?called?the?? risk?factor? ?of?having? z ?record?breaks?in?the? y ?years?following?year? w .?? ? 3.4?Decision?on?risk?taking? ? As? mentioned,? the? organizing? committee’s? financial? situation? and? the? risk? factor? are? the? most? important? criteria?for?determining?whether?the?committee?should?purchase?the?insurance.?? ? Suppose? the? organizing? committee? can? self?insure? for? a? maximum? of ? k times? at? most? without? facing? bankruptcy.?That? is,?if?the?committee?has? a?revenue?of?? $A? ,?which?can?be?written?as?a?function?of?year?number?and? past?profits,?then?? k + 1 > A ÷ C > k ? .? ? Say?the?income?of? the? organizing?committee? in?year? w ? is? T w .?According?to?our ?research? and ?analysis?of? data? collected,? if? a? record? is? broken? at? a? certain? competition,? top? athletes? will? be? incentivized? to? go? to? that? competition?in? hopes?of?breaking?another?record.?With?more ?top?athletes?and?more?advertisement,?the?income?will? increase.?It?is? therefore? reasonable?to? assume? that? for ?every?time? that? a?record?is?broken,?the?income?will?increase? by? ? N%? .? On? the? other? hand,? if? no?records? are?broken?during?the?competition,?its? credibility?and?consequently? its? income? will?decrease.?Suppose?that?whenever?a? record?is? not?broken,? the? income? will? decrease?by? ? M%?? per ?year.? Using?conditional?probability,?we?have?the? T (w, x) ,?the?expected?value?of?the?income?the? xth? year?from?year? w :? ?

T (w, x) = T w ∏ (p(w, i)(1 + N %) + (1 ? p(w, i))(1 ? M %)) ?
i=1

x

? The?total?expected?income?in? y years?from?year? w ?is?therefore:? ?

T ?total(w, y) = ∑ T (w, i)? ? ?
i=1

y

We? first? consider? a? simpler? case,? where? the? premium? is? assumed? to? be?constant?throughout? the? coverage? period? once? the?insurance? is?bought.? This? implies ?that ?we?only?need? to?consider? whether?the?insurance?should?be? bought? this? year,? because? if? there? are? any? future? changes? the ? organizing? committee? can? easily? start? buying? insurance.? Based? on? the? expected? income? of?the ?organizing? committee?and? its?current?assets,? we ?can?conclude?a? few?conditions?in?which?the?organizing?committee?should?purchase?the?insurance.?? ?

?

Team?#20160921?Page?17?of?20? 香港拔萃女書院 Hong Kong Diocesan Girls' School

First, ? if? the? organizing? committee? does? not ? have? enough? money? to? pay?the?bonus?(even? after?earning?the? yearly? income),? then ? it? is? a? must? for ? them?to? purchase? the? insurance? to?prevent?bankruptcy.? In?other?words,? the? organizing?committee?will?have?to?purchase?the?insurance?if:? ?

A + T (0, 1) ≤ C ? ??
? On?the?other?hand,?since?insurance?companies?may?choose?to?invest?in?other?places?for?profit?rather?than? directly?earn?from?an?added?cost,?the?expected?cost?the?organizing?committee?has?to?pay?may?be?higher?than?that? of?the?insurance?cost,?and?in?that?case?the?organizing?committee?should?also?purchase?the?insurance?to?save?money.? In?other?words,?the?organizing?committee?should?purchase?the?insurance?if:? ?

∑ i ? C ? P (0, y, i) = C ? ( ∑ p(0, i)) ≥ yΠ ?
i=0 i=1

y

y

?
Where Π ?is?the?premium?per? year?and?? C ?? is? the?bonus.?In?addition,?? y?is?any?value?from?1?to?the?longest?number?of? years? that? the? records? were? held? unbroken? in? the? past.? If,? for? some? ? y? ,? there? exists?a? number? ? z?that?satisfies?the? above?inequality,?then ?purchasing?the?plan?? (y,z)?would?be?a?wise?choice.?If?for?some?? y? ,?there?exists?more?than?one?z? that?satisfies? the?above?inequality,? then?the ?manager?should?note?that?the?larger?the?? z? ,?the ?safer?the?plan?is,?but?also? the? more?expensive??which ?? z? to?insure? for ?is? on?the?discretion?of?the?committee?as?to? whether? a?safer? or?cheaper? plan?is?preferred.?? ? Of? course,? if? ? y? and? ? z? are? decided? by? the?insurance?company?and?the?organizing? committee? only? needs?to? decide? whether? or? not? they? will? buy? the? offered? plan, ? the? organizing?committee?should?calculate? the?respective? values? for ?two ?sides?of?the?inequality.?If?the?inequality?holds,?then?the?organizing?committee?should?take?the?offer? ? else,?they?should?not.?? ? However,? in?real?life,?it?is?impossible?for?the?premium?to?stay?the?same?each?time?the?organizing?committee? decides? to? buy? insurance.? For ?instance,? when?the?insurance?company?knows?that?the?organizing?committee? is? in? tight?financial? situations,?it? is?likely?to?significantly?increase?the?premium,?knowing?that?the?organizing?committee? has ?no? choice? but? to?buy ?it.?This?complicates?the?problem?greatly, ?because?while?the?above?inequalities?still?apply,? they? are? not? the? only? considerations? ? say,? if? after ? a? few? years? the? organizing ? committee? is? forced? to? buy? the? insurance? no? matter? the? price,? they? may? end? up? paying? a? lot? more? than? the ?original?premium?they? would?have ? needed? to? pay? if? they? had? only? bought? the? insurance? a?few?years?earlier,?when? their? financial?situation?allowed? self?insurance.?To?build?a?more?realistic?model?for?this?situation,?we?have?the?following?assumptions:? ? ? When?circumstances?dictate?that?the?organizing?committee?buy?the?insurance,?the?insurance?company?will? set?the?price?to?the?highest?that?the?organizing?committee?can?afford?in?order?to?maximize?profit?according?to? the?law?of?demand?in?economics.?That?is,?if?the?organizing?committee?starts?to?buy?the?insurance?in?year?x,? the?price?per?year?will?be? T x .? ? The?yearly? income ?of?the?organizing?committee?is?not?comparable?to?the?bonus??else,?we?can?forgo?this?case? and? refer? to? the? previous? inequality? for? decision? scheming,? as? the? bonus? is? of? little? consequence? to? the? financial? situation? of? the? organizing ? committee? and? it? only? needs? to ? consider? whether? it ? can? save? more? money.? ? Let Y 0 be?the?largest?number?such?that??? T total(0, ?Y 0) + A < (k + 1) ? C .? ? Since?the? income?of?the?organizing? committee? is?not?comparable?to ?? C? ,?it?can?be?safely?assumed?that? Y 0 is?a? number? large? enough? to? consider? cases? for? the? foreseeable? future. ? (If? not? so,? then? ? A? must? be? very? close? to? (k + 1) ? C ,?implying?that?the?organizing?committee?can?refer?to?the?previously?mentioned?decision?scheme?until?A? reaches? (k + 1) ? C ,?then?consider?this?approach?again.)?

?

Team?#20160921?Page?18?of?20? 香港拔萃女書院 Hong Kong Diocesan Girls' School

? The? reasoning? of? this? approach? is?simple:?we?consider ?the? probability?of?the?record? being?broken ?k?times? before? the? year ?reaches? Y 0 ,?and?the?total?expected?bonus?caused?by?this?probability.?If?this?total?expected?bonus?is? higher? than? the? cost? of? buying? the? insurance? now,? the? insurance? should? be?purchased? ?otherwise,? the?insurance? should?not?be?purchased.?? ? Note? that? we? do? not? need? to? consider? alternatives? such? as? purchasing? the? insurance? after? a? number? of? record?breaks:? should?the?decision? now?be ?self?insure,?the?option? of?purchasing?in? the? future? is?always?open? and? can? be? considered?later?when? the? insurance? rates? and? new?competition?results?are?confirmed?? should?the?decision? now? be? purchase? insurance,? due? to ? changing? premiums? and? the? case?dependency? of? repeated? insurance? purchasing,?the?safer?option?will?be?to?purchase?now.?? ? The?problem?then?changes?to? calculating?the?expected?cost?of?purchasing?the?insurance?after?k?new ?records.? Since? the? ? kth ? ? new? record? can? be? set? at?any?time?from? the? ? kth ? ?to?the? Y 0th year? from?now,?we?will? sum?up ?all? the? possible?results?to?achieve?the?expected?value?of?the?total?asset?the?organizing?committee?will?have,?which?is:?

W (k, ?Y 0) = A ? C ? ∑ i ? P (0, Y 0, i) ? k ? C ? ∑ ?P (0, Y 0, i) ? + ∑ ∑ ∑ ... ∑ ((p(0, i1)p(0, i2)...p(0, ik)
i1=1 i2>i1 i3>i2 ik>ik?1 Y0 Y0 Y0 Y0 i=0

k?1

Y0



?

j= / i1,?j= / i2,...,j= / ik

(1 ? p(0, j))) ? ( ∑ T (0, 1)?(1 + N %) (1 ? M %)
l=1

i=k ik

α

? l?α?

?

Where?according?to?the?above?definitions,? T ik = T (0, 1)(1 + N %)k(1 ? M %)i?k ?and? α =?the?largest? u ?such?that? iu ≤ l ? .?If? W (k, Y 0) > A + T total(0, ?Y 0) ? Y 0Π ? ?where? Π is?the?premium?cost,?then?the?insurance?should?not?be?purchased? ? ? else,?the?insurance?should?be?purchased.??

+ T total(ik, Y 0 ? ik) ? (Y 0 ? ik)T ik)) ??? ? ?

?
3.5?Multiple?event?decision?? ? In? the?case?of ?multiple?events,?the?above?method?can?be ?used?for?multiple?times?to?decide?whether?insurance? should? be? purchased? for? each? event.?This?is ?because?the?probability?of? an? athlete? breaking? the? record?in? a?event? does?not?affect?that?of?another?event?and?therefore?all?events?can?be?considered?independently.? ? Since?it?is? assumed?that?the?profit?of?the ?organization?committee?each?year?is?enough?to?cover ?insurance?for? all?events,?and?that?the?income?of? each?event?is ?insignificant?compared ?to?that?of? the?award?money,? there?should? not? be? a? limit? on? the? number? of? events? for? which? insurance? is? purchased,? so? the?above?method ?can?be?used?to? determine? whether? purchasing? an? insurance? is? ideal? for? each? of? the? events. ? However,? should? the? organizing? committee?want? to?purchase?insurance?for?only?a?decided?number? of?events?(say? v ? events),?the?following?methods? can? be? used?to? rank? the? risk? of?self?insuring?for?each?event,?and? the?first? v ? events? with? the?highest?risk?should?be? purchased.?We?define?the?following?variables:?? ? Variable? Explanation? The?premium?per?year?for?the? n events,?respectively? The?bonus?for?the? n ?events,?respectively? The?expected?profit?by?the? n events?in?the? xth year? following?year? w ,?respectively?

Π1, ?Π2, ..., ?Πn ? C 1, ?C 2, ?..., ?C n ? T 1(w, x), ?T 2(w, x), ..., T n(w, x) ?

?

Team?#20160921?Page?19?of?20? 香港拔萃女書院 Hong Kong Diocesan Girls' School

T 1total(w, y), T 2total(w, y), ?..., ?T ntotal(w, y)? ? p1(w, x), ?p2(w, x), ..., pn(w, x) ? P 1(w, y, z), ?P 2(w, y, z), ..., ?P n(w, y, z) ?
? ?

The?expected?total?profit?by?the? n events?in?the? y years following?year? w ,?respectively? The?probability?of?the?record?being?broken?in?the? xth year after?year? w for?the? n events,?respectively? The?probability?of? z record?breaks?in?the? y years? following?year? w for?the? n events,?respectively?

Our? strategy? is? based? on? the? assumption? that? the ? most? important? aim? of? the? organizing? committee? is? to? maintain? a? stable? financial? status? for? future? events.? Naturally,? if? they? do? not? have? enough? assets? to?pay?for?the? bonuses? of? all?the?events,?they? will? want?to?purchase?insurance? for?some?events?and?save?their?assets?for?covering? the? bonuses? of? the? remaining? events? to ? avoid? bankruptcy.? Should? they? be? unable? to? purchase? all? the? required? insurance,?they?will?want? to?purchase?insurance?for?the?events?that?have?the?highest?expected?cost?(i.e.?the?highest? risk)?to?lower?the?chance?of?bankruptcy.?? ? After? short?term? considerations? for? avoiding? bankruptcy,? the ? organizing? committee? will? look? to? saving? money? in? the? long? term.? In? this? case,? the? model? presented?in? section?3.5?now? comes?into? view.?For?events? that? have? an? expected? higher? profit? for? buying? insurance ? now? than?later,?the?organizing? committee? should? purchase? insurance.?Naturally,?the?priority?goes?to?events?that?have?the?highest?difference?between ?the?expected?profit?from ? purchasing?now?than?that?of?later.?? ? Lastly,? after? all? the?above?is? considered,?if?the? organizing? committee?is?still?willing? to?purchase?insurance? for? more? events,? the? decisions? should? be?made? by?considering? whether?purchasing?an?insurance?for?a ?particular? event? will ? give?the?organizing? committee? a?profit? over? a?long?period? of ?time.? Although?this?choice?will?give?the? organizing? committee ? an? expected? profit,? it? does? not? contribute? to? maintaining? the? financial? stability? of? the? organizing?committee ?as?the?small?amounts?of?profit?here?cannot?provide?protection?from?bankruptcy?compared ?to? purchasing?insurance?for?the?events.?Thus?this?consideration?should?be?put?last.? ? In?light?of?the?above,?without?loss?of?generality,?suppose? C 1 ≥ C 2 ≥ ... ≥ C n .?The?organizing?committee ? ? should?decide?on?which?events?to?purchase?insurance?for?using?the?following?steps:? ? 1. If? A + T (0, 1) > ∑ C i ,? the? insurance? company? has?enough?assets?to? pay?for?the?total?bonus?for?all? events?
i=1 n

this? year? and?should?look?at? the? long? term,?and?should?refer?to?step?2.?If? that? is?not?the?case,?we?will?find? the? smallest? u ? such?that? A + T (0, 1) > ∑ C i + ∑ Π?i .? If? u ? 1 > v ? ,?then?the?organizing?committee ?should? calculate? the? average ? cost? C i ? pi(0, 1) for? each? event,? and? purchase? the? first? v ? events? with? the? largest? average?cost? ??although? in?this?case,?the?organizing?committee?is?highly?recommended?to?purchase?at ?least? u ? 1? insurances? for? all? the ? aforementioned? events.? If? u ? 1 ≤ v ?,? the? organizing? committee? should ? purchase?insurance?for?all? u ? 1 events,?then?go?to?step?2. ? ?? ? 2. We? now?assume?that?there?are? v1 events ? ? left?to? be?purchased.?We ?should?calculate? W i(ki, Y 0i)? for?each?of? the?events,?and? purchase? the? first? v1 events ? ?that?have?the?smallest? W i(ki, Y 0i) ? A ? T itotal(0, ?Y 0i) + Y 0iΠ?i ,? with?the?value?being?negative??should?there?be?any?quotas?for?purchasing?left,?step?3?can?be?referred?to.?? 3. For? each? of? the? events? that? no? insurance ? is? purchased,? the? C i ? (∑ pi(0, i)) ? should? be? calculated? for?
i=1 yi yi i=u i=1 n u

yi = the?longest?period?of ?years?no?record?is?broken?for?event?i ? .? If? C i ? (∑ pi(0, i)) ≥ yiΠ?i ,? then? the?
i=1

?

Team?#20160921?Page?20?of?20? 香港拔萃女書院 Hong Kong Diocesan Girls' School

insurance? should? be? purchased? for? those ? events? as? well?? in? particular,? the? events? with? the? largest?

C i ? (∑ pi(0, i)) ? yiΠ?i should?be?purchased? as ?long?as?the?difference?is?positive.?If?any?quotas?are?left?after?
step?3,?no?more?insurance?should?be?purchased?for?any?of?the?events?left.?? ? 3.6?General?decision?plan? ? The? general? decision? plan? under? any ? circumstance? is? very? similar? to? the? one? in? the? previous? section.? Although? there? may? be? variances,? the? general? approach? will? also?be?to?calculate? the?expected? bonus,?profit? and? premium?cost ?for?each? of?the?events,?then?analyze?first?the?short?term,?then?the?long?term?impacts?according?to?the? equations? developed? above,?and?purchase?insurance?for ?the?events?that?will?give?the?organizing?committee?a?better? protection?or?profit.?? ? Note:?One?major? question?that?may?be?posed?is?the?usability? of? the? above? model,?due?to?the?complexity?of? the? equations? in? the? above? decision? plan.? Indeed,? for? larger? numbers? human?hand?calculations?cannot?come? up? with? a? solution? within? a?reasonable? time? ?? but?computers? can.? Due ?to?the?fact?that?for?sports?events,?predictions? will? usually? not? go ? out? of? the? range?of? the?foreseeable?future?(say,?10? or?20? years),?the?plan?above?without?any? strategies? to?further?simplify? it?has? an?approximately? O(n ? 2y) complexity,? where? n is?the?total?number ?of?events? and? y is?the?number?of?years?for?consideration.?Since?in?general ? y ≤ 20 ? and? n should?be?within?a?reasonably?small? range ?(say,? n ≤ 1000 ), ? ? the?calculation? by?computer? can?be ?finished?in?a?very?reasonable?time?(within?seconds)?by? a ? program? in? accordance? to? the? method? utilizing? recursion.? Therefore,? the? above? method? can? be? used? by? any? organizing?committee?to?obtain?a?reasonable?prediction?as?to?whether?insurance?should?be?purchased.?? ?
i=1

yi

Strengths?and?Limitations?of?the?Model?
? Strengths? ? The?strength?of? our?model? is? that? it? is?easy?to? implement?after?inputting?the?data?for?the?first?time.?Through? inputting? the? correct ? data,? we? can? obtain? the ? result? desired? that? can? help? the? organizing? committee? and? the? insurance? company?to? make?a ?decision? that? could?benefit?them.?With?the?aid?of?programming?and?computer,?we? are? confident? that? the? the? model? we? developed? can? help ? the? insurer? and? the? insured? to? make ? their? decision? effectively?within?seconds.?? ? Apart? from? this,? by? considering?a? large ?variety? of? scenarios?and? variables,?for?example? the? three?different? scenarios ?including?the?worst?case?for?the?organizing?committee?in ?section?3,?as?well?as?some?of?the?more?practical? components? including? administrative?costs,?agent?commission?fees,?and?investment? to?ensure?profit ?gaining,? our? model?can?be?made?more?realistic?and?we?are?confident?that?it?can?be?used?in?real?life.?? ? Limitations? ? One? of?the ?limitations? of? this?model? is?that?there? are? a?large?number?of?variables?to? be?considered?in?each? equation? during? calculation.? The? large? number ? of? variables? thus? require? a? huge? amount ? of? information? for? calculation.? This? may? increase? the? cost? of?data? collections?for?both?parties,?causing ?variations? in?the?cost? of? the? premium?as?calculated?by?our?model.?? ? Also,? many?assumptions? are? made?when?we?develop?the?model.?The?large ?number?of ?assumptions?made?for? the?ease? of? our?calculation? may? potentially?have?significant?impacts? on?the?accuracy?of?the?output? of?our? model.? For?example,?it?is?assumed?that?all?the?top?runners?in?a?sport?event?will?go?to?the?same?competition?aiming?to?break? the? world? record ? and? that? there? would?be? no?accidents?that?may?hinder? their? performance? or? cause?them? to?pull? out,?which?may?not?be?true?in?real?life.?Thus?the?result?obtained?based?on?each?model?may?not?be?exactly?accurate.? ? ?

Team?#20160921?Page?21?of?20?

?

Conclusion?
? After?preliminary? literature? review,?data? collection?and?research,?we?have?developed ?three?different?models? in?this?paper?to?solve?the?problems?posed.?? ? Firstly,?to? find?the?average? cost?of? the? bonus? (i.e.?the?expected?cost ?of?the?premium),?we?use?the?concept?of? regression? to? predict? the? frequency? of? record?breaking? in? the? future? based? on? historical? data. ? Secondly,? to? determine? the ? price? premium,? we? include? a? number? of? components? that? are? included? in? the ? premium? while? considering? the? amount? of? reserve? and? profit? margin? for? the? insurance? company. ? Finally,? to? determine? if? the? organizing? committee ? should? purchase? insurance? in? different? circumstances,? we? make? use? of? the? concept? of? probability? and? inequalities?to? create?a?generalised?function?based?on ?their?expected?income?and?the?likelihood?for? record?breaking.?? ? Since?sections?2? and? 3?of?this?paper ?largely?depend ?on?the?model?in?section?1,?we?generate?a?test?case?using? real?life? data?for?15K? races? we? collected? to?assess? model?1.?The?result?is?reasonable?and?the?model ?can?be?used?in? real?life.? In? sections?2?and?3,?we? consider?multiple?scenarios? in?real ?life?situations.?With?this,?plus ?our?reasonable? approach?for?calculating?the?probability? of ?record?breaking,?we?are?confident?that?our?model? will?be?able?to?help? the? insurance? company? determine? the? premium? and? the? organizing? committee? decide? whether? to? purchase? insurance?or?not?for?their?own?benefit.? ?

References?? ?
● Olivieri,? A.,? &?Pitacco,? E.?(2011).?? Introduction?to?insurance?mathematics:?Technical?and?financial?features? of?risk?transfers? .?Berlin:?Springer?Verlag.?? ● Harrington,?S.?E.,?&?Niehaus,?G.?(2004).?? Risk?management?and?insurance? .?Boston,?MA:?McGraw?Hill.?? ● Karl?Borch?(1974).?Mathematical?Models?in?Insurance?.?ASTIN?Bulletin,?7,?pp?192?202? doi:10.1017/S0515036100006036? ● Zazanis,? M.? (2005,? July).? Stochastic? models? in? risk? theory.? Retrieved? March? 25,? 2016,? from? http://www.stat?athens.aueb.gr/~mzazanis/courses/MAP/notes?05.pdf? ● Taghavifard,?M.?T.,?Damghani,?K.?K.,?&?Moghaddam,?R.?T.?(2009).?Decision?Making?under?Uncertain?and? Risky?Situations.?Retrieved?March?25,?2016,?from?
http://www.soa.org/library/monographs/other?monographs/2009/april/mono?2009?m?as09?1?damghani.pdf? ?

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