Deeper Understanding, Faster Calculation --Exam C Insights & Shortcuts. by Yufeng Guo. The Missing Manual

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1 Deeper Uderstadg, Faster Calculato --Exam C Isghts & Shortcuts by Yufeg Guo The Mssg Maual Ths electroc book s teded for dvdual buyer use for the sole purpose of preparg for Exam C Ths book ca NOT be resold to others or shared wth others No part of ths publcato may be reproduced for resale or multple copy dstrbuto wthout the express wrtte permsso of the author Yufeg Guo /84

2 Table of Cotets Itroducto 4 Chapter Dog calculatos % correct % of the tme 5 6 strateges for mprovg calculato accuracy 5 6 powerful calculator shortcuts 6 # Solve ax + bx + c 6 # Keep track of your calculato #3 Calculate mea ad varace of a dscrete radom varable #4 Calculate the sample varace 9 #5 Fd the codtoal mea ad codtoal varace 3 #6 Do the least squares regresso 36 #7 Do lear terpolato 46 Chapter Maxmum lkelhood estmator 5 Basc dea 5 Geeral procedure to calculate the maxmum lkelhood estmator 53 Fsher Iformato 58 The Cramer-Rao theorem 6 Delta method 66 Chapter 3 Kerel smoothg 75 Essece of kerel smoothg 75 Uform kerel 77 Tragular kerel 8 Gamma kerel 9 Chapter 4 Bootstrap 95 Essece of bootstrappg 95 Recommeded supplemetal readg 96 Chapter 5 Bühlma credblty model Trouble wth black-box formulas Ratg challeges facg surers 3 prelmary cocepts for dervg the Bühlma premum formula 6 Prelmary cocept # Double expectato 6 Prelmary cocept # Total varace formula 8 Prelmary cocept #3 Lear least squares regresso Dervato of Bühlma s Credblty Formula Summary of how to derve the Bühlma credblty premum formulas 7 Specal case How to tackle Bühlma credblty problems 3 A example llustratg how to calculate the Bühlma credblty premum 3 Shortcut 6 Practce problems 6 Chapter 6 Bühlma-Straub credblty model 48 Cotext of the Bühlma-Straub credblty model 48 Assumptos of the Bühlma-Straub credblty model 49 Summary of the Bühlma-Straub credblty model 54 /84

3 Geeral Bühlma-Straub credblty model (more realstc) 55 How to tackle the Bühlma-Straub premum problem 58 Chapter 7 Emprcal Bayes estmate for the Bühlma model 68 Emprcal Bayes estmate for the Bühlma model 68 Summary of the estmato process for the emprcal Bayes estmate for the Bühlma model 7 Emprcal Bayes estmate for the Bühlma-Straub model 73 Sem-parametrc Bayes estmate 8 Chapter 8 Lmted fluctuato credblty 87 Geeral credblty model for the aggregate loss of r sureds 88 Key term formula: credblty for the aggregate loss 9 Fal formula you eed to memorze 9 Specal case 9 Chapter 9 Bayesa estmate Itutve revew of Bayes Theorem How to calculate the dscrete posteror probablty 6 Framework for calculatg the dscrete posteror probablty 8 How to calculate the cotuous posteror probablty 3 Framework for calculatg dscrete-pror Bayesa premums 9 Calculate Bayesa premums whe the pror probablty s cotuous 5 Posso-gamma model 6 Bomal-beta model 64 Chapter Clam paymet per paymet 68 Chapter LER (loss elmato rato) 74 Chapter Fd E(Y-M) + 76 About the author 84 3/84

4 Itroducto Ths maual s teded to be a mssg maual It skps what other mauals expla well It focuses o what other mauals do t expla or do t expla well Ths way, you get your moey s worth Chapter teaches you how to do maual calculato quckly ad accurately If you studed hard but faled Exam C repeatedly, chaces are that you are cocept strog, calculato weak The calculator techques wll mprove our calculato accuracy Chapter focuses o the varace of a maxmum lkelhood estmator (MLE), a dffcult topc for may Chapter 3 explas the essece of kerel smoothg ad teaches you how to derve complex kerel smoothg formulas for ky ( x) ad Ky ( x ) You should t have ay trouble memorzg complex kerel smoothg formulas after ths chapter May caddates do t kow the essece of bootstrap Chapter 4 s about bootstrap Chapter 5 explas the core theory behd the Bühlma credblty model Chapter 6 compares ad cotrasts the Bühlma-Straub credblty models wth the Bühlma credblty model May caddates are afrad of emprcal Bayes estmate problems The formulas are just too hard to remember Chapter 7 wll releve your pa May caddates fd that there are just too may lmted fluctuato credblty formulas to memorze To address ths, Chapter 8 gves you a ufed formula Chapter 9 presets a framework for quckly calculatg the posteror probablty (dscrete or cotuous) ad the posteror mea (dscrete or cotuous) May caddates ca recte Bayes theorem but ca t solve related problem the exam codto Ther calculato s log, tedous, ad proe to errors Ths chapter wll drastcally mprove your calculato effcecy Chapter s about clam paymet per paymet Chapter s about loss elmato rato Chapter s about how to quckly calculate E( Y M ) + 4/84

5 Chapter 5 Bühlma credblty model Trouble wth black-box formulas The Bühlma credblty premum formula s tested over ad over Course 4 ad Exam C However, may caddates do t have a good uderstadg of the er workgs of the Bühlma credblty premum model They just memorze a seres of black-box formulas: Z + k, k E ( ) Var X Var, ad µ ( ) µ ( ) P Z + ZX Rote memorzato of a formula wthout fully graspg the cocepts s tedous, dffcult, ad proe to errors Addtoally, a memorzed formula wll ot yeld the eeded uderstadg to grapple wth dffcult problems I ths chapter, we re gog to dg deep to Bühlma s credblty premum formula ad ga a crystal clear uderstadg of the cocepts Ratg challeges facg surers Let s start wth a smple example to llustrate oe major challege a surace compay faces whe determg premum rates Image you are the fouder ad the actuary of a auto surace compay Your compay s specalty s to provde auto surace for tax drvers Before you ope your busess, there are half of doze surace compaes your area that offer auto surace to tax drvers The world has bee gog o fe for may years wthout your start up It ca cotue gog o wthout your start up So t s tough for you to get customers Fally, you take out a bg porto of your savg accout ad buy TV advertsg, whch brgs your frst three customers: Adam, Bob, ad Collee Sce your corporate offce s your garage ad you have oly oe employee (you), you decde that three customers s good eough for you to start your busess Whe you ope your busess at t, you sell three auto surace polces to Adam, Bob, ad Collee The cotract of your surace polcy says that the premum rate s guarateed for oly two years Oce the two-year guaratee perod s over, you have the rght to set the reewal premum, whch ca be hgher tha the guarateed tal premum Whe you set your premum rate at t, you otce that Adam, Bob, ad Collee are smlar may ways They are all taxcab drvers They work at the same tax compay the same cty They are all 35 years old They all graduated from the same hgh school /84

6 They are all careful drvers Therefore, at t you treat Adam, Bob, ad Collee as detcal rsks ad charge the same premum for the frst two years To actually set the tal premum for the frst two years, you decde to buy a rate book from a cosultg frm Ths cosultg frm s well-kow the dustry Each year t publshes a rate maual that lsts the average clam cost of a tax drver by cty, by mleage ad by several other crtera Based o ths rate maual, you estmate that Adam, Bob, ad Collee may each cur $4 clam cost per year So at t, you charge Adam, Bob, ad Collee $4 each Ths premum rate s guarateed for two years Durg the -year guarateed perod, Adam, Bob, ad Collee have curred the followg clams: Year Clam Year Clam Total Clam Average clam per sured per year Adam $ $ $ $ / $ Bob $ $7 $8 $8 / $4 Collee $4 $9 $3 $3 / $65 Grad Total $ Average clam per perso per year (for the 3-perso group): $ / (3 ) $35 Now the two-year guaratee perod s over You eed to determe the reewal premum rate for Adam, Bob, ad Collee respectvely for the thrd year Oce you have determed the premum rates, you wll eed to fle these rates wth the surace departmet of the state where you do busess (called domcle state) Questo: How do you determe the reewal premum rate for the thrd year for Adam, Bob, ad Collee respectvely? Oe smple approach s to charge Adam, Bob, ad Collee a uform rate (e the group premum rate) After all, Adam, Bob, ad Collee are smlar rsks; they form a homogeeous group As such, they should pay a uform group premum rate, eve though ther actual clam patters for the past two years are dfferet You ca cotue chargg them the old rate of $4 per sured per year However, sce the average clam cost for the past two years s $35 per sured per year, you ca charge them $35 per perso for year three Uder the uform group rate of $35, Bob ad Collee wll probably uderpay ther premums; ther actual average aual clam for the past two years exceeds ths group premum rate Adam, o the other had, may overpay hs premums; hs average aual clam for the past two years s below the group premum rate Whe you charge each polcyholder the uform group premum rate, low-rsk polcyholders wll overpay ther premums ad the hgh-rsk polcyholders wll uderpay ther premums Your busess as whole, however, wll collect just eough premums to pay the clam costs 3/84

7 However, the real world, most lkely you wo t be able to charge Adam, Bob, ad Collee a uform rate of $35 Ay of your customers ca easly shop aroud, compare premum rates, ad buy a surace polcy elsewhere wth a better rate For example, Adam ca easly fd aother surer who sells a smlar surace polcy for less tha your $35 group rate Addtoally, the commssoer of your state surace departmet s ulkely to approve your uform rate The departmet wll wat to see that your low rsk customers pay lower premums Key pots to remember: Uder the classcal theory of surace, people wth smlar rsks form a homogeeous group to share the rsk Members of a homogeeous group are photocopes of each other The clam radom varable for each member s depedet detcally dstrbuted wth a commo desty fucto X of the homogeeous group should pay E( X ) f x The uform pure premum rate s E X Each member I realty, however, there s o such thg as a homogeeous group No two polcyholders, however smlar, have exactly the same rsks If you as a surer charge everybody a uform group rate, the low-rsk polcyholders wll leave ad buy surace elsewhere To stay busess, you have o choce but to charge dvdualzed premum rates that are proportoal to polcyholders rsks Now let s come back to our smple case We kow that uform ratg wo t work the real world We ll wat to set up a mathematcal model to calculate the far reewal premum rate for Adam, Bob, ad Collee respectvely Our model should reflect the followg observatos ad tuto: Adam, Bob, ad Collee are largely smlar rsks We ll eed to treat them as a ratg group Ths way, our reewal rates for Adam, Bob, ad Collee are somewhat related O the other had, we eed to dfferetate betwee Adam, Bob, ad Collee We mght wat to treat Adam, Bob, ad Collee as potetally dfferet sub-rsks wth a largely smlar rate group Ths way, our model wll produce dfferet reewal rates We hope the reewal rate calculated from our model wll agree wth our tuto that Adam deserves the lowest reewal rate, Bob a hgher rate, ad Collee the hghest rate To reflect the dea that Adam, Bob, ad Collee are dfferet sub-rsks wth a largely smlar rate group, we may wat to dvde the largely smlar rate group to four sub-rsks (or more sub-rsks f you lke): super preferred, preferred, stadard, ad sub-stadard So the rate group actually cossts of four sub-rsks Adam or Bob or Collee ca be ay oe of the four sub-rsks 4/84

8 Here comes a crtcal pot: we do t kow who belogs to whch sub-rsk We do t kow whether Adam s a super-preferred sub-rsk, or a preferred sub-rsk, a stadard sub-rsk, or a sub-stadard sub-rsk Nor do we kow to whch sub-rsk Bob or Collee belogs Ths s so eve f we have Adam s two-year clam data Judged from hs -year clam hstory, Adam seems to be a super preferred or at least a preferred sub-rsk However, a bad drver ca have o accdets for a whle due to good luck; a good drver ca have several bg accdets a row due to bad luck So we really ca t say for sure that Adam s deed a better rsk All we kow that Adam s sub-rsk class s a radom varable cosstg of 4 possble values: super preferred, preferred, stadard; ad substadard To vsualze that Adam s sub-rsk class s a radom varable, thk about rollg a 4-sded de Oe sde of the de s marked wth the letters SP (super preferred); aother sde s marked wth PF (preferred); the thrd sde s marked wth STD (stadard); ad the fourth sde s marked wth SUB (substadard) To determe Adam belogs to whch sub-class, we ll roll the de If the result s SP, the we ll assg Adam to the super preferred class If the result s PF, we ll assg hm to the preferred class Ad so o ad so forth Smlarly, we ca roll the de ad radomly assg Bob or Collee to oe of the four sub-classes: SP, PF, STD, ad SUB Now we are ready to come up wth a model to calculate the reewal premum rate: Let radom varable X jt represet the clam cost curred year t by the j -th sured, where t,,,, ad + ad j,,, ad m Here our example, (we have two years of clam data) ad m,, 3 (correspodg to Adam, Bob, ad Collee) For ay j,,, ad m j j, j, ad X j + are detcally dstrbuted wth a commo desty fucto fx, ( x, ), a commo mea µ E( X jt ), ad a commo varace Var ( X jt ) What we are sayg here s that all polcyholders j,,, ad m have detcal mea clam µ ad detcal clam varace s a realzato of s a radom varable (or a vector of radom varables) represetg the presece of multple sub-rsks X j j, j, ad X j +, whch represet the clam costs curred by the same polcyholder, belog to the same sub rsk class However, s ukow to us All we kow s that s a radom realzato of Here our example, { SP, PF, STD, SUB} Whe we say that s a realzato of, we mea that wth probablty p, SP ; wth probablty p, PF ; wth probablty p 3, STD ; wth probablty p4 ( p+ p + p3), SUB 5/84

9 Because X j j, j, ad X j + are clams geerated from the same (ukow) sub-rsk class, we assume that gve j j, j, ad X j + are depedet detcally dstrbuted That s j j, j j + are depedet detcally dstrbuted wth a commo codtoal mea E( X jt ) µ ( ) commo codtoal varace Var ( X jt ) ad a We have observed X j j, j Our goal s to estmate X j +, the clam cost year + by the j -th sured, usg hs pror -year average clam cost X j t X The estmated value of X j + s the pure reewal premum for year + Bühlma s approach s to use a j j Z X + j to approxmate j E a+ ZX X + s mmzed The fal result: ( ) a Z X j Z µ Z X j + +, X + subject to the codto that j t Z + k, ( jt) ( jt ) ( jt) E Var X E Var X k Var E X Var µ ( ) jt jt µ E X E E X E µ Next, we ll derve the above formulas However, before we derve the Bühlma premum formulas, let s go over some prelmary cocepts 3 prelmary cocepts for dervg the Bühlma premum formula Prelmary cocept # ( ) E X E E X Double expectato If X s dscrete, E( X) E E( X ) p E( X ) all + If X s cotuous, E X E E X E X f d 6/84

10 I ll expla the double expectato theorem assumg X s dscrete However, the same logc apples whe X s cotuous Let s use a smple example to uderstad the meag behd the above formula A class has 6 boys ad 4 grls These studets take a fal The average score of the 6 boys s 8; the average score of the 4 grls s 85 What s the average score of the whole class? Ths s a elemetary level math problem The average score of the whole class s: + Total score Average score 8 # of studets Now let s rearrage the above equato: 6 4 Average score If we express the above calculato usg the double expectato theorem, the we have: Geder E Score E E Score Geder P Geder E Score Geder P ( grl) E ( score grl) P boy E score boy ( 8) + ( 85) 8 So stead of drectly calculatg the average score for the whole class, we frst break dow the whole class to two groups based o geder We the calculate the average score of these two groups: boys ad grls Next, we calculate the weghted average of these two group averages Ths weghted average s the average of the whole class If you uderstad ths formula, you have uderstood the essece of the double expectato theorem The Double Expectato Theorem pla Eglsh: Istead of drectly calculatg the mea of the whole populato, you frst break dow the populato to several groups based o oe stadard (such as geder) You calculate the mea of each group Next, you calculate the mea of all the group meas Ths s the mea of the whole populato Problem A group of graduate studets ( wth o-math major ad 8 wth math major) have a total GRE score of,94 The GRE score dstrbuto by major s as follows: 7/84

11 Total GRE scores of o-math major 7,74 Total GRE scores of 8 math major 5, Total GRE score,94 Fd the average GRE score twce Frst tme, do ot use the double expectato theorem The secod tme, use the double expectato theorem Show that you get the same result Soluto () Fd the mea wthout usg the double expectato theorem The average GRE score for graduate studets s: Total score,94 Average score 647 # of studets () Fd the mea usg the double expectato theorem Major E GRE E E GRE Major P Major E GRE Major P ( o math) E ( GRE o math) + P ( math) E ( GRE math) 7,74 8 5, You ca see the two methods produce a detcal result Prelmary cocept # Total varace formula Y + Y Var X E Var X Y Var E X Y Proof ( ) Var X E X E X Put the double expectato theorem to use: Y E X E E X Y, E( X ) E E( X Y) Y However, E( X Y) Var( XY) E ( XY) + 8/84

12 { } Var ( X ) E ( X ) E ( X ) EY Var ( X Y ) + E ( X Y ) EY E ( X Y ) { } E Y Var XY + E Y E XY E Y E XY Y E Y Var X Y + Var E X Y If X s the lost amout of a polcyholder ad Y s the rsk class of the polcyholder, the Var ( X ) E Y Var ( X Y ) + Var Y E ( X Y ) meas that the total varace of the loss cossts of two compoets: EY E Y Var( X Y) Var Y E ( X Y ), the average varace by rsk class, the varace of the average loss by rsk class Var( X Y) s called the expected value of process varace Var Y E X Y called the varace of hypothetcal mea Y + Y Var X E Var X Y Var E X Y Total varace expected process varace varace of hypothetcal mea s Next, let s look at a comprehesve example usg double expectato ad total varace Example The umber of clams, N, curred by a polcyholder has the followg dstrbuto: 3! P p p!3! ( ) 3 P s uformly dstrbuted over [, ] Fd E( N) ad Var ( N ) Soluto If p s costat, N has a bomal dstrbuto wth mea ad varace: 3p, Var ( N ) 3p( p) E N However, p s also a radom varable So we caot drectly use the above formula 9/84

13 To fd E( N ), we dvde N to dfferet groups by p, just as we dvded the class to boys ad grls The oly dfferece s that ths tme we have a fte umber of groups ( p s a cotuous radom varable) Let s cosder a small group [ p, p+ dp] Each value of p s a separate group For each group, we wll calculate ts mea The we wll fd the weghted average mea of all the groups, wth weght beg the probablty of each group s p value The result should be E( N ) 3 p p 3 3 E( N) E P E( N p) E( N p) fp( p) dp pdp p Please ote that p s uform over [, ] Cosequetly, fp ( p ) 3 P P Alteratvely, E( N) E E( N p) E [ p] E( P) Next, we ll calculate Var ( N ) Oe method s to calculate Var ( N ) the stadard formula Var ( N ) E ( N ) E ( N ) theorem to calculate E( N ) ad E( N ) P E N E E N p E N p f p dp from scratch usg We ll use the double expectato E N p E N p + Var N p p + p p p + p E( N ) E( N p) f ( p) dp ( 6p + 3p) dp p + p Var ( N ) E ( N ) E ( N ) Alteratvely, you ca use the followg formula to calculate the varace: p + P Var N E Var N p Var E N p /84

14 Because N p s bomal wth parameter 3 ad p, we have: 3p, Var ( N p) 3p( p) E N p 3 ( ) ( 3 3 ) Ep( 3p) Ep( 3p ) 3Ep( p) 3Ep( p ) E Var N p E p p E p p p p p P( 3 ) 9 Var P E N p Var p Var p Var ( N ) E p Var ( N p) + Var P E ( N p) 3Ep( p) 3Ep( p ) + 9Var( p) Applyg the geeral formula: If X s uform over [, ] We have: + a+ b ab, the E( X), Var ( X ), Var ( P) E P ( ) 4 E( P ) E ( P) + Var( P) + ( b a) Var ( N ) E Var ( N p) + Var E ( N p) 3E ( p) 3E ( p ) + 9Var ( p) p P p p Prelmary cocept #3 Lear least squares regresso I a regresso aalyss, you try to ft a le (or a fucto) through a set of pots Wth least squares regresso, you get a better ft by mmzg the dstace squared of each pot to the ftted le Let s say you wat to fd out how a perso s come level affects how much lfe surace he buys Let X represet come Let Y represet the amout of lfe surace ths perso buys You have collected some data pars of ( XY, ) from a group of cosumers You suspect there s a lear relatoshp betwee X ad Y You wat to predct Y usg the fucto a+ bx, where a ad b are costat Wth least squares regresso, you wat to mmze the followg: /84

15 Q E a+ bx Y Next, we ll derve a ad b Q E( a+ bx Y) E! ( a+ bx Y) " a a # $ % a & E a+ bx Y a+ be X E Y Q Settg a a be( X) E( Y) + ( Equato I ) Q E( a+ bx Y) E! ( a+ bx Y) " b b # $ % b & E a + bx Y X ae X + be X E X Y Q Settg b ae ( X ) be ( X ) E ( XY ) (Equato II) - (Equato I) E( X ) : be( X ) E ( X) E( XY) E( X) E( Y) However, E( X ) E ( X) Var( X) + (Equato II ), E( XY) E( X) E( Y) Cov( X, Y) b Cov( X, Y ), a E( Y) be( X) Var ( X ) Dervato of Bühlma s Credblty Formula Now I m ready to gve you a quck proof of the Bühlma credblty formula To smplfy otatos, I m gog to fx o oe partcular sured (such as Adam) ad chage the symbol X jt to X t Remember, our goal s to estmate X +, the dvdualzed premum rate for year +, usg a+ ZX Z s the credblty factor assged to the mea of past clams X ( X+ X + + X ) We ll wat to fd a ad Z that mmze the followg: /84

16 ( + ) E a ZX X + Please ote that X,, ad X + are clams curred by the same polcyholder (whose rsk class s ukow to us) durg year,,,, ad + Applyg the formula developed prelmary cocept #3, we have: (, + ) Cov X X z Var X ( ) ( ) Cov X Cov X X X Cov X X X Cov( X + ) + Cov( X + ) + + Cov( X, X + ) Oe commo mstake s to assume that X, + are depedet detcally dstrbuted If deed X, + are depedet detcally dstrbuted, we would have Cov X Cov X Cov X Z (, + ) Cov X X Var X The result Z smply does t make sese What wet wrog s the assumpto that X, + are depedet detcally dstrbuted The correct statemet s that X,, ad X + are detcally dstrbuted wth a commo desty fucto f x,, where s ukow to us Or stated dfferetly,, ad X + are depedet detcally dstrbuted gve rsk class I other words, f we fx the sub-class varable at, the all the clams curred by the polcyholder who belogs to sub-class are depedet detcally dstrbuted Mathematcally, ths meas that X,, ad X + are depedet detcally dstrbuted Here s a tutve way to see why X ad X j have o-zero covarace X ad X j represet the clam amout curred at tme ad j by the polcyholder whose sub-class 3/84

17 s ukow to us So X ad X j are cotrolled by the same rsk-class factor If s a low rsk, the X ad X j both ted to be small O the other had, f s a hgh rsk, the X ad X j both ted to be bg So X ad X j are correlated ad have a o-zero varace Next, let s derve the formula: (, j) ( j) ( ) ( j) Cov X X E X X E X E X Var µ where ' j Usg the double expectato theorem, we have E( XX j) E E( XX j ) Because X ad codtoal mea µ ( ), we have: X j are depedet detcally dstrbuted wth a commo j j µ µ µ E X X E X E X j j µ E E X X E E X E X E Cov( X, X j) E µ ( ) { E µ ( ) } Var µ ( ) Cov X + Cov X X X + Cov X + Cov X + + Cov X { Var } Var µ µ ( ) ( ) Next, we ll calculate Var ( X ) Var ( X ) Var ( X+ X + + X ) Var ( X+ X + + X ) Oce aga, we have to be careful here Oe temptato s to wrte: ( ) Var X + X + + X Var X + Var X + + Var X Wrog! Ths s wrog because X, are ot depedet Istead, X, are depedet So we have to clude covarace amog X, X, The correct expresso s: 4/84

18 ( ) ( ) ( ) Var X X X Var X + Var X + + Var X + Cov X + Cov X + + Cov X 3 So we have varace terms Though X,, commo mea E( X) X µ ad commo varace Var X Var ( X ) Var ( X ) Var ( X ) Var ( X ) Next, let s look at the covarace terms: Cov X + Cov X + + Cov X 3 are ot depedet, they have a Out of X,, f you take out ay two tems X ad X j where ' j, you ll get a covarace Cov( X, X j) Var µ ( ) Sce there are C ways of takg out two tems X ad X j where ' j, the sum of the covarace terms becomes: Cov X Cov X Cov X 3 { Var µ } C Var µ ( ) ( ) ( ) Var µ ( ) { Var ( X) + Var ( X ) + + Var ( X ) + Cov( X ) + Cov( X 3) + + Cov( X, X ) } { Var ( X ) + ( ) Var µ ( ) } Var ( X ) + ( ) Var µ ( ) Var ( X ) Var ( X + X + + X ) Var µ ( ) Var X + Var µ ( ) Usg the total varace formula, we have: + Var X E Var X Var E X Var ( X ) Var µ E Var ( X ) { } 5/84

19 Var ( X ) Var µ + E Var ( X ) Fally, we have: Z (, ) Cov X X + Var µ Var µ Var X Var ( X ) Var µ + E Var ( X ) E Var( X ) + Var µ ( ) Let ( ) E Var X k The Var µ Var µ Z Var X + k Next, we eed to fd a E( X+ ) Z E( X) Remember,, X, though ot depedet, have a commo mea E( X ) µ ad a commo varace E( X) E ( X + X + + X ) E( X + X + + X ) E X µ + a E( X ) Z E( X) µ Zµ µ ( Z) + Var X µ µ a ZX ( Z) ZX ZX ( Z) + µ + + µ, where z + k 6/84

20 Summary of how to derve the Bühlma credblty premum formulas Z (, + ) Cov X X, ( ) Var X a Z µ (, ) (, j) µ ( ) Cov X X + Cov X X Var VE, where ' j Var X ( ) Var X X X ( ) + ( ) (, j) Var X X X Var X Cov X X { µ ( ) } µ ( ) Var X + Var µ Var X Var + Var ( ) µ ( ) E Var X + Var ( ) Var X X X Var X Var µ + E Var ( X ) VE + EV Cov X + Var µ Var µ Z Var ( X ) Var ( X ) E Var ( X ) + Var µ ( ) { } E Var( X ) k + + Var µ ( ), Or Z (, ) Cov X X + Var µ VE Var X Var ( X ) EV VE + EV + VE µ ( ) P a+ ZX Z + ZX Let s look at the fal formula: Reewal premum rsk-specfc sample mea ( ) P Z X + Z µ global mea 7/84

21 Here P s the reewal premum rate durg year + for a polcyholder whose sub-rsk s ukow to us X s the sample mea of the clams curred by the same polcyholder (hece the same sub-rsk class) durg year,,, µ s the mea clam cost of all the sub-rsks combed If we apply ths formula to set the reewal premum rate for Adam for Year 3, the the formula becomes: Adam Reewal premum Adam rsk-specfc sample mea,, ( ) Adam Bob Collee P Z X + Z µ global mea At frst, the above formula may seem couter-tutve If we are terested oly Adam s clam cost Year 3, why ot set Adam s reewal premum for Year 3 equal to hs pror two-year average clam X (so P X )? Why do we eed to drag µ, the global average, whch cludes the clam costs curred by Bob ad Collee? Actually, t s blessg that the reewal premum formula cludes µ X vares wdely based o your sample sze However, the state surace departmets geerally wat the reewal premum to be stable ad resposve to the past clam data If your reewal premum P s set to X, the P wll fluctuate wldly depedg o the sample sze The you ll have a dffcult tme gettg your reewal rates approved by state surace departmets I addto, you may have P X ; ths s the case for Adam You ll provde free surace to the polcyholder who has ot curred ay clam yet Ths certaly does t make ay sese By cludg the global mea µ, the reewal premum µ ( ) P Z + ZX s stablzed Adam At the same tme, P s stll resposve to X Sce X < X, the reewal Adam Bob P µ Z + ZX wll produce P < P premum formula There are other ways to derve the Bühlma credblty formula For example, stead of mmzg E( a+ ZX X ) +, we ca mmze Ea+ ZX µ Please ote that µ E( X ) example, µ ( ) has four possble values: Bob s a radom varable I our tax drver surace 8/84

22 E( X SP ), E( X PF ), E( X STD ), ad E( X SUB) The dea behd E a+ ZX µ s ths If we kow that a polcyholder belogs to sub-rsk, the we ca set our reewal premum for year + equal to hs codtoal µ E X E X E X E X However, we mea clam cost ( + ) ( ) ( ) ( ) do t kow As a result, we lst all the possble values of µ ( ) ad fd the least mea squared errors estmator of µ ( ) by mmzg E a+ ZX µ ( ) The usg Prelmary Cocept #3, we have: Z Cov X, µ Var X Cov X, µ Cov ( X+ X + + X ), µ ( ) Cov ( X+ X + + X ), µ ( ) { Cov X, µ ( ) + Cov X, µ ( ) + + Cov X, µ ( ) } For,,,, we have: Cov X, µ E X µ E X E µ { } E X E E X µ µ For a fxed, µ ( ) s a costat Hece E X µ µ E X µ ( ) { } EX µ E E X µ E µ µ ( ) µ ( ) µ ( ) { } E X E E E X E E, { } Cov X µ E µ E µ Var µ Cov X, µ ( ) + Cov X, µ ( ) + + Cov X, µ ( ) Var µ ( ) X, X, X, 9/84

23 ,,,, Cov X µ { Cov X µ + Cov X µ + + Cov X µ } Var µ ( ) Var ( X ) s the same whether E( a ZX X ) + mmzed: + or E a+ ZX µ ( ) X, s to be { ( ) + µ ( ) } Var X E Var X Var Oe aga, we get: Z (, ) Cov X X + Var µ Var X Var ( X ) E Var( X ) k + + Var µ ( ) µ µ µ a E X Z E X Z Z + ( ) ( ) a+ ZX µ Z + ZX ZX+ Z µ, There s a thrd approach to dervg Bühlma s credblty formula Istead of mmzg + or Ea+ ZX µ ( ) E a ZX X + we ca mmze E a+ ZX E X X, + Here X+ X,, X represets the clam cost at year + of the polcyholder who curred clams X, year,,, The otato X+ X,, X emphaszes that the clam amouts X, + are from the same sub-class Ths codto must hold for the Bühlma credblty formula to be vald For example, f X + comes from sub class ad X, from sub-class, the the Bühlma credblty formula wll ot hold true However, the requremet that the clam amouts X, + are from the same sub-class should t bother us at all At the very begg whe we preseted the Bühlma credblty formula, we already used X, + to refer to the clams curred by the same polcyholder whose sub-rsk s As a result, /84

24 ( +,,, ) E X X X X E X µ + So E a+ ZX E( X X, ) E a+ ZX µ ( ) Key Pots We ca derve the Bühlma credblty formula by mmzg ay of the followg three terms: +, Ea+ ZX µ ( ) E a ZX X +, Ea+ ZX E( X,,, ) + X X X The Bühlma credblty premum s the least squares lear estmator of ay of the followg three terms: X +, the clam amout year + curred by the polcyholder who has X, clams year,,, µ, the mea clam amout of the sub-class that has geerated X, (,,, ) E X + X X X, the Bayes posteror estmate of the mea clam year + gve we have observed that the same polcyholder has X, clam costs years,,, respectvely Eve though we have derved the Bühlma credblty formula assumg X s the clam cost, the Bühlma credblty formula works f X s ay other quatty such as loss rato, the aggregate loss amout, or the umber of clams Popularty of the Bühlma credblty formula The Bühlma credblty formula s popular due to ts smplcty The reewal premum s the weghted average of the uform group rate ad the sample mea of the past clams The reewal premum s easy to calculate ad easy to expla to clets I cotrast, Bayesa premums (the posteror meas) are ofte dffcult to calculate, requrg kowledge of pror dstrbutos ad volvg complex tegratos Next, let s derve a specal case of the Bühlma credblty formula Ths specal case s preseted Loss Models /84

25 Specal case If E( X ) µ, Var ( X ), ad for j ' Cov( X ) * j where correlato coeffcet * satsfes < * <, determe the Bühlma credblty premum Oce aga, the credblty premum s a+ Z X Z Cov( X + ), Cov( X + ) Var µ ( ) Cov( X, X j) * Var ( X ) Var X ( ) Var X X X ( ) + ( ) + ( ) Var X X X Var X Cov X j * Z (, + ) * * ( ) Cov X X Var X + * + * ( ) ( ) * * µ a ( Z) µ µ + ( )* + * The Bühlma credblty premum s ( ) ( ) * * µ * * µ ZX+ ( Z) µ X+ X + + * + * + * * + * * You do t eed to memorze the Bühlma credblty premum formula for ths specal case If you uderstad how to derve the geeral Bühlma credblty premum formula, you ca derve the specal case formula ay tme by settg Cov( X ) * Next, let s tur our atteto toward how to solve the Bühlma credblty problem o the exam j /84

26 How to tackle Bühlma credblty problems Step Dvde the polcyholders to sub-classes,, 3 Step For each sub-class, calculate the average clam cost (or loss rato, µ E X ; calculate the varace of the clam aggregate clam, etc) cost Var ( X ) Step 3 Calculate EV E Var( X ) combed Calculate VE Var E ( X ) clam for all sub-classes combed, the average varace for all sub-classes, the varace of the average Step 4 Calculate EV k, VE Z + k Step 5 Calculate µ E E( X ), the average clam cost for all sub-classes combed Ths s the uform group premum rate you would charge uder the classcal theory of surace Step 6 Calculate the sample clam of the past data X X Step 7 Calculate the Bühlma credblty premum ( ) ZX+ Z µ Ths s the weghted average of the sample mea ad the uform group rate A example llustratg how to calculate the Bühlma credblty premum (Nov 3 #3) You are gve: Two rsks have the followg severty dstrbutos: Amout of Clam Probablty of Clam Amout for Rsk Probablty of Clam Amout for Rsk 5 5 7,5 3 6, Rsk s twce as lkely to be observed as Rsk 3/84

27 A clam of 5 s observed Determe the Bühlma credblty estmate of the secod clam amout from the same rsk Soluto Ths s a typcal problem for Exam C Here polcyholders are from two rsk classes Eve though the problem does t say that Rsk ad Rsk are two sub-rsks of a smlar bgger rsk group (e homogeeous group), we should assume so Otherwse, the Bühlma credblty formula wo t work Remember the Bühlma credblty premum s the weghted average of the uform group rate µ ad the rsk specfc sample mea X If Rsk ad Rsk are ot sub-rsks of a homogeeous group, the the uform ZX+ Z µ group rate µ does t exst; we have o way of calculatg The problem says that a clam of 5 s observed Ths meas that a polcyholder of a ukow sub-class has curred a clam of X $5 Sce Rsk s twce as lkely as Rsk, the $5 clam has 3 chace of comg from Rsk ad 3 chace of from Rsk The questo asks us to estmate the ext clam amout X polcyholder curred by the same Amout of Clam Probablty of Clam Amout for Rsk Probablty of Clam Amout for Rsk 5 5 7,5 3 6, Let X represet the dollar amout of a clam radomly chose E( X rsk ) 5(5) +,5(3) + 6,(),875 E( X rsk ) 5(7) +,5() + 6,() 6,675 The uform group rate s: µ E X P X from rsk E X rsk + P X from rsk E X rsk (,875) + ( 6,675), The varace of the codtoal mea s: ( from rsk ) ( rsk ) ( from rsk ) ( rsk ) VE P X E X + P X E X µ 4/84

28 (,875) + ( 6,675),8833 8,54, 3 3 E( X rsk ) 5 (5) +,5 (3)+ 6, ()7,96,5 ( rsk ) E X 5 (7 )+,5 () + 6, () 36,93,75 Var X E X E X rsk rsk rsk 7,96, 5, ,4, 65 Var X E X E X rsk rsk rsk 36, 93, 75 6, , 738,5 The average codtoal varace s: ( from rsk ) ( rsk ) ( from rsk ) ( rsk ) EV P X Var X + P X Var X ( 556,4,65) + ( 36,738,5) 476,339, EV 476,339, 7967 k 5576 VE 8,54, Z 76% + k , P ZX + Z µ 76% %,8833, 65 Next, I wat to emphasze a mportat pot I the Bühlma credblty premum formula, what matters s the X ( X+ X + + X ), ot the dvdual clams data X, For example, for 3, ( X 3) (,3,6), (,, 3) (,7,) ( X 3) (,,5) have the same X 3 P ZX + ( Z) µ 3Z+ ( Z) µ X X X, ad ad wll produce the same reewal premum 5/84

29 Shortcut We ca rewrte the Bühlma credblty premum formula as kµ + X k kµ + X P ZX + ( Z) µ X + µ + k + k + k + k EV We ca terpret k as the umber of samples take out of the global mea µ VE Image we have two urs, A ad B A cotas a fte umber of detcal balls wth each ball marked wth the umber µ B cotas a fte umber of detcal balls wth each ball marked wth the umber X You take out k balls from Ur A ad balls from Ur B kµ + X kµ + X The the average value per ball s: P + k + k Ths s the reewal premum just for year + k µ X kµ + X kµ + X P + k + k A B Practce problems Q You are a actuary o group health surace prcg You wat to use the Bühlma credblty premum formula P ZX + ( Z ) µ to set the reewal premum rate for a polcy Oe day the vce presdet of your compay stops by He has a PhD degree statstcs ad s wdely regarded as a expert o the cetral lmt theorem He asks you to throw the formula P ZX + ( Z ) µ to the trash ca ad focus o µ All we care about s µ As log as we charge each polcyholder µ, we ll be okay, the vce presdet says The fudametal cocept of surace s that may people form a group to share the rsk If we charge µ, the law of large umbers wll work ts magc ad we ll be able to collect eough premums to pay our guaratees 6/84

30 Commet o the vce presdet s remarks Soluto Accordg to the law of large umbers, for a homogeeous group of polcyholders, we ca set the premum rate equal to the average clam cost µ E( X) Some polcyholders wll suffer losses greater tha E( X ), whle others wll suffer losses less tha E( X ) However, o average, surace compaes wll have collected just eough premums to offset the loss As log as each polcyholder pays µ, the the surer wll be solvet However, practce, surace compaes ca t charge X µ Members of a so-called homogeous rsk group are really dfferet rsks Polcyholders of dfferet rsks ca shop aroud ad compare premum rates If ay polcyholder beleves that hs premum s too hgh, he ca termate hs polcy ad buy cheaper surace elsewhere If a surer charges µ to smlar yet dfferet rsks, good rsks wll stop dog busess wth the surer ad buy cheaper surace elsewhere; oly bad rsks wll rema the surer s book of busess As more ad more good rsks leave the surer s book of busess, the actual expected clam cost wll exceed the orgal average premum rate µ The the surer has to crease µ, causg more polcyholders to termate ther polces Gradually, the surer s customer base wll shrk ad the surer wll go bakrupt Q Compare ad cotrast the classcal theory of surace ad the credblty theory of surace Soluto 7/84

31 The classcal theory vs the credblty theory: Classcal theory of surace Is there a homogeeous group? Are clam radom varables X of dfferet members of a group depedet detcally dstrbuted? What s the far premum rate? Yes Ths s the foudato of surace Idetcal rsks form a homogeeous group to share rsks Yes Sce each member of a homogeeous group has detcal rsk, each member s clam radom varable s depedet detcally dstrbuted at all tmes The far premum s E X µ, where X s the radom loss varable of ay polcyholder the homogeeous group Every member of a homogeeous group eeds to pay µ, the uform group pure premum rate Credblty Theory Each member of a seemgly homogeeous group belogs to a sub-class The surer does t kow who belogs to whch sub-class No Sce members of a smlar rsk group are actually of dfferet sub-rsk classes, oly clams curred by the same sub-class are depedet detcally dstrbuted The far premum s E( X ) µ ( ), whch s the mea clam cost of the sub-class Every member of the same sub-class eeds to pay µ ( ), Q3 Oe day you vsted your college statstcs professor He asked what you were dog your job You told hm that you used the Bühlma credblty premum formula to set the reewal premum for group health surace polces The Bühlma credblty theory was ew to the professor After lsteg to your explaato of the formula P ZX + Z µ, he looked puzzled He told you that for years he had bee tellg hs studets that X s the ubased estmator of E( X ) I do t get t Why do t you just set P X? Expla why t s ot a good dea to set P Soluto X Your stats professor s perfectly correct sayg that the sample mea s a ubased estmator of the populato mea If the umber of observatos s large (so we have observed X, clams), for ay polcyholder, settg hs reewal premum equal to hs pror average mea clam s a good dea 8/84

32 I realty, however, t s hard to mplemet the dea P X Ofte you, as a surer, have to set the reewal premum wth lmted data (so may be small) For a small, X may ot be a good estmate of E( X ) I addto, we may have a werd stuato where X I our tax drver surace example, f you use P X to set the reewal premum for Adam, you ll get P Ths clearly does t make ay sese Q4 Nov 5 #6 For each polcyholder, losses X,, detcally dstrbuted wth mea ad varace E( X j ) µ, j,,, v Var X j, j,,, X, codtoal o, are depedet You are gve: The Bühlma credblty factor assged for estmatg X 5 based o X, X 3 4 s Z 4 The expected value of the process varace s kow to be 8 Calculate (, j) Soluto Cov X X, ' j (, ) Cov X X + Var µ VE Z Var X Var ( X ) VE + EV We are told that 4 (we have four years of clam data), Z 4, ad VE 8 VE VE 4 8 VE + VE + 4, VE 33 So µ ( ) Cov X j Cov X + Var VE 33 9/84

33 Q5 Nov 5 #9 For a portfolo of depedet rsks, the umber of clams for each rsk a year follows a Posso dstrbuto wth meas gve the followg table: Class Mea # of clams per rsk # of rsks You observe x clams Year for a radomly selected rsk The Bühlma credblty estmate of the umber of clams for the same rsk Year s 983 Determe x Soluto The problem states that x clams Year have bee observed for a radomly selected rsk The wordg a radomly selected rsk s eeded because order for the Bühlma credblty formula to work, the rsk class must be ukow to us If we already kow the rsk class, we ca calculate the expected umber of clams Year ; we do t eed to estmate ay more Please also pay atteto to the wordg the Bühlma credblty estmate of the umber of clams for the same rsk Year s I order for the Bühlma credblty formula to work, the reewal premum (or the expected umber of clams ths problem) year + ad the pror year clams X, must refer to the same (ukow) rsk class Ad ow back to the problem Let Y represet the umber of clams curred a year by a radomly chose class Sce Y has s Posso radom varable, Var( Y ) E Y Class Mea # of clams per rsk E Y Var Y P ( ) # of rsks 9% 9 9% 9 3 % Total %, The global mea (usg the double expectato theorem): µ E Y E E Y P E Y 9% + 9% + % The average codtoal varace: 3/84

34 EV P Var Y 9% + 9% + % The varace of codtoal meas: { } VE Var E Y E E Y E E Y 9% + 9% + % 99 { } Alteratvely, VE Var E ( Y ) E E ( Y ) E E ( Y ) 9% + 9% + % 99 EV k VE 99 Y + kµ x + ( Y + kµ ) P , x 4 + k + k + 99 Q6 Nov 5 #7 For a portfolo of polces, you are gve: The aual clam amout o a polcy has probablty desty fucto x f ( x ), < x < The pror dstrbuto of has desty fucto: + 4, < < 3 A radomly selected polcy had clam amout Year Determe the Bühlma credblty estmate of the clam amout for the selected rsk Year Soluto The codtoal mea s: x x E X x f x dx x dx dx x Please do t wrte: 3/84

35 E X x f x d Wrog! E X s the expected value of X f we fx the radom varable So should be treated as a costat ad regardg x, ot d The correct calculato s to tegrate xf( x ) Codtoal varace: Var ( X ) E ( X ) E ( X ) 3 x x 4 4 E X x f x dx x dx dx x Var ( X ) E ( X ) E ( X ) The global mea: µ E( X) E E( X ) E E The expected codtoal varace: EV E Var X E E ( ) The varace of the codtoal mea: VE Var E ( X ) Var Var E d 4 3 d 4 4 d E d 4 3 d 4 5 d 4 4 Var E ( ) E EV E ( ) 4 µ E EV 8 6 k 35 VE 3 75, VE Var ( ), 3/84

36 The above fracto s complex We do t wat to bother expressg k a eat fracto; tryg to expressg k a eat fracto s proe to errors 8 kµ + X 35 + kµ + X k + k + 35 P Alteratve method of calculato Ths method s more complex x x E X x f x dx x dx dx x (as before) 3 5 E X E E X E X d 4 d 4 8 µ ( ) { } VE Var E X E E X E E X E E( X ) E( X ) + d + d d d { } VE VE E( X ) E E( X ) EV E Var X, Var ( X ) E ( X ) E ( X ) 3 x x 4 4 E X x f x dx x dx dx x Var ( X ) E ( X ) E ( X ) EV E Var X Var X d d 33/84

37 4 EV 8 6 k 35 VE 85 8 kµ + X 35 + kµ + X k + k + 35 P Q7 May 5, # For a partcular polcy, the codtoal probablty of the aual umber of clams gve,, ad the probablty dstrbuto of, are as follows: # of clams Probablty Probablty 8 Two clams are observed Year Calculate the Bühlma credblty estmate of the umber of clams Year Soluto Let X represet the aual umber of clams X Probablty 3 E X E X Var X [ 5 ] 5 ( 5) ( 5) 5 ( 9 5 ) 9 5 ( ) µ E E X E E VE Var E X V V Var EV E Var X E E E 5 3 Probablty 8 E ( ) 5( 8) + 3( ) E /84

38 Var ( ) µ 5E 5 5 VE 5Var 5 5 ( ) EV 9E 5E EV 4 k 6 VE 5 kµ + X 6 ( 5) kµ + X + P 69 + k + k + 6 Q8 May 5, #7 You are gve: The aual umber of clams o a gve polcy has a geometrc dstrbuto wth parameter - The pror dstrbuto of - has the Pareto desty fucto + - (- + ) +, < - < where s a kow costat greater tha A radomly selected polcy has x clams Year Calculate the Bühlma credblty estmate of the umber of clams for the selected polcy Year Soluto Let X represet the aual umber of clams o a radomly selected polcy Here the rsk factor s - The codtoal radom varable X - has geometrc dstrbuto If you look up Tables for Exam C/4, you ll fd geometrc radom varable N wth parameter - has mea ad varace as follows: E( N) -, Var ( N ) -( + -) Applyg the above mea ad varace formula, we have: The codtoal mea: E( X -) - The codtoal varace: Var ( X -) -( + -) 35/84

39 - ( -) - -( + -) - (-) + - (- ) E - ad E - (- ) as E (- ) So Typcally, we wrte E - (- ) EV E Var X E E E as ( -) -( -) (-) (- ) EV E Var X E + E + E - - VE V - E X - V - The global mea s: µ E E( X -) E( -) - We are told that the pror dstrbuto of - has the Pareto desty fucto + (-), < - < Here the phrase pror dstrbuto refers to the fact that we kow + (-) pror to our observato of x clams Year I other words, + (-) has t corporated our observato of x clams Year yet Please ote that the pror dstrbuto, ot the posteror dstrbuto, s used Bühlma s credblty estmate Frakly, I thk SOA s emphass that + (-) s pror (as opposed to posteror) dstrbuto s uecessary ad really meat to scare exam caddates Whe we talk about desty fucto, we always refer to pror dstrbuto So there s ever a eed to say pror dstrbuto If we wat to refer to a dstrbuto that has corporated our recet observatos, at that tme we say posteror dstrbuto Back to the problem We are told that - has Pareto dstrbuto Is t a oe-parameter Pareto or two-parameter Pareto? May caddates have trouble kowg whch oe to use Here s a smple rule: To decde whether to use oe-parameter Pareto or two-parameter Pareto, look at your radom varable X If X s greater tha a postve umber, the use sgle-parameter Pareto If X s greater tha zero, the use two-parameter Pareto: ; x If X > a postve costat, the use sgle-parameter Pareto f ( x) + If X >, the use two-parameter Pareto f ( x) ( x + ) + I ths problem, the Pareto radom varable - > So we should use the two-parameter Pareto formula Tables for Exam C/4 36/84

40 k E X k k! ( )( )( k ) Please ote that the deomator has k tems E( X), E( X ) ( )( ) Var ( X ) E ( X ) E ( X ) ( )( ) ( ) ( ) Sce the two-parameter Pareto s frequetly tested Exam C, you mght wat to memorze the followg formulas: E X, E( X ), Var ( X ) ( )( ) ( ) ( ) - s a two-parameter Pareto radom varable wth pdf + (-) parameters are ad So we have: (- + ) + So the two E (- ), E (- ), Var (- ) ( )( ) EV E (-) E (- ) ( ) ( ) + + (- ) VE V, µ E (- ) ( ) ( ) ( )( ) EV k VE ( ) ( ) kµ + X ( ) + x kµ + X x P + + k + k + 37/84

41 Q9 May 5 # You are gve: The umber of clams a year for a selected rsk follows Posso dstrbuto wth mea The severty of clams for the selected rsk follows expoetal dstrbuto wth mea The umber of clams s depedet of the severty of clams The pror dstrbuto of s expoetal wth mea The pror dstrbuto of s Posso wth mea A pror, ad are depedet Usg the Bühlma credblty for aggregate losses, determe k Soluto Let N represet the aual umber of clams for a radomly selected rsk Let X represet the loss dollar amout per loss cdet Let S represet the aggregate aual clam dollar amout curred by a rsk N The S X X+ X + + XN N s a Posso radom varable wth mea So N ( ) f e ( N,,, )! f e We have E ( ), E E + Var + Here s a expoetal radom varable wth pdf Var, ad X,, ( ) X N x fx x e E Var are depedet detcally dstrbuted wth a commo pdf Here s a Posso radom varable wth pdf f ( ) e have Hece E( ) E Var Here the rsk parameters are (, ) + + E( S) E( N) E( X), Var ( S ) E ( N ) Var ( X ) + Var ( N ) E ( X ) To remember that you eed to use E ( X ), ot E( X ), the Var ( S ) ote that Var ( S ) s dollar squared If you use squared As a result, you eed to use Var ( N ) E ( X ) For a fxed par of (, ), the codtoal mea s: We! formula, please Var N E X, you ll get dollar, ot dollar 38/84

42 (, ) E S E N E X The codtoal varace s: Var S (, ) E ( N ) Var ( X ) + Var ( N ) E ( X ) + Please ote that N s a Posso radom varable wth mea ; X s a expoetal radom varable wth mea { (, ) } ( ) EV E, VarS E, Sce ad are depedet, we have: EV E ( ) (, E E ) 4 { (, ) } ( ) ( ) ( ) VE Var, E S Var, E, E, ( ) ( ) ( ) ( ) E, E, E E 4 E, ( ) E( ) E( ) VE E ( ) E ( ),, 4 3 EV k VE 4 3 Q May 5 #6 You are gve: Clams are codtoally depedet ad detcally Posso dstrbuted wth mea The pror dstrbuto of s: 6 F ( ), > + Fve clams are observed Determe the Bühlma credblty factor Soluto 39/84

43 Let X represet the umber of clams The rsk factor s We are told that X s a Posso radom varable wth mea The codtoal mea s: E( X ) The codtoal varace s: Var ( X ), VE Var E ( X ) Var EV E Var X E EV E k VE Var ( ) ( ) To quckly calculate E ( ) ad Var ( ), you ll eed to recogze that: cdf for a two-parameter Pareto radom varable s: F ( x) E X, E( X ), Var ( X ) ( )( ) 6 x +, where x > ( ) ( ) Here we are gve that F ( ) So s a two-parameter Pareto radom + varable wth parameters ad 6 E X 6 6 So ( ) ( ) 6 6 Var X ad 6 6 ( 6 ) ( 6 ) 6 ( 6) EV E 6 6( 6) k 369 VE Var 6 6 Z k Q Nov 4 #9 You are gve: Clam couts follow a Posso dstrbuto wth mea Clam szes follow a logormal dstrbuto wth parameters µ ad Clam couts ad clam amouts are depedet The pror dstrbuto has jot pdf: 4/84

44 f, µ,, < <, < µ <, < < Calculate Bühlma s k credblty for aggregate losses Soluto Let N represet the clam couts the dollar amout of the -th clam, ad S the aggregate losses N has Posso dstrbuto wth mea of X µ, has logormal dstrbuto wth parameters µ ad I addto, for to N µ, s depedet detcally dstrbuted The aggregate loss s: N S X E( N) E( X), Var ( S ) E ( N ) Var ( X ) + Var ( N ) E ( X ) E S The rsk parameters are, µ, ad If we fx, µ, ad, the (, µ, ) ( ) ( µ, ) ( µ, ) E S E N E X E X (, µ, ) ( ) ( µ, ) + ( ) ( µ, ) Var ( X µ, ) + E ( X µ, ) E ( X µ, ) Var S E N Var X Var N E X From Tables for Exam C/4, we kow the logormal dstrbuto has the followg momets: E X k k k exp µ + E X exp µ +, ( ) E X expµ + exp( µ + ) E S, µ, E X µ, exp µ + 4/84

45 (, µ, ) ( µ, ) exp( µ + ) Var S E X The global mea s: µ E, µ, E S µ E, µ, µ + µ (,, ) exp exp µ + f (, µ, ) d dµ d expµ + d dµ d µ 5 µ e e d dµ d µ 5 e e µ 5 dµ d e e d µ Set 5 y The d dy 5 5 y 5 µ e e d ( e ) e dy ( e)( e ) y { } (, µ, ) (, µ, ) (, µ, ) VE Var,, E S µ E, µ, E S E, µ, E S E, µ, expµ + E, µ, exp µ + + µ µ exp µ f, µ, d dµ d + exp µ d dµ d exp( ) exp( µ ) d dµ d exp exp( ) µ 3 e d exp( ) exp 3 µ dµ d µ 3 e d 4/84

46 Set y The d dy y E, µ, expµ + ( e ) exp( ) d ( e ) e dy 3 3 y { E, µ, E( S, µ, ) } µ ( e) ( e ) ( e )( e ) ( e )( e ) { } VE E, µ, E( S, µ, ) E, µ, E( S, µ, ) ( ) ( 5 ) e e e e (, µ, ) exp( µ ) EV E, µ, Var S E, µ, + µ exp µ + f, µ, d dµ d µ exp µ + d dµ d exp( ) exp( µ ) d dµ d exp( ) exp( ) µ µ dµ d µ ( e ) exp( ) d ( e ) ( e ) ( e ) 53 8 EV 53 k 869 VE 587 Shortcut to avod the hard-core tegrato see above The jot pdf s f (, µ, ) a( ) b( µ ) c( ), where a ( ), c I addto,, µ, ad le the cube < <, µ b µ, ad < <, < < 43/84

47 Cosequetly,, µ, ad are depedet radom varables wth the followg margal pdf: f f, < < ; f µ µ, < µ < ;, < < µ 5 (,, ) exp µ E, µ, E S µ E, µ, µ + E E e E e E ( ), µ µ , E( e ) e d e ( e ) E e e du e µ E( ) E( e ) µ E e 5 ( e)( e 5 ) µ (, µ, ) exp( µ ) ( ) EV E, µ, Var S E, µ, + E E e E e µ µ µ E( e ) e du e ( e ) E e e d e e 53 8 µ EV E ( ) E ( e ) E ( e ) ( e ) ( e ) ( e ) (, µ, ) (, µ, ) (, µ, ) E, µ, E S, µ, µ { } VE Var,, E S µ E, µ, E S E, µ, E S, µ, exp µ +, µ, exp µ + µ E E E E e E e E d 3 µ, E( e ) ( e ), E e e d e e 5 VE E, µ, E S, µ, µ e e e e EV 53 k 869 VE /84