benefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ).


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1 REVIEW OF RISK MANAGEMENT CONCEPTS LOSS DISTRIBUTIONS AND INSURANCE Loss and nsurance: When someone s subject to the rsk of ncurrng a fnancal loss, the loss s generally modeled usng a random varable or some combnaton of random varables. The loss s often related to a partcular tme nterval  for example, an ndvdual may own property that mght suffer some damage durng the followng year. Someone who s at rsk of a fnancal loss may choose some form of nsurance protecton to reduce the mpact of the loss. An nsurance polcy s a contract between the party that s at rsk (the polcyholder) and an nsurer. Ths contract generally calls for the polcyholder to pay the nsurer some specfed amount, the nsurance premum, and n return, the nsurer wll remburse certan clams to the polcyholder. A clam s all or part of the loss that occurs, dependng on the nature of the nsurance contract. Modelng a loss random varable: There are a few ways of modelng a random loss/clam for a partcular nsurance polcy, dependng on the nature of the loss. Unless ndcated otherwse, we wll assume the amount pad to the polcyholder as a clam s the amount of the loss that occurs. Once the random varable representng the loss has been determned, the expected value of the loss,, µá s referred to as the pure premum for the polcy., µ s also the expected clam on the nsurer. Note that n general, mght be  t s possble that no loss occurs. The followng are the basc models used for descrbng the loss random varable. For a random varable a measure of the rsk s ~ = ]. The untzed rsk or coeffcent of varaton l = µ, µ for the random varable s defned to be ~. Models for descrbng a loss random varable : Case 1: The complete descrpton of s gven: In ths case, f s contnuous, the densty functon ²%³ or dstrbuton functon  ²%³ s gven. If s dscrete, the probablty functon (or possbly the dstrbuton functon) s gven. One typcal (and smple) example of the dscrete case s a loss random varable of the form ~F 2 wth probablty (ths mght arse n a one year term lfe nsurance n whch the death wth probablty c beneft s 2, pad f the polcyholder des wthn the year, and probablty of death wthn the year s ). Another example of a dscrete loss random varable (wth more than two ponts) s the followng example of dental expenses for a famly over a oneyear perod. 1
2 Amount of Dental Expense Probablty $ In some problems, all that s needed s the mean and varance of, and sometmes that s the only nformaton about that s gven (rather than the full descrpton of 's dstrbuton). Case 2: The probablty of a nonnegatve loss s gven, and the condtonal dstrbuton ) of loss amount gven that a loss has occurred s gven: The probablty of no loss occurrng s c, and the loss amount s 0 f no loss occurs. Thus, 7 ~µ~c and ~) f a loss does occur. The random varable ) s the loss amount gven that a loss has occurred, so that ) s really the condtonal dstrbuton of the loss amount gven that a loss occurs. The random varable ) mght be descrbed n detal, or only the mean and varance of ) mght be gven. Note that f, )µ and = )µ are gven, then, ) µ ~ = )µ b ², )µ³ (ths s needed n the formulaton of = µ ).. We formulate as a "mxture" of two random varables, > and >, where > ~ s the constant random varable (not really random at all), and > ~ ), and wth weghts ~ c and ~. Then the frst two moments of are, µ~²c³, >µbh, >µ~h, )µ, snce, >µ~, and, µ~²c³, > µbh, > µ~h, ) µ. Then, = µ~h, ) µc²h, )µ³. For example, the loss due to fre damagng a partcular property mght be modeled ths way. Suppose that ~À s the probablty that fre damage occurs, and gven that fre damage occurs, the amount of damage, ), has a unform dstrbuton between $ Á and $ Á. Keep n mnd that ) s the loss amount gven that a loss has occurred, whereas s the ÁÁ uncondtonal loss amount. Then,, )µ ~ $ Á and, ) µ ~. Usng the formulas above, ÁÁ ÁÁ, µ ~ ²À³²Á ³ ~ $, and = µ ~ ²À³² ³ c ²³ ~. 2
3 Example 135: For a oneyear dental nsurance polcy for a famly, we consder the followng two models for annual clams : () Amount of Dental Expense () Probablty $ () There s a probablty of Àthat no clam occurs, 7 ~µ~à, and f a clam occurs, the clam amount random varable ), has mean, )µ ~ À and varance = )µ ~ Á À. In each case, fnd, µ and = µ. Soluton: () In ths case the complete descrpton of s gven (Case 1 mentoned above)., µ ~ ²À³ b ²À³ b ²À³ b ²À³ b ²À³ ~ Á, µ ~ ²À³ b ²À³ b ²À³ b ²À³ b ²À³ ~ Á = µ ~ Á c ~ Á. () In ths case, the probablty of a clam occurrng s gven ² ~ À) along wth the mean and varance of the condtonal dstrbuton ) of clam amount gven that a clam occurs (Case 2 mentoned above)., µ ~ h, )µ ~ ²À³²À³ ~,, µ ~ h, ) µ ~ ²À³ = )µ b ², )µ³ µ ~ ²À³ Á À b ²À³ µ ~ Á À = µ ~, µ c ², µ³ ~ Á. Note that t s not a concdence that the mean and varance of turned out to be the same n () and (). Ths s true because the mean and varance of ) n () were chosen as the condtonal mean and varance of the dstrbuton n () gven that a clam occurs. Modelng the aggregate clams n a portfolo of nsurance polcesá The Indvdual Rsk Model The ndvdual rsk model assumes that the portfolo conssts of a specfc number, say, of nsurance polces, wth the clam for one perod on polcy beng the random varable (modeled n one of the ways descrbed above for an ndvdual polcy loss random varable). Unless mentoned otherwse, t s assumed that the 's are mutually ndependent random varables. Then the aggregate clam s the random varable : ~, wth, :µ ~, µ and = :µ ~ = µ. ~ ~ ~ 3
4 If, µ ~ and = µ ~ for each ~ Á Á ÀÀÀÁ, then the coeffcent of varaton of the l aggregate clam dstrbuton : s ~ ~, whch goes to 0 as SBÀ l = :µ = µ, : µ, µ l Example 136: An nsurer has a portfolo of 1000 oneyear term lfe nsurance polces just ssued to 1000 dfferent (ndependent) ndvduals. Each polcy wll pay $1000 n the event that the polcyholder des wthn the year. For 500 of the polces, the probablty of death s.01 per polcyholder, and for the other 500 polces the probablty of death s.02 per polcyholder. Fnd the expected value and the standard devaton of the aggregate clam that the nsurer wll pay. Soluton: The aggregate clam random varable s :~, where s the clam from polcy. Then, :µ ~, µ and snce the clams are ndependent, ~ = :µ~ = µ. If s one of the 500 polces wth death probablty.01, then ~ F prob..99 prob. 01 ~ S, µ ~ ²À³ ~, = µ ~, µ c ², µ³ ~. If s one of the 500 polces wth death probablty.02, then, µ~á= µ~á. Thus,, :µ ~ ²³ b ²³ ~ Á Á = :µ ~ ²³ b ²Á ³ ~ Á Á S l = :µ ~ Á. ~ Example 137: Two portfolos of ndependent nsurance polces have the followng characterstcs: Portfolo A: Probablty Number of Clam Clam Class n Class per Polcy Amount 1 2, Portfolo B: Probablty Clam Amount Number of Clam Dstrbuton Class n Class per Polcy Mean Varance 1 2, The aggregate clams n the portfolos are denoted by : and :. = : µ = : µ ( Fnd. ) ( ) 4
5 Soluton: In ths example, nformaton s gven n the followng form: for polcy, the probablty of a clam occurrng s gven,, and the mean and varance of the condtonal dstrbuton of clam amount gven a clam occurs s gven,, ) µ ~, = ) µ ~. (Note that for each polcy n Portfolo (, = )µ~.) Then for polcy,, µ ~ h, ) µ, and = µ ~ ² c ³², ) µ³ b h = ) µ, and for a portfolo of ndependent polces,, :µ ~, µ and = :µ ~ = µ. For Portfolo (, any polcy n Class 1 has = µ ~ ²À ³²À ³² ³ b ²À ³²³ ~ À and any polcy n Class 2 has = µ ~ ²À³²À³² ³ b ²À³²³ ~ À, so that = : ( µ ~ ²À ³ b ²À³ ~. For Portfolo ), any polcy n Class 1 has = µ ~ ²À ³²À ³² ³ b ²À ³²³ ~ À and any polcy n Class 2 has = : µ ~ ²À ³ b ²À³ ~ = : ( µ Then, ~ À. ). = : µ ) = µ ~ ²À³²À³² ³ b ²À³²³ ~ À, so that The normal approxmaton to aggregate clams: For an aggregate clams dstrbuton :, f the mean and varance of : are known (, :µ and = :µ ), t s possble to approxmate probabltes for : by usng the normal dstrbuton. The 95th percentle of aggregate clams s the number 8 for whch 7 : 8µ~À. If : s assumed to have a dstrbuton whch s approxmately normal, then by standardzng : we have :c, :µ 8c, :µ 8c, :µ = :µ = :µ = :µ l l l 7 : 8µ~7 µ~à, so that s equal to the 95th percentle of the standard normal dstrbuton (whch s found to be when referrng to the standard normal table), so that 8 can be found. If the nsurer collects total premum of amount 8, then (assumng that t s reasonable to use the approxmaton) there s a 95% chance (approxmately) that aggregate clams wll be less than the premum collected, and there s a 5% chance that aggregate clams wll exceed the premum. Snce : s a sum of many ndependent ndvdual polcy loss random varables, the Central Lmt Theorem suggests that the normal approxmaton to : s not unreasonable. Relatve securty loadng: If the aggregate clam dstrbuton s :, wth mean, :µ, the total premum collected, say 8, can be wrtten n the form 8 ~ (1+ ) h, :µ. The factor s sometmes referred to as the relatve securty loadng contaned wthn the premum. The premum conssts of, :µ (expected aggregate clam) plus h, :µ, a loadng on top of the expected aggregate clam. 5
6 Example 138: An nsurance company provdes nsurance to three classes of ndependent nsureds wth the followng characterstcs: For Each Insured Number Probablty Expected Varance of Class n Class of a Clam Clam AmountClam Amount For each class, the amount of premum collected s ² b ³ (expected clams), where s the same for all three classes. Usng the normal approxmaton to aggregate clams, fnd so that the probablty that total clams exceed the amount of premum collected s.05. Soluton: We wsh to fnd 8~²b ³, :µ, so that 7 : 8µ~À, or equvalently, 7 : 8µ~À. Applyng the normal approxmaton and standardzng :, ths can be wrtten n the form :c, :µ 7 : 8µ~7 8c, :µ µ~à, so that 8c, :µ µ~à (the 95th percentle of l l l = :µ = :µ = :µ the standard normal dstrbuton). Thus, once, :µ and = :µ are found, we can fnd 8~²b ³, :µ, and then fnd., :µ ~ ', µ ~ ' h, ) µ ~ ²À ³² ³ b ²À³²³ b ²À ³² ³ ~, snce there are 500 polces n class 1, each wth expected clam ²À ³² ³, and smlarly for classes 2 and 3. The polces are ndependent so that the varance of the sum of all polcy clams s the sum of the varances (no covarances when ndependence s assumed). The varance of clam for a polcy from class 1 s ² c ³ h ², )µ³ b h = )µ ~ ²À ³²À ³² ³ b ²À ³² ³, and there are 500 of those polces, and smlarly for classes 2 and 3. = :µ ~ ' = µ ~ ' ² c ³ h ², ) µ³ b h = ) µ µ ~ ²À ³²À ³² ³ b ²À ³² ³µ b ²À³²À³² ³ b ²À³²³µ b ²À ³²À ³² ³ b ²À ³² ³µ ~ Á À. Then, 8 ~ À~, and À. Note that s the relatve securty loadng. Mxture of Loss Dstrbutons 6
7 A portfolo of polces mght consst of two or more classes of polcyholder, as n the prevous example. In the prevous example, the number of polces n each class was known. It may be possble that the number of polces n each class s not known but the proporton of polces n each class s known. In such a stuaton, we mght be asked to descrbe the dstrbuton of the loss for a randomly chosen polcy from the portfolo of polces. The followng example llustrates ths dea. Example 139: The nsurer of a portfolo of automoble nsurance polces classfes each polcy as ether hgh rsk, medum rsk or low rsk. The portfolo conssts of 10% hgh rsk, 30% medum rsk and 60% low rsk polces. The clam means and varances for the three rsk classes are mean varance hgh rsk medum rsk 4 4 low rsk 1 1 A polcy s chosen at random from the portfolo. Fnd the mean and varance of ths polcy. Soluton: The dstrbuton of the randomly chosen polcy s the mxture of the three rsk class clam dstrbutons, usng the percentages as the mxng factors. If denotes the clam for the randomly chosen polcy, then all moments of (pdf and cdf also) are the weghted averages of the moments for the component dstrbutons n the mxture., µ ~ ²À³²³ b ²À³²³ b ²À³²³ ~ À s the mean. Snce = µ ~, µ c ², µ³, we need, µ n order to fnd the varance of. Let / denote the clam random varable for a hgh rsk polcy. Then ~ = / µ ~, µ c ², / µ³ ~, µ c ²³, from whch we get, µ ~ À / / / In a smlar way we get, µ~b²³ ~ and, µ~b²³ ~. Then 4 3, µ ~ ²À³²³ b ²À³²³ b ²À³²³ ~ À = µ ~ À c ²À³ ~ À. Note that the varance of s not the weghted average of the varances of /, 4 and 3., and Partal nsurance coverage: It s possble to construct an nsurance polcy n whch the clam pad by the nsurer s part, but not necessarly all, of the loss that occurs. There are a few standard types of partal nsurance coverage on a basc ground up loss random varable. () Excessofloss nsurance: An excessofloss nsurance specfes a deductble amount, say. If a loss of amount occurs, the nsurer pays nothng f the loss s less than, and pays the polcyholder the amount of the loss n excess of f the loss s greater than. The amount pad F f c f by the nsurer can be descrbed ~ ~ 4% c Á ¹. 7
8 The expected payment made by the nsurer per loss would be B ²% c ³ ²%³ % B n the contnuous case. Wth ntegraton by parts, ths can be shown to be equal to c ²%³µ %. Two varatons on the noton of deductble are (a) the franchse deductble: a franchse deductble of amount refers to the stuaton n whch the nsurer pays 0 f the loss s below but pays the full amount of loss f the loss s above ; the amount pad by the nsurer can be descrbed as F f f (b) the dsappearng deductble: a dsappearng deductble wth lower lmt and Z Z upper lmt (where ) refers to the stuaton n whch the nsurer pays 0 f the loss s below, the nsurer pays the full loss f the loss amount s above Z, and the deductble amount reduces lnearly from to 0 as the loss ncreases from to Z ; the amount pad by Z the nsurer can be descrbed as c Z J h Z c Z. Example 140: A dscrete loss random varable has the followng twopont dstrbuton: 7 ~µ~7 ~µ~à. An excessofloss nsurance polcy s set up for ths loss, wth deductble. It s found that the expected clam on the nsurer s. f c f Soluton: The excessofloss clam on the nsurer s. We proceed by "tralanderror". Suppose our ntal "guess" s that. Then the clam on the nsurer wll be ether c or c, each wth probablty À. so that the expected clam on the nsurer wll be ² c ³²À ³ b ² c ³²À ³ ~, whch mples that ~À. Ths contradcts our "guess" that. Ths ndcates that the guess was wrong. Thus,, so that the clam on the nsurer wll be (f ~) or c, each wth probablty À. The expected clam on the nsurer wll then be ² c ³²À ³ ~ S ~. () Polcy lmt: A polcy lmt of amount " ndcates that the nsurer wll pay a maxmum amount of " on a clam. Therefore, the amount pad by the nsurer s F f ". " f " " The expected payment made by the nsurer per loss would be %h ²%³%b"h c ²"³µ n the contnuous case. Ths can be shown to be equal to c ²%³µ %. " It s possble to combne an excessofloss partal nsurance wth a polcy lmt. If a polcy has a deductble of and a lmt of ", then the clam amount pad by the nsurer can be descrbed as 8
9 f H cf ". Note that "the deductble s appled after the polcy s appled". "c f " Ths means that a loss of amount greater than " trggers the polcy lmt of amount ", and then the deductble s appled to that lmt, resultng n a payment by the nsurer of amount "c. The expected payment made by the nsurer per loss would be " ²%c³h ²%³%b²"c³h c ²"³µ n the contnuous case. Ths can be shown to be equal to c ²%³µ %. " A varaton on the noton of polcy lmt s the maxmum clam amount, say nsurer pays the clam up to a maxmum amount of nsurance cap. An nsurance cap specfes a, that would be pad f a loss occurs on the polcy, so that the. If there s no deductble, ths s the same as a polcy lmt, but f there s a deductble of, then the maxmum amount pad by the nsurer s ~"c. In ths case, the polcy lmt of amount " s the same as an nsurance cap of amount "c. () Proportonal nsurance  Proportonal nsurance specfes a fracton ( ), and f a loss of amount occurs, the nsurer pays the polcyholder Á the specfed fracton of the full loss. When there are lmts on the amount pad by the nsurer (such as deductble, polcy lmt, etc.), t s possble to consder two random varables related to the basc ground up loss. The random s the amount pad per loss, and the random s the amount pad per payment made. If there s a deductble of Á the random wll be 0 f, ~ c f. The random s the condtonal random varable of amount pad gven that a payment s made. If there s a deductble of, ~ c f. If the pdf of s ²%³, and there s a deductble of and a polcy lmt of ", then the pdf's can be expressed n terms of the pdf of.  ²³ & ~ ²% ³ ²&b³ & "c² % E²&³ ~ J, and c  ²"³ & ~ " c ²% "³ & "c 9
10 ²&³ J ²&b³ c ²³ c ²"³ c ²³ 0 &"c(densty) &~"c(a pont of probablty). & "c Example 141: For a certan nsurance, ndvdual losses last year were unformly dstrbuted over the nterval ²Á ³. A deductble of 100 s appled to each loss (the nsurer pays the loss n excess of the deductble of 100). Ths year, ndvdual losses are unformly dstrbuted over the nterval ²Á ³ and a deductble of 100 s stll appled. Determne the percentage ncrease n the expected amount pad by the nsurer from last year to ths year. Also, for last year, fnd the pdf of the amount pad per loss, and fnd the pdf of the amount pad per payment. Soluton: Last year, for a loss of amount, the amount pad by the nsurer was F f c ~. Last year the p.d.f. of the loss random varable was ²%³ ~ (unform dstrbuton on the nterval ²Á ³ by the nsurer last year was ²% c ³ h % ~ À Ths year, for a loss of amount, the amount pad by the nsurer s stll F f c f ). The expected ~, but ths year the p.d.f. of the loss random varable s ²%³ ~ (unform dstrbuton on the nterval ²Á ³ by the nsurer ths year s ²% c ³ h % ~ À À À The percentage ncrease s ² c ³ ~ À. The pdf of the amount pad per loss last year s À & ~ ²% ³ ²&³ E H À & ² % ³, and 0 otherwse. Note µ ~ ²À³²³ b & h ²À³ & ~. ). The expected payment The pdf of the amount pad per payment last year s ²&b³ ²&b³ À ²&³ ~ ~ ~ ~ for & ( % c ²³ c ²³ µ µ & h ² ³ & ~ ~ À ~ c ²³ and 0 otherwse. Note that s the expected amount pad per payment made. Rensurance: In order to lmt the exposure to catastrophc clams that can occur, nsurers often set up rensurance arrangements wth other nsurers. The basc forms of rensurance are very smlar algebracally to the partal nsurances on ndvdual polces descrbed above, but they apply to the nsurer's aggregate clam random varable :. The clams pad by the cedng nsurer (the nsurer who purchases the rensurance) are referred to as retaned clams. 10
11 () Stoploss rensurance  A stoploss rensurance specfes a deductble amount. If the aggregate clam : s less than then the rensurer pays nothng, but f the aggregate clam s greater than then the rensurer pays the aggregate clam n excess of. F f : :c f : The amount pad by the rensurer can be descrbed as. () Rensurance cap  A rensurance cap specfes a maxmum amount pad by the rensurer, say. Ths would usually be set up n conjuncton wth a stoploss wth deductble. In that case. the clam pad by the rensurer can be descrbed as f : H :cf : b f : b Notce that ths stoploss rensurance wth a deductble of and a cap of s algebracally equal to a stop loss polcy wth deductble mnus a stoploss polcy wth a deductble of b. () Proportonal rensurance  Proportonal rensurance specfes a fracton ( ), and f aggregate clams of amount : occur, the rensurer pays :Á and the cedng nsurer pays ² c ³:. Example 142: A lfe nsurance company covers 16,000 mutually ndependent lves for 1year term lfe nsurance: Beneft Number Probablty Class Amount Covered of Clam The nsurance company's retenton lmt s 2 unts per lfe. Rensurance s purchased for 0.03 per unt. The cedng nsurer collects a premum of 8~, :µb l= :µb9, where : denotes the dstrbuton of retaned clams and 9 s the cost of rensurance. Fnd 8. Soluton: The cedng nsurer wll cover all clams from classes 1 and 2, and wll cover the frst 2 unts of clam from any polcy n class 3. The cedng nsurer purchases 2 unts of rensurance for each of the polces wth beneft amount 4, for a total of ²³ ~ unts rensured. The cost of the rensurance s 9 ~ ²À³ ~. The retaned clam dstrbuton : conssts of 8000 (Class 1) polces wth ~ À and, ) µ ~ and 8000 polces ( b, Classes 2 and 3 combned) wth ~ À and, )µ~. We are usng the notaton mentoned earler, ) s the condtonal clam from polcy 11
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