Reinsurance and the distribution of term insurance claims

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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

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

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

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

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

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

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