Reinsurance and the distribution of term insurance claims

Size: px
Start display at page:

Download "Reinsurance and the distribution of term insurance claims"

Transcription

1 Resurace ad the dstrbuto of term surace clams By Rchard Bruyel FIAA, FNZSA Preseted to the NZ Socety of Actuares Coferece Queestow - November 006 1

2 1 Itroducto Ths paper vestgates the effect of resurace o the dstrbuto of et clams arsg from a boo of term surace polces. The paper s dvded to several sectos. Secto troduces the stadard statstcal model of clams behavour, ad gves expressos for the momets of the clams dstrbuto. A optmal resurace strategy for a boo of polces s defed as the set of reteto levels (potetally dfferet for each polcy) that mmses a measure of the rs of excess et clams, for a gve level of expected et cost. Algebrac expressos for optmal resurace strateges are derved for three rs measures. It appears that these results are ot wdely ow, however the frst oe was derved, a geeral surace cotext, by de Fett Some mplcatos of these results are cosdered. Secto descrbes the use of Mote Carlo smulatos to model clams experece. Practcal ssues facg ths approach are dscussed. Secto 4 gves the result of modellg the dstrbuto of clams arsg from a hypothetcal boo of term surace. It cofrms that the dstrbuto of clams s postvely sewed, the sewess decreasg as ether the umber of polces rses or the reteto level falls. It s show that the lely rage of clams caot be relably estmated by assumg the clams dstrbuto s Normal. APRA s proposed oe year 00 stadard s used to estmate the level of captal requred uder each scearo. A method for determg the optmum reteto level, based o the margal retur o captal, s outled. Secto 5 wdes the stochastc model to clude the occurrece of a pademc. The possblty of a pademc lmts the dversfcato advatage a surer expects to eoy by creasg ts scale. May surers wll be uwllg to hold suffcet reserves to cope wth all the clams that may occur a severe pademc, as they are ulely to mae a adequate retur o ths volume of captal. A surer s free to choose ts ow styles of resurace ad reteto levels. These decsos wll be flueced by factors cludg the ease of admstrato, the level of resurace support requred uderwrtg ad clams, exstg resurace arragemets ad compay rs appette. The type of aalyss outled ths paper gves a obectve startg pot for resurace decsos.

3 Mmum Rs Resurace Strateges.1 Statstcal Model Cosder a surer wth term lfe polces. Polcy has sum sured S, ad probablty q of clamg the ext perod. Let X be the radom varable represetg the total clams uder polcy the ext perod. Thus X = S wth probablty q, X = 0 wth probablty 1 q. Let X = X = 1 perod. Clearly, the total clams uder the portfolo of polces the ext [ X ] = S q μ = E (1) = 1 ad, provded the clams uder the polces occur depedetly, further statstcs of the dstrbuto of total clams are as follows: [( X μ) ] = S q ( 1 q ) σ = E, () = 1 = ( X μ) = S q 1 q 1 = 1 4 = S q ( 1 q )( 1 6q + 6q ) = 1 [ ] ( )( q ) μ E, () γ. μ, the thrd cetral momet, s also ow as the thrd cumulat. γ s the fourth cumulat, ad from t the fourth cetral momet ca be foud as 4 4 [( μ) ] = γ μ = E X +. 4 σ From the cumulats the sewess ad urtoss of the clams dstrbuto ca be obtaed: u SKEW [ X ] = σ KURT [ X ] = γ σ 4.. Resurace Strategy The surer ow chooses that polcy has reteto R, where 0 R S. Let resurace strategy R be defed as R = { R : = 1,..., }, the set of retetos chose for each polcy. The resurer charges premum rate r for polcy. Hece the premum pad to the resurer s

4 r =1 ( S R ) ad the expected et cost to the surer (meag expected loss of proft) due to ths resurace arragemet s =1 ( r q )( S R ). Geerally a surer uses resurace to lmt the fluctuato aual clams experece, partcular to lmt the severty of a blow-out clams. A optmal resurace strategy s the resurace strategy that mmses a measure of the rs of excess et clams, across all the resurace strateges havg the same expected et cost. The most commo rs measure s the varace of the dstrbuto of et clams.. Resurace strategy mmsg varace We ow derve a expresso for the resurace strategy that mmses the varace of et clams for a gve level of expected et cost. Let X R be the radom varable of the total clams et of resurace uder resurace strategy R. We have E [ X R ] = R q Var = 1 [ X ] R q ( 1 q ) R = = 1 (4). (5) Let R be a optmal resurace strategy. Cosder the effect of movg to resurace strategy R ˆ = { Rˆ : = 1,..., } where Rˆ = R + ε Rˆ R + δ = R ˆ = R for,. Let X R ˆ be the radom varable of the total clams et of resurace uder resurace strategy Rˆ. We see that E X = E X + ε q + δq [ ] [ R ] Rˆ. The expected et cost of resurace strategy Rˆ s = 1 ( r q )( S R ) = ( r q )( S R ) ε ( r q ) δ ( r q ) ˆ. = 1 Now we choose ε ad δ so ther combed effect o the expected et cost of the resurace s zero, e 0 = ε r q + δ r q. (6) ( ) ( ) 4

5 Movg to strategy Rˆ also affects the varace of the portfolo et of resurace, t becomes Var X = Var X + R ε + ε q 1 q + R δ + δ q 1 q. [ ] [ ] ( ) ( ) ( ) ( ) Rˆ R As strategy R was optmal, movg to Rˆ caot reduce the varace, hece 0 ( Rε + ε ) q ( 1 q ) + ( R δ + δ ) q ( 1 q ), depedet of the choce of ε ad δ. Ths wll oly be true f ts frst order terms equal zero, e 0 = R ε q 1 q + R δq 1 q. ( ) ( ) Substtutg equato (6) to ths gves R ( ) ( ) q 1 q R q 1 q =. r q r q As ad were arbtrarly chose, ths expresso must be true for all values of ad, e the expresso R q ( 1 q ) r q should be costat for all. Ths meas that c( r q ) R = (7) q ( 1 q ) for some costat c. Ths result was frst publshed by de Fett The author has ot see the orgal paper (t s Itala!) but there are umerous dscussos of t o the teret (search for de Fett ad resurace ). The result s usually dscussed a geeral surace cotext but apples equally to lfe surace. Cosder that the resurace premum rate r ca be expressed as ( + λ ) q r = 1 where λ represets a marg for the resurer s expeses ad profts. Substtutg ths to equato (7) shows that uder a optmal resurace strategy cλ R = 1 q for some costat c. If the resurer s marg λ s costat for all the d R =. (8) 1 q for some costat d. Uder most polces the probablty q of clamg the ext perod s usually very small, so ( 1 q ) s very ear to oe. Mag ths approxmato reveals 5

6 that for the resurace strategy mmsg varace the reteto R must be costat for all. Ths occurs uder a Surplus Rs resurace treaty wth a fxed maxmum reteto for each polcy. Note that each dfferet reteto level uder a Surplus Rs treaty s a optmal resurace strategy for a dfferet level of expected et cost. Nothg the above aalyss helps a surer choose a approprate reteto level. Sectos of ths paper loos at the mplcatos of dfferet reteto levels o captal, ad outle a method of selectg reteto levels based o retur o captal requremets. It may happe that dfferet groups of polces do have dfferet resurer s margs. Typcally ths mght occur for dfferet product types, or betwee beeft types wth a product. I ths case the optmal reteto for each product or beeft type s proportoal to the resurer s marg for that product or beeft type. I other words f product group A has double the resurer s marg as product group B, the optmal reteto for product group A would be double that of product group B. The term ( 1 q ) expresso (8) s also worth examg. Cosder two polces, each wth $100,000 sum sured, the frst wth a 1% probablty of clamg, the secod wth a 10% probablty of clamg. Expresso (8) shows that a slghtly hgher reteto s preferred for the 10% polcy tha the 1% polcy. Oe way of seeg ths s to cosder a portfolo of te depedet polces, each wth $100,000 sum sured ad 1% probablty of clam. The expected total amout of clams uder the te 1% polces s $10,000, the same as uder the sgle 10% polcy. The sgle 10% polcy could cur ether $0 or $100,000 of clams. However uder the te 1% polces the total clams could be $0, $100,000, $00,000, or possbly up to $1,000,000. Clearly the group of te polces s slghtly rser tha the sgle 10% polcy. Expresso allows for ths by assgg the 1% polces a slghtly lower reteto tha the 10% polcy..4 Resurace Strategy mmsg μ or γ Varace s ust oe of a umber of possble measures of rs. The ma crtcsms of varace as a rs measure are: (a) That t pealses outcomes o ether sde of the mea, ot recogsg that low results are good ad hgh results are bad. (b) Its weghtg to extreme evets may be approprate. The advatages of varace are that t s wdely uderstood, ca be easly computed, ad s ofte algebracally tractable. Aggregate term surace clams are always postvely sewed, so there wll be more very hgh results tha very low results. Cosequetly the varace measure wll place most weght o hgh outcomes, whch gves some comfort that mmsg varace should produce a sesble resurace strategy. 6

7 However t would be terestg to see the resurace strategy obtaed by mmsg a dowsde-oly measure such as sem-varace. The ssue of the relatve weghtg placed o extreme results ultmately comes dow to the rs appette of the surer. For example, cosder a small surer that wll go broe f et clams are $5m over expected. I that case there s o pot gvg a hgher rs weghtg to the chace of $6m excess clams tha to the chace of $5m excess clams! Potetally each surer may have a dfferet atttude to extreme evets, select a dfferet rs measure ad cosequetly ed wth a dfferet resurace strategy. For most rs measures t s ot possble to derve useful expressos for the resultg optmal resurace strategy. However, by adaptg the method of secto., expressos for the resurace strateges mmsg μ, γ, or ay of the hgher cumulats ca be foud. The terested reader s vted to wor through the detals of ths. The μ statstc does ot suffer from crtcsm (a) of varace, as t recogses that outcomes below the mea are good (reduces the measure) ad that outcomes above the mea are bad (creases the measure). The resurace strategy mmsg μ satsfes r q R = c q ( 1 q )( 1 q ) whch s approxmately equvalet to q R c = 1 + λ. The fourth cumulat γ cotas some fourth order terms, so compared wth varace t places a relatvely hgher weghtg o the occurrece of extreme evets. The resurace strategy mmsg γ satsfes r ( 1 )( ) q R = c q q q q whch s approxmately equvalet to 7q R 1 = c + λ. 1 Comparg these results to the equvalet expressos for the resurace strategy mmsg varace shows: If the resurer s marg s costat the the Surplus Rs resurace strategy s very close to optmal, depedet of the choce of rs measure. Chages resurer s marg have less effect tha they dd whe varace was mmsed. If a product s resurace marg was doubled the optmal reteto would be 41% hgher to mmmse μ ad 6% hgher to mmse γ. 7

8 .5 Summary Surplus Rs resurace wth a fxed reteto s very close to the theoretcally optmal resurace strategy, depedet of what measure of rs s mmsed. It may be that a surace portfolo aturally subdvdes to sub-portfolos, such as dfferet beeft types, each wth a dfferet resurer s marg. I ths case t s optmal for each sub-portfolo to have ts ow reteto level, the retetos depedet o the relatve level of the resurer s marg o that sub-portfolo. Modellg Clams Experece.1 Smulato Process Outle The occurrece of clams o a term surace portfolo leds tself to aalyss by computer smulato. Ths secto outles how clams ca be modelled by a Mote Carlo method. Cosder a portfolo of term surace o lves sured. Let S be the sum sured o lfe ad q be the probablty of a clam o lfe the ext perod. Cosder a resurace strategy wth reteto R o lfe. For example uder a Surplus Rs resurace treaty the reteto s gve by R = m( R, S ) where R s the maxmum reteto o ay lfe sured. Step 1 Let α be a radom samplg from a Uform [0,1] dstrbuto. Thus Prob[ α z] z, for ay 0 z 1. = Step Repeat Step 1 for each lfe sured, esurg all the α s are depedet of each other. A clam occurs o lfe f α < q, ad that case the surer s et clam cost s R. Step Determe the aggregate et clams cost the surer faces uder ths samplg of the α. Ths s a sgle smulato of oe year s clams experece of the term surace portfolo. Step 4 Repeat steps 1- to buld up a pcture of the dstrbuto of aggregate term surace clams. A suffcetly large umber of smulatos must be ru to estmate the requred parameters of the dstrbuto. Depedg o the ature of the parameters ad the degree 8

9 of accuracy requred, the umber of smulatos eeded may be from 500 to 10,000 or more.. Issues to cosder A umber of practcal ssues should be cosdered before rug a smulato. Polcy vs. Lfe Isured data. A ey assumpto the statstcal model s that clams uder each polcy occur depedetly. Ths s certaly ot true whe two polces are wrtte o the same lfe sured! The obvous soluto s to create a record for each lfe sured rather tha for each polcy. I practce ths may ot be etrely straghtforward. The frst ssue s detfyg all staces where a dvdual s the sured uder two or more polces. Depedg o the avalable data ths may be aywhere from smple to complex or urelable. Oce ths s doe the aggregate sum sured for each lfe sured ca easly be foud. More problematc may be determg a sgle represetatve q for the customer. Health, smog ad other assessmets may legtmately vary betwee polces or wth layers o a polcy, ad potetally date of brth ad eve sex may vary due to data etry errors. Oe approach s to set q, S, q = S, wth the summatos tae across all the dfferet polces ad layers o that lfe sured, ad the q, beg the best estmate of clam probablty based o the detals of that polcy or layer. Rder beefts. May term surace polces have rder beefts uder whch a dfferet sum sured may be pad out. If requred these ca be hadled by modfyg the above algorthm. Oe way of dog ths s as follows: Let q, 1 be the probablty of a death clam o lfe Let q, be the probablty of a TPD clam but ot a death clam Choose α at radom the ormal way. If α < q, 1 the death clam occurs If q, 1 α < q,1 + q, the TPD clam occurs. Lves wth correlated rss. I ay surace portfolo there wll be groups of lves whose probablty of death s ot depedet. A smple example s a marred couple who may both de a car accdet. Geerally these effects are small eough to gore. If t s ecessary to model them t ca be doe the followg way. 9

10 Let customer 1 ad have probablty of dyg the ext perod q 1 ad q respectvely. Let the probablty of both dyg the ext perod be q 1 > q1q. Select α the ormal way. If α < q1 the both de If q 1 < α < q1 the lfe 1 des, lfe survves α < q + q the lfe des, lfe 1 survves. If 1 q1 Software pacage. The ecessary polcy data wll usually be obtaed drectly from the polcy admstrato platform or from a valuato extract. Ths data could be mported to Excel, wth oe row for each polcy. Excel has may advatages but does ted to loc up wth large data sets, ad s curretly lmted to 65,56 rows per sheet. Other database software, such as Access, may wor better for large data sets. 4 Smulato results 4.1 Portfolo detals I ths secto the dstrbuto of clams arsg from a hypothetcal portfolo of term surace s cosdered. The base portfolo cossts of 1,000 polces whose sums sured are gve the followg table. Sum Isured rage Number of polces m 50 1m m 150 m 5m 50 The dvdual polces are evely spaced wth each sum sured rage, eg S = 100,400, 00 1 S = 101,,, S 999 = 4,910, 000, S 1000 = 4,970, 000. The lves sured are assumed equally splt betwee me ad wome, ages radomly chose betwee 5 ad 60, ad mortalty rates tae from table NZ01. Cosequetly the assumed probabltes of a clam o a dvdual lfe vary from to O the base portfolo the expected umber of clams was 1.57, the expected total clam amout was $1.m, a stadard devato was $1.6m, ad the dstrbuto of clams had a sewess of 1.8. These values were computed drectly from the polcy data usg formulae (1) () of secto.1. Whe larger portfolos were requred the base portfolo was replcated as may tmes as ecessary. 10

11 4. Smulato Results Ths followg tables show the results of Mote Carlo smulatos ru o the portfolo of term surace polces. I each case 10,000 smulatos were ru. Note that as the largest polcy has a sum sured of $4.97m the le showg reteto of $5m s effect the gross portfolo. Base portfolo 1,000 polces Reteto Level Mea Clams Stadard devato Sewess 95%-le 99.5%-le $ $ $ $ $ $1m $m $5m ,000 polces Reteto Level Mea Clams Stadard devato Sewess 95%-le 99.5%-le $ $ $ $ $ $1m $m $5m

12 0,000 polces Reteto Level Mea Clams Stadard devato Sewess 95%-le 99.5%-le $ $ $ $ $ $1m $m $5m Observatos from Smulato Results Portfolo sze. The portfolo s clam results become more stable as the portfolo sze creases. The expected total clams creases proporto to, the umber of polces. However the stadard devato creases proporto to, ad the sewess of the dstrbuto s proportoate to 1. Thus as the umber of polces rses the expected magtude of ay blow-out clams falls relatve to the level of expected clams. For stace the total clams expected oce every 0 years are 7% of expected clams for the portfolo of 1,000 polces, 19% for the portfolo of 5,000 polces, ad 144% for the portfolo of 0,000 polces. Ths demostrates the atural advatage a surer eoys by creasg ts scale, through the poolg of depedet rss. The followg graph shows the dstrbutos of aggregate clams for the three portfolo szes. The mea of the dstrbutos have bee scaled to be the same so the dfferg shape of the dstrbutos ca be see o oe graph. 1

13 ,000 polces 5,000 polces 0,000 polces ,000,000,000,000,000,000 4,000,000 5,000,000 6,000,000 Aggregate Clams Ths shows the progresso from extremely log-taled aggregate clams dstrbuto for the portfolo of 1,000 polces, to the ear Normal shape for the portfolo of 0,000. Ths s essetally a demostrato of the Cetral Lmt Theorem. Reteto level. Resurace s effectve lmtg the total varato of a portfolo s clam results. It s much less effectve reducg the magtude of a clams blow-out relatve to the level of expected clams. Cosder the portfolo of 5,000 polces. The clams expected oce every 0 years s 19% of expected clams for the gross portfolo, 171% f reteto s $1m, ad 165% f reteto s $100. Eve f the reteto was oly $1 ths rato would ot fall below 165%. Accuracy of estmated parameters. Dfferet parameters requre very dfferet umbers of smulatos for estmato to the same degree of accuracy. For the portfolo of 5,000 polces, across all the reteto levels, the estmate of mea clams s always accurate to wth 0.15%. O the same portfolo the estmates of stadard devato are out by up to 1.4%, ad sewess estmates by up to 1%. Broadly ths s due to the relatve sestvty of the three measures to outlers. Ths ca be see by examg each polcy s cotrbuto to mea, varace ad sewess per the summatos formulae (1) - (). Of the 1,000 polces the base portfolo, the top 117 cotrbute half of μ, the top 6 cotrbute half of σ, ad the top 14 cotrbute half of μ. 4.4 Estmato of rage of clams The mea ad varace of the et clams dstrbuto ca be drectly calculated from the polcy lst usg formulae (4) ad (5). It s temptg to 1

14 estmate a cofdece terval for clams by assumg a Normal dstrbuto. For example the 95% cofdece terval s ofte stated as μ ± 1. 96σ, ad the 99% cofdece terval as μ ±. 58σ. However the smulato showed that the actual clams dstrbuto s postvely sewed, so the above rages are lely to be accurate. I partcular they are lely to uderestmate the extet of a blow-out of clams. The followg table gves the observed rages that 95% ad 99% of clams fell wth. The tals are of equal sze, for example the 95% cofdece teral s bouded by the.5%-le ad the 97.5%-le of the clams dstrbuto. The rages are expressed as umber of stadard devatos away from the mea. Portfolo Cofdece Reteto $100 Reteto $500 Reteto $5m Sze level 1,000 95% -1.7 to to to % -1.7 to to to ,000 95% to to to % -.1 to to to +.7 0,000 95% to to to % -.4 to to to +.99 The actual clams rages are most asymmetrc for the smaller portfolo szes ad the hghest retetos. Ths table demostrates the ecessty to ru smulatos to relably estmate the lely rage of clams. 4.5 Margal Captal Requremet Solvecy ad Captal Adequacy stadards gve detaled rules for determg the mmum captal a surer should hold. Here we are specfcally terested the effect o the total captal requremets of chagg the reteto level o a boo of term surace. For the purposes of ths paper, ths margal captal requremet s estmated by the Captal for Clams Fluctuato ( CFCF ). Ths s defed as 10% of the expected clams (et of resurace), plus the dfferece betwee the 99.5%-le of the et clams dstrbuto ad the mea of the et clams dstrbuto. Ths defto, whle very smplstc, taes to accout the rs of clams beg greater tha expected due to ms-estmatg the mea of the clams dstrbuto, ad of clams beg greater tha expected due to the radom occurrece of dvdual clams. The 10% fgure s based o GN5 s requremet to use 110% of best estmate mortalty for prudetal reservg. The 99.5% fgure was chose due to APRA s proposal to clude a oe year 00 requremet determg captal levels. It s acowledged that ths requremet s ot curretly cluded GN5. 14

15 If a surer totally resured ther boo the CFCF would be zero. As the reteto level creases the CFCF wll also grow. The followg table shows the CFCFs requred for the portfolos. CFCF Portfolo sze Reteto level 1,000 polces 5,000 polces 0,000 polces $ $ $ $ $ $1m $m $5m Ths cofrms that surers are advataged by creasg ther umber of depedet rss, as the captal requremets vary roughly wth the square root of the umber of polces. Ths s because, for the above portfolos, the CFCF s domated by the secod term, beg the allowace for the radom occurrece of clams. The portfolo would have to cota approxmately oe mllo depedet rss before the frst term becomes larger tha the secod. From that pot o the CFCF starts to crease more proporto to the umber of polces, rather tha to ts square root. I other words, oce a surer has tae o oe mllo depedet rss t has secured the maorty of the advatage ( terms of CFCF) t ca ever acheve by creasg ts scale. 4.6 Retur o captal ad selectg reteto level By resurg ts polces a surer s gvg away expected proft, the form of the resurer s marg. Tag o more rs creases a surer s et clams volatlty ad requres the surer to hold addtoal CFCF. The followg graph llustrates the trade-off betwee expected proft ad captal. Expected proft s calculated assumg a resurer s marg of 15% plus a 5% vestmet retur o the CFCF. The graph s for the portfolo of 5,000 polces. 15

16 Effect of Icreasg reteto 1 10 $5m reteto 8 CFCF 6 4 No reteto $00 reteto $500 reteto $1m reteto ,000 Expected Proft ($) The graph ca be regarded as a rs/retur pcture, as CFCF s effectvely a rs measure. Note how the curve becomes creasgly steep as reteto creases. Ths shows that creasgly large amouts of rs, ad addtoal captal, eed to be tae o to geerate every extra dollar of expected proft. I other words the margal retur o captal falls as reteto creases - a classc example of dmshg returs. 4.7 Optmal Reteto Level Most compaes operate wth a target retur o captal. To maxmse a surer s retur o captal, reteto should be set at the level where the margal retur o captal equals the target retur o captal. The followg table shows the margal retur o CFCF for the surace portfolos cosdered above. Margal retur o Portfolo sze CFCF Reteto level 1,000 polces 5,000 polces 0,000 polces 0 to $ % 18.% 0.5% $100 to $ % 18.4% 9.5% $00 to $ % 17.9% 8.9% $00 to $ % 17.% 9.0% $500 to $ % 15.5% 5.% $700 to $1m 8.6% 14.4%.% $1m to $m 7.5% 11.4% 18.6% $m to $5m 6.0% 8.% 1.7% Assume that the target retur o captal s 15%. The surer wth 1,000 polces should fully resure the boo, as the expected proft from retag rs ever provdes a adequate retur o the extra captal employed. The 16

17 surer wth 5,000 polces should set a $700 reteto for optmal captal usage. Smlarly the surer wth 0,000 polces should set a $m reteto. I practce a surer s decso about reteto levels wll be flueced by may factors, cludg ease of admstrato, the level of resurace support requred uderwrtg ad clams, exstg resurace arragemets ad compay rs appette. However the type of aalyss outled above gves a obectve startg pot for these decsos. 5 Modellg a Pademc 5.1 Pademc Scearos Pademcs of varyg testes have occurred for hudreds of years ad t s lely they wll cotue to occur. From a clams modellg vewpot a pademc ca be represeted by a perod whch the probablty of clam for every rs the portfolo s rased. I ths sese pademcs represet a addtoal rs that ca t be easly dversfed away. I ths secto the possblty of a pademc s cluded the clams modellg. The chage to the uderlyg model s as follows. I ay year there s a 97% chace that o pademc occurs. There s a % chace that a moderate pademc occurs ad a 1% chace that a severe pademc occurs. If a moderate pademc occurs the the probablty of clam o each polcy s creased by If a severe pademc occurs tha the probablty of a clam o each polcy s creased by I a year that a pademc does t occur, o the base portfolo of 1,000 polces we expect 1.6 clams totallg $1.m. If a moderate pademc occurs we expect.6 clams totallg $.01m. If a severe pademc occurs we expect 5.6 clams totallg $4.6m. Over the log term the total expected clams oly rses.% by cludg the possblty of pademcs. There s o mplcato that the parameters chose are a realstc estmate of ether the lelhood of a pademc or the actual effects a pademc may have o mortalty. However the chaces of a pademc occurrg are based o the occurrece of three flueza pademcs the 0 th cetury. The effect o mortalty s wth the rages gve for moderate ad severe pademcs Alex Sttt s excellet paper. 5. Smulato Results Icorporatg the possblty of pademcs s very dffcult a algebrac model of clams. I partcular the resultg clams dstrbutos are lely to be very dfferet from ay stadard dstrbuto. However t was smple to modfy the Mote Carlo smulato to clude pademcs. Aga 10,000 smulatos 17

18 were ru. The followg tables show the results of smulatos corporatg the pademc scearos. Base portfolo 1,000 polces Reteto Level Mea Clams Stadard devato Sewess 95%-le 99.5%-le $ $ $ $ $ $1m $m $5m ,000 polces Reteto Level Mea Clams Stadard devato Sewess 95%-le 99.5%-le $ $ $ $ $ $1m $m $5m ,000 polces Reteto Level Mea Clams Stadard devato Sewess 95%-le 99.5%-le $ $ $ $ $ $1m $m $5m

19 5. Observatos from Smulato Results Comparg wth the results secto 4., we see that cludg the pademc scearos had very lttle effect o the clam dstrbuto for the portfolo of 1,000 polces. Bascally the portfolo s so small that clams were hghly varable ayway, ad the possblty of a pademc has t made ths much worse. The clam dstrbuto for the portfolo of 5,000 polces s more strogly affected by cludg pademcs the modellg. Sewess s rased greatly, ad rocally sewess ow falls as reteto creases. The 95%-les are oly slghtly affected, creasg by aroud 5% from the o-pademc levels. However the 99.5%-les have creased by %. What s happeg s that the dstrbuto of clams s becomg tr-modal, oe mode for each pademc scearo. Ths patter s amplfed for the portfolo of 0,000 polces. Stadard devato s at least 50% above ts o-pademc levels, ad sewess has creased 10- to 0-fold. There s lttle chage to the 95%-le of the dstrbuto, but the 99.5%-le s more tha double ts pre-pademc level. The followg graph shows the dstrbuto of gross clams for the portfolo of 0,000 polces, based o the smulato results ,000,000 40,000,000 60,000,000 80,000, ,000,000 10,000,000 All of the occurreces of total clams over $80m were years whe a severe pademc occurred, ad t s these that have so greatly affected the summary statstcs. 19

20 5.4 Estmato of rage of clams The followg table gves the observed rages that 95% ad 99% of clams fell wth. The rages are expressed as umber of stadard devatos away from the mea. Portfolo Cofdece Reteto $100 Reteto $500 Reteto $5m Sze level 1,000 95% -1. to to to % -1. to to to ,000 95% to to to % to to to ,000 95% -1.1 to to to % -1. to to to Icluso of the pademc scearos has made the 99%-les eve more asymmetrc tha before, partcularly for the larger portfolos. I cotrast the 95%-le are tghter tha before. Ths s prmarly because the rages are expressed as stadard devatos away from the mea, ad the stadard devatos have grow sgfcatly. 5.5 Margal Captal Requremet The followg table shows the CFCF based o the portfolo. CFCF Portfolo sze Reteto level 1,000 polces 5,000 polces 0,000 polces $ $ $ $ $ $1m $m $5m Ths shows that the surer ow gets oly a small advatage by creasg the portfolo sze. Movg from 5,000 to 0,000 polces the CFCF has creased fourfold. 5.6 Optmal Reteto Level The followg table shows the margal retur o CFCF for the surace portfolos cosdered above. 0

21 Portfolo sze Reteto level 1000 polces 5,000 polces 0,000 polces 0 to $ % 11.0% 10.9% $100 to $ % 10.8% 10.8% $00 to $00 9.% 10.9% 10.9% $00 to $500 9.% 11.0% 10.8% $500 to $ % 10.6% 10.7% $700 to $1m 8.4% 1.0% 10.8% $1m to $m 7.7% 1.0% 10.5% $m to $5m 6.% 10.% 9.6% The table shows that oe of the surers acheves the target retur o captal by tag o surace rs. Ths s a complete chage from the o-pademc scearo, ad s caused by the eed to eep eormous levels of captal to cope wth a catastrophc evet that may occur oce a cetury. The results above have to be balaced wth practcal cosderatos. Resurers wo t geerally be wllg for a surer to cede all rs, so a compromse posto wll have to be reached. 5.7 Dscusso Pademcs lmt the ablty of surers to accept large umbers of rss ad eoy the beefts of dversfcato. By defto pademcs affect wde regos, but t s very lely that ay pademc wll be more severe some regos tha others. Ths gves surers, ad partcular resurers, the ablty to reduce ther rs cocetrato by acceptg rss across multple coutres ad cotets. Of course surers acheve dversty by tag o dfferet types of rss, such as property surace. The aalyss above dcates that lfe surers ca t geerally afford to hold suffcet captal to cope wth all the clams that would occur uder a partcularly severe pademc. To mata a vable surace dustry the captal requremets for lfe surers wll have to be set at lower levels tha those assumed ths paper. Ultmately ths reforces the pot that each compay s resources are fte, ad that ay surer wll fal to meet ts oblgatos uder the most extreme crcumstaces. 1

ANOVA Notes Page 1. Analysis of Variance for a One-Way Classification of Data

ANOVA Notes Page 1. Analysis of Variance for a One-Way Classification of Data ANOVA Notes Page Aalss of Varace for a Oe-Wa Classfcato of Data Cosder a sgle factor or treatmet doe at levels (e, there are,, 3, dfferet varatos o the prescrbed treatmet) Wth a gve treatmet level there

More information

An Effectiveness of Integrated Portfolio in Bancassurance

An Effectiveness of Integrated Portfolio in Bancassurance A Effectveess of Itegrated Portfolo Bacassurace Taea Karya Research Ceter for Facal Egeerg Isttute of Ecoomc Research Kyoto versty Sayouu Kyoto 606-850 Japa arya@eryoto-uacp Itroducto As s well ow the

More information

Average Price Ratios

Average Price Ratios Average Prce Ratos Morgstar Methodology Paper August 3, 2005 2005 Morgstar, Ic. All rghts reserved. The formato ths documet s the property of Morgstar, Ic. Reproducto or trascrpto by ay meas, whole or

More information

1. The Time Value of Money

1. The Time Value of Money Corporate Face [00-0345]. The Tme Value of Moey. Compoudg ad Dscoutg Captalzato (compoudg, fdg future values) s a process of movg a value forward tme. It yelds the future value gve the relevat compoudg

More information

Measures of Dispersion, Skew, & Kurtosis (based on Kirk, Ch. 4) {to be used in conjunction with Measures of Dispersion Chart }

Measures of Dispersion, Skew, & Kurtosis (based on Kirk, Ch. 4) {to be used in conjunction with Measures of Dispersion Chart } Percetles Psych 54, 9/8/05 p. /6 Measures of Dsperso, kew, & Kurtoss (based o Krk, Ch. 4) {to be used cojucto wth Measures of Dsperso Chart } percetle (P % ): a score below whch a specfed percetage of

More information

Chapter 3 3-1. Chapter Goals. Summary Measures. Chapter Topics. Measures of Center and Location. Notation Conventions

Chapter 3 3-1. Chapter Goals. Summary Measures. Chapter Topics. Measures of Center and Location. Notation Conventions Chapter 3 3- Chapter Goals Chapter 3 umercal Descrptve Measures After completg ths chapter, you should be able to: Compute ad terpret the mea, meda, ad mode for a set of data Fd the rage, varace, ad stadard

More information

ECONOMIC CHOICE OF OPTIMUM FEEDER CABLE CONSIDERING RISK ANALYSIS. University of Brasilia (UnB) and The Brazilian Regulatory Agency (ANEEL), Brazil

ECONOMIC CHOICE OF OPTIMUM FEEDER CABLE CONSIDERING RISK ANALYSIS. University of Brasilia (UnB) and The Brazilian Regulatory Agency (ANEEL), Brazil ECONOMIC CHOICE OF OPTIMUM FEEDER CABE CONSIDERING RISK ANAYSIS I Camargo, F Fgueredo, M De Olvera Uversty of Brasla (UB) ad The Brazla Regulatory Agecy (ANEE), Brazl The choce of the approprate cable

More information

STATISTICAL PROPERTIES OF LEAST SQUARES ESTIMATORS. x, where. = y - ˆ " 1

STATISTICAL PROPERTIES OF LEAST SQUARES ESTIMATORS. x, where. = y - ˆ  1 STATISTICAL PROPERTIES OF LEAST SQUARES ESTIMATORS Recall Assumpto E(Y x) η 0 + η x (lear codtoal mea fucto) Data (x, y ), (x 2, y 2 ),, (x, y ) Least squares estmator ˆ E (Y x) ˆ " 0 + ˆ " x, where ˆ

More information

The Gompertz-Makeham distribution. Fredrik Norström. Supervisor: Yuri Belyaev

The Gompertz-Makeham distribution. Fredrik Norström. Supervisor: Yuri Belyaev The Gompertz-Makeham dstrbuto by Fredrk Norström Master s thess Mathematcal Statstcs, Umeå Uversty, 997 Supervsor: Yur Belyaev Abstract Ths work s about the Gompertz-Makeham dstrbuto. The dstrbuto has

More information

APPENDIX III THE ENVELOPE PROPERTY

APPENDIX III THE ENVELOPE PROPERTY Apped III APPENDIX III THE ENVELOPE PROPERTY Optmzato mposes a very strog structure o the problem cosdered Ths s the reaso why eoclasscal ecoomcs whch assumes optmzg behavour has bee the most successful

More information

IDENTIFICATION OF THE DYNAMICS OF THE GOOGLE S RANKING ALGORITHM. A. Khaki Sedigh, Mehdi Roudaki

IDENTIFICATION OF THE DYNAMICS OF THE GOOGLE S RANKING ALGORITHM. A. Khaki Sedigh, Mehdi Roudaki IDENIFICAION OF HE DYNAMICS OF HE GOOGLE S RANKING ALGORIHM A. Khak Sedgh, Mehd Roudak Cotrol Dvso, Departmet of Electrcal Egeerg, K.N.oos Uversty of echology P. O. Box: 16315-1355, ehra, Ira sedgh@eetd.ktu.ac.r,

More information

Abraham Zaks. Technion I.I.T. Haifa ISRAEL. and. University of Haifa, Haifa ISRAEL. Abstract

Abraham Zaks. Technion I.I.T. Haifa ISRAEL. and. University of Haifa, Haifa ISRAEL. Abstract Preset Value of Autes Uder Radom Rates of Iterest By Abraham Zas Techo I.I.T. Hafa ISRAEL ad Uversty of Hafa, Hafa ISRAEL Abstract Some attempts were made to evaluate the future value (FV) of the expected

More information

Classic Problems at a Glance using the TVM Solver

Classic Problems at a Glance using the TVM Solver C H A P T E R 2 Classc Problems at a Glace usg the TVM Solver The table below llustrates the most commo types of classc face problems. The formulas are gve for each calculato. A bref troducto to usg the

More information

MEASURES OF CENTRAL TENDENCY

MEASURES OF CENTRAL TENDENCY MODULE - 6 Statstcs Measures of Cetral Tedecy 25 MEASURES OF CENTRAL TENDENCY I the prevous lesso, we have leart that the data could be summarsed to some extet by presetg t the form of a frequecy table.

More information

The Time Value of Money

The Time Value of Money The Tme Value of Moey 1 Iversemet Optos Year: 1624 Property Traded: Mahatta Islad Prce : $24.00, FV of $24 @ 6%: FV = $24 (1+0.06) 388 = $158.08 bllo Opto 1 0 1 2 3 4 5 t ($519.37) 0 0 0 0 $1,000 Opto

More information

The analysis of annuities relies on the formula for geometric sums: r k = rn+1 1 r 1. (2.1) k=0

The analysis of annuities relies on the formula for geometric sums: r k = rn+1 1 r 1. (2.1) k=0 Chapter 2 Autes ad loas A auty s a sequece of paymets wth fxed frequecy. The term auty orgally referred to aual paymets (hece the ame), but t s ow also used for paymets wth ay frequecy. Autes appear may

More information

Credibility Premium Calculation in Motor Third-Party Liability Insurance

Credibility Premium Calculation in Motor Third-Party Liability Insurance Advaces Mathematcal ad Computatoal Methods Credblty remum Calculato Motor Thrd-arty Lablty Isurace BOHA LIA, JAA KUBAOVÁ epartmet of Mathematcs ad Quattatve Methods Uversty of ardubce Studetská 95, 53

More information

T = 1/freq, T = 2/freq, T = i/freq, T = n (number of cash flows = freq n) are :

T = 1/freq, T = 2/freq, T = i/freq, T = n (number of cash flows = freq n) are : Bullets bods Let s descrbe frst a fxed rate bod wthout amortzg a more geeral way : Let s ote : C the aual fxed rate t s a percetage N the otoal freq ( 2 4 ) the umber of coupo per year R the redempto of

More information

Online Appendix: Measured Aggregate Gains from International Trade

Online Appendix: Measured Aggregate Gains from International Trade Ole Appedx: Measured Aggregate Gas from Iteratoal Trade Arel Burste UCLA ad NBER Javer Cravo Uversty of Mchga March 3, 2014 I ths ole appedx we derve addtoal results dscussed the paper. I the frst secto,

More information

RUSSIAN ROULETTE AND PARTICLE SPLITTING

RUSSIAN ROULETTE AND PARTICLE SPLITTING RUSSAN ROULETTE AND PARTCLE SPLTTNG M. Ragheb 3/7/203 NTRODUCTON To stuatos are ecoutered partcle trasport smulatos:. a multplyg medum, a partcle such as a eutro a cosmc ray partcle or a photo may geerate

More information

Simple Linear Regression

Simple Linear Regression Smple Lear Regresso Regresso equato a equato that descrbes the average relatoshp betwee a respose (depedet) ad a eplaator (depedet) varable. 6 8 Slope-tercept equato for a le m b (,6) slope. (,) 6 6 8

More information

6.7 Network analysis. 6.7.1 Introduction. References - Network analysis. Topological analysis

6.7 Network analysis. 6.7.1 Introduction. References - Network analysis. Topological analysis 6.7 Network aalyss Le data that explctly store topologcal formato are called etwork data. Besdes spatal operatos, several methods of spatal aalyss are applcable to etwork data. Fgure: Network data Refereces

More information

10.5 Future Value and Present Value of a General Annuity Due

10.5 Future Value and Present Value of a General Annuity Due Chapter 10 Autes 371 5. Thomas leases a car worth $4,000 at.99% compouded mothly. He agrees to make 36 lease paymets of $330 each at the begg of every moth. What s the buyout prce (resdual value of the

More information

Report 52 Fixed Maturity EUR Industrial Bond Funds

Report 52 Fixed Maturity EUR Industrial Bond Funds Rep52, Computed & Prted: 17/06/2015 11:53 Report 52 Fxed Maturty EUR Idustral Bod Fuds From Dec 2008 to Dec 2014 31/12/2008 31 December 1999 31/12/2014 Bechmark Noe Defto of the frm ad geeral formato:

More information

CHAPTER 2. Time Value of Money 6-1

CHAPTER 2. Time Value of Money 6-1 CHAPTER 2 Tme Value of Moey 6- Tme Value of Moey (TVM) Tme Les Future value & Preset value Rates of retur Autes & Perpetutes Ueve cash Flow Streams Amortzato 6-2 Tme les 0 2 3 % CF 0 CF CF 2 CF 3 Show

More information

Chapter 3. AMORTIZATION OF LOAN. SINKING FUNDS R =

Chapter 3. AMORTIZATION OF LOAN. SINKING FUNDS R = Chapter 3. AMORTIZATION OF LOAN. SINKING FUNDS Objectves of the Topc: Beg able to formalse ad solve practcal ad mathematcal problems, whch the subjects of loa amortsato ad maagemet of cumulatve fuds are

More information

SHAPIRO-WILK TEST FOR NORMALITY WITH KNOWN MEAN

SHAPIRO-WILK TEST FOR NORMALITY WITH KNOWN MEAN SHAPIRO-WILK TEST FOR NORMALITY WITH KNOWN MEAN Wojcech Zelńsk Departmet of Ecoometrcs ad Statstcs Warsaw Uversty of Lfe Sceces Nowoursyowska 66, -787 Warszawa e-mal: wojtekzelsk@statystykafo Zofa Hausz,

More information

of the relationship between time and the value of money.

of the relationship between time and the value of money. TIME AND THE VALUE OF MONEY Most agrbusess maagers are famlar wth the terms compoudg, dscoutg, auty, ad captalzato. That s, most agrbusess maagers have a tutve uderstadg that each term mples some relatoshp

More information

DETERMINISTIC AND STOCHASTIC MODELLING OF TECHNICAL RESERVES IN SHORT-TERM INSURANCE CONTRACTS

DETERMINISTIC AND STOCHASTIC MODELLING OF TECHNICAL RESERVES IN SHORT-TERM INSURANCE CONTRACTS DETERMINISTI AND STOHASTI MODELLING OF TEHNIAL RESERVES IN SHORT-TERM INSURANE ONTRATS Patrck G O Weke School of Mathematcs, Uversty of Narob, Keya Emal: pweke@uobacke ABSTART lams reservg for geeral surace

More information

Managing Interdependent Information Security Risks: Cyberinsurance, Managed Security Services, and Risk Pooling Arrangements

Managing Interdependent Information Security Risks: Cyberinsurance, Managed Security Services, and Risk Pooling Arrangements Maagg Iterdepedet Iformato Securty Rsks: Cybersurace, Maaged Securty Servces, ad Rsk Poolg Arragemets Xa Zhao Assstat Professor Departmet of Iformato Systems ad Supply Cha Maagemet Brya School of Busess

More information

Banking (Early Repayment of Housing Loans) Order, 5762 2002 1

Banking (Early Repayment of Housing Loans) Order, 5762 2002 1 akg (Early Repaymet of Housg Loas) Order, 5762 2002 y vrtue of the power vested me uder Secto 3 of the akg Ordace 94 (hereafter, the Ordace ), followg cosultato wth the Commttee, ad wth the approval of

More information

Numerical Methods with MS Excel

Numerical Methods with MS Excel TMME, vol4, o.1, p.84 Numercal Methods wth MS Excel M. El-Gebely & B. Yushau 1 Departmet of Mathematcal Sceces Kg Fahd Uversty of Petroleum & Merals. Dhahra, Saud Araba. Abstract: I ths ote we show how

More information

Commercial Pension Insurance Program Design and Estimated of Tax Incentives---- Based on Analysis of Enterprise Annuity Tax Incentives

Commercial Pension Insurance Program Design and Estimated of Tax Incentives---- Based on Analysis of Enterprise Annuity Tax Incentives Iteratoal Joural of Busess ad Socal Scece Vol 5, No ; October 204 Commercal Peso Isurace Program Desg ad Estmated of Tax Icetves---- Based o Aalyss of Eterprse Auty Tax Icetves Huag Xue, Lu Yatg School

More information

FINANCIAL MATHEMATICS 12 MARCH 2014

FINANCIAL MATHEMATICS 12 MARCH 2014 FINNCIL MTHEMTICS 12 MRCH 2014 I ths lesso we: Lesso Descrpto Make use of logarthms to calculate the value of, the tme perod, the equato P1 or P1. Solve problems volvg preset value ad future value autes.

More information

Capacitated Production Planning and Inventory Control when Demand is Unpredictable for Most Items: The No B/C Strategy

Capacitated Production Planning and Inventory Control when Demand is Unpredictable for Most Items: The No B/C Strategy SCHOOL OF OPERATIONS RESEARCH AND INDUSTRIAL ENGINEERING COLLEGE OF ENGINEERING CORNELL UNIVERSITY ITHACA, NY 4853-380 TECHNICAL REPORT Jue 200 Capactated Producto Plag ad Ivetory Cotrol whe Demad s Upredctable

More information

ADAPTATION OF SHAPIRO-WILK TEST TO THE CASE OF KNOWN MEAN

ADAPTATION OF SHAPIRO-WILK TEST TO THE CASE OF KNOWN MEAN Colloquum Bometrcum 4 ADAPTATION OF SHAPIRO-WILK TEST TO THE CASE OF KNOWN MEAN Zofa Hausz, Joaa Tarasńska Departmet of Appled Mathematcs ad Computer Scece Uversty of Lfe Sceces Lubl Akademcka 3, -95 Lubl

More information

Settlement Prediction by Spatial-temporal Random Process

Settlement Prediction by Spatial-temporal Random Process Safety, Relablty ad Rs of Structures, Ifrastructures ad Egeerg Systems Furuta, Fragopol & Shozua (eds Taylor & Fracs Group, Lodo, ISBN 978---77- Settlemet Predcto by Spatal-temporal Radom Process P. Rugbaapha

More information

n. We know that the sum of squares of p independent standard normal variables has a chi square distribution with p degrees of freedom.

n. We know that the sum of squares of p independent standard normal variables has a chi square distribution with p degrees of freedom. UMEÅ UNIVERSITET Matematsk-statstska sttutoe Multvarat dataaalys för tekologer MSTB0 PA TENTAMEN 004-0-9 LÖSNINGSFÖRSLAG TILL TENTAMEN I MATEMATISK STATISTIK Multvarat dataaalys för tekologer B, 5 poäg.

More information

Chapter Eight. f : R R

Chapter Eight. f : R R Chapter Eght f : R R 8. Itroducto We shall ow tur our atteto to the very mportat specal case of fuctos that are real, or scalar, valued. These are sometmes called scalar felds. I the very, but mportat,

More information

Proceedings of the 2010 Winter Simulation Conference B. Johansson, S. Jain, J. Montoya-Torres, J. Hugan, and E. Yücesan, eds.

Proceedings of the 2010 Winter Simulation Conference B. Johansson, S. Jain, J. Montoya-Torres, J. Hugan, and E. Yücesan, eds. Proceedgs of the 21 Wter Smulato Coferece B. Johasso, S. Ja, J. Motoya-Torres, J. Huga, ad E. Yücesa, eds. EMPIRICAL METHODS OR TWO-ECHELON INVENTORY MANAGEMENT WITH SERVICE LEVEL CONSTRAINTS BASED ON

More information

A New Bayesian Network Method for Computing Bottom Event's Structural Importance Degree using Jointree

A New Bayesian Network Method for Computing Bottom Event's Structural Importance Degree using Jointree , pp.277-288 http://dx.do.org/10.14257/juesst.2015.8.1.25 A New Bayesa Network Method for Computg Bottom Evet's Structural Importace Degree usg Jotree Wag Yao ad Su Q School of Aeroautcs, Northwester Polytechcal

More information

Report 19 Euroland Corporate Bonds

Report 19 Euroland Corporate Bonds Rep19, Computed & Prted: 17/06/2015 11:38 Report 19 Eurolad Corporate Bods From Dec 1999 to Dec 2014 31/12/1999 31 December 1999 31/12/2014 Bechmark 100% IBOXX Euro Corp All Mats. TR Defto of the frm ad

More information

The simple linear Regression Model

The simple linear Regression Model The smple lear Regresso Model Correlato coeffcet s o-parametrc ad just dcates that two varables are assocated wth oe aother, but t does ot gve a deas of the kd of relatoshp. Regresso models help vestgatg

More information

RQM: A new rate-based active queue management algorithm

RQM: A new rate-based active queue management algorithm : A ew rate-based actve queue maagemet algorthm Jeff Edmods, Suprakash Datta, Patrck Dymod, Kashf Al Computer Scece ad Egeerg Departmet, York Uversty, Toroto, Caada Abstract I ths paper, we propose a ew

More information

ANALYTICAL MODEL FOR TCP FILE TRANSFERS OVER UMTS. Janne Peisa Ericsson Research 02420 Jorvas, Finland. Michael Meyer Ericsson Research, Germany

ANALYTICAL MODEL FOR TCP FILE TRANSFERS OVER UMTS. Janne Peisa Ericsson Research 02420 Jorvas, Finland. Michael Meyer Ericsson Research, Germany ANALYTICAL MODEL FOR TCP FILE TRANSFERS OVER UMTS Jae Pesa Erco Research 4 Jorvas, Flad Mchael Meyer Erco Research, Germay Abstract Ths paper proposes a farly complex model to aalyze the performace of

More information

Mathematics of Finance

Mathematics of Finance CATE Mathematcs of ace.. TODUCTO ths chapter we wll dscuss mathematcal methods ad formulae whch are helpful busess ad persoal face. Oe of the fudametal cocepts the mathematcs of face s the tme value of

More information

We present a new approach to pricing American-style derivatives that is applicable to any Markovian setting

We present a new approach to pricing American-style derivatives that is applicable to any Markovian setting MANAGEMENT SCIENCE Vol. 52, No., Jauary 26, pp. 95 ss 25-99 ess 526-55 6 52 95 forms do.287/msc.5.447 26 INFORMS Prcg Amerca-Style Dervatves wth Europea Call Optos Scott B. Laprse BAE Systems, Advaced

More information

USEFULNESS OF BOOTSTRAPPING IN PORTFOLIO MANAGEMENT

USEFULNESS OF BOOTSTRAPPING IN PORTFOLIO MANAGEMENT USEFULNESS OF BOOTSTRAPPING IN PORTFOLIO MANAGEMENT Radovaov Bors Faculty of Ecoomcs Subotca Segedsk put 9-11 Subotca 24000 E-mal: radovaovb@ef.us.ac.rs Marckć Aleksadra Faculty of Ecoomcs Subotca Segedsk

More information

We investigate a simple adaptive approach to optimizing seat protection levels in airline

We investigate a simple adaptive approach to optimizing seat protection levels in airline Reveue Maagemet Wthout Forecastg or Optmzato: A Adaptve Algorthm for Determg Arle Seat Protecto Levels Garrett va Ryz Jeff McGll Graduate School of Busess, Columba Uversty, New York, New York 10027 School

More information

Performance Attribution. Methodology Overview

Performance Attribution. Methodology Overview erformace Attrbuto Methodology Overvew Faba SUAREZ March 2004 erformace Attrbuto Methodology 1.1 Itroducto erformace Attrbuto s a set of techques that performace aalysts use to expla why a portfolo's performace

More information

Optimal Packetization Interval for VoIP Applications Over IEEE 802.16 Networks

Optimal Packetization Interval for VoIP Applications Over IEEE 802.16 Networks Optmal Packetzato Iterval for VoIP Applcatos Over IEEE 802.16 Networks Sheha Perera Harsha Srsea Krzysztof Pawlkowsk Departmet of Electrcal & Computer Egeerg Uversty of Caterbury New Zealad sheha@elec.caterbury.ac.z

More information

Chapter 3 0.06 = 3000 ( 1.015 ( 1 ) Present Value of an Annuity. Section 4 Present Value of an Annuity; Amortization

Chapter 3 0.06 = 3000 ( 1.015 ( 1 ) Present Value of an Annuity. Section 4 Present Value of an Annuity; Amortization Chapter 3 Mathematcs of Face Secto 4 Preset Value of a Auty; Amortzato Preset Value of a Auty I ths secto, we wll address the problem of determg the amout that should be deposted to a accout ow at a gve

More information

Preprocess a planar map S. Given a query point p, report the face of S containing p. Goal: O(n)-size data structure that enables O(log n) query time.

Preprocess a planar map S. Given a query point p, report the face of S containing p. Goal: O(n)-size data structure that enables O(log n) query time. Computatoal Geometry Chapter 6 Pot Locato 1 Problem Defto Preprocess a plaar map S. Gve a query pot p, report the face of S cotag p. S Goal: O()-sze data structure that eables O(log ) query tme. C p E

More information

Statistical Pattern Recognition (CE-725) Department of Computer Engineering Sharif University of Technology

Statistical Pattern Recognition (CE-725) Department of Computer Engineering Sharif University of Technology I The Name of God, The Compassoate, The ercful Name: Problems' eys Studet ID#:. Statstcal Patter Recogto (CE-725) Departmet of Computer Egeerg Sharf Uversty of Techology Fal Exam Soluto - Sprg 202 (50

More information

Efficient Traceback of DoS Attacks using Small Worlds in MANET

Efficient Traceback of DoS Attacks using Small Worlds in MANET Effcet Traceback of DoS Attacks usg Small Worlds MANET Yog Km, Vshal Sakhla, Ahmed Helmy Departmet. of Electrcal Egeerg, Uversty of Souther Calfora, U.S.A {yogkm, sakhla, helmy}@ceg.usc.edu Abstract Moble

More information

ANNEX 77 FINANCE MANAGEMENT. (Working material) Chief Actuary Prof. Gaida Pettere BTA INSURANCE COMPANY SE

ANNEX 77 FINANCE MANAGEMENT. (Working material) Chief Actuary Prof. Gaida Pettere BTA INSURANCE COMPANY SE ANNEX 77 FINANCE MANAGEMENT (Workg materal) Chef Actuary Prof. Gada Pettere BTA INSURANCE COMPANY SE 1 FUNDAMENTALS of INVESTMENT I THEORY OF INTEREST RATES 1.1 ACCUMULATION Iterest may be regarded as

More information

The Analysis of Development of Insurance Contract Premiums of General Liability Insurance in the Business Insurance Risk

The Analysis of Development of Insurance Contract Premiums of General Liability Insurance in the Business Insurance Risk The Aalyss of Developmet of Isurace Cotract Premums of Geeral Lablty Isurace the Busess Isurace Rsk the Frame of the Czech Isurace Market 1998 011 Scetfc Coferece Jue, 10. - 14. 013 Pavla Kubová Departmet

More information

Speeding up k-means Clustering by Bootstrap Averaging

Speeding up k-means Clustering by Bootstrap Averaging Speedg up -meas Clusterg by Bootstrap Averagg Ia Davdso ad Ashw Satyaarayaa Computer Scece Dept, SUNY Albay, NY, USA,. {davdso, ashw}@cs.albay.edu Abstract K-meas clusterg s oe of the most popular clusterg

More information

Future Value of an Annuity

Future Value of an Annuity Future Value of a Auty After payg all your blls, you have $200 left each payday (at the ed of each moth) that you wll put to savgs order to save up a dow paymet for a house. If you vest ths moey at 5%

More information

LATERAL TRANSHIPMENT-A TECHNIQUE FOR INVENTORY CONTROL IN MULTI RETAILER SUPPLY CHAIN SYSTEM

LATERAL TRANSHIPMENT-A TECHNIQUE FOR INVENTORY CONTROL IN MULTI RETAILER SUPPLY CHAIN SYSTEM Iteratoal Joural of Iformato Techology ad Kowledge Maagemet July-December 200, Volume 2, No. 2, pp. 3-35 LATERAL TRANSHIPMENT-A TECHNIQUE FOR INVENTORY CONTROL IN MULTI RETAILER SUPPLY CHAIN SYSTEM Dharamvr

More information

Overview. Eingebettete Systeme. Model of periodic tasks. Model of periodic tasks. Echtzeitverhalten und Betriebssysteme

Overview. Eingebettete Systeme. Model of periodic tasks. Model of periodic tasks. Echtzeitverhalten und Betriebssysteme Overvew Egebettete Systeme able of some kow preemptve schedulg algorthms for perodc tasks: Echtzetverhalte ud Betrebssysteme 5. Perodsche asks statc prorty dyamc prorty Deadle equals perod Deadle smaller

More information

Integrating Production Scheduling and Maintenance: Practical Implications

Integrating Production Scheduling and Maintenance: Practical Implications Proceedgs of the 2012 Iteratoal Coferece o Idustral Egeerg ad Operatos Maagemet Istabul, Turkey, uly 3 6, 2012 Itegratg Producto Schedulg ad Mateace: Practcal Implcatos Lath A. Hadd ad Umar M. Al-Turk

More information

Optimal replacement and overhaul decisions with imperfect maintenance and warranty contracts

Optimal replacement and overhaul decisions with imperfect maintenance and warranty contracts Optmal replacemet ad overhaul decsos wth mperfect mateace ad warraty cotracts R. Pascual Departmet of Mechacal Egeerg, Uversdad de Chle, Caslla 2777, Satago, Chle Phoe: +56-2-6784591 Fax:+56-2-689657 rpascual@g.uchle.cl

More information

An Evaluation of Naïve Bayesian Anti-Spam Filtering Techniques

An Evaluation of Naïve Bayesian Anti-Spam Filtering Techniques Proceedgs of the 2007 IEEE Workshop o Iformato Assurace Uted tates Mltary Academy, West Pot, Y 20-22 Jue 2007 A Evaluato of aïve Bayesa At-pam Flterg Techques Vkas P. Deshpade, Robert F. Erbacher, ad Chrs

More information

Maintenance Scheduling of Distribution System with Optimal Economy and Reliability

Maintenance Scheduling of Distribution System with Optimal Economy and Reliability Egeerg, 203, 5, 4-8 http://dx.do.org/0.4236/eg.203.59b003 Publshed Ole September 203 (http://www.scrp.org/joural/eg) Mateace Schedulg of Dstrbuto System wth Optmal Ecoomy ad Relablty Syua Hog, Hafeg L,

More information

A Study of Unrelated Parallel-Machine Scheduling with Deteriorating Maintenance Activities to Minimize the Total Completion Time

A Study of Unrelated Parallel-Machine Scheduling with Deteriorating Maintenance Activities to Minimize the Total Completion Time Joural of Na Ka, Vol. 0, No., pp.5-9 (20) 5 A Study of Urelated Parallel-Mache Schedulg wth Deteroratg Mateace Actvtes to Mze the Total Copleto Te Suh-Jeq Yag, Ja-Yuar Guo, Hs-Tao Lee Departet of Idustral

More information

Optimal multi-degree reduction of Bézier curves with constraints of endpoints continuity

Optimal multi-degree reduction of Bézier curves with constraints of endpoints continuity Computer Aded Geometrc Desg 19 (2002 365 377 wwwelsevercom/locate/comad Optmal mult-degree reducto of Bézer curves wth costrats of edpots cotuty Guo-Dog Che, Guo-J Wag State Key Laboratory of CAD&CG, Isttute

More information

The premium for mandatory house insurance in Romania considerations regarding its financial solvability

The premium for mandatory house insurance in Romania considerations regarding its financial solvability Avalable ole at www.scecedrect.com Proceda Ecoomcs ad Face 3 ( 202 ) 829 836 Emergg Markets Queres Face ad Busess The premum for madatory house surace Romaa cosderatos regardg ts facal solvablty Raluca

More information

ISyE 512 Chapter 7. Control Charts for Attributes. Instructor: Prof. Kaibo Liu. Department of Industrial and Systems Engineering UW-Madison

ISyE 512 Chapter 7. Control Charts for Attributes. Instructor: Prof. Kaibo Liu. Department of Industrial and Systems Engineering UW-Madison ISyE 512 Chapter 7 Cotrol Charts for Attrbutes Istructor: Prof. Kabo Lu Departmet of Idustral ad Systems Egeerg UW-Madso Emal: klu8@wsc.edu Offce: Room 3017 (Mechacal Egeerg Buldg) 1 Lst of Topcs Chapter

More information

Network dimensioning for elastic traffic based on flow-level QoS

Network dimensioning for elastic traffic based on flow-level QoS Network dmesog for elastc traffc based o flow-level QoS 1(10) Network dmesog for elastc traffc based o flow-level QoS Pas Lassla ad Jorma Vrtamo Networkg Laboratory Helsk Uversty of Techology Itroducto

More information

The paper presents Constant Rebalanced Portfolio first introduced by Thomas

The paper presents Constant Rebalanced Portfolio first introduced by Thomas Itroducto The paper presets Costat Rebalaced Portfolo frst troduced by Thomas Cover. There are several weakesses of ths approach. Oe s that t s extremely hard to fd the optmal weghts ad the secod weakess

More information

Beta. A Statistical Analysis of a Stock s Volatility. Courtney Wahlstrom. Iowa State University, Master of School Mathematics. Creative Component

Beta. A Statistical Analysis of a Stock s Volatility. Courtney Wahlstrom. Iowa State University, Master of School Mathematics. Creative Component Beta A Statstcal Aalyss of a Stock s Volatlty Courtey Wahlstrom Iowa State Uversty, Master of School Mathematcs Creatve Compoet Fall 008 Amy Froelch, Major Professor Heather Bolles, Commttee Member Travs

More information

CHAPTER 13. Simple Linear Regression LEARNING OBJECTIVES. USING STATISTICS @ Sunflowers Apparel

CHAPTER 13. Simple Linear Regression LEARNING OBJECTIVES. USING STATISTICS @ Sunflowers Apparel CHAPTER 3 Smple Lear Regresso USING STATISTICS @ Suflowers Apparel 3 TYPES OF REGRESSION MODELS 3 DETERMINING THE SIMPLE LINEAR REGRESSION EQUATION The Least-Squares Method Vsual Exploratos: Explorg Smple

More information

Australian Climate Change Adaptation Network for Settlements and Infrastructure. Discussion Paper February 2010

Australian Climate Change Adaptation Network for Settlements and Infrastructure. Discussion Paper February 2010 Australa Clmate Chage Adaptato Network for Settlemets ad Ifrastructure Dscusso Paper February 2010 The corporato of ucertaty assocated wth clmate chage to frastructure vestmet apprasal Davd G. Carmchael

More information

DECISION MAKING WITH THE OWA OPERATOR IN SPORT MANAGEMENT

DECISION MAKING WITH THE OWA OPERATOR IN SPORT MANAGEMENT ESTYLF08, Cuecas Meras (Meres - Lagreo), 7-9 de Septembre de 2008 DECISION MAKING WITH THE OWA OPERATOR IN SPORT MANAGEMENT José M. Mergó Aa M. Gl-Lafuete Departmet of Busess Admstrato, Uversty of Barceloa

More information

Load and Resistance Factor Design (LRFD)

Load and Resistance Factor Design (LRFD) 53:134 Structural Desg II Load ad Resstace Factor Desg (LRFD) Specfcatos ad Buldg Codes: Structural steel desg of buldgs the US s prcpally based o the specfcatos of the Amerca Isttute of Steel Costructo

More information

Session 4: Descriptive statistics and exporting Stata results

Session 4: Descriptive statistics and exporting Stata results Itrduct t Stata Jrd Muñz (UAB) Sess 4: Descrptve statstcs ad exprtg Stata results I ths sess we are gg t wrk wth descrptve statstcs Stata. Frst, we preset a shrt trduct t the very basc statstcal ctets

More information

On Error Detection with Block Codes

On Error Detection with Block Codes BULGARIAN ACADEMY OF SCIENCES CYBERNETICS AND INFORMATION TECHNOLOGIES Volume 9, No 3 Sofa 2009 O Error Detecto wth Block Codes Rostza Doduekova Chalmers Uversty of Techology ad the Uversty of Gotheburg,

More information

AP Statistics 2006 Free-Response Questions Form B

AP Statistics 2006 Free-Response Questions Form B AP Statstcs 006 Free-Respose Questos Form B The College Board: Coectg Studets to College Success The College Board s a ot-for-proft membershp assocato whose msso s to coect studets to college success ad

More information

Curve Fitting and Solution of Equation

Curve Fitting and Solution of Equation UNIT V Curve Fttg ad Soluto of Equato 5. CURVE FITTING I ma braches of appled mathematcs ad egeerg sceces we come across epermets ad problems, whch volve two varables. For eample, t s kow that the speed

More information

Report 05 Global Fixed Income

Report 05 Global Fixed Income Report 05 Global Fxed Icome From Dec 1999 to Dec 2014 31/12/1999 31 December 1999 31/12/2014 Rep05, Computed & Prted: 17/06/2015 11:24 New Performace Idcator (01/01/12) 100% Barclays Aggregate Global Credt

More information

Report 06 Global High Yield Bonds

Report 06 Global High Yield Bonds Rep06, Computed & Prted: 17/06/2015 11:25 Report 06 Global Hgh Yeld Bods From Dec 2000 to Dec 2014 31/12/2000 31 December 1999 31/12/2014 New Bechmark (01/01/13) 80% Barclays Euro HY Ex Facals 3% Capped

More information

FINANCIAL FORMULAE. Amount of One or Future Value of One ($1, 1, 1, etc.)... 2. Present Value (or Present Worth) of One ($1, 1, 1, etc.)...

FINANCIAL FORMULAE. Amount of One or Future Value of One ($1, 1, 1, etc.)... 2. Present Value (or Present Worth) of One ($1, 1, 1, etc.)... Amout of Oe or Future Value of Oe ($,,, etc.)... 2 Preset Value (or Preset Worth) of Oe ($,,, etc.)... 2 Amout of Oe per Perod... 3 or Future Value of Oe per Perod Preset Value (or Preset Worth) of Oe

More information

Regression Analysis. 1. Introduction

Regression Analysis. 1. Introduction . Itroducto Regresso aalyss s a statstcal methodology that utlzes the relato betwee two or more quattatve varables so that oe varable ca be predcted from the other, or others. Ths methodology s wdely used

More information

Questions? Ask Prof. Herz, herz@ucsd.edu. General Classification of adsorption

Questions? Ask Prof. Herz, herz@ucsd.edu. General Classification of adsorption Questos? Ask rof. Herz, herz@ucsd.edu Geeral Classfcato of adsorpto hyscal adsorpto - physsorpto - dsperso forces - Va der Waals forces - weak - oly get hgh fractoal coerage of surface at low temperatures

More information

Modified Wilcoxon Signed-Rank Test

Modified Wilcoxon Signed-Rank Test Ope Joural o Statstcs,,, 7-7 http://dx.do.org/.43/ojs..9 Publshed Ole Aprl (http://www.scrp.org/joural/ojs) Moded Wlcoxo Sged-Rak est Ikewelugo Cypra Aaee Oyeka, Godday Uwawukoye Ebuh * Departmet o Statstcs,

More information

CH. V ME256 STATICS Center of Gravity, Centroid, and Moment of Inertia CENTER OF GRAVITY AND CENTROID

CH. V ME256 STATICS Center of Gravity, Centroid, and Moment of Inertia CENTER OF GRAVITY AND CENTROID CH. ME56 STTICS Ceter of Gravt, Cetrod, ad Momet of Ierta CENTE OF GITY ND CENTOID 5. CENTE OF GITY ND CENTE OF MSS FO SYSTEM OF PTICES Ceter of Gravt. The ceter of gravt G s a pot whch locates the resultat

More information

Fast, Secure Encryption for Indexing in a Column-Oriented DBMS

Fast, Secure Encryption for Indexing in a Column-Oriented DBMS Fast, Secure Ecrypto for Idexg a Colum-Oreted DBMS Tgja Ge, Sta Zdok Brow Uversty {tge, sbz}@cs.brow.edu Abstract Networked formato systems requre strog securty guaratees because of the ew threats that

More information

Impact of Interference on the GPRS Multislot Link Level Performance

Impact of Interference on the GPRS Multislot Link Level Performance Impact of Iterferece o the GPRS Multslot Lk Level Performace Javer Gozalvez ad Joh Dulop Uversty of Strathclyde - Departmet of Electroc ad Electrcal Egeerg - George St - Glasgow G-XW- Scotlad Ph.: + 8

More information

Aggregation Functions and Personal Utility Functions in General Insurance

Aggregation Functions and Personal Utility Functions in General Insurance Acta Polytechca Huarca Vol. 7, No. 4, 00 Areato Fuctos ad Persoal Utlty Fuctos Geeral Isurace Jaa Šprková Departmet of Quattatve Methods ad Iformato Systems, Faculty of Ecoomcs, Matej Bel Uversty Tajovského

More information

Hypothesis Testing on the Parameters of Exponential, Pareto and Uniform Distributions Based on Extreme Ranked Set Sampling

Hypothesis Testing on the Parameters of Exponential, Pareto and Uniform Distributions Based on Extreme Ranked Set Sampling Mddle-East Joural of Scetfc Research (9): 39-36, ISSN 99-933 IDOSI Publcatos, DOI:.589/dos.mejsr...9.87 Hypothess Testg o the Parameters of Expoetal, Pareto ad Uform Dstrbutos Based o Extreme Raed Set

More information

Mobile Agents in Telecommunications Networks A Simulative Approach to Load Balancing

Mobile Agents in Telecommunications Networks A Simulative Approach to Load Balancing Moble Agets Telecommucatos Networks A Smulatve Approach to Load Balacg Steffe Lpperts Departmet of Computer Scece (4), Uversty of Techology Aache Aache, 52056, Germay Ad Brgt Kreller Corporate Techology

More information

Probability, Statistics, and Reliability for Engineers and Scientists MULTIPLE RANDOM VARIABLES

Probability, Statistics, and Reliability for Engineers and Scientists MULTIPLE RANDOM VARIABLES CHAPTR Probablt, Statstcs, ad Relablt or geers ad Scetsts MULTIPL RANDOM VARIABLS Secod dto A. J. Clark School o geerg Departmet o Cvl ad vrometal geerg 6b Probablt ad Statstcs or Cvl geers Departmet o

More information

Preparation of Calibration Curves

Preparation of Calibration Curves Preparato of Calbrato Curves A Gude to Best Practce September 3 Cotact Pot: Lz Prchard Tel: 8943 7553 Prepared by: Vck Barwck Approved by: Date: The work descrbed ths report was supported uder cotract

More information

Green Master based on MapReduce Cluster

Green Master based on MapReduce Cluster Gree Master based o MapReduce Cluster Mg-Zh Wu, Yu-Chag L, We-Tsog Lee, Yu-Su L, Fog-Hao Lu Dept of Electrcal Egeerg Tamkag Uversty, Tawa, ROC Dept of Electrcal Egeerg Tamkag Uversty, Tawa, ROC Dept of

More information

MODELLING OF STOCK PRICES BY THE MARKOV CHAIN MONTE CARLO METHOD

MODELLING OF STOCK PRICES BY THE MARKOV CHAIN MONTE CARLO METHOD ISSN 8-80 (prt) ISSN 8-8038 (ole) INTELEKTINĖ EKONOMIKA INTELLECTUAL ECONOMICS 0, Vol. 5, No. (0), p. 44 56 MODELLING OF STOCK PRICES BY THE MARKOV CHAIN MONTE CARLO METHOD Matas LANDAUSKAS Kauas Uversty

More information

The Present Value of an Annuity

The Present Value of an Annuity Module 4.4 Page 492 of 944. Module 4.4: The Preset Value of a Auty Here we wll lear about a very mportat formula: the preset value of a auty. Ths formula s used wheever there s a seres of detcal paymets

More information

Conversion of Non-Linear Strength Envelopes into Generalized Hoek-Brown Envelopes

Conversion of Non-Linear Strength Envelopes into Generalized Hoek-Brown Envelopes Covero of No-Lear Stregth Evelope to Geeralzed Hoek-Brow Evelope Itroducto The power curve crtero commoly ued lmt-equlbrum lope tablty aaly to defe a o-lear tregth evelope (relatohp betwee hear tre, τ,

More information

10/19/2011. Financial Mathematics. Lecture 24 Annuities. Ana NoraEvans 403 Kerchof AnaNEvans@virginia.edu http://people.virginia.

10/19/2011. Financial Mathematics. Lecture 24 Annuities. Ana NoraEvans 403 Kerchof AnaNEvans@virginia.edu http://people.virginia. Math 40 Lecture 24 Autes Facal Mathematcs How ready do you feel for the quz o Frday: A) Brg t o B) I wll be by Frday C) I eed aother week D) I eed aother moth Aa NoraEvas 403 Kerchof AaNEvas@vrga.edu http://people.vrga.edu/~as5k/

More information

Numerical Comparisons of Quality Control Charts for Variables

Numerical Comparisons of Quality Control Charts for Variables Global Vrtual Coferece Aprl, 8. - 2. 203 Nuercal Coparsos of Qualty Cotrol Charts for Varables J.F. Muñoz-Rosas, M.N. Pérez-Aróstegu Uversty of Graada Facultad de Cecas Ecoócas y Epresarales Graada, pa

More information